Diagrams, equations and media

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Diagrams And Media · 326

• academic.cssacademic.css
• apr/app.jsapr/app.js
• apr/content.jsapr/content.js
• AIDA: four stages of a persuasive message · 1. Attention · Notice the headline or visual. · 2. Interest · Understand a relevant benefit. · 3. Desire · Develop a reason to prefer it. · 4. Action · Take the stated next steapr/fig/aida.291ff174c1.svg
• BDI and CDI: read both, then investigate the market · High BDI + high CDI · Strong brand concentration in a strong category · area. Consider sustaining the position. · Low BDI + high CDI · Category is strong; investigateapr/fig/bdi.3740d6dd92.svg
• Identity is intended; image is perceived · 1. Identity · Name, signs and intended associations. · 2. Touchpoints · Product, service, packaging and messages. · 3. Image · What the audience actually understands and · feelsapr/fig/brand.1a03b95528.svg
• Build a campaign around a defined problem · 1. Problem and publics · Identify what needs to change and for whom. · 2. Objective · Specify a measurable change and deadline. · 3. Actions and communication · Deliver the serapr/fig/campaign.e954dad25b.svg
• A complete case answer · 1. Facts · Name the event/campaign and verified context. · 2. Concept · Select the theory or planning principle that fits. · 3. Application · Point to the exact feature that illustrates it. · 4. apr/fig/case-method.b335ec8f82.svg
• Communication: meaning must reach the audience · 1. Sender and encoding · Choose words, images and sound for the · intended meaning. · 2. Message and channel · Send the message through a suitable medium. · 3. Receiver anapr/fig/communication.8f211923a1.svg
• DAGMAR: ACCA + a measurable objective · 1. Awareness · Know that the brand exists. · 2. Comprehension · Understand the offering and benefit. · 3. Conviction · Develop preference or belief. · 4. Action · Perform the definapr/fig/dagmar.e936f07148.svg
• DRIP: four functions, not a compulsory sequence · 1. Differentiate · Show a relevant difference. · 2. Remind / reinforce · Keep the brand in memory. · 3. Inform · Explain a useful fact. · 4. Persuade · Offer reasons for apr/fig/drip.19ea209b19.svg
• Course Ehrenberg model: ATRN · 1. Awareness · Recognise the familiar brand. · 2. Trial · Try and experience the product. · 3. Reinforcement · Support a satisfactory experience. · 4. Nudging · Prompt memory around the nexapr/fig/ehrenberg.26b8d4a608.svg
• Maslow: needs used as possible advertising appeals · 5 · Self-actualisation · Learning and potential · 4 · Esteem · Achievement and respect · 3 · Belonging · Friendship and connection · 2 · Safety · Protection and securiapr/fig/maslow.67c30abb5e.svg
• From market analysis to an evaluated media plan · 1. Market analysis · Identify people, opportunities and relevant · behaviour. · 2. Media objectives · Set audience, place, period and delivery targets. · 3. Strategy + imapr/fig/media-process.b35732e0a5.svg
• Match the format to the communication task · 1. Print · Read and reread details; text plus still image. · 2. Audio · Hear speech, sound and music; no visual · demonstration. · 3. Audio-visual · See motion and hear sound;apr/fig/media-types.d17fcb0c2c.svg
• Five steps of strategic advertising planning · 1. Target audience · Who must know, feel or do something · differently? · 2. Decision-making · Who suggests, advises, chooses, buys and uses? · 3. Positioning · What relevanapr/fig/planning.02823b5023.svg
• PR: a continuing process, not only a press release · 1. Research / fact-finding · Understand the issue and affected publics. · 2. Planning · Set objectives, strategy, tasks and resources. · 3. Implementation · Act, commuapr/fig/pr-process.94ce408616.svg
• Print advertisement: arrange the message in reading order · CYCLECARE · Know the repair cost · before the work begins. · Visual: estimate + bicycle · Show the customer reading the estimate before · approving the repair. apr/fig/print-ad-layout.a01c11cba2.svg
• Reach counts people; impressions count every exposure · A · Ad 1 · Ad 2 · Ad 3 · 3 seen · B · 2 seen · C · 1 seen · D · 0 seen · E · Unique reach: A, B, C = 3 people out of 5 = 60% · Impressions: 3 + 2 + 1 = 6 Averagapr/fig/reach.0b7f3f6fda.svg
• One purchase can involve five different roles · 1. Initiator · Suggests the purchase. · 2. Influencer · Advises or recommends. · 3. Decider · Makes the choice. · 4. Purchaser · Completes the transaction. · 5. User · Actuapr/fig/roles.25002db36b.svg
• Scheduling: steady, on/off, or base plus peaks · Continuous · 1 · 2 · 3 · 4 · 5 · 6 · 7 · 8 · Flighting · Pulsing · Illustrative equal time periods →apr/fig/schedule.517c850e91.svg
• Video storyboard: show the promise being carried out · 0–4 seconds · Problem · A loose chain. Student stops safely. · Cycle trouble before class? · 4–9 seconds · Estimate · Mechanic inspects and writes the proposed · worapr/fig/video-storyboard.60476eec4e.svg
• Design choices should help the audience understand · 1. Colour · Contrast, grouping and context-dependent · associations. · 2. Shape · Structure, symbols and direction. · 3. Form · Depth, texture and three-dimensional · apr/fig/visual.64f2529bb3.svg
• apr/pyq-analysis.jsonapr/pyq-analysis.json
• apr/style.cssapr/style.css
• Input · 227×227×3 · Conv1 · 11×11, 96 · Pool · 3×3, s2 · Conv2 · 5×5, 256 · Conv3 · 3×3, 384 · Conv4 · Conv5 · 3×3, 256 · Pool + flat · 9216 · Dense · 4096 · Output · 1000 · Read rows in order. Apply nonlinearities betweassets/figures/alexnet.svg
• INPUTS · WEIGHTED SUM · DECISION · x1 · x2 · w1 = 1 · w2 = 1 · z = x1 + x2 − 1.5 · step(z) · 1 when z ≥ 0 · 0 otherwise · Bias b = −1.5. Only input [1, 1] crosses the threshold.assets/figures/and_perceptron.svg
• Two outputs: add complete path contributions · h = wx + b = 0.5(2) + 0.1 = 1.1 · Shared hidden value · h = 1.1 · o₁ = 2h; target y₁ = 1 · o₁ = 2.2 · o₂ = −h; target y₂ = 0 · o₂ = −1.1 · × 2 · × (−1) · Output derivative: assets/figures/chain_branch_revision.svg
• One output: values and derivatives · Given: x = 2, y = 4, w = 1, b = 0 · FORWARD VALUES · z = wx + b · z = 2 · e = z − y · e = −2 · L = ½e² · L = 2 · BACKWARD DERIVATIVES · Loss derivative reaches the neuron · ∂L/∂z = (∂assets/figures/chain_linear_revision.svg
• Step 1: start (1,1) · 1 · 2 · 3 · 4 · 5 · 6 · 7 · 8 · 9 · K = [1, 0] · [0, −1] · 1×1 + 2×0 + 4×0 · + 5×(−1) = −4 · Step 2: start (1,2) · 2×1 + 3×0 + 5×0 · + 6×(−1) = −4 · Step 3: start (2,1) · 4×1 + 5×0 + 7×0 · + 8×(−1) assets/figures/convolution.svg
• Input · 10 neurons · Hidden 1 · 5 neurons · Hidden 2 · Hidden 3 · Output · 4 neurons · Every receiving neuron has one bias. All adjacent layers are fully connected.assets/figures/dense_10_5_5_5_4.svg
• Input · 6 neurons · Hidden 1 · 5 neurons · Hidden 2 · Output · 4 neurons · Every receiving neuron has one bias. All adjacent layers are fully connected.assets/figures/dense_6_5_5_4.svg
• assets/figures/first_training.pngassets/figures/first_training.png
• GIVEN VALUES — read each matrix row from left to right · W₁ · 0.2 · 0.1 · 0.3 · −0.1 · W₂ · 0.4 · 0.5assets/figures/given_0601bfe757.svg
• GIVEN VALUES — read each matrix row from left to right · W₁ · 0.2 · 0.15 · 0.28 · 0.3 · W₂ · 0.1 · 0.24 · 0.12 · 0.21assets/figures/given_23b4df4e5b.svg
• GIVEN VALUES — read each matrix row from left to right · W₁ · 0.8 · 0.2 · 0.3 · 0.4 · −0.5 · 0.1 · −0.1 · −0.2 · 0.5assets/figures/given_28edc44005.svg
• GIVEN VALUES — read each matrix row from left to right · W₁ · 0.1 · 0.4 · 0.2 · 0.3 · 0.5 · 0.7 · W₂ · 0.6 · 0.9assets/figures/given_2fad289da9.svg
• GIVEN VALUES — read each matrix row from left to right · W₁ · 0.2 · 0.4 · 0.3 · 0.7 · W₂ · 0.5 · 0.6assets/figures/given_412d332265.svg
• GIVEN VALUES — read each matrix row from left to right · A · 1 · 4 · 2 · 3 · 0 · 5 · 6 · 7 · 8assets/figures/given_563e4ee8f7.svg
• GIVEN VALUES — read each matrix row from left to right · W₁ · 0.2 · 0.4 · 0.3 · 0.7assets/figures/given_5f379b367f.svg
• GIVEN VALUES — read each matrix row from left to right · Matrix 1 · 1 · 2 · 3 · 5 · 4 · 6 · 7 · 8 · 9 · 10assets/figures/given_847554ea0e.svg
• GIVEN VALUES — read each matrix row from left to right · W₁ · 0.1 · 0.2 · −0.2 · 0.3 · W₂ · −0.1 · 0.4assets/figures/given_907c03b5dc.svg
• GIVEN VALUES — read each matrix row from left to right · Matrix 1 · 21 · 11 · 15 · 10 · 12 · 9 · 13 · 18 · 7 · 4assets/figures/given_915e1db14b.svg
• GIVEN VALUES — read each matrix row from left to right · W₁ · 0.5 · W₂ · 0.8assets/figures/given_bbb5f79ed7.svg
• GIVEN VALUES — read each matrix row from left to right · Matrix 1 · 1 · 0 · −1 · Matrix 2 · 2 · 3 · 4 · 5 · 6 · 7 · 8 · 9assets/figures/given_cd39cabe48.svg
• GIVEN VALUES — read each matrix row from left to right · Wxh · 0.6 · 0.4 · 0.3 · 0.7 · Whh · 0.5 · 0.2assets/figures/given_deac39f14c.svg
• GIVEN VALUES — read each matrix row from left to right · W₁ · 0.4 · W₂ · 0.6assets/figures/given_ec03263237.svg
• GIVEN VALUES — read each matrix row from left to right · W · 1 · 2 · −1 · 3assets/figures/given_ed1fcbfdf5.svg
• w · Loss · Negative slope · increase w · Positive slope · decrease w · New weight = old weight − learning rate × gradientassets/figures/gradient.svg
• h previous · Reset r × h · Candidate = tanh(...) · (1 − z) × h previous · z × candidate · + · New hidden state · Here z means the fraction of NEW content admitted.assets/figures/gru.svg
• Input · 1 × 1 · 1 × 1 reduce · 3 × 3 · 5 × 5 · 3 × 3 pool · Concatenate channels · All branch height/width values must match.assets/figures/inception.svg
• C previous · × · + · C new · tanh · h · new · σ: f · σ: i · tanh: C~ · σ: o · [h previous, x current] · Each gate has its own learned weights and bias.assets/figures/lstm.svg
• Read each receiving neuron as one row of weights · INPUT · HIDDEN · OUTPUT · x1=1 · x2=2 · h1 · h2 · o1 · target 1 · h1 incoming: x1 × 0.2 + x2 × 0.1; bias 0.1 · h2 incoming: x1 × 0.3 + x2 × -0.1; bias 0.2 · o1 incoming:assets/figures/net_drill13.svg
• Read each receiving neuron as one row of weights · INPUT · HIDDEN · OUTPUT · x1=0.2 · x2=0.4 · h1 · h2 · o1 · target 1 · o2 · target 0 · h1 incoming: x1 × 0.1 + x2 × 0.2; bias 0.1 · h2 incoming: x1 × -0.2 + x2 × 0.3; biaassets/figures/net_drill14.svg
• Read each receiving neuron as one row of weights · INPUT · HIDDEN · OUTPUT · x1=1 · x2=0 · x3=1 · h1 · h2 · o1 · target 1 · h1 incoming: x1 × 0.2 + x2 × 0.4 + x3 × -0.5; bias -0.4 · h2 incoming: x1 × -0.3 + x2 × 0.1 + x3assets/figures/net_f23.svg
• Read each receiving neuron as one row of weights · INPUT · HIDDEN · OUTPUT · x1=1 · x2=1 · h1 · h2 · o1 · target 1 · h1 incoming: x1 × 0.5 + x2 × 0.5; bias 0 · h2 incoming: x1 × 0.5 + x2 × 0.5; bias 0 · o1 incoming: h1 ×assets/figures/net_f25.svg
• Read each receiving neuron as one row of weights · INPUT · HIDDEN · OUTPUT · x1=2 · x2=3 · h1 · h2 · o1 · target 1 · h1 incoming: x1 × 0.11 + x2 × 0.21; bias 0 · h2 incoming: x1 × 0.12 + x2 × 0.08; bias 0 · o1 incoming: assets/figures/net_handwritten.svg
• Read each receiving neuron as one row of weights · INPUT · HIDDEN · OUTPUT · x1=0.4 · x2=0.3 · x3=0.5 · x4=-0.4 · h1 · h2 · h3 · h4 · o1 · target 0.9 · h1 incoming: x1 × 0.8 + x2 × -0.5 + x3 × 0.1 + x4 × -0.2; bias 0.6 ·assets/figures/net_r24.svg
• Read each receiving neuron as one row of weights · INPUT · HIDDEN · OUTPUT · x1=0.25 · x2=0.7 · h1 · h2 · o1 · target 0.7 · o2 · target 0.3 · h1 incoming: x1 × 0.2 + x2 × 0.15; bias 1 · h2 incoming: x1 × 0.28 + x2 × 0.3;assets/figures/net_r26.svg
• INPUT · HIDDEN · OUTPUT · x1 · x2 · h1 · h2 · o · Each hidden neuron receives both inputs and one bias. · Forward uses activations. Backward uses OLD weights.assets/figures/network.svg
• INPUTS · WEIGHTED SUM · ACTIVATION · x1 · w1 · x2 · w2 · x3 · w3 · z = Σwx + b · a = f(z) · Add one bias b · Multiply → add → activateassets/figures/neuron.svg
• assets/figures/notebooks_input.pngassets/figures/notebooks_input.png
• Loss · Updates · Full batch trend · SGD fluctuations · Illustration only: smoothness alone does not prove better learning.assets/figures/optimizers.svg
• Loss · Epochs · Validation · Training · Best validation region · Save the best validation checkpoint; do not choose by test loss.assets/figures/overfitting.svg
• INPUT 4 × 4 · 21 · 11 · 15 · 10 · 12 · 9 · 13 · 18 · 7 · 4 · MAX POOL · AVERAGE POOL · 13.25 · 11.5 · 8.75 · 14.25 · 2 × 2 windows · stride 2assets/figures/pooling.svg
• x · Learned layers F(x) · + · y · Shortcut S(x): identity or projection · y = F(x) + S(x) · S(x) = x for identity; S(x) = W_s x for projection. · Add matching shapes; do not concatenate.assets/figures/residual.svg
• RNN · Output p1 · Input x1 · t = 1 · h1 · Output p2 · Input x2 · t = 2 · h2 · Output p3 · Input x3 · t = 3 · h0 · Shared weights at all steps; forward → and backward through time ←assets/figures/rnn.svg
• One to many · h · x · y · Many to one · Many to many aligned · Encoder then decoder · Schematics show mapping patterns, not fixed sequence lengths.assets/figures/sequences.svg
• INPUTS · HIDDEN · OUTPUTS · x1 · x2 · h1 · h2 · o1 · o2 · 0.05 · 0.10 · w1 = .15 · w2 = .20 · w3 = .25 · w4 = .30 · w5 = .40 · w6 = .45 · w7 = .50 · w8 = .55 · target .01 · target .99 · Hidden biases = .35 Output bassets/figures/teams.svg
• One filter, nine window positions · Input: 5 × 5 · 1 · 0 · Shared filter: 3 × 3 · → · Output: 3 × 3 · 4 · 3 · 2 · Orange window → orange output cell = 4. Move one cell at a time.assets/figures/u4_binary_windows.svg
• Same output shape, different computation · Input · 14 × 14 × 480 · → · Direct 5 × 5 · 48 filters · Output · 14 × 14 × 48 · 112,896,000 MACs · 1 × 1 reduction · 480 → 16 channels · Reduced 5 × 5 · 16 → 48 channels · 1,505assets/figures/u4_bottleneck_480.svg
• Global average pooling keeps channels separate · Channel A · 1 · 3 · 5 · 7 · → · 4 · 16 ÷ 4 · Channel B · 2 · 6 · 8 · 20 ÷ 4 · 2 × 2 × 2 input → two channel means → [4, 5]assets/figures/u4_gap.svg
• Same four windows, three different summaries · Input: 4 × 4 · 12 · 20 · 30 · 0 · 8 · 2 · 34 · 70 · 37 · 4 · 112 · 100 · 25 · Max · Min · Average · 13 · 79 · 19.5 · Bottom-right average: (37 + 4 + 25 + 12) ÷ 4 = 19.5 · Poassets/figures/u4_pooling_windows.svg
• VGG16: 13 convolution layers + 3 dense layers · Block 1 · 2 convolutions · Block 2 · Block 3 · 3 convolutions · Block 4 · Block 5 · 112 × 112 · 56 × 56 · 28 × 28 · 14 × 14 · 7 × 7 · Sizes shown after each 2 × 2 pool, staassets/figures/u4_vgg16.svg
• See what each activation does · The horizontal axis is the raw score z; the vertical axis is the output. · Sigmoid · 0 · z · 0 < output < 1 · Tanh · −1 < output < 1 · ReLU · Negative → 0; positive → z · The shape of an aassets/figures/v_activations.svg
• AdaGrad accumulates all past squared gradients · Use the worked question: initial s = 0; gradients 2, then 1. · First gradient · s₁ = 0 + 2² = 4 · Second gradient · s₂ = 4 + 1² = 5 · Scale each step · Divide g by (√s + εassets/figures/v_adagrad.svg
• What happens to an old squared gradient? · Start with old memory 4. The current gradient is −3, so its square is 9. · AdaGrad: keep adding · Old sum G + new g² · 4 + 9 = 13 · An old contribution stays. · RMSProp: decay aassets/figures/v_adagrad_rms.svg
• Adam keeps two different memories · One describes signed direction. The other describes squared size. · Current gradient g · Example: g = −2 · First moment m · Average of signed gradients · Second moment s · Average of sassets/figures/v_adam.svg
• Correct the early zero-initialized memory · Example: β₁ = 0.9, first gradient = 2, initial m = 0. · Raw memory · m₁ = 0.9 × 0 + 0.1 × · 2 = 0.2 · Correction factor · 1 − 0.9¹ = 0.1 · Corrected memory · m̂₁ = 0.2 ÷ 0.1 = assets/figures/v_adam_correction.svg
• Deep learning belongs inside machine learning · These are nested areas, not three unrelated technologies. · ARTIFICIAL INTELLIGENCE · MACHINE LEARNING · DEEP LEARNING · Neural networks with many layers · AI is the broad assets/figures/v_ai_ml_dl.svg
• Three architecture ideas to remember · These are different designs, not three consecutive layers of one network. · AlexNet · Sequential · convolution and · pooling stages · Inception · Parallel branches; · concatenate thassets/figures/v_architectures.svg
• Carry a negative result as one bracketed value · Example: 0.4 − 0.1 × (−2). · Multiply first · 0.1 × (−2) = −0.2 · Replace product · 0.4 − (−0.2) · Subtract negative · 0.4 + 0.2 = 0.6 · The two minus signs have separate assets/figures/v_arithmetic.svg
• Same 6 samples, different update groups · Each outlined group produces one gradient update. · Batch GD · 1 · 2 · 3 · 4 · 5 · 6 · 1 update · Mini-batch, size 2 · 3 updates · SGD, size 1 · 6 updates · One epoch visits all assets/figures/v_batches.svg
• A bias adds a separate amount · A fixed packing fee is a useful picture of an added offset. · Notebook cost · 3 × 10 = 30 · Add packing · 30 + 2 · New output · Rs 32 · Multiply first. Add the bias once. 3 × 10 + 2 = 32.assets/figures/v_bias.svg
• Compare the two paths in the worked question · Input 28 × 28 × 192. Output 28 × 28 × 32. Count multiplications (MACs). · DIRECT · One 5 × 5 convolution · 28 × 28 × 32 × 25 × 192 · = 120,422,400 multiplications · REDUCE ·assets/figures/v_bottleneck.svg
• One shared weight receives contributions from many times · Unrolling draws repeated uses of one cell, not independent copies of its · weights. · Time 1 · Same weights W · Time 2 · Time 3 · Gradients flow backward throughassets/figures/v_bptt.svg
• The two-step RNN in the worked question · Shared weights u = 0.2, v = 0.5; bias 0; linear hidden activation. · Start · h₀ = 0 · Time 1: x₁ = 1 · h₁ = 0.2 · Time 2: x₂ = 2 · h₂ = 0.5 · δ₂ = 0.5 − 1 = −0.5 · Backward throuassets/figures/v_bptt_worked.svg
• Calculate inside the bracket first · Compare the two different expressions. · With brackets · 1 ÷ (1 + 3) · Finish inside · 1 ÷ 4 · Result · 0.25 · Without brackets: 1 ÷ 1 + 3 = 1 + 3 = 4. The expression has changed.assets/figures/v_brackets.svg
• Two output paths both affect the hidden neuron · Example: hidden activation h = 0.6, using sigmoid in the hidden neuron. · h = 0.6 · Output 1 delta = 0.2 · Connecting weight = 0.4 · Output 2 delta = −0.1 · Connecting weiassets/figures/v_branch_gradient.svg
• Build the calculator expression in pieces · Example: sigmoid at a raw score of 2. · Exponential · e⁻² ≈ 0.135335 · Denominator · 1 + 0.135335 · Reciprocal · 1 ÷ 1.135335 ≈ · 0.880797 · Enter 1 ÷ (1 + e^(−2)). The entire assets/figures/v_calculator.svg
• Linked changes multiply along a path · Example: z = 3w + 1 and L = z², evaluated at w = 1. · w = 1 · dz/dw = 3 · z = 4 · dL/dz = 2z = 8 · L = 16 · dL/dw = 8 × 3 = 24 · w affects z; z affects L. Multiply the two local sloassets/figures/v_chain_rule.svg
• A 1 × 1 filter still mixes all three channels · Use input [2, 5, 1], weights [0.1, −0.2, 0.5] and one bias 0.3. · Channel 1 · 2 × 0.1 = 0.2 · Channel 2 · 5 × (−0.2) = −1 · Channel 3 · 1 × 0.5 = 0.5 · One filter → one outassets/figures/v_channel_products.svg
• One filter combines all incoming channels · The three channel contributions join into one cell, not three separate maps. · Red channel · Weighted sum = 2 · Green channel · Weighted sum = −1 · Blue channel · Weighted sum assets/figures/v_channels.svg
• Follow the complete CNN question, one layer at a time · Start with 128 × 128 × 3. Shape and learned-parameter count are separate. · LAYER · OUTPUT SHAPE / LENGTH · PARAMETERS · 1. Convolution 1 · 126 × 126 × 16 · 448 · 2assets/figures/v_cnn_pipeline.svg
• Count connections separately from biases · A dense layer with 3 inputs and 2 receiving neurons. · x1 · x2 · x3 · a1 · + b1 · a2 · + b2 · 3 × 2 = 6 weights · Two receiving neurons also have 2 biases. Total = 6 + 2 = 8 parassets/figures/v_dense_count.svg
• Dropout temporarily removes selected activations · Example: keep probability q = 0.5. Inverted dropout during training. · Original · 0.8 · -0.4 · 0.6 · 0.2 · Mask · 1 · 0 · After mask ÷ q · 1.6 · 1.2 · The kept values doassets/figures/v_dropout.svg
• Why inverted dropout divides a kept value by q · The text uses activation 3 and keep probability q = 0.5. · If kept: probability 0.5 · Training output = 3 ÷ 0.5 = 6 · Weighted contribution = 3 · If dropped: probability 0assets/figures/v_dropout_expectation.svg
• Keep the model from the best validation point · Illustrative curves: lower loss is better. · Epoch · Loss · Validation · Training · Save this best checkpoint · Patience may wait for later epochs, but restore the model wiassets/figures/v_early_stop.svg
• Fully connected and convolutional layers connect differently · Both learn weights; the connection pattern changes. · Fully connected image layer · Every output sees every input. · Each connection has a weight. · Spatial assets/figures/v_fc_cnn.svg
• Each filter produces one output feature map · Three incoming channels; two filters; two output maps. · Input · H × W × 3 · Filter A · K × K × 3 · Filter B · Map A · Hout × Wout · Map B · Two filters mean output depth 2. assets/figures/v_filters.svg
• One whole, split into four equal parts · Shade one of the four parts. · 1 out of 4 · 25% · 1/4 = 1 ÷ 4 = 0.25 = 25%. These are three ways to name the same amount.assets/figures/v_fractions.svg
• A gate multiplies the value passing through it · Example old value = 8. Compare three gate settings. · Gate = 0 · 8 × 0 = 0 · Pass none. · Gate = 0.25 · 8 × 0.25 = 2 · Pass one quarter. · Gate = 1 · 8 × 1 = 8 · Pass all.assets/figures/v_gate.svg
• Keep the order of gate calculations straight · The cell equation specifies what each gate multiplies. · Forget path · Old cell × · forget gate · Input path · Candidate × · input gate · Cell update · Add those two · partsassets/figures/v_gate_order.svg
• Move against the slope to reduce local loss · Illustration: L = w²; start at w = 2 and use learning rate 0.1. · w · old w = 2 · new w = 1.6 · Gradient = 2w = 4 · Correction = 0.1 × 4 · New w = 2 − 0.4 · A positive gradieassets/figures/v_gd.svg
• Pixels are numerical entries · A small grayscale image can be stored as a matrix of brightness values. · 0 · 180 · Image · 4 rows × 4 columns · An RGB image has three such channels. A pixel location has a red, green and assets/figures/v_image_grid.svg
• Read height, width and channels separately · The text uses 4 × 5 × 3: four rows, five columns and three channels. · 1 · 2 · 3 · 4 · 5 · 3 stacked channels · H × W × C · H = 4 rows · W = 5 columns · C = 3 channels · Totalassets/figures/v_image_shape.svg
• Concatenation and addition produce different shapes · Use height and width 14 × 14 throughout. · CONCATENATE four branches · Channels: 16 + 32 + 8 + 8 · Result: 14 × 14 × 64 · Stack all 64 channels. · ADD two matching teassets/figures/v_join_add.svg
• A deep network connects several calculation stages · Numbers from one layer become inputs to the next layer. · 1 · 2 · 3 · Inputs · Hidden layer 1 · Hidden layer 2 · Output · Predictions are calculated forward · Circle nassets/figures/v_layers.svg
• Learning changes the adjustable number · Here we show a useful change; later, gradient descent calculates the · correction. · Before · 3 × 8 = 24 · Adjust weight · 8 becomes 10 · After · 3 × 10 = 30 · Input 3 and target assets/figures/v_learning.svg
• Match the loss to what the output means · A probability and an unrestricted numerical prediction need different · interpretations. · Predict a numerical quantity · Output: a price or temperature · Typical loss: MSE or MAassets/figures/v_loss_choice.svg
• Prediction error and its derivative · Keep the target y fixed when differentiating. · Half squared loss · L = ½(ŷ − y)² · Derivative = ŷ − y · The 2 cancels the half. · Mean squared error · L = sum of squares ÷ n · Derivassets/figures/v_loss_gradient.svg
• When the question supplies gate raw scores · A gate raw score is not yet a gate value. · Gate scores · f, i, o scores · = 0 · Apply sigmoid · Each gate · becomes 0.5 · Candidate · score · 0.5 → tanh → · 0.462117 · Then uassets/figures/v_lstm_gates_first.svg
• When the LSTM gate values are already supplied · Example: f = 0.4, old C = 0.5, i = 0.7, candidate = 0.9, o = 0.8. · Old part · 0.4 × 0.5 = · 0.2 · New part · 0.7 × 0.9 = · 0.63 · New cell · 0.2 + 0.63 = · 0.83 · Hidden assets/figures/v_lstm_given.svg
• Calculate cell memory before the hidden output · Example: old cell 0.8, forget gate 0.25, input gate 0.5, candidate −0.4. · Retain old content · 0.25 × 0.8 = 0.2 · Admit new content · 0.5 × (−0.4) = −0.2 · New cell · 0.2assets/figures/v_lstm_split.svg
• A vector is a list. A matrix has rows and columns. · The shape tells you how the values are arranged. · 2 · 1 · 3 · Vector · 3 × 1 · 4 · 5 · 6 · Matrix · 2 rows × 3 columns · Row 1 · Row 2 · A 2 × 3 matrix has six entrieassets/figures/v_matrix.svg
• Momentum remembers movement · Use signed velocity: v = βv_old − ηg; then w_new = w_old + v. · Keep movement · β × old velocity · Add correction · Subtract η × gradient · Move weight · Add the new velocity · The memory caassets/figures/v_momentum.svg
• Multiplication collects equal groups · Three notebooks. Each contributes the same Rs 10. · Rs 10 · + · = Rs 30 · 10 + 10 + 10 = 30. The shorter way to write it is 3 × 10 = 30.assets/figures/v_multiply.svg
• Negative numbers are positions and directions · Left means smaller. Right means larger. · -3 · -2 · -1 · 0 · 1 · 2 · 3 · 0 + (−2) = −2 · −2 + 2 = 0 · Subtracting a negative reverses its direction: 5 − (−2) = 5 + 2 = 7.assets/figures/v_negative.svg
• Nesterov checks the slope at the look-ahead point · Use the signed-velocity convention shown in the chapter. · Current w · Look-ahead · Updated w · Move by βv_old · Correct using gradient here · Look-ahead = w + βv_old ·assets/figures/v_nesterov.svg
• Two affine layers still make one affine rule · Without a nonlinear activation, these two stages can be combined. · Layer 1 and layer 2 · h = 2x + 1 · y = 3h + 4 · Substitute and simplify · y = 3(2x + 1) + 4 · y = 6x + 7 assets/figures/v_nonlinear.svg
• Read one update from old values to new values · The subscript is a step label, not multiplication. · OLD · w at step t − 1 · CURRENT · gradient g at step t · NEW · w at step t · An optimizer may also carry a memory from assets/figures/v_optimizer_notation.svg
• Follow the number through the machine · The units change: notebook quantity goes in; estimated rupees come out. · INPUT · 3 notebooks · MODEL · Multiply by 10 · OUTPUT · Rs 30 · 3 is the quantity. 30 is the bill predictiassets/figures/v_output.svg
• What is learned, and what is chosen? · Two different kinds of quantities appear in training. · PARAMETERS · Weights and biases · Changed by training · Example: w = 0.5 → 0.6 · HYPERPARAMETERS · Learning rate, batch size assets/figures/v_parameters.svg
• Learned parameters are not output activations · Example: a 5 × 5 × 1 input with one 3 × 3 filter, stride 1, no padding. · Learned filter · 3 × 3 × 1 = 9 weights · Plus one bias · Total parameters = 10 · Output feature maassets/figures/v_parameters_activations.svg
• Change one parameter. Hold the others fixed. · For z = wx + b, choose which parameter you are changing. · Change w; keep x and b fixed · ∂z/∂w = x · If x = 3, the slope is 3. · Change b; keep x and w fixed · ∂z/∂b = 1 · assets/figures/v_partial.svg
• A penalty adds a preference for smaller weights · Example weights: −2 and 1. The loss convention sets any λ or ½ factors. · L1: absolute values · |−2| + |1| = 2 + 1 = 3 · Linear cost with magnitude · Can promote sparse wassets/figures/v_penalties.svg
• A square and its square root · Three rows of three cells make nine cells. · 1 · 3² = 3 × 3 = 9 · √9 = 3 · The root returns the side length. · (−2)² = (−2) × (−2) = 4. Keep the minus sign inside the brackets.assets/figures/v_powers.svg
• Match the remedy to the observed problem · Compare training and validation behavior before changing the model. · Both losses high: underfitting · Model may be too simple. · Training may be insufficient. · Check optimizatassets/figures/v_remedies.svg
• RMSProp discounts older squared gradients · Use the worked question: initial s = 0; gradients 2, then 1; decay 0.9. · Step 1 memory · 0.9 × 0 + 0.1 × 4 = · 0.4 · Step 2 memory · 0.9 × 0.4 + 0.1 × 1 = · 0.46 · Scale each assets/figures/v_rmsprop.svg
• What each recurrent cell carries forward · Compare three different cell designs. These are alternatives, not consecutive · layers. · Vanilla RNN · One hidden state h · No memory gates · LSTM · Cell C and hidden h · Forgeassets/figures/v_rnn_comparison.svg
• Combine the current input and the previous state · The new state depends on both paths. · Previous state · h at time t − 1 · Current input · x at time t · Combine · Weighted sum + · bias · Activate · New state h at t · Tassets/figures/v_rnn_state.svg
• A sigmoid neuron: forward values, then one correction · Same example as the text: x = 2, w = 0, b = 0, target = 1; half squared loss. · Raw sum · 0 × 2 + 0 = 0 · Sigmoid · sigmoid(0) = 0.5 · Error · 0.5 − 1 = −0.5 · Sigmassets/figures/v_sigmoid_training.svg
• Follow this one-hidden-neuron example in both directions · x = 2, target = 1. Both activations are ReLU; both raw sums are positive. · 2 · INPUT · 1.1 · HIDDEN · 1.08 · OUTPUT · weight 0.5 · weight 0.8 · 0.5 × 2 + 0.1 = assets/figures/v_single_backprop.svg
• Shrink the interval to see the local slope · Use the same starting point every time: w = 3, L = 9. · Average slope = change in loss ÷ change in weight · Weight increase · New loss · Slope estimate · 1 · 16 · (16 − 9) ÷ 1assets/figures/v_slope_limit.svg
• Give each data split one job · Example: 1,000 independent samples split 70% / 15% / 15%. · 700 train · 150 validate · 150 test · Training · Adjust weights. · Validation · Choose settings. · Test · Final evaluation. · Fitassets/figures/v_splits.svg
• Padding adds a border. Stride moves the window. · Example: a 3 × 3 input becomes 5 × 5 after one zero border. · 0 · 1 · 2 · 3 · 4 · 5 · 6 · 7 · 8 · 9 · Padding P = 1 · Stride S = 2 · Starting positions: 0, 2, … · For eacassets/figures/v_stride_padding.svg
• Letters are labels for numbers · Read the meaning first, then put the number in its place. · x = input · x holds 3 · w = weight · w holds 10 · b = bias · b holds 2 · x × w + b = 3 × 10 + 2 = 32 · A letter does assets/figures/v_symbols.svg
• Prediction and target have different jobs · The target is the known correct answer for this training example. · Prediction · Machine says Rs 24 · Target · Actual bill is Rs 30 · Compare · 30 − 24 = 6 rupees too low · Preassets/figures/v_target.svg
• What kind of answer do you need? · The output decides whether this is regression or classification. · REGRESSION: a number · Input: house features · Output: predicted price · Example: Rs 50 lakh · CLASSIFICATION: a categassets/figures/v_tasks.svg
• A perceptron makes a threshold decision · Example threshold: return 1 when the weighted sum is greater than 0. · Inputs · x₁ and x₂ · Weighted sum · z = w₁x₁ + w₂x₂ + b · Threshold · z > 0: 1; otherwise: · 0 · Unlike sigassets/figures/v_threshold.svg
• The four operations in one training step · Repeat this sequence using the current parameters. · Predict · Forward pass · Measure error · Compute loss · Find · gradients · Backpropagation · Adjust · Optimizer · update · Cassets/figures/v_training_loop.svg
• Reuse learned features, then adapt the output · Example: start with an image model trained on a large dataset. · New images · Your smaller · labeled dataset · Feature extractor · Pretrained · convolutional layers · New hassets/figures/v_transfer.svg
• Repeated multiplication changes gradient size · These scalar examples illustrate one part of the BPTT mechanism. · Repeated factors below 1 · 0.5 × 0.5 × … × 0.5 · Ten factors: 0.0009765625 · Earlier learning signal is tassets/figures/v_vanishing.svg
• Same input. Change only the weight. · The multiplier controls the prediction. · Weight = 10 · 3 notebooks × 10 · Prediction = Rs 30 · Weight = 8 · 3 notebooks × 8 · Prediction = Rs 24 · The quantity stays 3 in both calcuassets/figures/v_weight.svg
• Slide the same filter; reuse its weights · A CNN does not learn a fresh filter for every image location. · 1 · 2 · 0 · 3 · -1 · Same four weights · -2 · Output map · Move the window to calculate the next output cell. · Lassets/figures/v_weight_sharing.svg
• x1 · x2 · 0 · 1 · XOR labels occupy opposite corners · One straight decision boundary cannot separate the two classes.assets/figures/xor.svg
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• ALOHA: too many attempts eventually reduce successful output · 0 · 0.5 · 1 · 1.5 · 2 · 2.5 · 3 · 0.1 · 0.2 · 0.3 · 0.4 · Slotted max ≈ 0.368 at G=1 · Pure max ≈ 0.184 at G=0.5 · G: aggregate attempts per frame time · Scn/fig/aloha-curve.58889b65f2.svg
• ARP resolves the selected local next hop · 1. Decide local destination or gateway · Compare the destination with the local prefix. · 2. Look in the ARP cache · If the next-hop mapping is missing, send a local broadcast rcn/fig/arp.c8af39ddb3.svg
• Bellman–Ford · given directed graph · A · B · 4 · C · 5 · -2 · D · 3cn/fig/bellman-1.1bf4232fcf.svg
• Bellman–Ford · given directed graph · A · B · 3 · D · 12 · C · 1 · 2cn/fig/bellman-2.9bfc43d501.svg
• Bellman–Ford · given directed graph · A · B · 1 · C · -2 · -1cn/fig/bellman-3.b876d89f79.svg
• Bellman–Ford · given directed graph · A · B · 5 · C · 8 · -3 · D · 7 · 4cn/fig/bellman-mock.47d822ce76.svg
• Bellman–Ford · final route and distances · A · 0 · B · 1 · 7 · C · 3 · 6 · D · 5 · E · -1 · -2 · F · 8 · G · 4cn/fig/bellman-pyq-solution.9ae8451e95.svg
• Bellman–Ford · given directed graph · A · B · 7 · C · 6 · D · 5 · E · -2 · 1 · F · 3 · G · 4cn/fig/bellman-pyq.8245b97635.svg
• Binary positions: each move left doubles the value · 128 · pos 7 · 1 · 64 · pos 6 · 0 · 32 · pos 5 · 16 · pos 4 · 8 · pos 3 · 4 · pos 2 · 2 · pos 1 · pos 0 · Bits × place values → 128 + 0 + 32 + 0 + 0 + 4 + 2 + 0 = 166 ·cn/fig/binary.f629a09a71.svg
• CSMA/CA: optional reservation, data, acknowledgement · Sender · Receiver · RTS (optional reservation) · CTS · DATA · ACK · Before data access: carrier sensing and a random backoff. · A busy medium freezes the eligible bacn/fig/ca-sequence.566280e50f.svg
• CSMA/CD: sender must still be sending after a round trip · Station A · Station B · Signal travels: Tp · Collision indication returns: Tp · Worst-case timing bound: Tframe ≥ 2Tp, so Lmin ≥ 2RTpcn/fig/collision-roundtrip.42358ec7bf.svg
• An idle observation is only local · 1. A starts transmitting · Its signal begins moving toward B. · 2. Before A’s signal arrives, B hears silence · B may start a transmission under its access rule. · 3. The signals overlcn/fig/csma-propagation.df55cdec46.svg
• DHCP initial allocation: DORA · Client · Server · DISCOVER · OFFER · REQUEST · ACK · Client asks → server offers → client chooses → server confirms.cn/fig/dhcp.510e8eca42.svg
• Dijkstra · given directed graph · A · B · 4 · C · 2 · D · 3 · 1 · 7cn/fig/dijkstra-1.973d01595e.svg
• Dijkstra · given undirected graph · A · B · 2 · C · 5 · 1 · D · 4 · E · 8 · 3cn/fig/dijkstra-2.170b001531.svg
• Dijkstra · given directed graph · A · B · 3 · C · D · 2 · Ecn/fig/dijkstra-3.312b4c0fd9.svg
• Dijkstra · given directed graph · A · B · 2 · C · 6 · 1 · D · 7cn/fig/dijkstra-mock.f988abf338.svg
• Dijkstra · final route and distances · A · 0 · B · 4 · C · 2 · 5 · D · 10 · 3 · E · 6 · 1 · F · 7 · G · 12cn/fig/dijkstra-pyq-solution.d92888ba14.svg
• Dijkstra · given directed graph · A · B · 4 · C · 2 · 5 · D · 10 · 3 · E · 1 · F · Gcn/fig/dijkstra-pyq.ffe6efac11.svg
• Distance vector: add your local cost to the advertisement · X · B · C · link cost 1 · link cost 2 · B advertises Z=4 · via B: 1+4=5 · C advertises Z=2 · via C: 2+2=4 · Choose C: total cost 4 is smaller than total cost 5.cn/fig/distance-vector.8f3c90bc87.svg
• Directed path cost: arrows determine permitted travel · A · B · C · 4 · 3 · 10 · Direct A→C = 10; via B = 4 + 3 = 7. Choose cost 7. · No reverse travel is implied by a directed arrow.cn/fig/graph-basics.ebd9ed3bc9.svg
• Hidden terminals: A and C cannot sense each other · A · B · C · Common receiver B hears both · B’s CTS can make a hidden sender defer. Collisions remain possible.cn/fig/hidden-terminal.1ffcf794d5.svg
• IPv4 header · field widths in bits · Version · 4 · IHL · DS field · 8 · Total Length · 16 · Identification · 16 · Flags · 3 · Fragment Offset · 13 · TTL · Protocol · Header Checksum · 16 · Source address · 32 bits · Destcn/fig/ipv4-header.5d41e2a297.svg
• IPv4 /26: 26 network bits and 6 host bits · Octet 1 · 192 · Octet 2 · 168 · Octet 3 · 10 · Octet 4 · 77 · 8 bits 8 bits 8 bits 2 + 6 bits · 26 network bits · 6 host bits · Thcn/fig/ipv4-prefix.d99075bfca.svg
• IPv6: 8 groups × 16 bits = 128 bits · 2001 · 16 bits · 0db8 · 0000 · 0001 · One hexadecimal digit = 4 bits; one four-digit group = 16 bits. · Compact form: 2001:db8::1cn/fig/ipv6-address.25497337ae.svg
• IPv6 base header · each horizontal row is 32 bits wide · Version · 4 · Traffic Class · 8 · Flow Label · 20 · Payload Length · 16 · Next Header · Hop Limit · Source address · 128 bits (four rows) · Destination address · 1cn/fig/ipv6-header.aef1bfbac7.svg
• One packet travels through several local links · 1. Host creates an IP packet · Destination IP identifies where the packet is going. · 2. Local frame carries it to the next hop · The next-hop MAC address is used on the ccn/fig/journey.a5c52083fd.svg
• Link-state routing: from local links to a shortest-path tree · 1. Discover neighbours and costs · Each router measures or configures its local link information. · 2. Flood link-state information · Routers share topology cn/fig/link-state.bbee3173ec.svg
• Bitwise AND: 77 AND 192 = 64 · IP octet 77 · 0 · 1 · Mask octet 192 · Network octet 64 · Only 1 AND 1 produces 1. Every host position becomes 0.cn/fig/mask-and.3b9a8ff9ea.svg
• 8 · 12 · 6 · 3 · 7 · 4 · 2 · z · x · y · w · v · t · ucn/fig/old-ai-cn-2022-11-16-ai-q3a.503e410dd7.svg
• 4 · 3 · 5 · 1 · 8 · 7 · 6 · u · v · x · w · y · zcn/fig/old-ai-cn-2023-01-27-ai-q3a.e2a89263bf.svg
• 8 · 12 · 6 · 3 · 7 · 4 · 2 · z · x · y · w · v · t · ucn/fig/old-ai-cn-2023-11-22-ai-q4a.23319ec754.svg
• 8 · 12 · 6 · 3 · 7 · 4 · 2 · z · x · y · w · v · t · ucn/fig/old-ai-cn-2024-02-06-ai-q2a.60ea5488ef.svg
• 4 · 3 · 6 · -1 · -4 · 2 · -2 · A · B · C · D · E · Fcn/fig/old-bellman-2023.5222c69942.svg
• 3 · 5 · 9 · 8 · 6 · 10 · A · B · C · D · E · Fcn/fig/old-dijkstra-2024.b1cf42a96f.svg
• 3 · 7 · 5 · 4 · 2 · 1 · 8 · A · B · C · D · E · Fcn/fig/old-dijkstra-2025.c24a5141a1.svg
• OSPFv2: five distinct packet types · 1. Hello · Discover and maintain neighbour relationships. · 2. Database Description · Summarize link-state database contents. · 3. Link State Request · Ask for particular missing or ncn/fig/ospf-packets.6093f0d178.svg
• CSMA persistence: what happens after carrier sensing? · 1-persistent · Busy: keep listening. Idle: transmit immediately. · Non-persistent · Busy: random wait, then sense again. Idle: transmit. · p-persistent · Eligible icn/fig/persistence.528dafcd8a.svg
• Packet pipeline: successive packets use different links · Equal serialization intervals; propagation omitted in this sketch. · Link 1 · P1 · P2 · P3 · Link 2 · Link 3 · For h links and k equal packets: h + k − 1 packet tcn/fig/pipeline.a023f04605.svg
• Pure ALOHA: arbitrary start times can overlap · Station A · Station B · Frame A · Frame B · Overlap: both fail in the basic collision model · A retry begins after a random waiting interval.cn/fig/pure-aloha.8aa7732ee2.svg
• Slotted ALOHA: one frame duration per slot · 0T · 1T · 2T · 3T · 4T · 5T · Station A · Station B · A1 · A2 · B1 · B2 · A2 and B1 choose the same slot: collision. Starts stay aligned.cn/fig/slotted-aloha.fbfbef9647.svg
• A /24 split into four /26 blocks · 1. /26 · First: .0 Last: .63 · Usable: .1 to .62 · 2. /26 · First: .64 Last: .127 · Usable: .65 to .126 · 3. /26 · First: .128 Last: .191 · Usable: .129 to .190 · 4. /26 · First: .192 Lcn/fig/subnet-blocks.67a24f46fc.svg
• Three switching techniques: what is forwarded? · 1. Circuit: reserve a path, then stream data · Setup → reserved capacity → data transfer → release. · 2. Message: store the entire message at each node · Large forwarding cn/fig/switching.2b51f856d9.svg
• Client-server and peer-to-peer organization · Client-server · Peer-to-peer · Server · C1 · C2 · P1 · P2 · P3 · Peers can supply and consume. A central service can be replicated.cn/fig/tee-client-peer.a7c6f1b535.svg
• Two congestion-control approaches · Open loop: prevent overload · Admission rules and retransmission/window/ACK policies reduce avoidable traffic before congestion · feedback. · Closed loop: observe and signal · Detect qcn/fig/tee-congestion.c27c2303bb.svg
• CRC sender and receiver · Message M and degree-r generator G · Write every coefficient, append r zeros to the message. · Sender: XOR long division · Divide the augmented message by G. Keep an r-bit remainder R. · Transmicn/fig/tee-crc.24705171b3.svg
• DNS namespace: full label paths are unique · Root . · com · org · example.com · example.org · www.example.com · www.example.org · Same local label “www”; different full names, so no collision.cn/fig/tee-dns.f5345fe9fb.svg
• FTP: separate control and data connections · Client → server TCP port 21 · Control: USER, PASS, CWD, TYPE, RETR/STOR/LIST, QUIT; server replies. · Active mode: server initiates data · Client provides a listening endpointcn/fig/tee-ftp.1b305fa997.svg
• GBN: a later cumulative ACK covers a lost ACK · Sender · Receiver · Frame 0 accepted; receiver expects 1 · ACK 1 is lost · Frame 1 accepted; receiver expects 2 · ACK 2 arrives before timeout · Sender advances base to 2; cn/fig/tee-gbn-ack.713dc9972e.svg
• GBN W=7: normal window movement · Sender · Receiver · Frame 0 from sender window [0…6] · ACK 1 → sender window [1…7] · Frame 1 arrives; receiver now expects 2 · ACK 2 → sender window [2…7,0] · Receiver window progresses cn/fig/tee-gbn-normal.144a24ea80.svg
• GBN W=7: loss and cumulative ACKs · Sender · Receiver · Frames 0,1 → accept in order · ACK 2: next expected is 2 · Frame 2 lost · Frames 3,4,5,6 → discard out of order · ACK 2 repeated · Timer for 2 expires: resend 2,3,4cn/fig/tee-gbn.5f577a6979.svg
• Hamming (7,4): even parity, positions left to right · p1 · 0 · 1 · p2 · 2 · d1 · 3 · p4 · 4 · d2 · 5 · d3 · 6 · d4 · 7 · p1: 1,3,5,7 p2: 2,3,6,7 p4: 4,5,6,7 · Data 1011 → codeword 0110011. Bit 5 flips → syndrcn/fig/tee-hamming.509c933bba.svg
• TCP: handshake, data and orderly close · Client · Server · SYN Seq=1000 · SYN+ACK Seq=5000, Ack=1001 · ACK Seq=1001, Ack=5001 · 200 data bytes: Seq=1001, last=1200 · ACK=1201 (next expected byte) · FIN Seq=1201 · ACK=120cn/fig/tee-handshake.e053246177.svg
• Opening a plain HTTP/1.1 URL · Browser parses http://www.example.com/ · Scheme=http, host=www.example.com, path=/, default port=80. · DNS resolves the host · A cached or newly resolved address identifies the target endpocn/fig/tee-http.de65bc75dd.svg
• Leaky-bucket data queue · Bursty arrivals enter finite buffer B · Accept packets that fit; the stated queue model drops overflow. · Queue drains at configured rate r · Send while data exists. An empty queue produces no pcn/fig/tee-leaky.77bdbcf957.svg
• Transmission media: guided and unguided · Guided: a physical path · Twisted pair: electrical pairs. Coaxial: shielded conductor. Fiber: light through a protected core. · Unguided: waves in free space · Radio, terrestrialcn/fig/tee-media.aff26cd0b5.svg
• Dijkstra · final route and distances · A · 0 · B · 3 · C · 2 · D · 7 · 4 · 1 · E · 9 · 10cn/fig/tee-mock-a-route-solution.c1a801d55e.svg
• Dijkstra · given directed graph · A · B · 3 · C · 2 · D · 4 · 1 · 7 · E · 10cn/fig/tee-mock-a-route.6f748058c9.svg
• Bellman–Ford · final route and distances · A · 0 · B · 4 · C · 2 · 5 · -2 · D · 6 · E · 7 · 8 · 3cn/fig/tee-mock-b-route-solution.f6bae119c4.svg
• Bellman–Ford · given directed graph · A · B · 4 · C · 5 · -2 · D · 6 · 2 · E · 8 · 3cn/fig/tee-mock-b-route.720ccc32b6.svg
• Dijkstra · final route and distances · A · 0 · B · 3 · 5 · C · 2 · D · 7 · 4 · 1 · E · 9 · 10cn/fig/tee-mock-c-route-solution.94d22e6eb0.svg
• Dijkstra · given directed graph · A · B · 5 · C · 2 · D · 4 · 1 · 7 · E · 10cn/fig/tee-mock-c-route.93b735aa12.svg
• OSI layers mapped to the four-layer TCP/IP model · 7 Application · 6 Presentation · 5 Session · 4 Transport · 3 Network · 2 Data link · 1 Physical · Application · HTTP, DNS, FTP, SMTP · Transport · Internet · Link / netwcn/fig/tee-models.5bff0518c8.svg
• Transport: from hosts to application processes · Host address selects the machine/interface · Example destination: 198.51.100.20. · Protocol + destination port select service · TCP port 443 and UDP port 53 identify diffecn/fig/tee-ports.f496c69028.svg
• SCTP association setup · Endpoint A · Endpoint B · INIT · INIT ACK + state cookie · COOKIE ECHO · COOKIE ACK · Stream 0 ordered messages; stream 1 separately ordered messages. · Multihoming supplies alternate paths, not cn/fig/tee-sctp.55048562bb.svg
• SMTP: envelope, message and completion · SMTP sender · Receiving server · 220 greeting · EHLO sender.example · 250 capabilities · MAIL FROM:<sender@example.com> · RCPT TO:<receiver@example.org> · DATA (after successful ecn/fig/tee-smtp.c6f99877c4.svg
• SR W=4: individual ACKs and buffering · Sender · Receiver · Frames 0,1 → accept and deliver · ACK 0 and ACK 1 individually · Frame 2 lost; frame 3 arrives · ACK 3; keep frame 3 in buffer · Timer for frame 2: resend only cn/fig/tee-sr.5ae2b52b45.svg
• Stop-and-Wait: lost ACK and duplicate rejection · Sender · Receiver · Frame 0 → accepted, delivered once · ACK 1 lost on return path · Timeout: resend frame 0 → duplicate · Repeat ACK 1; duplicate not delivered · Frame 1cn/fig/tee-stopwait.a99281ab96.svg
• TCP header: minimum 20 bytes · Source port · 16 · Destination port · 16 · Sequence number · 32 · Acknowledgement number · 32 · Offset / reserved / flags · 16 · Receive window · 16 · Checksum · 16 · Urgent pointer · 16 · cn/fig/tee-tcp.81a8dfb693.svg
• Token bucket: permission is separate from data · Token generator adds r tokens per second · Store at most B tokens; extra tokens disappear when the bucket is full. · Packet of L bytes requests L tokens · Enough tokens: dcn/fig/tee-token.c3ff1dbf76.svg
• Four physical network topologies · Star · Bus · Ring · Full mesh: 4 nodes, 6 links · S · Links show arrangement. Actual access rules are a separate concept.cn/fig/tee-topologies.d9e18f041b.svg
• UDP header: each row contains 32 bits · Source port · 16 · Destination port · 16 · Length · 16 · Checksum · 16 · Total 8 bytes. Length = header + UDP payload.cn/fig/tee-udp.b14cba3c28.svg
• Pure ALOHA: interfering starts from −T to +T · −T · 0 · +T · Our frame: 0 to T · Vulnerable start-time interval length = 2Tcn/fig/vulnerable.e2a01a5ab7.svg
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• Three plate-boundary movements · Convergent · Divergent · Transform · Separation / rising material · Compression · Extension · Sheardm/fig/boundaries.8b6b55444b.svg
• Coastal flood: four measures and the risk each reduces · 1 · Risk-informed land use · Keep new critical facilities outside mapped · surge / flood areas. · 2 · Drainage and protection · Maintain outfalls; use assessed engdm/fig/coastal-four.57a97d3a0e.svg
• A safe cordon follows the hazard, not a fixed circle · Sea / water · Restricted · hazard area · Authorised · teams only · Controlled access · Check entry; maintain rescue routes. · Safe reception / assembly · Outside assdm/fig/cordon.14f06142d3.svg
• Disaster scale: the faculty lecture framework · Level 01 · Own organisation or · community resources contain · the incident. · Level 02 · External help from nearby · authorities is required. · Level 03 · Community or regdm/fig/course-levels.26aad2f326.svg
• The management cycle continues after recovery · Risk reduction · Prevention and mitigation: avoid risk or · reduce losses. · Preparedness · Plans, warnings, training and resources to · act. · Response · Urgent protectiondm/fig/cycle.4dd3905659.svg
• Tropical cyclone: structure and hazards · Eye: relatively calm centre · Eyewall: strongest winds · Rainbands: heavy showers · Main hazards · Strong wind + storm surge + rainfall flooding · Schematic plan view; not a map dm/fig/cyclone.035d3a6b0d.svg
• Focus, epicentre and a slipping fault · Fault plane · Epicentre: surface above focus · Focus: rupture begins within Earth · Surface · Waves spread from the source; shaking effects vary with local conditions.dm/fig/earthquake.46adac0128.svg
• A drill tests the full evacuation process · Alarm · Recognise the signal · Safe movement · Use the assigned exit · Assembly · Count and report · Review · Correct the gaps · Include visitors and people needing assistance;dm/fig/evacuation.06122809c0.svg
• Flood management in India: central and state coordination · Central level: specialist roles and national support · NDMA: disaster policy, guidance and coordination · CWC: river observation, flood forecasts and technical dm/fig/flood-institutions.9a8cc4ec14.svg
• Four complementary flood-risk measures · 1 · Storage · Wetlands and detention hold water and reduce · a peak. · 2 · Drainage · Maintain flow paths, culverts and outfalls. · 3 · Land-use control · Keep vulnerable developmdm/fig/flood-measures.8571e23804.svg
• GIS combines location-based layers for decisions · Hazard layer · Flood extent / slope / drought · Exposure layer · People, homes, roads, hospitals · Resources layer · Shelters, teams, water supply · Decision supported ·dm/fig/gis.9072f68acf.svg
• From national policy to local action · National: NDMA and Union government · Policy, planning framework, guidance and national support · State: SDMA and state departments · State plans, coordination, sector responsibilitdm/fig/institutions.e7262187f0.svg
• Earthquake damage: a hazard-based cordon · Restricted damage area · Unstable building / debris · Controlled entry · Authorised rescue teams; structural · assessment and safe access. · Safe assembly / coordination · Outsidm/fig/quake-cordon.33e92aab0d.svg
• Why the same hazard causes different losses · Hazard · The potentially harmful · event · Exposure · People and assets in its path · Vulnerability · Conditions that increase · harm · Potential disaster losses · Serious didm/fig/risk-chain.ced919a55d.svg
• River flood: control access around the flood footprint · Restricted floodplain · River · Safe assembly area · Outside assessed inundation; · headcounts and support. · Controlled entry · Authorised rescue access; close · dm/fig/river-cordon.22669aea6f.svg
• Four complementary river-flood mitigation measures · 1 · Natural storage · Protect wetlands and floodplains; slow and · store part of the runoff. · 2 · Maintained flow paths · Keep drains, culverts and channels usable afdm/fig/river-four.7447872224.svg
• Landslide mechanisms require different solutions · Rotational slide · Topple · Curved slip surface · Drainage / stabilisation / safer siting · Block rotates forward · Assess joints, toe support and water · Measures must dm/fig/slope.38d4f3b001.svg
• Tsunami: displacement → propagation → inundation · 1 · Seafloor displacement · 2 · Waves propagate · 3 · Move to designated · safe high ground · The tsunami can be a series of waves; do not return after the first wave.dm/fig/tsunami.bc017e1e08.svg
• Urbanisation changes how rainfall becomes runoff · More permeable catchment · More paved catchment · More infiltration / natural storage · Faster surface runoff → drains · High rainfall + limited storage/conveyance + expdm/fig/urban-flood.03309d30e8.svg
• Pressure and release: reduce the causes of vulnerability · Root causes · Unequal access to resources · and decision-making · Dynamic pressures · Unsafe urban growth, weak · services and lost ecosystems · Unsafe conditiondm/fig/vulnerability.bbe86f2049.svg
• Warning must reach the person who needs to act · 1 · Know risk · Who is exposed? · 2 · Monitor · Detect and forecast · 3 · Communicate · Trusted action · message · 4 · Act · Safe route and · shelter · Continuous review: dm/fig/warning-chain.6950d12628.svg
• Seismic waves: compare particle motion · P wave · body wave · Push and pull parallel to travel: alternating compression and expansion. · S wave · body wave · Shearing motion perpendicular to the direction of wave travel.dm/fig/waves.3b808bf75c.svg
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• Why ants find the shorter path · Nest · Food · short path: ants return sooner, pheromone builds up · long path: fewer trips per minute, pheromone evaporatesis2/fig/aco_bridge.3fc4d22f6d.svg
• Final 2025 Q7b: distances (all pheromones start at 1) · 4 · 2 · 3 · 6 · A · B · C · Dis2/fig/aco_final_graph.ac3cd4aeac.svg
• Both ants: A → C → D → B → A, length 14 · 4 · 2 · 3 · 6 · A · B · C · Dis2/fig/aco_final_tour.1dbfdb8b98.svg
• Lecture example (Ant System): distances · 5 · 6 · 7 · 2 · 9 · 3 · A · B · C · Dis2/fig/aco_lecture_graph.629672bf56.svg
• Re-exam 2026 Q6a: both ants A → B → D → C → A, length 14 · 1 · 2 · 3 · 4 · 5 · 6 · A · B · C · Dis2/fig/aco_reexam_graph.02195c6ee9.svg
• Burglary network (Russell and Norvig) · Burglary · P(B) = 0.001 · Earthquake · P(E) = 0.002 · Alarm · P(A|B,E): .95 · .94 · .29 · .001 · (TT · TF · FT · FF) · JohnCalls · P(J|A) = .90, P(J|¬A) = .05 · MaryCalls · P(M|A) is2/fig/bn_burglary.9802d6d53d.svg
• Re-exam 2026 Q3a: weather–flu–cough–fever network · Weather · P(W) = 0.75 · Flu · P(F|W) = 0.2, P(F|¬W) = 0.5 · Cough · P(C|W) = 0.7, P(C|¬W) = 0.25 · Fever · P(V|F,C): .60 · .02 · .60 · .01 · (TT · TF · FT · FF)is2/fig/bn_flu.1d732f0fb6.svg
• Final 2025 Q3a: sprinkler network · Cloudy · P(C) = 0.5 · Sprinkler · P(S|C) = 0.1, P(S|¬C) = 0.5 · Rain · P(R|C) = 0.8, P(R|¬C) = 0.2 · Wet grass · P(W|S,R): .99 · .90 · .90 · .01 · (TT · TF · FT · FF)is2/fig/bn_sprinkler.a251583272.svg
• Fan output: trace the upper envelope before finding its centroid · High clipped at 0.3; Very High clipped at 0.5. Hidden overlap is counted once. · 0 · 0.3 · 0.5 · 50 · 54.5 · 74.5 · 77.5 · 92.5 · 100 · Centroid ≈77.087%is2/fig/deep_fan_centroid.98bf1c217d.svg
• Rows across. Columns down. Entry A₂₃ = 7. · Column 1 · Column 2 · Column 3 · Row 1 · 2 · 5 · 8 · Row 2 · 1 · 4 · 7 · Two rows × three columns = 2 × 3 matrixis2/fig/depth_matrix.a8870f4ab6.svg
• Read the direction before multiplying the pull · Best = 1 · Current = 4 · difference = 1 − 4 = −3 · coefficient 0.5 → pull 0.5 × (−3) = −1.5 (left)is2/fig/depth_pso.88e9c2a672.svg
• Probabilities become intervals on the same 0–1 line · A · 0.1 · 0 · B · 0.2 · C · 0.3 · D · 0.4 · 0.6 · 1 · r = 0.55 selects Cis2/fig/depth_roulette.12804497d2.svg
• Read x = 4: halfway from 2 to 6, so height = 0.5 · 0 · 0.25 · 0.5 · 1 · 2 · 3 · 4 · 6 · 8 · μ · Input x · Straight rising membership · x = 4is2/fig/depth_slope.c91a1030aa.svg
• Order crossover keeps a valid tour: copy B E from P1, fill A, D, C in P2's order · Parent 1 · A · B · E · C · D · length 28 · Parent 2 · length 31 · Child (OX) · length 23 · After swap · length 21 (shortest)is2/fig/ga_ox.8fd21cbc51.svg
• A 1-D Gaussian mixture: three weighted bells add up to p(x) · -8 · -4 · 0 · 4 · 8 · p · xis2/fig/gmm_mix.84e959698a.svg
• 0 · 10 · 20 · 30 · 40 · 50 · 60 · 70 · 80 · 90 · 100 · 0.25 · 0.5 · 0.75 · 1 · Young · Middle-aged · Old · μ(x)is2/fig/legacy24-age.140e92bbb3.svg
• 0.8 · 0.2 · 0.1 · 0.9 · A · Bis2/fig/legacy24-brands.f41eb4de5f.svg
• 0 · 1 · 2 · 3 · 4 · 5 · 6 · 7 · 8 · 9 · 10 · 0.25 · 0.5 · 0.75 · A1 h=0.5 · A2 h=0.3 · μ(x)is2/fig/legacy24-centre-sums.6ec254a338.svg
• Exam E · IQ I · Marks M · Aptitude S · Admit Ais2/fig/legacy24-exam-bn.9070a35002.svg
• 0 · 10 · 20 · 30 · 40 · 50 · 60 · 70 · 80 · 90 · 100 · 0.25 · 0.5 · 0.75 · 1 · Aggregated output · μ(x)is2/fig/legacy24-quality-aggregate.dae0d30118.svg
• 0 · 10 · 20 · 30 · 40 · 50 · 60 · 70 · 80 · 90 · 100 · 0.25 · 0.5 · 0.75 · 1 · Low · Medium · High · μ(x)is2/fig/legacy24-quality-inputs.c512a5765f.svg
• 0 · 10 · 20 · 30 · 40 · 50 · 60 · 70 · 80 · 90 · 100 · 0.25 · 0.5 · 0.75 · 1 · Very low defect · Low defect · Medium defect · High defect · Very high defect · μ(x)is2/fig/legacy24-quality-outputs.e8de6dab72.svg
• 0.5 · 0.25 · 0.333333 · 0.666667 · 1 · 2 · 3is2/fig/legacy24-three-state.d3ab43c06c.svg
• 0 · 10 · 20 · 30 · 40 · 50 · 60 · 70 · 80 · 90 · 100 · 0.25 · 0.5 · 0.75 · 1 · Aggregated output · μ(x)is2/fig/legacy24-wash-aggregate.b750073d41.svg
• 0 · 10 · 20 · 30 · 40 · 50 · 60 · 70 · 80 · 90 · 100 · 0.25 · 0.5 · 0.75 · 1 · Low · Medium · High · μ(x)is2/fig/legacy24-wash-inputs.c512a5765f.svg
• 0 · 10 · 20 · 30 · 40 · 50 · 60 · 70 · 80 · 90 · 100 · 0.25 · 0.5 · 0.75 · 1 · Very short · Short · Medium · High · Very high · μ(x)is2/fig/legacy24-wash-outputs.bb86080be7.svg
• 0.8 · 0.05 · 0.15 · 0.75 · 0.2 · 0.38 · 0.02 · 0.6 · Sunny · Cloudy · Rainyis2/fig/legacy24-weather.575ab21d0e.svg
• 0 · 0.7 · 1.4 · 2.1 · 2.8 · 3.5 · 4.2 · 4.9 · 5.6 · 6.3 · 7 · 0.25 · 0.5 · 0.75 · 1 · First graph · μ(x)is2/fig/legacy24r-defuzz-a.4cd4ea6810.svg
• 0 · 0.7 · 1.4 · 2.1 · 2.8 · 3.5 · 4.2 · 4.9 · 5.6 · 6.3 · 7 · 0.25 · 0.5 · 0.75 · 1 · Second graph · μ(x)is2/fig/legacy24r-defuzz-b.b1f504dbf4.svg
• 0.3 · 0.25 · 0.45 · 0.4 · 0.15 · 0.65 · 0.1 · Full F · Half H · None Nis2/fig/legacy24r-homework.4001dfd62e.svg
• 0.8 · 0.2 · 0.3 · 0.7 · 1 · 2is2/fig/legacy24r-stationary.41b587e21f.svg
• Smoking S · Lung cancer C · Cough Co · Pneumonia P · Fever Fis2/fig/legacy25-bayes.7f566dbf34.svg
• A · B · C · 0.50 · 0.10 · 0.20 · 0.30 · 0.60 · 0.80is2/fig/legacy25-bikes.7db03f5dcb.svg
• BestTV · CableCast · 0.40 · 0.30 · 0.60 · 0.70is2/fig/legacy25-cable.c15b41f741.svg
• Input value / allocation score · μ · 0 · 1 · 50 · 100 · Centroid z* =30.379571is2/fig/legacy25-centroid.6d2be43cdb.svg
• Sensors · temperature, occupancy · Fuzzification · membership degrees · Rule evaluation · IF–THEN strengths · Output calculation · one fan command · Membership functions + rule base define the controller · Mamdani: aggreis2/fig/legacy25-controller.80f7bae29e.svg
• Input value / allocation score · μ · 0 · 1 · 50 · 100 · Low / Conservative · Medium / Balanced · High / Aggressiveis2/fig/legacy25-memberships.904ed2cacf.svg
• A two-state MDP: actions branch, then the environment decides the next state · s₁ · s₂ · a = stay · a = go · p = 1, r = +1 · p = 0.8, r = +5 · p = 0.2, r = 0is2/fig/mdp_two.331bc03880.svg
• Nightly aircraft movement between hubs · 0.3 · 0.2 · 0.4 · 0.25 · 0.5 · A · B · Cis2/fig/mk_airline.779a8c5fd6.svg
• Savings (S) and Current (C) accounts per year · 0.2 · 0.1 · 0.8 · 0.9 · S · Cis2/fig/mk_bank.879499f6eb.svg
• Weekly booking channel: Website, App, Counter · 0.3 · 0.2 · 0.4 · 0.5 · W · A · Cis2/fig/mk_booking.a7176dff51.svg
• Brand switching each year · 0.25 · 0.4 · 0.75 · 0.6 · X · Yis2/fig/mk_brand.c2c4be3e6a.svg
• Daily guest movement: Paris, London, Rome · 0.25 · 0.15 · 0.3 · 0.2 · 0.6 · 0.5 · P · L · Ris2/fig/mk_hotel.55f5204329.svg
• Daily learner movement: Beginner, Intermediate, Advanced · 0.3 · 0.1 · 0.2 · 0.6 · 0.5 · 0.9 · B · I · Ais2/fig/mk_learning.61a789edc7.svg
• A three-state Markov reward process (rewards: Class −2, Lab −1, Done 0) · 0.5 · 0.4 · 0.6 · 1 · Class · Lab · Doneis2/fig/mrp_student.be69eb4eb6.svg
• Imagine 1000 people (disease 1 in 1000, test catches 99%, false alarm 2%) · 1000 people · 1 has disease · 0.001 × 1000 · 999 healthy · 0.999 × 1000 · 0.99 test + · 99% of 1 · 0.01 test − · 19.98 test + · 2% of 999 (falseis2/fig/prob_tree.201d46f630.svg
• AND = the overlap, OR = everything inside either circle · all students = 100 · only Cricket · 30 · both · 10 · only Football · 20 · neither = 40is2/fig/prob_venn.84491b9acc.svg
• One time-step of reinforcement learning · Agent · policy π(a | s) · Environment · P(s′ | s, a), R · action Aₜ · next state Sₜ₊₁ and reward Rₜ₊₁ · The loop repeats: S₀, A₀, R₁, S₁, A₁, R₂, S₂, …is2/fig/rl_loop_full.a9937cb961.svg
• Min-implication clips the triangle at strength 0.3 · 0 · 0.3 · 1 · 50 · 54.5 · 65 · 75.5 · 80 · μ · Output z · Original High · clip 0.3is2/fig/v2_clip.f101b5cfa9.svg
• Two active output sets · 0 · 0.5 · 1 · 50 · 54.5 · 65 · 70 · 75 · 77.5 · 85 · 92.5 · 100 · μ · Fan speed (%) · High · Very High · clip 0.3 · clip 0.5is2/fig/v2_fan_output.8708d0ea77.svg
• Chosen occupancy membership functions · 0 · 0.5 · 1 · 2 · 3 · 4 · 5 · 6 · 7 · 8 · 9 · 10 · μ · Number of people · Low · Medium · High · 7 peopleis2/fig/v2_fan_people.1861531dd1.svg
• Chosen temperature membership functions · 0 · 0.5 · 1 · 10 · 15 · 20 · 25 · 32 · 35 · 40 · μ · Temperature in °C · Low · Medium · High · 32°Cis2/fig/v2_fan_temp.434c9b6635.svg
• Chosen triangular output sets and their max-aggregate · 0 · 0.5 · 1 · 25 · 50 · 75 · 100 · μ · Output score · Low · Moderate · High · centroid ≈ 75is2/fig/v2_fire_aggregate.e0cab4ad42.svg
• One complete GA generation: read from top to bottom · Selected parents · 1010 × 1100 · Cut after bit 2 · Mark the cut · 10 | 10 11 | 00 · Keep prefix, exchange suffix · After crossover · 1000 1110 · Childris2/fig/v2_ga_generation.fa7bf86c05.svg
• Chosen triangular output sets and their max-aggregate · 0 · 0.5 · 1 · 25 · 50 · 75 · 100 · μ · Output score · Reject · Consider · Approve · centroid ≈ 42.78is2/fig/v2_loan_aggregate.2c5ffa32a7.svg
• 3 · 3.5 · 4 · 4.5 · 5 · 5.5 · 6 · 6.5 · 7 · 7.5 · 8 · 8.5 · Class PSO: positions after each iteration · 0 = initial, 1 = iteration 1, 2 = iteration 2 · A0 · A1 · A2 · B0 · B1 · B2 · C0 · C1 · C2 · First coordinate x₁ · Tis2/fig/v2_pso_class_paths.961831cda1.svg
• Best memory survives a worse move · Visit 1 · Position x = 3 · Score x² = 9 · Remember pbest = 3 · Visit 2 · Position x = 1 · Score x² = 1 · Remember pbest = 1 · Visit 3 · Position x = 2 · Score x² = 4 · Minimise x²: aftis2/fig/v2_pso_memory.fcc9783a95.svg
• A trapezoid has a fully belonging interval · 0 · 0.5 · 1 · 10 · 15 · 20 · 25 · 30 · 35 · 40 · 50 · μ · Input x · Comfortableis2/fig/v2_trapezoid.dbfff598ea.svg
• Triangle: left end 2, peak 5, right end 8 · 0 · 0.5 · 1 · 2 · 3 · 5 · 7 · 8 · 10 · μ · Input x · A · x = 3 · x = 7is2/fig/v2_triangle.4eb0b27c9d.svg
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Equations · 5,993

• \operatorname{BDI}=\frac{\%\text{ of national brand sales in area}}{\%\text{ of national population in area}}\times100apr/math/001ebcc5227e1401.svg
• \text{User index}=\frac{\%\text{ of users in segment}}{\%\text{ of population in segment}}\times100apr/math/01535b6a6a98534c.svg
• 0.75\times20000=15000apr/math/021eea82d8985f0e.svg
• 15000\times5.4=81000apr/math/06c5909ec8304957.svg
• 6000\times7=42000apr/math/0829f7beeea0528b.svg
• \bar f=180\div60apr/math/08ddc31fbaabaa44.svg
• \frac{20}{20}\times100=100\%apr/math/0a152ac23c9556e3.svg
• 45000\div300000=0.15apr/math/0ef3ae13eb510676.svg
• 2000\times1=2000apr/math/10b3210421b76f15.svg
• 1500\times3=4500apr/math/12d0eed4aaa616aa.svg
• \text{CDI}=\frac{20}{10}\times100apr/math/12e52a811cb9d3b6.svg
• 6000\times4=24000apr/math/145920655de87f8d.svg
• 0.3\times100=30\%apr/math/1659125940ddb33b.svg
• \text{BDI}=\frac{40}{25}\times100apr/math/1679e567ab216774.svg
• 60\div100=0.6apr/math/18774bf88f9c5c01.svg
• \text{CDI}=\frac{15}{25}\times100apr/math/1a3ab2a4f1faa3b1.svg
• \frac{30000}{200000}\times100=15\%apr/math/1bff8cb32cd6224e.svg
• \text{BDI}=\frac{\%\text{ of total brand sales in area}}{\%\text{ of population in area}}\times100apr/math/1c78a4292db1541b.svg
• 40-20=20\text{ percentage points}apr/math/1e87a320ffe788dd.svg
• \operatorname{GRP}=R_{\%}\times\bar f=\frac{I}{N}\times100apr/math/20a9ecb0980d2deb.svg
• n=\frac{R_{\%}}{100}N,\qquad I=n\bar fapr/math/2584ecc3590abe5f.svg
• 6000\div10000=0.6apr/math/27e6ae805e594de8.svg
• \bar f=\frac{81000}{15000}=5.4apr/math/293f00e886aa1b9f.svg
• 1.6\times100=160apr/math/2c7ec07abc5b960f.svg
• \bar f=\frac{6000}{3000}=2apr/math/2d8a41a9ced923a1.svg
• \text{GRP}=R_{\%}\bar f\quad\Rightarrow\quad\bar f=\frac{\text{GRP}}{R_{\%}}apr/math/2fc12a9dbc6536b5.svg
• 0.15\times1000=150apr/math/3ed58fa130bb8d36.svg
• \bar f=3apr/math/3f40b26205bf4630.svg
• I=\sum_i n_i f_i,\qquad n=\sum_i n_i,\qquad\bar f=\frac{I}{n}apr/math/40adadf3d5177697.svg
• 30000\div150000=0.2apr/math/40b980bf30409336.svg
• 6000+6000+3000=15000apr/math/42d9da6ffc99e407.svg
• 1500+1500=3000apr/math/43770216633296b8.svg
• \text{CPM}=\frac{C}{I}\times1000apr/math/4517470213e9f280.svg
• \text{Gross impressions}=\sum_i n_i f_iapr/math/45add4384b6536ac.svg
• \Delta_{\rm pp}=p_1-p_0,\qquad\text{relative increase}=\frac{p_1-p_0}{p_0}\times100apr/math/49c2ea796b9c19fe.svg
• \operatorname{CPM}=\frac{C}{I}\times1000apr/math/4a8b6a4a46aa3999.svg
• \bar f=\frac{I}{n}apr/math/4f1b167db0a42ea3.svg
• 1.5\times100=150apr/math/5111185c176f5626.svg
• R_{\%}=\frac{3000}{10000}\times100apr/math/513492c2e7e541c5.svg
• \frac{3000}{10000}=0.3apr/math/588eab3daa4a27cc.svg
• \bar f=\frac{\text{gross exposures}}{\text{unique people reached}}apr/math/5b72e4aceefa4b1f.svg
• 6000\times5=30000apr/math/5b9bf1b3b8c2a14b.svg
• 2000+1000=3000apr/math/5d1e1fd529353025.svg
• \text{CDI}=\frac{\%\text{ of total category sales in area}}{\%\text{ of population in area}}\times100apr/math/5d6f1afab96761c9.svg
• R_{\%}=\frac{8400}{12000}\times100apr/math/6367897498fbf4ac.svg
• \text{BDI}=\frac{B_{\%}}{P_{\%}}\times100,\qquad\text{CDI}=\frac{C_{\%}}{P_{\%}}\times100apr/math/64dd7de58503d27f.svg
• 40\times3=120apr/math/64e40459a60a4d11.svg
• 2\times100=200apr/math/6eaebc4a590da181.svg
• 24\div15=1.6apr/math/715b259f9c7e1c1e.svg
• \text{CDI}=\frac{30}{20}\times100apr/math/78c4ffa4fbd3fcb8.svg
• \frac{20}{10}\times100=200apr/math/790756f1d89d9534.svg
• \text{BDI}=\frac{B_{\%}}{P_{\%}}100,\quad\text{CDI}=\frac{C_{\%}}{P_{\%}}100apr/math/79a3bca69fe8beb7.svg
• \frac{15}{10}\times100=150apr/math/79bbfb1f227f9929.svg
• 40\times2.5=100\text{ GRPs}apr/math/7a151a29572eb514.svg
• \frac{8000}{20000}\times100=40\%apr/math/7bc6736dfeb42e51.svg
• 30000+42000+9000=81000apr/math/804e2697785396a0.svg
• 60\times4=240\text{ GRPs}apr/math/820733f9300af62c.svg
• \text{GRP}=R_{\%}\times\bar fapr/math/8b5701790501cc48.svg
• \text{BDI}=\frac{10}{20}\times100apr/math/8e1e326b44b9af67.svg
• 75\div100=0.75apr/math/8ffb1c2ef9f19bd2.svg
• \frac{12000}{24000}\times1000=500apr/math/90c3b4fa3f7a2821.svg
• \frac{120000}{600000}\times100=20\%apr/math/92fd84f9a8715726.svg
• |A\cup B|=|A|+|B|-|A\cap B|apr/math/9362b54b1d34475a.svg
• \text{CPM}=\frac{30000}{150000}\times1000apr/math/94cce74fdde287d9.svg
• \text{GRP}=\sum_j (\text{rating}_j\times\text{insertions}_j)apr/math/965a847e402eaf16.svg
• 24+24=48\text{ GRPs}apr/math/987e8372e09cf072.svg
• 18000\div60000=0.3apr/math/9f8fed66280c6329.svg
• \text{BDI}=\frac{15}{10}\times100apr/math/a3d0e86697b2a239.svg
• \operatorname{CDI}=\frac{\%\text{ of national category sales in area}}{\%\text{ of national population in area}}\times100apr/math/a4e9a29319a69fd9.svg
• \frac{8400}{12000}=0.7apr/math/aabb675ed2821842.svg
• 0.6\times10000=6000apr/math/ab8e9ccc6a7f4b9c.svg
• \frac{24}{15}\times100apr/math/ae64c7d7bcbed1f2.svg
• 40\div100=0.4apr/math/af5d450434c58368.svg
• \text{CPM}=\frac{18000}{60000}\times1000apr/math/b2215432e2005a71.svg
• 4000+3000=7000apr/math/b2933d9523bbc8c5.svg
• 1500\times6=9000apr/math/b368c3d09929cd03.svg
• 0.3\times1000=300apr/math/bb89894dcfbc5ed2.svg
• 8\times3=24apr/math/befd3877a212eb78.svg
• \text{Reach \%}=\frac{\text{unique people reached}}{\text{target population}}\times100apr/math/c354525fa526aa0d.svg
• 12\times2=24apr/math/c550117d481aa9d7.svg
• 3000\times3=9000apr/math/c589fc4eed5be14d.svg
• 0.6\times100=60\%apr/math/caa998ec158389a8.svg
• R_{\%}=\frac{|A\cup B|}{N}\times100apr/math/caf57942f6b25966.svg
• 2000+4000=6000apr/math/cb151bfd1ddd7f9a.svg
• 0.2\times1000=200apr/math/cffecd09f32e6cb9.svg
• \text{CPM}=\frac{\text{cost}}{\text{impressions}}\times1000apr/math/d113c4be05bd8e9c.svg
• \bar f=\frac{I}{n},\qquad I=\sum_j n_j f_japr/math/d2831504662dea8a.svg
• \frac{3750}{25000}=0.15apr/math/d4d0714195bc75e3.svg
• \text{GRP}=R_{\%}\bar fapr/math/d83014835cefa5d1.svg
• \frac{200000}{2000000}\times100=10\%apr/math/d8e7f2bb4a4eb3f7.svg
• 4500+9000=13500apr/math/d92a2312b0f59038.svg
• \frac{24000}{8000}=3apr/math/d992a55907ef3d1e.svg
• 75\times5.4=405\text{ GRPs}apr/math/da24fe5ce310e0a4.svg
• 0.4\times8000=3200apr/math/db504809d2ecfe70.svg
• 3200\times2.5=8000apr/math/dc678c06772e7d24.svg
• \bar f=\frac{13500}{3000}=4.5apr/math/e50dae9644ab93b3.svg
• R_{\%}=\frac{3750}{25000}\times100apr/math/e5184390da829581.svg
• 7000-1000=6000\text{ unique people}apr/math/e66e44e86b9f639a.svg
• \text{CPM}=\frac{45000}{300000}\times1000apr/math/e81a222a8134e98b.svg
• \text{Index}=\frac{\%\text{ users in segment}}{\%\text{ population in segment}}\times100apr/math/ed1997dfe9e240f9.svg
• 0.5\times100=50apr/math/ef59274b38abe172.svg
• 1000\times4=4000apr/math/efcd7cdf53c7b27f.svg
• R_{\%}=\frac{n}{N}\times100apr/math/f08d8cba0ad4c6c4.svg
• 0.7\times100=70\%apr/math/f6ba6258f10128aa.svg
• 0.6\times100=60apr/math/f77d9f4f8099c342.svg
• P_{\%}=\frac{P_a}{P_T}100,\ B_{\%}=\frac{B_a}{B_T}100,\ C_{\%}=\frac{C_a}{C_T}100apr/math/f864fd6c0d236dae.svg
• 0.15\times100=15\%apr/math/fc5d3b27b021f9af.svg
• b_{o_1,\mathrm{new}}=1-1(0.013687964)=0.986312036assets/equations/001388bc6f2ce55e.svg
• C_2=(0.3)\times(0.145933863)\approx 0.043780159assets/equations/0025ef2e4f3926ed.svg
• g_c=\frac{1}{HW}\sum_{i=1}^{H}\sum_{j=1}^{W}a_{i,j,c}assets/equations/002749b4b374cf54.svg
• f\prime(2)=(0.880797078)(1-0.880797078)\approx0.104993585assets/equations/00492ebf856c8aae.svg
• \mathrm{saving}=4{,}718{,}592-720{,}896=3{,}997{,}696assets/equations/005360b3f69dc10a.svg
• h_{1}=\frac{1}{1+e^{-(0.25)}}=0.562176501assets/equations/00585fcf1de8913d.svg
• g_{2,2}^{(1)}=(0.003509506)\times(0.7)\approx 0.002456654assets/equations/00759f830d2e59b2.svg
• J=J_{\mathrm{data}}+\lambda\sum_iw_i^2assets/equations/008650efadbf4e29.svg
• C_1=(0.7)\times(-0.093597941)\approx -0.065518559assets/equations/00d532669e6f8902.svg
• \hat y=wx,\quad L=\frac12(\hat y-y)^2,\quad g=(\hat y-y)xassets/equations/00e3dff87be9cc8c.svg
• z=wx+bassets/equations/00f49c1127dda70e.svg
• R_3=(2)\times(1)+(6)\times(1)+(7)\times(1)=15assets/equations/0136802d6b00a2fe.svg
• v_2=0.9(-0.1)-0.1(0.9)=-0.18assets/equations/0143acec3cdc1b06.svg
• W^{(1)}_{1,1,\mathrm{new}}=0.2-0.9(0.008724562)\approx 0.192147894assets/equations/0149f12f0acd3c01.svg
• C_1=(-0.3)\times(-0.131169078)\approx 0.039350723assets/equations/0153f0b9bcb4caa4.svg
• h_t=\tanh(W_{xh}x_t+W_{hh}h_{t-1}+b_h)assets/equations/015d839835ad8254.svg
• y_{XNOR}=1assets/equations/016b62d9d7436c82.svg
• 105-100=5assets/equations/018ad701e47e6473.svg
• b_{h_3,\mathrm{new}}=0.6-0.5(0.012994055)=0.593502972assets/equations/01910665e2cb2391.svg
• g_{batch}=\frac1B\sum_{i=1}^{B}g_i,\quad U_{epoch}=\left\lceil\frac NB\right\rceilassets/equations/01bdd22eb4365aa9.svg
• \tanh(C_t)=\tanh(0.83)\approx0.680476006assets/equations/01c5de218bff4e7f.svg
• C_2=(0.3)\times(0.145933863)\approx 0.043780159assets/equations/01c7679493ca990b.svg
• a_1=0.4+0.3+0=0.7assets/equations/01f98e63892e2cf5.svg
• P_{\mathrm{total}}=56+0+0+27=83assets/equations/021c0c327e9ed57d.svg
• (-2)(3)=-6assets/equations/02560299473dc26e.svg
• R_1=(1)\times(1)+(2)\times(0)=1assets/equations/029ae836b8cf695f.svg
• g_{2,2}^{(2)}=(-0.038098237)\times(0.596884378)\approx -0.022740242assets/equations/02d5efeb8dafde38.svg
• z_o=W_2h+b_2assets/equations/02e0daad7201e074.svg
• C_1=(0.1)\times(0.013687964)\approx 0.001368796assets/equations/02eca34bc6cd23b0.svg
• \left\lfloor\frac{105-2}{2}\right\rfloor+1assets/equations/02f6d3536f89b2ae.svg
• C_1=(0.2)\times(-0.093597941)\approx -0.018719588assets/equations/0303b36ec5b8423b.svg
• (f+g)\prime=f\prime+g\primeassets/equations/032a4e4c1f8b2817.svg
• \theta_2=-0.35+(0.27)=-0.08assets/equations/035af79f7cb5dc94.svg
• R_2=(0)\times(0)+(2)\times(-1)=-2assets/equations/03bf601080143dcc.svg
• W^{(1)}_{4,2,\mathrm{new}}=0.2-0.5(-0.001284824)\approx 0.200642412assets/equations/03e07f52a47c5589.svg
• \mathrm{saving}=4{,}718{,}592-720{,}896=3{,}997{,}696assets/equations/03fa81b05dc71237.svg
• L=\frac12(0.5-1)^2=0.125assets/equations/03fd42e4fe3cfc09.svg
• \theta_t=\theta_{t-1}-\frac{\eta g_t}{\sqrt{s_t}+\varepsilon}assets/equations/03fd4ff4e7ab2e6e.svg
• e^{2}-e^{-2}\approx7.389056099-0.135335283=7.253720816assets/equations/03ff82884c1feede.svg
• \text{weights}=3\times3\times1\times4=36assets/equations/04061505b018b4db.svg
• \delta_{h_1} = -0.11326 \times 1 = -0.11326assets/equations/0406bf395ccea595.svg
• \text{biases}=16,\qquad \text{total}=128+16=144assets/equations/040d4fe2bfbf253e.svg
• z=w_1x_1+w_2x_2+bassets/equations/04168cf0cec0712f.svg
• 1-0.9^2=0.19,\qquad1-0.999^2=0.001999assets/equations/0416e7f741fa3055.svg
• \frac{N}{B}=\frac{1000}{100}=10assets/equations/044096791068cfc7.svg
• 1+1.000000000\approx2.000000000assets/equations/048aaa762a498d29.svg
• e^{\ln a}=aassets/equations/04d92df0f222846e.svg
• R_2=(2)\times(0)+(3)\times(-1)=-3assets/equations/04e972a19789fc08.svg
• o_{1} = \max(0,0.191) = 0.191assets/equations/0510bc5bdad8a8b2.svg
• z=\sum_i w_i x_i+bassets/equations/051f16f4bf1744dd.svg
• g_{1,3}^{(1)}=(-0.009617203)\times(0.5)\approx -0.004808601assets/equations/053d52dc796832a9.svg
• \delta_t=\left(W_{hy}^{T}\delta_t^{out}+W_{hh}^{T}\delta_{t+1}\right)\odot(1-h_t^2)assets/equations/05416576e2064832.svg
• w_{1,\mathrm{new}}=0.5-0.1(0.2)=0.48assets/equations/055c0eb9628a74ae.svg
• P_7=28{,}800+120=28{,}920assets/equations/056fc82ce0b81134.svg
• e^{2z}-1=2.718281828-1=1.718281828assets/equations/057c3c395fc07789.svg
• W^{(1)}_{1,2,\mathrm{new}}=0.5-0.5(-0.004869072)\approx 0.502434536assets/equations/057d087cc10bb2a8.svg
• g_{2,\mathrm{total}}=0.3+(-0.04)=0.26assets/equations/0590c2b0e0fd2940.svg
• \delta_{h_1} = -0.049494292 \times 0.158684897 = -0.007853997assets/equations/059d2cfd4db853f1.svg
• Y_{1,1}=(1)+(-1)=0assets/equations/059edda10c2ac26e.svg
• a_{\mathrm{candidate}}=0.2(1)+0.5(0.2)+0=0.3assets/equations/05a1713a2609d0e7.svg
• a_2=0.5(2)+0.2(0.462117157)+0assets/equations/05a3c9bcda834dc2.svg
• R_1=(7)\times(1)+(6)\times(1)+(7)\times(1)=20assets/equations/05ae17650121d076.svg
• h_{\mathrm{new}}=0+(0.291312612)\approx0.291312612assets/equations/05b6b2f57ab16b9a.svg
• \begin{bmatrix}-1 & -1 & 1\\0 & 1 & -1\\0 & 1 & 1\end{bmatrix}assets/equations/05d6756a9472b44c.svg
• N_{\mathrm{weights}}=5\times5\times96\times256=614{,}400assets/equations/05edf5328cb35876.svg
• z=\frac{x-\mu}{\sigma}assets/equations/05f73c2bd4e79c0a.svg
• p=\frac1{1+e^{-z}}assets/equations/06236932ea7e53d0.svg
• wx=(1)(2)=2assets/equations/0690bd581db62ce5.svg
• w_{\mathrm{new}}=w-\eta gassets/equations/06b81f0d4da782b2.svg
• g_{1,2}^{(2)}=(-0.099059551)\times(0.731058579)\approx -0.072418335assets/equations/06ed406edd749bf9.svg
• \delta_{o_1} = 1 \times (0.473888899-1) \times 0.24931821 = -0.131169078assets/equations/06f4e7ac9debce46.svg
• g_{\mathrm{batch}}=\frac{g_1+g_2}{2},\qquad w_{\mathrm{new}}=w-\eta gassets/equations/070a9c1b54d9ac42.svg
• \delta_{o_1} = 1 \times (0.75136507-0.01) \times 0.186815602 = 0.138498562assets/equations/071d20abad3f9df8.svg
• \begin{bmatrix}4.111111111 & 5\\7.111111111 & 5.333333333\end{bmatrix}assets/equations/072844f2570c1168.svg
• (2)(3)=6assets/equations/0789e90228011be2.svg
• o_{1}=\frac{1}{1+e^{-(0.365529289)}}=0.590378257assets/equations/07b12d14b1f2a0ff.svg
• [A^T]_{ij}=A_{ji}assets/equations/07cc0861475dad67.svg
• 1(2)=2assets/equations/07ce1f7a3ced3979.svg
• \mathrm{MACs}_{1\times1}=16\times16\times8\times1\times1\times64=131{,}072assets/equations/07ded70ae81c3cd5.svg
• N_{\mathrm{biases}}=6assets/equations/07df0417eea506bc.svg
• \delta_2=(\hat y-y)\odot f_2'(z_2)assets/equations/07eda20c3b7d69f3.svg
• a=\frac{0.000000000}{2.000000000}\approx0.000000000assets/equations/081512ac7f7a1b51.svg
• \widetilde h=\tanh(Wx+U(r\odot h_{\mathrm{old}})+b)assets/equations/0823781665f17f77.svg
• 0.3(0)+0.7(1)=0+0.7=0.7assets/equations/0824814bcead44c1.svg
• C_1=(0.1)\times(0.013687964)\approx 0.001368796assets/equations/0831b555d8416461.svg
• v_2=0.999(0.001)+0.001(2)^2=0.004999assets/equations/0838089126c8ebdc.svg
• g_{1,1}^{(2)}=(0.013687964)\times(0.760423002)\approx 0.010408643assets/equations/084609daa967b1e5.svg
• W^{(2)}_{1,1,\mathrm{new}}=-0.3-0.9(-0.043523504)\approx -0.260828846assets/equations/08c4c4ecc81b818b.svg
• \delta_{o_1} = 1 \times (0.45-1) \times 1 = -0.55assets/equations/08c65e290585835f.svg
• g_{1,1}^{(2)}=(-0.093597941)\times(0.710949503)\approx -0.06654341assets/equations/08e2708ce0eb567a.svg
• \delta_{h_j}=\left(\sum_k W_{2,kj}\delta_{o_k}\right)f'(z_{h_j})assets/equations/08fe50471952c690.svg
• b_{h_1,\mathrm{new}}=1-1(0.002053687)=0.997946313assets/equations/08fecfb5b65de142.svg
• 8(8)=64assets/equations/0917529398667f13.svg
• \theta_2=0.9-\frac{0.1\times(1)}{\sqrt{5}}=0.85527864assets/equations/0919f3ad17401c14.svg
• J\prime(w)=2(1)w^{1-1}+0assets/equations/091a2e97f7d74170.svg
• w_2^{\mathrm{new}}=0.1+0.1(0)(2)=0.1assets/equations/097ad2a2a1d15af5.svg
• Y_{2,2}=(0)+(-4)+(-3)=-7assets/equations/0984e17e87f9bbc9.svg
• P_3=2{,}400+16=2{,}416assets/equations/0985f033fb9785fd.svg
• g_{1,1}^{(2)}=(-0.099059551)\times(0.731058579)\approx -0.072418335assets/equations/0995183f490671de.svg
• W^{(2)}_{1,4,\mathrm{new}}=0.2-0.5(-0.060432101)\approx 0.230216051assets/equations/09adce4276bdd712.svg
• \hat y_2=1.64(2)=3.28,\qquad L_2=\frac12(3.28-4)^2=0.2592assets/equations/09afd5ffebd67c9e.svg
• [\max(0,1),\max(0,-2),\max(0,1.5)]=[1,0,1.5]assets/equations/09e521346431d0a8.svg
• g_{2,\mathrm{total}}=-0.1+0.1=0assets/equations/09ef0e91f63c60f6.svg
• C=fC_{\mathrm{previous}}+i\widetilde C,\qquad h=o\tanh(C)assets/equations/09fa023c6943b010.svg
• [1(2),\;0(-1),\;1(0),\;1(3)]=[2,0,0,3]assets/equations/0a3991790719d436.svg
• g_{2,3}^{(1)}=(0.006542085)\times(1)\approx 0.006542085assets/equations/0a817131e93ad648.svg
• b_{h_1,\mathrm{new}}=0-0.5(-0.004869072)=0.002434536assets/equations/0aa5e626ccf3c903.svg
• \text{Sum}=C_1\approx -0.018719588assets/equations/0ab4749810e2fc16.svg
• 2z=2(-2)=-4assets/equations/0ab8a21d8aab882c.svg
• \theta_{\mathrm{look},1}=1+0.9(0)=1assets/equations/0abe86d8e2a6e122.svg
• \sigma(1)=\frac1{1+e^{-1}}\approx0.731058579assets/equations/0ac3af3fea75043b.svg
• L_{\mathrm{half}}=\frac{0.49}{2}=0.245,\qquad \mathrm{MSE}=\frac{0.49}{3}\approx0.163333333assets/equations/0add52f0daedefdc.svg
• \frac{dv}{du}=2assets/equations/0af363fd8a85c3ab.svg
• o=\operatorname{step}(h_1-2h_2-0.5)assets/equations/0af913266b36a0b9.svg
• h_{3}=\frac{1}{1+e^{-(0.56)}}=0.63645254assets/equations/0b006fd2ae332459.svg
• a^{(1)}_1=\max(0,0.3)=0.3assets/equations/0b16aa9984127202.svg
• -\ln(0.9)=-(-0.105360516)=0.105360516assets/equations/0b17db90c0b1eab0.svg
• g_i=(wx_i-y_i)x_iassets/equations/0b4939a48933eb0a.svg
• W^{(1)}_{1,1,\mathrm{new}}=0.2-0.9(0.008724562)\approx 0.192147894assets/equations/0b50d4b76d5185b7.svg
• N_{\mathrm{weights}}=120\times84=10{,}080assets/equations/0b5956346ee0f9d2.svg
• \sigma(z)=\frac{1}{1+e^{-z}},\quad \sigma'(z)=\sigma(z)[1-\sigma(z)]assets/equations/0b7ede87d0558048.svg
• v_2=0.9(0.15)-0.1(-1.35)=0.27assets/equations/0b81155b82e98b2d.svg
• z_o=w_2h+b_2,\qquad\hat y=g(z_o)assets/equations/0b90638b5849c176.svg
• z_o=1-2(0)-0.5=0.5\Rightarrow o=1assets/equations/0ba860b3391cf5c7.svg
• o=f_2(z_o)assets/equations/0badfa3430a92383.svg
• a=\frac1{1+e^{-z}}=\frac1{1.110803158}\approx0.900249511assets/equations/0bc1ac01a29b10b0.svg
• z=\begin{bmatrix}0.1\\3.8\end{bmatrix}assets/equations/0bcbcaa175e8ffb8.svg
• \mathrm{MACs}_{1\times1}=16\times16\times8\times1\times1\times64=131{,}072assets/equations/0bcc137de649f71d.svg
• s_t=\beta_2s_{t-1}+(1-\beta_2)g_t^2assets/equations/0be5e35b2aa1dc00.svg
• W^{(1)}_{1,2,\mathrm{new}}=0.2-0.5(0.000877135)\approx 0.199561432assets/equations/0bf05631c8b81136.svg
• R_2=(3)\times(1)+(3)\times(2)+(2)\times(1)=11assets/equations/0bf1f6754ed5f9c0.svg
• P_5=102{,}400+64=102{,}464assets/equations/0bf8643a16a94e64.svg
• \frac{dp}{dz}=p(1-p)assets/equations/0c0df88360e53bd3.svg
• g_{1,1}^{(1)}=(-0.22)\times(1)\approx -0.22assets/equations/0c1b77c7df67c595.svg
• \left\lfloor\frac{10-2}{2}\right\rfloor+1assets/equations/0c637a47bfb54036.svg
• e_2^2=(2)^2=4assets/equations/0c723f0acd2a3070.svg
• 4+1=5assets/equations/0c79fa7cad3c69e9.svg
• \sigma=\sqrt{0.666666667}\approx0.816496581assets/equations/0c7d94a64e886a59.svg
• z_{h_2} = (-0.2 \times 0.2) + (0.3 \times 0.4) + (0.1) = 0.18assets/equations/0c8577da1dc8776a.svg
• f_t=\sigma(W_f[h_{t-1},x_t]+b_f),\quad i_t=\sigma(W_i[h_{t-1},x_t]+b_i)assets/equations/0c991bf8824254bc.svg
• g_{1,1}^{(2)}=(-0.131169078)\times(0.331812228)\approx -0.043523504assets/equations/0cc7b6d9a9598b69.svg
• P_5=884{,}736+384=885{,}120assets/equations/0cca75bb5a08cdeb.svg
• h_2=0.2(2)+0.5(0.2)+0=0.4+0.1=0.5assets/equations/0cd838106937b9a7.svg
• g_{3,4}^{(1)}=(0.012994055)\times(-0.4)\approx -0.005197622assets/equations/0ce21e721ab1dd63.svg
• \frac{\partial L}{\partial o_1}=o_1-y_1=2.2-1=1.2assets/equations/0cee4ce79f255618.svg
• \tanh(0.14)=\frac{e^{0.28}-1}{e^{0.28}+1}=0.139092448assets/equations/0cf22c6bf34159ce.svg
• \begin{bmatrix}1 & 2 & 3\\4 & 5 & 6\\7 & 8 & 9\end{bmatrix}assets/equations/0cf23d37eabd00e4.svg
• N_{\mathrm{out}}=\left\lfloor\frac{227+2(0)-9}{1}\right\rfloor+1=219assets/equations/0d078197de388e4c.svg
• z_{h_2} = (0.25 \times 0.05) + (0.3 \times 0.1) + (0.35) = 0.3925assets/equations/0d0fc7f16047fdb4.svg
• \theta_2=0.79-0.131061=0.658939assets/equations/0d180bfeaebc1926.svg
• v_1=0.9(0)-0.1(1)=-0.1assets/equations/0d1a08b9ece0e5ac.svg
• h_{\mathrm{new}}=(0.500000000)(0.462117157)\approx0.231058579assets/equations/0d327c238e117d86.svg
• \theta_{new}=\theta-\eta g,\qquad g=\frac{\partial J}{\partial\theta}assets/equations/0d35498ffd4122e2.svg
• C_2=(0.21)\times(0.082533822)\approx 0.017332103assets/equations/0d410407b9e64f3c.svg
• 0.9\times4=3.6,\qquad(1-0.9)\times9=0.9assets/equations/0d4d3672b448c16a.svg
• 0^2+12^2=144assets/equations/0d5fe5de098f7a47.svg
• \delta_{\mathrm{output}}=\hat y-yassets/equations/0d69b7fe4a0f11cd.svg
• \frac{d}{dx}(cx^n)=cnx^{n-1},\quad\frac{d}{dx}c=0,\quad (f+g)\prime=f\prime+g\primeassets/equations/0d7c48cb1d9b9e1f.svg
• y=F(x)+xassets/equations/0d9d34020eaab826.svg
• y=3(2x+1)+4=6x+3+4=6x+7assets/equations/0da4ca75660c25f8.svg
• a'_j=\frac{m_ja_j}{q},\quad o=\sum_jw_ja'_j+b,\quad g_j=(o-y)a'_jassets/equations/0da6165a5d02d6ea.svg
• \frac{\partial L}{\partial W_{hh}}=\sum_t\delta_t h_{t-1}^Tassets/equations/0db4b816627efa5e.svg
• y_{\mathrm{avg}}=\frac{x_1+x_2+x_3+x_4}{4}assets/equations/0dcf475f3132c0a4.svg
• z_{o_2} = (0.3 \times 0.549833997) + (0.4 \times 0.544878892) + (0.1) = 0.482901756assets/equations/0e0191df53ed3741.svg
• i\widetilde C=(0.731058579)(0.761594156)\approx0.556769941assets/equations/0e17a5e53a3e2bd2.svg
• m_2=0.18+(-0.1)=0.08assets/equations/0e62fb9876ad9dac.svg
• U_{\mathrm{SGD}}=ENassets/equations/0eab2be6df842149.svg
• z_{h_2} = (0.3 \times 1) + (-0.1 \times 2) + (0.2) = 0.3assets/equations/0eb4dbe7db30c72c.svg
• g_t=J'(\theta_{\mathrm{look},t})assets/equations/0ec7b790dfa7e622.svg
• wx=(-0.5)(2)=-1assets/equations/0ee6e60703add42d.svg
• |2|=2,\qquad|-1|=1assets/equations/0f14566cc7516925.svg
• h=\max(0,-0.7)=0assets/equations/0f4049c82bf618fd.svg
• N_{\mathrm{weights}}=3\times3\times256\times384=884{,}736assets/equations/0f457ff5f3c6b74f.svg
• \left\lfloor\frac{28-2}{2}\right\rfloor+1assets/equations/0f4985548e8cd39b.svg
• u_t=\beta u_{t-1}+g_t,\qquad\theta_t=\theta_{t-1}-\eta u_tassets/equations/0f57b8541207a6df.svg
• \delta_o=\frac{\partial L}{\partial o}f_o\prime(z_o)assets/equations/0f696548e5b34122.svg
• e^{\ln x}=x\ (x>0),\qquad\ln(ab)=\ln a+\ln b\ (a,b>0)assets/equations/0f89b5131dee6b2d.svg
• b_{\mathrm{new}}=0-0.1(-0.75)=0.075assets/equations/0fa40afe6a0cd296.svg
• Y_{ij,o}=\sum_c\sum_u\sum_v K_{uvco}X_{iS+u-P,jS+v-P,c}+b_oassets/equations/0fcf0c4134fa9fe3.svg
• P_3=51{,}200+64=51{,}264assets/equations/101c0d0d89af6010.svg
• \sqrt{s_2}=\sqrt{0.46}\approx0.678232998assets/equations/10263e4a8476b1fa.svg
• 0.1(3)=0.3assets/equations/10346aae4a408d26.svg
• W^{(1)}_{1,2,\mathrm{new}}=0.4-0.9(0)\approx 0.4assets/equations/10496bc73c8c3b85.svg
• \left\lfloor\frac{14+2 \times 0-5}{1}\right\rfloor+1assets/equations/104aadf49e642e25.svg
• z_{o_2} = (0.12 \times 0.760423002) + (0.21 \times 0.782449776) + (1) = 1.255565213assets/equations/10506f88ee297ee7.svg
• o_t=\sigma(W_o[h_{t-1},x_t]+b_o),\quad h_t=o_t\odot\tanh(C_t)assets/equations/1081d04f622a619e.svg
• Y_{3,1}=(1)+(0)=1assets/equations/108e2edb8fe7afbe.svg
• R_1=(1)\times(1)+(2)\times(0)=1assets/equations/10a146899dccf86a.svg
• z=w_1x_1+w_2x_2+b,\qquad o=\begin{cases}1&z>0\\0&z\le0\end{cases}assets/equations/10b03fc6ae774e47.svg
• 0.6(0)+0.4(1)=0+0.4=0.4assets/equations/10e199f12bbe9b83.svg
• w_1=w_0-\eta g_1=1-0.1(-4)=1.4assets/equations/10fc5d59ff5f6fd3.svg
• W^{(1)}_{1,1,\mathrm{new}}=0.2-0.1(-0.22)\approx 0.222assets/equations/1103f1b43164b612.svg
• \sqrt{G_1}=\sqrt{1}=1assets/equations/1112f0b4d6224a38.svg
• z_{h_1} = (0.5 \times 2) + (0.1) = 1.1assets/equations/11165a2e9ad9cccb.svg
• \begin{aligned}\frac{(2+3)^2}{5}&=\frac{5^2}{5}\\&=\frac{25}{5}\\&=5\end{aligned}assets/equations/113124579583cb24.svg
• P_1=2{,}400+32=2{,}432assets/equations/113b757e4d085763.svg
• L=L_1+L_2=0.72+0.605=1.325assets/equations/115053bc42452eb3.svg
• p(1-p)=0.731058579\times0.268941421=0.196611933assets/equations/116080684bb307ac.svg
• \theta_2=-0.4-\frac{0.1\times(2)}{\sqrt{5}}=-0.489442719assets/equations/1161742779014e25.svg
• \sigma(z)=\frac{1}{1+e^{-z}},\quad \sigma\prime(z)=\sigma(z)[1-\sigma(z)]assets/equations/11660565b2daac98.svg
• 9-5=4assets/equations/116784e6eda9bc25.svg
• w_{\mathrm{new}}=w_{\mathrm{old}}-\eta\frac{\partial L}{\partial w}assets/equations/116cb3cc8bb58606.svg
• N_{\mathrm{out}}=\left\lfloor\frac{13-3}{2}\right\rfloor+1=6assets/equations/117abcc38db86957.svg
• J=L+\frac\lambda2\sum_jw_j^2assets/equations/119bfafa31acc3ba.svg
• g_{3,2}^{(1)}=\frac{\partial L}{\partial W^{(1)}_{3,2}}=\delta_{h_3}x_2assets/equations/11a3a8fd1c29457a.svg
• a=\tanh z=\frac{e^z-e^{-z}}{e^z+e^{-z}}assets/equations/11b2c40e39b0ee66.svg
• (ab^T)_{ij}=a_i b_jassets/equations/11b839db8ece52c1.svg
• \theta_2=0.79-0.131061=0.658939assets/equations/11da355fe098969f.svg
• 2^3=2\times2\times2=8assets/equations/11e21b993b310c0e.svg
• U_{\mathrm{batch}}=3\times1=3assets/equations/11f14a41e22f1916.svg
• [0.8(1),(-0.4)(0),0.6(1),0.2(0)]=[0.8,0,0.6,0]assets/equations/11f3e40a403e55d3.svg
• R_1=(2)\times(1)+(0)\times(0)=2assets/equations/11fea4ed5b78299e.svg
• e_3^2=(-1)^2=1assets/equations/1212dd80c80f5b96.svg
• \theta_{\mathrm{look}}=\theta_{t-1}+\beta v_{t-1},\quad g_t=J\prime(\theta_{\mathrm{look}})assets/equations/122d449c66d53648.svg
• f_1C_0=1(0)=0,\qquad i_1\widetilde C_1=0.5(0.8)=0.4assets/equations/12440f871df30d1a.svg
• 1+0.135335283\approx1.135335283assets/equations/1264450d54002695.svg
• w_{2,\mathrm{new}}=1-0.1(4)=0.6assets/equations/1268aa744d549860.svg
• q^2=q\times qassets/equations/12754a3d0dc07c58.svg
• \begin{bmatrix}4 & 3 & 4\\2 & 4 & 3\\2 & 3 & 4\end{bmatrix}assets/equations/12aa767f92286deb.svg
• g_{1,1}^{(1)}=(0.064)\times(2)\approx 0.128assets/equations/12beed29e7a45da4.svg
• g_{2,1}^{(1)}=(0.009954255)\times(0.05)\approx 0.000497713assets/equations/12dc3d7596ccb7c2.svg
• \hat{x}_i=\frac{x_i-\mu_B}{\sqrt{\sigma_B^2+\epsilon}}assets/equations/13126b94bcd9d415.svg
• n_{\mathrm{flat}}=HWCassets/equations/1321cd9ddfcf5b42.svg
• z_1=1(2)+(-2)(1)+0.1=0.1assets/equations/133f56723bb2a1d8.svg
• \frac{du}{dw}=3assets/equations/1379a65c2b33abc7.svg
• \frac14=1\div4=0.25assets/equations/13b13c0346dc6e67.svg
• R_1=(2)\times(1)+(0)\times(0)=2assets/equations/13b97b59677316ed.svg
• W^{(1)}_{4,4,\mathrm{new}}=0.8-0.5(0.001713099)\approx 0.799143451assets/equations/13c9d7893d4ecaff.svg
• z_{h_2} = (0.2 \times 0.4) + (0.4 \times 0.3) + (0.2 \times 0.5) + (0.5 \times -0.4) + (0.6) = 0.7assets/equations/13efea50d945325a.svg
• \mathrm{MACs}=H_{\mathrm{out}}W_{\mathrm{out}}C_{\mathrm{out}}K_hK_wC_{\mathrm{in}}assets/equations/13f4a3b9d3a03995.svg
• J_{L1}=L_{\mathrm{data}}+\lambda\sum_i|w_i|assets/equations/14053316f92ab54d.svg
• C_2=(0.4)\times(0.145933863)\approx 0.058373545assets/equations/14154a8890e8abb8.svg
• N_{\mathrm{weights}}=400\times120=48{,}000assets/equations/142c2516a0279849.svg
• R_1=(3)\times(1)+(2)\times(-1)=1assets/equations/142f16f0e77fc1bc.svg
• g_{1,1}^{(2)}=(-0.55)\times(0.5)\approx -0.275assets/equations/1452e508c28a6d93.svg
• U_{\mathrm{keep}}=E\left\lceil\frac NB\right\rceil,\qquad U_{\mathrm{drop}}=E\left\lfloor\frac NB\right\rfloorassets/equations/147e19d72acc4203.svg
• z=\sum_iw_ix_i+b,\qquad a=f(z)assets/equations/1489ef71177ccc4a.svg
• J=L+\lambda\sum_j|w_j|assets/equations/14b76928315b8569.svg
• \begin{bmatrix}0.6 & 0\\0.6 & 0\end{bmatrix}assets/equations/14d5a1ec1325dcc6.svg
• g_2=J'(0.9)=0.9assets/equations/14dc30bcccfa9d0f.svg
• \theta_t=\theta_{t-1}+v_tassets/equations/14de701208151b5b.svg
• v_t=\beta v_{t-1}-\eta g_tassets/equations/14e98f110476e8df.svg
• h_t=0.3+0.25\tanh(0.3)\approx0.372828153assets/equations/14fb84144802870c.svg
• b_{\mathrm{new}}=0.1-0.1(-0.3)=0.1-(-0.03)=0.13assets/equations/15129640706302ac.svg
• \tanh(z)=\frac{e^{2z}-1}{e^{2z}+1},\quad \tanh'(z)=1-\tanh^2(z)assets/equations/151466f4a43db268.svg
• \text{Sum}=C_1+C_2\approx 0.070517544assets/equations/1538baedb3bcff3c.svg
• \delta^{out}=p-yassets/equations/154ed49b87099426.svg
• f\prime(-2)=(0.119202922)(1-0.119202922)\approx0.104993585assets/equations/154ffcd0747b53aa.svg
• \frac{\partial L}{\partial o_2}=o_2-y_2=-1.1-0=-1.1assets/equations/1551245d19941245.svg
• \delta_{o_1} = 1 \times (1.08-1) \times 1 = 0.08assets/equations/156b50e8d480457c.svg
• z=s_R+s_G+s_B+bassets/equations/15d63177c46d68fd.svg
• W^{(1)}_{4,1,\mathrm{new}}=0.4-0.5(-0.001713099)\approx 0.400856549assets/equations/15f2ab91f5697a46.svg
• \sigma(z)=\frac{1}{1+e^{-z}},\qquad 0<\sigma(z)<1assets/equations/15f8d84de53f52c0.svg
• f_2C_1=0.9(0.4)=0.36,\qquad i_2\widetilde C_2=0(-0.2)=0assets/equations/1604cb4698139943.svg
• L=\frac12(o-y)^2,\quad \delta_o=o-yassets/equations/160d78635e6efcbf.svg
• H(x)=x+F(x),\qquad a_j=\max(0,H_j)assets/equations/16339c457d56baab.svg
• \begin{bmatrix}0 & 5 & 1\\0 & -2 & 2\\1 & 3 & 2\end{bmatrix}assets/equations/1659d27513ae9f09.svg
• 5(20)=100assets/equations/165b56a31834874f.svg
• H(p)\approx0.215761554+0.346573590=0.562335145assets/equations/1669c0167b94fcff.svg
• \text{Sum}=C_1+C_2\approx 0.036350306assets/equations/167389750e490b23.svg
• e_2=3-3=0assets/equations/168d50f2de11bd29.svg
• 1+7.389056099=8.389056099assets/equations/16b7ac365fcbbf5e.svg
• \ln(e^a)=a,\qquad e^{\ln a}=a\quad(a>0)assets/equations/16ba8aa03cb4b3ca.svg
• g_{2,1}^{(2)}=(0.145933863)\times(0.549833997)\approx 0.080239399assets/equations/16c40d7d34614dcb.svg
• a_t=W_{xh}x_t+W_{hh}h_{t-1}+b_hassets/equations/16c56e35b4a96bc6.svg
• h_2=0.2(2)+0.5(0.2)+0=0.5assets/equations/16e9c43648c8dc50.svg
• R_2=(7)\times(1)+(6)\times(2)+(7)\times(1)=26assets/equations/16eb873463fe23bb.svg
• z_{h_2} = (-0.3 \times 1) + (0.1 \times 0) + (0.2 \times 1) + (0.2) = 0.1assets/equations/16f057ecfbad99c5.svg
• -z=-(0)=0assets/equations/17055d0f6bb7d8cb.svg
• \theta_2=0.9-0.1(0.9)=0.81assets/equations/1708353d6ee40680.svg
• \frac{dv}{dw}=\frac{dv}{du}\frac{du}{dw}assets/equations/17086b5a41fb1edf.svg
• \sigma(0)=\frac1{1+e^0}=\frac12=0.5assets/equations/171834ffe9579c30.svg
• \delta_h=(-0.8)\times0.5\times0=0assets/equations/1721f576f8f22211.svg
• b_{o_1,\mathrm{new}}=0.1-0.9(-0.131169078)=0.218052170assets/equations/172a74d19c4bbc33.svg
• z=2+(-1.5)=0.5assets/equations/174899e28e234350.svg
• L_{\mathrm{new}}=\frac12(3-4)^2=0.5assets/equations/1762c7935645936f.svg
• C_1=(0.15)\times(-0.809)\approx -0.12135assets/equations/1763bcb69a11b8d9.svg
• a=0.2(1)+0.5(-0.1)+0=0.15assets/equations/177d51be8807173f.svg
• g_{1,1}^{(1)}=(0.004824633)\times(0.2)\approx 0.000964927assets/equations/1785a1b288336ba6.svg
• L\approx0.223143551assets/equations/178c3708a55e1ab7.svg
• L=-\ln(0.8)assets/equations/17a83c5bb645a7c5.svg
• R_3=(3)\times(1)+(2)\times(1)+(8)\times(1)=13assets/equations/17c18e38643140f3.svg
• i\widetilde C=0.5\times0.462117157=0.231058579assets/equations/17c2ce9389d35386.svg
• P=432+16=448assets/equations/17c5c3c49aac5f53.svg
• R_1=(10)\times(1)+(9)\times(1)+(5)\times(1)=24assets/equations/17e6fe356ebb57a3.svg
• g_{1,2}^{(1)}=(0.008771355)\times(0.1)\approx 0.000877135assets/equations/1850aa156783fa4e.svg
• c=\min(1,3/12)=0.25assets/equations/185f7122d90f817c.svg
• w_{\mathrm{new}}=0.2-0.1(-0.9)=0.2-(-0.09)=0.29assets/equations/187fe0232f317977.svg
• 0.1\times(-3)=-0.3assets/equations/189f052516957693.svg
• s_1=0+2^2=4assets/equations/18b84cdbfe918936.svg
• \sum_iw_i^2=4+1+0.25=5.25assets/equations/18bd7cc9173340e8.svg
• 100\frac{107627520}{112896000}\approx95.333333\%assets/equations/18ccfe0746a80b04.svg
• 15+10+10+11=46assets/equations/18d73c8a689f86bd.svg
• z_{h_2} = (-0.3 \times 1) + (0.1 \times 0) + (0.2 \times 1) + (0.2) = 0.1assets/equations/18df325704d974aa.svg
• a^2\approx0.213552267,\qquad1-a^2\approx0.786447733assets/equations/19090a10397c086b.svg
• \int x^n\,dx=\frac{x^{n+1}}{n+1}+C\quad(n\ne-1)assets/equations/19176aa5b80f6365.svg
• \text{weights}=75\times8=600assets/equations/19223176d175b12b.svg
• f=\sigma(z_f),\quad i=\sigma(z_i),\quad o=\sigma(z_o),\quad \widetilde C=\tanh(z_C)assets/equations/19237ddb83267bd8.svg
• z_{\mathrm{new}}=0.5\times1+0.5\times1+0.5=1.5assets/equations/195cc54e0942b6c7.svg
• R_2=(3)\times(0)+(2)\times(1)=2assets/equations/1986dc904d99f31e.svg
• N_{\mathrm{weights}}=400\times120=48{,}000assets/equations/19a0a131878ed831.svg
• \frac{\partial L}{\partial w}=\frac{\partial L}{\partial h}xassets/equations/19ebe8d0b832533d.svg
• A=\begin{bmatrix}1&2&3\\4&5&6\end{bmatrix}assets/equations/19f2e9948f5cc39f.svg
• \hat m_t=\frac{m_t}{1-\beta_1^t},\quad\hat v_t=\frac{v_t}{1-\beta_2^t}assets/equations/19f778fcb8d9f7b4.svg
• R_3=(3)\times(1)+(2)\times(0)+(1)\times(-1)=2assets/equations/19fc18a5d8a877eb.svg
• b_{h_2,\mathrm{new}}=0-0.05(-0.121350000)=0.006067500assets/equations/1a16f67956817315.svg
• Y_{ij}=\sum_{u=0}^{K_h-1}\sum_{v=0}^{K_w-1}X_{iS+u,jS+v}K_{uv}+bassets/equations/1a1aa2b10ea1816e.svg
• H_1=2+(-1)=1assets/equations/1a26d4fa253f0177.svg
• M=\max_j z_jassets/equations/1a279f991a1a98b5.svg
• \theta_{\mathrm{look}}=\theta_{t-1}+\beta v_{t-1}assets/equations/1a4a295c7aac3f17.svg
• r_th_{t-1}=0.5(0.4)=0.2assets/equations/1a4ee6cc7b942d74.svg
• g_b=\frac{\partial L}{\partial b}=z-yassets/equations/1a5a6192e644e19b.svg
• w_2=w_1-\eta g_2=1.4-0.1(-2.4)=1.64assets/equations/1a64bdc673ed180f.svg
• 54.598150033+1\approx55.598150033assets/equations/1a7460c08ec0eb4e.svg
• W^{(1)}_{1,3,\mathrm{new}}=-0.5-0.9(0.008724562)\approx -0.507852106assets/equations/1a852512507633c5.svg
• a_t=W_{xh}x_t+W_{hh}h_{t-1}+b,\qquad h_t=\tanh(a_t)assets/equations/1a9a9845cda07cae.svg
• \mathrm{MAE}=\frac{|1-2|+|6-4|}{2}=\frac{1+2}{2}=1.5assets/equations/1aaa963ebac2df98.svg
• \left\lfloor\frac{61-2}{2}\right\rfloor+1assets/equations/1ac02853de1445dd.svg
• f(z)=\max(0,z)assets/equations/1ad762a28c8a1a87.svg
• 0.5(0)=0assets/equations/1ad7c98370e2604f.svg
• 0.5(0)+0.3(1)=0+0.3=0.3assets/equations/1ae989326f224964.svg
• \tanh(C)=\tanh(0.331058579)\approx0.319471636assets/equations/1b0373022abd784e.svg
• w_{\mathrm{new}}=w-\eta g_{\mathrm{clip}}assets/equations/1b334e0c9202e5b4.svg
• \frac{\partial L}{\partial W_{hh}}=\sum_t\delta_t h_{t-1}^{T},\quad \frac{\partial L}{\partial W_{xh}}=\sum_t\delta_t x_t^{T},\quad \frac{\partial L}{\partial b_h}=\sum_t\delta_tassets/equations/1b44f91dfbf3ef75.svg
• Y_{1,1}=(24)+(25)+(8)=57assets/equations/1b718b308893f322.svg
• f\prime(2)=1assets/equations/1b767f92de440262.svg
• g_{4,3}^{(1)}=(-0.004282747)\times(0.5)\approx -0.002141373assets/equations/1b83f252b55ba36d.svg
• R_1=(1)\times(1)+(0)\times(0)=1assets/equations/1b9538ffab8a7d30.svg
• W^{(1)}_{1,2,\mathrm{new}}=0.15-1(0.001437581)\approx 0.148562419assets/equations/1b9c2ae970f21b09.svg
• R_2=(0)\times(0)+(2)\times(-1)=-2assets/equations/1baae78cba2e069c.svg
• J=J_{\mathrm{data}}+\lambda\sum_i|w_i|assets/equations/1baed4b772bbe05f.svg
• L_{\mathrm{half}}=\frac{\mathrm{SSE}}2assets/equations/1bdd5bbb9ca64f61.svg
• z_{h_1} = (0.15 \times 0.05) + (0.2 \times 0.1) + (0.35) = 0.3775assets/equations/1be2ce6f8e5a6ae7.svg
• \frac{\partial\mathrm{SSE}}{\partial\hat y_i}=2e_i,\quad \frac{\partial L_{\mathrm{half}}}{\partial\hat y_i}=e_i,\quad \frac{\partial\mathrm{MSE}}{\partial\hat y_i}=\frac2n e_iassets/equations/1bf294550797614e.svg
• P=55+30+30+24=139assets/equations/1c0d23dc90da3211.svg
• e_1^2=(3)^2=9assets/equations/1c1774a6ed0f013e.svg
• [3-5,3-5,7-5,7-5]=[-2,-2,2,2]assets/equations/1c77fe3ea3cb5cf4.svg
• \Delta w_2=-0.1(-0.25)=0.025assets/equations/1c844e3b7026fa3a.svg
• s=1+1=2assets/equations/1c8553f39604f98c.svg
• N_{\mathrm{out}}=\left\lfloor\frac{227+2(0)-11}{4}\right\rfloor+1=55assets/equations/1ca643e3f874dc98.svg
• e^{\ln x}=x\ (x>0),\qquad\ln(ab)=\ln a+\ln b\ (a,b>0)assets/equations/1cb05479c1668460.svg
• N_{\mathrm{biases}}=32assets/equations/1cbdbfdd12c82e10.svg
• 1-0.9=0.1assets/equations/1d3a9c6b5c275e6a.svg
• w_{2,\mathrm{new}}=0.8-0.1(0.088)=0.7912assets/equations/1d57ae029d4adfac.svg
• \theta_1=-0.5-\frac{0.1\times(-1)}{\sqrt{0.1}}=-0.183772234assets/equations/1da7d1cdeda02890.svg
• U(rh_{\mathrm{previous}})=0.5\times0.2=0.1assets/equations/1db56b816164bc83.svg
• 1\div1+3=1+3=4assets/equations/1dbb10cc9ab1aacc.svg
• D_{\mathrm{KL}}(p\Vert q)=H(p,q)-H(p)assets/equations/1dc0b474b12a2bf8.svg
• h_2=0assets/equations/1de867d02667e910.svg
• s_2=4+1^2=5assets/equations/1e29c54438050499.svg
• w_{2,\mathrm{new}}=0+0.5\times1\times1=0.5assets/equations/1e2f50640ebb5c1e.svg
• L=-[0\ln p_1+1\ln p_2+0\ln p_3]=-\ln p_2assets/equations/1e41b81b1fc86ad6.svg
• h_{2}=\frac{1}{1+e^{-(0.7)}}=0.668187772assets/equations/1e4c0a3ecd05bfd3.svg
• \frac{(w+\Delta w)^2-w^2}{\Delta w}=2w+\Delta wassets/equations/1e547bc2dd023797.svg
• \frac{d}{dz}\tanh z=\frac{2e^{2z}(e^{2z}+1)-2e^{2z}(e^{2z}-1)}{(e^{2z}+1)^2}=\frac{4e^{2z}}{(e^{2z}+1)^2}assets/equations/1eaf1c302e07c3ae.svg
• g_{1,1}^{(1)}=(-0.009617203)\times(0.4)\approx -0.003846881assets/equations/1ec98954f2916f01.svg
• h=0.5\times0.319471636=0.159735818assets/equations/1edcaccecc620bf5.svg
• e^{-a}=\frac1{e^a}assets/equations/1ee2a4da74df0c61.svg
• \sigma(0)=\frac1{1+e^0}=\frac1{1+1}=0.5assets/equations/1ef72ba6a1deae0f.svg
• \theta_1=1-0.1(1)=0.9assets/equations/1f113e7fc6ecc1f4.svg
• W^{(2)}_{1,2,\mathrm{new}}=-0.1-0.2(-0.066170086)\approx -0.086765983assets/equations/1f30020d32cef10b.svg
• b_{h_3,\mathrm{new}}=0.6-0.5(0.012994055)=0.593502972assets/equations/1f4b70cb7e5f9e49.svg
• R_1=(3)\times(1)+(2)\times(0)+(1)\times(-1)=2assets/equations/1f506dd6c963dbdb.svg
• g_t=J\prime(\theta_{t-1}),\quad \theta_t=\theta_{t-1}-\eta g_tassets/equations/1fa856180f9f092b.svg
• N_{\mathrm{out}}=\left\lfloor\frac{N+P_{\mathrm{before}}+P_{\mathrm{after}}-K}{S}\right\rfloor+1assets/equations/1fb9e63f94b876bf.svg
• w_i^{\mathrm{new}}=w_i+\eta(t-o)x_iassets/equations/1fccc40e029a1d4e.svg
• e\odot e=[0.04,0.09,0.36]assets/equations/1fddc4677249bcbb.svg
• p_i=\frac{e^{z_i-c}}{\sum_j e^{z_j-c}},\quad c=\max_jz_j,\quad L=-\ln p_{\mathrm{correct}}assets/equations/1ffdc701f4381a15.svg
• A=55(55)(96)=290400assets/equations/200bca238c47286b.svg
• a-(-b)=a+bassets/equations/204904c78b9dfc36.svg
• \delta_{h_1} = 0.019492161 \times 0.247516573 = 0.004824633assets/equations/206e786abddd6488.svg
• R_1=(1)\times(1)+(0)\times(0)=1assets/equations/206ee4c111bd14b3.svg
• y_{\max}=\max(x_1,x_2,x_3,x_4)assets/equations/208abbe18f11b701.svg
• z_{h_2}=1+1-1.5=0.5\Rightarrow h_2=1assets/equations/208b1fc802d6494c.svg
• R_3=(0)\times(1)+(2)\times(0)+(3)\times(-1)=-3assets/equations/20986846d13e4484.svg
• \sigma'(0)=0.5(1-0.5)=0.5\times0.5=0.25assets/equations/20af7f249f7a71f8.svg
• \frac{\partial L}{\partial w_2}=0.08(1.1)=0.088assets/equations/20d536b8c74dbceb.svg
• g_{\mathrm{data}}=(-0.2)\times2=-0.4assets/equations/20e7a6699528fa05.svg
• z_h=w_1x+b_1,\quad h=\max(0,z_h),\quad o=w_2h+b_2assets/equations/2107914ba4f232f1.svg
• \tanh(z)=\frac{e^{2z}-1}{e^{2z}+1},\quad \tanh\prime(z)=1-\tanh^2(z)assets/equations/210ffd0354b9c9a9.svg
• L=\frac12(-0.3)^2=\frac12(0.09)=0.045assets/equations/21397d642a37130f.svg
• g_{4,3}^{(1)}=(-0.004282747)\times(0.5)\approx -0.002141373assets/equations/2192ca5fcdf60ef8.svg
• h_{1}=\frac{1}{1+e^{-(0.3775)}}=0.593269992assets/equations/2196ffb4d15fecbe.svg
• (2-2)=0assets/equations/21c8488d16b7e891.svg
• 1+0.135335283\approx1.135335283assets/equations/21d0f7a7fd447a52.svg
• f_2C_1=0.9(0.4)=0.36,\qquad i_2\widetilde C_2=0(-0.2)=0assets/equations/21e9e3dc7975b5fa.svg
• w_{1,\mathrm{new}}=1-0.1(3)=0.7assets/equations/22080c63aaa74ec0.svg
• z_{o_1} = (0.6 \times 0.562176501) + (0.9 \times 0.627147766) + (0.5) = 1.40173889assets/equations/221339217d406589.svg
• \text{Sum}=C_1+C_2\approx 0.020617214assets/equations/22192ace89a11d90.svg
• G_t=G_{t-1}+g_t^2assets/equations/221c2b3d2079d996.svg
• \operatorname{Var}(X)=\mathbb E[(X-\mu)^2]=\mathbb E[X^2]-\mu^2assets/equations/2228ef692c83a513.svg
• \sigma(z)=\frac1{1+e^{-z}},\quad C=fC_{\mathrm{previous}}+i\widetilde C,\quad h=o\tanh(C)assets/equations/2240d8201e156ff4.svg
• C_{\mathrm{combined}}=C_A+C_B+C_Cassets/equations/2245536fdd10a080.svg
• g_{2,2}^{(2)}=(0.082533822)\times(0.782449776)\approx 0.06457857assets/equations/2264f32e74104528.svg
• b_{o_2,\mathrm{new}}=0.6-0.5(-0.038098237)=0.619049118assets/equations/2267211e36f3ce48.svg
• R_2=(0.5)\times(1)+(0.5)\times(0)+(0.3)\times(-1)=0.2assets/equations/227859c9722c8d9d.svg
• R_1=(9)\times(1)+(5)\times(1)+(6)\times(1)=20assets/equations/227f73214fffc2b3.svg
• g_1=1assets/equations/228fc3b3267b30c9.svg
• a=\frac{53.598150033}{55.598150033}\approx0.964027580assets/equations/22a5642fac7a0997.svg
• N_{\mathrm{pool}}=\left\lfloor\frac{N+2P-K}{S}\right\rfloor+1assets/equations/22a9854190bb7c9a.svg
• \widetilde h=\tanh(0.15)\approx0.148885034assets/equations/22b0ab9fe31c5dbb.svg
• \delta_{o_1} = 1 \times (0.779684759-0.7) \times 0.171776436 = 0.013687964assets/equations/22b99147ed30be36.svg
• g_{1,2}^{(2)}=\frac{\partial L}{\partial W^{(2)}_{1,2}}=\delta_{o_1}h_2assets/equations/2314d47ce6468a48.svg
• e=[0.2-0,\ 0.7-1,\ 0.4-1]=[0.2,-0.3,-0.6]assets/equations/23245cd97ea478bc.svg
• C_1=(0.8)\times(0.08)\approx 0.064assets/equations/234bc238f68f3454.svg
• R_3=(0.5)\times(1)+(0.5)\times(0)+(0.3)\times(-1)=0.2assets/equations/23643e567273ca91.svg
• 2+1=3assets/equations/23688a205b4ece73.svg
• Y_{1,1}=(-1)+(1)=0assets/equations/23722ac152853b27.svg
• v_2=0.9(0.15)-0.1(-1.35)=0.27assets/equations/23765d6232402c77.svg
• N_{\mathrm{biases}}=240assets/equations/237f8d226a0fb66a.svg
• R_1=(3)\times(1)+(2)\times(0)=3assets/equations/239b4ccf3eb098a0.svg
• W^{(1)}_{1,3,\mathrm{new}}=0.1-0.5(-0.004808601)\approx 0.102404301assets/equations/239fd2aae5bb89b7.svg
• 1+2+3=6assets/equations/23a1d761557fba99.svg
• z=\sum_iw_ix_i+bassets/equations/23c659c8111d9781.svg
• z_{o_1} = (0.4 \times 0.5) + (0.5 \times 0.3) + (0.1) = 0.45assets/equations/24202b454fb4dad2.svg
• \delta_{o_1} = 1 \times (0.191-1) \times 1 = -0.809assets/equations/24763fc6860afd3e.svg
• z=Wx+b,\qquad a=f(z)assets/equations/247718acabda6724.svg
• o_{1}=\frac{1}{1+e^{-(0.05547891)}}=0.513866171assets/equations/247adbbc93580b44.svg
• R_1=(1)\times(1)+(0)\times(-1)=1assets/equations/248757728423ee8c.svg
• a=-0.2assets/equations/24b81e63d794388e.svg
• \frac{\partial L}{\partial z}=0.731058579-1=-0.268941421assets/equations/24cc736ea2ce6146.svg
• U_{epoch}=\left\lceil\frac NB\right\rceilassets/equations/24d8d40909055c78.svg
• \widetilde h_t=\tanh(W_hx_t+U_h(r_t\odot h_{t-1})+b_h),\quad h_t=(1-z_t)\odot h_{t-1}+z_t\odot\widetilde h_tassets/equations/24dba2ef5a8c9844.svg
• N_{\mathrm{weights}}=28{,}800\times240=6{,}912{,}000assets/equations/24e6116ce6d82f93.svg
• \begin{aligned}(-0.3)^2&=(-0.3)(-0.3)\\&=0.09\end{aligned}assets/equations/24f2cf37fb12267b.svg
• a=\max(0,-1.3)=0assets/equations/2503c315df3664fa.svg
• W^{(1)}_{2,1,\mathrm{new}}=-0.2-0.2(0.003497471)\approx -0.200699494assets/equations/251cd8c98d77e90c.svg
• g_{1,1}^{(1)}=(0.008771355)\times(0.05)\approx 0.000438568assets/equations/2525430dc0c2af1f.svg
• \text{Sum}=C_1+C_2\approx 0.036350306assets/equations/2537e0a69ea64761.svg
• g_{2,2}^{(1)}=(0.017487356)\times(0.4)\approx 0.006994942assets/equations/253913dce1e316cf.svg
• N_{\mathrm{weights}}=9{,}216\times4{,}096=37{,}748{,}736assets/equations/253a54d671a02309.svg
• \theta_{\mathrm{look},2}=0.9+0.9(-0.1)=0.81assets/equations/25448224eab650c3.svg
• o=(0.2\times3)+0.1=0.6+0.1=0.7assets/equations/259368438902dc69.svg
• \sigma'(z)=-(1+e^{-z})^{-2}(-e^{-z})=\frac{e^{-z}}{(1+e^{-z})^2}assets/equations/25b5ff83cfc93c4e.svg
• \theta_t=\theta_{t-1}-\frac{\eta g_t}{\sqrt{G_t}+\epsilon}assets/equations/2633d5c0ed034026.svg
• \lambda w_1=0.2\times0.5=0.1assets/equations/26564fe78f783309.svg
• L_{\mathrm{CCE}}=-\ln p_{\mathrm{correct}}assets/equations/266c4f8c9aa94e08.svg
• v_1=0.9(0)-0.1(-1.5)=0.15assets/equations/2679c9e024274dac.svg
• g_{1,1}^{(2)}=\frac{\partial L}{\partial W^{(2)}_{1,1}}=\delta_{o_1}h_1assets/equations/267f6c273b5a3588.svg
• N_{\mathrm{weights}}=120\times5=600assets/equations/26857a3f5e70fd9d.svg
• \tanh z=\frac{e^z-e^{-z}}{e^z+e^{-z}},\qquad\frac{d}{dz}\tanh z=1-\tanh^2zassets/equations/268c7169f8f53160.svg
• 2-0.1=1.9assets/equations/26a467850584ea86.svg
• \delta_{h_2} = -0.065518559 \times 0.221712873 = -0.014526308assets/equations/26b1106bf05e76a6.svg
• s=1+0=1assets/equations/2707337aabb06170.svg
• R_1=(2)\times(1)+(1)\times(-1)=1assets/equations/271ba4631f61b76a.svg
• R_3=(2)\times(1)+(6)\times(1)+(7)\times(1)=15assets/equations/2745b26251055027.svg
• g_{1,1}^{(2)}=(-0.082490487)\times(0.802183889)\approx -0.066172539assets/equations/275f10a4f3768fee.svg
• C_1=(0.24)\times(0.013687964)\approx 0.003285111assets/equations/276b2a8c520bda99.svg
• w_1^{\mathrm{new}}=0.55+(-0.03)=0.52assets/equations/278794747f889246.svg
• f\prime(2)=(0.880797078)(1-0.880797078)\approx0.104993585assets/equations/278a4098b6b446b4.svg
• m_t=\beta m_{t-1}+(1-\beta)g_t,\qquad0\le\beta<1assets/equations/278e38d8879b5e28.svg
• \delta_{h_1} = 0.036350306 \times 0.241300709 = 0.008771355assets/equations/2792e998f36b387c.svg
• e=[0.2-0,\ 0.7-1,\ 0.4-1]=[0.2,-0.3,-0.6]assets/equations/27a2e2729d22f591.svg
• g_{2,2}^{(1)}=(-0.275)\times(2)\approx -0.55assets/equations/27b6c549a9eafc2d.svg
• R_1=(0.5)\times(1)+(0.5)\times(0)+(0.3)\times(-1)=0.2assets/equations/27b98fade189b3cb.svg
• g_{1,3}^{(1)}=(0.008724562)\times(1)\approx 0.008724562assets/equations/27c65cc6a73f6ef7.svg
• h=(1-z)\odot h_{\mathrm{old}}+z\odot\widetilde hassets/equations/27ea37a678ab26e6.svg
• z_{o_1} = (0.25 \times 0.731058579) + (0.25 \times 0.731058579) + (0) = 0.365529289assets/equations/28292662ef4f2232.svg
• \begin{aligned}\sqrt9&=3\\\sqrt{0.09}&=0.3\end{aligned}assets/equations/2838a7934ce26d9d.svg
• z^{(l)}=W^{(l)}a^{(l-1)}+b^{(l)},\qquad a^{(l)}=f(z^{(l)})assets/equations/2845b5af7a9f9056.svg
• z_1=1+1-0.5=1.5assets/equations/285947e47dfebe51.svg
• g_{2,2}^{(1)}=(0.009954255)\times(0.1)\approx 0.000995425assets/equations/28799f80ac592874.svg
• \sigma=\sqrt{0.666666667}\approx0.816496581assets/equations/28845fc5f4238747.svg
• \delta_{o_2} = 1 \times (0.778261739-0.3) \times 0.172570404 = 0.082533822assets/equations/288db260700d28bd.svg
• g_{1,2}^{(2)}=(-0.809)\times(0.48)\approx -0.38832assets/equations/289592bf5d88c4d2.svg
• \widetilde C=\tanh(0)\approx0.000000000assets/equations/289a06589f8d4dfc.svg
• w_{ji,\mathrm{new}}=w_{ji}-\eta\delta_j a_i,\quad b_{j,\mathrm{new}}=b_j-\eta\delta_jassets/equations/289f74bc5d59b047.svg
• g_{1,2}^{(2)}=(-0.121439988)\times(0.544878892)\approx -0.066170086assets/equations/28a4d3d30f2657c9.svg
• w_{\mathrm{new}}=2-0.007=1.993assets/equations/28a66f5f8853795a.svg
• 2z=2(2)=4assets/equations/28a6e73374f6be65.svg
• (-2)(1)=-2assets/equations/28ad4ad477ec80b7.svg
• z=Wx+bassets/equations/28b5d89094e7d00d.svg
• W^{(2)}_{2,1,\mathrm{new}}=0.12-1(0.062760617)\approx 0.057239383assets/equations/28cd106ebaec1141.svg
• m_t=\beta_1m_{t-1}+(1-\beta_1)g_t,\quad v_t=\beta_2v_{t-1}+(1-\beta_2)g_t^2assets/equations/28d34b30a47142d0.svg
• z=\sum_{i=1}^{d}w_ix_i+b=w^Tx+bassets/equations/28f8b2ebee0ebbf6.svg
• J=J_{\mathrm{data}}+\frac\lambda2\sum_iw_i^2assets/equations/29419cadd5578572.svg
• 1+1.000000000=2.000000000assets/equations/29756201dd5e46c6.svg
• g_t=\left.\frac{\partial L}{\partial w}\right|_{w_{t-1}}=\frac{\partial L}{\partial\hat y}\frac{\partial\hat y}{\partial w}=(\hat y-y)xassets/equations/297f87410a0b6ab4.svg
• rh_{\mathrm{old}}=0.5(0.4)=0.2assets/equations/298840de1c01f101.svg
• U=5+1=6assets/equations/298ed637938b4bf6.svg
• x-\mu=3-5=-2assets/equations/29b71ab0c556c6dc.svg
• P_3=614{,}400+256=614{,}656assets/equations/29c935e56c01ee26.svg
• \theta_{look}=\theta+\beta v,\quad v_{new}=\beta v-\eta g(\theta_{look}),\quad\theta_{new}=\theta+v_{new}assets/equations/29d3ddbf9730bc3a.svg
• z=\sum_iw_ix_i+b,\qquad p=\frac1{1+e^{-z}},\qquad L=-\ln passets/equations/2a2f870df03533ca.svg
• \widetilde h=\tanh(Wx+U(rh_{\mathrm{old}})+b)assets/equations/2a33ca0d61aaaa89.svg
• z=(0.2)(1)+(0.1)(2)+0=0.4assets/equations/2a3bdd44838a1ac8.svg
• \sum_i|w_i|=2+1+0.5=3.5assets/equations/2a71081b4845e558.svg
• W_{ji,\mathrm{new}}=W_{ji}-\eta\frac{\partial L}{\partial W_{ji}}assets/equations/2a87ae98981ac797.svg
• \mathrm{saving\ percent}=\frac{3{,}997{,}696}{4{,}718{,}592}\times100=84.722222\%assets/equations/2a96913f36caca06.svg
• z^{(1)}_1=0.2(1)+0.4(0)+0.1=0.3assets/equations/2aa7feda8e9b9e38.svg
• g(\theta)=0.7(3)\theta^2=2.1\theta^2assets/equations/2aadd5dc4445f1ba.svg
• x_2w_2=1(-0.2)=-0.2assets/equations/2aecb1c03e00ecf3.svg
• C_2=(0.55)\times(-0.038098237)\approx -0.02095403assets/equations/2b032a83e37967d3.svg
• N_{\mathrm{weights}}=5\times5\times64\times128=204{,}800assets/equations/2b0e3ca2da50e61a.svg
• c_t=f_tc_{t-1}+i_t\widetilde c_tassets/equations/2b17315e0dde1dde.svg
• H(p,q)\approx0.519860385+0.173286795\approx0.693147181\ \text{nats}assets/equations/2b18f4e26d74ffd0.svg
• \widehat m_2=\frac{0.28}{1-0.9^{2}}=1.473684211assets/equations/2b1a3823c4815fe2.svg
• a'_1=\frac{1\times1}{0.5}=2,\qquad a'_2=\frac{0\times2}{0.5}=0assets/equations/2b32d1a861daf9ff.svg
• H_2=-3+1=-2assets/equations/2b33025f78e725f0.svg
• a=\max(0,0.2)=0.2assets/equations/2b62624e146d1934.svg
• h_{1} = \max(0,0.3) = 0.3assets/equations/2b99f7dd17297569.svg
• b_{h_1,\mathrm{new}}=0-0.05(-0.113260000)=0.005663000assets/equations/2bb2563809fc624e.svg
• h=o\odot\tanh Cassets/equations/2bb414a00be2c088.svg
• \frac{\partial L}{\partial z}=p-yassets/equations/2bbef5a5d1c7047f.svg
• w_1^{\mathrm{new}}=0.2+0.1(-1)(1)=0.1assets/equations/2bd36e593c9db7e2.svg
• R_2=(2)\times(2)+(1)\times(0)+(2)\times(-2)=0assets/equations/2be91abc57b10ecb.svg
• z=\sum_iw_ix_i+b,\qquad a=f(z)assets/equations/2beefda557917f96.svg
• g_{2,2}^{(2)}=(0.145933863)\times(0.544878892)\approx 0.079516282assets/equations/2bf8871d8771b0c8.svg
• w_2^{\mathrm{new}}=0.65+0.025=0.675assets/equations/2bfe89025b1bd27a.svg
• \theta_1=-0.5-0.1(-1.5)=-0.35assets/equations/2c2dd4efda1f26eb.svg
• g_{\mathrm{total}}=-0.4+0.04=-0.36assets/equations/2c43044f63438eb7.svg
• P_{\mathrm{total}}=40+0+0+195=235assets/equations/2c4f62d57a39f2bf.svg
• b_{h_1,\mathrm{new}}=0.35-0.5(0.008771355)=0.345614322assets/equations/2c688c2612d04439.svg
• v_1=0-0.1=-0.1assets/equations/2c6e04e58352c2ce.svg
• s_2=0.9(0.1)+0.1(2)^2=0.49assets/equations/2c7113e75687c23e.svg
• g_{2,1}^{(1)}=(0.017487356)\times(0.2)\approx 0.003497471assets/equations/2c7569fc93ce3696.svg
• 54.598150033+1\approx55.598150033assets/equations/2c86d55c9549cedd.svg
• \theta_2=0.9+(-0.171)=0.729assets/equations/2c96483424ef8929.svg
• \begin{aligned}-3-(-2)&=-3+2\\&=-1\end{aligned}assets/equations/2cc0c9e475076c06.svg
• \hat y=0.4\times2=0.8,\qquad \hat y-y=0.8-1=-0.2assets/equations/2cdfc715227663d5.svg
• b_{h_4,\mathrm{new}}=0.6-0.5(-0.004282747)=0.602141373assets/equations/2d5702d767818373.svg
• g_{1,2}^{(2)}=(0.013687964)\times(0.782449776)\approx 0.010710144assets/equations/2d59dee332a4bda3.svg
• C_{12}=1(0)+2(3)=0+6=6assets/equations/2d6dd50233caecfb.svg
• a=\frac1{1+e^{0}}=\frac1{1+1}=0.5assets/equations/2d7874ea2511447b.svg
• 2-(-0.3)=2+0.3=2.3assets/equations/2d796303b17c0e32.svg
• \mathrm{MAE}=\frac{1+2}{2}=1.5assets/equations/2d85cc8a7db473e8.svg
• z_{XOR}=1-2(0)-0.5=0.5assets/equations/2d91f383d484e960.svg
• C_1=(0.5)\times(-0.55)\approx -0.275assets/equations/2daae1e05e9f3afa.svg
• b_{h_2,\mathrm{new}}=0-0.05(-0.121350000)=0.006067500assets/equations/2db77709ff0c80b8.svg
• e^{0}-e^{0}\approx1.000000000-1.000000000=0.000000000assets/equations/2df025bea6def52b.svg
• 0.6/0.5=1.2assets/equations/2e2dc3409b6b4b5c.svg
• \frac{dL}{dz}=p-yassets/equations/2e2f1b40461ded43.svg
• \widetilde h_t=\tanh(0.3)\approx0.291312612assets/equations/2e3e84d154de9f24.svg
• \frac{\partial L}{\partial h}=\sum_j(o_j-y_j)\frac{\partial o_j}{\partial h}assets/equations/2e472e733a927b24.svg
• N_{\mathrm{weights}}=28{,}800\times240=6{,}912{,}000assets/equations/2e815f655162e42b.svg
• P_5=884{,}736+384=885{,}120assets/equations/2e9338e40808aff5.svg
• N_{\mathrm{weights}}=3\times3\times3\times16=432assets/equations/2eb50e4698b8f7ed.svg
• \frac{\partial L}{\partial z_i}=p_i-y_iassets/equations/2f0494c85fef2342.svg
• z^{(1)}_1=0.2(1)+0.4(0)+0.1=0.3assets/equations/2f1a7c307bfbc37e.svg
• W^{(2)}_{1,2,\mathrm{new}}=-0.2-0.9(-0.068861036)\approx -0.138025068assets/equations/2f20144355441ad1.svg
• z_1=1(2)+(-2)(1)+0.1=0.1assets/equations/2f3707d9acaff1b9.svg
• y=[1.1,1.8,3.3]assets/equations/2f3d1f51931e7a8d.svg
• a_{\mathrm{candidate}}=0.2(1)+0.5(0.2)+0=0.3assets/equations/2f40a657a883a8d8.svg
• p=\frac1{1.367879441}\approx0.731058579assets/equations/2f55393e129c6628.svg
• \theta_t=\theta_{t-1}-\frac{\eta\hat m_t}{\sqrt{\hat v_t}+\varepsilon}assets/equations/2f5e5116ef55c3b3.svg
• J(\theta+\Delta\theta)\approx J(\theta)+\nabla J(\theta)^T\Delta\thetaassets/equations/2f6a587716d3d081.svg
• z_{h_1} = (0.2 \times 1) + (0.4 \times 0) + (-0.5 \times 1) + (-0.4) = -0.7assets/equations/2f70d876265d1c94.svg
• \mathrm{MSE}=\frac{0.49}{3}=0.163333333\ldotsassets/equations/2f7b20630e72d239.svg
• \sigma=\sqrt{\sigma^2},\qquad x_{\mathrm{std}}=\frac{x-\mu}{\sigma}assets/equations/2f7fcc7ba3415fbf.svg
• \theta_{\mathrm{look}}=0.9+0.9(-0.1)=0.81assets/equations/2f8c51c39237a8f3.svg
• g_{2,2}^{(1)}=(-0.12135)\times(3)\approx -0.36405assets/equations/2f8d27aa1e3f5627.svg
• N_{\mathrm{out}}=\left\lfloor\frac{61-2}{2}\right\rfloor+1=30assets/equations/2fa03284017c373e.svg
• \widehat y=\operatorname{step}(w^Tx+b),\quad w_{new}=w+\eta(y-\widehat y)x,\quad b_{new}=b+\eta(y-\widehat y)assets/equations/2fd235815e859770.svg
• z^{(2)}=0.15+0.30+0.10=0.55assets/equations/2fda741950753a37.svg
• \lfloor1.5\rfloor=1assets/equations/2ff7c16eb83493b5.svg
• W^{(1)}_{2,2,\mathrm{new}}=0.3-1(0.002456654)\approx 0.297543346assets/equations/300691e5cdc3e69a.svg
• a=\frac{-0.981684361}{1.018315639}\approx-0.964027580assets/equations/300fc0089c5516db.svg
• \begin{bmatrix}1 & 0 & 1\\0 & 1 & 0\\1 & 0 & 1\end{bmatrix}assets/equations/305a3f29286c4e55.svg
• \bar x_w=\frac{\sum_i a_ix_i}{\sum_i a_i}assets/equations/305c0c88eaed50a3.svg
• G_t=G_{t-1}+g_t^2,\quad\theta_t=\theta_{t-1}-\frac{\eta g_t}{\sqrt{G_t}+\epsilon}assets/equations/3060020ef7a35ed4.svg
• g_v=\sum_t\delta_th_{t-1}assets/equations/3065785a32d83c15.svg
• h=0.5\times0.319471636=0.159735818assets/equations/3071cb7f6661d372.svg
• N_{\mathrm{out}}=\left\lfloor\frac{252-2}{2}\right\rfloor+1=126assets/equations/3083aa89c3311d57.svg
• v_2=0.999(0.004)+0.001(1)^2=0.004996assets/equations/308792cf26a22040.svg
• 0.9(0.2)=0.18assets/equations/30b407cc303a0e19.svg
• z_{o_1} = (0.14 \times 0.85) + (0.15 \times 0.48) + (0) = 0.191assets/equations/30c6287eca7be224.svg
• W^{(2)}_{1,3,\mathrm{new}}=-0.6-0.5(-0.059570648)\approx -0.570214676assets/equations/30d92749f17fd5d6.svg
• u_{\mathrm{new}}=0.2-0.1(-1.25)=0.325assets/equations/30e22de1103f6dec.svg
• W^{(1)}_{2,2,\mathrm{new}}=0.3-0.2(0.006994942)\approx 0.298601012assets/equations/30eae54a2d6333bf.svg
• g_{1,1}^{(2)}=(-0.099059551)\times(0.731058579)\approx -0.072418335assets/equations/30f7bf8aa04d2bf2.svg
• L_2=\frac12(-1.1-0)^2=0.605assets/equations/31043c1e17e75c9f.svg
• \text{biases}=8,\qquad \text{total}=600+8=608assets/equations/311dd5c76d87d54c.svg
• m_1=(1-\beta)g_1assets/equations/311f8a66ba1558a5.svg
• g_{4,1}^{(1)}=(-0.004282747)\times(0.4)\approx -0.001713099assets/equations/313f27074af89747.svg
• \tanh(0)\approx\frac{0.000000000}{2.000000000}\approx0.000000000assets/equations/3179ee482059cbdd.svg
• a=\frac1{1+e^{-z}}assets/equations/3188e642a44baa47.svg
• J_{L2}=0.4+0.2625=0.6625assets/equations/318e776fa7639b85.svg
• z_1=1+0-0.5=0.5assets/equations/318ee2a6640d10f0.svg
• b_{h_1,\mathrm{new}}=0.1-0.1(-0.220000000)=0.122000000assets/equations/31940cedcf20decc.svg
• \delta_{h_2} = 0.041370323 \times 0.240613417 = 0.009954255assets/equations/31d2f2dbdaea47ab.svg
• N_{\mathrm{out}}=\left\lfloor\frac{126-2}{2}\right\rfloor+1=63assets/equations/31d9e5cfd413812f.svg
• \widetilde C_t=\tanh(W_c[h_{t-1},x_t]+b_c),\quad C_t=f_t\odot C_{t-1}+i_t\odot\widetilde C_tassets/equations/31dbb5df72a8ba34.svg
• \|g\|_2=\sqrt{25}=5assets/equations/31e01f04f62bf975.svg
• W^{(1)}_{2,1,\mathrm{new}}=0.28-1(0.000877376)\approx 0.279122624assets/equations/31e044948722082e.svg
• \hat y=2+(0)=2assets/equations/31e3d42796c5dd84.svg
• N_{flat}=HWCassets/equations/31ebc79afc790262.svg
• \mathrm{ReLU}(-2)=\max(0,-2)=0assets/equations/31ec7384f72f0b9e.svg
• h_2=\operatorname{step}(x_1+x_2-1.5)assets/equations/31fad4d0ea8f08ac.svg
• \max(0,-3)=0,\quad \max(0,0)=0,\quad \max(0,2)=2assets/equations/3202f6e374a86c4c.svg
• \widetilde h_t=\tanh(W_hx_t+U_h(r_t\odot h_{t-1})+b_h),\quad h_t=z_t\odot h_{t-1}+(1-z_t)\odot\widetilde h_tassets/equations/3206de1fb9907f41.svg
• W^{(2)}_{2,2,\mathrm{new}}=0.4-0.2(0.079516282)\approx 0.384096744assets/equations/320cca8c034f0c86.svg
• \frac{\partial L}{\partial b}=\frac{\partial L}{\partial h}(1)=3.5assets/equations/327ca2cf6e5be3c3.svg
• z=w_1x_1+w_2x_2+\cdots+w_nx_n+b,\qquad a=f(z)assets/equations/32a736852ff7774e.svg
• w_1=w_0-\eta g_1=1-0.1(-4)=1.4assets/equations/32a8eeeef9367ad9.svg
• Y_{2,1}=(0.2)+(0.2)+(0.2)=0.6assets/equations/32aaa578d4b2a675.svg
• z_{o_2} = (0.12 \times 0.760423002) + (0.21 \times 0.782449776) + (1) = 1.255565213assets/equations/32b3910a4ac203c7.svg
• h_3=0.7\tanh(-0.228)=0.7(-0.224129674)=-0.156890772assets/equations/32c2f7dd58700393.svg
• \delta_{o_1} = 1 \times (0.513866171-1) \times 0.249807729 = -0.121439988assets/equations/32cdbe21f65e2b06.svg
• N_{\mathrm{out}}=\left\lfloor\frac{128+2(0)-3}{1}\right\rfloor+1=126assets/equations/32d2786097210b6e.svg
• s_t=s_{t-1}+g_t^2assets/equations/32f05266b341eaa2.svg
• 0.9^2=0.9(0.9)=0.81assets/equations/3338c806844083ae.svg
• Y_{ij}=\sum_{u=0}^{K_h-1}\sum_{v=0}^{K_w-1}X_{iS+u,jS+v}K_{uv}+bassets/equations/33407dafb559f8d6.svg
• \delta_{h_1} = 0.019492161 \times 0.247516573 = 0.004824633assets/equations/3348b1ba7e091393.svg
• W^{(2)}_{2,2,\mathrm{new}}=0.21-1(0.06457857)\approx 0.14542143assets/equations/334d93272bd11b36.svg
• w_2=1.1-0.1(-3.6)=1.46assets/equations/337ee1d4f3248df4.svg
• \frac{d(z^2)}{dz}=2zassets/equations/33825b3d129b23da.svg
• \frac{\partial L}{\partial w_2}=\delta_o h,\qquad\frac{\partial L}{\partial b_2}=\delta_oassets/equations/338a3082f0298cb5.svg
• \begin{bmatrix}0.5 & 0.3\\0.2 & 0.4\end{bmatrix}assets/equations/339cb87bfb0225e4.svg
• N_{flat}=HWC,\qquad P_{dense}=(N_{in}+1)N_{out}assets/equations/33a05d43681713eb.svg
• b_{1,\mathrm{new}}=0.1-0.1(0.064)=0.0936assets/equations/33bdb6cbe9ce338c.svg
• \sqrt{G_2}=\sqrt{5}=2.236067977assets/equations/33c414269f4796af.svg
• w_{\mathrm{new}}=w-\eta g_wassets/equations/33df8d5a70c57c14.svg
• \begin{bmatrix}12 & 20 & 30 & 0\\8 & 12 & 2 & 0\\34 & 70 & 37 & 4\\112 & 100 & 25 & 12\end{bmatrix}assets/equations/33f58372f0707267.svg
• \sigma^2=\frac{1+0+1}{3}\approx0.666666667assets/equations/33fa105b83cd3564.svg
• m_t=(1-\beta)\sum_{k=1}^{t}\beta^{t-k}g_kassets/equations/3400900697b48b50.svg
• \hat y=0.4\times2=0.8,\qquad \hat y-y=0.8-1=-0.2assets/equations/3414961b88098bc1.svg
• e^{2z}+1=2.718281828+1=3.718281828assets/equations/343ff32a5ddf3a45.svg
• \frac{\partial J}{\partial w_i}=g_{\mathrm{data},i}+2\lambda w_iassets/equations/3465dcd568b96ffb.svg
• \frac{\partial\mathrm{SSE}}{\partial\hat y_j}=2e_j,\qquad \frac{\partial L_{\mathrm{half}}}{\partial\hat y_j}=e_j,\qquad \frac{\partial\mathrm{MSE}}{\partial\hat y_j}=\frac{2}{n}e_jassets/equations/3476c309d4e1d90b.svg
• z^{(2)}=0.5(0.3)+0.6(0.5)+0.1assets/equations/3480e69d1dd395dc.svg
• w_{new}=w-\eta(g_{data}+\lambda w)assets/equations/349319f842d12440.svg
• N_{\mathrm{weights}}=107{,}648\times84=9{,}042{,}432assets/equations/3496f06ddb91fc91.svg
• z=\sum_i w_ix_i+bassets/equations/3497cbcbfa494dd1.svg
• (1-0.8)(0.148885034)\approx0.029777007assets/equations/349c3a8b18523729.svg
• g_2=(-1.8)(2)=-3.6assets/equations/352a3c332b7e819d.svg
• \frac{dL}{dz}=(-1.25)(0.16)=-0.2assets/equations/352e50cd272ef421.svg
• g_{2,2}^{(1)}=(0.009954255)\times(0.1)\approx 0.000995425assets/equations/3534da4dd6514b21.svg
• \theta_2=-0.35-0.1(-1.35)=-0.215assets/equations/354b5f7978875963.svg
• Y_{ij}=\sum_{u,v}X_{iS+u,jS+v}K_{uv}+bassets/equations/357d6cd4ecbfd331.svg
• \text{parameters}=\text{weights}+\text{biases}assets/equations/35882bbb02ddb8aa.svg
• z=(-0.2)(1)+(0.1)(2)+0=0assets/equations/358f13c8da68054b.svg
• 6(16)=96assets/equations/35a3fea4574170f0.svg
• w_1^{\mathrm{new}}=-0.2+0.1(0)(1)=-0.2assets/equations/35b1f47fa8ef32d6.svg
• e_1=4-1=3assets/equations/35c4c4ba3ef807a6.svg
• C_{\mathrm{retained}}=f\odot C_{\mathrm{old}}assets/equations/35cedc4f6bc20c8a.svg
• g_{2,1}^{(2)}=(-0.038098237)\times(0.593269992)\approx -0.02260254assets/equations/35e5314b87f154a8.svg
• v_2=0.9(-0.1)-0.1(0.81)=-0.171assets/equations/35f06a75a728a22a.svg
• 0.3(0)+0.7(1)=0+0.7=0.7assets/equations/360d1cebae87086f.svg
• h_t=z_th_{t-1}+(1-z_t)\widetilde h_tassets/equations/360ddf17a95ce9ee.svg
• \mathrm{SSE}=0.04+0.09+0.36=0.49assets/equations/36128683498bbba3.svg
• p_2=\frac{1}{1.503214724}=0.665240956assets/equations/361efa91829285c7.svg
• \sigma(z)=\frac{1}{1+e^{-z}},\qquad 0<\sigma(z)<1assets/equations/36359a5421bdb1af.svg
• P_11=16{,}777{,}216+4{,}096=16{,}781{,}312assets/equations/3652163d527344ad.svg
• \hat{x}_1=\frac{-1}{d}\approx-0.99999500004assets/equations/3676422778896944.svg
• L=-\ln(0.731058579)\approx0.313261688assets/equations/36829459a4aa6b4b.svg
• H_o=W_o=\left\lfloor\frac{N+2P-K}{S}\right\rfloor+1assets/equations/368322f7f93c9e79.svg
• w^Tx=(2)+(-6)=-4assets/equations/368a39aeec6c928d.svg
• a^{(1)}=\begin{bmatrix}0.3\\0.5\end{bmatrix}assets/equations/368b9c59527b8200.svg
• R_3=(0)\times(1)+(2)\times(0)+(3)\times(-1)=-3assets/equations/369116c3da370aea.svg
• \tanh(z)=\frac{e^z-e^{-z}}{e^z+e^{-z}},\qquad -1<\tanh(z)<1assets/equations/369af2e85f2204d4.svg
• z-c=\begin{bmatrix}-2&-1&0\end{bmatrix}assets/equations/36a969786afa504d.svg
• h_2=0.7\tanh(0.36)=0.7(0.345214034)=0.241649824assets/equations/36c9d733b0dbfd49.svg
• z=0.1(2)+(-0.2)(5)+0.5(1)+0.3assets/equations/36e8e2cc2919ceb8.svg
• \text{parameters}=\text{weights}+\text{biases}assets/equations/37224dd4ef25f22f.svg
• \widetilde C=\tanh(1)\approx0.761594156assets/equations/37268273eb5c85e7.svg
• N_{\mathrm{weights}}=3\times3\times384\times384=1{,}327{,}104assets/equations/3729f79bf2b69fab.svg
• \theta_1=-0.5-\frac{0.1\times(-1)}{\sqrt{1}}=-0.4assets/equations/37395f4e33d5e1e8.svg
• J=\frac{L_1+L_2+L_3}{N}assets/equations/3763c395ed6a5c3c.svg
• \widetilde h_t=\tanh(Wx_t+U(r_t\odot h_{t-1})+b)assets/equations/37663e895cb447b3.svg
• o_{2}=\frac{1}{1+e^{-(1.224921404)}}=0.772928465assets/equations/378ac86a046b16fc.svg
• \begin{bmatrix}3 & 5\\3 & -7\end{bmatrix}assets/equations/3797d8870cefe33e.svg
• 1(0)=0assets/equations/3798c4898b2e9ea7.svg
• s=0.6+0.6=1.2assets/equations/379a82d19e479abf.svg
• \text{Sum}=C_1\approx 0.064assets/equations/37af0a1e41171eca.svg
• b_{o_1,\mathrm{new}}=0.6-0.5(0.138498562)=0.530750719assets/equations/37ec194c9837cd70.svg
• R_1=(0)\times(1)+(1)\times(-1)=-1assets/equations/37f1662710810d02.svg
• N_{\mathrm{weights}}=4{,}096\times4{,}096=16{,}777{,}216assets/equations/37f33b938916c485.svg
• y_{\max}=13assets/equations/37fcb9070f6a3c0a.svg
• J\prime(1)=0.7(3)(1)^{2}=2.1assets/equations/38040daaa10cb32e.svg
• \frac{\partial h_T}{\partial h_t}=\prod_{k=T}^{t+1}\frac{\partial h_k}{\partial h_{k-1}}assets/equations/382feba776054adf.svg
• y_2=2(0.99999500004)+0.5\approx2.49999000008assets/equations/3848a4f079b59c74.svg
• \frac{\partial L}{\partial b}=-0.052877093assets/equations/385f9ff4016d4aa5.svg
• \begin{bmatrix}0.25 & 0.75\\0.75 & 0.75\end{bmatrix}assets/equations/386a0646a3bb55b5.svg
• \begin{bmatrix}0.11 & 0.21\\0.12 & 0.08\end{bmatrix}assets/equations/386eee5b3bdd13ab.svg
• \Delta w_i=\eta(t-o)x_iassets/equations/3870c6199f4ce08e.svg
• L_{\mathrm{half}}=\frac12\sum_i(\hat y_i-y_i)^2assets/equations/38b125f3e664a9e5.svg
• R_1=(0)\times(1)+(1)\times(0)=0assets/equations/38b1e54a3b119ea9.svg
• 0.018315639-1\approx-0.981684361assets/equations/38bd067833181d6c.svg
• q=1-0.5=0.5assets/equations/39070c1933e29986.svg
• \mathrm{MACs}_{\mathrm{bottleneck}}=2408448+10035200=12443648assets/equations/3917ad1b6a4cc0eb.svg
• \text{weights}=K^2C_{\mathrm{in}}C_{\mathrm{out}},\quad \text{biases}=C_{\mathrm{out}}assets/equations/391b54fdb0c0ef2b.svg
• \widehat v_2=\frac{0.004996}{1-0.999^{2}}=2.499249625assets/equations/3922f5c0c13f3037.svg
• z_h=W_1x+b_1assets/equations/3929c1e524affa01.svg
• g_{2,1}^{(2)}=(0.082533822)\times(0.760423002)\approx 0.062760617assets/equations/393cb494838a4f33.svg
• \widetilde C=\tanh(0.5)=0.462117157assets/equations/396b8090ffcbdc51.svg
• \hat y_0=w_0x=1(2)=2,\qquad e_0=\hat y_0-y=2-4=-2assets/equations/396cd8262d595e6d.svg
• v=2(4)=8assets/equations/396fa5f1dfbfd157.svg
• h_2=1assets/equations/397f5515899ffae5.svg
• g_{3,1}^{(1)}=(0.012994055)\times(0.4)\approx 0.005197622assets/equations/3982bafceb3d1f9a.svg
• \sigma(2)=\frac1{1.135335283}\approx0.880797078assets/equations/398e028b924c49e3.svg
• 3(4)=12,\qquad3(5)=15assets/equations/399713a01b02cfa8.svg
• \hat s_t=\frac{s_t}{1-\beta_2^t}assets/equations/39aff9d162818e1c.svg
• g_{4,4}^{(1)}=(-0.004282747)\times(-0.4)\approx 0.001713099assets/equations/39b46eb4072b2e58.svg
• h_t=0.5\tanh(C_t)\approx0.159735818assets/equations/39bac92ae8b9ef62.svg
• w_{2,\mathrm{new}}=0.3-0.2(0)=0.3assets/equations/39fa74bf610f2797.svg
• f_3C_2=0.2(0.36)=0.072,\qquad i_3\widetilde C_3=0.5(-0.6)=-0.3assets/equations/39fc048515ad2602.svg
• H(p,q)=-[0.75\ln(0.5)+0.25\ln(0.5)]assets/equations/3a2a7c8cae7d7ea4.svg
• \hat y=3+(1)=4assets/equations/3a2f92dc89e2b2d2.svg
• \frac{\partial J}{\partial w}=(\hat y-y)x+\lambda w,\qquad w_{\mathrm{new}}=w-\eta\frac{\partial J}{\partial w}assets/equations/3a34fcc434834bc2.svg
• g_{1,2}^{(1)}=(-0.004869072)\times(1)\approx -0.004869072assets/equations/3a5cd9a960e8e0d8.svg
• P_10=37{,}748{,}736+4{,}096=37{,}752{,}832assets/equations/3a6d14bbf5757207.svg
• P_8=9{,}042{,}432+84=9{,}042{,}516assets/equations/3a73b69ea184fcd0.svg
• \frac{dL}{dz}=2z=2(4)=8assets/equations/3a7684d15448f2e4.svg
• 16(8)=128assets/equations/3a81b7540b48b829.svg
• o_{1}=\frac{1}{1+e^{-(1.105905967)}}=0.75136507assets/equations/3a8725cabdafc869.svg
• g_{1,1}^{(2)}=(-0.55)\times(0.5)\approx -0.275assets/equations/3a95215144e52b49.svg
• N_{\mathrm{weights}}=5\times5\times64\times64=102{,}400assets/equations/3a9b9e818c6252de.svg
• \frac{\partial z}{\partial w}=3+0=3assets/equations/3ab15e4d1d60b4eb.svg
• \mathrm{SSE}=9=9assets/equations/3acb15eecc0c073c.svg
• W^{(1)}_{2,2,\mathrm{new}}=0.4-0.5(-0.004357892)\approx 0.402178946assets/equations/3acb31488ce3b963.svg
• g_{1,2}^{(1)}=(-0.009617203)\times(0.3)\approx -0.002885161assets/equations/3ad09496c853dfe2.svg
• P_12=4{,}096{,}000+1{,}000=4{,}097{,}000assets/equations/3adeefbc3dbe4d6b.svg
• -z=0assets/equations/3aebd020c0eccec6.svg
• W^{(2)}_{1,1,\mathrm{new}}=0.4-0.5(0.082167041)\approx 0.35891648assets/equations/3afaa45378006799.svg
• C_t=f_tC_{t-1}+i_t\widetilde C_t,\qquad h_t=o_t\tanh(C_t)assets/equations/3aff429544aa57f2.svg
• \frac{\partial L}{\partial o_j}=\frac2n(o_j-y_j)assets/equations/3b0ae494a9901408.svg
• 7-3=4assets/equations/3b2ddd3dd39f8f0c.svg
• L=-\sum_j y_j\ln p_jassets/equations/3b5ae8d980bc0fd4.svg
• P_8=840+10=850assets/equations/3b9609ba48ce42f1.svg
• N_{\mathrm{conv}}=\left\lfloor\frac{7+2(1)-3}{2}\right\rfloor+1=3+1=4assets/equations/3ba361586be8312e.svg
• w_{1,\mathrm{new}}=0+0.5\times1\times1=0.5assets/equations/3bbcbc135cd52b17.svg
• f(z)=\max(0,z),\quad f\prime(z)=\begin{cases}1&z>0\\0&z<0\end{cases}assets/equations/3bc788a99b96d468.svg
• L=\frac12(\hat y-y)^2,\qquad \frac{\partial L}{\partial \hat y}=\hat y-yassets/equations/3bd0a5a4ecdf16a4.svg
• a_t=W_{xh}x_t+W_{hh}h_{t-1}+b_h,\qquad h_t=\tanh(a_t)assets/equations/3bf188b606a0ff69.svg
• z_h=w_1x+b_1,\qquad h=f(z_h)assets/equations/3c0624453c85b214.svg
• R_3=(0.5)\times(1)+(0.5)\times(0)+(0.3)\times(-1)=0.2assets/equations/3c0f1258f7cc1b3a.svg
• W^{(1)}_{2,2,\mathrm{new}}=-0.1-0.1(-0.55)\approx -0.045assets/equations/3c1258873b38aaf8.svg
• N_{\mathrm{out}}=\left\lfloor\frac{126-2}{2}\right\rfloor+1=63assets/equations/3c15021631c4c384.svg
• W^{(2)}_{2,1,\mathrm{new}}=0.12-1(0.062760617)\approx 0.057239383assets/equations/3c6ebfdbda545915.svg
• \begin{bmatrix}57 & 59\\40 & 50\end{bmatrix}assets/equations/3c78c4fdcdcef59f.svg
• i_t\widetilde C_t=0.5\tanh(0.5)\approx0.231058579assets/equations/3c7c75923a5c216c.svg
• g_{1,1}^{(1)}=(0.002053687)\times(0.25)\approx 0.000513422assets/equations/3c850e9fc715149c.svg
• \text{biases}=16,\qquad \text{total}=128+16=144assets/equations/3c90598459083061.svg
• z=-1.5+(0.2)=-1.3assets/equations/3c99c8f72adf2141.svg
• \hat{x}_2=\frac{1}{d}\approx0.99999500004assets/equations/3c9c2142072c8bc9.svg
• h_{2} = \max(0,0.3) = 0.3assets/equations/3ca4b1ac8771a1a8.svg
• wx=(1.5)(2)=3assets/equations/3cafd271941bad15.svg
• \widetilde h_t=\tanh(W_hx_t+U_h(r_t\odot h_{t-1})+b_h)assets/equations/3cb42e5f674e4241.svg
• W^{(1)}_{2,2,\mathrm{new}}=0.1-0.9(0)\approx 0.1assets/equations/3ce2a12d83343b99.svg
• P_9=10{,}080+84=10{,}164assets/equations/3cfadc0698f826a8.svg
• L_{\mathrm{half}}=\frac{0.49}{2}=0.245assets/equations/3d10f476bd1cf13c.svg
• \hat s_2=\frac{0.004996}{0.001999}\approx2.499249625assets/equations/3d1151d9e31acb34.svg
• \hat m_2=\frac{0.28}{0.19}\approx1.473684211assets/equations/3d16280e17e8b649.svg
• P_{\mathrm{pool}}=0,\quad P_{\mathrm{flatten}}=0,\quad P_{\mathrm{dense}}=(n_{\mathrm{in}}+1)n_{\mathrm{out}}assets/equations/3d36d7d9ab25e138.svg
• N_{\mathrm{weights}}=36{,}864\times120=4{,}423{,}680assets/equations/3d52ede2004cebfb.svg
• \eta_{\mathrm{effective},2}=\frac{0.1}{\sqrt5}\approx0.044721360assets/equations/3d6533ca9b2ac114.svg
• \theta_1=-0.5-\frac{0.1\times(-1)}{\sqrt{0.1}}=-0.183772234assets/equations/3d6651cada170e8a.svg
• N_{\mathrm{weights}}=4{,}096\times1{,}000=4{,}096{,}000assets/equations/3d7876f0d5fdb4fa.svg
• \frac{d}{dz}e^{-z}=-e^{-z},\qquad\frac{d}{dp}[-\ln p]=-\frac1passets/equations/3d8e93cb87192a72.svg
• \delta_{h_1} = 0.036350306 \times 0.241300709 = 0.008771355assets/equations/3d990b0c410497e3.svg
• \theta_1=-0.5+(0.15)=-0.35assets/equations/3de31f4085d1f9eb.svg
• \frac{\partial L}{\partial b}=-0.25-0.5=-0.75assets/equations/3e07d50d2b197b5f.svg
• C_3=0.072-0.3=-0.228assets/equations/3e11138e5892ca48.svg
• W^{(1)}_{1,2,\mathrm{new}}=0.21-0.05(-0.33978)\approx 0.226989assets/equations/3e18578432e574df.svg
• a'_i=\frac{m_i a_i}{q},\qquad m_i\in\{0,1\}assets/equations/3e23910228005a23.svg
• \theta_2=-0.4-\frac{0.1\times(0.578947368)}{\sqrt{2.500750375}}=-0.436610353assets/equations/3e2be14fb02aba94.svg
• h_{4}=\frac{1}{1+e^{-(0.6)}}=0.645656306assets/equations/3e375a5e30dabd6f.svg
• W^{(1)}_{1,2,\mathrm{new}}=0.4-0.9(0)\approx 0.4assets/equations/3e63507e62c558f8.svg
• 2(4)=8,\qquad2(5)=10assets/equations/3e7e9d7e4e1bc080.svg
• D=0.135335283+0.367879441+1.000000000\approx1.503214724assets/equations/3e8d536144b8581d.svg
• J_{L1}=L+\lambda\sum_j|w_j|,\qquad J_{L2}=L+\frac{\lambda}{2}\sum_jw_j^2assets/equations/3eb3323790998696.svg
• z_{h_2}=0+1-1.5=-0.5\Rightarrow h_2=0assets/equations/3eb9b9d929e96417.svg
• z_{h_2} = (0.25 \times 0.05) + (0.3 \times 0.1) + (0.35) = 0.3925assets/equations/3ec93d9dd746e448.svg
• 0.1(2.1)=0.21assets/equations/3ed37be87afc8239.svg
• N_{\mathrm{out}}=\left\lfloor\frac{13+2(1)-3}{1}\right\rfloor+1=13assets/equations/3eda49dc7f4fca09.svg
• N_{\mathrm{out}}=\left\lfloor\frac{219-2}{2}\right\rfloor+1=109assets/equations/3f1c898ff7560508.svg
• z_h=w_1x+b_1,\qquad h=f(z_h)assets/equations/3f2141c2576fb066.svg
• C_{\mathrm{new}}=0.500000000+(0.000000000)\approx0.500000000assets/equations/3f297224345b33d3.svg
• A^T=\begin{bmatrix}1&0\\-2&3\end{bmatrix}assets/equations/3f6121853d314156.svg
• E_k=\frac{N-C_k}{N}\times100\%assets/equations/3fb17785121a672d.svg
• 2+2+2=6assets/equations/3fc38a48a0a6427f.svg
• a_t=W_{xh}x_t+W_{hh}h_{t-1}+b_h,\qquad h_t=\tanh(a_t)assets/equations/3fed90dc3ade6a15.svg
• \frac{\partial L}{\partial w_2}=(-0.8)\times0=0assets/equations/400d3f8e82bf09fc.svg
• 1+7.389056099\approx8.389056099assets/equations/4026cb0f8944f24f.svg
• 1.000000000-1\approx0.000000000assets/equations/40327f28206586d2.svg
• s_{\mathrm{new}}=3.6+0.9=4.5assets/equations/4044336c67492dab.svg
• \tanh(C_{\mathrm{new}})\approx0.188942847assets/equations/404fb8d8c547f3b7.svg
• g_{3,4}^{(1)}=(0.012994055)\times(-0.4)\approx -0.005197622assets/equations/406ff2da076d59bc.svg
• \begin{bmatrix}1 & 0\\1 & 0\end{bmatrix}assets/equations/40c2bacd9336c7bb.svg
• h_t=(1-z_t)\odot h_{t-1}+z_t\odot\widetilde h_tassets/equations/40cc13a7254b2881.svg
• \delta_{h_1} = 0.011272855 \times 0.18217986 = 0.002053687assets/equations/40cdf85c2c162114.svg
• \mathrm{MACs}=H_{\mathrm{out}}W_{\mathrm{out}}C_{\mathrm{out}}K_hK_wC_{\mathrm{in}}assets/equations/40f8ac20e40ebe3e.svg
• R_2=(2)\times(0)+(3)\times(1)=3assets/equations/410d185792d0d4e1.svg
• 1-0.81=0.19assets/equations/41335385ba7e04c7.svg
• R_3=(0.3)\times(1)+(0.3)\times(0)+(0.3)\times(-1)=0assets/equations/41420636bdf1878d.svg
• \frac{\partial L}{\partial b}=-0.25-0.5=-0.75assets/equations/41657b19e6fd16c8.svg
• 1+e^{-z}=1+0.110803158=1.110803158assets/equations/417b75e5d36172d1.svg
• \left\lfloor\frac{219-2}{2}\right\rfloor+1assets/equations/4185cd16a03f14e2.svg
• a=0.2+0.1+0=0.3assets/equations/41862a6072aeaa37.svg
• \tanh z=\frac{e^z-e^{-z}}{e^z+e^{-z}},\qquad\frac{d}{dz}\tanh z=1-\tanh^2zassets/equations/419a6fb08fcd40a6.svg
• z_t=W_{hy}h_t+b_y,\qquad p_t=\operatorname{softmax}(z_t)assets/equations/41eb927502818778.svg
• \text{Sum}=C_1+C_2\approx 0.070517544assets/equations/41fc299f66c265e9.svg
• (w+\Delta w)^2=w^2+2w\Delta w+(\Delta w)^2assets/equations/4203633eb5975d8b.svg
• \beta(1-\beta)+(1-\beta)=(1-\beta)(1+\beta)=1-\beta^2assets/equations/420fad46aabea894.svg
• a'_j=\frac{m_j a_j}{q}assets/equations/4210eb7af01249ba.svg
• a'_1=\frac2{0.75}=2.666666667assets/equations/4257cce51e38bfe0.svg
• \frac1{1+3}=\frac14=0.25assets/equations/425b33a05c6633ee.svg
• W^{(1)}_{1,1,\mathrm{new}}=0.1-0.2(0.000964927)\approx 0.099807015assets/equations/429125d2eee69ed4.svg
• N_{\mathrm{weights}}=3\times3\times384\times256=884{,}736assets/equations/42a81620f94360dd.svg
• g_{1,2}^{(1)}=(-0.009617203)\times(0.3)\approx -0.002885161assets/equations/42a98609ebae9895.svg
• L_{\mathrm{MAE}}=\frac1n\sum_i|o_i-y_i|assets/equations/42ae3654ddbdf33e.svg
• g_{2,3}^{(1)}=(-0.014526308)\times(0.5)\approx -0.007263154assets/equations/42b60f11f5e3cb58.svg
• 8(6)=48assets/equations/42c41d8c1f1be799.svg
• b_{o_1,\mathrm{new}}=0.1-0.1(-0.550000000)=0.155000000assets/equations/42cf2912fda00eff.svg
• f_t=\sigma(W_f[h_{t-1},x_t]+b_f)assets/equations/42d259e3cca6ee0a.svg
• P_10=37{,}748{,}736+4{,}096=37{,}752{,}832assets/equations/42e41d5d625e4b33.svg
• \delta_{h_4} = -0.018719588 \times 0.22878424 = -0.004282747assets/equations/432e3ebd056288aa.svg
• C_t=0.1+0.5\tanh(0.5)\approx0.331058579assets/equations/4334a43ffdec3984.svg
• \delta_t=(W_{hy}^{T}\delta_t^{out}+W_{hh}^{T}\delta_{t+1})\odot(1-h_t^2),\quad \frac{\partial L}{\partial W_{hh}}=\sum_t\delta_t h_{t-1}^{T}assets/equations/43466db031266f9c.svg
• \frac{dL}{dw}=\frac{dL}{dz}\frac{dz}{dw}assets/equations/434e48fc84f871b4.svg
• \Delta w_2=-0.1(-0.25)=0.025assets/equations/43503938e7c174a7.svg
• 0.9(0)=0assets/equations/4351a4e3442a023e.svg
• 18+12+15+12=57assets/equations/437a8b87df7ffce7.svg
• 1-p=1-0.8=0.2assets/equations/438398f064595bb7.svg
• a_1=\max(0,4.5)=4.5,\qquad a_2=\max(0,0.5)=0.5assets/equations/43cee11d40484528.svg
• \delta_{h_3} = 0.056158765 \times 0.231380704 = 0.012994055assets/equations/43d6f061bf15172b.svg
• Y_{3,2}=(1)+(-2)=-1assets/equations/43f244bc80788fd1.svg
• h_{2,\mathrm{new}}=0.325(2)+0.51(0.4)+0.075=0.929assets/equations/43f5d3f6c6303f7b.svg
• y_{XOR}=1assets/equations/43f7e4fce3fb5771.svg
• N_{\mathrm{out}}=\left\lfloor\frac{227+2(0)-11}{4}\right\rfloor+1=55assets/equations/441a061296efda7a.svg
• s_2=0.999(0.004)+0.001(1^2)=0.004996assets/equations/442e6bd08bfece88.svg
• C_1=(-0.6)\times(-0.093597941)\approx 0.056158765assets/equations/44316b3d49dde0a3.svg
• (-0.2)(1)=-0.2assets/equations/44552ee4afe034fa.svg
• a'_1=\frac{1\times1}{0.5}=2,\qquad a'_2=\frac{0\times2}{0.5}=0assets/equations/446657a723f0a489.svg
• \theta_{\mathrm{look}}=1+0.9(0)=1assets/equations/44a5155fc196e4c6.svg
• N_{\mathrm{out}}=\left\lfloor\frac{14+2(0)-5}{1}\right\rfloor+1=10assets/equations/44a93c57d72f0b31.svg
• w_1^{\mathrm{new}}=-0.2+0.1(0)(1)=-0.2assets/equations/44ce80449df859db.svg
• \delta_2=\frac{\partial L}{\partial h_2}=h_2-1=-0.5assets/equations/44dd9c1e67b5f92d.svg
• b_{o_2,\mathrm{new}}=1-1(0.082533822)=0.917466178assets/equations/44e91760f0bb8195.svg
• \frac{\partial L}{\partial b}=\frac{\partial L}{\partial h}assets/equations/44f3dd28940be744.svg
• \theta_2=0.9+(-0.18)=0.72assets/equations/4502fcbdb27f5deb.svg
• N_{\mathrm{weights}}=5\times5\times1\times6=150assets/equations/4503fe259db6cc31.svg
• \left\lfloor\frac{4+2 \times 0-3}{1}\right\rfloor+1assets/equations/451621c0173b1850.svg
• g_2^{\mathrm{clip}}=12(0.25)=3assets/equations/451a7aae4fe2ab76.svg
• q=1-p,\quad a_{train}=\frac{m\odot a}{q},\quad a_{test}=aassets/equations/455d85a70cf8258a.svg
• \text{Sum}=C_1+C_2\approx 0.020617214assets/equations/4564965b0a02c263.svg
• P_6=1{,}327{,}104+384=1{,}327{,}488assets/equations/4565bf3714918ad3.svg
• N_{\mathrm{pool}}=\left\lfloor\frac{8-2}{2}\right\rfloor+1=3+1=4assets/equations/4578ff92c09a54ec.svg
• \text{Sum}=C_1\approx -0.046798971assets/equations/457a7a6b8a317393.svg
• 1-0.9^2=0.19,\qquad1-0.999^2=0.001999assets/equations/457ac955a89f3be2.svg
• f\prime(z)=\begin{cases}1&z>0\\0&z<0\end{cases}assets/equations/458083af5d5ba2ef.svg
• -\ln p=\ln\frac1passets/equations/45890ca9be189629.svg
• z_{XNOR}=-0+2(0)+0.5=0.5assets/equations/458da5139ca95df2.svg
• s_t=\rho s_{t-1}+(1-\rho)g_t^2,\quad\theta_t=\theta_{t-1}-\frac{\eta g_t}{\sqrt{s_t}+\epsilon}assets/equations/459c86785f7d6c4f.svg
• a_2\approx1+0.2(0.462117157)\approx1.092423431assets/equations/45ba45a94914183c.svg
• D_{\mathrm{KL}}(p\Vert q)\approx0.693147181-0.562335145=0.130812036assets/equations/45d882f13b18b109.svg
• z_{h_2}=0+1-1.5=-0.5\Rightarrow h_2=0assets/equations/45e0fb48b7159e24.svg
• z=\frac{x-\mu_{\mathrm{train}}}{\sigma_{\mathrm{train}}},\qquad \sigma_{\mathrm{train}}>0assets/equations/45ffeb13784592e1.svg
• \mathrm{saving\ percent}=\frac{3{,}997{,}696}{4{,}718{,}592}\times100=84.722222\%assets/equations/46344c94632c2247.svg
• \delta_{h_1} = 0.039350723 \times 0.221712873 = 0.008724562assets/equations/464805fe24d02ce3.svg
• 3(1)=3assets/equations/464c4958d256b6c8.svg
• y_1=2(-0.99999500004)+0.5\approx-1.49999000008assets/equations/4655c2949211af8b.svg
• C_{ij}=\sum_k A_{ik}B_{kj}assets/equations/46672bbce5b57e87.svg
• (-1)^2=1assets/equations/46782fbf983b8482.svg
• b_{2,\mathrm{new}}=0.2-0.1(0.08)=0.192assets/equations/4678bc376cf0d02b.svg
• P_{\mathrm{total}}=40+0+0+195=235assets/equations/468553004a2eb33c.svg
• \operatorname{MSE}=\frac1n\sum_{i=1}^n(\hat y_i-y_i)^2,\qquad \operatorname{MAE}=\frac1n\sum_{i=1}^n|\hat y_i-y_i|assets/equations/46872a64dc44481e.svg
• R_1=(0.5)\times(1)+(0.5)\times(0)+(0.3)\times(-1)=0.2assets/equations/46893fc8d069d1a9.svg
• \lambda w_2=0.2\times(-0.2)=-0.04assets/equations/469562aa281fbfba.svg
• z^{(l)}=W^{(l)}a^{(l-1)}+b^{(l)},\qquad a^{(l)}=f(z^{(l)})assets/equations/4698521353358290.svg
• 1.000000000+1\approx2.000000000assets/equations/46b371b545cc8bc8.svg
• W^{(1)}_{2,1,\mathrm{new}}=0.25-0.5(0.000497713)\approx 0.249751144assets/equations/46b3c1d000945dc9.svg
• h_{1}=\frac{1}{1+e^{-(1.155)}}=0.760423002assets/equations/46c27da6983edf63.svg
• w_{\mathrm{new}}=w-\eta\delta x,\quad b_{\mathrm{new}}=b-\eta\deltaassets/equations/46cbd30453d64730.svg
• \theta_1=1-0.316227766\approx0.683772234assets/equations/46ce5315f0e4b144.svg
• x-\mu=5-5=0assets/equations/46cf71d500f243a5.svg
• z_1=(1\times2)+(2\times1)+0.5assets/equations/46e8f36136a4c4d2.svg
• R_2=(2)\times(2)+(1)\times(0)+(0)\times(-2)=4assets/equations/47348f4aaa64e235.svg
• \theta_{look}=\theta+\beta vassets/equations/47358875d94f2558.svg
• (1)(0.5)=0.5assets/equations/47470f92bf2adcc4.svg
• z_{h_2} = (0.28 \times 0.25) + (0.3 \times 0.7) + (1) = 1.28assets/equations/474f99259d04d172.svg
• (1-0.8)(0.148885034)\approx0.029777007assets/equations/47661b22efc0daeb.svg
• g_{3,4}^{(1)}=\frac{\partial L}{\partial W^{(1)}_{3,4}}=\delta_{h_3}x_4assets/equations/4791ee899924c031.svg
• C_1=(-0.6)\times(-0.093597941)\approx 0.056158765assets/equations/47a1c511453df266.svg
• \begin{bmatrix}1 & 2\\-1 & 3\end{bmatrix}assets/equations/47c671c1553de484.svg
• z_{h_2} = (0.28 \times 0.25) + (0.3 \times 0.7) + (1) = 1.28assets/equations/47dad759cc7cca83.svg
• \sigma^2=\frac{16}{4}=4assets/equations/47e0cbf4557b9859.svg
• (1-z)\widetilde h=0.75(0.291312612)\approx0.218484459assets/equations/47e740aad633e99c.svg
• (AB)_{ij}=\sum_k A_{ik}B_{kj}assets/equations/47f627ca6464a106.svg
• h_2=\tanh(a_2)\approx0.797760954assets/equations/47f73b96c961ac1f.svg
• v_{\mathrm{new}}=0.5-0.1(-0.1)=0.51assets/equations/4822c14a12ff6b68.svg
• b_{h_2,\mathrm{new}}=0.2-0.9(0.006542085)=0.194112124assets/equations/4826d8c0344d9683.svg
• \frac\lambda2=\frac{0.1}2=0.05assets/equations/482c5bfd76d38e6c.svg
• M=H_oW_oC_oK^2C_iassets/equations/48494d663392b9d5.svg
• C_1=(0.25)\times(-0.099059551)\approx -0.024764888assets/equations/486b61a3d9f85a8a.svg
• N_{\mathrm{weights}}=120\times84=10{,}080assets/equations/4874c9639ae8a286.svg
• g_t=\left.\frac{\partial L}{\partial w}\right|_{w_{t-1}}=\frac{\partial L}{\partial\hat y}\frac{\partial\hat y}{\partial w}=(\hat y-y)xassets/equations/48779dd96502307c.svg
• \mathrm{SSE}=0.04+0.09+0.36=0.49assets/equations/4890329b3b657194.svg
• o_{1}=\frac{1}{1+e^{-(0.05547891)}}=0.513866171assets/equations/4890faea42fbcddd.svg
• i_t=\sigma(W_i[h_{t-1},x_t]+b_i)assets/equations/48984d6eb2de43b5.svg
• f=\sigma(z_f),\quad i=\sigma(z_i),\quad o=\sigma(z_o),\quad \widetilde C=\tanh(z_C)assets/equations/48a155da6647af6e.svg
• Y_{1,3}=(-1)+(2)=1assets/equations/48b778ac3ad46a58.svg
• w_{\mathrm{new}}=0.5-0.1(-0.105754186)=0.510575419assets/equations/48c8e5f15b7994a8.svg
• g_{2,3}^{(1)}=\frac{\partial L}{\partial W^{(1)}_{2,3}}=\delta_{h_2}x_3assets/equations/48d08524866113b9.svg
• J(\theta)=\frac1N\sum_{i=1}^{N}L_i(\theta)assets/equations/48f2afe9071bd1c9.svg
• \widetilde h=\tanh(0.3)=0.291312612assets/equations/49097e56740c770e.svg
• N_{\mathrm{weights}}=3\times3\times256\times384=884{,}736assets/equations/492d70515e0406f9.svg
• P=(n_{\mathrm{in}}+1)n_{\mathrm{out}}assets/equations/494312d0893d20d2.svg
• \frac{\partial L}{\partial W_{xh}}=\sum_t\delta_t x_t^Tassets/equations/4953486e2063f6f0.svg
• p=\frac1{1+e^{-1}}=0.731058579assets/equations/496f72f75e4bdba9.svg
• b_{o_1,\mathrm{new}}=0.1-0.9(-0.131169078)=0.218052170assets/equations/4993d13feab65d1f.svg
• \theta_t=\theta_{t-1}-\frac{\eta g_t}{\sqrt{G_t}+\varepsilon}assets/equations/49a0512d42c905d8.svg
• w_{1,\mathrm{new}}=-0.5-0.2(0.1)=-0.5-0.02=-0.52assets/equations/49b338a1efee1ae5.svg
• h_{1}=\frac{1}{1+e^{-(0.25)}}=0.562176501assets/equations/49beb90377a79a2a.svg
• R_2=(3)\times(0)+(2)\times(-1)=-2assets/equations/49c3d4aac1f50f22.svg
• \delta_2=h_2-y,\qquad \delta_1=v\delta_2assets/equations/49cb0da4f4956b69.svg
• \sigma(z)=\frac1{1+e^{-z}},\qquad \sigma'(z)=\sigma(z)[1-\sigma(z)]assets/equations/4a542f0b5de210cf.svg
• \sigma(0)=\frac1{1+e^{-0}}\approx0.500000000assets/equations/4a59e93b5287041f.svg
• g_{1,2}^{(1)}=(0.002053687)\times(0.7)\approx 0.001437581assets/equations/4a66a34e7f770bc0.svg
• g_{1,1}^{(2)}=(-0.131169078)\times(0.331812228)\approx -0.043523504assets/equations/4a6b58e5f34c0b65.svg
• N_{\mathrm{out}}=\left\lfloor\frac{N+2P-K}{S}\right\rfloor+1assets/equations/4a7024fc5224e01a.svg
• \tanh(z)=\frac{e^z-e^{-z}}{e^z+e^{-z}},\qquad -1<\tanh(z)<1assets/equations/4a9360ff9640949f.svg
• W^{(2)}_{1,2,\mathrm{new}}=0.45-0.5(0.082667628)\approx 0.408666186assets/equations/4aaec1199c9864bb.svg
• R_2=(0)\times(2)+(3)\times(0)+(0)\times(-2)=0assets/equations/4ab2e486b72e8236.svg
• H(p)=-\sum_i p_i\ln p_iassets/equations/4ac39baa15925718.svg
• W^{(2)}_{1,2,\mathrm{new}}=0.45-0.5(0.082667628)\approx 0.408666186assets/equations/4ad573638df0109c.svg
• e^{2z}+1=2.718281828+1=3.718281828assets/equations/4ae221248ca27fec.svg
• \widehat v_2=\frac{0.004996}{1-0.999^{2}}=2.499249625assets/equations/4ae265791f4d129d.svg
• R_2=(5)\times(1)+(7)\times(2)+(6)\times(1)=25assets/equations/4ae5e24b45a652e7.svg
• s=0+1=1assets/equations/4ae62ee9eceeaffe.svg
• v_1=0.99(0)+0.01(2)^2=0.04assets/equations/4af117593c0c60f1.svg
• R_2=(2)\times(0)+(3)\times(-1)=-3assets/equations/4b10b62665a7718a.svg
• wx=(1.5)(3)=4.5assets/equations/4b2ec1addfdbb9fd.svg
• e_i=\hat y_i-y_iassets/equations/4b31e4fcfd52ba58.svg
• g_1=(\hat y_0-y)x=(-2)(2)=-4assets/equations/4b34bd0ab6469482.svg
• w^Tx=(8)+(3)=11assets/equations/4b366d83f24fd5dd.svg
• \delta_2=(h_2-y)(1-h_2^2),\qquad \delta_1=v\delta_2(1-h_1^2)assets/equations/4b5081051ef8a38b.svg
• a=\tanh(0.5)=\frac{1.718281828}{3.718281828}\approx0.462117157assets/equations/4b6a824a25e2320c.svg
• a_{\mathrm{train}}=\frac{m\odot a}{q}\quad\text{and}\quad a_{\mathrm{test}}=a\quad\text{in inverted dropout}assets/equations/4b6a9d814bc5c7bb.svg
• \widetilde C=\tanh(0.5)=0.462117157assets/equations/4b709e4b1fd2b287.svg
• \delta_{o_1} = 1 \times (0.641368858-1) \times 0.230014846 = -0.082490487assets/equations/4b919c902c0118b6.svg
• \tanh(-2)\approx\frac{-7.253720816}{7.524391382}\approx-0.964027580assets/equations/4bb67612e76e38e6.svg
• y_{\mathrm{avg}}=\frac{46}4=11.5assets/equations/4bcedeb9a26a70b6.svg
• H(p,q)=-\sum_i p_i\ln q_i,\qquad\log_2q=\frac{\ln q}{\ln2}assets/equations/4be60e1a1e5d9ec8.svg
• L=-(-0.693147181)=0.693147181assets/equations/4bf6a8c1047d0f4a.svg
• R_1=(0)\times(1)+(1)\times(-1)=-1assets/equations/4bff5878f34559b0.svg
• U_{\mathrm{SGD}}=103\times3=309assets/equations/4c08a8b839ca4b99.svg
• i_t\widetilde C_t=0.5\tanh(0.5)\approx0.231058579assets/equations/4c11c7ce4e04ba62.svg
• \widehat m_t=\frac{m_t}{1-\beta_1^t}assets/equations/4c24c9160cc537a1.svg
• g_2=(0.2-2)\times2=(-1.8)\times2=-3.6assets/equations/4c4f586bc3f0f97e.svg
• \beta(1-\beta)+(1-\beta)=(1-\beta)(1+\beta)=1-\beta^2assets/equations/4c68b61c5bdf0798.svg
• W^{(2)}_{1,1,\mathrm{new}}=0.2-0.2(-0.066771834)\approx 0.213354367assets/equations/4c9c4df3cf58176a.svg
• z_t=\sigma(W_zx_t+U_zh_{t-1}+b_z),\quad r_t=\sigma(W_rx_t+U_rh_{t-1}+b_r)assets/equations/4c9f5914ab749b31.svg
• z_t\widetilde h_t=0.25\tanh(0.3)\approx0.072828153assets/equations/4cab7af53e3adf95.svg
• \delta_j^h=f\prime(z_j^h)\sum_k w_{kj}^{out}\delta_k^{out},\quad\frac{\partial L}{\partial w_{ji}}=\delta_j a_i,\quad\frac{\partial L}{\partial b_j}=\delta_jassets/equations/4cb07890e7ce7a95.svg
• z_{h_2}=0+0-1.5=-1.5\Rightarrow h_2=0assets/equations/4cb4887fc9401848.svg
• [0.8(1),(-0.4)(0),0.6(1),0.2(0)]=[0.8,0,0.6,0]assets/equations/4ccd5e8324645240.svg
• \sqrt{3^2+4^2}=\sqrt{9+16}=5assets/equations/4cd7b08ae98b6682.svg
• 5\times5\times3=75\text{ weights per filter}assets/equations/4ce380d19eaf1a9e.svg
• g_{2,2}^{(1)}=(-0.004869072)\times(1)\approx -0.004869072assets/equations/4d0ac3ffd181b903.svg
• \left\lfloor\frac{61 - 2}{2}\right\rfloor+1assets/equations/4d2232cedf251bfa.svg
• \eta_{\mathrm{effective},2}=\frac{0.1}{\sqrt5}\approx0.044721360assets/equations/4d33a28de17065dc.svg
• C_1=(0.7)\times(-0.093597941)\approx -0.065518559assets/equations/4d3e0ecf8502df13.svg
• \text{mean}=\frac{\text{sum}}{\text{count}}assets/equations/4d51e471133d25a2.svg
• \ln(0.8)\approx-0.223143551assets/equations/4d6b8e2a24a05d6f.svg
• \theta_2=0.9-\frac{0.1\times(1.473684211)}{\sqrt{2.499249625}}=0.806782036assets/equations/4d6c4212f9eead9c.svg
• g_t=J\prime(\theta_{\mathrm{look}})assets/equations/4d83df079bf260cc.svg
• b_{h_1,\mathrm{new}}=0.1-0.2(0.004824633)=0.099035073assets/equations/4d95f499c8cf92af.svg
• \text{Sum}=C_1\approx -0.018719588assets/equations/4daca8885b8234c0.svg
• 2z=2(0.5)=1,\qquad e^{2z}=e^1\approx2.718281828assets/equations/4ded3cb512b92053.svg
• z_{o_2} = (0.3 \times 0.549833997) + (0.4 \times 0.544878892) + (0.1) = 0.482901756assets/equations/4e7177927a4b11d4.svg
• G_2=1+(2)^2=5assets/equations/4e7c36432792b91c.svg
• g_{4,2}^{(1)}=\frac{\partial L}{\partial W^{(1)}_{4,2}}=\delta_{h_4}x_2assets/equations/4e7cc212944346db.svg
• p_2=\frac{1.000000000}{3.000000000}\approx0.333333333assets/equations/4e80a74382faa499.svg
• C_1=(0.4)\times(0.138498562)\approx 0.055399425assets/equations/4e8c5af2cd60942b.svg
• (w+\Delta w)^2=w^2+2w\Delta w+(\Delta w)^2assets/equations/4e8dd2d8556610b9.svg
• J=J_{\mathrm{data}}+\frac{\lambda}{2}\sum_iw_i^2assets/equations/4e906f5f903db890.svg
• 1+0.135335283=1.135335283assets/equations/4e918224e5cd8eaa.svg
• W=\begin{bmatrix}1&2\\-1&3\end{bmatrix},\quad x=\begin{bmatrix}2\\1\end{bmatrix},\quad b=\begin{bmatrix}0.5\\-0.5\end{bmatrix}assets/equations/4e990284b43290d2.svg
• n_{\mathrm{flat}}=2\times2\times2=8assets/equations/4eb23baff49f82cf.svg
• \operatorname{LeakyReLU}(z)=\begin{cases}z&z\ge0\\\alpha z&z<0\end{cases}assets/equations/4edc237fa121ff9b.svg
• \mathrm{MACs}=H_{\mathrm{out}}W_{\mathrm{out}}C_{\mathrm{out}}K^2C_{\mathrm{in}}assets/equations/4ef70226a6e5e478.svg
• P_{\mathrm{total}}=36+438+18=492assets/equations/4f06023c81396b20.svg
• h_{4}=\frac{1}{1+e^{-(0.6)}}=0.645656306assets/equations/4f0e84b4e9fe0752.svg
• R_2=(0.3)\times(1)+(0.3)\times(0)+(0.3)\times(-1)=0assets/equations/4f1d831d541f9cae.svg
• \frac{\partial L}{\partial v}=\delta_1h_0+\delta_2h_1assets/equations/4f377c9f2aefec11.svg
• \begin{bmatrix}0.6 & 0.4\\0.3 & 0.7\end{bmatrix}assets/equations/4f3aa9485d9c39f6.svg
• \delta_t=0.3(0.64)=0.192assets/equations/4f408ded05ecb892.svg
• m_2=0.9(0.2)+0.1(1)=0.28assets/equations/4f47b7aabd4f81c4.svg
• H_3=1+0.5=1.5assets/equations/4f50029ce8c5be41.svg
• n_{\mathrm{flat}}=29\times29\times128=107{,}648assets/equations/4f51a385169358d4.svg
• p_i=\frac{e^{z_i-c}}{\sum_j e^{z_j-c}},\qquad c=\max_j z_jassets/equations/4f5f1a315cf3ec0f.svg
• a=\frac{e^{2z}-1}{e^{2z}+1}assets/equations/4f6e15283db79ff2.svg
• W^{(1)}_{1,1,\mathrm{new}}=0.15-0.5(0.000438568)\approx 0.149780716assets/equations/4f7a0b344b0cf201.svg
• g_{1,1}^{(2)}=(0.08)\times(1.1)\approx 0.088assets/equations/4f9573bd5025e29a.svg
• \text{weights}=K^2C_{\mathrm{in}}C_{\mathrm{out}},\quad \text{biases}=C_{\mathrm{out}}assets/equations/4fb085a2eda022c5.svg
• \begin{bmatrix}1 & 2\\2 & 2\end{bmatrix}assets/equations/4fb14ffabb3fef49.svg
• h=0.3+0.072828153=0.372828153assets/equations/4fbe2a7f0d77a4fd.svg
• h=f^{(1)}(W^{(1)}x+b^{(1)})assets/equations/4ff7d7b3e6546d64.svg
• N_{\mathrm{out}}=\lfloor(N+2P-K)/S\rfloor+1assets/equations/500258811ea3eb2f.svg
• z_1=2+2+0.5=4.5assets/equations/50386fe91979c412.svg
• \begin{aligned}\text{mean}&=\frac{2+4+9}{3}\\&=\frac{15}{3}\\&=5\end{aligned}assets/equations/503bca869d213251.svg
• \widetilde h_t=\tanh(W_hx_t+U_h(r_t\odot h_{t-1})+b_h),\quad h_t=z_t\odot h_{t-1}+(1-z_t)\odot\widetilde h_tassets/equations/50489aea6d050084.svg
• a_{train}=\frac{m\odot a}{q}assets/equations/504d04d32df8a91c.svg
• \frac{\partial y}{\partial x}=\frac{\partial F}{\partial x}+Iassets/equations/505f1041667e53bb.svg
• \mathrm{MAE}=\frac{|1-2|+|6-4|}{2}=\frac{1+2}{2}=1.5assets/equations/506b26dfbefb2185.svg
• p_i=\frac{e^{z_i}}{\sum_j e^{z_j}}=\frac{e^{z_i-c}}{\sum_j e^{z_j-c}},\qquad c=\max_jz_jassets/equations/50824a322c80361f.svg
• R_1=(1)\times(1)+(3)\times(0)=1assets/equations/50ba62bd05697a22.svg
• h_1=\frac{2.718281828-1}{2.718281828+1}=0.462117157assets/equations/50c20979a353d66c.svg
• Y_{3,2}=(1)+(2)=3assets/equations/50d4d84e0be67f18.svg
• \delta_2=0.5-1=-0.5assets/equations/50ef5b3a75d08159.svg
• L_{\mathrm{data}}=\frac12(-0.2)^2=\frac12(0.04)=0.02assets/equations/5140459142337abc.svg
• \mu=\frac{1}{n}\sum_i x_iassets/equations/515c6d49d5a0ee05.svg
• 1-q^2=1-(q\times q)assets/equations/5166132378f964d6.svg
• D_{\mathrm{KL}}(p\Vert q)=\sum_i p_i\ln\frac{p_i}{q_i}=H(p,q)-H(p)assets/equations/5174a1ccffce0c30.svg
• g_{1,4}^{(1)}=\frac{\partial L}{\partial W^{(1)}_{1,4}}=\delta_{h_1}x_4assets/equations/51881657a5cb91ef.svg
• L_{\mathrm{BCE}}=-[y\ln p+(1-y)\ln(1-p)]assets/equations/519958f461b019f9.svg
• \frac{\partial L}{\partial w_2}=\delta_o h,\quad \delta_h=\delta_o w_2\operatorname{ReLU}'(z_h),\quad \frac{\partial L}{\partial w_1}=\delta_h xassets/equations/51a8b01e1bee8834.svg
• g_1=1-(0)=1assets/equations/51e0e1de442cadbf.svg
• h_{1} = \max(0,1.1) = 1.1assets/equations/51f4d0c29cd3e471.svg
• y_{XNOR}=0assets/equations/51f578d6708881af.svg
• \begin{bmatrix}0 & 0 & 0\\0 & 1 & 2\\2 & 0 & 0\end{bmatrix}assets/equations/5203ca394b677586.svg
• h=\operatorname{ReLU}(1.1)=1.1assets/equations/5206a5b2ac9abfe4.svg
• W=\begin{bmatrix}0.11&0.21\\0.12&0.08\end{bmatrix},\quad x=\begin{bmatrix}2\\3\end{bmatrix},\quad Wx=\begin{bmatrix}0.11(2)+0.21(3)\\0.12(2)+0.08(3)\end{bmatrix}=\begin{bmatrix}0.85\\0.48\end{bmatrix}assets/equations/5226c7a9750fc29a.svg
• J_{L1}=L+\lambda\sum_j|w_j|,\qquad J_{L2}=L+\frac{\lambda}{2}\sum_jw_j^2assets/equations/522dfd9873b52e27.svg
• \theta_t=\theta_{t-1}-\eta_{\mathrm{effective},t}g_tassets/equations/52438222ad8d94df.svg
• Y_{3,3}=(-1)+(3)=2assets/equations/5248561d79f5d37a.svg
• e_1^2=(1)^2=1assets/equations/527befb1696e696d.svg
• g_{1,1}^{(1)}=(0.008724562)\times(1)\approx 0.008724562assets/equations/527d49eeeec9b622.svg
• s_2=0.999(0.004)+0.001(1^2)=0.004996assets/equations/528b23a17d8f719c.svg
• J_{L1}=J_{\mathrm{data}}+\lambda\sum_i|w_i|assets/equations/52bb091c05d16021.svg
• \mathrm{MSE}=\frac1n\sum_i(\hat y_i-y_i)^2assets/equations/52c8252c5d3ef107.svg
• f\prime(2)=1-(0.964027580)^2\approx0.070650825assets/equations/52d252bd261b679f.svg
• g_t=J\prime(\theta_{t-1})assets/equations/52ef83d9d4ea4381.svg
• \operatorname{MSE}=\frac1n\sum_{i=1}^n(\hat y_i-y_i)^2,\qquad \operatorname{MAE}=\frac1n\sum_{i=1}^n|\hat y_i-y_i|assets/equations/52f47ec283865e9e.svg
• \mathrm{SSE}=\sum_{i=1}^{n}e_i^2assets/equations/5312f21f7180b663.svg
• 64(8)(144)=73728assets/equations/531722215214eb67.svg
• C_1=(0.14)\times(-0.809)\approx -0.11326assets/equations/53405206a97657fd.svg
• q=1-passets/equations/536be85758e6d02d.svg
• s=0.6+0=0.6assets/equations/5398d7a271c52472.svg
• a_t=W_{xh}x_t+W_{hh}h_{t-1}+b,\qquad h_t=\tanh(a_t)assets/equations/53aa19b3509081a2.svg
• \text{Sum}=C_1+C_2\approx 0.011272855assets/equations/53b10cae0b234be6.svg
• -3-2=-5assets/equations/53bef08286131079.svg
• \frac{\partial L}{\partial w}=\frac{\partial L}{\partial h}x=3.5(2)=7assets/equations/54071389e64f7570.svg
• R_1=(1)\times(1)+(3)\times(-1)=-2assets/equations/540cabd018cf9dd5.svg
• W^{(2)}_{1,2,\mathrm{new}}=-0.2-0.9(-0.068861036)\approx -0.138025068assets/equations/54531eae8fb17595.svg
• P_{\mathrm{head}}=64(3)+3=(64+1)3=195assets/equations/5453c1fee1f9ccef.svg
• Wx=0.2\times1=0.2assets/equations/545ce783984e0414.svg
• w_2^{\mathrm{new}}=0.1+0.1(-1)(2)=-0.1assets/equations/5463d220fc6276c5.svg
• g_{1,1}^{(1)}=(0.008724562)\times(1)\approx 0.008724562assets/equations/548224ebc74d73ba.svg
• g_{4,4}^{(1)}=\frac{\partial L}{\partial W^{(1)}_{4,4}}=\delta_{h_4}x_4assets/equations/54a191df99677201.svg
• [(-2)^2,(-2)^2,2^2,2^2]=[4,4,4,4]assets/equations/54a284752c898da0.svg
• R_1=(0.3)\times(1)+(0.3)\times(0)+(0.3)\times(-1)=0assets/equations/54a80ceab4517447.svg
• \begin{bmatrix}8 & 0\\34 & 4\end{bmatrix}assets/equations/54afd252549c8a81.svg
• e^{2}+e^{-2}\approx7.389056099+0.135335283=7.524391382assets/equations/54c21815de76e465.svg
• p_3=\frac{0.135335283}{1.503214724}=0.090030573assets/equations/54c91e5ed1921172.svg
• b_{o_2,\mathrm{new}}=0.1-0.2(0.145933863)=0.070813227assets/equations/54d821197ed88717.svg
• \begin{bmatrix}0 & 0 & 0\\0 & 156 & 155\\0 & 153 & 154\end{bmatrix}assets/equations/54e18a92743f2660.svg
• \left\lfloor\frac{61+2 \times 1-5}{1}\right\rfloor+1assets/equations/54ee85e0468271d4.svg
• z_t=\sigma(W_zx_t+U_zh_{t-1}+b_z),\quad r_t=\sigma(W_rx_t+U_rh_{t-1}+b_r)assets/equations/552178cdb39642a9.svg
• 0.8/0.5=1.6assets/equations/552d192c8691910e.svg
• N_{\mathrm{weights}}=240\times120=28{,}800assets/equations/55419bb786cbe655.svg
• \mu=\frac1N\sum_i x_i,\qquad\sigma^2=\frac1N\sum_i(x_i-\mu)^2assets/equations/554920a13a606ed9.svg
• \frac{dL}{dp}=\frac{0.8-1}{0.16}=-1.25assets/equations/55554414edc61afc.svg
• b_{o_1,\mathrm{new}}=0-0.5(-0.099059551)=0.049529775assets/equations/55595ef27eaeb48f.svg
• \widetilde C_t=\tanh(W_c[h_{t-1},x_t]+b_c),\quad C_t=f_t\odot C_{t-1}+i_t\odot\widetilde C_tassets/equations/555b66da5e34fae2.svg
• \theta_2=0.683772234-\frac{0.1\times(1)}{\sqrt{0.46}}=0.536330278assets/equations/555de02648791e71.svg
• g_{\mathrm{clipped}}=0.5\begin{bmatrix}6\\8\end{bmatrix}=\begin{bmatrix}3\\4\end{bmatrix}assets/equations/55639775a78bd93d.svg
• \left\lfloor\frac{27-3}{2}\right\rfloor+1assets/equations/55802f10aac3668d.svg
• N_{\mathrm{biases}}=384assets/equations/558f11bf6ad2aa25.svg
• N_{\mathrm{out}}=\lfloor7.5\rfloor+1=7+1=8assets/equations/55b23544c4917d4f.svg
• b_{2,\mathrm{new}}=0.2-0.1(0.08)=0.192assets/equations/55b6e2f442ea2fb4.svg
• p_1=\frac{0.367879441}{1.503214724}=0.244728471assets/equations/55ba241642d89ad8.svg
• e_2=2-4=-2assets/equations/55c62638d290dff9.svg
• \mathbb E\left[\frac{ma}{q}\right]=\frac aq\mathbb E[m]=\frac aq\,q=aassets/equations/55c82e5345c55ebe.svg
• a=\frac1{2.000000000}\approx0.500000000assets/equations/55f56d862c8a637f.svg
• \tanh(C_{\mathrm{new}})\approx0.462117157assets/equations/55fd6c15079562c2.svg
• z_h=0.5(2)+0.1=1.1assets/equations/5613b3868c5f8d85.svg
• W^{(1)}_{2,1,\mathrm{new}}=0.5-0.5(-0.004869072)\approx 0.502434536assets/equations/5614079c42cc22b4.svg
• \nabla_w J=\nabla_w J_{\mathrm{data}}+2\lambda wassets/equations/56481f4684193b6b.svg
• W^{(2)}_{2,1,\mathrm{new}}=0.3-0.2(0.080239399)\approx 0.28395212assets/equations/5660f8c2c8b62a6f.svg
• C_{\mathrm{combined}}=4+6+2=12assets/equations/567c2c4512c2cb27.svg
• w_{\mathrm{batch}}=0-0.1(-2.5)=0.25assets/equations/568002eaa70aedb2.svg
• g_{1,1}^{(1)}=(-0.009617203)\times(0.4)\approx -0.003846881assets/equations/568ab83eab81feec.svg
• x_1w_1=2(0.4)=0.8assets/equations/568d8f8a50e883d1.svg
• \text{Sum}=C_1\approx -0.049494292assets/equations/569b8ebcbf1272c9.svg
• \widehat m_2=\frac{0.38}{1-0.9^2}=\frac{0.38}{0.19}=2assets/equations/569c2f898bf0d2e3.svg
• \delta_{h_2} = -0.024764888 \times 0.196611933 = -0.004869072assets/equations/569e06b7bb3c113c.svg
• R_1=(1)\times(1)+(0)\times(-1)=1assets/equations/56ab2f7173dad1a0.svg
• (1-z_t)h_{t-1}=(1-0.25)(0.4)=0.3assets/equations/56b9a2466f5d57e7.svg
• h_{\mathrm{new}}=o\tanh(C_{\mathrm{new}})assets/equations/56c0f06510c31697.svg
• N_{flat}=HWC,\qquad P_{dense}=(N_{in}+1)N_{out}assets/equations/56dad01ac3222ddd.svg
• 5\times5\times3=75\text{ weights per filter}assets/equations/56e79229382e122b.svg
• \lambda\operatorname{sign}(-0.5)=0.1(-1)=-0.1assets/equations/56f95d7675b65e54.svg
• Y_{1,1}=(24)+(25)+(8)=57assets/equations/572c32730d6fe617.svg
• z_{XNOR}=-1+2(0)+0.5=-0.5assets/equations/573087efcdde0646.svg
• b_{h_2,\mathrm{new}}=0.1-0.2(0.017487356)=0.096502529assets/equations/574987d80e93f51f.svg
• g_{1,4}^{(1)}=(-0.009617203)\times(-0.4)\approx 0.003846881assets/equations/576777ca605fe30f.svg
• g_{1,2}^{(1)}=(0.008724562)\times(0)\approx 0assets/equations/579a2b920b0ef6cb.svg
• g_{2,2}^{(2)}=\frac{\partial L}{\partial W^{(2)}_{2,2}}=\delta_{o_2}h_2assets/equations/57d7e754e94fe533.svg
• \frac{\partial L}{\partial b_2}=0.08assets/equations/57f3bbde46d88ee8.svg
• g_{2,1}^{(1)}=(0.006542085)\times(1)\approx 0.006542085assets/equations/5846229bdca31634.svg
• R_2=(3)\times(1)+(3)\times(2)+(2)\times(1)=11assets/equations/5861059147729994.svg
• z_{o_1} = (0.4 \times 0.593269992) + (0.45 \times 0.596884378) + (0.6) = 1.105905967assets/equations/587365feb9b3026e.svg
• C_{concat}=\sum_b C_b,\qquad y_{res}=F(x)+xassets/equations/588297b667afd636.svg
• N_{\mathrm{out}}=\left\lfloor\frac{52+2(0)-5}{1}\right\rfloor+1=48assets/equations/58a6dd2bccbabd6a.svg
• g_{1,1}^{(2)}=(0.08)\times(1.1)\approx 0.088assets/equations/58dabafbc78878c9.svg
• b^{\mathrm{new}}=b+\eta(t-o)assets/equations/58dd8fcb5fcd562f.svg
• \delta_1=\frac{\partial L}{\partial h_1}=\frac{\partial L}{\partial h_2}\frac{\partial h_2}{\partial h_1}assets/equations/5911c2f4d8e13a0a.svg
• \begin{bmatrix}0.2 & 0.4\\0.3 & 0.7\end{bmatrix}assets/equations/59303f63a5ad2b9b.svg
• b_{h_2,\mathrm{new}}=0-0.5(-0.004869072)=0.002434536assets/equations/594dcb090f7d4418.svg
• e^{-\ln a}=\frac1aassets/equations/5953def5a7690c0e.svg
• P_1=7{,}776+32=7{,}808assets/equations/59616c98ab8fffe7.svg
• e^{2}+e^{-2}\approx7.389056099+0.135335283=7.524391382assets/equations/597fbd34bf357417.svg
• R_2=(1)\times(0)+(3)\times(-1)=-3assets/equations/59a1357910dce5f4.svg
• \delta_1=\delta_2v=(-0.5)(0.5)=-0.25assets/equations/59ca9544049d91dc.svg
• b_{o_1,\mathrm{new}}=0.6-0.5(0.138498562)=0.530750719assets/equations/59d69997bb013f94.svg
• h_t=o_t\odot\tanh(C_t)assets/equations/5a01a031c432f962.svg
• h=f_1(z_h)assets/equations/5a067fbd2b571e3c.svg
• W^{(1)}_{1,2,\mathrm{new}}=0.21-0.05(-0.33978)\approx 0.226989assets/equations/5a3350f101c32e1b.svg
• \tanh(-2)\approx\frac{-7.253720816}{7.524391382}\approx-0.964027580assets/equations/5a3af3ace5989f73.svg
• g_{2,1}^{(1)}=(0.017487356)\times(0.2)\approx 0.003497471assets/equations/5a519701097922e5.svg
• P=(K_hK_wC_{\mathrm{in}}+1)C_{\mathrm{out}}assets/equations/5a603987eb773068.svg
• W^{(2)}_{1,1,\mathrm{new}}=0.14-0.05(-0.68765)\approx 0.1743825assets/equations/5a6b892831ec0aa5.svg
• z_o=0.8(1.1)+0.2=1.08assets/equations/5a7535982ea7d2cd.svg
• P_{\mathrm{head}}=64(3)+3=(64+1)3=195assets/equations/5a75359a035a6d76.svg
• \sigma=\sqrt4=2assets/equations/5a8d993f2bb3889d.svg
• |-1|=1,\qquad |2|=2assets/equations/5a8f4cb66d218e6f.svg
• m_1=0.9(0)+0.1(2)=0.2assets/equations/5ac027abcdfcdb75.svg
• g_{2,1}^{(2)}=(0.145933863)\times(0.549833997)\approx 0.080239399assets/equations/5b06ea5d37905422.svg
• 2-0.1\times(-3)assets/equations/5b11a126e768858f.svg
• \text{biases}=3,\qquad P_{\mathrm{dense}}=192+3=195assets/equations/5b290fab1b1da9d6.svg
• f_t=i_t=o_t=\sigma(0)=\frac1{1+e^0}=0.5assets/equations/5b2fa847b2d2be9a.svg
• C_1=(0.2)\times(-0.093597941)\approx -0.018719588assets/equations/5b437d314c196916.svg
• \int x^n\,dx=\frac{x^{n+1}}{n+1}+C\quad(n\ne-1)assets/equations/5b516266e8ee3cea.svg
• A^T=\begin{bmatrix}-1&4&2\end{bmatrix}assets/equations/5b69eb45dcd67397.svg
• \widetilde C=\tanh(1)\approx0.761594156assets/equations/5b79535c0abe19aa.svg
• \left\lfloor\frac{126+2 \times 0-5}{1}\right\rfloor+1assets/equations/5b79f5df9ff78236.svg
• \theta_2=0.9+(-0.171)=0.729assets/equations/5b7b6422b337bab9.svg
• (1-0)(0.291312612)\approx0.291312612assets/equations/5b85f83251c9025c.svg
• N_{\mathrm{weights}}=5\times5\times32\times64=51{,}200assets/equations/5b861b3fc768032c.svg
• v=\frac{1}{n}\sum_i(x_i-\mu)^2assets/equations/5b9364e6a03439b4.svg
• W^{(2)}_{1,1,\mathrm{new}}=0.4-0.1(-0.275)\approx 0.4275assets/equations/5bb285fe6a5d5ca6.svg
• W^{(1)}_{4,3,\mathrm{new}}=0.2-0.5(-0.002141373)\approx 0.201070687assets/equations/5bb6f3c8fa38e377.svg
• g_{1,4}^{(2)}=(-0.093597941)\times(0.645656306)\approx -0.060432101assets/equations/5bd5e9e2c424a3de.svg
• W^{(1)}_{2,1,\mathrm{new}}=-0.3-0.9(0.006542085)\approx -0.305887877assets/equations/5c15a33a77f594c0.svg
• C_t=f_tC_{t-1}+i_t\widetilde C_tassets/equations/5c31e372051573ef.svg
• \text{Sum}=C_1+C_2\approx 0.041370323assets/equations/5c3c002092c7eca9.svg
• J(1)=0.7(1)^3+0=0.7assets/equations/5c598d8efe6ecd14.svg
• (1)(0.2)=0.2assets/equations/5c759a37e815fafb.svg
• W^{(1)}_{2,2,\mathrm{new}}=0.3-1(0.002456654)\approx 0.297543346assets/equations/5c8769134fc1a5f1.svg
• \begin{bmatrix}1 & 0\\0 & -1\end{bmatrix}assets/equations/5c899cfe4dee6d06.svg
• g_{1,2}^{(1)}=(0.008724562)\times(0)\approx 0assets/equations/5c9199d39ef465bf.svg
• h=f(Wx+b)assets/equations/5c938cf38724d627.svg
• D=1.000000000+1.000000000+1.000000000\approx3.000000000assets/equations/5cbcea061b92c91b.svg
• \ln(0.1)\approx-2.302585093assets/equations/5ccbedb0aa22add9.svg
• o_{1}=\frac{1}{1+e^{-(0.770465929)}}=0.683621675assets/equations/5ce3c8d47df25f40.svg
• \lfloor4/1\rfloor=4assets/equations/5ce7e863527f4bb8.svg
• \eta_{\mathrm{effective},t}=\frac{\eta}{\sqrt{s_t}+\varepsilon}assets/equations/5cf38c4d809aeefb.svg
• \lVert g\rVert_2=\sqrt{6^2+8^2}=\sqrt{36+64}=10assets/equations/5cf7611fc86c5694.svg
• \text{weights}=8\times3=24,\quad\text{biases}=3,\quad P_{\mathrm{dense}}=27assets/equations/5d064152bee760f3.svg
• y_{max}=\max_{(i,j)\in W}x_{ij},\quad y_{avg}=\frac1{|W|}\sum_{(i,j)\in W}x_{ij}assets/equations/5d156947c613cf69.svg
• N_{\mathrm{weights}}=5\times5\times6\times16=2{,}400assets/equations/5d2e007b43373429.svg
• g_{2,3}^{(1)}=(-0.014526308)\times(0.5)\approx -0.007263154assets/equations/5d328fe04a37cfc6.svg
• b_{h_1,\mathrm{new}}=0.6-0.5(-0.009617203)=0.604808601assets/equations/5d88aea5b1073764.svg
• W^{(1)}_{2,2,\mathrm{new}}=0.08-0.05(-0.36405)\approx 0.0982025assets/equations/5d899a39e45c0338.svg
• \frac{\partial L}{\partial v}=(-0.25)(0)+(-0.5)(0.2)=-0.1assets/equations/5d9bcac5813d6549.svg
• 5-(-2)=5+2=7assets/equations/5db46d4affd9d09b.svg
• p=\frac{1}{1+e^{-z}}assets/equations/5dd16ba0b4d79641.svg
• v_1=0.999(0)+0.001(2)^2=0.004assets/equations/5ddaed21d0d6628c.svg
• \ln(0.9)\approx-0.105360516assets/equations/5deadd8f45d24fdd.svg
• f\prime(0)=(0.500000000)(1-0.500000000)\approx0.250000000assets/equations/5e112d2ee69475de.svg
• \frac{\partial L}{\partial W_{ji}}=\delta_ja_i,\qquad\frac{\partial L}{\partial b_j}=\delta_jassets/equations/5e2e128efb837fa9.svg
• R_2=(7)\times(1)+(6)\times(2)+(7)\times(1)=26assets/equations/5e2ed47a500a084d.svg
• h_2=0.2(2)+0.5(0.2)+0=0.4+0.1=0.5assets/equations/5e340a1f975faf8c.svg
• L_{\mathrm{half}}=\frac{6}{2}=3assets/equations/5e41d17ab2e98c3d.svg
• n_{\mathrm{flat}}=24\times24\times64=36{,}864assets/equations/5ee055840a7e6f06.svg
• W^{(1)}_{2,3,\mathrm{new}}=0.2-0.5(-0.007263154)\approx 0.203631577assets/equations/5eeed9f16670c55b.svg
• R_3=(1)\times(1)+(0)\times(0)+(1)\times(-1)=0assets/equations/5f03457a49dcba68.svg
• h_{2}=\frac{1}{1+e^{-(1)}}=0.731058579assets/equations/5f1426f29e3d87e6.svg
• \frac{\partial (a-y)^2}{\partial a}=2(a-y)\times1assets/equations/5f66834fe766593d.svg
• 27+1=28assets/equations/5f8acbeb6a1db0ac.svg
• W^{(1)}_{2,3,\mathrm{new}}=0.2-0.9(0.006542085)\approx 0.194112123assets/equations/5f93f3b16b7e9a42.svg
• C_{\mathrm{out}}=\sum_k C_kassets/equations/5fcf90049e052b72.svg
• h_1=0.2(1)+0.5(0)+0=0.2assets/equations/600933d472001379.svg
• h_1=1assets/equations/6025713e0c4b56ee.svg
• w_{2,\mathrm{new}}=0+0.5\times1\times1=0.5assets/equations/6067f1cf6b435026.svg
• N_{\mathrm{out}}=N-K+1assets/equations/6074916da488a8bc.svg
• \delta_{h_j}=\left(\sum_kw_{kj}\delta_{o_k}\right)f'(z_{h_j})assets/equations/6086310bbc701ed1.svg
• P_10=32+1=33assets/equations/6095954f97f48924.svg
• \theta_2\approx0.9-0.093217964\approx0.806782036assets/equations/60970554f64b5b60.svg
• \delta_{h_2} = 0.020617214 \times 0.170222124 = 0.003509506assets/equations/60a157af571eaf0a.svg
• N_{\mathrm{out}}=\left\lfloor\frac{128+2(0)-3}{1}\right\rfloor+1=126assets/equations/60bad070a07deb57.svg
• \delta_{o_2} = 1 \times (0.778261739-0.3) \times 0.172570404 = 0.082533822assets/equations/60c2dd77da13f6a8.svg
• P_1=150+6=156assets/equations/60d683ee579f75b5.svg
• p_1=\frac{0.135335283}{1.503214724}\approx0.090030573assets/equations/60d7f2c1aed7f3ca.svg
• g_{batch}=\frac1B\sum_{i=1}^{B}g_i,\quad U_{epoch}=\left\lceil\frac NB\right\rceilassets/equations/60db920018eb907a.svg
• 8(8)(8)(3)(3)(4)=18432assets/equations/60dc7f05c85051f9.svg
• z_2=0+1-1.5=-0.5assets/equations/60ef65c2ea0ecf5f.svg
• z_o=w_2h+b_2,\qquad\hat y=g(z_o)assets/equations/60f589c099da8571.svg
• \sigma(-2)=\frac1{8.389056099}\approx0.119202922assets/equations/610916657193c755.svg
• \tanh(z)=\frac{e^{2z}-1}{e^{2z}+1},\quad \tanh'(z)=1-\tanh^2(z)assets/equations/610c0fbb4970deec.svg
• R=\frac{0.1}{2}(0.4)^2=0.05(0.16)=0.008assets/equations/61163a2b02bc5b66.svg
• R_3=(3)\times(1)+(3)\times(1)+(2)\times(1)=8assets/equations/612c6e731f469d2a.svg
• N_b=C_{\mathrm{out}}assets/equations/614d1f9d70b9928b.svg
• w_{1,\mathrm{new}}=0+0.5\times1\times1=0.5assets/equations/61526c5e9b2167a4.svg
• N_{\mathrm{out}}=\left\lfloor\frac{122-2}{2}\right\rfloor+1=61assets/equations/61527941fb172a39.svg
• p_3=\frac{1.000000000}{1.503214724}\approx0.665240956assets/equations/6153a5efe8276ec2.svg
• g_{\mathrm{batch}}=\frac{-1+(-4)}2=-2.5assets/equations/6159bfc1e25aba9f.svg
• w_i^{\mathrm{new}}=w_i-\eta g_iassets/equations/616c8cbb90430248.svg
• N_{\mathrm{out}}=\left\lfloor\frac{105-2}{2}\right\rfloor+1=52assets/equations/616d6791b658eeb9.svg
• z_{h_2} = (-0.2 \times 0.2) + (0.3 \times 0.4) + (0.1) = 0.18assets/equations/6178667245d07427.svg
• g_{3,1}^{(1)}=(0.012994055)\times(0.4)\approx 0.005197622assets/equations/618f330eb0013565.svg
• P=N_w+N_bassets/equations/61924cb4898d8799.svg
• y_{\min}=\min(x_1,x_2,x_3,x_4)assets/equations/619bb1c028de4233.svg
• \delta_T=h_T-yassets/equations/61b2086787d829b8.svg
• \begin{aligned}\frac12&=0.5\\\frac1{10}&=0.1\\\frac1{100}&=0.01\end{aligned}assets/equations/61c8b62ceed00f18.svg
• f\prime(-2)=(0.119202922)(1-0.119202922)\approx0.104993585assets/equations/61cf4a9551a25fd6.svg
• g_{1,2}^{(2)}=(-0.131169078)\times(0.524979187)\approx -0.068861036assets/equations/61ef33f1a6d1ba26.svg
• P_C=0+(1\times1\times8+1)\times2=18assets/equations/61f74ab4b2624099.svg
• \frac{\partial L}{\partial h}=2.4+1.1=3.5assets/equations/61ff2149ac92a571.svg
• W^{(l)}\in\mathbb R^{m\times n},\qquad b^{(l)},z^{(l)},a^{(l)}\in\mathbb R^{m\times1}assets/equations/6213a7d0a0d572f3.svg
• \delta_o=\frac{\partial L}{\partial z_o}=(\hat y-y)g'(z_o)assets/equations/6215ea243129f507.svg
• W^{(2)}_{2,1,\mathrm{new}}=0.5-0.5(-0.02260254)\approx 0.51130127assets/equations/62292012970cdab4.svg
• m_1=0+0.2=0.2assets/equations/624684fe1c8cf03d.svg
• z_{h_1} = (0.4 \times 3) + (0.2) = 1.4assets/equations/624d0aa704cb40ed.svg
• o_{2}=\frac{1}{1+e^{-(1.224921404)}}=0.772928465assets/equations/624e9e7741da0cb6.svg
• \widehat m_2=\frac{0.28}{1-0.9^{2}}=1.473684211assets/equations/62512ed956356eba.svg
• J(-1)=2(-1)^1+4=2assets/equations/625a082643eb2a43.svg
• a^{(1)}_2=\max(0,0.5)=0.5assets/equations/6263aed38871ad59.svg
• g_{1,4}^{(2)}=\frac{\partial L}{\partial W^{(2)}_{1,4}}=\delta_{o_1}h_4assets/equations/62648b24044cc2ea.svg
• R_1=(1)\times(1)+(2)\times(-1)=-1assets/equations/62842635b6eb75f8.svg
• C_1=(0.15)\times(-0.809)\approx -0.12135assets/equations/628dab575705c1e4.svg
• C_{\mathrm{new}}=-0.365529289+(0.556769941)\approx0.191240652assets/equations/62b242322f0e2241.svg
• z h_{\mathrm{previous}}=0.25(0.4)=0.1assets/equations/62b46633101b4f4e.svg
• \frac{\partial L}{\partial z_j}=-\sum_i\frac{y_i}{p_i}p_i(\delta_{ij}-p_j)=-y_j+p_j\sum_i y_i=p_j-y_jassets/equations/630851ea15af9c6b.svg
• e^{-z}\approx0.135335283assets/equations/63363cd12105c12d.svg
• b_{h_2,\mathrm{new}}=0.2-0.1(-0.275000000)=0.227500000assets/equations/6350b7de1a7fd962.svg
• s_2=0.9(0.1)+0.1(2)^2=0.49assets/equations/6351a5fcafc4c5e8.svg
• D_{\mathrm{KL}}(p\Vert q)=H(p,q)-H(p)assets/equations/636a195581e18582.svg
• \hat m_t=\frac{m_t}{1-\beta_1^t},\qquad \hat v_t=\frac{v_t}{1-\beta_2^t} · \hat m_t=\frac{m_t}{1-\beta_1^t},\quad\hat v_t=\frac{v_t}{1-\beta_2^t} · \hat m_t=\frac{m_t}{1-\beta_1^t},\qquad\hat v_t=\frac{v_t}{1-\beta_2^t}assets/equations/637a35db2d99492a.svg
• J_{L2}=0.4+0.2625=0.6625assets/equations/6399f9c64b91d83f.svg
• z=1.2+(-0.8)=0.4assets/equations/63a2763d058f9aec.svg
• h_{2}=\frac{1}{1+e^{-(0.18)}}=0.544878892assets/equations/63a52c4f6d59b8f1.svg
• N_{\mathrm{weights}}=32\times1=32assets/equations/63bcab1d4c5ad97d.svg
• \begin{bmatrix}20 & 30\\112 & 37\end{bmatrix}assets/equations/63c5529d08234533.svg
• -\ln(0.1)=-(-2.302585093)=2.302585093assets/equations/63eac781aee3c744.svg
• N_{\mathrm{weights}}=107{,}648\times84=9{,}042{,}432assets/equations/63fdb30c55665f91.svg
• \delta_h=\delta_o w_2 f'(z_h)assets/equations/6409774c2652b14e.svg
• x-\mu=9-5=4assets/equations/640d8d05be10b7d3.svg
• R_3=(3)\times(1)+(3)\times(1)+(2)\times(1)=8assets/equations/640dbb668111d247.svg
• (N_{\mathrm{in}}+1)N_{\mathrm{out}}assets/equations/641d99b5a929f5f6.svg
• W^{(2)}_{1,1,\mathrm{new}}=0.1-1(0.010408643)\approx 0.089591357assets/equations/642d828dd8261846.svg
• \frac{\partial z}{\partial w}=x,\qquad \frac{\partial z}{\partial b}=1,\qquad \frac{\partial z}{\partial x}=wassets/equations/6437deea893fce68.svg
• N_{\mathrm{weights}}=84\times32=2{,}688assets/equations/6439e320bc2235fb.svg
• U_{\mathrm{keep}}=6\times3=18assets/equations/64422d140052bf5c.svg
• h_1=\tanh(0.7)\approx0.604367777assets/equations/644bf14870d92481.svg
• \theta_{\mathrm{look},t}=\theta_{t-1}+\beta v_{t-1}assets/equations/64563c533a276a7d.svg
• v_{1,\mathrm{new}}=0.9(0.1)-0.1(-1.2)=0.09+0.12=0.21assets/equations/6463456a76ac5311.svg
• C_{11}=1(2)+2(1)=2+2=4assets/equations/646952b8a744b648.svg
• a=\frac1{1.135335283}\approx0.880797078assets/equations/6480e62e9ef0f8a2.svg
• \frac{\partial L}{\partial z_3}=0.1-0=0.1assets/equations/64941f713e55940c.svg
• N_{\mathrm{out}}=\left\lfloor\frac{48-2}{2}\right\rfloor+1=24assets/equations/64a91a8e536e0765.svg
• \tanh(C)=\tanh(0.331058579)\approx0.319471636assets/equations/64c9a601580cf0bc.svg
• g_{2,2}^{(1)}=(0.006542085)\times(0)\approx 0assets/equations/64d7fdc83e81a56d.svg
• g_{1,2}^{(2)}=(0.013687964)\times(0.782449776)\approx 0.010710144assets/equations/64e131d9d2fc4b37.svg
• d=\sqrt{1+0.00001}\approx1.00000499999assets/equations/64e8bea52d19d8d8.svg
• R_2=(1)\times(0)+(3)\times(1)=3assets/equations/651ec43291a6ceee.svg
• g_{1,1}^{(1)}=(-0.11326)\times(2)\approx -0.22652assets/equations/6540c5b7941f90d3.svg
• C=\begin{bmatrix}4&6\\10&12\end{bmatrix}assets/equations/65427879b2654199.svg
• \delta_{o_2} = 1 \times (0.618432848-0) \times 0.235973661 = 0.145933863assets/equations/6546421fef60e0c5.svg
• g_{\mathrm{batch}}=\frac1B\sum_i g_iassets/equations/657f63b5043d3907.svg
• \delta_{o_2} = 1 \times (0.772928465-0.99) \times 0.175510053 = -0.038098237assets/equations/65a80e7107d22398.svg
• L=-(-0.223143551)=0.223143551assets/equations/65af868a518fd08e.svg
• g_1=(0-1)\times1=-1assets/equations/65b3616ae354de6a.svg
• fC_{\mathrm{old}}=(0.500000000)(1)\approx0.500000000assets/equations/65b5c67b809b7ce0.svg
• R_1=(2)\times(1)+(0)\times(-1)=2assets/equations/65b69a57c4f3db18.svg
• z_{XOR}=0-2(0)-0.5=-0.5assets/equations/65c6f2e86202561b.svg
• \left\lfloor\frac{3-2}{1}\right\rfloor+1assets/equations/65d999fa66cd37e1.svg
• L=-\ln(0.665240956)=0.407605964assets/equations/65db0db7f14d7ab8.svg
• \mathrm{MACs}_{\mathrm{direct}}=8192\times576=4{,}718{,}592assets/equations/660ea75dfc92d56c.svg
• p=\frac1{1.367879441}\approx0.731058579assets/equations/6617ec66ea10120e.svg
• \left\lfloor\frac{52+2 \times 0-5}{1}\right\rfloor+1assets/equations/6619d9b65db6ea0f.svg
• \nabla_{\hat y}\mathrm{MSE}=\frac23[0.2,-0.3,-0.6]=[0.133333333,-0.2,-0.4]assets/equations/661d59f4a5205fa6.svg
• s_1=0.999(0)+0.001(2^2)=0.004assets/equations/664a8100f810d926.svg
• Y_{ij}=\sum_{u,v}X_{iS+u,jS+v}K_{uv}+bassets/equations/6652d02289b39594.svg
• z_{h_1}=0+0-0.5=-0.5\Rightarrow h_1=0assets/equations/6669d48b7635634e.svg
• \frac{\partial L}{\partial w}=\delta x,\quad \frac{\partial L}{\partial b}=\deltaassets/equations/6678df97903ca342.svg
• \frac{\partial L}{\partial h_t}=0.1+0.2=0.3assets/equations/6685e4ec204aeb2d.svg
• \text{Sum}=C_1+C_2\approx 0.019492161assets/equations/66b0d378cdd2503f.svg
• N_{\mathrm{pool}}=\left\lfloor\frac{8-2}{2}\right\rfloor+1=3+1=4assets/equations/66d12a448520cead.svg
• \frac{\partial e}{\partial z}=\frac{\partial(z-y)}{\partial z}=1assets/equations/66db443def7ab596.svg
• a=f(z)assets/equations/66df67478858e480.svg
• W^{(2)}_{1,2,\mathrm{new}}=0.25-0.5(-0.072418335)\approx 0.286209167assets/equations/66e7552de1ba0f15.svg
• v_{2,\mathrm{new}}=0.9(-0.1)-0.1(0)=-0.09assets/equations/66eba48d514886f3.svg
• \sigma'(z)=-(1+e^{-z})^{-2}(-e^{-z})=\frac{e^{-z}}{(1+e^{-z})^2}assets/equations/67121a49478eada8.svg
• W^{(1)}_{3,2,\mathrm{new}}=0.3-0.5(0.003898216)\approx 0.298050892assets/equations/671d11febf8e19cd.svg
• \left\lfloor\frac{256+2 \times 0-5}{1}\right\rfloor+1assets/equations/6723e51cb832856a.svg
• \mathrm{MACs}_{\mathrm{direct}}=8192\times576=4{,}718{,}592assets/equations/673cd31676ab87d1.svg
• \frac{\partial L}{\partial w_1}=\delta_h x,\qquad\frac{\partial L}{\partial b_1}=\delta_hassets/equations/675e596602704d40.svg
• P_{\mathrm{pool}}=0,\quad P_{\mathrm{flatten}}=0,\quad P_{\mathrm{dense}}=(n_{\mathrm{in}}+1)n_{\mathrm{out}}assets/equations/676bf495296f4651.svg
• W^{(1)}_{1,2,\mathrm{new}}=0.1-0.1(-0.44)\approx 0.144assets/equations/679cfd3e82742ab1.svg
• z=\frac{-2}{2}=-1assets/equations/679dccde63fa3152.svg
• a_2\approx1+0.2(0.462117157)\approx1.092423431assets/equations/67ad1cca6bc26681.svg
• \theta_1=1-0.316227766\approx0.683772234assets/equations/67ba0f8d71ab9792.svg
• N_{\mathrm{weights}}=240\times120=28{,}800assets/equations/67cf05b2fc828555.svg
• P_3=614{,}400+256=614{,}656assets/equations/67ebf4d50d295918.svg
• \operatorname{ReLU}(z)=\max(0,z)assets/equations/67fab45c729a152a.svg
• W^{(1)}_{1,1,\mathrm{new}}=0.8-0.5(-0.003846881)\approx 0.801923441assets/equations/681392cf40c3207a.svg
• g_{3,3}^{(1)}=(0.012994055)\times(0.5)\approx 0.006497027assets/equations/682fcf2989244bd9.svg
• N_{\mathrm{pool}}=\left\lfloor\frac{4-2}{2}\right\rfloor+1=1+1=2assets/equations/6833d7ee27072935.svg
• P_{\mathrm{conv}}=(K^2C_{\mathrm{in}}+1)C_{\mathrm{out}},\qquad P_{\mathrm{dense}}=(n_{\mathrm{in}}+1)n_{\mathrm{out}}assets/equations/683bcf7c089c902f.svg
• 0.1(2)=0.2assets/equations/687125898d5f60b7.svg
• L=\frac12(\hat y-y)^2,\qquad \frac{\partial L}{\partial \hat y}=\hat y-yassets/equations/689f9f8b396b9fab.svg
• h_{1}=\frac{1}{1+e^{-(1)}}=0.731058579assets/equations/68fa3eedfd721c67.svg
• 10\longrightarrow5\longrightarrow5\longrightarrow5\longrightarrow4assets/equations/68fe56c651575649.svg
• fC_{\mathrm{previous}}=0.8\times0.3=0.24assets/equations/692dcfa59ab03f38.svg
• h_t=o_t\tanh(C_t)assets/equations/694789d6807a2931.svg
• (a\odot b)_i=a_i b_iassets/equations/696e2da1033ab6fa.svg
• o_{1}=\frac{1}{1+e^{-(1.40173889)}}=0.802459679assets/equations/69ae9d8fc60db8fd.svg
• m_2=0.9(0.2)+0.1(-1)=0.08assets/equations/69b54ca313d8829c.svg
• \frac{\partial L}{\partial x}=\frac{\partial L}{\partial y}\left(\frac{\partial F}{\partial x}+I\right)assets/equations/69be13fbc705ad29.svg
• 4(8)=32assets/equations/69e92d1e9a3f1ea7.svg
• x_{scaled}=\frac{x-\mu_{train}}{s_{train}}assets/equations/6a0a4f4a38fa7007.svg
• y_{\max}=15assets/equations/6a17ef02b89e67e2.svg
• Y_{2,2}=(-2)+(0)=-2assets/equations/6a1f7599e33d0692.svg
• L_{\mathrm{CCE}}=-\sum_i y_i\ln p_iassets/equations/6a21d413c89b5a80.svg
• g_b=\frac{\partial L}{\partial b}=\delta_z\frac{\partial z}{\partial b}=(-2)(1)=-2assets/equations/6a2c0a4ed56c043f.svg
• R_2=(0.5)\times(1)+(0.5)\times(0)+(0.3)\times(-1)=0.2assets/equations/6a410c53ff471ca9.svg
• b_{h_1,\mathrm{new}}=0-0.05(-0.113260000)=0.005663000assets/equations/6a689b75bd65601f.svg
• w_{1,\mathrm{new}}=0.5-0.1(0.128)=0.4872assets/equations/6a6ad843a78157c3.svg
• D_{\mathrm{KL}}(p\Vert q)\approx0.693147181-0.562335145=0.130812036assets/equations/6a763ffee09c5ad1.svg
• g_2=(\hat y_1-y)x=(2.8-4)(2)=-2.4assets/equations/6a7b033dfd881b35.svg
• b_{o_1,\mathrm{new}}=0-0.05(-0.809000000)=0.040450000assets/equations/6a95d59624552356.svg
• b_{o_2,\mathrm{new}}=0.1-0.2(0.145933863)=0.070813227assets/equations/6ab34abf185daf06.svg
• h_1=0.7\tanh(0.4)=0.7(0.379948962)=0.265964274assets/equations/6ab753ebe65dd463.svg
• \text{Sum}=C_1+C_2\approx 0.019492161assets/equations/6abb0857809f8db8.svg
• p_i=\frac{e^{z_i}}{\sum_j e^{z_j}}=\frac{e^{z_i-c}}{\sum_j e^{z_j-c}},\qquad c=\max_jz_jassets/equations/6ad7e7d9830e70ac.svg
• L=L_1+L_2=0.72+0.605=1.325assets/equations/6adf6f563023ced5.svg
• -z=-2.2assets/equations/6ae6042702919060.svg
• \tanh(0.83)\approx0.680476006assets/equations/6aee7e26a8e96885.svg
• s_t=\beta s_{t-1}+(1-\beta)g_t^2assets/equations/6b2de5dc1fbe8c99.svg
• g_{3,2}^{(1)}=(0.012994055)\times(0.3)\approx 0.003898216assets/equations/6b2e81bef7a212bd.svg
• U_{\mathrm{drop}}=5\times3=15assets/equations/6b34106919fd6639.svg
• e^{0}\approx1.000000000assets/equations/6b377e92ad2fd68c.svg
• \tanh(0)\approx\frac{0.000000000}{2.000000000}\approx0.000000000assets/equations/6b66e253736ed179.svg
• \frac{da}{dz}=1-a^2assets/equations/6b6ac5e0c1df9b26.svg
• W^{(2)}_{1,2,\mathrm{new}}=0.5-0.1(-0.165)\approx 0.5165assets/equations/6b95f80af95c8ca3.svg
• \frac{\partial z}{\partial w}=x=2assets/equations/6badd88b171bea97.svg
• 10(5)=50assets/equations/6bd3c4c12c1c454b.svg
• z_{o_1} = (0.1 \times 0.760423002) + (0.24 \times 0.782449776) + (1) = 1.263830247assets/equations/6bdace1303796a35.svg
• z_{h_2} = (0.3 \times 1) + (0.7 \times 0) + (0.2) = 0.5assets/equations/6bf10ef9c5abf8e9.svg
• e^{0}+e^{0}\approx1.000000000+1.000000000=2.000000000assets/equations/6c3fb9efcb35a389.svg
• \tanh(2)\approx\frac{7.253720816}{7.524391382}\approx0.964027580assets/equations/6c415e8a033259d4.svg
• J(\theta-\eta\nabla J)\approx J(\theta)-\eta\|\nabla J\|_2^2assets/equations/6c50025de1d032a9.svg
• \theta_2\approx0.9-0.093217964\approx0.806782036assets/equations/6c55d4ff26b9e4b3.svg
• W^{(2)}_{1,3,\mathrm{new}}=-0.6-0.5(-0.059570648)\approx -0.570214676assets/equations/6c5ddaf1b80c1d4b.svg
• L_{\mathrm{half}}=\frac{9}{2}=4.5assets/equations/6c789702092108e1.svg
• \widetilde C_t=\tanh(W_C[h_{t-1},x_t]+b_C)assets/equations/6c797c7201f58bbb.svg
• \mathrm{MACs}=H_{\mathrm{out}}W_{\mathrm{out}}C_{\mathrm{out}}K^2C_{\mathrm{in}}assets/equations/6c847f45a17553cd.svg
• L=-[0\ln(0.1)+1\ln(0.8)+0\ln(0.1)]assets/equations/6cc766252350138b.svg
• L=\frac12(-0.3)^2=\frac12(0.09)=0.045assets/equations/6ccc7a83cb90d797.svg
• W^{(1)}_{2,2,\mathrm{new}}=0.08-0.05(-0.36405)\approx 0.0982025assets/equations/6cd428530ca786f9.svg
• \widetilde h=\tanh(0.3)=0.291312612assets/equations/6ce7040fd0d1872b.svg
• \sigma=\sqrt{\sigma^2},\qquad x_{\mathrm{std}}=\frac{x-\mu}{\sigma}assets/equations/6d0a9fe98061d681.svg
• z_2=1+0-1.5=-0.5assets/equations/6d1cd15a27d9572e.svg
• 4(1)=4assets/equations/6d1e399cd2e46091.svg
• w_{1,\mathrm{new}}=-0.5-0.2(0.1)=-0.5-0.02=-0.52assets/equations/6d4d841ab160b740.svg
• a_1=0.5(1)+0.2(0)+0=0.5assets/equations/6d7c2d3322af892b.svg
• \frac{\partial L}{\partial z}=\frac{\partial L}{\partial p}\frac{\partial p}{\partial z}=\frac{p-y}{p(1-p)}p(1-p)=p-yassets/equations/6d9c2d8649b6476d.svg
• \frac{dL}{dp}=\frac{p-y}{p(1-p)}assets/equations/6db0182710f5e684.svg
• L_0=\frac12(-2)^2=2assets/equations/6de0a03e9715872f.svg
• g_{j,\mathrm{total}}=g_{j,\mathrm{data}}+\lambda w_jassets/equations/6dece7c528660c2d.svg
• \theta_t=\theta_{t-1}-\eta g_tassets/equations/6df9cee9010e92ee.svg
• y_{\max}=18assets/equations/6e005f373e798e70.svg
• g_1=2.1(1)^2=2.1assets/equations/6e21de54ead2d325.svg
• W^{(1)}_{4,2,\mathrm{new}}=0.2-0.5(-0.001284824)\approx 0.200642412assets/equations/6e488fc21bed8f11.svg
• J(\theta)=\frac12(\theta-1)^2,\qquad J\prime(\theta)=\theta-1assets/equations/6e49b01175757fe7.svg
• \tanh(z)=\frac{e^{2z}-1}{e^{2z}+1},\quad \tanh\prime(z)=1-\tanh^2(z)assets/equations/6e538ee72147c838.svg
• a'_i=\frac{m_i a_i}{q},\qquad m_i\in\{0,1\}assets/equations/6e936750aa2b4998.svg
• h=0.8(0.680476006)\approx0.544380805assets/equations/6ea5c9d366d636a3.svg
• \mu=\frac{20}{4}=5assets/equations/6eafa89b652a5378.svg
• 0.6(1)=0.6assets/equations/6eb084da38b1ada3.svg
• z_{o_1} = (0.6 \times 0.802183889) + (0.1) = 0.581310333assets/equations/6ee49ebab88ca242.svg
• z_{XOR}=1-2(0)-0.5=0.5assets/equations/6ee6edf14f171730.svg
• z=0\times1+0\times1+0=0,\qquad o=0assets/equations/6eff381fa8ae5b02.svg
• w_3x_3=(-0.2)\times0.5=-0.1assets/equations/6f257d4f869d8fa2.svg
• f\prime(2)=1-(0.964027580)^2\approx0.070650825assets/equations/6f2d0c822236e436.svg
• z=0\times1+0\times1+0=0,\qquad o=0assets/equations/6f31e8743e028b72.svg
• \eta_{effective}=\frac{\eta}{\sqrt{G_t}+\epsilon}assets/equations/6f335dbb7b791f6d.svg
• z=\frac{x-\mu_{\mathrm{train}}}{\sigma_{\mathrm{train}}},\qquad \sigma_{\mathrm{train}}>0assets/equations/6f3798eefa1e56a4.svg
• N_{\mathrm{out}}=1+1=2assets/equations/6f3dfc60f0024b2c.svg
• z=Wx+b,\qquad a_j=\max(0,z_j)assets/equations/6f48223c352868b0.svg
• \mu=\frac{6}{3}=2assets/equations/6f48ae6fcd2c250e.svg
• g_2=(-2)(2)=-4assets/equations/6f6d46df3c042284.svg
• \begin{bmatrix}4 & 6\\10 & 12\end{bmatrix}assets/equations/6f769fba9b2febe4.svg
• J\prime(w)=0.7(3)w^{3-1}+0assets/equations/6f7d282bf1129194.svg
• \frac{\partial L}{\partial w_{ji}}=\delta_j a_i,\qquad \frac{\partial L}{\partial b_j}=\delta_jassets/equations/6f883fcc3e95816b.svg
• z_o=1-2(1)-0.5=-1.5\Rightarrow o=0assets/equations/6f8b6a8f26883852.svg
• p_2=\frac{1.000000000}{3.000000000}\approx0.333333333assets/equations/6f93f83cf16d42a1.svg
• 21+11+12+9=53assets/equations/6f9a0a96d6f9e048.svg
• L_2=\frac12(-1.1-0)^2=0.605assets/equations/6fa32649e91893bc.svg
• \operatorname{Var}(X)=\mathbb E[(X-\mu)^2]=\mathbb E[X^2]-\mu^2assets/equations/6fbba36f3ebff68d.svg
• z_{h_3} = (0.3 \times 0.4) + (0.3 \times 0.3) + (-0.1 \times 0.5) + (0.5 \times -0.4) + (0.6) = 0.56assets/equations/6fe54eb13873171d.svg
• \frac{\partial L}{\partial w}=(o-y)x,\qquad \frac{\partial L}{\partial b}=o-yassets/equations/6ffe92ced0000c23.svg
• v_2=0.999(0.001)+0.001(2)^2=0.004999assets/equations/6ffede766a41d6bd.svg
• e_2=6-4=2assets/equations/7004e5d2962bc876.svg
• a_2=1+0.092423431=1.092423431assets/equations/7019e7bc4f1316c8.svg
• h_{2}=\frac{1}{1+e^{-(1.28)}}=0.782449776assets/equations/70222d9bfd6af500.svg
• N_{\mathrm{out}}=\left\lfloor\frac{28}{2}\right\rfloor+1=14+1=15assets/equations/70537488f5dc8b8a.svg
• 1505280+3763200=5268480assets/equations/7061d250d8255604.svg
• \frac{d(cw^n+k)}{dw}=cnw^{n-1}assets/equations/706ea9b2290fe843.svg
• h_{2}=\frac{1}{1+e^{-(0.1)}}=0.524979187assets/equations/708107445814199b.svg
• u_{\mathrm{new}}=0.2-0.1(-1.25)=0.325assets/equations/7085a94b1a865ca8.svg
• a=0.2(1)+0.5(0.2)+0=0.3assets/equations/708a856537eccf89.svg
• r_t=\sigma(W_rx_t+U_rh_{t-1}+b_r)assets/equations/70d5c123a1a8d5b7.svg
• \left\lfloor\frac{27+2 \times 2-5}{1}\right\rfloor+1assets/equations/71046f85d0cc0ca5.svg
• \delta_{o_1} = 1 \times (1.08-1) \times 1 = 0.08assets/equations/71054ce9ec8e2e00.svg
• L=-(-0.223143551)=0.223143551assets/equations/710d02f73efd4006.svg
• \text{future contribution}=0.5(0.4)=0.2assets/equations/71202ea5331ced5a.svg
• C_1=(0.5)\times(-0.55)\approx -0.275assets/equations/7150a7153bb561f5.svg
• \begin{bmatrix}4 & 5\\7 & 8\end{bmatrix}assets/equations/71895dac90bdedf0.svg
• z_{h_2} = (0.12 \times 2) + (0.08 \times 3) + (0) = 0.48assets/equations/71f2cfea44a277ff.svg
• \frac{224-2}{2}+1=111+1=112assets/equations/71feab89feae610a.svg
• \widehat m_1=\frac{-0.1}{1-0.9^{1}}=-1assets/equations/721803042ed359b9.svg
• \text{L2 penalty}=0.05\times5.25=0.2625assets/equations/722ddb9c14ea7480.svg
• 0.9\times4=3.6,\qquad(1-0.9)\times9=0.9assets/equations/722eae46e0545987.svg
• \|g\|_2=\sqrt{144}=12assets/equations/723310d10d7d6089.svg
• z_{h_1}=1+1-0.5=1.5\Rightarrow h_1=1assets/equations/7236f1129a18b80e.svg
• z=wx+b,\quad p=\frac1{1+e^{-z}},\quad L=\frac12(p-y)^2assets/equations/72569e3d1cf18215.svg
• (1)(2)=2assets/equations/7259499711e389d2.svg
• z_{o_1} = (0.5 \times 0.3) + (0.6 \times 0.5) + (0.1) = 0.55assets/equations/725a43999780ebd1.svg
• e_1^2=(-1)^2=1assets/equations/725d81fce4be12d0.svg
• o_{1}=\frac{1}{1+e^{-(1.40173889)}}=0.802459679assets/equations/728269faf184ac73.svg
• J=0.02+0.008=0.028assets/equations/72c65e08a565a4c1.svg
• o_{1}=\frac{1}{1+e^{-(0.581310333)}}=0.641368858assets/equations/72c83495e26a383d.svg
• [1-2,\;2-2,\;0-2]=[-1,0,-2]assets/equations/72ceba57b734c648.svg
• h_t=\tanh(W_{xh}x_t+W_{hh}h_{t-1}+b_h),\quad p_t=\operatorname{softmax}(W_{hy}h_t+b_y)assets/equations/72d09ad3983b6b52.svg
• 1+e^{-z}=1+0.110803158=1.110803158assets/equations/72d833375eab0d1c.svg
• z=0.6+(-0.4)+(-0.1)+0.1assets/equations/72dd56dcc4bb45b6.svg
• \frac{d}{dz}\tanh z=\frac{2e^{2z}(e^{2z}+1)-2e^{2z}(e^{2z}-1)}{(e^{2z}+1)^2}=\frac{4e^{2z}}{(e^{2z}+1)^2}assets/equations/730661d0a7189944.svg
• g_{1,2}^{(1)}=(-0.22)\times(2)\approx -0.44assets/equations/7306e7ed77d36dd1.svg
• \delta_{h_2} = -0.12135 \times 1 = -0.12135assets/equations/73149a9e0e57e742.svg
• \sqrt{s_1}=\sqrt{0.4}\approx0.632455532assets/equations/73269d6fb6447a99.svg
• P_9=2{,}688+32=2{,}720assets/equations/737742757d7487f8.svg
• L=\frac12\sum_j(o_j-y_j)^2assets/equations/737ddfb3a05bc433.svg
• \text{weights}=64\times3=192assets/equations/738d2c475096e79a.svg
• h_1=0assets/equations/738e2d766424863e.svg
• W^{(1)}_{1,2,\mathrm{new}}=0.2-0.5(0.000877135)\approx 0.199561432assets/equations/738f6089731cd94e.svg
• J_{L1}=J_{\mathrm{data}}+\lambda\sum_i|w_i|,\qquad J_{L2}=J_{\mathrm{data}}+\lambda\sum_iw_i^2assets/equations/73a39a472fd87b8c.svg
• z=wx+b=1(2)+0=2assets/equations/73aa46b2ed8f2b55.svg
• \left\lfloor\frac{13-3}{2}\right\rfloor+1assets/equations/73ab07c24eb836b2.svg
• \theta_{new}=\theta+v_{new}assets/equations/73afc93b70062b2d.svg
• z_2=0.5(2)+3(1)-0.2=3.8assets/equations/73b17a005abd5aba.svg
• h_t=\tanh(W_{xh}x_t+W_{hh}h_{t-1}+b_h),\quad p_t=\operatorname{softmax}(W_{hy}h_t+b_y)assets/equations/73b94d9abc839fd5.svg
• 18=N-3+1=N-2assets/equations/73b9e54fad32ca52.svg
• Y_{3,3}=(0)+(-3)=-3assets/equations/73f07f2c01400713.svg
• N_{\mathrm{out}}=\left\lfloor\frac{219-2}{2}\right\rfloor+1=109assets/equations/73ff985869e15a08.svg
• \frac{\partial L}{\partial W_2}=\delta_2a_1^T,\qquad\frac{\partial L}{\partial b_2}=\delta_2assets/equations/7401adc017e3686d.svg
• z=1+(-1.5)=-0.5assets/equations/740ecc8b0234d38c.svg
• 2P=2(1)=2assets/equations/741118212eb25f73.svg
• z_{h_2} = (0.5 \times 1) + (0.5 \times 1) + (0) = 1assets/equations/74149e3e776dce9d.svg
• \mathrm{MACs}_{\mathrm{bottleneck}}=131{,}072+589{,}824=720{,}896assets/equations/741f21ae6e5bfc83.svg
• N_w=K_hK_wC_{\mathrm{in}}C_{\mathrm{out}}assets/equations/74209557d398075e.svg
• \left\lfloor\frac{227+2 \times 0-11}{4}\right\rfloor+1assets/equations/74235148b2b7312a.svg
• z_{h_1} = (0.2 \times 1) + (0.4 \times 0) + (-0.5 \times 1) + (-0.4) = -0.7assets/equations/743c8cd16a50ddc1.svg
• \frac{\partial L_2}{\partial h}=(-1.1)(-1)=1.1assets/equations/74549258b69e068b.svg
• w_{t+1}=w_t-\eta(\nabla_w J_{\mathrm{data}}+2\lambda w_t)=(1-2\eta\lambda)w_t-\eta\nabla_w J_{\mathrm{data}}assets/equations/745f5ba3fa313836.svg
• \frac{dv}{dw}=2(3)=6assets/equations/746afadc90e4f6c9.svg
• \mathrm{SSE}=\sum_i e_i^2,\qquad L_{\mathrm{half}}=\frac12\sum_i e_i^2,\qquad \mathrm{MSE}=\frac1n\sum_i e_i^2assets/equations/74a2e5af6d9c93b2.svg
• Y_{2,1}=(0)+(-1)=-1assets/equations/74f7aefb8b0d75bd.svg
• g=0.3+0.4=0.7assets/equations/74fdb43956c5f462.svg
• C=0.24+(-0.1)=0.14assets/equations/752dd644f97def6b.svg
• R_2=(1)\times(0)+(0)\times(1)=0assets/equations/75388bfbb51edc5c.svg
• g_{1,2}^{(1)}=(0.008771355)\times(0.1)\approx 0.000877135assets/equations/753fcf33c45e8537.svg
• (-2)^2=(-2)(-2)=4assets/equations/756026084f28a30c.svg
• a=\frac1{1+e^{-z}}=\frac1{1.110803158}\approx0.900249511assets/equations/7567a6adfafdbad0.svg
• b_{h_2,\mathrm{new}}=0.6-0.5(-0.014526308)=0.607263154assets/equations/758b03c4be06d181.svg
• C_t=f_t\odot C_{t-1}+i_t\odot\widetilde C_t,\quad h_t=o_t\odot\tanh(C_t)assets/equations/758fd5519435e456.svg
• N_{\mathrm{pool}}=\left\lfloor\frac{4-2}{2}\right\rfloor+1=1+1=2assets/equations/75bb76e8c3f74574.svg
• g_{1,3}^{(1)}=\frac{\partial L}{\partial W^{(1)}_{1,3}}=\delta_{h_1}x_3assets/equations/75cb25a10b4d1d19.svg
• b_{o_1,\mathrm{new}}=0.2-0.5(-0.093597941)=0.246798971assets/equations/75d8de25e1127fac.svg
• 103=5\times20+3assets/equations/7600fce77b11dfac.svg
• e_2^2=(0)^2=0assets/equations/760944c761b54733.svg
• g_{1,\mathrm{total}}=0.2+(-0.1)=0.1assets/equations/7626b740cec3ed4d.svg
• g_{2,1}^{(1)}=(0.009954255)\times(0.05)\approx 0.000497713assets/equations/76611b6fffb6eae0.svg
• \delta_{o_1} = 2 \times (0.683621675-0.9) \times 0.216283081 = -0.093597941assets/equations/7665056831c42066.svg
• v_2=0.99(0.04)+0.01(-1)^2=0.0496assets/equations/76708a6ba327fc21.svg
• g_{2,2}^{(1)}=(-0.014526308)\times(0.3)\approx -0.004357892assets/equations/7674dd635ddd9ccc.svg
• N_{\mathrm{weights}}=3\times3\times16\times32=4{,}608assets/equations/767ba18d56e40888.svg
• \sigma(0)=\frac1{1+e^{-0}}\approx0.500000000assets/equations/76974c84c0527257.svg
• \text{future contribution}=0.5(0.4)=0.2assets/equations/76c508722fc577c7.svg
• \delta_{o_2} = 1 \times (0.618432848-0) \times 0.235973661 = 0.145933863assets/equations/76cba5dae26e3af0.svg
• \theta_1=-0.5+(0.15)=-0.35assets/equations/76cd873d8955bf17.svg
• \text{biases}=4,\qquad P_{\mathrm{conv}}=36+4=40assets/equations/76ea5b945615119b.svg
• W^{(1)}_{2,2,\mathrm{new}}=0.3-0.5(0.000995425)\approx 0.299502287assets/equations/770fd618f5ab4f7d.svg
• h_t=\tanh(W_{xh}x_t+W_{hh}h_{t-1}+b_h),\quad o_t=W_{hy}h_t+b_yassets/equations/77145310b191a975.svg
• \theta_1=-0.5-\frac{0.1\times(-1)}{\sqrt{1}}=-0.4assets/equations/771de6f5ff86a0f6.svg
• \sigma(z)=\frac1{1+e^{-z}},\qquad \sigma'(z)=\sigma(z)[1-\sigma(z)]assets/equations/7727ce70c548ec20.svg
• R_2=(0)\times(2)+(3)\times(0)+(0)\times(-2)=0assets/equations/7737415f7a101340.svg
• e_2=5-3=2assets/equations/776a0d96be6de464.svg
• g_{\mathrm{batch}}=\frac{g_1+g_2}{2},\qquad w_{\mathrm{new}}=w-\eta gassets/equations/7783293f367f0a09.svg
• Y_{1,3}=(0)+(-2)=-2assets/equations/77839b654a5d343b.svg
• W^{(2)}_{1,2,\mathrm{new}}=0.24-1(0.010710144)\approx 0.229289856assets/equations/7790608d36a6c436.svg
• \widehat v_2=\frac{0.0496}{1-0.99^{2}}=2.492462312assets/equations/77a755525cf83c35.svg
• \delta_t=\left[W_{hy}^T(p_t-y_t)+W_{hh}^T\delta_{t+1}\right]\odot(1-h_t^2)assets/equations/77be07478b9cbea0.svg
• D=1.000000000+1.000000000+1.000000000\approx3.000000000assets/equations/77bfb08512787403.svg
• 96+16=112assets/equations/77dd73646865b98e.svg
• \left\lfloor\frac{32+2 \times 0-5}{1}\right\rfloor+1assets/equations/77e2ec6d75d55591.svg
• H(p,q)\approx0.519860385+0.173286795\approx0.693147181\ \text{nats}assets/equations/77f2dae5c0c0431a.svg
• P_7=28{,}800+120=28{,}920assets/equations/780340973d3c0ce4.svg
• W^{(2)}_{1,2,\mathrm{new}}=0.15-0.05(-0.38832)\approx 0.169416assets/equations/780b675c44e8ec0a.svg
• 0.018315639-1\approx-0.981684361assets/equations/782210c69a20fd3b.svg
• H(p)=-[0.75\ln(0.75)+0.25\ln(0.25)]assets/equations/78338077753f61a2.svg
• \text{L2 penalty}=0.05\times5.25=0.2625assets/equations/785c143e9ad28b18.svg
• \frac{\partial h_t}{\partial h_{t-1}}=\operatorname{diag}(1-h_t^2)W_{hh}assets/equations/78674f482f3097e9.svg
• \text{Sum}=C_1\approx -0.024764888assets/equations/786870ca2f319d3c.svg
• \begin{bmatrix}0 & -1 & -2\\-1 & 1 & 2\\2 & -1 & -3\end{bmatrix}assets/equations/78836b0ad068fd57.svg
• C_1=(0.45)\times(0.138498562)\approx 0.062324353assets/equations/78866731f335909d.svg
• [\max(0,1),\max(0,-2),\max(0,1.5)]=[1,0,1.5]assets/equations/78b039e2c7b066b2.svg
• A^T=\begin{bmatrix}1&4\\2&5\\3&6\end{bmatrix}assets/equations/78bde861f7b64a93.svg
• \mathrm{saving}\%=100\frac{M_{\mathrm{direct}}-M_{\mathrm{bottleneck}}}{M_{\mathrm{direct}}}assets/equations/78cf62ab96d05d20.svg
• h_{2}=\frac{1}{1+e^{-(0.3925)}}=0.596884378assets/equations/78d6d0a3e42830ce.svg
• H(x)=x+F(x),\qquad a_j=\max(0,H_j)assets/equations/78deb7e05ee3749d.svg
• h_{\mathrm{new}}=-0.16+(0.029777007)\approx-0.130222993assets/equations/78f9973806e42ffa.svg
• \sqrt{G_1}=\sqrt{4}=2assets/equations/7923d1e84dd761ae.svg
• g_{1,3}^{(1)}=(0.008724562)\times(1)\approx 0.008724562assets/equations/79371253b0a7529c.svg
• W^{(1)}_{4,3,\mathrm{new}}=0.2-0.5(-0.002141373)\approx 0.201070687assets/equations/7947b1af682a8c01.svg
• \ln(e^a)=a,\qquad e^{\ln a}=a\quad(a>0)assets/equations/794f58bf694147ad.svg
• N_{\mathrm{conv}}=\frac{10+0-3}{1}+1=7+1=8assets/equations/79853ff0cb463822.svg
• \mathrm{MACs}_{3\times3}=16\times16\times32\times3\times3\times8=589{,}824assets/equations/798ae9e57f70e0ad.svg
• y_{max}=\max_{(i,j)\in W}x_{ij}assets/equations/79973536a2057db8.svg
• m_t=\beta m_{t-1}+(1-\beta)g_tassets/equations/7999b979f742c872.svg
• g_{1,1}^{(2)}=(-0.082490487)\times(0.802183889)\approx -0.066172539assets/equations/79c1b8202fcf991a.svg
• g_i=g_{\mathrm{data},i}+\lambda w_iassets/equations/79fc02963f72332f.svg
• \hat y=\max(0,0.55)=0.55assets/equations/79fc763b55540917.svg
• e=1-0=1assets/equations/7a3b896434f77ddd.svg
• W^{(2)}_{1,1,\mathrm{new}}=0.25-0.5(-0.072418335)\approx 0.286209167assets/equations/7a3beb590205510e.svg
• g_1=-1assets/equations/7a43ae4bf3e05e38.svg
• i\widetilde C=0.7(0.9)=0.63assets/equations/7a48bf51c7d9b9c7.svg
• \max(0,-3)=0,\quad \max(0,0)=0,\quad \max(0,2)=2assets/equations/7a53989c39d25faa.svg
• g_{2,1}^{(2)}=(0.082533822)\times(0.760423002)\approx 0.062760617assets/equations/7a7372df9b0b6f47.svg
• \delta_j^h=f\prime(z_j^h)\sum_k w_{kj}^{out}\delta_k^{out}assets/equations/7a85cefe361f399b.svg
• \theta_2\approx0.683772234-0.147441956\approx0.536330278assets/equations/7aa9a16e4306403f.svg
• \delta_{h_2} = -0.275 \times 1 = -0.275assets/equations/7ad50eed6d4c12e2.svg
• L=\frac12e^2=\frac12(-2)^2=2assets/equations/7adb597b6b18fec3.svg
• \sigma(z)=\frac{1}{1+e^{-z}},\quad \sigma'(z)=\sigma(z)[1-\sigma(z)]assets/equations/7addd035227c8e01.svg
• \frac{\partial L}{\partial o}=o-y,\quad \frac{\partial o}{\partial w}=x,\quad \frac{\partial o}{\partial b}=1assets/equations/7ae2da1d297dac40.svg
• h_{2}=\frac{1}{1+e^{-(0.1)}}=0.524979187assets/equations/7b09cff87f366114.svg
• 3(3)(16)=144assets/equations/7b0c2720e21a1506.svg
• \begin{bmatrix}1 & 0 & 0\\1 & -1 & -1\\1 & 0 & -1\end{bmatrix}assets/equations/7b136c068c1d807b.svg
• \sigma=\sqrt{\sigma^2}assets/equations/7b5afbd2f5b393f2.svg
• R_1=(5)\times(1)+(7)\times(1)+(6)\times(1)=18assets/equations/7ba28bdc7c806323.svg
• Y_{2,1}=(18)+(11)+(11)=40assets/equations/7ba895c690daff56.svg
• \sqrt{s_2}=\sqrt{0.46}=0.678232998assets/equations/7bc15fc56ff7b9f2.svg
• N_{\mathrm{conv}}=\frac{10+0-3}{1}+1=7+1=8assets/equations/7bc6460a33d41247.svg
• \mu=\frac{1}{n}\sum_i x_i,\quad v=\frac{1}{n}\sum_i(x_i-\mu)^2,\quad s=\sqrt{v}assets/equations/7bccad9589b45084.svg
• \hat y_2=1(2)=2assets/equations/7be604573d864f92.svg
• C_1=(0.2)\times(-0.121439988)\approx -0.024287998assets/equations/7be9181956492e01.svg
• z_{h_1}=1+0-0.5=0.5\Rightarrow h_1=1assets/equations/7c04e117ce2375c9.svg
• b_{o_1,\mathrm{new}}=1-1(0.013687964)=0.986312036assets/equations/7c1e6deff768e7e1.svg
• h_1=\tanh(0.5)\approx0.462117157assets/equations/7c336c24f0f94322.svg
• w_{2,\mathrm{new}}=-0.2-0.1(0.26)=-0.226assets/equations/7c525ffa9dc17255.svg
• 0.5(0)+0.3(1)=0+0.3=0.3assets/equations/7c5775eea93e5ad0.svg
• h_2=\tanh(a_2)\approx0.797760954assets/equations/7c7eb453d84ea5ea.svg
• a_{\mathrm{train}}=\frac{m\odot a}{q}\quad\text{and}\quad a_{\mathrm{test}}=a\quad\text{in inverted dropout}assets/equations/7c8360afe0eaec6b.svg
• h_{1}=\frac{1}{1+e^{-(1.155)}}=0.760423002assets/equations/7c8842866c444dfb.svg
• 9408(12000)=112896000assets/equations/7c9631645fc9930b.svg
• h=(1-z)h_{\mathrm{previous}}+z\widetilde hassets/equations/7c9d53d54875e282.svg
• Y_{2,1}=(-1)+(1)=0assets/equations/7cb2a77b737bf455.svg
• z=6+(1)=7assets/equations/7cd7ec47b25d8ce7.svg
• 0.1(1)=0.1assets/equations/7cde65d80bbbce53.svg
• z_{XNOR}=-1+2(1)+0.5=1.5assets/equations/7ce164dbb1fd1549.svg
• \delta_h=0.08(0.8)(1)=0.064assets/equations/7cf97525165cb969.svg
• z_{XNOR}=-1+2(1)+0.5=1.5assets/equations/7d050e174f67d846.svg
• \widehat v_2=\frac{0.0496}{1-0.99^{2}}=2.492462312assets/equations/7d078eb9618846c0.svg
• \begin{bmatrix}13 & 8\\79 & 19.5\end{bmatrix}assets/equations/7d1100eaf9980bd3.svg
• MAC=H_{out}W_{out}K_hK_wC_{in}C_{out}assets/equations/7d20bfcc57cb122e.svg
• C=fC_{\mathrm{previous}}+i\widetilde C,\qquad h=o\tanh(C)assets/equations/7d40da391fed6757.svg
• \tanh(C_t)=\tanh(0.83)\approx0.680476006assets/equations/7d47d77e77a829e1.svg
• C_1=0+0.4=0.4assets/equations/7d6404b116e9cbf8.svg
• s=(6)=6assets/equations/7d688f2957123f9f.svg
• e^{-2}-e^{2}\approx0.135335283-7.389056099=-7.253720816assets/equations/7d6baa3ddebec66e.svg
• o=(0.2\times3)+0.1=0.6+0.1=0.7assets/equations/7d8d618e9d2cfc7d.svg
• R_2=(3)\times(1)+(2)\times(2)+(8)\times(1)=15assets/equations/7daf395d533e53c4.svg
• -0.25\ln(0.5)\approx0.173286795assets/equations/7dbaf11af5ca2bb7.svg
• w_{2,\mathrm{new}}=0.3-0.09=0.21assets/equations/7dc7d2ca77c91e6c.svg
• \frac{dL}{dp}=\frac{0.8-0}{0.16}=5assets/equations/7dcb217487ddffdd.svg
• \sigma_B^2=\frac{1}{m}\sum_{i=1}^{m}(x_i-\mu_B)^2assets/equations/7dcfc8cc5fd3adc8.svg
• N_{\mathrm{biases}}=120assets/equations/7dde36966a653a38.svg
• b_{o_2,\mathrm{new}}=1-1(0.082533822)=0.917466178assets/equations/7e14e96554640753.svg
• z-c=\begin{bmatrix}0&0&0\end{bmatrix}assets/equations/7e35e2c8f440c3bf.svg
• \widetilde C=\tanh(0)\approx0.000000000assets/equations/7e41db25271d6f77.svg
• z_{h_4} = (0.4 \times 0.4) + (0.2 \times 0.3) + (0.2 \times 0.5) + (0.8 \times -0.4) + (0.6) = 0.6assets/equations/7e4d728bbba36941.svg
• C_1=(-0.1)\times(-0.121439988)\approx 0.012143999assets/equations/7e59b3ad426aa9b6.svg
• y_{\mathrm{avg}}=\frac1n\sum_{v\in V}vassets/equations/7e5c101541d074f2.svg
• \theta_2=-0.35+(0.2565)=-0.0935assets/equations/7e5dfb5f2581a338.svg
• f\prime(z)=1-a^2assets/equations/7e7efe34a06943be.svg
• g_2=-0.35-(1)=-1.35assets/equations/7e8a70b34a625603.svg
• b_{h_2,\mathrm{new}}=0.1-0.2(0.017487356)=0.096502529assets/equations/7e8c76ed543b6526.svg
• \mathrm{MACs}_{\mathrm{direct}}=25088(4800)=120422400assets/equations/7f08b61aa51f8fc4.svg
• z_{o_1} = (0.8 \times 1.1) + (0.2) = 1.08assets/equations/7f3c69d6493dcb74.svg
• g_1=(-0.6)(2)=-1.2,\qquad g_2=(-0.6)(0)=0assets/equations/7f44ece1dbabee9e.svg
• \begin{bmatrix}-4 & -4\\-4 & -4\end{bmatrix}assets/equations/7f45e7b63383ada2.svg
• q_k=\frac{e^{z_k}}{\sum_{j=1}^{3}e^{z_j}},\qquad L=-\sum_{k=1}^{3}y_k\ln q_kassets/equations/7f4b48b92a27eca9.svg
• N_{\mathrm{out}}=\left\lfloor\frac{27+2(2)-5}{1}\right\rfloor+1=27assets/equations/7f4e102905e6bf9f.svg
• A=\begin{bmatrix}2&1&3\end{bmatrix}assets/equations/7f5189f9c2294a43.svg
• g_{2,1}^{(1)}=(0.003509506)\times(0.25)\approx 0.000877376assets/equations/7f5370d39d2a07d5.svg
• 0.1(1.31061)=0.131061assets/equations/7f5b4b6bf69894db.svg
• 1-(0.5)^2=0.75assets/equations/7f60b4793b26f591.svg
• b_{o_1,\mathrm{new}}=0-0.2(-0.121439988)=0.024287998assets/equations/7f891f57c5f4c76f.svg
• f_t=i_t=o_t=\sigma(0)=\frac1{1+e^0}=0.5assets/equations/7fb3fdabf106eb0d.svg
• y_i=\gamma\hat{x}_i+\betaassets/equations/7fd3e426f48f08b9.svg
• g_{2,1}^{(1)}=(-0.12135)\times(2)\approx -0.2427assets/equations/80079b2d3df7b0fb.svg
• P_{\mathrm{total}}=56+0+0+27=83assets/equations/8013f82d7c04ad92.svg
• p_3=\frac{0.135335283}{1.503214724}=0.090030573assets/equations/8024840665a8cd1c.svg
• z=\sum_{i=1}^{d}w_ix_i+b=w^Tx+bassets/equations/8032854fc489fa3f.svg
• W^{(2)}_{2,1,\mathrm{new}}=0.5-0.5(-0.02260254)\approx 0.51130127assets/equations/80442c4c79c12529.svg
• W^{(1)}_{1,2,\mathrm{new}}=-0.5-0.5(-0.002885161)\approx -0.49855742assets/equations/8044a6006ab0ca77.svg
• z_{h_2} = (0.3 \times 0.5) + (0.5 \times 0.2) + (0.7 \times 0.1) + (0.2) = 0.52assets/equations/805e34a1908652d7.svg
• p_1=\frac{0.135335283}{1.503214724}\approx0.090030573assets/equations/806ad5813b3ff8c0.svg
• L\approx-(-0.223143551)=0.223143551assets/equations/8076883238c541d3.svg
• W^{(1)}_{1,4,\mathrm{new}}=-0.2-0.5(0.003846881)\approx -0.201923441assets/equations/80909f1382fb6dde.svg
• a=\frac{-0.981684361}{1.018315639}\approx-0.964027580assets/equations/809bd6f34ad5f6a6.svg
• W^{(1)}_{3,3,\mathrm{new}}=-0.1-0.5(0.006497027)\approx -0.103248514assets/equations/80c1a9c61f216679.svg
• \hat y_1=1.4(2)=2.8,\qquad L_1=\frac12(2.8-4)^2=0.72assets/equations/80c9306a0dfccc42.svg
• \tanh(0.14)=\frac{e^{0.28}-1}{e^{0.28}+1}=0.139092448assets/equations/80cec30ff5e2f701.svg
• \frac{\partial L}{\partial b}=(-0.3)\times1=-0.3assets/equations/80efe080f6cbf4d3.svg
• f\odot C_{\mathrm{old}}=[0.2,0.9]assets/equations/813a6e1d025fa14c.svg
• w_3x_3=(-0.2)\times0.5=-0.1assets/equations/8173a34611c8e936.svg
• b_{\mathrm{new}}=b-\eta g_b=0-0.1(-2)=0.2assets/equations/8186eb873e399f7a.svg
• \frac{227-11}{4}+1=\frac{216}{4}+1=54+1=55assets/equations/8187570cd03b5f03.svg
• 0.2(0)+0.4(1)=0+0.4=0.4assets/equations/818979c823c2927e.svg
• P_5=204{,}800+128=204{,}928assets/equations/81ad3f9ee9d3ccc2.svg
• h_1=0.2(1)+0.5(0)+0=0.2assets/equations/81bb99ddeeaa7079.svg
• s_1=0.999(0)+0.001(2^2)=0.004assets/equations/81bd015faf03cb0a.svg
• 0(0.4)=0assets/equations/81c714ea0d0f3d30.svg
• z_h=(-0.4)\times2+0.1=-0.8+0.1=-0.7assets/equations/81ce8e4429aab67a.svg
• \delta_{h_1} = -0.046798971 \times 0.205500307 = -0.009617203assets/equations/81cfb49d2e12d4ca.svg
• s_1=0.9(0)+0.1(-1)^2=0.1assets/equations/81d06c696eada47a.svg
• \theta_2=0.9-\frac{0.1(1)}{\sqrt5}\approx0.855278640assets/equations/81f6101b82b353cb.svg
• N_{\mathrm{biases}}=10assets/equations/8203d983cccce3a9.svg
• h=0.75(0.4)+0.25(0.291312612)assets/equations/82108e6e910af0fd.svg
• W^{(2)}_{1,1,\mathrm{new}}=0.25-0.5(-0.072418335)\approx 0.286209167assets/equations/82203dea1f21db8c.svg
• R_1=(1)\times(1)+(0)\times(0)+(1)\times(-1)=0assets/equations/8247d99177a36658.svg
• f=\sigma(z_f),\quad i=\sigma(z_i),\quad o=\sigma(z_o)assets/equations/82486bbd71caa406.svg
• z_{XNOR}=-0+2(0)+0.5=0.5assets/equations/824ad51bf0e571df.svg
• \mathrm{SSE}=1+4+1=6assets/equations/8252220326cba08a.svg
• R_2=(2)\times(0)+(1)\times(-1)=-1assets/equations/825cd1ca9fbe27a2.svg
• h_{2}=\frac{1}{1+e^{-(0.3925)}}=0.596884378assets/equations/8275f180adb30096.svg
• h_{2}=\frac{1}{1+e^{-(1)}}=0.731058579assets/equations/8276b57ee8bf3bfd.svg
• g_{1,2}^{(1)}=(-0.22)\times(2)\approx -0.44assets/equations/828d4390f7c40527.svg
• w_{new}=w+\eta(y-\widehat y)xassets/equations/82b6f20b5710456c.svg
• \begin{bmatrix}0.15 & 0.2\\0.25 & 0.3\end{bmatrix}assets/equations/82d4501fc485058b.svg
• z=\sum_i w_i x_i+b,\qquad a=f(z)assets/equations/8316d459b29940dc.svg
• N_{\mathrm{out}}=\left\lfloor\frac{61+2(1)-5}{1}\right\rfloor+1=59assets/equations/8321e7609e3b2886.svg
• w_{\mathrm{new}}=w-\eta g_w=1-0.1(-4)=1.4assets/equations/8330219a13690888.svg
• g_{1,1}^{(2)}=(-0.121439988)\times(0.549833997)\approx -0.066771834assets/equations/8335141cc6234897.svg
• \sigma'(z)=\frac{1}{1+e^{-z}}\frac{e^{-z}}{1+e^{-z}}=\sigma(z)[1-\sigma(z)]assets/equations/8342957fe889eca9.svg
• i_t\widetilde C_t=0.7(0.9)=0.63assets/equations/83577513b17d4dfa.svg
• L_{\mathrm{half}}=\frac12\sum_{i=1}^{n}e_i^2=\frac{\mathrm{SSE}}2assets/equations/8370703d0e22f365.svg
• 0.5(6)+0.5(0)=3assets/equations/83765955bdb2c901.svg
• W^{(1)}_{2,3,\mathrm{new}}=0.2-0.5(-0.007263154)\approx 0.203631577assets/equations/83ac46e79dd0ac76.svg
• 2z=2(0)=0assets/equations/83fefb855224c584.svg
• z_{h_2} = (0.3 \times 1) + (-0.1 \times 2) + (0.2) = 0.3assets/equations/840732725d47f1fe.svg
• [(-2)^2,(-2)^2,2^2,2^2]=[4,4,4,4]assets/equations/8423c755b10f2fe0.svg
• z=w_1x_1+w_2x_2+\cdots+w_nx_n+b,\qquad a=f(z)assets/equations/8477fc1ddc03ffc3.svg
• 14(14)(16)(480)=1505280assets/equations/8486a56714848659.svg
• R_3=(3)\times(1)+(2)\times(1)+(6)\times(1)=11assets/equations/849173dcb9ca1a0f.svg
• h_3=0.7\tanh(-0.228)=0.7(-0.224129674)=-0.156890772assets/equations/84a567774fa02539.svg
• \theta_{new}=\theta-\eta gassets/equations/84b9c6c86f29aa06.svg
• t-o=0-1=-1assets/equations/84c2ca9c5cec76e2.svg
• \theta_1=1-\frac{0.1\times(2)}{\sqrt{0.4}}=0.683772234assets/equations/84ca434a6eea16db.svg
• z_t\widetilde h_t=0.25\tanh(0.3)\approx0.072828153assets/equations/8512106d268f635a.svg
• \mathrm{ReLU}(0)=\max(0,0)=0assets/equations/8512edeebdecaad0.svg
• g_{1,2}^{(1)}=(-0.11326)\times(3)\approx -0.33978assets/equations/851d3df89190bcd1.svg
• 1-z=1-0.25=0.75assets/equations/854154216ffe0396.svg
• s=0+0.6=0.6assets/equations/8544b9e49d354ea3.svg
• N_{\mathrm{out}}=\left\lfloor\frac{14+2(0)-5}{1}\right\rfloor+1=10assets/equations/85506bf5dfa9eb3a.svg
• W^{(1)}_{3,4,\mathrm{new}}=0.5-0.5(-0.005197622)\approx 0.502598811assets/equations/85589f5b6f5c0202.svg
• \widehat m_2=\frac{0.38}{1-0.9^2}=\frac{0.38}{0.19}=2assets/equations/8560767b69d9e851.svg
• W^{(1)}_{1,1,\mathrm{new}}=0.11-0.05(-0.22652)\approx 0.121326assets/equations/857c0ad228d64fc5.svg
• P_{\mathrm{conv}}=(K_hK_wC_{\mathrm{in}}+1)C_{\mathrm{out}}assets/equations/859b60e888192c5a.svg
• s_1=0.9(0)+0.1(-1)^2=0.1assets/equations/859c9536cc17d749.svg
• \log_2 q=\frac{\ln q}{\ln 2}assets/equations/85b0a11172f30b61.svg
• \begin{bmatrix}2 & 1\\1 & 3\end{bmatrix}assets/equations/85c116687119a006.svg
• b_{h_2,\mathrm{new}}=0.2-0.9(0.006542085)=0.194112124assets/equations/85c1ef6621c6cf83.svg
• a^{(l)}=f(z^{(l)})assets/equations/85d817a60006abc5.svg
• m_2=0.9(0.2)+0.1(-1)=0.08assets/equations/85e3c1c423ab1151.svg
• g_{1,3}^{(2)}=\frac{\partial L}{\partial W^{(2)}_{1,3}}=\delta_{o_1}h_3assets/equations/85e8a967a0a12f92.svg
• 25+5=30assets/equations/862c71dcbfedc68d.svg
• \text{Deep learning}\subset\text{Machine learning}\subset\text{Artificial intelligence}assets/equations/862fde9ce2f6cd54.svg
• N_{\mathrm{weights}}=9{,}216\times4{,}096=37{,}748{,}736assets/equations/864d5792c1506a4b.svg
• h_1=\frac{2.718281828-1}{2.718281828+1}=0.462117157assets/equations/8658aac9b96930a3.svg
• Y_{2,1}=(18)+(11)+(11)=40assets/equations/867dfee50107ebc8.svg
• P_8=9{,}042{,}432+84=9{,}042{,}516assets/equations/868b3136771d3432.svg
• h_{1} = \max(0,0.85) = 0.85assets/equations/86a3de78168e02c1.svg
• w_{1,\mathrm{new}}=0.5-0.1(0.2)=0.48assets/equations/86b47a5bd5373b33.svg
• z_2=1+1-1.5=0.5assets/equations/86c5a45ae836809c.svg
• W^{(1)}_{1,3,\mathrm{new}}=0.1-0.5(-0.004808601)\approx 0.102404301assets/equations/86d9f58aa097fd71.svg
• \frac{dL}{dz}=(5)(0.16)=0.8assets/equations/86de3ab6d9deb754.svg
• o_{2}=\frac{1}{1+e^{-(1.255565213)}}=0.778261739assets/equations/86e75776a23fe7d0.svg
• W^{(1)}_{1,2,\mathrm{new}}=0.2-0.2(0.001929853)\approx 0.199614029assets/equations/86e7e16686d8ad69.svg
• z=0.1(2)+(-0.2)(5)+0.5(1)+0.3assets/equations/86f1a8b54476b2cb.svg
• MAC=H_{out}W_{out}K_hK_wC_{in}C_{out}assets/equations/86ffc171b254aab9.svg
• p_{\mathrm{max}}=\max(a_1,a_2,a_3,a_4)assets/equations/870fc5e76c2ff285.svg
• W^{(1)}_{3,2,\mathrm{new}}=0.3-0.5(0.003898216)\approx 0.298050892assets/equations/87125a5db71e09ef.svg
• \left\lfloor\frac{122-2}{2}\right\rfloor+1assets/equations/8713b9890edc970c.svg
• \sqrt{s_1}=\sqrt{0.4}=0.632455532assets/equations/8724fbf30b88d3b5.svg
• e^{-x}=\frac1{e^x},\qquad\ln(e^x)=xassets/equations/8726504f24b7b8cc.svg
• z^{(l)}=W^{(l)}a^{(l-1)}+b^{(l)}assets/equations/87472012ebbb4a9d.svg
• C_t=f_t\odot C_{t-1}+i_t\odot\widetilde C_tassets/equations/8751dee634f91753.svg
• p-y=0.731058579-1=-0.268941421assets/equations/8773c0aada00e672.svg
• o=0.5\times0+0.2=0.2assets/equations/8775ad02ed7eaf6c.svg
• N_{\mathrm{out}}=\left\lfloor\frac{52+2(0)-5}{1}\right\rfloor+1=48assets/equations/87809c641f196e58.svg
• \begin{bmatrix}0 & 0 & 0\\0 & 163 & 162\\0 & 160 & 161\end{bmatrix}assets/equations/8798bafaa1889aed.svg
• z=0.5\times2+0=1assets/equations/88322b08817273b7.svg
• \frac{\partial L}{\partial z}=\frac{\partial L}{\partial p}\frac{\partial p}{\partial z}=\frac{p-y}{p(1-p)}p(1-p)=p-yassets/equations/88355caa5908a259.svg
• \delta_{h_1} = -0.046798971 \times 0.205500307 = -0.009617203assets/equations/88603ee1250e709b.svg
• h=W_1x+b_1,\qquad \hat y=W_2h+b_2=W_2W_1x+(W_2b_1+b_2)assets/equations/886e2a40c2cd8e26.svg
• C_1=(0.45)\times(0.138498562)\approx 0.062324353assets/equations/88812c8e2d3784df.svg
• h_t=o_t\tanh(c_t)assets/equations/88b1929e93128b1e.svg
• R_2=(1)\times(0)+(0)\times(-1)=0assets/equations/88c7ac8976f026b8.svg
• g_t=J'(\theta_{\mathrm{look}})assets/equations/88dc28f2501b464e.svg
• h_{\mathrm{new}}=zh_{\mathrm{old}}+(1-z)\widetilde hassets/equations/88df46d24a37114e.svg
• \frac{\partial L}{\partial W_1}=\delta_1x^T,\qquad\frac{\partial L}{\partial b_1}=\delta_1assets/equations/88df80b8a0e03820.svg
• \left\lfloor\frac{252-2}{2}\right\rfloor+1assets/equations/88e7931306349767.svg
• 1+1.000000000\approx2.000000000assets/equations/88f353ee1ae5e616.svg
• g_{2,2}^{(2)}=(-0.038098237)\times(0.596884378)\approx -0.022740242assets/equations/890a2f895bb237f9.svg
• \delta_{h_1} = 0.039350723 \times 0.221712873 = 0.008724562assets/equations/890b9a76a0888667.svg
• N_{\mathrm{biases}}=5assets/equations/8917be67939f2d89.svg
• D_{\mathrm{KL}}(p\Vert q)=\sum_i p_i\ln\frac{p_i}{q_i}=H(p,q)-H(p)assets/equations/8933903c2570cd47.svg
• \text{Sum}=C_1\approx -0.065518559assets/equations/893a1456a543073d.svg
• R_2=(3)\times(0)+(2)\times(-1)=-2assets/equations/893c6acf067ed4a9.svg
• v_{2,\mathrm{new}}=0.9(-0.1)-0.1(0)=-0.09assets/equations/896aef66e0912dce.svg
• z_{o_1} = (0.2 \times 0.549833997) + (-0.1 \times 0.544878892) + (0) = 0.05547891assets/equations/897314316dee8982.svg
• C_2=(0.5)\times(-0.038098237)\approx -0.019049118assets/equations/899ef2a84df9b3cd.svg
• R_2=(1)\times(0)+(3)\times(1)=3assets/equations/89af9e3096091f5c.svg
• ab^T=\begin{bmatrix}8&10\\12&15\end{bmatrix}assets/equations/89b24f1f7b777962.svg
• U_{\mathrm{batch}}=Eassets/equations/89b7eb4a772465f4.svg
• \ln(0.5)\approx-0.693147181assets/equations/89c211e38f586f86.svg
• W^{(2)}_{1,1,\mathrm{new}}=0.4-0.5(0.082167041)\approx 0.35891648assets/equations/89c2c818c7a2e7dd.svg
• fC_{\mathrm{old}}=(0.500000000)(1)\approx0.500000000assets/equations/89caa918473d927f.svg
• z_{h_1} = (0.11 \times 2) + (0.21 \times 3) + (0) = 0.85assets/equations/89d0bf0bfbd0eee8.svg
• m_2=0.9(-0.1)+0.1(2)=0.11assets/equations/89d9785f6f04be8b.svg
• \begin{bmatrix}21 & 15\\13 & 18\end{bmatrix}assets/equations/89ec596cf1c6ec68.svg
• \widetilde h=\tanh(0.15)\approx0.148885034assets/equations/89f3eabbc08b8f86.svg
• \mathrm{MSE}=\frac{1}{2}\approx0.500000000assets/equations/89fe10c4fdc1b2d9.svg
• b_{o_1,\mathrm{new}}=0-0.5(-0.099059551)=0.049529775assets/equations/8a047fabd725efcd.svg
• P_{conv}=(K_hK_wC_{in}+1)C_{out}assets/equations/8a253f24fdf99d8e.svg
• \left\lfloor\frac{227+2 \times 0-9}{1}\right\rfloor+1assets/equations/8a2a9eb5376c2557.svg
• z=3(1)+1=4assets/equations/8a300d26d67e84f4.svg
• \hat y=\operatorname{ReLU}(1.08)=1.08assets/equations/8a3b4bb2871113ef.svg
• \sqrt{s_1}=\sqrt{0.1}=0.316227766assets/equations/8a4a542960803c14.svg
• \theta_{\mathrm{new}}=\theta_{\mathrm{old}}-\eta gassets/equations/8a6ef020e7632784.svg
• L=-(-1.609437912)=1.609437912assets/equations/8a78ffe2df23393c.svg
• \mathbb E\left[\frac{ma}{q}\right]=\frac aq\mathbb E[m]=\frac aq\,q=aassets/equations/8a9a35419aebc53b.svg
• \theta_1=1-0.05(2)=0.9assets/equations/8aa23a9e3a9c5875.svg
• P_{\mathrm{projection}}=(K^2C_{\mathrm{in}}+1)C_{\mathrm{out}}assets/equations/8abfcf2c3c8a28be.svg
• \eta_{\mathrm{effective},1}=\frac{0.1}{\sqrt4}=0.05assets/equations/8acc57815fd4951a.svg
• z_{h_1}=0+0-0.5=-0.5\Rightarrow h_1=0assets/equations/8b21d73077236089.svg
• R_2=(0)\times(0)+(2)\times(1)=2assets/equations/8b2b9a5baccb2ea6.svg
• W^{(2)}_{1,2,\mathrm{new}}=0.5-0.1(-0.165)\approx 0.5165assets/equations/8b302d0b5078734a.svg
• \sigma(0)=\frac1{1+e^0}=\frac1{1+1}=0.5assets/equations/8b3dd0cfc097c6b9.svg
• b_{h_1,\mathrm{new}}=0.6-0.5(-0.009617203)=0.604808601assets/equations/8b40f970f91aece5.svg
• 4+4+4+4=16assets/equations/8b51d1e9b84e8106.svg
• z^{(1)}_2=0.3(1)+0.7(0)+0.2=0.5assets/equations/8b6a4f4bbfa44dae.svg
• C_1=(-0.2)\times(-0.131169078)\approx 0.026233816assets/equations/8b80255c0daf41f8.svg
• z_{o_1} = (0.6 \times 0.562176501) + (0.9 \times 0.627147766) + (0.5) = 1.40173889assets/equations/8b8732deca8578d9.svg
• C_{concat}=\sum_b C_b,\qquad y_{res}=F(x)+xassets/equations/8b8cdd9b96177407.svg
• \mu=\frac{1}{N}\sum_{i=1}^{N}x_i,\qquad \sigma^2=\frac{1}{N}\sum_{i=1}^{N}(x_i-\mu)^2,\qquad \sigma=\sqrt{\sigma^2}assets/equations/8b9cec002314f2de.svg
• z_2=-2+3-0.5=0.5assets/equations/8bdba182228536bc.svg
• R_2=(0)\times(0)+(1)\times(1)=1assets/equations/8c103dd6ecdec24c.svg
• z=0.2-1+0.5+0.3=0assets/equations/8c377dccccbc387a.svg
• W^{(1)}_{2,1,\mathrm{new}}=0.25-0.5(0.000497713)\approx 0.249751144assets/equations/8c4ebc4a08766591.svg
• w_{\mathrm{new}}=w-\eta\frac{\partial L}{\partial w},\qquad b_{\mathrm{new}}=b-\eta\frac{\partial L}{\partial b}assets/equations/8c595e12402e35b1.svg
• \text{Sum}=C_1+C_2\approx 0.011272855assets/equations/8c5cd57b66d241cf.svg
• \theta_2=-0.183772234-\frac{0.1\times(2)}{\sqrt{0.49}}=-0.46948652assets/equations/8c6d3f7be7bc6289.svg
• p_i=\frac{e^{z_i-c}}{\sum_j e^{z_j-c}}assets/equations/8c7e5aa437b15aae.svg
• p=0.5assets/equations/8c8d4b00d7d4788e.svg
• A=H_{\mathrm{out}}W_{\mathrm{out}}C_{\mathrm{out}}assets/equations/8c9b86a3e71336c4.svg
• m_2=\beta(1-\beta)g_1+(1-\beta)g_2assets/equations/8ca813db07725b2e.svg
• m_1=0.9(0)+0.1(-1)=-0.1assets/equations/8cb1bdd325f9978f.svg
• g=\frac{-1+(-4)}2=-2.5assets/equations/8cfa6ee1614fe8e3.svg
• \frac{\eta g_2}{\sqrt{s_2}}=\frac{0.1(1)}{\sqrt{0.46}}\approx0.147441956assets/equations/8d403055f469e062.svg
• z_{h_1} = (0.4 \times 3) + (0.2) = 1.4assets/equations/8d4aa9c483de249e.svg
• o_{1}=\frac{1}{1+e^{-(-0.104539506)}}=0.473888899assets/equations/8d4f0a07993ddded.svg
• 0.8(-0.2)=-0.16assets/equations/8d5d62808c22a4a6.svg
• C_1=(0.8)\times(0.08)\approx 0.064assets/equations/8d62e6bfc12bb1ba.svg
• g_{1,1}^{(2)}=(0.138498562)\times(0.593269992)\approx 0.082167041assets/equations/8d844a8676791dfd.svg
• P_6=1{,}327{,}104+384=1{,}327{,}488assets/equations/8d8b08cc6480d083.svg
• n_{\mathrm{flat}}=6\times6\times256=9{,}216assets/equations/8d9921c92554e599.svg
• L_{\mathrm{new}}=\frac12(0.929-1)^2=0.0025205assets/equations/8da79c9221b5024a.svg
• \delta_{o_1} = 1 \times (0.779684759-0.7) \times 0.171776436 = 0.013687964assets/equations/8de071d3c6435068.svg
• \theta_t=\theta_{t-1}-\frac{\eta g_t}{\sqrt{s_t}+\epsilon}assets/equations/8de35fe2ee69fe63.svg
• \sum_i w_ix_i\ge\theta\quad\Longleftrightarrow\quad\sum_i w_ix_i-\theta\ge0assets/equations/8deda7d73444e933.svg
• h_{1}=\frac{1}{1+e^{-(0.9)}}=0.710949503assets/equations/8df07c79c171fe96.svg
• C_1=(0.6)\times(-0.082490487)\approx -0.049494292assets/equations/8e21ad6a02297b75.svg
• w_{j,\mathrm{new}}=w_j-\eta g_{j,\mathrm{total}}assets/equations/8e2c2fa59e985f4e.svg
• e^{-2}+e^{2}\approx0.135335283+7.389056099=7.524391382assets/equations/8e33913f50550c4f.svg
• \sqrt{s_1}=\sqrt{0.1}=0.316227766assets/equations/8e43eda4dcf6bcb7.svg
• \text{Sum}=C_1\approx -0.049494292assets/equations/8e5ced5b8d72aefa.svg
• \frac{\partial z}{\partial w}=x,\qquad\frac{\partial z}{\partial x}=w,\qquad\frac{\partial z}{\partial b}=1assets/equations/8e90f73571d42f2e.svg
• \operatorname{ReLU}(z)=\max(0,z),\quad f'(z)=\begin{cases}1&z>0\\0&z<0\end{cases}assets/equations/8ed42aad8099504b.svg
• \widetilde h=\tanh(0.3)\approx0.291312612assets/equations/8ef1fca1e3e9a662.svg
• \delta_{h_1} = 0.011272855 \times 0.18217986 = 0.002053687assets/equations/8f0662ebcf4e93d4.svg
• z_{o_2} = (0.5 \times 0.593269992) + (0.55 \times 0.596884378) + (0.6) = 1.224921404assets/equations/8f06ed5dcff2563f.svg
• fC_{old}=0.4(0.5)=0.2assets/equations/8f3bf24a2e1d90a0.svg
• \hat y_0=w_0x=1(2)=2,\qquad e_0=\hat y_0-y=2-4=-2assets/equations/8f7cd4947851e2db.svg
• N_{\mathrm{out}}=\left\lfloor\frac{126+2(0)-5}{1}\right\rfloor+1=122assets/equations/8f94d1b01f536cf3.svg
• \frac{\partial L}{\partial o_j}=o_j-y_jassets/equations/8faacff5464dcaf1.svg
• g_{1,1}^{(1)}=(-0.007853997)\times(3)\approx -0.02356199assets/equations/8fbb5c252349f4b6.svg
• \theta_t=\theta_{t-1}-\eta\frac{\widehat m_t}{\sqrt{\widehat v_t}+\epsilon}assets/equations/8fbedcf992dd719a.svg
• W^{(1)}_{2,1,\mathrm{new}}=0.5-0.5(-0.004869072)\approx 0.502434536assets/equations/8fd73e44677515a6.svg
• R_1=(0.3)\times(1)+(0.3)\times(0)+(0.3)\times(-1)=0assets/equations/8fffc5f555fb2f95.svg
• W^{(1)}_{2,4,\mathrm{new}}=0.5-0.5(0.005810523)\approx 0.497094738assets/equations/901765ceedbe9376.svg
• o=wx+b,\qquad L=\frac12(o-y)^2assets/equations/90379cab01aa101e.svg
• m_t=\beta_1m_{t-1}+(1-\beta_1)g_t,\quad v_t=\beta_2v_{t-1}+(1-\beta_2)g_t^2assets/equations/903aa08b8d83755c.svg
• \frac{\partial L}{\partial w}=(-0.052877093)\times2=-0.105754186assets/equations/904a6dac1e6b4e5a.svg
• \hat y_2=0.1\times2=0.2assets/equations/905f0951780ee22a.svg
• J(\theta+\Delta\theta)\approx J(\theta)+\nabla J(\theta)^T\Delta\thetaassets/equations/9060e5490f15d100.svg
• z_o=0-2(0)-0.5=-0.5\Rightarrow o=0assets/equations/90628ef67242e4d9.svg
• W^{(1)}_{1,2,\mathrm{new}}=-0.5-0.5(-0.002885161)\approx -0.49855742assets/equations/9076174f0a478139.svg
• \mathrm{SSE}=\sum_i e_i^2,\qquad L_{\mathrm{half}}=\frac12\sum_i e_i^2,\qquad \mathrm{MSE}=\frac1n\sum_i e_i^2assets/equations/90823ff8e0467213.svg
• o_{1}=\frac{1}{1+e^{-(0.581310333)}}=0.641368858assets/equations/909595bd0fd39ab1.svg
• C_1=(-0.1)\times(-0.121439988)\approx 0.012143999assets/equations/909a9296285664eb.svg
• \frac{\partial L}{\partial w}=(-0.052877093)\times2=-0.105754186assets/equations/909d6c4ba3b4dba5.svg
• Y_{2,2}=(20)+(15)+(15)=50assets/equations/90b52e79a3bf1954.svg
• \sigma\prime(z)=\sigma(z)[1-\sigma(z)]assets/equations/90cd2771a42dc9ef.svg
• \frac{dL}{dw}=8(3)=24assets/equations/90d0d059a0fe1d53.svg
• N_{\mathrm{weights}}=9\times9\times3\times32=7{,}776assets/equations/90e2d92ad9867cbd.svg
• rh_{\mathrm{previous}}=0.5\times0.4=0.2assets/equations/90ee4b754abc5317.svg
• z_1=0+1-0.5=0.5assets/equations/90f88a492c8d4d66.svg
• y_{\mathrm{avg}}=\frac{35}4=8.75assets/equations/90fc093dce5b3464.svg
• v_t=\beta v_{t-1}-\eta g_t,\quad \theta_t=\theta_{t-1}+v_tassets/equations/91375a23b0e2fbcf.svg
• s_t=\rho s_{t-1}+(1-\rho)g_t^2,\quad\theta_t=\theta_{t-1}-\frac{\eta g_t}{\sqrt{s_t}+\epsilon}assets/equations/914e44e2b8e8e8f9.svg
• N_{\mathrm{biases}}=1assets/equations/9172e32c44e88eec.svg
• i\widetilde C=(0.500000000)(0.000000000)\approx0.000000000assets/equations/9178374ae7a038b8.svg
• \frac{\eta g_1}{\sqrt{s_1}}=\frac{0.1(2)}{\sqrt{0.4}}\approx0.316227766assets/equations/918a6ff758f0307c.svg
• g_{1,1}^{(2)}=(-0.121439988)\times(0.549833997)\approx -0.066771834assets/equations/918d32ea9955cc69.svg
• \frac{\partial J}{\partial w_i}=g_{\mathrm{data},i}+\lambda w_iassets/equations/919b9b7755f3c860.svg
• P_12=4{,}096{,}000+1{,}000=4{,}097{,}000assets/equations/91b7bd4147e4b8d6.svg
• R_3=(1)\times(1)+(0)\times(0)+(1)\times(-1)=0assets/equations/91d68511c5df7991.svg
• \frac{\eta g_1}{\sqrt{s_1}}=\frac{0.1(2)}{\sqrt{0.4}}\approx0.316227766assets/equations/91debda43e9c7015.svg
• b_{\mathrm{new}}=0+0.1(0)=0assets/equations/920794cfde95f28a.svg
• \mu=\frac{1}{N}\sum_{i=1}^{N}x_i,\qquad \sigma^2=\frac{1}{N}\sum_{i=1}^{N}(x_i-\mu)^2,\qquad \sigma=\sqrt{\sigma^2}assets/equations/9231d9bbf6a6372d.svg
• \mathrm{MSE}=\frac{1+4}{2}=2.5assets/equations/9258cd0363de1324.svg
• R_2=(0)\times(0)+(2)\times(1)=2assets/equations/925c2ca7e5d512dc.svg
• \widehat m_t=\frac{m_t}{1-\beta_1^t},\quad\widehat v_t=\frac{v_t}{1-\beta_2^t},\quad\theta_t=\theta_{t-1}-\eta\frac{\widehat m_t}{\sqrt{\widehat v_t}+\epsilon}assets/equations/92678247acd594b0.svg
• o_{1}=\frac{1}{1+e^{-(0.770465929)}}=0.683621675assets/equations/928c85e2f67a9c66.svg
• s=0.8+(-0.2)+1.5=0.6+1.5=2.1assets/equations/928ca4d8a84ae46e.svg
• f_1C_0=1(0)=0,\qquad i_1\widetilde C_1=0.5(0.8)=0.4assets/equations/92909a33a68a6d6c.svg
• 20+4=24assets/equations/92df7685dd83d921.svg
• R_1=(2)\times(1)+(0)\times(-1)=2assets/equations/92e95617002bb2ff.svg
• \lVert g\rVert_2=\sqrt{6^2+8^2}=\sqrt{36+64}=10assets/equations/92ed9c7434bcec37.svg
• N_{\mathrm{out}}=\left\lfloor\frac{N+2P-K}{S}\right\rfloor+1assets/equations/930fa72046a69449.svg
• w_2=1.1-0.1(-3.6)=1.46assets/equations/93143771f8225850.svg
• g_{1,2}^{(2)}=(-0.55)\times(0.3)\approx -0.165assets/equations/9325b59e9bc80f3d.svg
• p_3=\frac{1.000000000}{3.000000000}\approx0.333333333assets/equations/932805d4a7ff7186.svg
• o_t=\sigma(W_o[h_{t-1},x_t]+b_o),\qquad h_t=o_t\odot\tanh(C_t) · o_t=\sigma(W_o[h_{t-1},x_t]+b_o),\quad h_t=o_t\odot\tanh(C_t)assets/equations/932f49e3f8a32474.svg
• a'_2=\frac0{0.75}=0,\qquad a'_3=\frac0{0.75}=0assets/equations/93312a45b7b42272.svg
• Y_{2,2}=(20)+(15)+(15)=50assets/equations/9333688e57e199f1.svg
• c=\min(1,10/5)=1assets/equations/93428ecf1a9721ce.svg
• 48+6=54assets/equations/935190190dc4ddae.svg
• a_i^{\mathrm{train}}=\frac{m_ia_i}{q}assets/equations/93807008ce7b52a8.svg
• R_2=(2)\times(2)+(1)\times(0)+(0)\times(-2)=4assets/equations/93a6f5b5b20f2f23.svg
• w_{\mathrm{new}}=0.4-0.2(-0.36)=0.4+0.072=0.472assets/equations/93c291aebdd73770.svg
• g_{1,2}^{(1)}=(-0.11326)\times(3)\approx -0.33978assets/equations/93e0a3007b972754.svg
• R_3=(0.3)\times(1)+(0.3)\times(0)+(0.3)\times(-1)=0assets/equations/9405479984ca544c.svg
• R_1=(3)\times(1)+(2)\times(-1)=1assets/equations/941c8bddd3a4fd90.svg
• o_{1}=\frac{1}{1+e^{-(-0.104539506)}}=0.473888899assets/equations/942e59b171acdc83.svg
• J_{L2}=J_{\mathrm{data}}+\lambda\sum_iw_i^2assets/equations/9438fe7d9420663e.svg
• N_{out}=4+1=5assets/equations/94437545b203f62e.svg
• \frac{\partial L}{\partial u}=\delta_1x_1+\delta_2x_2assets/equations/94543a0839df0ed3.svg
• \log_2q=\frac{\ln q}{\ln2}assets/equations/945fabaced090f8f.svg
• \begin{bmatrix}13.25 & 11.5\\8.75 & 14.25\end{bmatrix}assets/equations/94682b0fab3f0b97.svg
• h_t=(1-z_t)\odot h_{t-1}+z_t\odot\widetilde h_tassets/equations/949c126edafa8f0a.svg
• h_{1}=\frac{1}{1+e^{-(0.2)}}=0.549833997assets/equations/94bdd56fd67f70d6.svg
• b_{h_1,\mathrm{new}}=1-1(0.002053687)=0.997946313assets/equations/94f695ae06ea20c8.svg
• W^{(1)}_{2,1,\mathrm{new}}=0.3-0.1(-0.275)\approx 0.3275assets/equations/950cba298eb25531.svg
• N_{\mathrm{weights}}=5\times5\times6\times16=2{,}400assets/equations/95205b29ca5fb5db.svg
• \widehat m_1=\frac{0.2}{1-0.9}=2assets/equations/95360064605e4687.svg
• g_w=\frac{\partial L}{\partial w}=(z-y)xassets/equations/955c63d58ce8cb9b.svg
• D=0.367879441+1+0.135335283=1.503214724assets/equations/95814cf2d0de3a7e.svg
• p_2=\frac{0.367879441}{1.503214724}\approx0.244728471assets/equations/958d44e6661ca616.svg
• R_2=(0)\times(2)+(1)\times(0)+(2)\times(-2)=-4assets/equations/959c04625a9a0df4.svg
• p-y=0.731058579-1=-0.268941421assets/equations/95ba7a4e62bbaee0.svg
• g_{1,4}^{(2)}=(-0.093597941)\times(0.645656306)\approx -0.060432101assets/equations/95d6468f8fc70889.svg
• R_1=(0)\times(1)+(1)\times(0)=0assets/equations/9602643188947584.svg
• \eta g=0.01(0.7)=0.007assets/equations/9603f5d8320b5539.svg
• b_{h_1,\mathrm{new}}=-0.4-0.9(0.008724562)=-0.407852106assets/equations/96118c608a159aef.svg
• N_{\mathrm{weights}}=5\times5\times3\times32=2{,}400assets/equations/963130ef1e615735.svg
• 1(16)=16assets/equations/96319009caa148f5.svg
• C_{concat}=\sum_b C_bassets/equations/963754238ceb3b1d.svg
• o_{1} = \max(0,1.08) = 1.08assets/equations/965f3f37dce04d5d.svg
• \left\lfloor\frac{5+2 \times 0-3}{2}\right\rfloor+1assets/equations/966c8fb493880239.svg
• g_{3,2}^{(1)}=(0.012994055)\times(0.3)\approx 0.003898216assets/equations/96944db9c899fc7e.svg
• z_h=w_1x+b_1,\quad h=\max(0,z_h),\quad o=w_2h+b_2assets/equations/96964269705ef1b5.svg
• \begin{bmatrix}2 & 0\\1 & 3\end{bmatrix}assets/equations/969e894914a74419.svg
• o_{1}=\frac{1}{1+e^{-(1.263830247)}}=0.779684759assets/equations/96a03b52510301b3.svg
• z_o=0-2(0)-0.5=-0.5\Rightarrow o=0assets/equations/96a5d17e0b8dd052.svg
• L=-\ln p_cassets/equations/96bd39e07641cf89.svg
• N_{\mathrm{biases}}=256assets/equations/96c4c1de3aa70d55.svg
• f\prime(z)=a(1-a)assets/equations/97012822a1c7e305.svg
• W^{(1)}_{3,3,\mathrm{new}}=-0.1-0.5(0.006497027)\approx -0.103248514assets/equations/9712a05089c95dbd.svg
• \frac{\partial\mathrm{SSE}}{\partial\hat y_j}=2e_j,\qquad \frac{\partial L_{\mathrm{half}}}{\partial\hat y_j}=e_j,\qquad \frac{\partial\mathrm{MSE}}{\partial\hat y_j}=\frac{2}{n}e_jassets/equations/972f5a96b91fe930.svg
• W^{(1)}_{2,2,\mathrm{new}}=0.5-0.5(-0.004869072)\approx 0.502434536assets/equations/9733a71bb8ca6e2e.svg
• g_{1,1}^{(1)}=(-0.004869072)\times(1)\approx -0.004869072assets/equations/9759c3655229ab0e.svg
• w^Tx=w_1x_1+w_2x_2+w_3x_3assets/equations/97799e8f51bd1031.svg
• z_{h_1} = (0.11 \times 2) + (0.21 \times 3) + (0) = 0.85assets/equations/977db4dfa37a6afe.svg
• P_B=(3\times3\times8+1)\times6=73\times6=438assets/equations/97eb7526b53ae379.svg
• \hat y=wx,\qquad J=\frac12(\hat y-y)^2+\frac\lambda2w^2assets/equations/9807919ec463b336.svg
• \frac{1}{1+e^{1}}assets/equations/983a154bf3030fc0.svg
• L_{\mathrm{half}}=\frac{0.49}{2}=0.245,\qquad \mathrm{MSE}=\frac{0.49}{3}\approx0.163333333assets/equations/98422634bd86c12f.svg
• z_{h_1}=1+0-0.5=0.5\Rightarrow h_1=1assets/equations/986114a7a93ab89f.svg
• W^{(2)}_{1,2,\mathrm{new}}=0.7-0.5(-0.062541)\approx 0.7312705assets/equations/988e1d1b7e014f5c.svg
• \delta_1=0.5(-0.5)=-0.25assets/equations/98a077f6210fe712.svg
• \hat y=wx,\qquad L(w)=\frac12(wx-y)^2,\qquad w_t=w_{t-1}-\eta g_tassets/equations/991254a061f484f9.svg
• N_{\mathrm{out}}=\left\lfloor\frac{126+2(0)-5}{1}\right\rfloor+1=122assets/equations/991cd2a9cf06a3a0.svg
• (3)(1)=3assets/equations/9934bcd789a00e7d.svg
• |-2|=2,\quad |1|=1,\quad |0.5|=0.5assets/equations/9936fa64e265469b.svg
• w_2^{\mathrm{new}}=0.65+0.025=0.675assets/equations/99431f20567e072c.svg
• \tanh(a)=\frac{e^{2a}-1}{e^{2a}+1}assets/equations/994b9a9f57dad438.svg
• 0.1(-1)=-0.1assets/equations/995079ed548eec78.svg
• N_{\mathrm{weights}}=84\times10=840assets/equations/995f161a94dc7ae1.svg
• \tanh(2)\approx\frac{7.253720816}{7.524391382}\approx0.964027580assets/equations/99823d0144ee439b.svg
• \widehat v_2=\frac{0.004999}{1-0.999^{2}}=2.500750375assets/equations/998deb4908102974.svg
• \Delta b=\eta(t-o)assets/equations/998f813758f64544.svg
• P_8=4{,}423{,}680+120=4{,}423{,}800assets/equations/99a27e646d35bbda.svg
• a_t=W_{xh}x_t+W_{hh}h_{t-1}+b_hassets/equations/99a78e2772b9da6e.svg
• \left\lfloor\frac{4+2 \times 0-2}{1}\right\rfloor+1assets/equations/99ba828c16946a06.svg
• h_t=0.3+0.25\tanh(0.3)\approx0.372828153assets/equations/99c980a8ec789fec.svg
• s_2=0.9(0.4)+0.1(1)^2=0.46assets/equations/99dd3340957517b7.svg
• \tanh(W_{xh}x+W_{hh}h_{t-1}+b)assets/equations/99df7aa8ced09e2c.svg
• N_{\mathrm{weights}}=11\times11\times3\times96=34{,}848assets/equations/9a11e5db4a09d338.svg
• h_1=ux_1+vh_0+bassets/equations/9a1296eaa212dfb3.svg
• s=(0.2)+(-0.2)=0assets/equations/9a1dc06bff5e31d8.svg
• g_{1,2}^{(2)}=(-0.093597941)\times(0.668187772)\approx -0.062541assets/equations/9a4ef8c75729d56e.svg
• N_{\mathrm{out}}=\left\lfloor\frac{5+0+1-2}{1}\right\rfloor+1=4+1=5assets/equations/9a5597c2b0d54bcf.svg
• o_{2}=\frac{1}{1+e^{-(0.482901756)}}=0.618432848assets/equations/9a8b37a15ece040e.svg
• \frac{\partial L}{\partial W_{hh}}=\sum_t\delta_t h_{t-1}^T,\quad\frac{\partial L}{\partial W_{xh}}=\sum_t\delta_t x_t^Tassets/equations/9aa073a86bb67d16.svg
• 6(5)=30assets/equations/9aa0cefe0fe7172a.svg
• h=(1-z)h_{t-1}+z\widetilde hassets/equations/9abbc158fb98ec74.svg
• z_{h_1} = (0.2 \times 1) + (0.1 \times 2) + (0.1) = 0.5assets/equations/9b009885daaa5779.svg
• s_t=\rho s_{t-1}+(1-\rho)g_t^2assets/equations/9b03d623d9793c84.svg
• L_{\mathrm{new}}=\frac12(0.929-1)^2=0.0025205assets/equations/9b3580237637a521.svg
• \delta_t=(q_t+u\delta_{t+1})(1-h_t^2)assets/equations/9b43734482c48bdd.svg
• f=i=o=\sigma(z)assets/equations/9b53d0d62e0b3dde.svg
• z_{\mathrm{new}}=0.5\times1+0.5\times1+0.5=1.5assets/equations/9b6e487add8bda9e.svg
• h_{\mathrm{new}}=(0.500000000)(0.462117157)\approx0.231058579assets/equations/9bcf24dd3ef65cfd.svg
• C_1=(0.14)\times(-0.809)\approx -0.11326assets/equations/9be822c811879a17.svg
• fC_{\mathrm{previous}}=0.8\times0.3=0.24assets/equations/9bf864ebb414856e.svg
• I=\begin{bmatrix}1&2&1\\0&1&3\\2&1&0\end{bmatrix},\quad K=\begin{bmatrix}1&-1\\0&1\end{bmatrix}assets/equations/9bf8a81efcf4cdbd.svg
• g_{j,\mathrm{total}}=g_{j,\mathrm{data}}+\lambda\operatorname{sign}(w_j)assets/equations/9bff63bbdd0e51df.svg
• U_{\mathrm{batch}}=E,\qquad U_{\mathrm{SGD}}=NEassets/equations/9c0c1479389c6c19.svg
• \frac{\partial\mathrm{SSE}}{\partial\hat y_i}=2e_i,\quad \frac{\partial L_{\mathrm{half}}}{\partial\hat y_i}=e_i,\quad \frac{\partial\mathrm{MSE}}{\partial\hat y_i}=\frac2n e_iassets/equations/9c169d2ae1cd10a3.svg
• g_2^{\mathrm{clip}}=4(1)=4assets/equations/9c36bcaa7f2c9e3b.svg
• e^{-4}\approx0.018315639assets/equations/9c3b8c152fd312b9.svg
• \widehat y=\operatorname{step}(w^Tx+b)assets/equations/9c6e513ba240f8dc.svg
• w_{\mathrm{new}}=w-\eta\frac{\partial L}{\partial w},\qquad b_{\mathrm{new}}=b-\eta\frac{\partial L}{\partial b}assets/equations/9cca82416f4ee903.svg
• L_1=\frac12(2.2-1)^2=0.72assets/equations/9cca9af1e961da75.svg
• R_2=(1)\times(0)+(3)\times(-1)=-3assets/equations/9cf98567181f7899.svg
• g_{1,1}^{(1)}=(0.008771355)\times(0.05)\approx 0.000438568assets/equations/9d2ed0236b62ab2a.svg
• 0.1+0=0.1assets/equations/9d31df4938e5db9a.svg
• z_{h_2} = (0.5 \times 1) + (0.5 \times 1) + (0) = 1assets/equations/9d50f49ceec1f464.svg
• a=\frac{53.598150033}{55.598150033}\approx0.964027580assets/equations/9d5e297333003136.svg
• g_{3,3}^{(1)}=(0.012994055)\times(0.5)\approx 0.006497027assets/equations/9d6717c29bdbe2b4.svg
• m_t=\beta m_{t-1}+(1-\beta)g_t,\qquad0\le\beta<1assets/equations/9d6a2ee32d9c0644.svg
• Y_{2,2}=(0)+(0)+(0)=0assets/equations/9d6dfc0f38e45b75.svg
• v_2=0.9(-0.1)-0.1(0.9)=-0.18assets/equations/9d76f7ff1692b41e.svg
• \mathrm{SSE}=1+0=1assets/equations/9d7eadaf7257dae1.svg
• h_{1}=\frac{1}{1+e^{-(1)}}=0.731058579assets/equations/9d9a8c14314f06ad.svg
• R_2=(3)\times(1)+(2)\times(2)+(8)\times(1)=15assets/equations/9dcb85c5c6766eb3.svg
• g_{1,2}^{(2)}=(-0.55)\times(0.3)\approx -0.165assets/equations/9dd543c1a4cfe977.svg
• h_t=\tanh(W_{xh}x_t+W_{hh}h_{t-1}+b)assets/equations/9dd7210eacaa003c.svg
• \frac{\partial L}{\partial v}=\delta_1h_0+\delta_2h_1,\qquad \frac{\partial L}{\partial b}=\delta_1+\delta_2assets/equations/9e07d1c7fc0741ba.svg
• w_{\mathrm{new}}=w-\eta\delta x,\quad b_{\mathrm{new}}=b-\eta\deltaassets/equations/9e125df8665bd715.svg
• 3^2+4^2=25assets/equations/9e1425dca608e703.svg
• Y_{2,1}=(0.2)+(0.2)+(0.2)=0.6assets/equations/9e16ceceb6a0a438.svg
• z_o=1-2(0)-0.5=0.5\Rightarrow o=1assets/equations/9e3e9239cb2ed70d.svg
• v_1=0.99(0)+0.01(2)^2=0.04assets/equations/9e506bfe9fd357f5.svg
• \left\lfloor\frac{126 - 2}{2}\right\rfloor+1assets/equations/9e62fef4c3e773c7.svg
• v_1=0.9(0)-0.1(-1.5)=0.15assets/equations/9e63b276a509c75c.svg
• e_i=\hat y_i-y_i,\qquad \mathrm{SSE}=\sum_{i=1}^{n}e_i^2,\qquad L_{\mathrm{half}}=\frac12\sum_{i=1}^{n}e_i^2,\qquad \mathrm{MSE}=\frac1n\sum_{i=1}^{n}e_i^2assets/equations/9e6b354fa6fa3c49.svg
• \delta_{o_2} = 1 \times (0.772928465-0.99) \times 0.175510053 = -0.038098237assets/equations/9e7235b42f1e0b22.svg
• g_1=J'(1)=1assets/equations/9e80b135dd68a0a3.svg
• \delta_o=(1.08-1)(1)=0.08assets/equations/9ead62e2abd6d370.svg
• L_{\mathrm{BCE}}=-[y\ln p+(1-y)\ln(1-p)],\qquad L_{\mathrm{CCE}}=-\sum_k y_k\ln p_kassets/equations/9ec44fbf24f3156c.svg
• g=g_{\mathrm{data}}+g_{\mathrm{penalty}}assets/equations/9ed52defd9fef6cb.svg
• W^{(2)}_{2,2,\mathrm{new}}=0.21-1(0.06457857)\approx 0.14542143assets/equations/9edb4302c3d8076f.svg
• \sigma=\sqrt{0.000000000}\approx0.000000000assets/equations/9ee45ab162936835.svg
• \begin{bmatrix}1 & 3\\1 & 0\end{bmatrix}assets/equations/9f01f837180dd993.svg
• b_{h_1,\mathrm{new}}=0.1-0.1(-0.220000000)=0.122000000assets/equations/9f4d72c657ffcbf4.svg
• e^{-2}\approx0.135335283assets/equations/9f6c4ea0634c6cc7.svg
• z_1=2+(-2)+0.1=0.1assets/equations/9f6eacd9d3a609cc.svg
• \frac{\partial L}{\partial b}=\delta_1+\delta_2=-0.25-0.5=-0.75assets/equations/9f8bea284b02b5e1.svg
• R_1=(7)\times(1)+(6)\times(1)+(7)\times(1)=20assets/equations/9f8fed3c271dceb7.svg
• \Delta w=\eta t x,\quad\Delta b=\eta t\quad\text{on a mismatch}assets/equations/9f96d47c421b72ff.svg
• \text{biases}=2,\qquad P_{\mathrm{conv}}=54+2=56assets/equations/9fa35646db3c8aec.svg
• v_2=0.99(0.04)+0.01(-1)^2=0.0496assets/equations/9fa8fd0cf3a03d97.svg
• W\in\mathbb R^{m\times d},\quad x\in\mathbb R^{d\times1},\quad b,z\in\mathbb R^{m\times1}assets/equations/9fb325b4580ff43a.svg
• w_{\mathrm{new}}=1-0.1(-2.5)=1.25assets/equations/9fb7a46ddd37e16c.svg
• R_1=(10)\times(1)+(9)\times(1)+(5)\times(1)=24assets/equations/9fe1b0c406527aa8.svg
• R_1=(3)\times(1)+(2)\times(0)=3assets/equations/9fed257738d47bdb.svg
• n_{\mathrm{flat}}=30\times30\times32=28{,}800assets/equations/9fedbcb21e866a38.svg
• y=F(x)+W_sxassets/equations/9ffeadda67cc02f9.svg
• o_{1}=\frac{1}{1+e^{-(1.105905967)}}=0.75136507assets/equations/a028b62b987f7076.svg
• h_{1}=\frac{1}{1+e^{-(-0.7)}}=0.331812228assets/equations/a03d842e36f6059c.svg
• \ln(0.75)\approx-0.287682072,\qquad\ln(0.25)\approx-1.386294361assets/equations/a0496aac07524e4b.svg
• s_1=0.9(0)+0.1(2)^2=0.4assets/equations/a0575b9596aea909.svg
• o_1=2(1.1)=2.2,\qquad o_2=-(1.1)=-1.1assets/equations/a05e20f353481e0c.svg
• C_1=(0.4)\times(-0.55)\approx -0.22assets/equations/a0677d80f1fb0659.svg
• g_{2,1}^{(1)}=(-0.275)\times(1)\approx -0.275assets/equations/a0689c05816a3a3a.svg
• -\ln(0.9)=-(-0.105360516)=0.105360516assets/equations/a07489f729a63131.svg
• C_{\mathrm{new}}=fC_{\mathrm{old}}+i\widetilde Cassets/equations/a0929338a45c9e2f.svg
• N+2P=5+2=7assets/equations/a0c02a700e3aef2c.svg
• N_{\mathrm{out}}=\left\lfloor\frac{63+2(0)-3}{1}\right\rfloor+1=61assets/equations/a0caa4135f45ed9c.svg
• a_1=0.5(1)+0.2(0)+0=0.5assets/equations/a11d26405f8ffbf6.svg
• \frac{\partial L}{\partial w_1}=0\times2=0assets/equations/a131ac401273e09f.svg
• R_1=(5)\times(1)+(7)\times(1)+(6)\times(1)=18assets/equations/a148f40de653ff23.svg
• W^{(1)}_{1,1,\mathrm{new}}=0.5-0.5(-0.004869072)\approx 0.502434536assets/equations/a169e16b705d6d2b.svg
• o_{1}=\frac{1}{1+e^{-(0.365529289)}}=0.590378257assets/equations/a199e77a25ea274a.svg
• \sigma^2=\frac{1}{N}\sum_{i=1}^{N}(x_i-\mu)^2assets/equations/a1a03dea0d7cf33e.svg
• \theta_1=1-\frac{0.1\times(2)}{\sqrt{0.4}}=0.683772234assets/equations/a1a1a24a56913ad6.svg
• W^{(2)}_{1,1,\mathrm{new}}=0.14-0.05(-0.68765)\approx 0.1743825assets/equations/a1a6aafaa6af94cf.svg
• g_{2,2}^{(1)}=(-0.12135)\times(3)\approx -0.36405assets/equations/a1b23a6eb1610029.svg
• \frac{\partial L}{\partial p}=-\frac yp+\frac{1-y}{1-p}=\frac{p-y}{p(1-p)}assets/equations/a1bc86d47cf16a06.svg
• b_{h_2,\mathrm{new}}=0.2-0.1(-0.275000000)=0.227500000assets/equations/a1c182cfafd9e6ab.svg
• w_2^{\mathrm{new}}=0.1+0.1(0)(2)=0.1assets/equations/a1cc84ebf8de3e99.svg
• \widetilde h_t=\tanh(0.3)\approx0.291312612assets/equations/a1e77c483544b002.svg
• v_2=0.999(0.004)+0.001(1)^2=0.004996assets/equations/a1ecf2cf2c6e9e7e.svg
• J=L+\frac\lambda2\sum_jw_j^2,\quad w_{new}=w-\eta(g_{data}+\lambda w)assets/equations/a21ef40f9211941d.svg
• N_{\mathrm{weights}}=5\times5\times32\times64=51{,}200assets/equations/a22447ce668f8dbf.svg
• \hat s_2=\frac{0.004996}{0.001999}\approx2.499249625assets/equations/a22ef3b407f486c3.svg
• N_{\mathrm{conv}}=\left\lfloor\frac{7+2(1)-3}{2}\right\rfloor+1=3+1=4assets/equations/a2592712257363b0.svg
• g_{2,1}^{(1)}=(0.003509506)\times(0.25)\approx 0.000877376assets/equations/a274b0a4de386476.svg
• R_2=(0.3)\times(1)+(0.3)\times(0)+(0.3)\times(-1)=0assets/equations/a27c355bf724dc1d.svg
• \mathrm{SSE}=\sum_i(o_i-y_i)^2assets/equations/a288e1ba23bc9347.svg
• b_{h_2,\mathrm{new}}=0-0.5(-0.004869072)=0.002434536assets/equations/a2b6bc6b29544f77.svg
• N_{\mathrm{out}}=\left\lfloor\frac{109+2(0)-5}{1}\right\rfloor+1=105assets/equations/a2cc5fced7b8ab11.svg
• h_t=\tanh(W_{xh}x_t+W_{hh}h_{t-1}+b_h),\quad o_t=W_{hy}h_t+b_yassets/equations/a2d32a1185192d80.svg
• G_1=0+(2)^2=4assets/equations/a2f78c9b3d52d653.svg
• N_{\mathrm{weights}}=3\times3\times384\times384=1{,}327{,}104assets/equations/a2fb560ecf5f18c3.svg
• \frac{\partial L}{\partial w_1}=0.064(2)=0.128assets/equations/a31fd300551ec8e1.svg
• o=0.2(2)+0.3(0)+0=0.4assets/equations/a323109c2640591d.svg
• W^{(1)}_{2,1,\mathrm{new}}=-0.3-0.9(0.006542085)\approx -0.305887877assets/equations/a32b4e1a94dcddab.svg
• L=\frac12\sum_{j=1}^{k}(\hat y_j-y_j)^2assets/equations/a3370e1f71500a0e.svg
• \theta_2=0.9-\frac{0.1\times(0.421052632)}{\sqrt{2.492462312}}=0.873330057assets/equations/a3614192f516725e.svg
• P_6=6{,}912{,}000+240=6{,}912{,}240assets/equations/a36eb80ee2f332fd.svg
• W\in\mathbb R^{m\times d},\quad x\in\mathbb R^{d\times1},\quad b,z\in\mathbb R^{m\times1}assets/equations/a38b418b5411afa4.svg
• N_{\mathrm{biases}}=96assets/equations/a3b19ed994769f54.svg
• z_{o_2} = (0.5 \times 0.593269992) + (0.55 \times 0.596884378) + (0.6) = 1.224921404assets/equations/a3d7358ccdf52d31.svg
• y_{\mathrm{avg}}=\frac{53}4=13.25assets/equations/a3fb67b8edc22789.svg
• g_{2,2}^{(1)}=(0.003509506)\times(0.7)\approx 0.002456654assets/equations/a4233a0dbb4825a8.svg
• e_2^2=2^2=4assets/equations/a4282b80c2e07509.svg
• \theta_{\mathrm{look}}=-0.5+0.9(0)=-0.5assets/equations/a4363d91dbba8cc0.svg
• P_6=6{,}912{,}000+240=6{,}912{,}240assets/equations/a4474a3c31a68206.svg
• g_{1,2}^{(1)}=(0.004824633)\times(0.4)\approx 0.001929853assets/equations/a45450793aa1fbee.svg
• N_{\mathrm{weights}}=4{,}096\times1{,}000=4{,}096{,}000assets/equations/a47853490ec706c7.svg
• b^T=\begin{bmatrix}4&5\end{bmatrix}assets/equations/a491055be3cc3700.svg
• z_{o_1} = (0.4 \times 0.593269992) + (0.45 \times 0.596884378) + (0.6) = 1.105905967assets/equations/a499c079904c4335.svg
• \int\frac1x\,dx=\ln|x|+C,\qquad\int e^x\,dx=e^x+Cassets/equations/a4a9ce610fff6be4.svg
• g_{1,4}^{(1)}=(-0.009617203)\times(-0.4)\approx 0.003846881assets/equations/a4af21673941e601.svg
• (1)^2=1assets/equations/a4b8daac35aec1bc.svg
• g_{1,1}^{(2)}=(-0.093597941)\times(0.710949503)\approx -0.06654341assets/equations/a4c5e0af98c4e800.svg
• N_{\mathrm{out}}=\left\lfloor\frac{31+2(1)-5}{2}\right\rfloor+1assets/equations/a4fa49b6e311d65e.svg
• \frac{\partial L}{\partial h}=\frac{\partial L_1}{\partial h}+\frac{\partial L_2}{\partial h}assets/equations/a4fb627a77ae8284.svg
• a_2=0.5(2)+0.2h_1+0assets/equations/a530e8ec85bf260b.svg
• h=0.1+0.218484459=0.318484459assets/equations/a53355ba328d8b0b.svg
• h_{\mathrm{new}}=-0.16+(0.029777007)\approx-0.130222993assets/equations/a559a88735fecf3e.svg
• z_{\mathrm{new}}=\frac42=2assets/equations/a563909e15c191b2.svg
• W^{(2)}_{1,2,\mathrm{new}}=0.7-0.5(-0.062541)\approx 0.7312705assets/equations/a56876d47ec41774.svg
• h_1=\tanh(0.5)=\frac{e^1-1}{e^1+1}assets/equations/a56af3f825290564.svg
• z_{h_1} = (0.5 \times 1) + (0.5 \times 1) + (0) = 1assets/equations/a57d08e71109e92c.svg
• N_{\mathrm{out}}=\left\lfloor\frac{32+2(0)-5}{1}\right\rfloor+1=28assets/equations/a58485fe718e957c.svg
• \nabla_{\hat y}\mathrm{MSE}=\frac23[0.2,-0.3,-0.6]=[0.133333333,-0.2,-0.4]assets/equations/a584bba4763cda9f.svg
• z=\sum_iw_ix_i+b,\qquad p=\frac1{1+e^{-z}},\qquad L=-\ln passets/equations/a593381e0c37ce0b.svg
• \theta_{\mathrm{new}}=\theta-\eta\frac{\partial L}{\partial\theta}assets/equations/a5c419cc72b98572.svg
• 2+0.5+0.5=3assets/equations/a5dc1556ae91ba13.svg
• \hat m_t=\frac{m_t}{1-\beta_1^t},\qquad\hat v_t=\frac{v_t}{1-\beta_2^t}assets/equations/a5e0dae9a0fb7261.svg
• w_{\mathrm{batch}}=0-0.1(-2.5)=0.25assets/equations/a5ec534d58a905c7.svg
• 64(4)(1)(1)(16)=4096assets/equations/a619ccae82b20a14.svg
• a_2=0.5(2)+0.2(0.462117157)+0assets/equations/a620346ee9bd0632.svg
• \frac{\partial L}{\partial w}=(o-y)x,\qquad \frac{\partial L}{\partial b}=o-yassets/equations/a63ae410089093d5.svg
• \widehat v_2=\frac{0.004999}{1-0.999^{2}}=2.500750375assets/equations/a6410e8fada39e27.svg
• \begin{bmatrix}0.4 & 0.45\\0.5 & 0.55\end{bmatrix}assets/equations/a6699cc1064bfaf3.svg
• \left\lfloor\frac{126-2}{2}\right\rfloor+1assets/equations/a6979fd0d1f6b483.svg
• \sum_i p_i=1,\qquad\mathbb E[X]=\sum_i p_i x_iassets/equations/a6a90dd79f192150.svg
• m_2=0.9(0.2)+0.1(1)=0.28assets/equations/a6a9a858d7c18aeb.svg
• U_{\mathrm{keep}}=E\left\lceil\frac NB\right\rceil,\qquad U_{\mathrm{drop}}=E\left\lfloor\frac NB\right\rfloorassets/equations/a6ec99ec225775f5.svg
• C_{\mathrm{new}}=-0.365529289+(0.556769941)\approx0.191240652assets/equations/a705b1677cd4e456.svg
• a=\max(0,7)=7assets/equations/a730b8a58ff43cba.svg
• U=\lceil N/B\rceilassets/equations/a73af57990a3f346.svg
• b_{new}=b+\eta(y-\widehat y)assets/equations/a75f55d01930a8a4.svg
• \widetilde h=\tanh\left(Wx+U(rh_{\mathrm{previous}})+b\right)assets/equations/a7608696c8363c58.svg
• 2z=2(0.5)=1,\qquad e^{2z}=e^1\approx2.718281828assets/equations/a761ae42b593cfbc.svg
• s_2=0.9(0.4)+0.1(1^2)=0.46assets/equations/a780b29bab6859eb.svg
• g_{1,1}^{(2)}=(-0.809)\times(0.85)\approx -0.68765assets/equations/a7b158dc1aab8c85.svg
• R_1=(3)\times(1)+(1)\times(0)+(2)\times(-1)=1assets/equations/a7b5121489bd31f6.svg
• C_1=(0.4)\times(0.138498562)\approx 0.055399425assets/equations/a7ba541a2ae61050.svg
• w_{ji,\mathrm{new}}=w_{ji}-\eta\delta_j a_i,\quad b_{j,\mathrm{new}}=b_j-\eta\delta_jassets/equations/a7c0c7e8cef31635.svg
• g_{1,2}^{(1)}=\frac{\partial L}{\partial W^{(1)}_{1,2}}=\delta_{h_1}x_2assets/equations/a7e2552eb055cb6f.svg
• 54.598150033-1\approx53.598150033assets/equations/a7f29872524e69fc.svg
• P_7=884{,}736+256=884{,}992assets/equations/a803766772104e69.svg
• \ln(0.2)\approx-1.609437912assets/equations/a82bb66e18d1bf7c.svg
• p_j=\frac{e^{z_j-M}}{\sum_k e^{z_k-M}}assets/equations/a8375be5e6a8189d.svg
• p_2=\frac{0.367879441}{1.503214724}\approx0.244728471assets/equations/a839a3e6b8973897.svg
• \hat m_t=\frac{m_t}{1-\beta_1^t},\qquad\hat s_t=\frac{s_t}{1-\beta_2^t}assets/equations/a84b9734c9141df3.svg
• 112896000-5268480=107627520assets/equations/a866c1ac7a2fce39.svg
• g_{1,1}^{(1)}=(0.004824633)\times(0.2)\approx 0.000964927assets/equations/a8abe88d39c4a4c9.svg
• \left\lfloor 2.5\right\rfloorassets/equations/a8acf65164593cb2.svg
• (0)^2=0assets/equations/a8b9442558db09d6.svg
• g_{1,1}^{(1)}=(0.064)\times(2)\approx 0.128assets/equations/a8c9a6cdcee54a64.svg
• \mathrm{SSE}=\sum_i e_i^2assets/equations/a8dd5fa4e080919c.svg
• N_{\mathrm{out}}=\lfloor(N+2P-K)/S\rfloor+1assets/equations/a8df1422782a8067.svg
• J_{L1}=0.5+0.3=0.8assets/equations/a8ed142793124641.svg
• \tanh(z)=\frac{e^{2z}-1}{e^{2z}+1}assets/equations/a8ee72519b086209.svg
• W^{(1)}_{2,1,\mathrm{new}}=0.28-1(0.000877376)\approx 0.279122624assets/equations/a8f3aa782e00c9f2.svg
• 0.9(-0.1)=-0.09assets/equations/a8fd30bce103b617.svg
• \widetilde C_t=\tanh(W_C[h_{t-1},x_t]+b_C),\quad o_t=\sigma(W_o[h_{t-1},x_t]+b_o)assets/equations/a90c778fa86aa08a.svg
• \widetilde C=\tanh(z)assets/equations/a911824e1fcf0c02.svg
• \mu_B=\frac{1}{m}\sum_{i=1}^{m}x_iassets/equations/a91b1c9216cbd799.svg
• R_2=(0)\times(0)+(1)\times(-1)=-1assets/equations/a91d42dac9ba1633.svg
• g_{1,2}^{(2)}=(-0.131169078)\times(0.524979187)\approx -0.068861036assets/equations/a93072cdabc372b8.svg
• P_{\mathrm{conv}}=(363+1)(96)=34944assets/equations/a95b022f4ca93164.svg
• g_{2,2}^{(1)}=(-0.275)\times(2)\approx -0.55assets/equations/a96b50c26c1fb262.svg
• \frac{\partial L}{\partial u}=(-0.25)(1)+(-0.5)(2)=-1.25assets/equations/a96bfa656867b634.svg
• g_{1,1}^{(2)}=(0.138498562)\times(0.593269992)\approx 0.082167041assets/equations/a9b1122ca44cd5cd.svg
• P_{\mathrm{total}}=P_{\mathrm{before}}+P_{\mathrm{after}}=K-1assets/equations/a9c76650bca9e58b.svg
• h_{\mathrm{new}}=(0.731058579)(0.188942847)\approx0.138128289assets/equations/a9cc310c7b05f01a.svg
• v_{new}=\beta v-\eta g(\theta_{look})assets/equations/a9d756d38e5b7933.svg
• \frac{\partial L}{\partial w_2}=\delta_o h,\quad \delta_h=\delta_o w_2\operatorname{ReLU}'(z_h),\quad \frac{\partial L}{\partial w_1}=\delta_h xassets/equations/a9eda432984ec8da.svg
• \widehat v_t=\frac{v_t}{1-\beta_2^t}assets/equations/a9f81435f55f92a6.svg
• v_{\mathrm{new}}=0.5-0.1(-0.1)=0.51assets/equations/a9f8b51dd6b5c81d.svg
• R_1=(3)\times(1)+(1)\times(0)+(2)\times(-1)=1assets/equations/a9fce55fa8f0fbec.svg
• -z=-(-2)=2assets/equations/a9feed82e056cf24.svg
• 1+1.000000000=2.000000000assets/equations/aa21aa600ebc8e6e.svg
• R_1=(2)\times(1)+(2)\times(0)+(1)\times(-1)=1assets/equations/aa4766cd874d5e7f.svg
• z=Wx+b,\qquad a_j=\max(0,z_j)assets/equations/aa52f339dd0d5a91.svg
• h_2=ux_2+vh_1+bassets/equations/aa5a122e6af7b01d.svg
• p_1=\frac{0.367879441}{1.503214724}=0.244728471assets/equations/aa686bb14848b0d1.svg
• z=0.6+(-0.4)+(-0.1)+0.1assets/equations/aa77eba1df26bb28.svg
• \mu=\frac{1}{N}\sum_{i=1}^{N}x_iassets/equations/aa827d67a05a9234.svg
• L(w)=w^2,\qquad \frac{dL}{dw}=2wassets/equations/aab6f621e9833b07.svg
• w_{\mathrm{new}}=0.4-0.1(-2)=0.6assets/equations/aab9aaca5264ba06.svg
• C_1=(0.5)\times(-0.093597941)\approx -0.04679897assets/equations/aabc39607d847229.svg
• p_1=\frac{1.000000000}{3.000000000}\approx0.333333333assets/equations/aac567888a814ba9.svg
• z_1=0+0-0.5=-0.5assets/equations/aac6904086092425.svg
• z_{h_1} = (0.5 \times 2) + (0.1) = 1.1assets/equations/aac90742e2057dd1.svg
• \begin{bmatrix}1 & -2\\0.5 & 3\end{bmatrix}assets/equations/aad1d6a46cb2c7f1.svg
• C_2=(0.12)\times(0.082533822)\approx 0.009904059assets/equations/aad6a8cccacf85e0.svg
• z_{h_1}=0+1-0.5=0.5\Rightarrow h_1=1assets/equations/aadd7c29e1061db5.svg
• \Delta w=\eta t x,\quad\Delta b=\eta t\quad\text{on a mismatch}assets/equations/aae3c82602ee4336.svg
• \mathrm{MACs}_{1\times1}=28(28)(16)(192)=2408448assets/equations/aaefb7a933a188a0.svg
• y_{\mathrm{avg}}=\frac{57}4=14.25assets/equations/aaffd26053e275ce.svg
• 2+(-0.2)=1.8assets/equations/ab49a448719b780e.svg
• W_{\mathrm{padded}}=5+0+1=6assets/equations/ab52e1fac6848205.svg
• N_{\mathrm{weights}}=5\times5\times3\times32=2{,}400assets/equations/ab73b23ba038b015.svg
• h_{3}=\frac{1}{1+e^{-(0.56)}}=0.63645254assets/equations/ab920f7e27d63556.svg
• p_1=\frac{1.000000000}{3.000000000}\approx0.333333333assets/equations/abb38634253b454f.svg
• a=\max(0,0.9)=0.9assets/equations/abc54eeca8cf7516.svg
• L=-\sum_i y_i\ln p_i=-\ln p_{true}assets/equations/abecd65b7d7b5753.svg
• h_{2} = \max(0,0.5) = 0.5assets/equations/ac1a985150f1b35b.svg
• \begin{aligned}0.2\times0.3&=\frac2{10}\times\frac3{10}\\&=\frac6{100}\\&=0.06\end{aligned}assets/equations/ac4c9d847d5a49fc.svg
• g=\frac{\partial J}{\partial\theta}assets/equations/ac6a050d5a887603.svg
• \frac{1}{1+e^{-6}}assets/equations/ac8062a4bfd609d2.svg
• h_{1}=\frac{1}{1+e^{-(0.2)}}=0.549833997assets/equations/acdd5499eb345eef.svg
• \begin{bmatrix}1 & 1 & 1 & 0 & 0\\0 & 1 & 1 & 1 & 0\\0 & 0 & 1 & 1 & 1\\0 & 0 & 1 & 1 & 0\\0 & 1 & 1 & 0 & 0\end{bmatrix}assets/equations/acdfbb90974d971f.svg
• g_{1,2}^{(1)}=(0.002053687)\times(0.7)\approx 0.001437581assets/equations/acf691feadca0305.svg
• g_2=0.9-(0)=0.9assets/equations/acfaeec4e6af0c51.svg
• g_i=g_{\mathrm{data},i}+\lambda\operatorname{sign}(w_i)assets/equations/ad178fc1be6044c9.svg
• z_2=0+0-1.5=-1.5assets/equations/ad20a476f045fd25.svg
• \left\lfloor\frac{4-2}{2}\right\rfloor+1assets/equations/ad46d0c1b3646c71.svg
• \widehat m_2=\frac{0.08}{1-0.9^{2}}=0.421052632assets/equations/ad611d8dfc7121b6.svg
• R_1=(1)\times(1)+(3)\times(-1)=-2assets/equations/ad79fb60fa82034f.svg
• z=(-0.2)(1)+(0.1)(2)+0=0assets/equations/ad825aadcb48f967.svg
• w_{1,\mathrm{new}}=0.2+0.21=0.41assets/equations/ad8445391f1a6df2.svg
• g_u=\sum_t\delta_tx_tassets/equations/ad879502d644f051.svg
• J(\theta)=\frac12(\theta-0)^2,\qquad J\prime(\theta)=\theta-0assets/equations/ad9f147031605f82.svg
• \theta_t=\theta_{t-1}-\frac{\eta\hat m_t}{\sqrt{\hat s_t}+\varepsilon}assets/equations/ada7a51c96cfce92.svg
• \delta_{h_2} = 0.020617214 \times 0.170222124 = 0.003509506assets/equations/adbef55caefb8d30.svg
• \frac{\partial L}{\partial z_j}=-\sum_i\frac{y_i}{p_i}p_i(\delta_{ij}-p_j)=-y_j+p_j\sum_i y_i=p_j-y_jassets/equations/add665e2f73a094d.svg
• R_3=(3)\times(1)+(2)\times(1)+(8)\times(1)=13assets/equations/adea8f6e7cdc7e9c.svg
• \sum_iw_i^2=4+1+0.25=5.25assets/equations/ae0c24324eadcb63.svg
• e_i=\hat y_i-y_i,\qquad \mathrm{SSE}=\sum_{i=1}^{n}e_i^2,\qquad L_{\mathrm{half}}=\frac12\sum_{i=1}^{n}e_i^2,\qquad \mathrm{MSE}=\frac1n\sum_{i=1}^{n}e_i^2assets/equations/ae0cc9bf3a93ca00.svg
• W^{(2)}_{1,2,\mathrm{new}}=0.15-0.05(-0.38832)\approx 0.169416assets/equations/ae18879446b6ea9d.svg
• 0.6(0)+0.4(1)=0+0.4=0.4assets/equations/ae37ee7dd9298b47.svg
• g_{\mathrm{batch}}=\frac{-1+(-4)}2=-2.5assets/equations/ae3a9f41f9f31680.svg
• P_1=34{,}848+96=34{,}944assets/equations/ae51bc20750a112f.svg
• \frac{\partial L}{\partial W_1}=\delta_1x^T,\qquad\frac{\partial L}{\partial b_1}=\delta_1assets/equations/ae5ad96746c9621d.svg
• \sqrt{s_2}=\sqrt{0.46}\approx0.678232998assets/equations/ae7e6b23e0aacbac.svg
• z_{h_2} = (0.12 \times 2) + (0.08 \times 3) + (0) = 0.48assets/equations/ae9a1b94e41b8118.svg
• Y_{2,2}=(0)+(-4)+(-3)=-7assets/equations/aec0342788a9224e.svg
• J(\theta)=\frac12(\theta-0)^2,\qquad J\prime(\theta)=\theta-0assets/equations/aec670e56abfcf00.svg
• g_1^{\mathrm{clip}}=3(1)=3assets/equations/af049506394d9137.svg
• W^{(2)}_{2,2,\mathrm{new}}=0.55-0.5(-0.022740242)\approx 0.561370121assets/equations/af1284429e824dd6.svg
• \frac{\partial L}{\partial u}=(-0.25)(1)+(-0.5)(2)=-0.25-1=-1.25assets/equations/af14a42daa22ee7b.svg
• g_{batch}=\frac1B\sum_{i=1}^{B}g_iassets/equations/af742484e9a4fffc.svg
• \begin{bmatrix}1 & 2\\3 & 4\end{bmatrix}assets/equations/af9cca7044f770c7.svg
• h_{1,\mathrm{new}}=0.325(1)+0.51(0)+0.075=0.4assets/equations/af9ceb62c0e1f59e.svg
• v_2=0.9(0.15)-0.1(-1.215)=0.2565assets/equations/afa161b36559b605.svg
• v_{1,\mathrm{new}}=0.9(0.1)-0.1(-1.2)=0.09+0.12=0.21assets/equations/afa6505a2a4ce39d.svg
• g_{2,1}^{(1)}=(-0.014526308)\times(0.4)\approx -0.005810523assets/equations/afb5bb784d05f461.svg
• U(rh_{\mathrm{previous}})=0.5\times0.2=0.1assets/equations/afd69509f26a8283.svg
• \sigma(z)=\frac1{1+e^{-z}},\qquad\sigma\prime(z)=\sigma(z)[1-\sigma(z)]assets/equations/afe493539b707a8a.svg
• W^{(1)}_{4,4,\mathrm{new}}=0.8-0.5(0.001713099)\approx 0.799143451assets/equations/b007fcf05f829e6c.svg
• v_1=0.999(0)+0.001(2)^2=0.004assets/equations/b00fce2cf8ae5a5b.svg
• \frac{\partial L_1}{\partial h}=(1.2)(2)=2.4assets/equations/b03138eedef68e56.svg
• \widetilde C_t=\tanh(0.5)\approx0.462117157assets/equations/b03e93e6315982fa.svg
• \hat y=wxassets/equations/b0668b91b6f454d0.svg
• 5(4)=20assets/equations/b06f6eedc0b0c61f.svg
• 7-2=5assets/equations/b07e3025ba7ec8bf.svg
• z_2=((-1)\times2)+(3\times1)+(-0.5)assets/equations/b0b4c769cba96d49.svg
• e_1=2-1=1assets/equations/b0c87b502c9af9cc.svg
• \sigma^2=\frac{0+0+0}{3}\approx0.000000000assets/equations/b0fc9edc8362e2a5.svg
• Y_{1,1}=(1)+(0)+(2)=3assets/equations/b108d091066c8f7c.svg
• \delta_{h_1} = 0.064 \times 1 = 0.064assets/equations/b10ee7f796aaf918.svg
• u=3(1)+1=4assets/equations/b14c00d5dab41947.svg
• \int\frac1x\,dx=\ln|x|+C,\qquad\int e^x\,dx=e^x+Cassets/equations/b17aa54fb4500d95.svg
• C_2=(0.4)\times(0.145933863)\approx 0.058373545assets/equations/b190ef59d9695988.svg
• h_2=\tanh(1.092423431)=0.797760954assets/equations/b1963e3f6225d34f.svg
• \frac{1}{1+e^{-1}}assets/equations/b1ad72f023278d5b.svg
• \theta_t=\theta_{t-1}-\frac{\eta\hat m_t}{\sqrt{\hat s_t}+\epsilon}assets/equations/b1b0801ea4c5e3b1.svg
• \frac{\partial L}{\partial z_1}=0.1-0=0.1assets/equations/b1d662617a540534.svg
• N_{\mathrm{out}}=\left\lfloor\frac{13+2(1)-3}{1}\right\rfloor+1=13assets/equations/b1fa159221b64b09.svg
• A=\begin{bmatrix}-1\\4\\2\end{bmatrix}assets/equations/b21656f3402af964.svg
• \frac{\partial L}{\partial W_2}=\delta_2a_1^T,\qquad\frac{\partial L}{\partial b_2}=\delta_2assets/equations/b217a2191f391cf2.svg
• \theta_2=-0.35-0.1(-1.35)=-0.215assets/equations/b21cdf85d50680f5.svg
• 32+8=40assets/equations/b223db7a6cdbf384.svg
• z_{h_1} = (0.2 \times 0.25) + (0.15 \times 0.7) + (1) = 1.155assets/equations/b2652e077761f90d.svg
• x_3w_3=3(0.5)=1.5assets/equations/b26c8cde278f5783.svg
• h=0.1+0.218484459=0.318484459assets/equations/b283dd6b6dd4cf79.svg
• \hat y=4.5+(1)=5.5assets/equations/b28b202a19e96ace.svg
• 3+3+7+7=20assets/equations/b29c1c44847515f7.svg
• \sigma(2)=\frac1{1.135335283}\approx0.880797078assets/equations/b29d72f7254a4877.svg
• p_2=\frac{1}{1.503214724}=0.665240956assets/equations/b2c010d4d592ad18.svg
• e^{-1}=0.367879441,\quad e^0=1,\quad e^{-2}=0.135335283assets/equations/b2c277f81e3d9881.svg
• z_{h_2} = (0.3 \times 0.5) + (0.5 \times 0.2) + (0.7 \times 0.1) + (0.2) = 0.52assets/equations/b2e10076eac3269f.svg
• N_{\mathrm{out}}=\left\lfloor\frac{32+2(0)-5}{1}\right\rfloor+1=28assets/equations/b2fe45e42e65423c.svg
• \frac{(w+\Delta w)^2-w^2}{\Delta w}=2w+\Delta wassets/equations/b33822670f9d55b8.svg
• z_2=1+3-0.2=3.8assets/equations/b340e43d0e100bc4.svg
• C=0.1+0.231058579=0.331058579assets/equations/b3473959824ebef9.svg
• z_{h_1} = (0.1 \times 0.2) + (0.2 \times 0.4) + (0.1) = 0.2assets/equations/b34f23de35a7cf0c.svg
• \text{Sum}=C_1\approx 0.056158765assets/equations/b369b4600b5c9b60.svg
• \text{Sum}=C_1\approx -0.22assets/equations/b39852d5faaf11fb.svg
• \widetilde C=\tanh(z_c)assets/equations/b3be31d6f9a03979.svg
• g_t=J\prime(\theta_{t-1}),\quad \theta_t=\theta_{t-1}-\eta g_tassets/equations/b3d142fa7d569702.svg
• g_{4,2}^{(1)}=(-0.004282747)\times(0.3)\approx -0.001284824assets/equations/b3de47e4aca426e5.svg
• f\prime(0)=1-(0.000000000)^2\approx1.000000000assets/equations/b3f22ac5fb036f65.svg
• (1-z_t)h_{t-1}=(1-0.25)(0.4)=0.3assets/equations/b3f54d1f31eaa225.svg
• 6+2(0)-3=3assets/equations/b4027b9bc8f6fe22.svg
• \frac{\partial L}{\partial W_{ji}}=\delta_ja_iassets/equations/b40acc83c437110d.svg
• g_{1,1}^{(2)}=(0.013687964)\times(0.760423002)\approx 0.010408643assets/equations/b41423f571bf9199.svg
• 0.6(0)=0assets/equations/b4145d1d293ebdf3.svg
• g_{\mathrm{total}}=-0.4+0.04=-0.36assets/equations/b422d0258ed7d40f.svg
• n_{\mathrm{flat}}=4\times4\times4=64assets/equations/b42325b09a6929a4.svg
• w_{\mathrm{new}}=0.5-0.1(-0.105754186)=0.510575419assets/equations/b4289d1b3f149e0a.svg
• P_8=600+5=605assets/equations/b4289ff8bd02992b.svg
• \hat y_2=1.1(2)=2.2assets/equations/b434bd09ecf3d6fa.svg
• a=\frac1{8.389056099}\approx0.119202922assets/equations/b44984dd703daaf9.svg
• L=\frac12(a-y)^2assets/equations/b45e01da7c114be3.svg
• \text{weights}=1\times1\times8\times16=128assets/equations/b4a1699aef64acc8.svg
• 0.5(2)=1assets/equations/b4b945ff5abf26e2.svg
• o_{2}=\frac{1}{1+e^{-(0.482901756)}}=0.618432848assets/equations/b4bdac71b195a0cc.svg
• v_{j,\mathrm{new}}=\beta v_{j,\mathrm{old}}-\eta g_j,\qquad w_{j,\mathrm{new}}=w_j+v_{j,\mathrm{new}}assets/equations/b4d3d4646fcd808d.svg
• \widehat v_1=\frac{0.001}{1-0.999^{1}}=1assets/equations/b4d865fbfcf2b328.svg
• R_2=(0)\times(2)+(1)\times(0)+(2)\times(-2)=-4assets/equations/b50d1e82d7e6e8b7.svg
• g_w=\frac{\partial L}{\partial w}=\delta_z\frac{\partial z}{\partial w}=(-2)(2)=-4assets/equations/b51cdfbc8d705d9a.svg
• w_{2,\mathrm{new}}=-0.2-0.1(0.26)=-0.226assets/equations/b559be4dc46e47e0.svg
• \frac{dL}{dz}=(-1.25)(0.16)=-0.2assets/equations/b564b0f8f85410b8.svg
• h=0.75(0.4)+0.25(0.291312612)assets/equations/b56e2871d1cdeae5.svg
• R_2=(1)\times(0)+(0)\times(1)=0assets/equations/b595ad69d60edb10.svg
• r h_{\mathrm{previous}}=0.5(0.4)=0.2assets/equations/b5a08a736e0f6884.svg
• h_{2} = \max(0,0.48) = 0.48assets/equations/b5c17f7904703c18.svg
• \frac{d}{dx}(cx^n)=cnx^{n-1}assets/equations/b5c78e62c5b2a236.svg
• g_{2,2}^{(1)}=\frac{\partial L}{\partial W^{(1)}_{2,2}}=\delta_{h_2}x_2assets/equations/b5ce6325e440d113.svg
• W^{(1)}_{1,3,\mathrm{new}}=-0.5-0.9(0.008724562)\approx -0.507852106assets/equations/b5e18accd983b7d3.svg
• g_2=(0-2)\times2=-4assets/equations/b5e54f46939905d8.svg
• e^{-\ln2}=\frac1{e^{\ln2}}=\frac12=0.5assets/equations/b612f5289fa6fa44.svg
• J(2)=3(2)^2+5=17assets/equations/b64892fad56f04e4.svg
• W^{(1)}_{1,1,\mathrm{new}}=0.2-1(0.000513422)\approx 0.199486578assets/equations/b64d0783616b88f9.svg
• \delta_{h_2} = 0.070517544 \times 0.247985885 = 0.017487356assets/equations/b6511e93ffa2060f.svg
• g_w=\frac{\partial L}{\partial w}=\delta_z\frac{\partial z}{\partial w}=(-2)(2)=-4assets/equations/b65d42a6fa860d4a.svg
• h_2=\tanh(1.092423431)=0.797760954assets/equations/b65e6ee603919e32.svg
• Y_{1,2}=(20)+(26)+(13)=59assets/equations/b674e79df8f34858.svg
• z_j=\sum_iw_{ji}a_i+b_j,\quad a_j=f(z_j),\quad L=\frac12\sum_k(o_k-y_k)^2assets/equations/b6818e3243c5ede2.svg
• Y_{1,2}=(2)+(-3)=-1assets/equations/b6828ab91aa100a9.svg
• e_3=1-2=-1assets/equations/b685cf877781a273.svg
• \widehat m_2=\frac{0.08}{1-0.9^{2}}=0.421052632assets/equations/b68cf26945a449a6.svg
• L_{\mathrm{half}}=\frac{1}{2}=0.5assets/equations/b6b038fb864ecaa7.svg
• a_{\mathrm{train}}=m\odot a\quad\text{and}\quad a_{\mathrm{test}}=q a\quad\text{in classical dropout}assets/equations/b6b21f6d03d2989f.svg
• C_1=(0.4)\times(-0.55)\approx -0.22assets/equations/b6b2e13098fb66e7.svg
• [1-2,\;2-2,\;0-2]=[-1,0,-2]assets/equations/b6bd92da9301ed16.svg
• \hat m_t=\frac{m_t}{1-\beta_1^t}assets/equations/b6ee8d213ad96014.svg
• \left\lfloor\frac{48-2}{2}\right\rfloor+1assets/equations/b6ef920332c6684a.svg
• \mathrm{MACs}_{5\times5}=28(28)(32)(25)(16)=10035200assets/equations/b6f81bb572b38322.svg
• W^{(1)}_{3,1,\mathrm{new}}=0.3-0.5(0.005197622)\approx 0.297401189assets/equations/b709fa905c40f2d3.svg
• L\approx-(-0.223143551)=0.223143551assets/equations/b71a143e9ddc291e.svg
• R_2=(2)\times(2)+(1)\times(0)+(2)\times(-2)=0assets/equations/b71bdd3aed98c346.svg
• N_{\mathrm{weights}}=5\times5\times96\times256=614{,}400assets/equations/b720c78ffc948913.svg
• \frac{\partial z}{\partial w}=x,\qquad\frac{\partial z}{\partial x}=w,\qquad\frac{\partial z}{\partial b}=1assets/equations/b7574a6d14a06a47.svg
• J(\theta)=\frac1N\sum_{i=1}^{N}L_i(\theta)+\lambda\Omega(\theta)assets/equations/b7857d96c4edb1c7.svg
• h_{1} = \max(0,0.5) = 0.5assets/equations/b78961f402336b21.svg
• b_{\mathrm{new}}=0-0.1(-0.052877093)=0.005287709assets/equations/b7af577aaec3d7cf.svg
• \delta_{o_1} = 1 \times (0.590378257-1) \times 0.241831771 = -0.099059551assets/equations/b7b0310d2c3be2ab.svg
• N_{out}=\left\lfloor\frac{N+2P-K}{S}\right\rfloor+1assets/equations/b7c680fad37aede4.svg
• g_{1,3}^{(2)}=(-0.093597941)\times(0.63645254)\approx -0.059570648assets/equations/b7d94cc71bc1bf29.svg
• e^{-\ln 2}assets/equations/b7f5712356b73942.svg
• C_1=(-0.2)\times(-0.131169078)\approx 0.026233816assets/equations/b7f5cea4501981c5.svg
• n_{\mathrm{flat}}=5\times5\times16=400assets/equations/b7fdfd1b9a608954.svg
• b_{h_2,\mathrm{new}}=0.35-0.5(0.009954255)=0.345022872assets/equations/b81e2513786c90a9.svg
• \begin{aligned}z_{o_1}={}&0.5(0.710949503)+0.7(0.668187772)\\&-0.6(0.63645254)+0.2(0.645656306)+0.2\\={}&0.770465929\end{aligned}assets/equations/b82d291c09c8255f.svg
• e^{2}\approx7.389056099assets/equations/b83d9532e6445b5a.svg
• w^Tx=(2)+(2)+(-3)=1assets/equations/b85e0c042b7b8396.svg
• 2(0.1)(2)=0.4assets/equations/b8811e22460d0c91.svg
• \hat y=wx+bassets/equations/b896c82c08881db8.svg
• C=0.2+0.63=0.83assets/equations/b89883a392a4e3ec.svg
• P_{\mathrm{projection}}=(K^2C_{\mathrm{in}}+1)C_{\mathrm{out}}assets/equations/b8afb47cd18fe03e.svg
• g_2=-0.215-(1)=-1.215assets/equations/b8b2f5a400acaace.svg
• H(p)\approx0.215761554+0.346573590=0.562335145assets/equations/b8c34bbe332a549f.svg
• \text{biases}=2,\qquad P_{\mathrm{conv}}=54+2=56assets/equations/b8f0882262171dff.svg
• \frac{d}{dz}e^{-z}=-e^{-z},\qquad\frac{d}{dp}[-\ln p]=-\frac1passets/equations/b8f3afdc22e0eaaf.svg
• \delta_{h_4} = -0.018719588 \times 0.22878424 = -0.004282747assets/equations/b8fde02c1bff3e4f.svg
• p_i=\frac{e^{z_i-c}}{\sum_j e^{z_j-c}},\qquad c=\max_jz_jassets/equations/b901ae0600ed0d36.svg
• z=\frac{4}{2}=2assets/equations/b913089938e80c1f.svg
• i\widetilde C=0.2\times(-0.5)=-0.1assets/equations/b92f4b1761bc1ff0.svg
• z=\begin{bmatrix}4.5\\0.5\end{bmatrix},\qquad a=\begin{bmatrix}4.5\\0.5\end{bmatrix}assets/equations/b93dfe6da27f9705.svg
• z_h=(-0.4)\times2+0.1=-0.8+0.1=-0.7assets/equations/b94d4768d9ba8e6f.svg
• N_{\mathrm{weights}}=36{,}864\times120=4{,}423{,}680assets/equations/b963fc1e5d67938a.svg
• C_1=(0.25)\times(-0.099059551)\approx -0.024764888assets/equations/b990515cad5dba89.svg
• C_{22}=3(0)+4(3)=0+12=12assets/equations/b9ab897087e15f9b.svg
• \delta_1=(W_2^T\delta_2)\odot f_1'(z_1)assets/equations/b9b8719d9f5e5e0f.svg
• C_1=(0.24)\times(0.013687964)\approx 0.003285111assets/equations/b9be444cf3dd3958.svg
• P_{\mathrm{dense}}=mn+n=(m+1)nassets/equations/b9cd556f84e19c2e.svg
• z_{h_2}=1+0-1.5=-0.5\Rightarrow h_2=0assets/equations/b9f2126aba5c12cc.svg
• \frac{\partial L}{\partial b_j}=\delta_jassets/equations/b9f5adb924b1f09e.svg
• N_{\mathrm{out}}=\left\lfloor\frac{61+2(1)-5}{1}\right\rfloor+1=59assets/equations/b9ff7e28d1cec2ae.svg
• \theta_{look}=\theta+\beta v,\quad v_{new}=\beta v-\eta g(\theta_{look}),\quad\theta_{new}=\theta+v_{new}assets/equations/ba02f58e5e55b487.svg
• -0.75\ln(0.5)\approx0.519860385assets/equations/ba03d4ba50c7c4e4.svg
• 0.9(1)=0.9assets/equations/ba3003dba6d22d16.svg
• a=0.2(1)+0.5(-0.1)+0=0.15assets/equations/ba304bd37c97d1fe.svg
• \frac{\partial L}{\partial e}=\frac12(2e)=eassets/equations/ba371707e98b2bc9.svg
• \theta_{\mathrm{look},2}=0.9+0.9(-0.1)=0.81assets/equations/ba38208dd4ceff26.svg
• 11+13+7+4=35assets/equations/ba50b6f7e4060a25.svg
• h_t=ux_t+vh_{t-1}+b,\qquad L=\frac12(h_2-y)^2assets/equations/ba5840cf8abc2eb8.svg
• \delta_t=v\delta_{t+1}assets/equations/ba589b262a27bc88.svg
• \operatorname{LeakyReLU}(z)=\begin{cases}z&z\ge0\\az&z<0\end{cases},\qquad a=0.01\ \text{unless specified}assets/equations/ba63451671574a0f.svg
• \mu=\frac{1}{n}\sum_i x_i,\quad v=\frac{1}{n}\sum_i(x_i-\mu)^2,\quad s=\sqrt{v}assets/equations/ba649f65f75f4057.svg
• Y_{2,3}=(1)+(1)=2assets/equations/ba68e9967c8a5ec6.svg
• (2)(-0.1)=-0.2assets/equations/ba87aedd127c080a.svg
• h_{1,\mathrm{new}}=0.325(1)+0.51(0)+0.075=0.4assets/equations/bac6924667182d4d.svg
• C=f\odot C_{\mathrm{old}}+i\odot\widetilde Cassets/equations/baccbadbb58af40d.svg
• -z=2assets/equations/baed17557936141e.svg
• P_A=(1\times1\times8+1)\times4=9\times4=36assets/equations/baf7f3bc0580d3c4.svg
• W^{(l)}\in\mathbb R^{m\times n},\qquad b^{(l)},z^{(l)},a^{(l)}\in\mathbb R^{m\times1}assets/equations/bb1ec5bee5b7117c.svg
• f\prime(-2)=1-(-0.964027580)^2\approx0.070650825assets/equations/bb2d4ce554c14198.svg
• 0.4(0.3)=0.12assets/equations/bb48ef38d6d22273.svg
• a=\frac1{2.000000000}\approx0.500000000assets/equations/bb9808b4290f77c2.svg
• G_1=0+(-1)^2=1assets/equations/bba453205f160315.svg
• h_2=0.7\tanh(0.36)=0.7(0.345214034)=0.241649824assets/equations/bbded31a747f8c6f.svg
• R_1=(1)\times(1)+(0)\times(0)+(1)\times(-1)=0assets/equations/bbe676b3e02a351a.svg
• h_1=\operatorname{step}(x_1+x_2-0.5)assets/equations/bbffb2524956d0e5.svg
• \frac{\partial L}{\partial b}=\delta_1+\delta_2=-0.25-0.5=-0.75assets/equations/bc0231e316221e07.svg
• -\ln(0.1)=-(-2.302585093)=2.302585093assets/equations/bc081502a0ccb2a9.svg
• \frac{\partial L}{\partial p}=-\frac yp+\frac{1-y}{1-p}=\frac{p-y}{p(1-p)}assets/equations/bc392cea6dae6dda.svg
• g_{\mathrm{clipped}}=g\min\left(1,\frac{\tau}{\lVert g\rVert_2}\right)assets/equations/bc3f87fdb9a197b4.svg
• g_{1,3}^{(2)}=(-0.093597941)\times(0.63645254)\approx -0.059570648assets/equations/bc44ca9541b76f51.svg
• g_2=0.9assets/equations/bc4532aff9a9a2da.svg
• b_{\mathrm{new}}=0-0.1(-0.052877093)=0.005287709assets/equations/bc4fa856481c3256.svg
• h_1=\tanh(0.7)\approx0.604367777assets/equations/bc5f2bb1f8425e2b.svg
• h_1=0.7\tanh(0.4)=0.7(0.379948962)=0.265964274assets/equations/bc61a66b7daaff60.svg
• h_1=\tanh(0.5)\approx0.462117157assets/equations/bc76aa14d3af5acb.svg
• \tanh\prime(z)=1-\tanh^2(z)assets/equations/bc7dbcac123b9229.svg
• Y_{2,3}=(3)+(-1)=2assets/equations/bc81876afa80a85e.svg
• R_2=(2)\times(0)+(1)\times(1)=1assets/equations/bd0efbee152c1e5c.svg
• \theta_2=0.9-\frac{0.1\times(0.421052632)}{\sqrt{2.492462312}}=0.873330057assets/equations/bd194d13466767bb.svg
• y_{avg}=\frac1{|W|}\sum_{(i,j)\in W}x_{ij}assets/equations/bd2034a9c3d7a387.svg
• \frac{\partial L}{\partial v}=(-0.25)(0)+(-0.5)(0.2)=-0.1assets/equations/bd2ee36e0bc151c4.svg
• N_{\mathrm{pool}}=\left\lfloor\frac{N+2P-K}{S}\right\rfloor+1assets/equations/bd33b42a8814468f.svg
• H_{\mathrm{padded}}=5+0+1=6assets/equations/bd37f3052feb0554.svg
• C=0.1+0.231058579=0.331058579assets/equations/bd4c66227139045d.svg
• \sum_i w_ix_i\ge\theta\quad\Longleftrightarrow\quad\sum_i w_ix_i-\theta\ge0assets/equations/bd6e61f8065942de.svg
• P_{conv}=(K_hK_wC_{in}+1)C_{out}assets/equations/bd7e05cdd2c763eb.svg
• P_C=0+(1\times1\times8+1)\times2=18assets/equations/bdab800ad8bc0347.svg
• \delta_{h_2} = 0.041370323 \times 0.240613417 = 0.009954255assets/equations/bdbc44ac317050d2.svg
• W^{(1)}_{1,4,\mathrm{new}}=-0.2-0.5(0.003846881)\approx -0.201923441assets/equations/bdc8b7088a5ab57a.svg
• \mathrm{MSE}=\frac1n\sum_{i=1}^{n}e_i^2=\frac{\mathrm{SSE}}nassets/equations/bde24be7fa074ec1.svg
• L_{\mathrm{MSE}}=\frac{\mathrm{SSE}}nassets/equations/bdee7885d635e78c.svg
• N_{\mathrm{weights}}=3\times3\times384\times256=884{,}736assets/equations/be0b63ec1f7ffb8c.svg
• 73728-22528=51200assets/equations/be3a4d13f46b02ba.svg
• h_t=o_t\odot\tanh(c_t)assets/equations/be4b02b5e68c7633.svg
• P_10=840+10=850assets/equations/be85fac4796ebc34.svg
• P_5=102{,}400+64=102{,}464assets/equations/be86652891fb9307.svg
• \delta_{o_j}=\frac{\partial L}{\partial o_j}f_o\prime(z_{o_j})assets/equations/bea7700a623139b7.svg
• \sqrt{(-4)^2}=\sqrt{16}=4assets/equations/bed469d925782c9b.svg
• \text{Sum}=C_1\approx -0.065518559assets/equations/bed646f2d5a5b17a.svg
• \widehat m_2=\frac{0.11}{1-0.9^{2}}=0.578947368assets/equations/bee1e67cf69a8631.svg
• e^{2}-e^{-2}\approx7.389056099-0.135335283=7.253720816assets/equations/beea5fae2dfeb83f.svg
• C_{12}=1(0)+2(3)=0+6=6assets/equations/bf3f58226647b592.svg
• n_{\mathrm{flat}}=HWC,\qquad P_{\mathrm{dense}}=(n_{\mathrm{in}}+1)n_{\mathrm{out}}assets/equations/bf6a8c697b8e11ab.svg
• w^Tx=(2)+(2)+(-3)=1assets/equations/bf7dc79763479e35.svg
• U_{\mathrm{drop}}=E\left\lfloor\frac NB\right\rfloorassets/equations/bf7e65981e0892c7.svg
• \frac{\partial L}{\partial h_j}=W_{2,1j}\delta_{o_1}+W_{2,2j}\delta_{o_2}assets/equations/bf829204c0f41f49.svg
• 0.1(0.9)=0.09assets/equations/bf93339d0945c688.svg
• W^{(1)}_{2,2,\mathrm{new}}=0.5-0.5(-0.004869072)\approx 0.502434536assets/equations/bfabbdcd726cbd9c.svg
• \delta_{h_2} = -0.12135 \times 1 = -0.12135assets/equations/bff8528433d7519f.svg
• q_k=\frac{e^{z_k}}{\sum_{j=1}^{3}e^{z_j}},\qquad L=-\sum_{k=1}^{3}y_k\ln q_kassets/equations/c00adcf16ff90eab.svg
• h_{2}=\frac{1}{1+e^{-(1.28)}}=0.782449776assets/equations/c014b8bfa585b6d4.svg
• W^{(1)}_{1,1,\mathrm{new}}=0.2-1(0.000513422)\approx 0.199486578assets/equations/c0273447c0bc7aba.svg
• \left\lfloor\frac{N+2P-K}{S}\right\rfloor+1assets/equations/c0351298d1be04e6.svg
• 32+4=36assets/equations/c052c15ffd718757.svg
• \begin{aligned}20\%&=\frac{20}{100}=0.2\\50\%&=\frac{50}{100}=0.5\end{aligned}assets/equations/c062ee6009f1f875.svg
• g_{2,4}^{(1)}=(-0.014526308)\times(-0.4)\approx 0.005810523assets/equations/c06fd2203e85c0d9.svg
• rh_{\mathrm{old}}=0.5(-0.2)=-0.1assets/equations/c07f11824db74c75.svg
• \|g\|_2=\sqrt{\sum_i g_i^2}assets/equations/c087afaaeb69ebf5.svg
• \widetilde C_t=\tanh(W_c[h_{t-1},x_t]+b_c)assets/equations/c08c8ddc953628c6.svg
• P_8=4{,}423{,}680+120=4{,}423{,}800assets/equations/c09e1443401b8c28.svg
• \frac{7-5}{2}=1assets/equations/c0a2851f2b573cc2.svg
• C_1=(-0.3)\times(-0.131169078)\approx 0.039350723assets/equations/c0b7949b10b38e6f.svg
• b_{o_1,\mathrm{new}}=0-0.2(-0.121439988)=0.024287998assets/equations/c0bf69b79bc6be7c.svg
• P_{\mathrm{dense}}=(n_{\mathrm{in}}+1)n_{\mathrm{out}}assets/equations/c0e2ff46bbfc4867.svg
• N_{\mathrm{out}}=\left\lfloor\frac{28}{2}\right\rfloor+1=14+1=15assets/equations/c0ed9983d10f2a99.svg
• g_{2,4}^{(1)}=\frac{\partial L}{\partial W^{(1)}_{2,4}}=\delta_{h_2}x_4assets/equations/c111f40d015517f1.svg
• Y_{1,1}=(0.2)+(0.2)+(0.2)=0.6assets/equations/c15a1021698c03e8.svg
• z_{o_1} = (-0.3 \times 0.331812228) + (-0.2 \times 0.524979187) + (0.1) = -0.104539506assets/equations/c16074919b105b06.svg
• e^{-1}=0.367879441,\quad e^0=1,\quad e^{-2}=0.135335283assets/equations/c1751b34781aca62.svg
• R_2=(2)\times(0)+(3)\times(1)=3assets/equations/c18dea868e99c1c8.svg
• h=0.3+0.072828153=0.372828153assets/equations/c192a426278a53e6.svg
• g_t=J'(\theta_{t-1})assets/equations/c19372684b38c7e4.svg
• N_{\mathrm{weights}}=5\times5\times64\times64=102{,}400assets/equations/c1baccc90afa895b.svg
• z_{h_1} = (0.2 \times 0.25) + (0.15 \times 0.7) + (1) = 1.155assets/equations/c1e1a5f6adaaf15c.svg
• h_{1}=\frac{1}{1+e^{-(1.4)}}=0.802183889assets/equations/c20e39027aee1353.svg
• N_{\mathrm{out}}=\left\lfloor\frac{10-2}{2}\right\rfloor+1=5assets/equations/c2393768f5c7cc6f.svg
• g_{2,3}^{(1)}=(0.006542085)\times(1)\approx 0.006542085assets/equations/c2728e830590acf7.svg
• P_{dense}=(N_{in}+1)N_{out}assets/equations/c28314915c655018.svg
• b_{\mathrm{new}}=b-\eta g_bassets/equations/c2843cb0410cd77b.svg
• W^{(2)}_{1,2,\mathrm{new}}=-0.1-0.2(-0.066170086)\approx -0.086765983assets/equations/c29bb818f965c973.svg
• w_{2,\mathrm{new}}=0.8-0.1(0.088)=0.7912assets/equations/c2a2db66205fe592.svg
• \theta_2=0.9-\frac{0.1\times(1)}{\sqrt{5}}=0.85527864assets/equations/c2a79e598e1b306b.svg
• R_3=(1)\times(1)+(1)\times(0)+(0)\times(-1)=1assets/equations/c2dca2b82f9aac21.svg
• \frac{\partial L}{\partial o_2}=o_2-y_2=-1.1-0=-1.1assets/equations/c2e5f710a0148c3f.svg
• P_7=884{,}736+256=884{,}992assets/equations/c304d00633ff2322.svg
• \sigma(z)=\frac{1}{1+e^{-z}}assets/equations/c3083d9958c3932d.svg
• R_1=(1)\times(1)+(3)\times(0)=1assets/equations/c318e658016b1fbd.svg
• -0.75\ln(0.5)\approx0.519860385assets/equations/c34f15079f0e23f0.svg
• y_{XOR}=0assets/equations/c3624761b5a00605.svg
• z=wx+b,\quad p=\frac1{1+e^{-z}},\quad L=\frac12(p-y)^2assets/equations/c39b7cba136e59b8.svg
• J_{L2}=L_{\mathrm{data}}+\frac\lambda2\sum_iw_i^2assets/equations/c3cf9af2ddfa197c.svg
• W^{(1)}_{1,1,\mathrm{new}}=0.5-0.5(-0.004869072)\approx 0.502434536assets/equations/c3d90c909dab8209.svg
• b_{\mathrm{new}}=0.1-0.1(-0.3)=0.1-(-0.03)=0.13assets/equations/c3e5531ea9063ace.svg
• \sigma'(z)=\frac{1}{1+e^{-z}}\frac{e^{-z}}{1+e^{-z}}=\sigma(z)[1-\sigma(z)]assets/equations/c3ea50ce5747d8ce.svg
• \widehat s_t=\frac{s_t}{1-\beta_2^t}assets/equations/c3ebf23cb24a812d.svg
• c=\max_j z_jassets/equations/c40885bb51eddb21.svg
• g_{1,2}^{(2)}=(0.138498562)\times(0.596884378)\approx 0.082667628assets/equations/c40f5279cb6e8a57.svg
• C_{22}=3(0)+4(3)=0+12=12assets/equations/c42e9c75451d67af.svg
• \sigma=\sqrt{0.000000000}\approx0.000000000assets/equations/c43ce89c0112ae99.svg
• \widehat m_t=\frac{m_t}{1-\beta_1^t},\quad\widehat v_t=\frac{v_t}{1-\beta_2^t},\quad\theta_t=\theta_{t-1}-\eta\frac{\widehat m_t}{\sqrt{\widehat v_t}+\epsilon}assets/equations/c443fc0367c152b1.svg
• z_{o_1} = (0.25 \times 0.731058579) + (0.25 \times 0.731058579) + (0) = 0.365529289assets/equations/c44792db3abbdcd0.svg
• \frac{\partial z}{\partial w}=xassets/equations/c448c09e6087f1d5.svg
• \mathrm{ReLU}(2)=\max(0,2)=2assets/equations/c454b416d112784d.svg
• p_i=\frac{e^{z_i-c}}{\sum_j e^{z_j-c}},\qquad c=\max_j z_jassets/equations/c492ec558e0b1e23.svg
• \sigma(-2)=\frac1{8.389056099}\approx0.119202922assets/equations/c492fd878121169c.svg
• \delta_t=(0.1)(0.96)=0.096assets/equations/c494eb0b53453545.svg
• C_{21}=3(2)+4(1)=6+4=10assets/equations/c49b369448ab5419.svg
• \frac{\partial L}{\partial w_{ji}}=\delta_j a_iassets/equations/c4e0f7ded9389700.svg
• \begin{bmatrix}1.75 & 2.75\\4.25 & 3.5\end{bmatrix}assets/equations/c4e510493e329d60.svg
• 128+8=136assets/equations/c4ed13ec1c4089a9.svg
• R=\frac{0.1}{2}(0.4)^2=0.05(0.16)=0.008assets/equations/c4fc429067a3f156.svg
• f\prime(-2)=1-(-0.964027580)^2\approx0.070650825assets/equations/c4ff92106fa31931.svg
• \delta_{h_2} = 0.026233816 \times 0.24937604 = 0.006542085assets/equations/c52b54d99a7b69d0.svg
• N_{\mathrm{weights}}=5\times5\times64\times128=204{,}800assets/equations/c52ed6d2bdad242d.svg
• W^{(1)}_{2,2,\mathrm{new}}=0.1-0.9(0)\approx 0.1assets/equations/c531dfb193e72d45.svg
• h=0.5(2)+0.1=1.1assets/equations/c54d1626a0bf10fc.svg
• C_t=0.1+0.5\tanh(0.5)\approx0.331058579assets/equations/c56ba22ea129ddab.svg
• h_t=\phi(W_{xh}x_t+W_{hh}h_{t-1}+b_h)assets/equations/c5a9455cdb766031.svg
• 0.79^2=0.6241assets/equations/c5b6ff4c2963f39a.svg
• N_{\mathrm{out}}=\left\lfloor\frac{105-2}{2}\right\rfloor+1=52assets/equations/c5c0ee2fb6dc698d.svg
• g_{2,1}^{(1)}=(-0.12135)\times(2)\approx -0.2427assets/equations/c5d0a80d06a8382c.svg
• z_{o_1} = (0.14 \times 0.85) + (0.15 \times 0.48) + (0) = 0.191assets/equations/c5d0ce28abb02731.svg
• (-1)(3)=-3assets/equations/c5d478557eb834e1.svg
• \widetilde h=\tanh(0.3)\approx0.291312612assets/equations/c5d5bc478f8a3566.svg
• C_2=(0.21)\times(0.082533822)\approx 0.017332103assets/equations/c5de7487aca6930f.svg
• w_1^{\mathrm{new}}=0.55+(-0.03)=0.52assets/equations/c5de9a5a93170dac.svg
• 1.000000000+1\approx2.000000000assets/equations/c5e16c3256057084.svg
• \frac{\eta\hat m_2}{\sqrt{\hat s_2}}\approx\frac{0.1(1.473684211)}{\sqrt{2.499249625}}\approx0.093217964assets/equations/c636739e003f73e5.svg
• z_{h_1}=1+1-0.5=1.5\Rightarrow h_1=1assets/equations/c6382db6dca562bf.svg
• g_{3,1}^{(1)}=\frac{\partial L}{\partial W^{(1)}_{3,1}}=\delta_{h_3}x_1assets/equations/c647740724db2342.svg
• \sigma(z)=\frac1{1+e^{-z}},\quad C=fC_{\mathrm{previous}}+i\widetilde C,\quad h=o\tanh(C)assets/equations/c660725ff6ab02eb.svg
• \text{Sum}=C_1\approx 0.039350723assets/equations/c677dcae787d564f.svg
• m_t=\beta_1m_{t-1}+(1-\beta_1)g_tassets/equations/c684b3ce4733fb3b.svg
• P_{\mathrm{total}}=\sum_k P_kassets/equations/c6958d2f8ded49d5.svg
• a\odot b=\begin{bmatrix}8\\15\end{bmatrix}assets/equations/c69bdb17e45bd1d3.svg
• L=-\ln(0.665240956)=0.407605964assets/equations/c6ddf889d37419c3.svg
• \sigma'(0)=0.5(1-0.5)=0.5\times0.5=0.25assets/equations/c6f6bc13772ea829.svg
• N_{\mathrm{out}}=\lfloor7.5\rfloor+1=7+1=8assets/equations/c73f4efe4e78e925.svg
• N=20assets/equations/c758e056cb2f1688.svg
• W^{(1)}_{1,1,\mathrm{new}}=0.8-0.5(-0.003846881)\approx 0.801923441assets/equations/c75e592a0b5226d2.svg
• R_1=(9)\times(1)+(5)\times(1)+(6)\times(1)=20assets/equations/c760b3950eb840ed.svg
• 0.018315639+1\approx1.018315639assets/equations/c7713b25655beb72.svg
• N_{\mathrm{out}}=\left\lfloor\frac{59-2}{2}\right\rfloor+1=29assets/equations/c7e04e5a610fd348.svg
• W^{(1)}_{2,2,\mathrm{new}}=-0.1-0.1(-0.55)\approx -0.045assets/equations/c7fab0773a9a18b8.svg
• w_{\mathrm{new}}=0.2-0.1(-0.9)=0.2-(-0.09)=0.29assets/equations/c80322ecffe9a16b.svg
• \frac{\partial L}{\partial W_{ji}}=\delta_ja_i,\qquad\frac{\partial L}{\partial b_j}=\delta_jassets/equations/c82c3235c49eee6f.svg
• 1+7.389056099=8.389056099assets/equations/c833999cba87b62d.svg
• R_3=(3)\times(1)+(2)\times(1)+(6)\times(1)=11assets/equations/c85bf61f7da39979.svg
• h=0.8(0.680476006)\approx0.544380805assets/equations/c85eca49bc37b1e7.svg
• z=3(2)+1=6+1=7assets/equations/c86ffb173c8d136d.svg
• W^{(1)}_{2,1,\mathrm{new}}=0.3-0.1(-0.275)\approx 0.3275assets/equations/c874099b770b95c2.svg
• \begin{bmatrix}2 & 4\\6 & 8\end{bmatrix}assets/equations/c890243a56df9f91.svg
• fC_{\mathrm{old}}=(0.731058579)(-0.5)\approx-0.365529289assets/equations/c8908718c1a2b52e.svg
• e^{-1}\approx0.367879441assets/equations/c8b081af8baa5f1b.svg
• f_tC_{t-1}=0.4(0.5)=0.2assets/equations/c8c446684892171e.svg
• \ln(0.75)\approx-0.287682072,\qquad\ln(0.25)\approx-1.386294361assets/equations/c8cc7ed1293c1673.svg
• R_1=(2)\times(1)+(1)\times(0)=2assets/equations/c8ce43d27f6afa3e.svg
• z=0+(0)=0assets/equations/c8d9f0d00314027b.svg
• z_{h_1} = (0.1 \times 0.2) + (0.2 \times 0.4) + (0.1) = 0.2assets/equations/c91412b204993ed6.svg
• \hat y=wx,\qquad L(w)=\frac12(wx-y)^2,\qquad w_t=w_{t-1}-\eta g_tassets/equations/c9459723553604be.svg
• b_{h_1,\mathrm{new}}=0-0.5(-0.004869072)=0.002434536assets/equations/c94badc259c5a62b.svg
• \delta=(-0.268941421)\times0.196611933=-0.052877093assets/equations/c963a470be9b7811.svg
• N_{\mathrm{weights}}=9\times9\times3\times32=7{,}776assets/equations/c97378e50b8b5ac8.svg
• g_2=2.1(0.6241)=1.31061assets/equations/c986b4d77e3de9d0.svg
• W^{(2)}_{1,1,\mathrm{new}}=0.4-0.1(-0.275)\approx 0.4275assets/equations/c98fe26955db1400.svg
• g_{2,1}^{(1)}=(-0.014526308)\times(0.4)\approx -0.005810523assets/equations/c99ccbe66229f64d.svg
• N_{\mathrm{out}}=\left\lfloor\frac{5}{1}\right\rfloor+1=5+1=6assets/equations/c9e9b01d59dde654.svg
• a=0.2(1)+0.5(0.2)+0=0.2+0.1=0.3assets/equations/c9ea266a45504d10.svg
• \delta_{h_2} = -0.024764888 \times 0.196611933 = -0.004869072assets/equations/c9f11eb5cc7cb0d3.svg
• \frac{\partial L}{\partial W_{hh}}=\sum_t\delta_t h_{t-1}^T,\quad\frac{\partial L}{\partial W_{xh}}=\sum_t\delta_t x_t^Tassets/equations/c9fa6409d743c860.svg
• P_7=10{,}080+84=10{,}164assets/equations/c9fa89f780aa10aa.svg
• s=\sqrt{v}assets/equations/c9fd8a46ad5f2e60.svg
• h_2=0.2(2)+0.5(0.2)+0=0.5assets/equations/ca0b29f546ae57d0.svg
• P(A\mid B)=\frac{P(A\cap B)}{P(B)}=\frac{P(B\mid A)P(A)}{P(B)}assets/equations/ca109f70fdd2c998.svg
• \begin{bmatrix}0 & 0 & 0\\0 & 167 & 166\\0 & 164 & 165\end{bmatrix}assets/equations/ca174e19344d6ef9.svg
• -0.25\ln(0.5)\approx0.173286795assets/equations/ca2ff9da024bd22a.svg
• g_{2,2}^{(2)}=(0.145933863)\times(0.544878892)\approx 0.079516282assets/equations/ca3eeac86363862f.svg
• P_11=16{,}777{,}216+4{,}096=16{,}781{,}312assets/equations/ca42e0122e881f05.svg
• N_{\mathrm{out}}=\left\lfloor\frac{27-3}{2}\right\rfloor+1=13assets/equations/ca522f15a3632877.svg
• H(p,q)=-\sum_{i=1}^{K}p_i\log q_iassets/equations/ca68a6ddd1ac17cc.svg
• \delta=(-0.268941421)\times0.196611933=-0.052877093assets/equations/ca75a0d20742f652.svg
• \frac{\partial L}{\partial o}=o-y,\quad \frac{\partial o}{\partial w}=x,\quad \frac{\partial o}{\partial b}=1assets/equations/ca7badc8e4620a3c.svg
• R_1=(2)\times(1)+(1)\times(0)=2assets/equations/ca82f5af7cbb2e71.svg
• \begin{bmatrix}0 & 4\\0 & 0\end{bmatrix}assets/equations/ca987c6e03219656.svg
• a_{\mathrm{candidate}}=W_hx_t+U_h(r_th_{t-1})+b_hassets/equations/ca995973009650d3.svg
• a_t=W_{xh}x_t+W_{hh}h_{t-1}+bassets/equations/cac638ba5ee98e1c.svg
• 50+5=55assets/equations/cacbd65e2c06f4f4.svg
• v_t=\beta_2v_{t-1}+(1-\beta_2)g_t^2assets/equations/caedeeee27912e58.svg
• b_{\mathrm{new}}=0-0.1(-0.75)=0.075assets/equations/cb22ad4d5122d6cb.svg
• b_{h_2,\mathrm{new}}=0.6-0.5(-0.014526308)=0.607263154assets/equations/cb4635077c8c6a0a.svg
• g_{1,1}^{(1)}=\frac{\partial L}{\partial W^{(1)}_{1,1}}=\delta_{h_1}x_1assets/equations/cb514e27c4d52804.svg
• \theta_2=-0.4-\frac{0.1\times(2)}{\sqrt{5}}=-0.489442719assets/equations/cb523a64bc5840f4.svg
• a_{\mathrm{train}}=m\odot a\quad\text{and}\quad a_{\mathrm{test}}=q a\quad\text{in classical dropout}assets/equations/cb674fd2e32b87a1.svg
• N_{\mathrm{out}}=\left\lfloor\frac{N+2P_{\mathrm{pad}}-K}{S}\right\rfloor+1assets/equations/cb685a42551287a1.svg
• w_{\mathrm{new}}=w-\eta g_w=1-0.1(-4)=1.4assets/equations/cb9c6fe7ef91a2db.svg
• -z=-(2)=-2assets/equations/cbbb55e23c6ba6a0.svg
• e_2=2.2-4=-1.8assets/equations/cbcb5d7c79c1f6bd.svg
• E_5=\frac{8}{100}\times100\%=8\%assets/equations/cbf5783f9d42ea19.svg
• z_{h_3} = (0.3 \times 0.4) + (0.3 \times 0.3) + (-0.1 \times 0.5) + (0.5 \times -0.4) + (0.6) = 0.56assets/equations/cbf6b0463b666856.svg
• 8(8)(4)(1)(1)(16)=4096assets/equations/cbfbed2568dc51ee.svg
• z=(0.2)(1)+(0.1)(2)+0=0.4assets/equations/cc07c4020befc673.svg
• (2)(-1)=-2assets/equations/cc29fd0da04569d8.svg
• h=0.6\times0.139092448=0.083455469assets/equations/cc330c5bb2af2ffc.svg
• g_b=\sum_t\delta_tassets/equations/cc44b288566e90ca.svg
• i\widetilde C=0.5\times0.462117157=0.231058579assets/equations/cc4de2d727b7f825.svg
• \hat m_t=\frac{m_t}{1-\beta_1^t},\qquad \hat v_t=\frac{v_t}{1-\beta_2^t}assets/equations/cc72454f862696ef.svg
• L=-[0\ln p_1+1\ln p_2+0\ln p_3]=-\ln p_2assets/equations/cc77914badc24790.svg
• fC_{\mathrm{previous}}=0.5\times0.2=0.1assets/equations/cc79f89f1b075375.svg
• \left\lfloor\frac{5-3}{2}\right\rfloor+1assets/equations/cc7bf1ef4d7c44f8.svg
• \theta_2=-0.35+(0.2565)=-0.0935assets/equations/cc919e70a6113df8.svg
• N_{\mathrm{biases}}=4{,}096assets/equations/cc9a0e4319a0da43.svg
• w^Tx=w_1x_1+w_2x_2+w_3x_3assets/equations/ccb1c24c2967c1fa.svg
• z^{(2)}=0.15+0.30+0.10=0.55assets/equations/ccbd4dc810f4e3fd.svg
• \lambda\operatorname{sign}(0.3)=0.1(1)=0.1assets/equations/cce8304f0261d4a5.svg
• P_{\mathrm{pool}}=P_{\mathrm{flatten}}=0assets/equations/cd31b9774832f122.svg
• N_{\mathrm{weights}}=3\times3\times3\times16=432assets/equations/cd36df36d5f6f027.svg
• 0.2(0)+0.4(1)=0+0.4=0.4assets/equations/cd403f4588367553.svg
• g_{2,1}^{(1)}=(-0.004869072)\times(1)\approx -0.004869072assets/equations/cd47a26ddf1d4480.svg
• z_{h_1} = (0.2 \times 1) + (0.4 \times 0) + (0.1) = 0.3assets/equations/cd4f127dab262e74.svg
• e=t-o,\qquad w_{j,\mathrm{new}}=w_j+\eta e x_j,\qquad b_{\mathrm{new}}=b+\eta eassets/equations/cd56f1cf8ec70869.svg
• f=\sigma(z_f),\quad i=\sigma(z_i),\quad o=\sigma(z_o)assets/equations/cd80ed46001ad4fc.svg
• \left\lfloor\frac{55-3}{2}\right\rfloor+1assets/equations/cdac8f1c682bc40a.svg
• 1-h_t^2=1-(0.6)^2=1-0.36=0.64assets/equations/cdb26bda41e90caa.svg
• (1-2)=-1assets/equations/cde1ff63a64aa7ce.svg
• \hat y_2=1.64(2)=3.28,\qquad L_2=\frac12(3.28-4)^2=0.2592assets/equations/cde5a74260a032c4.svg
• W^{(1)}_{2,1,\mathrm{new}}=0.12-0.05(-0.2427)\approx 0.132135assets/equations/ce1785dad16e9ba5.svg
• z_{h_1} = (0.1 \times 0.5) + (0.4 \times 0.2) + (0.2 \times 0.1) + (0.1) = 0.25assets/equations/ce1c6f120d2c62ed.svg
• \delta_{h_1} = -0.11326 \times 1 = -0.11326assets/equations/ce1e6b90f6c99a43.svg
• g_1=-0.5-(1)=-1.5assets/equations/ce4034cb1a64ebf6.svg
• b_{o_1,\mathrm{new}}=0.1-0.1(-0.550000000)=0.155000000assets/equations/ce690df86543a5bc.svg
• \frac{\eta g_2}{\sqrt{s_2}}=\frac{0.1(1)}{\sqrt{0.46}}\approx0.147441956assets/equations/ce8864a56ffdc251.svg
• a=\begin{cases}z&z\ge0\\0.1z&z<0\end{cases}assets/equations/ce8d71b3a5eb9dfc.svg
• \frac{\partial L}{\partial v}=(-0.25)(0)+(-0.5)(0.2)=0-0.1=-0.1assets/equations/ce956c0f17770985.svg
• w^Tx=\sum_i w_ix_iassets/equations/ce95abc5189aaf50.svg
• \text{Sum}=C_1\approx 0.026233816assets/equations/cea4b9d0baabf5b5.svg
• h_2=\tanh(1.1)\approx0.800499022assets/equations/ceaf167b0d164375.svg
• L_{\mathrm{BCE}}=-[y\ln p+(1-y)\ln(1-p)],\qquad L_{\mathrm{CCE}}=-\sum_k y_k\ln p_kassets/equations/ceba44b43aaab755.svg
• 1+0.135335283=1.135335283assets/equations/cec6cfa7eb635dc5.svg
• C_t=f_tC_{t-1}+i_t\widetilde C_t,\qquad h_t=o_t\tanh(C_t)assets/equations/cec99860eadf260d.svg
• w_2^{\mathrm{new}}=0.1+0.1(-1)(2)=-0.1assets/equations/ced16326d7d6ffec.svg
• \delta_{o_1} = 1 \times (0.641368858-1) \times 0.230014846 = -0.082490487assets/equations/cee1321a8d736f56.svg
• h_{2,\mathrm{new}}=0.325(2)+0.51(0.4)+0.075=0.929assets/equations/cef917ae20dca74f.svg
• H_2(p,q)=-[0.75(-1)+0.25(-1)]=1\ \text{bit}assets/equations/cf0bfa788dbc4b07.svg
• y_{\max}=21assets/equations/cf103da2c4c32945.svg
• \frac{\partial L}{\partial o_1}=o_1-y_1=2.2-1=1.2assets/equations/cf26e13f912b7730.svg
• N_{\mathrm{biases}}=128assets/equations/cf36568f06144792.svg
• h_t=0.8\tanh(0.83)\approx0.544380805assets/equations/cf4e11309aadd94a.svg
• p_{\mathrm{avg}}=\frac{a_1+a_2+a_3+a_4}{4}assets/equations/cf572bd2a565d88a.svg
• \delta=(p-y)p(1-p)assets/equations/cf8f40120fc7de9a.svg
• (K^2C_{\mathrm{in}}+1)C_{\mathrm{out}}assets/equations/cfa2f04f36764d25.svg
• \delta_{o_1} = 1 \times (0.590378257-1) \times 0.241831771 = -0.099059551assets/equations/cfa9f3b565ded315.svg
• 18+2=N-2+2assets/equations/cfbfbce776e71cb1.svg
• v_t=\beta v_{t-1}-\eta g_t,\quad \theta_t=\theta_{t-1}+v_t · v_t=\beta v_{t-1}-\eta g_t,\qquad\theta_t=\theta_{t-1}+v_tassets/equations/cfd2c8e117828e21.svg
• N_{\mathrm{biases}}=1{,}000assets/equations/d0008f88a44995b2.svg
• \delta_{o_1} = 1 \times (0.191-1) \times 1 = -0.809assets/equations/d0231cff1e5d205d.svg
• n_{\mathrm{flat}}=29\times29\times128=107{,}648assets/equations/d04e40df15cb4a0b.svg
• v_1=0.999(0)+0.001(-1)^2=0.001assets/equations/d05bfcaee435c205.svg
• R_2=(0)\times(0)+(1)\times(1)=1assets/equations/d060bcb31ccbcc5c.svg
• g_{1,2}^{(2)}=(-0.809)\times(0.48)\approx -0.38832assets/equations/d09d2ce4b3621339.svg
• W^{(1)}_{1,2,\mathrm{new}}=0.1-0.1(-0.44)\approx 0.144assets/equations/d0bf76ef563b87e1.svg
• R_1=(1)\times(1)+(2)\times(-1)=-1assets/equations/d0c50421371c4116.svg
• b_{h_1,\mathrm{new}}=0.35-0.5(0.008771355)=0.345614322assets/equations/d0caa37a084caf21.svg
• h_t=0.25(0.4)+0.75\tanh(0.3)\approx0.318484459assets/equations/d0cde1d1f4ff9640.svg
• \frac32=1.5assets/equations/d103d3b0cd150006.svg
• 3+0.3=3.3assets/equations/d117d7aaf6f65395.svg
• z_{h_1} = (0.2 \times 1) + (0.1 \times 2) + (0.1) = 0.5assets/equations/d12052a5268cb30c.svg
• z=0+(-0.8)=-0.8assets/equations/d15ca461c07dd695.svg
• 5(5)=25assets/equations/d16b109d87f9088b.svg
• \delta_{o_1} = 2 \times (0.683621675-0.9) \times 0.216283081 = -0.093597941assets/equations/d17a9957b64c0cce.svg
• o_t=\sigma(W_o[h_{t-1},x_t]+b_o),\qquad h_t=o_t\odot\tanh(C_t)assets/equations/d1817c5fc6ff06c8.svg
• \frac{\partial z}{\partial w}=x,\qquad \frac{\partial z}{\partial b}=1,\qquad \frac{\partial z}{\partial x}=wassets/equations/d1922f2b91d23e75.svg
• o_t=W_{hy}h_t+b_yassets/equations/d195662850af1e7e.svg
• \hat y_1=1(1)=1assets/equations/d19c8d7124e25bc5.svg
• h_{\mathrm{new}}=(0.731058579)(0.188942847)\approx0.138128289assets/equations/d1aa738ae9a7c24f.svg
• N_{\mathrm{out}}=\left\lfloor\frac{63+2(0)-3}{1}\right\rfloor+1=61assets/equations/d1b1c42ec3b8b44b.svg
• \lambda w_2=0.2\times(-0.2)=-0.04assets/equations/d1d1a2ba29bb3862.svg
• \delta_{h_2} = 0.070517544 \times 0.247985885 = 0.017487356assets/equations/d2104a629894d5d4.svg
• \text{weights}=3\times3\times1\times4=36assets/equations/d2206cecca6115a8.svg
• o_{1} = \max(0,0.55) = 0.55assets/equations/d22e4c171ba2abbb.svg
• \begin{aligned}5+2&=7\\5+(-2)&=3\end{aligned}assets/equations/d25cc2f4afc0b12d.svg
• \delta=\frac{\partial L}{\partial z}=(a-y)a(1-a)assets/equations/d269ffed64d3f80c.svg
• z=w_1x_1+w_2x_2+w_3x_3+b,\qquad a=\max(0,z)assets/equations/d2b18ce5f8c75557.svg
• n_{\mathrm{flat}}=24\times24\times64=36{,}864assets/equations/d2c7c9ed057461d8.svg
• g_{1,1}^{(1)}=(-0.22)\times(1)\approx -0.22assets/equations/d2c8214211374554.svg
• \frac{\partial z}{\partial b}=1assets/equations/d2edf1641e728713.svg
• a=2assets/equations/d3043f41a0385109.svg
• H(x)=F(x)+xassets/equations/d30a7ff979462f45.svg
• W^{(1)}_{2,1,\mathrm{new}}=-0.2-0.2(0.003497471)\approx -0.200699494assets/equations/d31db2b9d2bae391.svg
• 0.1(5)=0.5assets/equations/d3298085ce085f2d.svg
• 9408(400)=3763200assets/equations/d32e7e22a58c8e10.svg
• Y_{3,1}=(2)+(0)=2assets/equations/d3322f48fabaf515.svg
• \begin{bmatrix}0.14 & 0.15\end{bmatrix}assets/equations/d33649adcaa2e9c5.svg
• \text{Sum}=C_1\approx -0.275assets/equations/d346b0df4b69bd95.svg
• y_{res}=F(x)+xassets/equations/d362a7b729c4f87f.svg
• W^{(2)}_{1,4,\mathrm{new}}=0.2-0.5(-0.060432101)\approx 0.230216051assets/equations/d362d593396fe5bf.svg
• \sqrt{s_1}=\sqrt{0.4}\approx0.632455532assets/equations/d388f002971175a6.svg
• f\prime(0)=1-(0.000000000)^2\approx1.000000000assets/equations/d3899b5f76509b8c.svg
• w_2x_2=0.4\times(-1)=-0.4assets/equations/d39443d0dbfaaa2b.svg
• R_1=(3)\times(1)+(2)\times(0)+(1)\times(-1)=2assets/equations/d3a3d0394f9567ec.svg
• z_{XOR}=0-2(0)-0.5=-0.5assets/equations/d3b2998fcd8ceee6.svg
• N_{\mathrm{out}}=\left\lfloor\frac{227+2(0)-9}{1}\right\rfloor+1=219assets/equations/d3b5ab84e61f126d.svg
• f(z)=\begin{cases}z&z\geq0\\az&z<0\end{cases}assets/equations/d3faffb967fafdf3.svg
• n_{\mathrm{flat}}=HWC,\qquad P_{\mathrm{dense}}=(n_{\mathrm{in}}+1)n_{\mathrm{out}}assets/equations/d4064c7b5aff30cf.svg
• N_{\mathrm{out}}=\left\lfloor\frac{55-3}{2}\right\rfloor+1=27assets/equations/d42806d80c07005c.svg
• z=w_1x_1+w_2x_2+w_3x_3+b,\qquad a=\max(0,z)assets/equations/d429a2592c8f926e.svg
• B=\begin{bmatrix}2&0\\1&3\end{bmatrix}assets/equations/d45fb6a21c1dc67e.svg
• g_1^{\mathrm{clip}}=0(0.25)=0assets/equations/d479ef7213e9c007.svg
• C_{11}=1(2)+2(1)=2+2=4assets/equations/d47e20af5264ff61.svg
• p=\frac1{1+e^{-1}}=0.731058579assets/equations/d48b6096bc280c6f.svg
• \mathrm{MSE}=\frac{0.49}{3}=0.163333333\ldotsassets/equations/d4cd1db7c5158eb6.svg
• J\prime(2)=3(2)(2)^{1}=12assets/equations/d4fbd4bd0e97e8fb.svg
• g_{1,1}^{(1)}=(-0.11326)\times(2)\approx -0.22652assets/equations/d4fe1c7bccab7769.svg
• q=1-p,\quad a_{train}=\frac{m\odot a}{q},\quad a_{test}=aassets/equations/d4fe7ae66a3ded01.svg
• \frac{\partial L}{\partial z}=0.731058579-1=-0.268941421assets/equations/d503567052ec89af.svg
• W^{(2)}_{1,1,\mathrm{new}}=0.5-0.5(-0.06654341)\approx 0.533271705assets/equations/d5193daabf661dea.svg
• fC_{\mathrm{old}}=(0.731058579)(-0.5)\approx-0.365529289assets/equations/d526a95c3a3f61ff.svg
• b_{\mathrm{new}}=b-\eta g_b=0-0.1(-2)=0.2assets/equations/d5507ad8ee01624d.svg
• g_2=(\hat y_1-y)x=(2.8-4)(2)=-2.4assets/equations/d5601224799e33a1.svg
• C_{ij}=\sum_{k=1}^{n}A_{ik}B_{kj}assets/equations/d566b516b84686ee.svg
• \frac{d}{dx}c=0assets/equations/d569e096ba9d6dba.svg
• z_{h_1} = (0.8 \times 0.4) + (-0.5 \times 0.3) + (0.1 \times 0.5) + (-0.2 \times -0.4) + (0.6) = 0.9assets/equations/d580103bada11b64.svg
• a_2=0.7+0.4+0=1.1assets/equations/d5abcbc4ff3b8b0f.svg
• a'_4=\frac3{0.75}=4assets/equations/d5f3e01a5c059a83.svg
• H(p,q)=-\sum_i p_i\ln q_i,\qquad\log_2q=\frac{\ln q}{\ln2}assets/equations/d607248d18cc3f42.svg
• g_{2,2}^{(1)}=(-0.004869072)\times(1)\approx -0.004869072assets/equations/d63a8ff35a6906b2.svg
• J\prime(-1)=2(1)(-1)^{0}=2assets/equations/d63d0415964167b7.svg
• \sigma(a)=\frac1{1+e^{-a}}assets/equations/d65b06a34ee57b5a.svg
• N_{\mathrm{out}}=\left\lfloor\frac{28-2}{2}\right\rfloor+1=14assets/equations/d66b872d3a8b7d76.svg
• C_{\mathrm{new}}=0.500000000+(0.000000000)\approx0.500000000assets/equations/d66b879a42e6911e.svg
• w_{\mathrm{new}}=0.4-0.1(-2)=0.6assets/equations/d677f9f3ff8120b5.svg
• \sqrt{s_2}=\sqrt{0.46}=0.678232998assets/equations/d68789eef851ad06.svg
• P_B=(3\times3\times8+1)\times6=73\times6=438assets/equations/d6970788f463cb74.svg
• b_{\mathrm{new}}=0+0.1(-1)=-0.1assets/equations/d6c4895766052790.svg
• v_t=\beta v_{t-1}-\eta g_t,\qquad\theta_t=\theta_{t-1}+v_tassets/equations/d6d609593a44d53f.svg
• W^{(2)}_{2,2,\mathrm{new}}=0.4-0.2(0.079516282)\approx 0.384096744assets/equations/d6e1bdf461c23702.svg
• c_t=f_t\odot c_{t-1}+i_t\odot\widetilde c_tassets/equations/d70555acf8f3a0de.svg
• \text{Sum}=C_1\approx -0.024764888assets/equations/d708f3df447c37c4.svg
• R_2=(0)\times(0)+(1)\times(-1)=-1assets/equations/d7380bbaffbc2948.svg
• 1+(-0.2)=0.8assets/equations/d74c1d06c5bc866b.svg
• N_{\mathrm{weights}}=5\times5\times1\times6=150assets/equations/d75bc00144d70f4e.svg
• b_{1,\mathrm{new}}=0.1-0.1(0.064)=0.0936assets/equations/d766af9fb7ee76f0.svg
• w_{1,\mathrm{new}}=0.5-0.1(0.128)=0.4872assets/equations/d781e32010d9fb09.svg
• \sigma^2=\frac1N\sum_i(x_i-\mu)^2assets/equations/d7a9d7e061fca1d7.svg
• W^{(1)}_{2,1,\mathrm{new}}=0.2-0.5(-0.005810523)\approx 0.202905262assets/equations/d7bc835c6d4c639d.svg
• g_{1,1}^{(1)}=(-0.007853997)\times(3)\approx -0.02356199assets/equations/d7d37c7aaeaeb2f8.svg
• \widetilde C_t=\tanh(0.5)\approx0.462117157assets/equations/d7e58ebe85cb8fb2.svg
• \frac{\eta\hat m_2}{\sqrt{\hat s_2}}\approx\frac{0.1(1.473684211)}{\sqrt{2.499249625}}\approx0.093217964assets/equations/d8416409081efc8e.svg
• \sigma(1)=\frac1{1+e^{-1}}\approx0.731058579assets/equations/d87096cb2c998d15.svg
• h_{1}=\frac{1}{1+e^{-(-0.7)}}=0.331812228assets/equations/d8bf113d37695d28.svg
• C_1=(0.6)\times(-0.082490487)\approx -0.049494292assets/equations/d8c0fdc1c33b51f1.svg
• z=\frac{0}{2}=0assets/equations/d8d3a4b4364ff836.svg
• z_{h_2}=1+0-1.5=-0.5\Rightarrow h_2=0assets/equations/d8ee9449e01e7c58.svg
• g_{2,1}^{(1)}=\frac{\partial L}{\partial W^{(1)}_{2,1}}=\delta_{h_2}x_1assets/equations/d8f0b089bd0fe450.svg
• W^{(2)}_{2,1,\mathrm{new}}=0.3-0.2(0.080239399)\approx 0.28395212assets/equations/d900aa21d0b17c7e.svg
• W^{(1)}_{2,3,\mathrm{new}}=0.2-0.9(0.006542085)\approx 0.194112123assets/equations/d92699928835e1d2.svg
• h_{2}=\frac{1}{1+e^{-(0.7)}}=0.668187772assets/equations/d9473d1a5522dcc7.svg
• o=0.2(2)+0.3(0)+0=0.4assets/equations/d947f0c3c40b6311.svg
• \delta_{h_1} = -0.049494292 \times 0.158684897 = -0.007853997assets/equations/d94f5a36d64f33c1.svg
• \mu=\frac1N\sum_i x_iassets/equations/d96a5eb55c670639.svg
• \text{biases}=4,\qquad P_{\mathrm{conv}}=36+4=40assets/equations/d97de20b29f5a9fb.svg
• e^{0}-e^{0}\approx1.000000000-1.000000000=0.000000000assets/equations/d9a65f4b4641a1c1.svg
• \mathrm{MSE}=\frac1n\sum_i e_i^2assets/equations/d9bd32ee9a4320cb.svg
• J(\theta)=\frac12(\theta-1)^2,\qquad J\prime(\theta)=\theta-1assets/equations/d9f043ffb887fd14.svg
• W^{(1)}_{1,2,\mathrm{new}}=0.5-0.5(-0.004869072)\approx 0.502434536assets/equations/da0b9b9064b8dcd6.svg
• C_1=(0.2)\times(-0.121439988)\approx -0.024287998assets/equations/da1d2225b6cf0b3e.svg
• \theta_2=0.9-\frac{0.1\times(1.473684211)}{\sqrt{2.499249625}}=0.806782036assets/equations/da23812bc6e746bd.svg
• \mathrm{MACs}_{3\times3}=16\times16\times32\times3\times3\times8=589{,}824assets/equations/da27ae2a69680cb6.svg
• a^{(l)}=f^{(l)}(z^{(l)})assets/equations/da3b4bdac78b839b.svg
• a=\frac{0.000000000}{2.000000000}\approx0.000000000assets/equations/da7a029e0ae3e61f.svg
• \mathrm{MAE}=\frac1n\sum_i|\hat y_i-y_i|assets/equations/da84fa0919b7d150.svg
• N_{\mathrm{out}}=\left\lfloor\frac{27+2(2)-5}{1}\right\rfloor+1=27assets/equations/da9a62eaf99ea1e8.svg
• \hat m_1=\frac{0.2}{1-0.9^1}=\frac{0.2}{0.1}=2assets/equations/daa0ad010127bd94.svg
• h=W_1x+b_1,\qquad \hat y=W_2h+b_2=W_2W_1x+(W_2b_1+b_2)assets/equations/dac24947cfd3e2d0.svg
• \frac{\partial L}{\partial v}=(-0.25)(0)+(-0.5)(0.2)=0-0.1=-0.1assets/equations/dadd294afa7bf158.svg
• \theta_{\mathrm{look}}=-0.35+0.9(0.15)=-0.215assets/equations/dade5906dabc1018.svg
• A^T=\begin{bmatrix}2\\1\\3\end{bmatrix}assets/equations/dafe6e5e80b48abd.svg
• 1+0.1=1.1assets/equations/db025849e060d366.svg
• W^{(2)}_{1,2,\mathrm{new}}=0.24-1(0.010710144)\approx 0.229289856assets/equations/db1d78d9cd004855.svg
• \begin{bmatrix}8 & 10\\12 & 15\end{bmatrix}assets/equations/db73fdbe57c2c901.svg
• W^{(2)}_{1,1,\mathrm{new}}=0.5-0.5(-0.06654341)\approx 0.533271705assets/equations/db770dae3f630bdc.svg
• W^{(1)}_{1,1,\mathrm{new}}=0.15-0.5(0.000438568)\approx 0.149780716assets/equations/db7d742025f4d68e.svg
• e_1=1-2=-1assets/equations/dba84d8eacba9175.svg
• \left\lfloor\frac{128+2 \times 0-3}{1}\right\rfloor+1assets/equations/dbb0832b53a96b49.svg
• \theta_2=-0.183772234-\frac{0.1\times(2)}{\sqrt{0.49}}=-0.46948652assets/equations/dbbd9b6c37d99e9e.svg
• f_tC_{t-1}=0.5(0.2)=0.1assets/equations/dbc91b31be7f6188.svg
• R_3=(1)\times(1)+(1)\times(0)+(0)\times(-1)=1assets/equations/dbc9736155e0a1ec.svg
• z_2=0.5(2)+3(1)-0.2=3.8assets/equations/dbd8ceb08ae05e33.svg
• g_{\mathrm{clip}}=g\min\left(1,\frac\tau{\lVert g\rVert_2}\right)assets/equations/dbdee9367720972d.svg
• \frac{dz}{dw}=3assets/equations/dbe8fc0fd6788a9a.svg
• A=\begin{bmatrix}21&11&15&10\\12&9&10&11\\11&13&18&12\\7&4&15&12\end{bmatrix}assets/equations/dbeeead3eba02270.svg
• N_{\mathrm{biases}}=16assets/equations/dc0c940eb69b41c0.svg
• \delta^{out}=(o-y)f\prime(z)assets/equations/dc1948e719d0e70b.svg
• m_2=0.9(-0.1)+0.1(2)=0.11assets/equations/dc1a680eebaa1164.svg
• z_j=\sum_iw_{ji}a_i+b_j,\quad a_j=f(z_j),\quad L=\frac12\sum_k(o_k-y_k)^2assets/equations/dc1bda021c501564.svg
• f_3C_2=0.2(0.36)=0.072,\qquad i_3\widetilde C_3=0.5(-0.6)=-0.3assets/equations/dc1de2dfd9757248.svg
• \frac{\partial L}{\partial W_{hh}}=\sum_t\delta_t h_{t-1}^{T},\quad \frac{\partial L}{\partial W_{xh}}=\sum_t\delta_t x_t^{T},\quad \frac{\partial L}{\partial b_h}=\sum_t\delta_tassets/equations/dc28790fa408d967.svg
• z_{h_1} = (0.2 \times 1) + (0.4 \times 0) + (0.1) = 0.3assets/equations/dc2c7898e4942415.svg
• N_{\mathrm{biases}}=64assets/equations/dc3a74202b89cbd8.svg
• z_j=\sum_i W_{ji}x_i+b_jassets/equations/dc40cdac8ad4c81c.svg
• D=0.367879441+1+0.135335283=1.503214724assets/equations/dc4d03dcae0355c5.svg
• e^{2z}-1=2.718281828-1=1.718281828assets/equations/dc5e5624414a3d95.svg
• f_t=\sigma(W_f[h_{t-1},x_t]+b_f),\quad i_t=\sigma(W_i[h_{t-1},x_t]+b_i)assets/equations/dc6d5a7033a71923.svg
• C_2=(0.12)\times(0.082533822)\approx 0.009904059assets/equations/dc757bbb5573b8af.svg
• W^{(1)}_{2,1,\mathrm{new}}=0.2-0.5(-0.005810523)\approx 0.202905262assets/equations/dc852cb894d5794e.svg
• \hat y_1=1.4(2)=2.8,\qquad L_1=\frac12(2.8-4)^2=0.72assets/equations/dc92a06893eca5f4.svg
• 1-h_t^2=1-(0.6)^2=1-0.36=0.64assets/equations/dca805bc28733c27.svg
• W^{(1)}_{1,2,\mathrm{new}}=0.15-1(0.001437581)\approx 0.148562419assets/equations/dcaf6fc9df70c8e9.svg
• \mu_B=\frac{2+4}{2}=3assets/equations/dcbc368ca301a562.svg
• e^{4}\approx54.598150033assets/equations/dcc7f746c4400033.svg
• z_{h_2}=0+0-1.5=-1.5\Rightarrow h_2=0assets/equations/dcc9f2d8a503f369.svg
• p_i=\frac{e^{z_i-c}}{\sum_j e^{z_j-c}},\qquad c=\max_jz_jassets/equations/dd0c8ddf71d7e5e7.svg
• z_{XOR}=1-2(1)-0.5=-1.5assets/equations/dd1914e888d3b3fb.svg
• \theta_1=1-\frac{0.1(2)}{\sqrt4}=0.9assets/equations/dd2c0d46e60db2c0.svg
• \theta_2\approx0.683772234-0.147441956\approx0.536330278assets/equations/dd5064d2dd1b316c.svg
• a_2=1+0.092423431=1.092423431assets/equations/dd63fd4bc0c56e9d.svg
• Y_{1,2}=(1)+(4)+(0)=5assets/equations/dd6e5a7547d2ee64.svg
• G_2=4+(1)^2=5assets/equations/dd892c081c748bf2.svg
• p_3=\frac{1.000000000}{3.000000000}\approx0.333333333assets/equations/dd8dca9605ca9fba.svg
• \text{average slope}=\frac{\text{new loss}-\text{old loss}}{\text{new weight}-\text{old weight}}assets/equations/ddbf51217582c207.svg
• J=\frac33=1assets/equations/ddda757808debff5.svg
• g_{2,1}^{(2)}=\frac{\partial L}{\partial W^{(2)}_{2,1}}=\delta_{o_2}h_1assets/equations/dde4ee7e9e8f997c.svg
• a^2\approx0.213552267,\qquad1-a^2\approx0.786447733assets/equations/de782b0d8cb8f7bb.svg
• a_i^{\mathrm{test}}=a_iassets/equations/de8c0e93f06e9aa8.svg
• o_{2}=\frac{1}{1+e^{-(1.255565213)}}=0.778261739assets/equations/de94c544219f4fc9.svg
• \theta_2=0.683772234-\frac{0.1\times(1)}{\sqrt{0.46}}=0.536330278assets/equations/dec405b6648724b8.svg
• R_2=(5)\times(1)+(7)\times(2)+(6)\times(1)=25assets/equations/dec76766281c84de.svg
• \delta_{h_3} = 0.056158765 \times 0.231380704 = 0.012994055assets/equations/ded30217d8f2c313.svg
• 0.018315639+1\approx1.018315639assets/equations/dee5ac5546f1b2f2.svg
• g_{1,2}^{(2)}=(0.138498562)\times(0.596884378)\approx 0.082667628assets/equations/df479afdae4fe00d.svg
• \begin{aligned}1-0.9^2&=1-0.81\\&=0.19\end{aligned}assets/equations/df575ab27bdf9309.svg
• L=\frac12(\hat y-y)^2assets/equations/df5fc0c0a9ad8296.svg
• \nabla_{\hat y}L_{\mathrm{half}}=[0.2,-0.3,-0.6]assets/equations/dfa71d7fa81c823c.svg
• \text{weights}=1\times1\times8\times16=128assets/equations/dfae1039bf042104.svg
• \left\lfloor 29.5\right\rfloorassets/equations/dfb80f08b9b0c4e1.svg
• 8(8)(8)(3)(3)(16)=73728assets/equations/dffc8608ebe7e697.svg
• \frac{d}{dx}(cx^n)=cnx^{n-1},\quad\frac{d}{dx}c=0,\quad (f+g)\prime=f\prime+g\primeassets/equations/dffe83aa3b00d40c.svg
• \hat y=wx,\quad L=\frac12(\hat y-y)^2,\quad g=(\hat y-y)xassets/equations/e02d92789136f341.svg
• \sigma(z)=\frac{1}{1+e^{-z}},\quad \sigma\prime(z)=\sigma(z)[1-\sigma(z)]assets/equations/e03a4bc95b918203.svg
• \begin{bmatrix}0 & 1\\2 & 1\end{bmatrix}assets/equations/e06165b70c0dec06.svg
• P_{\mathrm{conv}}=(K^2C_{\mathrm{in}}+1)C_{\mathrm{out}},\qquad P_{\mathrm{dense}}=(n_{\mathrm{in}}+1)n_{\mathrm{out}}assets/equations/e06582d1ab100452.svg
• P_{\mathrm{conv}}=(K_hK_wC_{\mathrm{in}}+1)C_{\mathrm{out}}assets/equations/e06cb7b0e9f45658.svg
• Y_{1,2}=(2)+(3)=5assets/equations/e073412e219f2310.svg
• J=L+\frac\lambda2\sum_jw_j^2,\quad w_{new}=w-\eta(g_{data}+\lambda w)assets/equations/e0797baa70350e25.svg
• h_t=0.8\tanh(0.83)\approx0.544380805assets/equations/e0a7ec476bae68f2.svg
• \widetilde h_t=\tanh(W_hx_t+U_h(r_t\odot h_{t-1})+b_h),\quad h_t=(1-z_t)\odot h_{t-1}+z_t\odot\widetilde h_tassets/equations/e0ad3b2f8e3f246b.svg
• L=-(-0.693147181)=0.693147181assets/equations/e0e8fd22135c9d7b.svg
• U_{\mathrm{batch}}=E,\qquad U_{\mathrm{SGD}}=NEassets/equations/e0f5f4a6b7d3af9c.svg
• \theta_1=1-\frac{0.1\times(2)}{\sqrt{4}}=0.9assets/equations/e1122171b0bd236e.svg
• -z=-2assets/equations/e116d9c93aba3d51.svg
• 2^2=4,\qquad(-1)^2=1assets/equations/e11b2a71ed1c964f.svg
• g_{2,4}^{(1)}=(-0.014526308)\times(-0.4)\approx 0.005810523assets/equations/e120af498da6af3d.svg
• a_{test}=aassets/equations/e1251aa7fa418d32.svg
• b_{h_2,\mathrm{new}}=1-1(0.003509506)=0.996490494assets/equations/e12e4a571f964fad.svg
• \sqrt{s_2}=\sqrt{0.49}=0.7assets/equations/e14a6fd4eb271def.svg
• \begin{aligned}\frac{0.8}{0.5}&=\frac{0.8\times10}{0.5\times10}\\&=\frac85\\&=1.6\end{aligned}assets/equations/e15debe515ad70bf.svg
• w_{t+1}=w_t-\eta(\nabla_w J_{\mathrm{data}}+2\lambda w_t)=(1-2\eta\lambda)w_t-\eta\nabla_w J_{\mathrm{data}}assets/equations/e175e10221a61929.svg
• v_2=0.9(0.15)-0.1(-1.215)=0.2565assets/equations/e1b4633f6988ed67.svg
• p(1-p)=0.8(0.2)=0.16assets/equations/e1d74c616a26b88e.svg
• C_1=(0.5)\times(-0.093597941)\approx -0.04679897assets/equations/e1dda3f8681ba484.svg
• z_{o_1} = (-0.3 \times 0.331812228) + (-0.2 \times 0.524979187) + (0.1) = -0.104539506assets/equations/e1f248930dc68c45.svg
• \mathrm{MSE}=\frac{9}{1}\approx9.000000000assets/equations/e1fcd30496290844.svg
• E_1=\frac{22}{100}\times100\%=22\%assets/equations/e21c77c5c190c14b.svg
• g_{\mathrm{penalty}}=2\lambda wassets/equations/e2245fb17ab44d83.svg
• h_{2}=\frac{1}{1+e^{-(0.18)}}=0.544878892assets/equations/e227b5ddfa70fc91.svg
• z_{o_1} = (0.5 \times 0.3) + (0.6 \times 0.5) + (0.1) = 0.55assets/equations/e22a0b62445991e6.svg
• \mathrm{saving\ percentage}=100\frac{107978752}{120422400}\approx89.666667\%assets/equations/e22e38a0d749a968.svg
• v_1=0.9(0)-0.1(1)=-0.1assets/equations/e23af655c3740539.svg
• w_{new}=w-\eta\left[g_{data}+\lambda\operatorname{sign}(w)\right]assets/equations/e25593d3f219dc71.svg
• \sigma(0)=\frac1{2.000000000}\approx0.500000000assets/equations/e2631bc2ce7ebc9e.svg
• 30+5=35assets/equations/e27ba4eb6d98f55b.svg
• N_{\mathrm{padded}}=5+2=7assets/equations/e288268af2ced78c.svg
• W^{(2)}_{1,1,\mathrm{new}}=-0.3-0.9(-0.043523504)\approx -0.260828846assets/equations/e28a7e27c4f74611.svg
• g_{1,1}^{(1)}=(-0.004869072)\times(1)\approx -0.004869072assets/equations/e290def3a1c4ea90.svg
• C_t=f_t\odot C_{t-1}+i_t\odot\widetilde C_t,\quad h_t=o_t\odot\tanh(C_t)assets/equations/e295aa726de62b85.svg
• 8(4)=32assets/equations/e2ee029f7f3db879.svg
• \left\lfloor\frac{13+2 \times 1-3}{1}\right\rfloor+1assets/equations/e30499da6b8e9d4a.svg
• \text{Sum}=C_1\approx -0.11326assets/equations/e31db28d376c594d.svg
• \theta_{\mathrm{look}}=0.9+0.9(-0.1)=0.81assets/equations/e3234cc25d1508de.svg
• a=\max(0,z)assets/equations/e3267d2a19366c7b.svg
• \widehat y=\operatorname{step}(w^Tx+b),\quad w_{new}=w+\eta(y-\widehat y)x,\quad b_{new}=b+\eta(y-\widehat y)assets/equations/e3355300bf932c93.svg
• h_{2}=\frac{1}{1+e^{-(0.52)}}=0.627147766assets/equations/e33e2e29f7e69ace.svg
• (0.5)(4)=2assets/equations/e3430445c8f6a121.svg
• W^{(1)}_{1,1,\mathrm{new}}=0.1-0.2(0.000964927)\approx 0.099807015assets/equations/e34dab3cfb2bef31.svg
• Y_{2,2}=(1)+(0)=1assets/equations/e353ed5c5344d9f0.svg
• -3+2=-1assets/equations/e35736ce1cfc9727.svg
• \theta_1=-0.5-0.1(-1.5)=-0.35assets/equations/e397e226e8c7c566.svg
• p_i=\frac{e^{z_i-c}}{\sum_j e^{z_j-c}},\quad c=\max_jz_j,\quad L=-\ln p_{\mathrm{correct}}assets/equations/e3c5269ae1be7e5e.svg
• J_{L1}=0.4+0.35=0.75assets/equations/e3ee4cac7da2daf6.svg
• R_3=(3)\times(1)+(2)\times(0)+(1)\times(-1)=2assets/equations/e404b4617ef0e7d0.svg
• \mathrm{saving\ percentage}=100\frac{\mathrm{direct}-\mathrm{reduced}}{\mathrm{direct}}assets/equations/e413f16bb5ba7714.svg
• v_1=0.999(0)+0.001(-1)^2=0.001assets/equations/e44b2300355f9498.svg
• e=t-o,\qquad w_{j,\mathrm{new}}=w_j+\eta e x_j,\qquad b_{\mathrm{new}}=b+\eta eassets/equations/e4615db5fadb580f.svg
• z=\sum_i w_i x_i+b,\qquad a=f(z)assets/equations/e4620ff07e410bce.svg
• 4096+18432=22528assets/equations/e4821e959cb7a516.svg
• a_1=\max(0,4.5)=4.5,\qquad a_2=\max(0,0.5)=0.5assets/equations/e48d03ac9b49ea16.svg
• \mathrm{MSE}=\frac{6}{3}\approx2.000000000assets/equations/e4955c778fd95a4f.svg
• \sigma(z)=(1+e^{-z})^{-1}assets/equations/e4d8f5baacfd2120.svg
• L=\frac12(1.08-1)^2=0.0032assets/equations/e51ef56449f25818.svg
• \delta_{o_1} = 1 \times (0.45-1) \times 1 = -0.55assets/equations/e54a4bff1059c242.svg
• R_2=(3)\times(0)+(2)\times(1)=2assets/equations/e54c044f72c7530b.svg
• P=55+30+30+24=139assets/equations/e55d02bddb539fc5.svg
• L=-[y\ln p+(1-y)\ln(1-p)]assets/equations/e55d3d5e343976d3.svg
• \delta_{o_k}=(o_k-y_k)f'(z_{o_k})assets/equations/e56bc6ef11f21beb.svg
• h_t=\tanh(a_t)assets/equations/e57a9d6af94ad1bb.svg
• \text{biases}=8,\qquad \text{total}=600+8=608assets/equations/e580536b54e8c233.svg
• h_{1}=\frac{1}{1+e^{-(0.3775)}}=0.593269992assets/equations/e586b2530e4f9cf0.svg
• \text{Sum}=C_1+C_2\approx 0.041370323assets/equations/e59e8769f280fbd2.svg
• Y_{ij,o}=\sum_c\sum_u\sum_v K_{uvco}X_{iS+u-P,jS+v-P,c}+b_oassets/equations/e5afb61a6d5f3879.svg
• g_{4,1}^{(1)}=\frac{\partial L}{\partial W^{(1)}_{4,1}}=\delta_{h_4}x_1assets/equations/e5b6b4e6329ece21.svg
• \theta_2=-0.35+(0.27)=-0.08assets/equations/e5bc5d38d05f5667.svg
• m_2=\beta(1-\beta)g_1+(1-\beta)g_2assets/equations/e5bd41b82ae93376.svg
• \text{L1 penalty}=0.1\times3.5=0.35assets/equations/e5d16adbe508b8b5.svg
• J_{L1}=J_{\mathrm{data}}+\lambda\sum_i|w_i|,\qquad J_{L2}=J_{\mathrm{data}}+\lambda\sum_iw_i^2assets/equations/e5d7bc3aa72256c3.svg
• \text{average slope}=\frac{\text{new loss}-\text{old loss}}{\text{new weight}-\text{old weight}}assets/equations/e5eede05db6e20c3.svg
• N_{\mathrm{out}}=\left\lfloor\frac{122-2}{2}\right\rfloor+1=61assets/equations/e5f058816466e626.svg
• z_{o_1} = (0.2 \times 0.549833997) + (-0.1 \times 0.544878892) + (0) = 0.05547891assets/equations/e5fd2ad81db5cb12.svg
• w_1x_1=0.3\times2=0.6assets/equations/e6092f2a718fe356.svg
• \hat s_1=\frac{0.004}{1-0.999^1}=\frac{0.004}{0.001}=4assets/equations/e60fd90ed8957afb.svg
• g_{1,\mathrm{total}}=0.1+0.1=0.2assets/equations/e611707728d90ba8.svg
• o-y=0.4-1=-0.6assets/equations/e63213278aa7de98.svg
• R_2=(2)\times(0)+(1)\times(1)=1assets/equations/e63832c06fcb7aed.svg
• g_{2,1}^{(1)}=(0.006542085)\times(1)\approx 0.006542085assets/equations/e657c5ac09391365.svg
• \delta_{h_2} = -0.065518559 \times 0.221712873 = -0.014526308assets/equations/e65ba09e76cd6492.svg
• L=-\left[y\ln p+(1-y)\ln(1-p)\right]assets/equations/e6b608fdfba7ba2f.svg
• N_{\mathrm{padded}}=N+2Passets/equations/e6bbc5ec5f7e71fe.svg
• \text{weights}=3\times3\times3\times2=54assets/equations/e6c2c8a35c15f9ce.svg
• g_{1,2}^{(2)}=(-0.121439988)\times(0.544878892)\approx -0.066170086assets/equations/e6cb2fb79ea00bc6.svg
• (-3)+(-2)=-5assets/equations/e6f43c9051ebbeef.svg
• o_{1} = \max(0,0.45) = 0.45assets/equations/e6f7822c28a4fdd9.svg
• z=0+(-1.5)=-1.5assets/equations/e7029152ef49c01f.svg
• \frac{\partial L}{\partial v}=\delta_1h_0+\delta_2h_1,\qquad \frac{\partial L}{\partial b}=\delta_1+\delta_2assets/equations/e7070aed32f74b9a.svg
• \mu=\frac1N\sum_i x_i,\qquad\sigma^2=\frac1N\sum_i(x_i-\mu)^2assets/equations/e7193fbf5e735f5d.svg
• s=0+0=0assets/equations/e725afda84fc9ed3.svg
• \delta_t=(0.32)(0.75)=0.24assets/equations/e726c1bb32ed233e.svg
• 1(1)=1assets/equations/e75a800ad85088f5.svg
• C_2=0.36+0=0.36assets/equations/e78e9762a4457abe.svg
• L=4^2=16assets/equations/e7a8c4341fdaf1f8.svg
• i\widetilde C=(0.500000000)(0.000000000)\approx0.000000000assets/equations/e7c8fe1c4efb3692.svg
• g_{1,1}^{(2)}=(-0.809)\times(0.85)\approx -0.68765assets/equations/e7ee971a6dfd8bcc.svg
• w_1^{\mathrm{new}}=0.2+0.1(-1)(1)=0.1assets/equations/e7f1347350326319.svg
• \left\lfloor\frac{59-2}{2}\right\rfloor+1assets/equations/e80bd4fe22509760.svg
• P_1=(27+1)(64)=1792assets/equations/e82763edab004507.svg
• o_t=\sigma(W_o[h_{t-1},x_t]+b_o)assets/equations/e82eb2b55af9249e.svg
• g_2=J'(0.81)=0.81assets/equations/e8429bc60dff5c5c.svg
• P_{\mathrm{dense}}=mn+n=(m+1)nassets/equations/e846c0a1de0fb579.svg
• \begin{bmatrix}1 & 3\\5 & 7\end{bmatrix}assets/equations/e857333332fbdd17.svg
• N_{\mathrm{weights}}=11\times11\times3\times96=34{,}848assets/equations/e8a26a7b03e38af6.svg
• J(\theta)=\frac1N\sum_{i=1}^{N}L_i(\theta)+\lambda\Omega(\theta)assets/equations/e8f2cd5bb532bd92.svg
• \begin{aligned}2\div0.5&=2\div\frac12\\&=2\times2\\&=4\end{aligned}assets/equations/e8f8b190ccce77f7.svg
• h_t=0.25(0.4)+0.75\tanh(0.3)\approx0.318484459assets/equations/e8fba95c9469ec76.svg
• \sigma^2=\frac{1+0+1}{3}\approx0.666666667assets/equations/e913fcfda4146fb6.svg
• W^{(1)}_{1,1,\mathrm{new}}=0.2-0.1(-0.22)\approx 0.222assets/equations/e91b335304219c5c.svg
• \left\lfloor\frac{109+2 \times 0-5}{1}\right\rfloor+1assets/equations/e91c4793e36ecf45.svg
• a_{\mathrm{train}}=[1.6,0,1.2,0]assets/equations/e92cb4c6084d0ae8.svg
• v_2=0.9(-0.1)-0.1(0.81)=-0.171assets/equations/e92dee4c49c58b06.svg
• a=0assets/equations/e93d3c89fb4a0a4b.svg
• \theta_1=1-0.21=0.79assets/equations/e9423903ada8f563.svg
• g_{2,2}^{(1)}=(0.006542085)\times(0)\approx 0assets/equations/e95a4cf560314569.svg
• \theta_1=1+(-0.1)=0.9assets/equations/e96386ed9eb17f75.svg
• \widehat m_2=\frac{0.11}{1-0.9^{2}}=0.578947368assets/equations/e970513a5c078389.svg
• y_{\max}=\max Vassets/equations/e971be750f6cd770.svg
• \delta_z=\frac{\partial L}{\partial z}=e(1)=-2assets/equations/e9802e836c671214.svg
• g_{2,1}^{(1)}=(-0.004869072)\times(1)\approx -0.004869072assets/equations/e99b8cf0ca1131d6.svg
• L=\frac12(o-y)^2,\quad \delta_o=o-yassets/equations/e9baab27f7f33ee2.svg
• b_{o_1,\mathrm{new}}=0.2-0.5(-0.093597941)=0.246798971assets/equations/e9c198f623dedcbc.svg
• s_2=0.9(0.4)+0.1(1^2)=0.46assets/equations/e9da8c8b2ad0696a.svg
• o=wx+b,\qquad L=\frac12(o-y)^2assets/equations/e9fdca67a77fbb78.svg
• N_{\mathrm{out}}=\left\lfloor\frac{252-2}{2}\right\rfloor+1=126assets/equations/ea4835a0c4e1225d.svg
• L=-\sum_i y_i\ln p_i=-\ln p_{true}assets/equations/ea93bf47748eb958.svg
• \delta_{h_i}=f_h\prime(z_{h_i})\sum_jW_{ji}^{(2)}\delta_{o_j}assets/equations/ea9470bc5f0a15d2.svg
• s_1=0.9(0)+0.1(2^2)=0.4assets/equations/ea96f08ec1d03a73.svg
• \mathrm{saving\ percentage}=100\frac{\mathrm{direct}-\mathrm{reduced}}{\mathrm{direct}}assets/equations/ea97c94764a3090c.svg
• z_1=(1\times2)+(2\times1)+0.5assets/equations/eaa341baa6e16b1f.svg
• f\prime(-2)=0.1assets/equations/eaab86572902924a.svg
• J(\theta)=0.7\theta^3assets/equations/eaad5a7869c46a71.svg
• z_{XOR}=1-2(1)-0.5=-1.5assets/equations/eac20b8fec07829f.svg
• \Delta w_1=-0.1(0.3)=-0.03assets/equations/eaf1d1bad63a2309.svg
• R_1=(2)\times(1)+(2)\times(0)+(1)\times(-1)=1assets/equations/eaf8c509090f51f1.svg
• \begin{bmatrix}4\end{bmatrix}assets/equations/eaf8e658e6675411.svg
• \mathrm{ReLU}(-2)=\max(0,-2)=0assets/equations/eb663510f8ab435b.svg
• P_3=4{,}608+32=4{,}640assets/equations/ebc484850d4827e3.svg
• z_{h_1} = (0.5 \times 1) + (0.5 \times 1) + (0) = 1assets/equations/ebc8dda9737caeb3.svg
• P_{\mathrm{total}}=2-1=1assets/equations/ebca391be173fb93.svg
• \frac{\partial L}{\partial u}=(-0.25)(1)+(-0.5)(2)=-0.25-1=-1.25assets/equations/ebd6a8221b328c2e.svg
• a=\tanh(0.5)=\frac{1.718281828}{3.718281828}\approx0.462117157assets/equations/ebe33ccb05db4c46.svg
• (3-2)=1assets/equations/ebe985168f035d36.svg
• \lambda\operatorname{sign}(-0.5)=0.1(-1)=-0.1assets/equations/ebeb8e1558332839.svg
• g_{1,2}^{(1)}=(-0.004869072)\times(1)\approx -0.004869072assets/equations/ebfca408333d760d.svg
• W^{(1)}_{2,2,\mathrm{new}}=0.4-0.5(-0.004357892)\approx 0.402178946assets/equations/ec08f8a3c878658a.svg
• W^{(1)}_{2,4,\mathrm{new}}=0.5-0.5(0.005810523)\approx 0.497094738assets/equations/ec111f9ae9bb85b7.svg
• g_{3,3}^{(1)}=\frac{\partial L}{\partial W^{(1)}_{3,3}}=\delta_{h_3}x_3assets/equations/ec47dc9f5403d781.svg
• y=x+F(x)assets/equations/ec48bb473e2d7934.svg
• t-o=1-1=0assets/equations/ec562da655acc385.svg
• N_{\mathrm{biases}}=84assets/equations/ec58135349b68114.svg
• s_2=0.9(0.4)+0.1(1)^2=0.46assets/equations/ec584873dea573e6.svg
• \mathrm{MACs}_{\mathrm{bottleneck}}=131{,}072+589{,}824=720{,}896assets/equations/ec8abadaa9257a67.svg
• g_1=(\hat y_0-y)x=(-2)(2)=-4assets/equations/eca6d4b6240460b4.svg
• \operatorname{step}(z)=\begin{cases}1&z\ge0\\0&z<0\end{cases}assets/equations/ecb2a58d23e67edf.svg
• N_{\mathrm{out}}=\left\lfloor\frac{16+0-1}{2}\right\rfloor+1assets/equations/ecbcadd42ab07705.svg
• \widehat v_1=\frac{0.04}{1-0.99^{1}}=4assets/equations/ecdf53b308191b04.svg
• f\odot C=\begin{bmatrix}0.1(2)\\0.9(1)\end{bmatrix}=\begin{bmatrix}0.2\\0.9\end{bmatrix}assets/equations/ece27e6278683163.svg
• g_{\mathrm{penalty}}=0.1\times0.4=0.04assets/equations/ed23175a058690ac.svg
• \frac{\partial L}{\partial z_2}=0.8-1=-0.2assets/equations/ed27397a5b5cfeec.svg
• z_{h_4} = (0.4 \times 0.4) + (0.2 \times 0.3) + (0.2 \times 0.5) + (0.8 \times -0.4) + (0.6) = 0.6assets/equations/ed52dc7ef4c0c34d.svg
• a=0.2(1)+0.5(0.2)+0=0.2+0.1=0.3assets/equations/ed7061874c7df25f.svg
• \frac{\partial L}{\partial w}=(-0.3)\times3=-0.9assets/equations/ed97f79ab4475569.svg
• R_1=(2)\times(1)+(1)\times(-1)=1assets/equations/ed9de22182a739e4.svg
• \frac{\partial z}{\partial b}=0+1=1assets/equations/edaff0abc794c10d.svg
• \hat m_t=\frac{m_t}{1-\beta_1^t},\qquad\hat s_t=\frac{s_t}{1-\beta_2^t}assets/equations/eddaf3b9789092ac.svg
• h_{1}=\frac{1}{1+e^{-(1.4)}}=0.802183889assets/equations/eddfa3ffedc7ea27.svg
• y_{max}=\max_{(i,j)\in W}x_{ij},\quad y_{avg}=\frac1{|W|}\sum_{(i,j)\in W}x_{ij}assets/equations/edf81d51873a0dfc.svg
• g_{j,\mathrm{total}}=g_{j,\mathrm{data}}+\lambda\operatorname{sign}(w_j)assets/equations/edfc42b95078bb2c.svg
• P_6=48{,}000+120=48{,}120assets/equations/ee00c5a42d90e348.svg
• 3(3)(3)=27assets/equations/ee13fc57d61624d3.svg
• z^{(1)}_2=0.3(1)+0.7(0)+0.2=0.5assets/equations/ee1bd9d85fa15d84.svg
• e^{-2}+e^{2}\approx0.135335283+7.389056099=7.524391382assets/equations/ee253631806c17e2.svg
• R_2=(2)\times(0)+(1)\times(-1)=-1assets/equations/ee2b941345fd8cb4.svg
• e^{-z}\approx7.389056099assets/equations/ee2ca92f60ab98c1.svg
• o_{1}=\frac{1}{1+e^{-(1.263830247)}}=0.779684759assets/equations/ee7303ec679c99fc.svg
• h_{2}=\frac{1}{1+e^{-(0.52)}}=0.627147766assets/equations/ee84a1f5182572dc.svg
• g_{1,\mathrm{total}}=0.2+(-0.1)=0.1assets/equations/ee9041d822361870.svg
• z_{h_1} = (0.8 \times 0.4) + (-0.5 \times 0.3) + (0.1 \times 0.5) + (-0.2 \times -0.4) + (0.6) = 0.9assets/equations/ee9cf86ec848950c.svg
• g_{2,\mathrm{total}}=0.3+(-0.04)=0.26assets/equations/eeb80cfd582d91ed.svg
• \theta_{\mathrm{look}}=\theta_{t-1}+\beta v_{t-1},\quad g_t=J\prime(\theta_{\mathrm{look}})assets/equations/eebc0b956dd1a540.svg
• z_{h_1} = (0.1 \times 0.5) + (0.4 \times 0.2) + (0.2 \times 0.1) + (0.1) = 0.25assets/equations/eecbe3a4008b0943.svg
• \begin{bmatrix}1 & 2\\0 & 1\end{bmatrix}assets/equations/eed57e81393ab435.svg
• 1000/100=10assets/equations/eee877d70ea14115.svg
• \begin{aligned}0.9(10)+0.1(20)&=9+2\\&=11\end{aligned}assets/equations/eef093497c3c04cc.svg
• z_{o_1} = (0.8 \times 1.1) + (0.2) = 1.08assets/equations/ef13baba40338fd0.svg
• (-2)^2=4,\quad 1^2=1,\quad 0.5^2=0.25assets/equations/ef16358fc81a0cdc.svg
• w_2=w_1-\eta g_2=1.4-0.1(-2.4)=1.64assets/equations/ef24d99282ed8b6f.svg
• e=z-y=2-4=-2assets/equations/ef2733b3de84bfed.svg
• b_{\mathrm{new}}=0+0.5\times1=0.5assets/equations/ef3a575874682e63.svg
• H_2(p,q)=-[0.75(-1)+0.25(-1)]=1\ \text{bit}assets/equations/ef4d3bcee474b612.svg
• C_{21}=3(2)+4(1)=6+4=10assets/equations/ef4f6c3c29a39671.svg
• \begin{aligned}2+3^2\times4&=2+9\times4\\&=2+36\\&=38\end{aligned}assets/equations/ef5577a23a205b11.svg
• \frac{\partial L}{\partial u}=(-0.25)(1)+(-0.5)(2)=-1.25assets/equations/ef5c8122a0e849c2.svg
• \nabla_\theta J=\begin{bmatrix}\partial J/\partial\theta_1\\\vdots\\\partial J/\partial\theta_d\end{bmatrix}assets/equations/ef6399f58ca20acf.svg
• W^{(1)}_{4,1,\mathrm{new}}=0.4-0.5(-0.001713099)\approx 0.400856549assets/equations/ef6569978fef20ab.svg
• b_{h_2,\mathrm{new}}=0.35-0.5(0.009954255)=0.345022872assets/equations/ef67240d0fdef037.svg
• z_{h_2}=1+1-1.5=0.5\Rightarrow h_2=1assets/equations/ef884a3d2271fe60.svg
• L_{\mathrm{half}}=\frac12\sum_i e_i^2assets/equations/ef8982ea86700e61.svg
• N_{\mathrm{out}}=\left\lfloor\frac{N+2P_{\mathrm{pad}}-K}{S}\right\rfloor+1assets/equations/ef936ccc7475be1a.svg
• e^{-2}-e^{2}\approx0.135335283-7.389056099=-7.253720816assets/equations/efa746faa7f9257e.svg
• a'_j=\frac{m_ja_j}{q},\quad o=\sum_jw_ja'_j+b,\quad g_j=(o-y)a'_jassets/equations/efaf3726aa8b79f8.svg
• g_{1,3}^{(1)}=(-0.009617203)\times(0.5)\approx -0.004808601assets/equations/efb4f1bd62923583.svg
• o_1=2(1.1)=2.2,\qquad o_2=-(1.1)=-1.1assets/equations/efc23fa7e963abec.svg
• g_{2,1}^{(1)}=(-0.275)\times(1)\approx -0.275assets/equations/efc4ca15325664fa.svg
• \delta_{h_1} = -0.024764888 \times 0.196611933 = -0.004869072assets/equations/efc565ce467804a9.svg
• P_5=204{,}800+128=204{,}928assets/equations/f00380ddc73ffc90.svg
• b_{o_1,\mathrm{new}}=0-0.05(-0.809000000)=0.040450000assets/equations/f01c357a6fdbd536.svg
• L_{\mathrm{data}}=\frac12(-0.2)^2=\frac12(0.04)=0.02assets/equations/f025cfcfa02c965d.svg
• g_{2,1}^{(2)}=(-0.038098237)\times(0.593269992)\approx -0.02260254assets/equations/f032d0a14a103c60.svg
• h_t=z_t\odot h_{t-1}+(1-z_t)\odot\widetilde h_tassets/equations/f047371c1b66f6fe.svg
• z_t=ux_t+vh_{t-1}+b,\qquad h_t=\tanh(z_t)assets/equations/f064c725161475e9.svg
• Y_{2,1}=(2)+(0)+(1)=3assets/equations/f077b3678b0e1f35.svg
• Y_{1,2}=(20)+(26)+(13)=59assets/equations/f077bed2b9d89eff.svg
• a=\frac1{1.135335283}\approx0.880797078assets/equations/f08d0c34f5e2fdf7.svg
• P=(3\times3\times C_{\mathrm{in}}+1)\times12assets/equations/f0b530e411679c54.svg
• o-y=0.7-1=-0.3assets/equations/f0bd8184760ad600.svg
• z_o=1-2(1)-0.5=-1.5\Rightarrow o=0assets/equations/f0db03c68af4ca9e.svg
• \lVert g\rVert_2=\sqrt{\sum_i g_i^2}assets/equations/f0dbe784a7be0b2f.svg
• W^{(1)}_{2,2,\mathrm{new}}=0.3-0.5(0.000995425)\approx 0.299502287assets/equations/f0ef9e4142c40173.svg
• J(\theta-\eta\nabla J)\approx J(\theta)-\eta\|\nabla J\|_2^2assets/equations/f0fe0e6b83fb6f66.svg
• [1(2),\;0(-1),\;1(0),\;1(3)]=[2,0,0,3]assets/equations/f0ff08cea46f6163.svg
• g_{4,3}^{(1)}=\frac{\partial L}{\partial W^{(1)}_{4,3}}=\delta_{h_4}x_3assets/equations/f10479c14422cd43.svg
• z=0.6+(-0.8)=-0.2assets/equations/f10c0127832440d1.svg
• \sum_i p_i=1,\qquad\mathbb E[X]=\sum_i p_i x_iassets/equations/f1386430c4600e23.svg
• 1+e^{-1}\approx1.367879441assets/equations/f1586fb666c5061e.svg
• \widehat v_1=\frac{0.004}{1-0.999^{1}}=4assets/equations/f16a835e4241c7ed.svg
• g_1=(-0.6)(2)=-1.2,\qquad g_2=(-0.6)(0)=0assets/equations/f17168c45fd84cbe.svg
• g_{2,2}^{(1)}=(0.017487356)\times(0.4)\approx 0.006994942assets/equations/f1825dbaa53d6e64.svg
• v_2=-0.09-0.09=-0.18assets/equations/f18fc7f5dc715d05.svg
• \theta_{\mathrm{look}}=-0.5+0.9(0)=-0.5assets/equations/f19658c31004feb6.svg
• h_2=\tanh(1.1)\approx0.800499022assets/equations/f19c8c78d2c06e2b.svg
• (-2)^2=4,\quad 1^2=1,\quad 0.5^2=0.25assets/equations/f1d2b6bccfe10f10.svg
• e^{0}+e^{0}\approx1.000000000+1.000000000=2.000000000assets/equations/f1f2e3760e822d0f.svg
• \text{Sum}=C_1\approx -0.046798971assets/equations/f1f4d5705234aba2.svg
• \delta_{o_1} = 1 \times (0.513866171-1) \times 0.249807729 = -0.121439988assets/equations/f1f5b1787fc59def.svg
• z_{o_1} = (0.4 \times 0.5) + (0.5 \times 0.3) + (0.1) = 0.45assets/equations/f22e357a250f5762.svg
• W^{(2)}_{2,2,\mathrm{new}}=0.55-0.5(-0.022740242)\approx 0.561370121assets/equations/f2313d50d5e6909b.svg
• s=(-2)+(0.5)=-1.5assets/equations/f2345e5dfa56c33e.svg
• \hat y=-1+(3)=2assets/equations/f2369bf69098b26c.svg
• g_2=(0.2-2)\times2=(-1.8)\times2=-3.6assets/equations/f262bc5d6c22d8bb.svg
• w_{\mathrm{new}}=1-0.1(-2.5)=1.25assets/equations/f263219d999d6318.svg
• z_{\mathrm{new}}=1.4(2)+0.2=3assets/equations/f27e43a3756292cb.svg
• w_1=1-0.1(-1)=1.1assets/equations/f2a537796fbcbe31.svg
• w_1=0-0.1(-1)=0.1assets/equations/f2b5721e0e1c67b4.svg
• \begin{bmatrix}0 & 4\\0 & -2\end{bmatrix}assets/equations/f2f63d345efa8255.svg
• A=\begin{bmatrix}1&2\\3&4\end{bmatrix}assets/equations/f32200eba37c0ea5.svg
• g_{4,4}^{(1)}=(-0.004282747)\times(-0.4)\approx 0.001713099assets/equations/f326d58085d05315.svg
• J=L+\lambda\sum_j|w_j|,\quad w_{new}=w-\eta\left[g_{data}+\lambda\operatorname{sign}(w)\right]assets/equations/f35555b2b4b6691c.svg
• g_b=\frac{\partial L}{\partial b}=\delta_z\frac{\partial z}{\partial b}=(-2)(1)=-2assets/equations/f357d7ca8a5a76dc.svg
• z=0.2-0.1+0.1=0.2assets/equations/f3991f5f0c7df172.svg
• \operatorname{ReLU}(z)=\max(0,z)=\begin{cases}z&z\geq0\\0&z<0\end{cases}assets/equations/f3a2f94af1e35f2f.svg
• Y_{1,2}=(0)+(0)+(0)=0assets/equations/f3a79dadbe7e581b.svg
• e^{-z}=e^{-2.2}\approx0.110803158assets/equations/f3e8120863f30069.svg
• P_{\mathrm{conv}}=(K^2C_{\mathrm{in}}+1)C_{\mathrm{out}}assets/equations/f3eea0634c9f3976.svg
• b_{h_4,\mathrm{new}}=0.6-0.5(-0.004282747)=0.602141373assets/equations/f3f5c6e854f64acf.svg
• z^{(2)}=0.5(0.3)+0.6(0.5)+0.1assets/equations/f3fff5a6e4a523e4.svg
• L=-\ln(0.731058579)\approx0.313261688assets/equations/f41a1bed5052539d.svg
• \frac{dL}{dw}=\frac{dL}{dz}\frac{dz}{dw}=(2z)(3)=8\times3=24assets/equations/f43476deed69b900.svg
• \delta_{h_1} = -0.22 \times 1 = -0.22assets/equations/f452f757c3f0207e.svg
• \begin{aligned}10^{-3}&=\frac1{10^3}\\&=\frac1{1000}\\&=0.001\end{aligned}assets/equations/f45c43d6b073f1bb.svg
• P(A\mid B)=\frac{P(A\cap B)}{P(B)}=\frac{P(B\mid A)P(A)}{P(B)}assets/equations/f464138921aade18.svg
• N_{out}=\left\lfloor\frac{N+2P-K}{S}\right\rfloor+1assets/equations/f467d926f905df6d.svg
• H_2(p,q)=\frac{H(p,q)}{\ln2}assets/equations/f4684a5e966eb42a.svg
• g_{4,2}^{(1)}=(-0.004282747)\times(0.3)\approx -0.001284824assets/equations/f46fd9e254c79580.svg
• 64(8)(3)(3)(4)=18432assets/equations/f478849ec0c64569.svg
• W^{(2)}_{1,1,\mathrm{new}}=0.1-1(0.010408643)\approx 0.089591357assets/equations/f48683b41d55c168.svg
• \delta_1=\delta_2v=(-0.5)(0.5)=-0.25assets/equations/f49beec63f88f3a8.svg
• a=\frac1{8.389056099}\approx0.119202922assets/equations/f4a3cb96f1098e79.svg
• N_{\mathrm{out}}=\left\lfloor\frac{256+2(0)-5}{1}\right\rfloor+1=252assets/equations/f4b85116af3d067a.svg
• w_{\mathrm{new}}=0.4-0.2(-0.36)=0.4+0.072=0.472assets/equations/f4be65ea94b49c52.svg
• W^{(1)}_{2,2,\mathrm{new}}=0.3-0.2(0.006994942)\approx 0.298601012assets/equations/f4cac41f24ea1d22.svg
• \sigma^2=\frac{0+0+0}{3}\approx0.000000000assets/equations/f4e20a1682fadce3.svg
• 1.9\times(-3)=-5.7assets/equations/f4e294dd1ab7a8aa.svg
• L=\frac12(z-y)^2assets/equations/f50d7e1cde6462d5.svg
• y=3(2x+1)+4=6x+3+4=6x+7assets/equations/f54676c512498a38.svg
• a_{\mathrm{candidate}}=W_hx_t+U_h(r_th_{t-1})+b_hassets/equations/f549a71ed6a573d3.svg
• L=\frac1n\sum_j(o_j-y_j)^2assets/equations/f5544f3d0af79c26.svg
• [3-5,3-5,7-5,7-5]=[-2,-2,2,2]assets/equations/f5609e02cf210acd.svg
• e^{-z}\approx1.000000000assets/equations/f5ab9de96f693147.svg
• b_{h_2,\mathrm{new}}=1-1(0.003509506)=0.996490494assets/equations/f5d3af77f03c0547.svg
• (1-z)\widetilde h=0.75(0.291312612)\approx0.218484459assets/equations/f5dda0f46575aa22.svg
• g_2=0.81-(0)=0.81assets/equations/f61619e280f936a0.svg
• z_{h_2} = (0.2 \times 0.4) + (0.4 \times 0.3) + (0.2 \times 0.5) + (0.5 \times -0.4) + (0.6) = 0.7assets/equations/f62211e96d9d3780.svg
• \theta_{\mathrm{look}}=-0.35+0.9(0.15)=-0.215assets/equations/f623b6f68ec5b35a.svg
• W^{(2)}_{1,1,\mathrm{new}}=0.2-0.2(-0.066771834)\approx 0.213354367assets/equations/f6338ce1a52bb766.svg
• 54.598150033-1\approx53.598150033assets/equations/f63e045301bde055.svg
• i\widetilde C=(0.731058579)(0.761594156)\approx0.556769941assets/equations/f64edf0a473f6a00.svg
• \frac{dL}{dw}=\frac{dL}{dz}\frac{dz}{dw}=(2z)(3)=8\times3=24assets/equations/f69ed1214540da40.svg
• a=0.2(1)+0.5(0.2)+0=0.3assets/equations/f6b1201a9f34684a.svg
• z_t=W_{hy}h_t+b_y,\qquad p_t=\operatorname{softmax}(z_t)assets/equations/f6b889b0139771ed.svg
• C_t=0.2+0.63=0.83assets/equations/f6bfe5d9884a48c2.svg
• W^{(1)}_{1,1,\mathrm{new}}=0.11-0.05(-0.22652)\approx 0.121326assets/equations/f709770d92b5a6d7.svg
• W^{(1)}_{3,1,\mathrm{new}}=0.3-0.5(0.005197622)\approx 0.297401189assets/equations/f7218a8971ce6230.svg
• N_{\mathrm{out}}=\left\lfloor\frac{31+2(1)-5}{2}\right\rfloor+1assets/equations/f73307dcf9d893e8.svg
• z_{o_1} = (0.6 \times 0.802183889) + (0.1) = 0.581310333assets/equations/f73d73bef40a3e53.svg
• \text{Sum}=C_1\approx -0.12135assets/equations/f74169ed59976f22.svg
• h_{1}=\frac{1}{1+e^{-(0.9)}}=0.710949503assets/equations/f76f739a80a9361b.svg
• g_2=-0.215-(1)=-1.215assets/equations/f77fca68d44fde4d.svg
• 27(16)=432assets/equations/f7cebde5567e220a.svg
• R_2=(1)\times(0)+(0)\times(-1)=0assets/equations/f7d66e56559f9de7.svg
• \hat y=wx,\qquad J=\frac12(\hat y-y)^2+\frac\lambda2w^2assets/equations/f7ed754c852735b2.svg
• a'_2=\frac0{0.75}=0,\qquad a'_3=\frac0{0.75}=0assets/equations/f7f43ac5cd2d312a.svg
• \theta_2=0.9-\frac{0.1(1)}{\sqrt5}\approx0.855278640assets/equations/f835d09c27f0313a.svg
• b_{h_1,\mathrm{new}}=-0.4-0.9(0.008724562)=-0.407852106assets/equations/f8478e0f34b4c2ea.svg
• \widehat m_1=\frac{0.2}{1-0.9^{1}}=2assets/equations/f8975b50fed0845b.svg
• z_{h_2} = (0.3 \times 1) + (0.7 \times 0) + (0.2) = 0.5assets/equations/f8a809dfb5349555.svg
• 11(11)(3)=363assets/equations/f8b467d436e9d44e.svg
• b_{o_2,\mathrm{new}}=0.6-0.5(-0.038098237)=0.619049118assets/equations/f8b71ed80001f62a.svg
• \sigma_B^2=\frac{(-1)^2+1^2}{2}=\frac{1+1}{2}=1assets/equations/f8c2f38c1c4da922.svg
• w_2=0.1-0.1(-3.6)=0.1+0.36=0.46assets/equations/f8c5ce8387c2e254.svg
• n_{\mathrm{flat}}=30\times30\times32=28{,}800assets/equations/f8d6a23f5595005e.svg
• g=(\hat y-y)xassets/equations/f90a893376d444b0.svg
• W^{(1)}_{2,1,\mathrm{new}}=0.12-0.05(-0.2427)\approx 0.132135assets/equations/f91b7787569a2882.svg
• \text{biases}=3,\qquad P_{\mathrm{dense}}=192+3=195assets/equations/f927a7fa1a913d20.svg
• z_{h_1}=0+1-0.5=0.5\Rightarrow h_1=1assets/equations/f9475f1991e07af3.svg
• \delta_{h_j}=\left(\sum_k W_{2,kj}\delta_{o_k}\right)f'(z_{h_j})assets/equations/f94e1705f6027563.svg
• \delta_j^h=f\prime(z_j^h)\sum_k w_{kj}^{out}\delta_k^{out},\quad\frac{\partial L}{\partial w_{ji}}=\delta_j a_i,\quad\frac{\partial L}{\partial b_j}=\delta_jassets/equations/f960780a672401ea.svg
• g_{2,2}^{(1)}=(-0.014526308)\times(0.3)\approx -0.004357892assets/equations/f974f6695dcb0ab0.svg
• J\prime(w)=3(2)w^{2-1}+0assets/equations/f98b439990365e5d.svg
• P_{\mathrm{total}}=36+438+18=492assets/equations/f9a71aaf4ed27273.svg
• U_{\mathrm{keep}}=E\left\lceil\frac NB\right\rceilassets/equations/f9d09d1e5734cccd.svg
• \left\lfloor\frac{63+2 \times 0-3}{1}\right\rfloor+1assets/equations/f9e5ee2c7c7e1bbb.svg
• g_{2,2}^{(2)}=(0.082533822)\times(0.782449776)\approx 0.06457857assets/equations/fa16e98e65914abf.svg
• \text{weights}=3\times3\times3\times2=54assets/equations/fa20139330cc7a23.svg
• \delta_{o_1} = 1 \times (0.473888899-1) \times 0.24931821 = -0.131169078assets/equations/fa2e026fc3449bb0.svg
• 1.000000000-1\approx0.000000000assets/equations/fa618ba26a236aab.svg
• \operatorname{LeakyReLU}(z)=\begin{cases}z&z\geq0\\\alpha z&z<0\end{cases},\qquad 0<\alpha<1assets/equations/fa7213ea51b43af7.svg
• s=0.8+(-0.2)+1.5=0.6+1.5=2.1assets/equations/fa72e2cee0620f1c.svg
• h_t=z_t\odot h_{t-1}+(1-z_t)\odot\widetilde h_tassets/equations/fa76004105af78bd.svg
• \delta_{h_1} = -0.024764888 \times 0.196611933 = -0.004869072assets/equations/faa954804a7ac56f.svg
• z_t=\sigma(W_zx_t+U_zh_{t-1}+b_z)assets/equations/facdae4e8c80fb1f.svg
• W^{(1)}_{3,4,\mathrm{new}}=0.5-0.5(-0.005197622)\approx 0.502598811assets/equations/facfaa0bc7aa72e7.svg
• \frac{3-5}{2}=-1assets/equations/fb029ad3b6f0a91a.svg
• C_2=(0.5)\times(-0.038098237)\approx -0.019049118assets/equations/fb089b7c1d8f6873.svg
• h_{\mathrm{new}}=0+(0.291312612)\approx0.291312612assets/equations/fb12f65fb3a6f99d.svg
• \hat y=f^{(2)}(W^{(2)}h+b^{(2)})assets/equations/fb13b3127fe27394.svg
• \sum_i|w_i|=2+1+0.5=3.5assets/equations/fb36584739f81886.svg
• (2)(4)=8assets/equations/fb39ee363580ed9c.svg
• (1-0)(0.291312612)\approx0.291312612assets/equations/fb448a828d5eb818.svg
• 1-(-0.2)^2=0.96assets/equations/fb671dc68ac70d16.svg
• 1+7.389056099\approx8.389056099assets/equations/fb7c0cac9ced67f0.svg
• Y_{1,1}=(0.2)+(0.2)+(0.2)=0.6assets/equations/fb7d1b049e078f58.svg
• f\prime(-2)=0assets/equations/fbc65c947d4ed97f.svg
• z_{XNOR}=-1+2(0)+0.5=-0.5assets/equations/fbcbb7c54a55ba6f.svg
• X=\begin{bmatrix}1&2&3\\4&5&6\\7&8&9\end{bmatrix},\quad K=\begin{bmatrix}1&0\\0&-1\end{bmatrix}assets/equations/fbe5e0aa8714ed4e.svg
• g_{\mathrm{clip}}=g\min(1,\tau/\|g\|_2)assets/equations/fbf72036323a5940.svg
• \theta_2=-0.4-\frac{0.1\times(0.578947368)}{\sqrt{2.500750375}}=-0.436610353assets/equations/fc06a7b542215246.svg
• N_{\mathrm{out}}=\left\lfloor\frac{109+2(0)-5}{1}\right\rfloor+1=105assets/equations/fc0be4ce48988be6.svg
• b_{h_1,\mathrm{new}}=0.1-0.2(0.004824633)=0.099035073assets/equations/fc185daa7ea868a7.svg
• g_{1,1}^{(1)}=(0.002053687)\times(0.25)\approx 0.000513422assets/equations/fc19919c700c68af.svg
• z=s+b=2.1+0.1=2.2assets/equations/fc2fbdf24fc7fcdc.svg
• h_t=0.5\tanh(C_t)\approx0.159735818assets/equations/fc38db30848509e4.svg
• \sigma(z)=\frac1{1+e^{-z}},\qquad\sigma\prime(z)=\sigma(z)[1-\sigma(z)]assets/equations/fc390cc0a677c2cc.svg
• \delta_{h_2} = 0.026233816 \times 0.24937604 = 0.006542085assets/equations/fc3edfca69eaeef5.svg
• \sigma(0)=\frac1{2.000000000}\approx0.500000000assets/equations/fc3feba9268d5c6f.svg
• g_{1,2}^{(2)}=(-0.093597941)\times(0.668187772)\approx -0.062541assets/equations/fc6c904cdf89e2ca.svg
• \text{L1 penalty}=0.1\times3.5=0.35assets/equations/fc6e588ea36fd61a.svg
• \frac{\partial L}{\partial b_1}=0.064assets/equations/fc73ac9e8421c88b.svg
• v_{j,\mathrm{new}}=\beta v_{j,\mathrm{old}}-\eta g_j,\qquad w_{j,\mathrm{new}}=w_j+v_{j,\mathrm{new}}assets/equations/fc93fbe5bcc88056.svg
• P_{\mathrm{block}}=1792+36928=38720assets/equations/fc9e657e6b2a7ee5.svg
• L=-\left[1\ln(0.731058579)+(1-1)\ln(1-0.731058579)\right]assets/equations/fca94e6b25348fdc.svg
• u_t=\beta u_{t-1}+g_t,\qquad\theta_t=\theta_{t-1}-\eta u_tassets/equations/fcab484ef321b3b2.svg
• p=0.8assets/equations/fcb072c6f4519069.svg
• H(p,q)=-[0.75\ln(0.5)+0.25\ln(0.5)]assets/equations/fcc9ac503722bcd6.svg
• \frac{\partial J}{\partial w}=(\hat y-y)x+\lambda w,\qquad w_{\mathrm{new}}=w-\eta\frac{\partial J}{\partial w}assets/equations/fccbb24d2fd5ed97.svg
• \begin{bmatrix}0 & 1 & 1\\0 & 1 & 0\\1 & -1 & 1\end{bmatrix}assets/equations/fccc1fad371104fc.svg
• |-2|=2,\quad |1|=1,\quad |0.5|=0.5assets/equations/fcceddf63f9341b2.svg
• N_{\mathrm{weights}}=3\times3\times16\times32=4{,}608assets/equations/fcf6aecafd4a3651.svg
• g_{1,2}^{(2)}=(-0.099059551)\times(0.731058579)\approx -0.072418335assets/equations/fd02f5bab6bb789a.svg
• P_6=48{,}000+120=48{,}120assets/equations/fd12e231aa432c56.svg
• J=L+\lambda\sum_j|w_j|,\quad w_{new}=w-\eta\left[g_{data}+\lambda\operatorname{sign}(w)\right]assets/equations/fd296afd4f88edb6.svg
• A=H_oW_oC_{\mathrm{out}}assets/equations/fd2a21cf67217e86.svg
• N_{\mathrm{out}}=\left\lfloor\frac{256+2(0)-5}{1}\right\rfloor+1=252assets/equations/fd4c812f057cfbdc.svg
• h_t=ux_t+vh_{t-1}+b,\qquad L=\frac12(h_2-y)^2assets/equations/fd56e04419aef1fc.svg
• W^{(2)}_{1,2,\mathrm{new}}=0.25-0.5(-0.072418335)\approx 0.286209167assets/equations/fd5e1c3ecdebae0c.svg
• a'=\frac{ma}{q}assets/equations/fda2e84f024891bd.svg
• g_1=(-1)(1)=-1assets/equations/fdbcda3ad0050efb.svg
• 0.2+0.12=0.32assets/equations/fdbe3cda286c3c16.svg
• p_3=\frac{1.000000000}{1.503214724}\approx0.665240956assets/equations/fdcbdba7b8490dd2.svg
• J_{L2}=0.5+0.5=1.0assets/equations/fde60160d7428d14.svg
• \delta_t=(W_{hy}^{T}\delta_t^{out}+W_{hh}^{T}\delta_{t+1})\odot(1-h_t^2),\quad \frac{\partial L}{\partial W_{hh}}=\sum_t\delta_t h_{t-1}^{T}assets/equations/fdf1ee48fdcfd79f.svg
• f\prime(0)=(0.500000000)(1-0.500000000)\approx0.250000000assets/equations/fe0302f0c6aecdca.svg
• P_1=432+16=448assets/equations/fe2e71f3a8efffa4.svg
• L=-(-1.609437912)=1.609437912assets/equations/fe3e4a4ff07bc0e0.svg
• h=0.6\times0.139092448=0.083455469assets/equations/fe422cf3cbda4007.svg
• G_t=G_{t-1}+g_t^2,\quad\theta_t=\theta_{t-1}-\frac{\eta g_t}{\sqrt{G_t}+\epsilon}assets/equations/fe45aacbe9d8627a.svg
• N_{\mathrm{weights}}=4{,}096\times4{,}096=16{,}777{,}216assets/equations/fe4cfd667cc7b7ce.svg
• \mathrm{saving}=120422400-12443648=107978752assets/equations/fe529a7c8d14263e.svg
• z=0.8+0.1=0.9assets/equations/fe5a369324074a4c.svg
• p(1-p)=0.731058579\times0.268941421=0.196611933assets/equations/fe81a8ae8e93cc3b.svg
• z_2=((-1)\times2)+(3\times1)+(-0.5)assets/equations/fe996afd770f8834.svg
• C_2=(0.55)\times(-0.038098237)\approx -0.02095403assets/equations/fea577940f5fb0f2.svg
• H(p,q)=-\sum_i p_i\ln q_iassets/equations/feb21e9c89a912d7.svg
• \nabla_{\hat y}\mathrm{SSE}=[0.4,-0.6,-1.2]assets/equations/fec22e50bd202c79.svg
• z_{h_1} = (0.15 \times 0.05) + (0.2 \times 0.1) + (0.35) = 0.3775assets/equations/feca30d72e6cdb12.svg
• g_{4,1}^{(1)}=(-0.004282747)\times(0.4)\approx -0.001713099assets/equations/fed96721b701f460.svg
• g_{1,2}^{(1)}=(0.004824633)\times(0.4)\approx 0.001929853assets/equations/fef8b9d9db76a45d.svg
• m_1=0.9(0)+0.1(-1)=-0.1assets/equations/ff0acbc11515087e.svg
• W^{(1)}_{1,2,\mathrm{new}}=0.2-0.2(0.001929853)\approx 0.199614029assets/equations/ff165bc32cce96bf.svg
• P_A=(1\times1\times8+1)\times4=9\times4=36assets/equations/ff251b0ed25a7ae7.svg
• \delta_o=0.2-1=-0.8assets/equations/ff295007085f430d.svg
• D=0.135335283+0.367879441+1.000000000\approx1.503214724assets/equations/ff6afad7c399e8cd.svg
• \text{weights}=8\times3=24,\quad\text{biases}=3,\quad P_{\mathrm{dense}}=27assets/equations/ff79309a11966950.svg
• P_2=(576+1)(64)=36928assets/equations/ff96ebed9174c6f1.svg
• w_2=0.1-0.1(-3.6)=0.1+0.36=0.46assets/equations/ff99526a93e31696.svg
• z_{o_1} = (0.1 \times 0.760423002) + (0.24 \times 0.782449776) + (1) = 1.263830247assets/equations/ffac566fd3026595.svg
• \sqrt{s_1}=\sqrt{0.4}=0.632455532assets/equations/ffb5003b52e07ea4.svg
• A=\begin{bmatrix}1&-2\\0&3\end{bmatrix}assets/equations/ffb51554b200bfc3.svg
• \hat s_1=\frac{0.004}{1-0.999^1}=\frac{0.004}{0.001}=4assets/equations/ffb98ca5eb20d577.svg
• \delta_{o_1} = 1 \times (0.75136507-0.01) \times 0.186815602 = 0.138498562assets/equations/ffde7979414e34c0.svg
• \widetilde C_t=\tanh(W_C[h_{t-1},x_t]+b_C),\quad o_t=\sigma(W_o[h_{t-1},x_t]+b_o)assets/equations/ffe858586bf2d707.svg
• p_i=\frac{e^{z_i-m}}{\sum_je^{z_j-m}},\quad m=\max_j z_j;\qquad\sigma(z)=\frac1{1+e^{-z}}assets/equations/recovered-051b60342e33ce99.svg
• \mathrm{ReLU}(z)=\max(0,z)assets/equations/recovered-1c6ec1b1ec321f01.svg
• \begin{bmatrix}2&3&8\\2&2&4\\1&2&1\end{bmatrix}assets/equations/recovered-217f7d6e6c6e566c.svg
• \begin{bmatrix}2&3&8&2&3\\2&2&4&4&6\\1&2&1&2&9\\7&5&3&3&2\\2&4&3&5&4\end{bmatrix}assets/equations/recovered-26a644e9131b783f.svg
• z=w^Tx+b,\quad w\leftarrow w+(t-y)x,\quad b\leftarrow b+(t-y)assets/equations/recovered-31e9ee08177b99d9.svg
• \begin{bmatrix}0.7&0.8&0.9\\0.75&0.85&0.95\end{bmatrix}assets/equations/recovered-327612e026ce8f89.svg
• h_j=\sigma(\sum_iW_{ji}x_i+0.6),\quad o_k=\sigma(\sum_jV_{kj}h_j+0.6),\quad E=\frac12\sum_k(o_k-t_k)^2assets/equations/recovered-356678101dc6b13b.svg
• \frac{\partial J}{\partial w}=\frac1m\sum_ie_ix_i,\quad\frac{\partial J}{\partial b}=\frac1m\sum_ie_i,\quad e_i=wx_i+b-y_iassets/equations/recovered-4886822a6f85a414.svg
• \begin{bmatrix}0.1&0.25&0.4&0.55\\0.15&0.3&0.45&0.6\\0.2&0.35&0.5&0.65\end{bmatrix}assets/equations/recovered-4cd053daebb13329.svg
• \begin{bmatrix}38&67\\70&74\end{bmatrix}assets/equations/recovered-58e9e7ec888c133f.svg
• g=w+2,\quad v_1=\beta v_0+\eta g,\quad w_1=w_0-v_1assets/equations/recovered-5b00da4482787868.svg
• P_{conv}=(F_hF_wC_{in}+1)C_{out},\quad P_{dense}=(n_{in}+1)n_{out}assets/equations/recovered-69756abf2018c02c.svg
• h_t=\operatorname{ReLU}(x_tW_{xh}+h_{t-1}W_{hh}+b)assets/equations/recovered-88904d5970c9803f.svg
• \begin{bmatrix}1&2&9\\3&3&2\\3&5&4\end{bmatrix}assets/equations/recovered-99c4e3c248d2285d.svg
• W_{xh}=\begin{bmatrix}0.2&0.4\\0.1&0.3\end{bmatrix}assets/equations/recovered-a3abd76a293a016b.svg
• \begin{bmatrix}0.2&0.4&-0.5\\-0.3&0.1&0.2\end{bmatrix}assets/equations/recovered-af190f1cb4fd05fd.svg
• L_{class}=-\sum_ky_k\log p_k,\qquad L_{reg}=\frac1n\sum_i(\hat y_i-y_i)^2assets/equations/recovered-bb4226c392bbf68f.svg
• E=\frac12(o-t)^2,\quad\delta_o=(o-t)o(1-o),\quad\frac{\partial E}{\partial w_k}=\delta_o h_j,\quad w_k^{new}=w_k-\eta\frac{\partial E}{\partial w_k}assets/equations/recovered-bca4073f798d2cce.svg
• \begin{bmatrix}1&2&1\\7&5&3\\2&4&3\end{bmatrix}assets/equations/recovered-c03180114d7dc748.svg
• W_{hh}=\begin{bmatrix}0.5&0.6\\0.3&0.4\end{bmatrix}assets/equations/recovered-c38f9bda7b0fbd25.svg
• O=\left\lfloor\frac{N+2P-F}{S}\right\rfloor+1assets/equations/recovered-cbf0b1f38ebaaccb.svg
• p=\operatorname{softmax}(Wh+b),\qquad L=-\sum_{k=1}^3y_k\log p_kassets/equations/recovered-ce62d9e3abab42e3.svg
• g=w-3,\quad v_1=\beta v_0+\eta g,\quad w_1=w_0-v_1assets/equations/recovered-d398e0121f17d787.svg
• \begin{bmatrix}1&0&1\\2&4&0\\6&4&2\end{bmatrix}assets/equations/recovered-dfae566e77e4351c.svg
• \begin{bmatrix}8&2&3\\4&4&6\\1&2&9\end{bmatrix}assets/equations/recovered-fd5db35a9f88c331.svg
• P=T-Hcn/math/00cfb5dffef469d2.svg
• T_{new}=\min(B,T+r\Delta t)cn/math/017cf4ec943dd0f5.svg
• S=G e^{-G}cn/math/01e46e63e1b256b9.svg
• U_W\approx\min\left(1,\frac{WT_t}{T_t+2T_p+T_a}\right)cn/math/041cbb2263043e29.svg
• T_f=L/R,\quad G=\lambda T_f,\quad S=Ge^{-2G},\quad \lambda_{success}=S/T_f=\lambda e^{-2G}cn/math/043fb89e4a6dc8db.svg
• T_{\mathrm{frame}}=\frac{L}{R}cn/math/045768b5f250c7fa.svg
• N=2^{32-p}cn/math/06561cae3cdafd6c.svg
• R(x)=x^rM(x)\bmod G(x),\quad T=x^rM\oplus Rcn/math/08451acce5e9e980.svg
• d(a,b)=\mathrm{weight}(a\oplus b)cn/math/08c67b82aac87a74.svg
• T_f=L/R,\quad G=\lambda T_f,\quad S=Ge^{-G},\quad \lambda_{success}=S/T_f=\lambda e^{-G}cn/math/094350050f7316f3.svg
• \mathrm{IPv4\ header\ bytes}=4\,\mathrm{IHL}cn/math/0abf62f351ee6ca7.svg
• 2^h-2\geq H,\quad p=32-hcn/math/0d2799e2eb681b5f.svg
• \text{ordinary usable hosts}=2^h-2cn/math/0d9d5151a6681f69.svg
• \frac{dS}{dG}=e^{-2G}(1-2G)cn/math/0e9a0d697e50f118.svg
• U_{SW}=\frac{T_t}{T_t+2T_p+T_a}cn/math/0ed523dd39640c27.svg
• k=\min(n,10),\qquad K\in\{0,1,\ldots,2^k-1\}cn/math/0f0e79cc7f4deaac.svg
• W_{new}=\max(0,\min(rwnd,cwnd)-F)cn/math/13bf9aaf2ed709a3.svg
• d(a,b)=\operatorname{weight}(a\oplus b)cn/math/15901a0538322dc0.svg
• \text{candidate via neighbour}=\text{link cost}+\text{advertised distance}cn/math/16a3dc8e6e4edd60.svg
• T_{object}=T_{setup}+RTT+T_{transmission}cn/math/174653a5aa0dead0.svg
• N=2^m,\quad W_{GBN}\le N-1,\quad W_{R,GBN}=1cn/math/17ff91ae191f528e.svg
• L_{mesh}=n(n-1)/2cn/math/1aaf8e0c5d7c5d53.svg
• h=32-p,\quad N=2^hcn/math/1b1ff40e2008c7a0.svg
• \text{candidate}=\min(16,1+\text{advertised metric})cn/math/2010d425a3ad7305.svg
• T_{\mathrm{packet}}=(h+k-1)L/R+h\taucn/math/209da411e5cae6a2.svg
• T_j=RTT+\text{position of the last frame of object }jcn/math/20bdb51428a09b4e.svg
• 2^s\geq K,\quad p_{new}=p_{old}+scn/math/20f9f1a5957fbd76.svg
• T_{non}=n(2RTT+T_o)cn/math/21c4c4bee22560eb.svg
• T_{persistent}=RTT+n(RTT+T_{object})cn/math/257c101efa25bb1f.svg
• T=\min(R_s,R_c,R/n)cn/math/26062ae8bce5ae5e.svg
• F_{\mathrm{success}}=S/T_fcn/math/2656297b585ffbee.svg
• T=N_{hit}T_{hit}+N_{miss}(RTT+T_{object})+T_{setup}cn/math/277a57e2fbd98438.svg
• S_{\mathrm{pure}}=G e^{-2G}cn/math/28ca9624d26f78f7.svg
• C_v=c(X,v)+D_v(Z)cn/math/2bb516a29dc6ed94.svg
• p=32-\log_2(\text{addresses per customer}),\quad A_{group}=n\times A_{customer}cn/math/2cb46cf5eb7fdc4d.svg
• \text{Decimal value}=\sum_{k=0}^{7} b_k2^kcn/math/2d0b01052ad0f77f.svg
• T_p=(h+k-1)L/R+h\taucn/math/2fccb0c108e66b5b.svg
• P=E+D,\qquad T=40+Pcn/math/30fbafad36710670.svg
• \text{omitted groups}=8-\text{visible groups}cn/math/38a8a52c7013e253.svg
• h=32-p,\quad N=2^h,\quad U=N-2cn/math/391b44ac8c0c63db.svg
• \text{Network address}=\text{IP address AND subnet mask}cn/math/3a94cf83bf5040ba.svg
• p_{\mathrm{new}}=p_{\mathrm{old}}+scn/math/3c981479a80689de.svg
• R=(M\,2^r)\bmod_2 G,\qquad T=(M\,2^r)\oplus Rcn/math/3db4ba7cff91f15a.svg
• W_{S,SR}=W_{R,SR}\le N/2cn/math/413c503dfd6c2a68.svg
• B_{\mathrm{success}}=SRcn/math/4164452cf4f984f6.svg
• W_s=W_r\le 2^{m-1}cn/math/454034d348f17623.svg
• C=T_t+2T_p+T_a,\quad U_1=T_t/C,\quad U_W=\min(1,WT_t/C)cn/math/46d5773e25e12379.svg
• T=L/Rcn/math/4ac7eb7c34287111.svg
• H_{TCP}=4\times\mathrm{data\ offset}cn/math/4b6e848089433565.svg
• T_{\mathrm{frame}}=L/Rcn/math/4bcb5bb2e87de6c2.svg
• T_{\mathrm{frame}}\geq 2T_pcn/math/4f522413155daf19.svg
• \text{successful frame bits/s}=SRcn/math/58e5c110cc978ab9.svg
• T_c=T_{\mathrm{setup}}+M/R+h\taucn/math/59666d3c7abd8bb9.svg
• L=\frac{n(n-1)}{2},\qquad p=n-1cn/math/5de641c579576dbf.svg
• T(x)=x^rM(x)\oplus R(x),\quad R(x)=x^rM(x)\bmod G(x)cn/math/5edc4316a910e4aa.svg
• d[v]\leftarrow\min(d[v],d[u]+w(u,v))cn/math/651d18e4e3d7619d.svg
• S=G e^{-2G}cn/math/66b7a6b6082e223e.svg
• \mathrm{RIP\ metric}=\min(16,\mathrm{link\ cost}+\mathrm{advertisement})cn/math/6790d72650e66c1e.svg
• S_{\mathrm{slotted}}=Ge^{-G}cn/math/679e9767406c13ac.svg
• D(t)\le B+rtcn/math/6a390266ecc7c5fa.svg
• N_{data}=2^k,\quad N_{possible}=2^n,\quad N_{valid}=2^kcn/math/6ac9b03072926d25.svg
• S_{\mathrm{slotted}}=G e^{-G}cn/math/6d136a965636bb20.svg
• \text{block step}=256-\text{changing mask octet}cn/math/6d26c03e47eb5b6c.svg
• e^{-2} \approx 0.135335283cn/math/6d4c1915157a6bc5.svg
• T=\sum_{i=1}^{3}(L/R_i+d_{prop,i})cn/math/73c28ce2abb77cac.svg
• U_W=\min(1,WT_t/(T_t+2T_p+T_a))cn/math/73d665b598b96583.svg
• \mathrm{admitted\ data}(t)\le B+rtcn/math/750b0148fd39654a.svg
• \text{successful frames/s}=S/Tcn/math/751670fd3de42086.svg
• \text{network}=\text{address AND mask}cn/math/75b831766f4f99cd.svg
• \mathrm{offset}=\mathrm{preceding\ payload\ bytes}/8cn/math/76b072d173a1b0cc.svg
• G=\lambda T_fcn/math/777410ed8c9656d0.svg
• p=32-\log_2(N)cn/math/796d561e6716df45.svg
• S_{\mathrm{pure}}=Ge^{-2G}cn/math/7b496d33240552f3.svg
• N_{\mathrm{department}}=N_{\mathrm{labs}}N_{\mathrm{addresses\ per\ lab}}cn/math/830cf354ee16974f.svg
• H=4\,\mathrm{IHL},\qquad P=T-Hcn/math/830ed738bb3329f8.svg
• T=L/R,\qquad G=\lambda Tcn/math/85fb8d2da8040e20.svg
• e^{-0.5} \approx 0.606530660cn/math/871b71246f6a730f.svg
• T_{\mathrm{message}}=hM/R+h\taucn/math/88eb2b8ca0c2e1fd.svg
• ACK=\mathrm{first\ byte\ sequence}+Lcn/math/8d1628ee7de8ff31.svg
• e^{-0.25} \approx 0.778800783cn/math/8ed0d2b7d5ee0b1f.svg
• s_p=\bigoplus_{i\,\&\,p\ne0}b_i,\quad \text{error position}=4s_4+2s_2+s_1cn/math/915c318ac72ec415.svg
• U=2^h-2cn/math/92ad06adcbf23a21.svg
• t=\frac{B}{P-r},\quad D=Ptcn/math/931f9e3ab8969d1f.svg
• L_{UDP}=8+L_{data}cn/math/940bfc5f08006c8a.svg
• \text{total addresses}=\sum_i N_icn/math/96879d7c45f4939b.svg
• p=32-\log_2(N_{\mathrm{addresses\ per\ lab}})cn/math/96f141ad847f8a75.svg
• d[v]\leftarrow\min\bigl(d[v],d[u]+w(u,v)\bigr)cn/math/96f918d5ce249dc4.svg
• D_x(y)=\min_{v\in N(x)}\{c(x,v)+D_v(y)\}cn/math/9703c520036a4b0a.svg
• T_m=hM/R+h\taucn/math/9a18147a7dd3f73d.svg
• T_p=d/vcn/math/9c2209baf93e272e.svg
• 2^h-2\geq H,\qquad p=32-hcn/math/9d320d9c17125391.svg
• T=KT_{\mathrm{slot}}cn/math/9da1be02953867d3.svg
• T_f=L/Rcn/math/9e384aa108fc34e3.svg
• \text{step}=256-Mcn/math/9f548fe5bbbe07ef.svg
• \text{missing groups}=8-\text{visible groups}cn/math/a013f99fcbb0987c.svg
• \mathrm{IPv6\ total}=40+\mathrm{payload\ length}cn/math/a05ef7dd84fc645e.svg
• P(\mathrm{success\ per\ attempt})=S/Gcn/math/a1083530dd6239fa.svg
• T=(h+k-1)L/R+h\taucn/math/a1dee7ec7d280062.svg
• \mathrm{syndrome}=s_1+2s_2+4s_4+\cdotscn/math/a38011b407dfa547.svg
• D_x(u)=\min(2+5,5+6)cn/math/a570618da07d1f4e.svg
• e^{-0.6} \approx 0.548811636cn/math/aa9f70bc612091c4.svg
• h=32-p,\qquad N=2^hcn/math/ab3d8502169db5ce.svg
• 2^h-2\geq Hcn/math/ab881b9b5531cc3b.svg
• W_{need}=\lceil(T_t+2T_p+T_a)/T_t\rceilcn/math/acc548bc5c4e4387.svg
• d(U)=d(\operatorname{pred}(U))+c(\operatorname{pred}(U),U)cn/math/af3d37f6202e501e.svg
• F_{\mathrm{success}}=S/T,\qquad B_{\mathrm{success}}=SRcn/math/afb39a0ed04909f3.svg
• a_{accepted}=\min(a,B-q_{before})cn/math/b022488f5ef91a76.svg
• s=d_{\min}-1,\quad t=\lfloor(d_{\min}-1)/2\rfloorcn/math/b0c374242e18e388.svg
• \Delta=\max(0,L-T),\quad t_{wait}=\Delta/rcn/math/b2d0e2858bb8f8cc.svg
• q_{before}=\max(0,q_{previous}-r\Delta t)cn/math/b431990cb4021a43.svg
• T_{\mathrm{backoff}}=K T_{\mathrm{slot}}cn/math/b5a8dc2b6f87ba95.svg
• L_{UDP}=8+P,\quad L_{IP}=H_{IP}+L_{UDP}cn/math/b72774caf03e04db.svg
• T_p=\frac{d}{v}cn/math/b9c12990b79f75b6.svg
• q_b=\max(0,q-r\Delta t)cn/math/b9d2a7d69c31ca42.svg
• D_z(d)=\min\{6+D_v(d),\ 2+D_x(d)\}cn/math/b9f38cdb21f5d397.svg
• s_{\max}=d_{\min}-1,\qquad t_{\max}=\left\lfloor\frac{d_{\min}-1}{2}\right\rfloorcn/math/bd55e600f9b980f0.svg
• r=n-k,\quad R_c=k/ncn/math/beca5466928acbfc.svg
• k=\min(n,10),\quad K_{\max}=2^k-1cn/math/c06068d72176a7c3.svg
• S=\operatorname{endAroundCarry}(\text{sum of words}),\qquad C=\operatorname{NOT}(S)cn/math/c15ebbb2cede607a.svg
• W_{\mathrm{required}}=\left\lceil\frac{T_t+2T_p+T_a}{T_t}\right\rceilcn/math/c54f515ee0c2f8e8.svg
• T_{\mathrm{backoff}}=KT_{\mathrm{slot}}cn/math/c833ce43529e0206.svg
• N=2^m,\quad W_{GBN}\le 2^m-1,\quad W_{SR}\le 2^{m-1}cn/math/c9f05985613df1dc.svg
• 2^s\geq N_{\mathrm{subnets}}cn/math/ca4fab7e8dac4ad7.svg
• T_1=0.5L,\qquad T_2=0.875Lcn/math/ca97cfd9b6235ae7.svg
• \text{step}=256-\text{changing mask octet}cn/math/cb1467f42fc2ae0f.svg
• 2^r\ge m+r+1,\qquad m=8cn/math/ced24dba71859ec0.svg
• k=M/P,\quad L=P+Hcn/math/cf2d3cfa77719992.svg
• T_{non}=n(2RTT+T_{object})cn/math/d5549e76446992ce.svg
• R_c=k/n,\quad r=n-kcn/math/d973fda532c4ffe6.svg
• T_{service}=N/r,\quad T_{travel}=d/vcn/math/d9a16d08c4aea6f6.svg
• L_{\min}=\lceil 2RT_p\rceilcn/math/dc84d9f4c66a26e3.svg
• \mathrm{IPv4\ payload}=\mathrm{Total\ Length}-4\,\mathrm{IHL}cn/math/dfefa084951b704c.svg
• \mathrm{new\ allowance}=\max(0,\min(rwnd,cwnd)-\mathrm{bytes\ in\ flight})cn/math/e04fbf2fea600eab.svg
• d(v)\leftarrow\min(d(v),d(u)+w(u,v))cn/math/e31cde29472d2c2d.svg
• a_{ok}=\min(a,B-q_b),\quad drop=a-a_{ok}cn/math/e3b5b2a12bccb581.svg
• \frac{dS}{dG}=e^{-G}(1-G)cn/math/e448a50c7be7dbba.svg
• \text{broadcast}=\text{network}+(N-1)cn/math/e5e4b592ee8fd45e.svg
• \mathrm{omitted\ groups}=8-\mathrm{visible\ groups}cn/math/e61302b7dd479191.svg
• t_{burst}=B/(P-r),\quad P>rcn/math/e637610146069ae8.svg
• c=d[U]+w(U,V),\quad d[V]_{\mathrm{new}}=\min(d[V]_{\mathrm{old}},c)cn/math/e662a1e6810b8636.svg
• e^{-1.6} \approx 0.201896518cn/math/e7173ebbd6128f2f.svg
• T_{persistent}=RTT+n(RTT+T_o)cn/math/e9189d93499eafbc.svg
• \mathrm{first}=ISN+1,\quad \mathrm{next}=\mathrm{first}+Lcn/math/e95efbc97b48115e.svg
• L_{\min}\geq 2RT_pcn/math/eae5a57f0575ef7d.svg
• 2^r\ge m+r+1cn/math/ecdbdeb0433d59c5.svg
• k=\min(n,10),\quad 0\leq K\leq 2^k-1cn/math/f340420d788785f5.svg
• e^{-1} \approx 0.367879441cn/math/f3e24ac70fcf4336.svg
• G=\lambda Tcn/math/f8d7da692e922bd4.svg
• t_{wait}=\max(0,L-T)/rcn/math/f9ae1e0794f3ac3b.svg
• P_{\max}=8\left\lfloor\frac{\mathrm{MTU}-H}{8}\right\rfloorcn/math/fd129cf4e0fe3540.svg
• Q=A_{flow}\,vdm/math/00d281d05c08dc32.svg
• 67500-37500=30000\text{ m}^3/\text{h}dm/math/0798b79a0aebbdaf.svg
• 10^1=10dm/math/0af507209207dea9.svg
• A_{flow}=2\times1.5=3\text{ m}^2dm/math/0cc00d61ae3996f0.svg
• Q=10CIA/3600\quad(A\text{ in ha})dm/math/0efa2b2cbab987ad.svg
• Q=1000CIA;\quad \%\text{increase}=\frac{Q_{new}-Q_{old}}{Q_{old}}\times100dm/math/11a5141046de3b9b.svg
• 0.1\times20=2dm/math/14642cf48b9242c4.svg
• A=10000\times5000=5\times10^7\text{ m}^2dm/math/1a171e4702297407.svg
• R_{new}=(6\times4)/6=4dm/math/1af4ac096ef5510e.svg
• R=HV/Cdm/math/1b6152ff862c4a33.svg
• 2.4-1.6=0.8\text{ m}^3/\text{s}dm/math/1c7763f4811e74bd.svg
• M_{02}/M_{01}=2dm/math/24470bbc86f455b5.svg
• 10^{1.5\times1}=10^{1.5}\approx31.62dm/math/24f3381a18f46f60.svg
• Q=1000CIA\ (\text{m}^3/\text{h},\ A\text{ in km}^2)dm/math/2ca1c999c2316cff.svg
• M_0=(3\times10^{10})(5\times10^7)(1)=1.5\times10^{18}\text{ N m}dm/math/2da4e5a2d9ff2e10.svg
• A_2/A_1=10^{\Delta M};\quad E_2/E_1\approx10^{1.5\Delta M}dm/math/2ee73d43e47f749b.svg
• E_2/E_1\approx10^{1.5}=31.6228dm/math/3149c35129ac03cc.svg
• 67500/3600=18.75\text{ m}^3/\text{s}dm/math/3244e07a3e264368.svg
• Q\approx0.27778\times15=4.1667\,\mathrm{m^3/s}dm/math/331c6894f0bbb745.svg
• Q=A_c vdm/math/33ca538060b25ce8.svg
• 1\,\mathrm{N\,m}=10^7\,\mathrm{dyne\,cm}dm/math/3546153e872b3ebd.svg
• Q=37500/3600=10.416667\text{ m}^3/\text{s}dm/math/3a73f4f0dbc5f221.svg
• Q=10(0.6)(50)(20)=6000\text{ m}^3/\text{h}dm/math/3ce753cc67fcdbbc.svg
• A_{flow}=6/1.5dm/math/40cac43c63c6d88d.svg
• (30000/37500)\times100=80\%dm/math/4661d095c1d3e1fa.svg
• R=24/3=8dm/math/4ad1286ee895343a.svg
• 10^{1.5\times2}=10^3=1000dm/math/4b1c2ee0f5650ece.svg
• M_w=\tfrac23\times25-10.7=16.666667-10.7=5.966667dm/math/4d349b27d1851a47.svg
• (4/8)\times100=50\%dm/math/502da78c811bf4f0.svg
• 8-4=4dm/math/51c30ed45a5104d6.svg
• (0.8/2.4)\times100=33.333\%dm/math/5349ba33291c9315.svg
• L=10\times1000=10000\text{ m};\quad W=5000\text{ m}dm/math/5701d41826a1a891.svg
• M_2=4.3+2=6.3dm/math/577422d0457cf4f8.svg
• R\propto\frac{H\times V}{C}dm/math/5bf776e6d67541cc.svg
• \Delta M=7-5=2dm/math/5facb9cbf54fb977.svg
• \log_{10}(1.5\times10^{18})=18.176091dm/math/5fcce3b65b1452e6.svg
• M_w=\tfrac23\log_{10}M_0-10.7dm/math/63508c1a7c23719a.svg
• 1000\times75\times1=75000\text{ m}^3/\text{h}dm/math/63e08167804c9bbb.svg
• Q=6000/3600=1.666667\text{ m}^3/\text{s}dm/math/674b12c6893ae725.svg
• A_{flow}=2\times1=2\text{ m}^2dm/math/68f58e503c39eaaf.svg
• A_{flow}=4\text{ m}^2dm/math/69132fe50a468b91.svg
• 1\,\mathrm{km}=1000\,\mathrm m,\qquad1\,\mathrm{km^2}=10^6\,\mathrm{m^2}dm/math/6b541cea66b65aa1.svg
• A_2/A_1=10^{\Delta M},\qquad E_2/E_1\approx10^{1.5\Delta M}dm/math/6c0f9697b5805581.svg
• Q=A_{flow}v\quad\Rightarrow\quad A_{flow}=Q/vdm/math/6d569abd32dc88d5.svg
• Q_{\rm hour}=1000CIA,\quad Q_{\rm second}=Q_{\rm hour}/3600dm/math/71f29f4c3994b34c.svg
• M_w=\frac{2}{3}(\log_{10}M_0-9.1)dm/math/78939271fff1c920.svg
• M_w=\frac{2}{3}(\log_{10}M_0-9.1)\quad(M_0\text{ in N m})dm/math/7d270c60cc26a642.svg
• \frac{E_2}{E_1}\approx10^{1.5(M_2-M_1)}dm/math/7e52223c33fb504b.svg
• p=1/T;\quad T=100\Rightarrow p=0.01=1\%dm/math/828dbd3c6cb5e913.svg
• M_0=\mu A D=\mu LWDdm/math/8374142fa666a89f.svg
• A_2/A_1=10^1=10dm/math/91b5b617edf787c3.svg
• Q=A_{flow}vdm/math/9385947feec50c2c.svg
• M_0=\mu A Ddm/math/9a57955fc6ad7423.svg
• R_{index}=\frac{H\times E\times V}{C}\quad\text{if exposure is explicitly included}dm/math/9b1a267f652f98c1.svg
• 30\times0.5=15dm/math/9b6e747a4100c69b.svg
• Q_{new}=1000(0.9)(75)(1)=67500\text{ m}^3/\text{h}dm/math/a13c2e075fc32a79.svg
• \text{Expected loss}=p\times\text{conditional loss}dm/math/a14ec0148e3e5e45.svg
• HV=6\times4=24dm/math/a3421d2f279465c5.svg
• M_0=\mu AD;\quad \Delta M_w=\tfrac23\log_{10}(M_{02}/M_{01})dm/math/a7c8c4dd0bbb8636.svg
• M_w\approx\frac{2}{3}\log_{10}M_0-10.7\quad(M_0\text{ in dyn cm})dm/math/aca9c65eae708651.svg
• \Delta M=M_2-M_1,\qquad \frac{A_2}{A_1}=10^{\Delta M}dm/math/ad6b0eb73f6fb365.svg
• M_w=\tfrac23(18.176091-9.1)=6.050727dm/math/af09d884d44575bd.svg
• \log_{10}(2)=0.301030dm/math/b1c7f44e76f4dbd5.svg
• \Delta M=6-5=1dm/math/b9d9abb2ae049297.svg
• \Delta M=\log_{10}(A_2/A_1),\quad M_2=M_1+\Delta Mdm/math/bcb15b8d83186a34.svg
• M_w=\frac{2}{3}\left(\log_{10}M_0-9.1\right)\quad(M_0\text{ in N m})dm/math/be11d6630d2edf0c.svg
• A_{flow}=\text{width}\times\text{depth},\quad Q=A_{flow}vdm/math/bfe1e11b735d5920.svg
• Q=0.27778\,C I A\quad[Q:\text{m}^3/\text{s},\ I:\text{mm/h},\ A:\text{km}^2]dm/math/c51028188b91b5ab.svg
• 10^2=100dm/math/c6153b80efda738b.svg
• \Delta M_w=\tfrac23\times0.301030=0.200687dm/math/c75b9c1b8eae94d0.svg
• 0.6\times50=30dm/math/cd0452c9c459e2b7.svg
• \frac{E_2}{E_1}\approx10^{1.5\Delta M}dm/math/cdde7fc3d085a3ad.svg
• Q=10CIA\ (\text{m}^3/\text{h},\ A\text{ in hectares})dm/math/d24ecdcf63f2c435.svg
• Q=3\times0.8=2.4\text{ m}^3/\text{s}dm/math/d752a5e1627e58c1.svg
• M_0=\mu LWD;\quad M_w=\tfrac23(\log_{10}M_0-9.1)dm/math/dcb2c14b7f2d8270.svg
• Q=0.5\times75000=37500\text{ m}^3/\text{h}dm/math/dd68b109ea14a04a.svg
• \Delta M=\log_{10}(100)=2dm/math/de8bef1adcae3c49.svg
• R=HV/C;\quad \%\text{reduction}=\frac{R_{old}-R_{new}}{R_{old}}100dm/math/df4fc500039d5833.svg
• \frac{A_2}{A_1}=10^{M_2-M_1}dm/math/e8492e4cbc3776f4.svg
• 1\,\mathrm{ha}=10^4\,\mathrm{m^2},\qquad1\,\mathrm h=3600\,\mathrm sdm/math/eb607cc1e0420d0e.svg
• Q=2\times0.8=1.6\text{ m}^3/\text{s}dm/math/f551bf77d2962265.svg
• \log_{10}(10^{25})=25dm/math/fc1db69285d09308.svg
• Q=0.27778CIAdm/math/ffea534526275274.svg
• P(R\mid W)=0.18/0.26=9/13\approx0.692307692is2/math/000872ced145bc03.svg
• v^{(1)}=0+0+(-1)=-1is2/math/000e26bf52e8a50e.svg
• R=\begin{bmatrix}0.7&0\\0.4&0\end{bmatrix}is2/math/00a9ec4b6529a3a5.svg
• q_2=0.5\frac{e^{-(1-4)^2/2}}{\sqrt{2\pi}}\approx0.002215924is2/math/00ad8a0accc6492e.svg
• f(2)=2^2=4=4is2/math/00c58d851ae8132e.svg
• P(\text{Reputation}=\text{Good}\mid C=\text{Yes})=\frac{3}{4}=0.75is2/math/00e0bac1e035a9f7.svg
• A=\begin{bmatrix}2&5&8\\1&4&7\end{bmatrix}is2/math/01196c6c529af391.svg
• \text{Grand total}=0.90+0.30=1.20\ne1is2/math/017ee19616d63679.svg
• d_g=2-(3.5)=-1.5is2/math/01ad6e560abc36e6.svg
• 1\times(-1)=-1is2/math/01d91edc60cbc82c.svg
• v^{(2)}=(-0.75)+0+(-0.75)=-1.5is2/math/02342391a534b540.svg
• d_g=7.21-(8.1428)=-0.9328is2/math/024d71da7fe2cc3c.svg
• I=0.5\times(-3)=-1.5is2/math/02f316319733ea90.svg
• x_{t+1}(j)=\sum_i x_t(i)\,P_{ij}is2/math/030f0c9b46caf90b.svg
• P(W)=\frac3{20}+\frac5{16}=\frac{12+25}{80}=\frac{37}{80}=0.4625is2/math/0310d0eb2cb65a99.svg
• \pi=[\pi_B,\pi_I,\pi_A]is2/math/031664350e5c144f.svg
• z^*=(\sum A_jc_j)/(\sum A_j)is2/math/033d1d87c88bf6a8.svg
• \begin{bmatrix}0.3&0.3&0.3\\0.4&0.3&0.4\\0.6&0.3&0.6\end{bmatrix}is2/math/035832180398151c.svg
• W_{new}=0.2+0.14+0.05=0.39is2/math/0367611df103826d.svg
• S=\begin{bmatrix}0.2&0.3&0.3\\0.1&0.1&0.1\\0.2&0.4&0.3\end{bmatrix}is2/math/03c1b332a4b4facf.svg
• \alpha_r=\min(a_r,b_r)\quad\mu_r(z)=\min(\alpha_r,\mu_{C_r}(z))\quad\mu_{out}(z)=\max_r\mu_r(z)is2/math/043f511911bca932.svg
• 30-10=20is2/math/04505a3de0e682e9.svg
• \max\{\min(0.2,0.2),\min(0.3,0.2),\min(0.3,0.1)\}is2/math/04a79caa9a98c1c4.svg
• N_2=0.01798621+0.5+0.98201379=1.5is2/math/04b9a1a1ff3b0c25.svg
• \sum_k\pi_k\mathcal N(\mu_k,\Sigma_k)is2/math/04ce4e1e89065414.svg
• x^{(2)}=7.36+(-0.033)=7.327is2/math/04ced9552ec0a2fc.svg
• x^{(2)}=3.19+(-0.011)=3.179is2/math/04d374313aaab72d.svg
• \sum A=35.58is2/math/04fe6e4450abbfe1.svg
• \frac14=p_1P_{12}+p_2P_{22}+p_3P_{32}=\frac14\frac14+0+\frac12p_3is2/math/052bbb970339e754.svg
• A=2+1=3\qquad M=4+6=10is2/math/0582db5cb8a7455e.svg
• x_1=[1,0]P=[0.75,\;0.25]is2/math/05a31cec6643ce3f.svg
• v_\pi(s_1)is2/math/05a49a1a1019f0f7.svg
• X\cup Y:\max(0.4,0.7)=0.7is2/math/05b7177b2b3fbf81.svg
• X\cup Y:\max(0.9,0.3)=0.9is2/math/05dee517321dfca3.svg
• S=0.12\times(-1.01)=-0.1212is2/math/060e9302bdac4457.svg
• P(X_2=N\mid X_1=H)=P_{HN}=0.15is2/math/061a644323cfaa64.svg
• \mu_{Medium}(55)=\frac{75-55}{75-50}=0.8is2/math/0625f25570fc580f.svg
• s_c=P(c)P(Young\mid c)P(High\mid c)P(PreviousNo\mid c)is2/math/0629d7338a1ecd30.svg
• c=0.2(a+b)+0.5cis2/math/063ad466582b38bf.svg
• P^2=\begin{bmatrix}5/8&3/8\\9/16&7/16\end{bmatrix}is2/math/063d8bbaa02b11f7.svg
• \alpha=1/\text{(sum)}is2/math/0666d2fea1f0f562.svg
• \max(0.2,0.4)=0.4is2/math/06c831ab89f196ab.svg
• \boxed{P(A\mid B)=\frac{P(A\text{ and }B)}{P(B)}}\qquad P(F\mid C)=\frac{0.1}{0.4}=0.25is2/math/06d91b8a011bacbf.svg
• \log L(\theta)=\sum_i\log\Big(\sum_k\lambda_k^i\,\frac{\pi_k\mathcal N(x_i\mid\mu_k,\sigma_k^2)}{\lambda_k^i}\Big)\;\ge\;\sum_i\sum_k\lambda_k^i\log\frac{\pi_k\mathcal N(x_i\mid\mu_k,\sigma_k^2)}{\lambda_k^i}is2/math/06dff5bae3d2cfe1.svg
• \neg X\cup\neg Y:\max(0.3,0.8)=0.8is2/math/0712a083103a814e.svg
• (2)^2=4\qquad2\sigma^2=2(2^2)=8is2/math/0776e93b48581bdc.svg
• (1,1)=\max[\min(.4,.4),\min(.5,.4),\min(.3,.1)]=\max(.4,.4,.1)=0.4is2/math/07a16abae3efd072.svg
• \bar f_{old}=(11+19+0-1)/4=7.25is2/math/07b0b7549d04f350.svg
• \mu(3)=\frac{1}{4}=0.25is2/math/07bbe7a121e1fdbb.svg
• \alpha_r=\min(\mu_{A_r}(x),\mu_{B_r}(y)),\quad z^*=\frac{\int z\mu_{out}(z)\,dz}{\int\mu_{out}(z)\,dz}is2/math/07c525af02eb813b.svg
• T_{21}=\max\{\min(0.6,0.5),\min(0.3,0.7)\}is2/math/07f583df1f70440b.svg
• z_{MOM}=\frac{a+b}{2},\qquad z_{COA}=\frac{\int x\mu(x)dx}{\int\mu(x)dx}=\frac{\sum A_ic_i}{\sum A_i}is2/math/083457b6b2635a4c.svg
• (1,3)=\max[\min(.2,.3),\min(.3,.3),\min(.3,.1)]=0.3is2/math/0857c1f332a31d0c.svg
• 38.6747>37.1117is2/math/0858b2bce4269ef8.svg
• q_\pi(s,a)=\mathbb E_\pi[G_t\mid S_t=s,A_t=a]is2/math/0885e6b22d6c67c2.svg
• d_p=7.21-(7.36)=-0.15is2/math/08869d3b9cefb65b.svg
• P(x_j=v\mid c)=\frac{N_{jvc}+1}{N_c+K_j}is2/math/08b794e505b1bd84.svg
• v^{(1)}=0.25+0+(-0.1212)=0.1288is2/math/08fa8ab93dce4fd3.svg
• \mu_{\neg X}(c)=1-\mu_X(c)is2/math/094fb09a37beb92f.svg
• P(Co\mid\neg C,S)=\sum_{p\in\{T,F\}}P(Co\mid\neg C,S,P=p)P(P=p\mid\neg C,S)is2/math/09e203ebfdf53e2b.svg
• nis2/math/0a03c1424e475329.svg
• \alpha=\beta=1,\;\rho=0.5,\;Q=1,\;\tau_0=1is2/math/0a0a3aed7677e436.svg
• \sum M=1575is2/math/0a14da79e18490bb.svg
• \mu_{Low}(80)=0is2/math/0a1689e705282396.svg
• \mathcal N(x\mid\mu,\sigma^2)=\frac1{\sqrt{2\pi}\sigma}e^{-\frac{(x-\mu)^2}{2\sigma^2}}is2/math/0a27dae1a77ea038.svg
• S=\begin{bmatrix}0.6&0.7&0.4\\0.3&0.3&0.3\\0.5&0.5&0.4\end{bmatrix}is2/math/0a4c516a6ffc1e2a.svg
• q_\pi(s,a)=\sum_{s'}P(s'\mid s,a)\,\Big[R(s,a,s')+\gamma\sum_{a'}\pi(a'\mid s')q_\pi(s',a')\Big]is2/math/0aaca96cab3cf147.svg
• f(4.129264, 7.32764)=4.129264^2+7.32764^2-(4.129264)(7.32764)is2/math/0ac428234bdb63ad.svg
• x^{(2)}=(-2)+1.5=-0.5is2/math/0af67fb865c5f5c7.svg
• \neg X:1-0.4=0.6\qquad\neg Y:1-0.7=0.3is2/math/0b5f5f89785edd94.svg
• v=0.5(1+0.9v)+0.5(4+0.18v)=2.5+0.54v\Rightarrow0.46v=2.5\Rightarrow v_\pi(s_1)=5.4348is2/math/0b711572c2407738.svg
• X=\begin{bmatrix}3.1472&-4.0246\\4.0579&-2.2150\end{bmatrix},\;V=\begin{bmatrix}0.2194&0.2449\\0.1908&0.2228\end{bmatrix}is2/math/0ba0d9983a41cf83.svg
• 0+19.5+42.5=62is2/math/0bdf70679aafe819.svg
• v_{11}=0.9(0.2194)+0+0.12825(4.0579-3.1472)=0.19746+0.116797=0.314257is2/math/0c6669f7602148c0.svg
• \mu=\frac{2+4}{2}=3is2/math/0c66b5b185711660.svg
• Q(s,a)is2/math/0caba73ef360f772.svg
• \tau_{ij}\leftarrow(1-\rho)\tau_{ij}+\sum_{k:\,ij\in tour_k} Q/L_kis2/math/0cc34ae2992158a9.svg
• 0+0.3+0.6+0.1=1is2/math/0cd832e4cf2e8857.svg
• V^*(s)=\max_a\sum_{s\prime}P(s\prime\mid s,a)\left[R(s,a,s\prime)+\gamma V^*(s\prime)\right]is2/math/0d0ba7664438689d.svg
• \pi_B+2\pi_B+7\pi_B=1\;\Rightarrow\;\pi_B=0.1,\quad\pi_I=0.2,\quad\pi_A=0.7is2/math/0d4ea6d007ce74ae.svg
• \mu_{B^c}(u_1)=1-0.3=0.7is2/math/0d89f5d2d8ca13ed.svg
• z^*=\frac{\sum z_i\mu_i}{\sum\mu_i}is2/math/0daa1f2e296038c3.svg
• \mu_L(7)=0\qquad\mu_M(7)=0is2/math/0dc9da58647d7498.svg
• 6-2=4is2/math/0e4232a9d761c2cf.svg
• L_k=\sum_{(i,j)\in tour_k}d_{ij},\qquad \tau_{ij}^{new}=(1-\rho)\tau_{ij}^{old}+\sum_k\Delta\tau_{ij}^{k},\quad \Delta\tau_{ij}^{k}=Q/L_kis2/math/0e8184d1b87733c1.svg
• P_{ss'}=P(S_{t+1}=s'\mid S_t=s)is2/math/0ebf21d8b931d2d0.svg
• \frac{1}{2}=0.5is2/math/0ef4a9ef78e1b7d5.svg
• q(\text{stay})=1+0.9(4.878)=5.390>4.878is2/math/0eff67e823e28a45.svg
• \min(0.7,0.2)=0.2is2/math/0f24b8a13d9d75d1.svg
• z^*=\frac{\sum_r\alpha_r c_r}{\sum_r\alpha_r}is2/math/0fb4cda8fcd0e1d8.svg
• Q/L_kis2/math/0fe8d82420c3adda.svg
• 50(0.15)+30(0.20)+20(0.50)=7.5+6+10=23.5is2/math/1005a6b946cdda94.svg
• 8+0+2+1=11is2/math/1006619e9b2d19e6.svg
• (8-x)/3=0.5is2/math/101167a6754cbdc0.svg
• P=\begin{bmatrix}0.75&0.25\\0.40&0.60\end{bmatrix}is2/math/1028784b72801945.svg
• 2\times4=8is2/math/104d4bd59aaafd44.svg
• (\sigma_k^2)^{new}=\frac{\sum_i\gamma_{ik}(x_i-\mu_k^{new})^2}{N_k},\quad\pi_k^{new}=N_k/nis2/math/10983e175fb2b145.svg
• \tau_0is2/math/10a4957527d9bc35.svg
• \mathbf P(A\mid e)=\alpha\,\mathbf P(A,e)is2/math/10d12eaae749bfb3.svg
• A=\frac{(b-a)(u+v)}2,\qquad M=\frac{b-a}{6}[(2a+b)u+(a+2b)v]is2/math/10dac72a97306eb6.svg
• F_1=9.904868+12.666221+16.197405+9.4416-16.0984+5=37.1117is2/math/10f900e5ecd1d222.svg
• R_{ij}=\min(\mu_A(a_i),\mu_B(b_j))is2/math/1132359c819cab8a.svg
• d_p=3-(3)=0is2/math/114e8d23bb26d062.svg
• S=0.12\times(-1.1388)=-0.136656is2/math/115797e01c1867dd.svg
• 2x^3=8192\quad 3x^2=768\quad 8x=128is2/math/116d99ec87e1c7e9.svg
• \pi=[\pi_1,\dots,\pi_n]is2/math/118db8b67dc77bc6.svg
• a=0.2,\;b=0.1\Rightarrow\pi=[1/3,\;2/3]is2/math/119a700d02c295a5.svg
• 8-5=3is2/math/11fb664a99823789.svg
• Q(s,a)=4.75is2/math/12b09999f026b089.svg
• 0.5\times1=0.5is2/math/12ea92d29476784c.svg
• R=\begin{bmatrix}0.3&0.6&0.9\end{bmatrix}is2/math/12f5f2edbde65b9d.svg
• f(5)=5^2=25=25is2/math/12fb2bbb149af9d6.svg
• gbest^0=(-2)is2/math/130e21267aeccc61.svg
• A^T=\begin{bmatrix}2&1\\5&4\\8&7\end{bmatrix}is2/math/1317eeff1ada1bd8.svg
• \sum\pi_k=1is2/math/131aa1c87311a5f9.svg
• \alpha=\frac1{0.5185},\quad P(hd\mid bp)=0.803,\quad P(\lnot hd\mid bp)=0.197is2/math/13277c3f7f4e8b01.svg
• v^{(1)}=(-0.5)+0+(-1.5)=-2is2/math/1329839a69ba9e13.svg
• \mu=\frac{4+4+4}{3}=4is2/math/133ab42db9ac4f4c.svg
• S=0.5\times(-3)=-1.5is2/math/13ae8508c1795085.svg
• \mu=\frac{\sum_i x_i}{n}\quad\sigma^2=\frac{\sum_i(x_i-\mu)^2}{n}\quad\sigma=\sqrt{\sigma^2}is2/math/13c92c5c73ee1c2f.svg
• x^2=14\times14=196\qquad x^3=196\times14=2744is2/math/13e67d57f1606f3f.svg
• d_p=6.17-(6.17)=0is2/math/13f27d08182e6e02.svg
• A\to B\to D\to C\to A,\qquad L=1+5+6+2=14\;\text{(both ants)}is2/math/1409076465c0fee4.svg
• L_{new}=12.5+15+6=33.5is2/math/141ce72082fe6ed8.svg
• q_1=0.5\frac{e^{-(5-2)^2/2}}{\sqrt{2\pi}}\approx0.002215924is2/math/1433e6810e632ac7.svg
• x^{(2)}=2+(-1)=1is2/math/144d4f69ce96cfe2.svg
• \mu_H(32)=\frac{32-25}{35-25}=\frac7{10}=0.7is2/math/1456bdcfcdae3dcf.svg
• T_{21}=\max\{\min(0.4,0.2),\min(0.7,0.8)\}is2/math/145badb953d03e7a.svg
• \frac{z_L-50}{65-50}=0.3is2/math/145de080f101d4ed.svg
• v^{(1)}=0+0+(-2)=-2is2/math/146db9adf4c34c65.svg
• score(Yes)=0.046875\qquad score(No)=0is2/math/14926697646660d6.svg
• 25-10=15is2/math/14c6ec773492ed50.svg
• x^{(t+1)}=x^{(t)}-1is2/math/14d3646dbde001b1.svg
• v(s_2)=0is2/math/14eba4db728727d6.svg
• \mu_{High}(35)=0is2/math/1509e75effd66d75.svg
• \overline f_{old}=\frac{81+225+225+81}{4}=\frac{612}{4}=153is2/math/1526400ff09c5372.svg
• \neg X\cup\neg Y:\max(0.6,0.3)=0.6is2/math/1568254a4bf6e796.svg
• x=(4.0579,\,-2.2150)is2/math/1588eb472c257b22.svg
• I=0.5\times1=0.5is2/math/15c0369c89ebba10.svg
• v_{ij}\leftarrow wv_{ij}+c_1r_1(p^b_{ij}-x_{ij})+c_2r_2(p^g_j-x_{ij}),\qquad x_{ij}\leftarrow x_{ij}+v_{ij}is2/math/15c9c9495dc886c9.svg
• q(\text{go})=4+0.18(10)=5.8<10=q(\text{stay})is2/math/15cfa2584493bce2.svg
• \pi_A=0.1\pi_B+0.3\pi_I+0.9\pi_A\;\Rightarrow\;0.1\pi_A=0.1\pi_B+0.3(2\pi_B)=0.7\pi_B\;\Rightarrow\;\pi_A=7\pi_Bis2/math/15df96f89f27c195.svg
• f(15)=6750-1125+105=5730is2/math/1609c9dac811fd7a.svg
• \text{Ant 2: }A\to D\to C\to B\to A,\quad L_2=7+3+2+5=17is2/math/161908e94cfc42c7.svg
• p-x=(1-3,\;8-7)=(-2,1)is2/math/162ee1a9497444ec.svg
• f(x)=x(15-x)+1is2/math/163a21622d906b2b.svg
• v_\pi(s),\;q_\pi(s,a)is2/math/16476f59173bd3a8.svg
• S=0.12\times(0)=0is2/math/16804d70bddc0ccc.svg
• 0.5+0.25=0.75is2/math/16a735913edad702.svg
• c_1r_1=1.2\times0.5=0.6is2/math/16c273adcc74f25a.svg
• I=0.5\times(-1.5)=-0.75is2/math/16e3f0369a50c03f.svg
• \pi_P=0.6\pi_P+0.3\pi_L+0.2\pi_R\;\Rightarrow\;\pi_P=0.75\pi_L+0.5\pi_Ris2/math/1708781ef111507a.svg
• 6\times10^{16}is2/math/1746ec49ffd70b58.svg
• P(A\cap B)=P(B\mid A)P(A)=P(A\mid B)P(B)is2/math/174a2d1dd45a6fd5.svg
• z^*_{COA}=\frac{\int z\mu_{out}(z)\,dz}{\int\mu_{out}(z)\,dz}is2/math/175fa0720f954403.svg
• f(x)=(x-3)^2is2/math/17ba9de444962fb9.svg
• P=\begin{bmatrix}0.3&0.25&0.45\\0.45&0.4&0.15\\0.65&0.25&0.1\end{bmatrix}is2/math/17d3fd5cb7fb6fac.svg
• (6)_{10}=(0110)_2is2/math/17dc4cb6ab18cf42.svg
• x^{(2)}=8.1428+(-0.175536)=7.967264is2/math/17e63ccf7082af52.svg
• V=\begin{bmatrix}0.314257&0.452491\\0.17172&0.20052\end{bmatrix}is2/math/181c38752c9bafe9.svg
• S=\begin{bmatrix}0.8&0.1\\0.2&0.7\\0.5&0.4\end{bmatrix}is2/math/1823a7e672a05e76.svg
• P(yes\mid x)=\frac{1/189}{1/189+18/875}\approx0.204582651,\qquad P(no\mid x)\approx0.795417349is2/math/183b5657f9957d15.svg
• \sum_{i=1}^{3}a_i=a_1+a_2+a_3is2/math/187790b2a1d823b4.svg
• \frac{1}{\sqrt{2\pi}}\approx0.39894228is2/math/187fc723b4884bed.svg
• P=\begin{bmatrix}0.30&0.50&0.20\\0.10&0.60&0.30\\0.10&0.10&0.80\end{bmatrix}is2/math/18c74fdbd43cdfe0.svg
• s+c=1\quad\Rightarrow\quad s+2s=1\quad\Rightarrow\quad3s=1is2/math/191d2d48b18264ce.svg
• s=0.8s+0.1cis2/math/193b341ebcb587b5.svg
• d_g=3.14-(4.15)=-1.01is2/math/1978fefc1be6f223.svg
• =\max\{0.2,0.1\}=0.2is2/math/197ec23ba477e3ac.svg
• f(4.15, 7.31)=4.15^2+7.31^2-(4.15)(7.31)is2/math/1980ac03eb9e462c.svg
• c_1r_1=1\times0.5=0.5is2/math/19e8c86525a0eadb.svg
• VH:\quad z_L=70+0.5(85-70)=77.5,\quad z_R=100-0.5(100-85)=92.5is2/math/19f28361b506bad3.svg
• z^*=\frac{637971/686}{1500/49}=\frac{212657}{7000}\approx\boxed{30.379571}is2/math/19f8c5be600656c4.svg
• T_{ik}=\max_j\min(S_{ij},R_{jk})is2/math/1a39e95150df91b7.svg
• P_{+1}(Y\mid x)=\frac{1/24}{1/24+8/735}=\frac{245}{309}\approx0.792880is2/math/1a4ef59913a7687a.svg
• \sum A=0.675+6+1.2+7.5+1.875=17.25is2/math/1a4fac5b0fc5e065.svg
• x^2=16\times16=256\qquad x^3=256\times16=4096is2/math/1a7d6636e629e211.svg
• v^{(2)}=(-0.0636)+0+(-0.111936)=-0.175536is2/math/1a832b53cf0e106d.svg
• \neg X\cup\neg Y:\max(0.3,0.4)=0.4is2/math/1aca856a2f38fa09.svg
• P(\lnot a)=1-P(a)is2/math/1afe967401ee19dd.svg
• P(\neg R,W)=(1-0.2)\times0.1=0.8\times0.1=0.08is2/math/1b0b61dc6cadc7a7.svg
• 2+12+12=26is2/math/1b63f80f68e4f903.svg
• \eta=\tfrac12,\tfrac13\Rightarrow P_{CB}=0.6,\;P_{CD}=0.4is2/math/1b6803f7bb0eede8.svg
• P(a\mid b)=P(a)is2/math/1b8c2e9773a67bd2.svg
• (11)_{10}=(1011)_2is2/math/1b92e7546fc83d0f.svg
• =18.30812944+53.993104-31.4406224=40.86061104is2/math/1baa4a09739c8180.svg
• A=\frac m2(b^2-a^2)+c(b-a),\qquad M=\frac m3(b^3-a^3)+\frac c2(b^2-a^2)is2/math/1c36be6d1316a809.svg
• c_2r_2=1\times0.5=0.5is2/math/1cc595b44ce899fa.svg
• 0.4(0.05)=0.02is2/math/1cee8e50d6767b0f.svg
• \langle\mathcal S,P,R,\gamma\rangleis2/math/1cf28b140fd79e47.svg
• z^*_{WA}=\frac{19.5+42.5}{0.3+0.5}=\frac{62}{0.8}=77.5is2/math/1cf8861b16222d21.svg
• X\cap Y:\min(0.4,0.7)=0.4is2/math/1cfa1c76382cbe6b.svg
• R_{new}=7.5+6+10=23.5is2/math/1cfa32a9ab8dc27f.svg
• x+v=(3+(-1),\;7+2)=(2,9)is2/math/1d4e7e4fb543eeb3.svg
• \gamma_{ik}=\frac{\pi_k\mathcal N(x_i\mid\mu_k,\sigma_k^2)}{\sum_j\pi_j\mathcal N(x_i\mid\mu_j,\sigma_j^2)}is2/math/1d66a1a364cc52ce.svg
• d_p=-2-(-2)=0is2/math/1d782a69e1d0fcb8.svg
• d_g=3.14-(3.19)=-0.05is2/math/1dbfd156a9aafbb3.svg
• kis2/math/1dc6f9b1a0213681.svg
• 2^kis2/math/1e088370ad90412f.svg
• v=1+0.9v\Rightarrow v=10is2/math/1f1b525d33de57d2.svg
• P=\begin{bmatrix}1/2&1/4&1/4\\1/3&0&2/3\\1/2&1/2&0\end{bmatrix}is2/math/1f388f49cad4444e.svg
• Q^*(s,a)=\sum_{s',r}p(s',r\mid s,a)\left[r+\gamma\max_{a'}Q^*(s',a')\right]is2/math/1f89cb0381d5b714.svg
• (P^2)_{ij}=\sum_kP_{ik}P_{kj}is2/math/1fa028a29cde1ac6.svg
• \rhois2/math/210f47ee7d9c0bf6.svg
• P(\text{Experience}=\text{Mid}\mid C=\text{No})=\frac{0}{4}=0is2/math/210f557f91613842.svg
• \max\{\min(0.4,0.4),\min(0.5,0.4),\min(0.3,0.1)\}is2/math/21233f6c94b2b173.svg
• (1,1)=\max[\min(.2,.2),\min(.3,.2),\min(.3,.1)]=\max(.2,.2,.1)=0.2is2/math/2134a14de60dd459.svg
• v(s)is2/math/214f0238bdf479cc.svg
• \alpha=\beta=1,\;\rho=0.5,\;Q=1is2/math/215be7d8bc049557.svg
• v(yes)=\tfrac9{14}\cdot\tfrac29\cdot\tfrac39\cdot\tfrac39\cdot\tfrac39\approx0.005291005,\qquad v(no)=\tfrac5{14}\cdot\tfrac35\cdot\tfrac15\cdot\tfrac45\cdot\tfrac35\approx0.020571429is2/math/21600d7754c6d954.svg
• x^{(2)}=2+0=2is2/math/2189c0d7cea22f3c.svg
• \bar z=\frac{50}{1}=50is2/math/21b878898a4a0815.svg
• P(A\mid B)=\dfrac{P(B\mid A)P(A)}{P(B\mid A)P(A)+P(B\mid\lnot A)P(\lnot A)}is2/math/21c4330d83c41c29.svg
• \text{Remaining}=3-2=1is2/math/21e8a95558e5db30.svg
• D=\begin{array}{c|cccc}&A&B&C&D\\\hline A&0&1&2&3\\B&1&0&4&5\\C&2&4&0&6\\D&3&5&6&0\end{array},\qquad\tau_{ij}=1\;(i\ne j)is2/math/21fbc23ee8a38afa.svg
• \pi_1^{new}=1.5/3=0.5is2/math/221d1a6298a7ea65.svg
• p_1^{(2)}=\frac14\frac58+\frac34\frac9{16}=\frac{10+27}{64}=\frac{37}{64}=0.578125is2/math/223d07fbdc1aa402.svg
• P(x_1,\dots,x_n)=\prod_{i=1}^{n}P\big(x_i\mid parents(X_i)\big)is2/math/228b443a51a8c577.svg
• \overline f_{old}=\frac{27+51+55+37}{4}=\frac{170}{4}=42.5is2/math/22945ccb0391cac7.svg
• =\max\{0.6,0\}=0.6is2/math/234802688b1bc952.svg
• f(2)=16-20+14=10is2/math/235889b9921ff9b5.svg
• s(N)=\frac4{10}\frac14\frac14\frac14=\frac1{160}=0.00625is2/math/237fd941cadfe143.svg
• (\sigma_1^2)^{new}=\frac{1.521735533}{1.5}\approx1.014490355is2/math/23888cceffe7f895.svg
• P(X_{t+n}=j\mid X_t=i)=(P^n)_{ij}is2/math/23d2321daf991110.svg
• n_{free}=Kd+K\frac{d(d+1)}2+(K-1)is2/math/24096c582fbc12b9.svg
• C=0.6\times(-0.038)=-0.0228is2/math/241c822838613f93.svg
• \mu(7)=1/3\approx0.333333is2/math/245668271fba02af.svg
• =11.981686467+12.364701993+12.759961279+10.384371825+(-14.2884352)+5\approx38.202286364is2/math/246cb1e4add2ab6e.svg
• X\cap Y:\min(0.2,0.8)=0.2is2/math/247907e255730431.svg
• b_{next}=0.50a+0.60b+0.10cis2/math/24b806632ba7642d.svg
• c_1r_1=1.5\times0.5949=0.89235,\qquad c_2r_2=1.5\times0.0855=0.12825is2/math/24c25a9332d3129e.svg
• \neg X:1-0.2=0.8\qquad\neg Y:1-0.8=0.2is2/math/24c8a43adb44c324.svg
• x^{(2)}=3.5+(-1.5)=2is2/math/24dbc384ce5eb239.svg
• P(X_1=3,X_2=2,X_3=1)=P(X_1=3)P_{32}P_{21}is2/math/250a6ac8bef84a28.svg
• 8-6=2is2/math/2511936ca0026cfc.svg
• \pi=\left[\frac{b}{a+b},\;\frac{a}{a+b}\right]is2/math/2511ae4b7c84ed79.svg
• P(E,I,M,S,A)=P(E)P(I)P(M\mid E,I)P(S\mid I)P(A\mid M)is2/math/2567b59d7d2c83ad.svg
• v^{(1)}=2+0+0=2is2/math/25c7782eb44a678a.svg
• \tau_{\text{new}}=\begin{array}{c|cccc}&A&B&C&D\\\hline A&0&0.6429&0.6429&0.5\\B&0.6429&0&0.5&0.6429\\C&0.6429&0.5&0&0.6429\\D&0.5&0.6429&0.6429&0\end{array}is2/math/25cf4d02bee20e2a.svg
• v^{(2)}=0+0+1.5=1.5is2/math/25e29c2b2380aded.svg
• x_{new}=(-2)+2.5=0.5is2/math/264c82a55bf4620a.svg
• q_*(s_1,\text{stay})=10is2/math/26abf1418e7a596a.svg
• P_{new}=30+9+4=43is2/math/26b7dba609bb77e3.svg
• M=(14.4/6)(0+19.08)=2.4\times19.08=45.792is2/math/26cef63984606712.svg
• (P^{\,n})_{ij}is2/math/270ba67bd7a265ef.svg
• 0.4\times0=0is2/math/2711ae2397112a4c.svg
• =34.88556096+66.30519184-48.09463392=53.09611888is2/math/273655b1059b7aef.svg
• X\cap Y:\min(0.8,0.5)=0.5is2/math/276fcc7ea7945a48.svg
• Fis2/math/277a47abb82a8255.svg
• s=1/3\qquad c=2s=2/3is2/math/2788e62b0b2007f8.svg
• P(\neg C,S)=0.9\times0.3=0.27is2/math/27b16cf13dc13276.svg
• f(-3)=18+1=19is2/math/27fc155eedcc5888.svg
• x=-1is2/math/2854d883b15c7a60.svg
• \mu_{B^c}(u_1)=1-1=0is2/math/2890d976bd930ef0.svg
• v^{(1)}=0+0+(-0.1272)=-0.1272is2/math/28af9633e743719d.svg
• f(2)=(-1)^2=(-1)\times(-1)=1is2/math/28ec806f80fe7e4c.svg
• \overline f_{old}=\frac{64+225+225+64}{4}=\frac{578}{4}=144.5is2/math/290d9d3cf2ac1342.svg
• S\circ R=\begin{bmatrix}0.4&0.4&0.3\\0.2&0.2&0.2\\0.4&0.4&0.3\end{bmatrix}is2/math/291bb354e326dcfa.svg
• (1-0.54)v=2.5\quad0.46v=2.5\quad v=2.5/0.46\approx5.434782609is2/math/294b5e0b255011dd.svg
• \text{Remaining}=6-4=2is2/math/2a4c16cf6f52dfc6.svg
• \neg X\cup\neg Y:\max(0.1,0.9)=0.9is2/math/2a8b87917a1340f9.svg
• \langle\mathcal S,\mathcal A,P,R,\gamma\rangleis2/math/2a8c7332f3030a47.svg
• \alpha=0.8is2/math/2a9a9be0e9cd69dc.svg
• \neg X\cap\neg Y:\min(0.5,0.4)=0.4is2/math/2acca2b0b18cfada.svg
• x^{(1)}=4+(-2)=2is2/math/2b24c838f14c9776.svg
• v(C)=\frac{-2.45}{0.838}=-2.9236,\qquad v(L)=-1+0.36(-2.9236)=-2.0525is2/math/2b295ea148a86b91.svg
• 2/14=0.1429is2/math/2b49c5f9ee4dd7a9.svg
• 0.5(p-x)=(0.5(-2),\;0.5(1))=(-1,0.5)is2/math/2b6de1af24a02bfa.svg
• P(\text{Temperature}=\text{Normal}\mid C=\text{Yes})=\frac{0}{3}=0is2/math/2bcd33037561157b.svg
• 2^5-1=31is2/math/2bd250ce49e6fced.svg
• S=\begin{bmatrix}0.3&0.3&0.3\\0.4&0.3&0.4\\0.6&0.3&0.6\end{bmatrix}is2/math/2c582f7ae6099f7e.svg
• v_{NB}=\arg\max_cP(c)\prod_iP(a_i\mid c)is2/math/2c5a9cfc42c5e274.svg
• (1011)_2=(11)_{10}is2/math/2c7d9feaea569d4c.svg
• =\max\{0.1,0.1,0.1\}=0.1is2/math/2caac33b19e2f71f.svg
• P_i=F_i/\sum_jF_jis2/math/2cf98c22ed8b8331.svg
• z^*=\frac{1525.000000}{36.000000}\approx 42.361111is2/math/2cfff2284cb1a5a3.svg
• x=-2+(5/7)(7)=3is2/math/2d086e3a5cf5f29f.svg
• ais2/math/2d4a1257492be872.svg
• 1\times 16=16is2/math/2d5a7498be08cfdd.svg
• P(X=1)=\frac{0.30}{1.20}=0.25is2/math/2d7f62f0b50223d0.svg
• P(\neg A)=1-0.6=0.4is2/math/2d95ba220d546868.svg
• d_p=4.15-(4.15)=0is2/math/2ddd2b101f36cd26.svg
• f(\sum\lambda a)\ge\sum\lambda f(a)is2/math/2ddf537dc44d74a6.svg
• \sum_\omega P(\omega)=1is2/math/2e04a098f6f00102.svg
• \alpha_r=\min(\mu_{T,r},\mu_{O,r})\quad\mu_{out}(z)=\max_r\min(\alpha_r,\mu_{C_r}(z))is2/math/2e1abd53c3392e1f.svg
• a+0.5a+\frac5{12}\frac27=1-\frac27=\frac57is2/math/2e4dfec7416e59d8.svg
• s_{+1}(N)=\frac4{10}\frac{1+1}{4+3}\frac{1+1}{4+3}\frac{1+1}{4+2}=\frac8{735}\approx0.0108844is2/math/2e5b140a14fedb05.svg
• d_g=1-(1.5)=-0.5is2/math/2e6ff91ff778f21f.svg
• 0\times(4)=0is2/math/2e86b7550d7cec9c.svg
• \mu_{out}(z)=\max\{\min(0.3,\mu_H(z)),\min(0.5,\mu_{VH}(z))\}is2/math/2ead49cb7f0674dc.svg
• \frac{0.12}{0.3}=0.4is2/math/2ef396ae4f7c0a1c.svg
• (1011)_2is2/math/2f07200d2566e6c6.svg
• (0)^2=0\qquad2\sigma^2=2(2^2)=8is2/math/2fa85e59ecadded4.svg
• d_g=7.21-(7.31)=-0.1is2/math/2fc0a8e88db078cd.svg
• P(b)=P(b\mid a)P(a)+P(b\mid\lnot a)P(\lnot a)is2/math/2fd503c9ba9e8f6f.svg
• d_g=1-(2)=-1is2/math/2fe1a995f4b4afef.svg
• P(\text{Fever}=\text{Yes}\mid C=\text{No})=\frac{2}{5}=0.4is2/math/2feb71df5b426bfc.svg
• \sum A=21is2/math/2ffb3645f0d4cc76.svg
• x=L+\frac{d}{2^n-1}(U-L)is2/math/3013a3fc845c64e5.svg
• \sum A=1500/49\approx30.612244898is2/math/302b0a0b55fb1874.svg
• d_p=5-(5)=0is2/math/302db7bc64d6f9e9.svg
• T=\begin{bmatrix}0.2&0.6\\0.7&0.4\end{bmatrix}is2/math/303c2bbbbf128653.svg
• \mu(12)=e^{-0.5}\approx0.60653066is2/math/3045651368f264c4.svg
• 0.9\times(5)=4.5is2/math/307a4fc23f4ba8e0.svg
• R_{.5}=\begin{bmatrix}1&0&0&0\\0&1&1&1\\0&1&1&0\\0&0&0&0\end{bmatrix}is2/math/3083e2884959ed29.svg
• \bar a=15/3=5is2/math/30b09f1f2d830b9a.svg
• X\cup Y:\max(0.5,0.6)=0.6is2/math/30d486e82175d463.svg
• f(4)=4^2=16=16is2/math/312433d364d9cfab.svg
• L=\sum_{i=1}^{n-1}d(c_i,c_{i+1})+d(c_n,c_1)is2/math/316d88916534af37.svg
• \Delta\tau_1=1/15\approx0.0666667,\qquad\Delta\tau_2=1/10=0.1is2/math/31ae3d58c7494e4d.svg
• z^*=\frac{62}{0.8}=77.5is2/math/31b2865a5ddd122a.svg
• v^{(2)}=(-0.5)+0+(-0.5)=-1is2/math/31b640bb301f8dce.svg
• 0+5+5=10is2/math/31c26d8043dbf781.svg
• -4/8=-0.5is2/math/31d0827fcff0f487.svg
• 10-10=0is2/math/321dce230aba9ce3.svg
• x_{t+n}=x_tP^nis2/math/3229af8e13e5a4dd.svg
• R_{ij}=\min(D_i,I_j),\quad t_i=\max_j\min(R_{ij},V_j),\quad u_i=\max_j(R_{ij}V_j)is2/math/322f826c4ddc94dd.svg
• F_2=16.466552+8.988249+4.906225+12.1737-8.86+5=38.6747is2/math/323572bf8ccf630b.svg
• 65\times0.3=19.5is2/math/3285138cffe8477f.svg
• \mu_{\mathrm{rule}}(z)=\min(\alpha,\mu_{\mathrm{consequent}}(z))is2/math/32c896372ab93b2e.svg
• \alpha=0.5,\;\gamma=0.9is2/math/32cf597702a3bb21.svg
• X\cap Y:\min(0.3,0.9)=0.3is2/math/33c018c6e76860e7.svg
• f(17)=9826-867+136=9095is2/math/33f7934e87e644ca.svg
• A=14.4\times0.6/2=8.64/2=4.32is2/math/344db3e57e5c18d1.svg
• \gamma_{ik}=\frac{\pi_k\mathcal N(x_i\mid\mu_k,\Sigma_k)}{\sum_j\pi_j\mathcal N(x_i\mid\mu_j,\Sigma_j)}is2/math/3463c7734eab2353.svg
• P(premium)=\frac{0.302}{1.002}\approx0.301397is2/math/3465c1e9a29a79c1.svg
• P(hd)=0.7(0.25\times0.25+0.45\times0.75)+0.3(0.55\times0.25+0.75\times0.75)=0.7(0.4)+0.3(0.7)=0.49is2/math/3485875632aa8ada.svg
• X\cup Y:\max(0.8,0.5)=0.8is2/math/34ae0707b407acaa.svg
• p(x_i\mid\theta)=\sum_{k=1}^{K}\pi_k\mathcal N(x_i\mid\mu_k,\Sigma_k),\quad \pi_k\geq0,\quad\sum_k\pi_k=1is2/math/34e74a21f3e93c71.svg
• A_\alpha=\{x:\mu_A(x)\ge\alpha\}is2/math/34e8388304c6f4b2.svg
• p(x)=\sum_{k=1}^K\pi_k\mathcal N(x\mid\mu_k,\Sigma_k)is2/math/3599d1d0f37ccb77.svg
• 8-x=1.5is2/math/35ee639a9616fa80.svg
• P(A\mid B)=\frac{0.0612}{0.1004}\approx0.609561753is2/math/35fed556321f4dd0.svg
• z^*=\frac{\int_1^{100}z\mu_{agg}(z)\,dz}{\int_1^{100}\mu_{agg}(z)\,dz}is2/math/3619dbb6f7443588.svg
• F=\frac{1}{1+f(x)}is2/math/361a654c14fe241e.svg
• 60\times0.2=12\qquad40\times0.9=36is2/math/365204994bb40e62.svg
• 15+16+12=43is2/math/368823bf9efe0162.svg
• b=1-\frac{25}{63}-\frac{18}{63}=\frac{20}{63}is2/math/369b239ecbe26cbf.svg
• P_{AB}=0.5/0.75=2/3is2/math/371fe53440644f73.svg
• S=\begin{bmatrix}0.2&0.6\\0.8&0.3\end{bmatrix}is2/math/377016d2bce333c2.svg
• \mu_A(7)=\frac{8-7}{8-5}=\frac{1}{3}=0.333333333is2/math/379229dc05ca4e37.svg
• v_*(s_1)=10is2/math/379f03e15709665d.svg
• c_1r_1=0.89235is2/math/37a512f090105dd3.svg
• 1\times 1=1is2/math/37d06720ad44993d.svg
• \mu(x)=\frac{1}{1+|(x-c)/a|^{2b}}is2/math/37da3dd6f213b8e7.svg
• e^{-2}\approx0.135335283is2/math/37f9ca8ef53fee4b.svg
• w=(4/7)/1.5=8/21is2/math/38376c6530974a73.svg
• 6+0=6\qquad0.5\times6=3is2/math/386c4e1d4519186c.svg
• \mu_{Reject}=(50-z)/25\approx0.288764474,\quad\mu_{Consider}=(z-25)/25\approx0.711235526,\quad\mu_{Approve}=0is2/math/388fb9433b0d08db.svg
• \gamma\in[0,1]is2/math/389866bfd285136c.svg
• S=0.5\times(-1)=-0.5is2/math/38f7adbc6f0de36b.svg
• v_0(s_1)=0is2/math/392369c0ba50f924.svg
• F(2.5,-3)=2(6.25)-3(2.5)(-3)+9+5(2.5)+2(-3)+7=12.5+22.5+9+12.5-6+7=57.5is2/math/3938546a658fac5b.svg
• 20\times0.2=4is2/math/396c5c156cb14965.svg
• 50(0.60)+30(0.30)+20(0.20)=30+9+4=43is2/math/396ed931211b6fc9.svg
• (1-0)^2=1is2/math/39a0d42c254cae5b.svg
• z_L-50=0.3\times15=4.5is2/math/39a20fc95a439ada.svg
• A_2=\tfrac12\times4\times0.5=1is2/math/39c5ce3ae9dbf13b.svg
• c=0.2w+0.2a+0.5cis2/math/39cb7355944fbe4a.svg
• x^{(1)}=3+(-2)=1is2/math/39edc010c99575f0.svg
• \mu(25)=1is2/math/3a0f8e229e9dd73e.svg
• v(s)=\mathbb E\,[\,G_t\mid S_t=s\,]is2/math/3a1a21a071450128.svg
• 0.4b=0.2i\quad\Rightarrow\quad i=2bis2/math/3a1b2778c1d42930.svg
• x^{t+1}=x^t+v^{t+1}is2/math/3a35d97963567317.svg
• x^2=2\times2=4\qquad x^3=4\times2=8is2/math/3a35f3beb07d2871.svg
• U=R\odot S^T=\begin{bmatrix}0.7&0.3&0.5\\0.5&0.3&0.5\\0.2&0.2&0.2\end{bmatrix}is2/math/3a687cf430317d18.svg
• \sum F=0.038461538+0.1+0.5+0.038461538\approx0.676923077is2/math/3a777f87cb72ab3e.svg
• \mu(x)=e^{-(x-c)^2/(2\sigma^2)}is2/math/3a8ce5565c9dd1ff.svg
• f(5.9064, 8.1428)=5.9064^2+8.1428^2-(5.9064)(8.1428)is2/math/3a9fac3b191a1ffa.svg
• \pi = \pi Pis2/math/3ac2346fee4256a7.svg
• f(3.19, 7.36)=3.19^2+7.36^2-(3.19)(7.36)is2/math/3ae29bc527e20248.svg
• N_k=\sum_i\gamma_{ik},\quad\mu_k^{new}=\frac{\sum_i\gamma_{ik}x_i}{N_k}is2/math/3aff5f591828cac6.svg
• x^2=13\times13=169\qquad x^3=169\times13=2197is2/math/3b2d51c91ac2cf33.svg
• P(A\mid E)=\frac{P(A\cap E)}{P(E)},\quad P(E)>0is2/math/3b42af132ac8057a.svg
• q(s_1,\text{go})=0.8(5+0.9\cdot0)+0.2(0+0.9\,v(s_1))=4+0.18\,v(s_1)is2/math/3b7159339f0c3987.svg
• \pi=[38/97,\;34/97,\;25/97]is2/math/3b7300b6febd286b.svg
• \mu_{A\setminus B}(u_1)=\min(0.2,0.3)=0.2is2/math/3b878090c37dc92f.svg
• v_{k+1}(s)\leftarrow\sum_a\pi(a\mid s)\sum_{s'}P(s'\mid s,a)\,[R+\gamma v_k(s')]is2/math/3bbeb8acbcb4f4f5.svg
• \pi(a\mid s)=P(A_t=a\mid S_t=s)is2/math/3bcff52e9ca32562.svg
• v=(0.1,\,0.25)is2/math/3bf1b4b0830b2535.svg
• iis2/math/3c3a2b02c199adb2.svg
• \text{Fault total}=0.05+0.10+0.08+0.07=0.30is2/math/3c42285ce038d0e3.svg
• x_1^2=9.904868,\;-x_1x_2=+12.666221,\;x_2^2=16.197405,\;3x_1=9.4416,\;4x_2=-16.0984is2/math/3c47a0a1f1465e31.svg
• I=0.5\times0.2=0.1is2/math/3c5f4d0f2cda8136.svg
• \sum F=27+51+55+37\approx170is2/math/3c9eb5a0f63c3e40.svg
• q_*is2/math/3ccaa9f377c85da8.svg
• v_\pi=\sum_a\pi(a\mid s)q_\pi(s,a)is2/math/3ce6dda6fd8fe296.svg
• v(C),\;v(L)is2/math/3d1e99a2714e55b1.svg
• u=-\frac{4}{2}=-2is2/math/3d2cdafe17449912.svg
• x_A^1=(0+0.12825,0+0.12825)=(0.12825,0.12825)is2/math/3d974cd8e38814b0.svg
• v^{(1)}=0.05+0+(-0.012)=0.038is2/math/3dca081cd95c2dd0.svg
• X\cap Y:\min(0.7,0.2)=0.2is2/math/3de60cd86ef1a257.svg
• s-0.8s=0.1c\quad\Rightarrow\quad0.2s=0.1cis2/math/3dfa7c84f62b2e86.svg
• \mu_{B^c}(u_1)=1-0.9=0.1is2/math/3e895fda5c8f4e29.svg
• A_1c_1=2\times2=4is2/math/3e993e836c87a393.svg
• P=\begin{bmatrix}1-a&a\\b&1-b\end{bmatrix}is2/math/3eacaa69615e9607.svg
• \mu_A(32)=\frac{35-32}{35-25}=\frac{3}{10}=0.3is2/math/3ed9e45724bceb7e.svg
• 0^2\times(8)=0is2/math/3f0045412e0c9af2.svg
• 2x^3=9826\quad 3x^2=867\quad 8x=136is2/math/3f446d5c18f6525b.svg
• G_t=\sum_{k\ge0}\gamma^kR_{t+k+1}is2/math/3f482d8ea694678d.svg
• \max_{a'}Q(s',a')=5is2/math/3f677ab805df68d2.svg
• \sum M=35.775+387+91.35+637.5+178.125=1329.75is2/math/3f79a707c0d7b969.svg
• v^{(1)}=0.1+0+(-0.3636)=-0.2636is2/math/3fdb1b4bffd3874d.svg
• F(4.22962,-2.01448)=(4.22962)^2-(4.22962)(-2.01448)+(-2.01448)^2+3(4.22962)+4(-2.01448)+5is2/math/3fe27d20d1a3f2c2.svg
• P(a)is2/math/3ff7ba191ece3ffd.svg
• g(10)=8/4=2is2/math/40041b1514c36954.svg
• \mu_A(x)=\begin{cases}0,&x\le a\text{ or }x\ge c\\\dfrac{x-a}{b-a},&a<x\le b\\\dfrac{c-x}{c-b},&b<x<c\end{cases}is2/math/4010d79d1ea394d4.svg
• 60+80=140is2/math/4012d4bc25614db2.svg
• \text{Remaining}=19-16=3is2/math/401912fa7840eae2.svg
• 10b=1\quad b=0.1\quad i=0.2\quad a=0.7is2/math/401c3cb4b3606dfc.svg
• 2x^3=432\quad 5x^2=180\quad 7x=42is2/math/402fc08edced6189.svg
• x^{(2)}=3-1=2is2/math/40962d708fa42c89.svg
• (1.52+1.36+1)\pi_R=3.88\,\pi_R=1\Rightarrow\pi_R=0.2577,\;\pi_L=0.3505,\;\pi_P=0.3918is2/math/40a070a0bf109760.svg
• 3\times3=9is2/math/40a9936a2387e5b6.svg
• f(12)=12^2is2/math/40c9415d3dcdd694.svg
• \prod_{i=1}^{3}a_i=a_1a_2a_3is2/math/40f044c5860eb614.svg
• x^{(1)}=6.17+(-0.2636)=5.9064is2/math/40f8a4a790d7094f.svg
• S_2=52\times0.8+48\times0.1=41.6+4.8=46.4is2/math/413733c9cae5c40e.svg
• S=0.5\times(-0.5)=-0.25is2/math/418a15cd3fa3bbcc.svg
• G_tis2/math/418bfe298764c6bc.svg
• (-2)^2=4is2/math/4193a49c3f887499.svg
• L(\theta)=\prod_{i=1}^{n}\Big[\sum_{k=1}^{K}\pi_k\frac{1}{\sqrt{2\pi}\,\sigma_k}e^{-\frac{(x_i-\mu_k)^2}{2\sigma_k^2}}\Big]is2/math/41dc390b2652cd78.svg
• \neg X\cup\neg Y:\max(0.8,0.2)=0.8is2/math/41e12e8e143ca16d.svg
• ABEDC\to ABDEC,\quad L=2+4+1+5+9=21,\quad F=0.0476is2/math/421251db4f0614de.svg
• d_g=-1-(3)=-4is2/math/4224cdc3d1fd7742.svg
• P(cavity)=0.2is2/math/422e41a24d087e55.svg
• e^u=e^{0}\approx 1is2/math/4263edec66681677.svg
• v_{21}=0.9(0.1908)=0.17172,\quad x_{21}=4.22962;\qquad v_{22}=0.9(0.2228)=0.20052,\quad x_{22}=-2.01448is2/math/42d42b7358982c8c.svg
• 2x^3=16\quad 5x^2=20\quad 7x=14is2/math/42f55d86ed2302e7.svg
• \Sigma_k^{new}=\frac{\sum_i\gamma_{ik}(x_i-\mu_k^{new})(x_i-\mu_k^{new})^T}{N_k}is2/math/433755f43e0968ba.svg
• P(W,R)=0.0396+0.324+0.0495+0.045=0.4581is2/math/43750abdd00f3218.svg
• v=(0.2,\,0.3)is2/math/438166430b0200d4.svg
• P(\neg A\cap B)=0.4\times0.01=0.004is2/math/43a91fadefff9097.svg
• Pis2/math/43eba13e48b4ac63.svg
• l=1.36(25/97)=34/97\quad p=1.52(25/97)=38/97is2/math/442f9f9b9a48c0ce.svg
• \pi_B=0.6\pi_B+0.2\pi_I+0\cdot\pi_A\;\Rightarrow\;0.4\pi_B=0.2\pi_I\;\Rightarrow\;\pi_I=2\pi_Bis2/math/445f3c8d74cadf82.svg
• P(W,B_2)=\frac58\frac12=\frac5{16}=0.3125is2/math/44aff1e67025be81.svg
• P(x_1..x_n)=\prod_iP(x_i\mid parents(X_i))is2/math/44d1eaaf40faab74.svg
• P(A\cap B)=0.001\times0.99=0.00099is2/math/44ef59701ff8e69a.svg
• \max\{\min(0.2,0.3),\min(0.3,0.3),\min(0.3,0.1)\}is2/math/4535d46543709132.svg
• w(premium)=0.08+0.10+0.072+0.05=0.302is2/math/455884f740d33473.svg
• s'is2/math/45840389386c62f8.svg
• 1+2+3+4=10is2/math/45a76e7db8947072.svg
• I=0.5\times0.038=0.019is2/math/45baaa3ae928b8b4.svg
• \alpha=0.5is2/math/45f07f09a68169c8.svg
• x_t\to\piis2/math/45f2d2d1b7d524a8.svg
• R=\begin{bmatrix}0.4&0.6&0.3\\0.4&0.4&0.3\\0.1&0.1&0.1\end{bmatrix},\qquad S=\begin{bmatrix}0.4&0.5&0.3\\0.2&0.2&0.2\\0.4&0.4&0.3\end{bmatrix}is2/math/463247054a5e2b55.svg
• 0.6b=0.3a+0.25cis2/math/463b4822655ff527.svg
• 23=(2\times 10)+(3\times 1)=20+3is2/math/46c502cb15486e6a.svg
• 1\times 2=2is2/math/46c6b9d4e71ea84d.svg
• v_\pi(s)=\sum_a\pi(a\mid s)q_\pi(s,a)is2/math/46eb24952a8c1ab7.svg
• \mathbf P(Cavity\mid toothache)=\alpha\langle0.12,\,0.08\rangle=\langle0.6,\,0.4\rangleis2/math/473e1cb48309ce75.svg
• A_1=4\times0.5=2\qquad c_1=(0+4)/2=2is2/math/478220c1039e3c18.svg
• q_\pi(s,a)=\sum_{s'}P(s'\mid s,a)\,\Big[\,R(s,a,s')+\gamma\sum_{a'}\pi(a'\mid s')\,q_\pi(s',a')\,\Big]is2/math/4789f0f7def0f243.svg
• \mu_{A\setminus B}(u_1)=\min(0,0)=0is2/math/478aba1f930ec419.svg
• \ln L=\ln(0.1)\approx-2.302585093is2/math/47a26d7f62e313c6.svg
• \overline f_{new}=\frac{15+57+57+37}{4}=\frac{166}{4}=41.5is2/math/4832c61f6dee4ab2.svg
• \sum_j P_{ij}=1is2/math/4864b97aaf5ebb33.svg
• Ris2/math/486a3031e7ac622b.svg
• \begin{bmatrix}0&7.566667&2.6&2.166667\\7.566667&0&2.166667&4.1\\2.6&2.166667&0&0.566667\\2.166667&4.1&0.566667&0\end{bmatrix}is2/math/4905d8df69aaed31.svg
• 0.5(3-(4))=-0.5is2/math/4914c1155f2d3a69.svg
• 0.5\times(-2)=-1is2/math/493f21e1dcf9ae65.svg
• P(X\mid e)=\alpha\sum_{\text{hidden}}\prod P(\cdot\mid parents)is2/math/49b2c852ff5f290f.svg
• b-a=15.4-1=14.4\qquad u+v=0.6is2/math/49c4a9b06649aac9.svg
• R_1=-2,\;R_2=-1,\;R_3=-2,\;R_4=0is2/math/49e74a4d062c7fe7.svg
• c=0.2(1-c)+0.5c=0.2+0.3cis2/math/49f2613d4b58dc75.svg
• x^2=6\times6=36\qquad x^3=36\times6=216is2/math/4a05c63ed6e04319.svg
• P_{ij}=\frac{\tau_{ij}\,\eta_{ij}}{\sum_{l\in\text{allowed}}\tau_{il}\,\eta_{il}},\qquad\eta_{ij}=\frac1{d_{ij}}is2/math/4a0efb39e84f4c4e.svg
• X\cap Y:\min(0.9,0.3)=0.3is2/math/4a2bc31233f313e7.svg
• v_i^{new}=0.5(-2-x_i)is2/math/4a6d06df6529bc16.svg
• T=\begin{bmatrix}0.6&0.3&0.6\\0.2&0.2&0.2\\0.6&0.3&0.6\end{bmatrix}is2/math/4a88f859d09d91ef.svg
• f(5)=(2)^2=(2)\times(2)=4is2/math/4ad5b97ffc47b26c.svg
• s_{+1}(Y)=\frac6{10}\frac{2+1}{6+3}\frac{2+1}{6+3}\frac{4+1}{6+2}=\frac1{24}\approx0.0416667is2/math/4ae4ba086320b944.svg
• P(Z{=}1)=0.576+0.072+0.008+0.012=0.668is2/math/4ae537c83ce53b71.svg
• x=8-1.5=6.5is2/math/4ae644eed157e001.svg
• 0.25\times0.2=0.05is2/math/4b7931095eb1884c.svg
• gbest^0=(-1)is2/math/4bb3271e5427b8e5.svg
• f(14)=5488-980+98=4606is2/math/4be8ca3a4ac49622.svg
• \frac12p_3=\frac14-\frac1{16}=\frac3{16}\quad\Rightarrow\quad p_3=\frac38is2/math/4c1768a409855602.svg
• \max\{\min(0.4,0.4),\min(0.4,0.4),\min(0.3,0.1)\}is2/math/4c61c54bcac1075e.svg
• 16+0+0+2+1=19is2/math/4c61d1192553e0c2.svg
• f(0.5)=0.5^2=0.25=0.25is2/math/4c640081b99e1be6.svg
• O(2^n)is2/math/4c7e79dfd318e20b.svg
• f(3.5)=3.5^2=12.25=12.25is2/math/4c8cdadcd9350fe6.svg
• z^*=\frac{\sum_i z_i\mu(z_i)}{\sum_i\mu(z_i)}is2/math/4ca780545740bc9c.svg
• V=\begin{bmatrix}0.30825&0.462375\\0.09&0.225\end{bmatrix}is2/math/4cf56ad79b60f4ff.svg
• F(3.5,-1.5)=2(12.25)-3(3.5)(-1.5)+2.25+17.5-3+7=24.5+15.75+2.25+17.5-3+7=64is2/math/4d48c40dfb866c17.svg
• (10011)_2is2/math/4d6378ea1db035a2.svg
• I=0.5\times2=1is2/math/4d6486404a7509ba.svg
• \gamma_1=\frac{0.120985362}{0.120985362+0.120985362}\approx0.5is2/math/4db037db96f3bfa7.svg
• S=0.5\times(-6)=-3is2/math/4dd7f304ffa4f233.svg
• u=-\frac{0}{2}=0is2/math/4e1e416350f910b4.svg
• v^{(2)}=(-1.5)+0+0=-1.5is2/math/4e2008e440fe556c.svg
• v^{(1)}=0+0+(-1.5)=-1.5is2/math/4e26de872fe8bfca.svg
• a=0.7b/0.1=7bis2/math/4e3c76ca7c5b3101.svg
• x^{(1)}=(-2)+0=-2is2/math/4e58cfc2f623e47e.svg
• v^{(2)}=0.025+(-0.03)+(-0.006)=-0.011is2/math/4e6287c93cb39658.svg
• z_{MOM,2}=(3+4)/2=3.5is2/math/4ea913e911135351.svg
• \gamma<1is2/math/4ec13e93147870ec.svg
• \mu_{A\setminus B}(u_1)=\min(0.1,0.1)=0.1is2/math/4ec160ba0bee68fe.svg
• \gamma_2=1-\gamma_1\approx0.5is2/math/4ed7d74583102629.svg
• f(x)=2x^2+1is2/math/4ee7030c8a3d9d3a.svg
• \neg X\cap\neg Y:\min(0.3,0.4)=0.3is2/math/4ef33b6f553800c5.svg
• I=0.5\times0.1=0.05is2/math/4f87680527f659f2.svg
• x=\sum_{i=1}^{n}b_i2^{n-i}is2/math/4fb94e1436769fba.svg
• \min(0.4,0.6)=0.4\ne0is2/math/4fcb041eeb76aeb4.svg
• \lambda_k\ge0,\;\sum\lambda_k=1is2/math/5006fcb64f6c3cf6.svg
• \neg X\cap\neg Y:\min(0.2,0.5)=0.2is2/math/503c7cc18cb10c65.svg
• R_{ij}=\min(A_i,B_j)is2/math/50b8f8fcf2c15000.svg
• \gamma_2=1-\gamma_1\approx0.98201379is2/math/50f16fe4d69df396.svg
• f(5.442632, 7.967264)=5.442632^2+7.967264^2-(5.442632)(7.967264)is2/math/515e8ddbaa301792.svg
• P(Co,\neg C,S)=0.27\times0.28=0.0756is2/math/51760d268143a4b5.svg
• \mu(x)=\exp\left(-\frac{(x-c)^2}{2\sigma^2}\right)is2/math/518c1dcbf29cffeb.svg
• f(3)=3^2=9=9is2/math/519522f49346438a.svg
• x^{(1)}=2+(-0.5)=1.5is2/math/51c2ec0b6a29b186.svg
• F(x_1,x_2)=2x_1^2-3x_1x_2+x_2^2+5x_1+2x_2+7is2/math/51d7b4f8ba214898.svg
• P(a_i\mid c)=is2/math/51ff85a257c034ba.svg
• R=\begin{bmatrix}0.6&0.8&0.4\\0.5&0.5&0.4\\0.2&0.2&0.2\end{bmatrix}is2/math/52316a1420d445ac.svg
• F(x)=x_1^2-x_1x_2+x_2^2+3x_1+4x_2+5is2/math/523616224185a5ce.svg
• \max\{\min(0.2,0.3),\min(0.2,0.3),\min(0.2,0.1)\}is2/math/524a01bd31cd2a0e.svg
• -0/(2\times1)=0is2/math/52577cebaa3ae73f.svg
• \mu_H(7)=\frac{7-6}{8-6}=\frac12=0.5is2/math/527793f08bd07193.svg
• C\to B=\tfrac13=0.3333,\quad C\to D=\tfrac12=0.5;\quad\text{sum}=0.8333\Rightarrow P_{CB}=0.4,\;P_{CD}=0.6is2/math/52a7763a8065747e.svg
• G_0is2/math/5322d8ff87ec1002.svg
• \overline f_{new}=\frac{64+225+225+64}{4}=\frac{578}{4}=144.5is2/math/534a2fd2aa176655.svg
• \bar f_{new}=((20)+(10)+(0)+(11)+(9.5)+(9.5))/6=10is2/math/539cf61121bb4c13.svg
• L(ABECD)=2+3+5+8+10=28,\quad F=1/28=0.0357is2/math/5403cdb93ef730d5.svg
• X\cap Y:\min(0.9,0.1)=0.1is2/math/5461384d5ad733c2.svg
• \mu_{A^c}(x)=1-0.4=0.6is2/math/54b240c60f89d754.svg
• \eta_{ij}is2/math/54ca51b8bfad2b71.svg
• G_0=-4.52is2/math/54ecfef1a5e374f5.svg
• z_{MOM,1}=(2+3)/2=2.5is2/math/54f0d91e056c6e0b.svg
• \mathcal Sis2/math/54f9d81745f0a6da.svg
• P(a\mid c)=\dfrac{\text{count}+1}{N_c+k}is2/math/55066f62fb8b94f2.svg
• P=\begin{array}{c|cc} & X & Y\\\hline X & 0.75 & 0.25\\ Y & 0.40 & 0.60\end{array}\qquad\text{i.e. } P=\begin{bmatrix}0.75&0.25\\0.40&0.60\end{bmatrix}is2/math/550d8027e65ec3eb.svg
• R=\begin{bmatrix}1&0\\0.4&0.7\end{bmatrix}is2/math/5525680a26d9df18.svg
• 1-0.4=0.6is2/math/553cb087421dc8d7.svg
• \pi(a\mid s)=P(A_t=a\mid S_t=s)is2/math/556c05bd826349e2.svg
• M_1=\frac{4.5}{6}[(100+54.5)0+(50+109)0.3]=35.775is2/math/5570b4038f564dfd.svg
• \mu_{max}=0.8is2/math/55721303722e693a.svg
• =38.0689+68.3929-51.0259=55.4359is2/math/558e7a3b9a40d34e.svg
• T_{11}=\max\{\min(0.3,0.8),\min(0.6,0.2),\min(0.9,0.5)\}is2/math/559127d73e85d555.svg
• \frac{35-32}{35-25}is2/math/55a4e1a39f439f8c.svg
• x-3=2-3=-1is2/math/55d4a2b861701691.svg
• f(15)=15^3-4(15^2)+6(15)=3375-900+90=2565is2/math/5600d463abeea9e2.svg
• \frac{7-6}{8-6}is2/math/5627768e8762bda4.svg
• S\circ R=\begin{bmatrix}0.2&0.3&0.3\\0.1&0.1&0.1\\0.2&0.3&0.3\end{bmatrix}is2/math/564480fc64ab9d52.svg
• P(b\mid j,m)=\alpha\,P(b)\sum_aP(j\mid a)P(m\mid a)\sum_eP(a\mid b,e)P(e)is2/math/5648415bb994a47e.svg
• x_{ij}^{t+1}=x_{ij}^t+v_{ij}^{t+1}is2/math/566cb0a85349654b.svg
• gbest^0=(3.14, 7.21)is2/math/567a650862d29da5.svg
• \mu_k^{new}=(0.8+1)/1=1.8is2/math/5699fad0064a323f.svg
• T_{ik}=\max_j\min(R_{ij},S_{jk})is2/math/56cc213f59fd4c68.svg
• \sigma=\sqrt{0}\approx0is2/math/56fce38f6a89699b.svg
• \mu(x)=\frac1{1+\left|\frac{x-c}{a}\right|^{2b}}is2/math/5710690d4b19691e.svg
• x^{(2)}=2+(-1.5)=0.5is2/math/5721079871bdab79.svg
• P(\lnot b\mid j,m)=\alpha\,(0.999)\big[0.9(0.7)(0.29\times0.002+0.001\times0.998)+0.05(0.01)(0.71\times0.002+0.999\times0.998)\big]=\alpha\times0.001492is2/math/57244f2b9ccfa07b.svg
• 0.5(2-(4))=-1is2/math/572c7c6589c1b745.svg
• \alpha=\frac1{0.000592+0.001492}=479.8\;\Rightarrow\;P(b\mid j,m)=0.284,\;P(\lnot b\mid j,m)=0.716is2/math/57f47697add29181.svg
• d_g=1-(3)=-2is2/math/57f4bea8a1c6b79f.svg
• X\cap Y:\min(0.6,0.4)=0.4is2/math/580180c90f774b30.svg
• 2x^3=16000\quad 3x^2=1200\quad 8x=160is2/math/581d9ac708a9edf4.svg
• v=(0.2194,\,0.2449)is2/math/582ec15d8b22479d.svg
• x-2=0.5\times3=1.5is2/math/58646302eb803fe5.svg
• R=\begin{bmatrix}0.2&0.1\\0.6&0.1\\0.5&0.1\end{bmatrix}is2/math/587e0aaf2cb10cc7.svg
• P(B)=0.00099+0.01998=0.02097is2/math/588fc60be824b4ec.svg
• 0.838v_C=-2.45\quad\Rightarrow\quad v_C=-2.45/0.838\approx-2.923627685is2/math/591d6eadcfd21cd8.svg
• P(credit)=\frac{0.08+0.10+0.10+0.15}{1.002}=\frac{0.43}{1.002}\approx0.429142is2/math/594c62fd5c571144.svg
• v_\piis2/math/59ce33a75ce0983e.svg
• S=0.5\times(-1.5)=-0.75is2/math/59d60f341dbf8e9b.svg
• R_{23}=\min(A_2,B_3)=\min(0.2,0.5)=0.2is2/math/59fc3a309a7ab813.svg
• \bar f=\frac{f_1+\cdots+f_N}{N}is2/math/5a16aa84fc40bbd3.svg
• \ln(0.2)+\ln(0.5)\approx-1.609437912-0.693147181=-2.302585093is2/math/5a43c9ee348bb3a2.svg
• P(b)>0is2/math/5a47a3fc657b0d7c.svg
• Hot=\{0/20,\;0.2/25,\;0.6/30,\;1/35\}is2/math/5a7f23351df5307b.svg
• z^*=\frac{\int z\mu_{out}(z)\,dz}{\int\mu_{out}(z)\,dz}is2/math/5aa4acd6aab4215a.svg
• \mu(25)=\frac{15}{20}=0.75is2/math/5aad4f1600b806e2.svg
• \gamma_{ik}=q_{ik}/\sum_jq_{ij}\qquad\mu_k^{new}=\frac{\sum_i\gamma_{ik}x_i}{\sum_i\gamma_{ik}}is2/math/5add87e9c47f7469.svg
• \overline f_{new}=\frac{57+55+57+27}{4}=\frac{196}{4}=49is2/math/5b337e1a852cf7fa.svg
• 80\times1=80is2/math/5b5619f67f240a03.svg
• P_{ij}=\frac{\tau_{ij}^{\alpha}(1/d_{ij})^\beta}{\sum_{k\in allowed}\tau_{ik}^{\alpha}(1/d_{ik})^\beta}is2/math/5b888e9e5a3c73eb.svg
• z^*=\frac{55}{0.8}=68.75is2/math/5b92e77608da1213.svg
• z^*_{MoM}=\frac{a+b}{2}is2/math/5be124f1bdbedfca.svg
• \bar a=\frac{a_1+a_2+a_3}{3}is2/math/5c0dc23066626f45.svg
• 0.75\times7=5.25is2/math/5c142a10f628b6eb.svg
• q_2=0.5\frac{e^{-(3-4)^2/2}}{\sqrt{2\pi}}\approx0.120985362is2/math/5c2d2f7bb6375554.svg
• A_1=(54.5-50)(0+0.3)/2=0.675is2/math/5c3f2b7a4bd76f0b.svg
• \tau=1is2/math/5c811732c72735df.svg
• \sum M=637971/686\approx929.986880466is2/math/5cd3d9be4d829cea.svg
• z_L=54.5is2/math/5ce2529d6b5b0ef5.svg
• L=d_{AB}+d_{BC}+d_{CD}+d_{DA},\qquad fitness=1/L\quad(L>0)is2/math/5d153218d012483f.svg
• R^{a}_{s}=\mathbb E[R_{t+1}\mid S_t=s,A_t=a]is2/math/5d475467732358d0.svg
• 2K+(K-1)=3K-1=3(5)-1=\mathbf{14}is2/math/5d57bdfd8f83239c.svg
• =\max\{0.2,0.7\}=0.7is2/math/5d7c26e3f57345f9.svg
• =\max\{0.5,0.3\}=0.5is2/math/5d84612aa0c3dabf.svg
• d=1(4)+0(2)+1(1)=5is2/math/5d98008e8ca61fcc.svg
• x=1.5+2=3.5is2/math/5def47a62d44d4a8.svg
• 0.6a=0.3w+0.3c\quad a=0.5w+0.5cis2/math/5e60a8a8d562c8fe.svg
• \tau_{ij}^{new}=(1-\rho)\tau_{ij}^{old}+\sum_k Q/L_k\;\text{on edges used by ant }kis2/math/5e92452bfde69641.svg
• P(J)=\sum_AP(J\mid A)\sum_MP(M\mid A)\sum_BP(B)\underbrace{\sum_EP(A\mid B,E)P(E)}_{f_1(A,B)}is2/math/5ea704a1fb57ef3a.svg
• \max\{\min(0.2,0.6),\min(0.2,0.4),\min(0.2,0.1)\}is2/math/5eeb6f5598722d4c.svg
• \pi_L=0.25\pi_P+0.5\pi_L+0.3\pi_R\;\Rightarrow\;0.5\pi_L=0.25\pi_P+0.3\pi_Ris2/math/5f029acce8c76d6c.svg
• A_{new}=0.12+0.14+0.075=0.335is2/math/5f59f5d8bf197b8d.svg
• 2x^3=3456\quad 3x^2=432\quad 8x=96is2/math/5f6a8611548b73ce.svg
• 0.375\times1=0.375is2/math/5f8249f6f69de64a.svg
• C=0.5\times(-1)=-0.5is2/math/5fb29aabbbcf9b90.svg
• \bar f_{age}=\frac{509+378}{10}=88.7is2/math/5fdd0bc72d39766b.svg
• a=0.25,\;b=0.40\Rightarrow\pi=[0.40/0.65,\;0.25/0.65]=[0.6154,\;0.3846]is2/math/6003dba221d13feb.svg
• x^{(2)}=3+(-1)=2is2/math/60104aa04f06aad8.svg
• \sum F=1+2+3+4=10is2/math/601b2d3d469972c2.svg
• \neg X\cap\neg Y:\min(0.1,0.9)=0.1is2/math/601ec64fa8926431.svg
• z^*=\frac{43}{1.2}=35.833333333is2/math/603521823aee6db6.svg
• B_{new}=30+6+0=36is2/math/60af91b5a47e34d2.svg
• (x-2)/3=0.5is2/math/60bd77c1b7954114.svg
• 1\times 4=4is2/math/60e4619429362147.svg
• x_{i,j}^{t+1}=x_{i,j}^{t}+v_{i,j}^{t+1}is2/math/60fe3c0d98c51b4c.svg
• \overline f_{old}=\frac{4+49+64+169}{4}=\frac{286}{4}=71.5is2/math/6171658d65ed4b49.svg
• -16/8=-2is2/math/6189f75f7af84207.svg
• G_t=R_{t+1}+\gamma R_{t+2}+\gamma^2R_{t+3}+\dots=\sum_{k=0}^{\infty}\gamma^kR_{t+k+1}is2/math/61d2488b0a270706.svg
• P(female\mid premium)=\frac{0.08+0.072}{0.302}=\frac{0.152}{0.302}\approx0.503311is2/math/61d38ae71dd8a06e.svg
• 0.5(-1-(-2))=0.5is2/math/61d7bcc89fa9fd8a.svg
• \mu(x)=\max\left(0,\min\left(\frac{x-a}{b-a},1,\frac{d-x}{d-c}\right)\right)is2/math/61dfda51f019f3c9.svg
• C_2=52\times0.2+48\times0.9=10.4+43.2=53.6is2/math/61e8c23afa57bd0c.svg
• d_g=3.14-(3.14)=0is2/math/6212c336dbecc343.svg
• d_g=-1-(-1)=0is2/math/6223039cb154c72e.svg
• \sum_MP(M\mid A)=1is2/math/6230fbd1828345e4.svg
• U_{11}=\max\{\min(0.6,0.6),\min(0.8,0.7),\min(0.4,0.4)\}=0.7is2/math/62612d777f597842.svg
• \neg X\cap\neg Y:\min(0.3,0.8)=0.3is2/math/6274e157a87eaf7b.svg
• P(C,S,R,W)=P(C)\,P(S\mid C)\,P(R\mid C)\,P(W\mid S,R)is2/math/62a8884ca57ac2e6.svg
• \tau_{ji}is2/math/62cdaada6c1142ff.svg
• 0.2+0.6+0.2=1is2/math/62f59e75ba76247e.svg
• A_2=0.15+1.2+0.15=1.5,\quad M_2=0.55+7.2+1.25=9,\quad c_2=6is2/math/6325190477396ab2.svg
• x^2=5\times5=25\qquad x^3=25\times5=125is2/math/63703f64d73bd95d.svg
• v_C=-2+0.45(-1+0.36v_C)is2/math/637ce97b6476a63f.svg
• S_1=48+4=52is2/math/637f4b18747624bc.svg
• \text{No-fault total}=0.40+0.25+0.20+0.05=0.90is2/math/6388bf0e029e7de3.svg
• 2s=cis2/math/63aa27a487626a55.svg
• G_0=(2)+(0)+(0)=2is2/math/6433acabcea5b54f.svg
• x^{(1)}=7.31+0.038=7.348is2/math/644604a12b5bcc31.svg
• v=1+0.9v\Rightarrow v=10is2/math/64606eb39a2dfd24.svg
• 0.25\times0.25=0.0625is2/math/646a455657cc6928.svg
• z^*=\frac{1575}{21}\approx75is2/math/647a203082aabc5c.svg
• 2.5+52.5=55is2/math/648cd602080c422e.svg
• 8+0+0+1=9is2/math/64fd3b60b066ebf6.svg
• \mu_A(5)=\frac{5-2}{5-2}=\frac{3}{3}=1is2/math/650bc1ad2a8788e3.svg
• 0.25\times3=0.75is2/math/653f73dba2be3445.svg
• f_{mix}(x_i)=\sum_k\pi_k\mathcal N(x_i\mid\mu_k,\sigma_k^2)is2/math/6550a4aaf9e87286.svg
• z_{COA,2}=\frac{3+3.5+7.5}{1.5+1+1.5}=\frac{14}{4}=3.5is2/math/656c9873a2d32c03.svg
• P(A\mid E)=\frac{\text{sum of weights for }A\cap E}{\text{sum of weights for }E}is2/math/656f97e9f0aa20fc.svg
• \tau_{ij}^{new}=(1-\rho)\tau_{ij}^{old}+\sum_k\Delta\tau_{ij}^{k}is2/math/657b62b2f2dbf71a.svg
• C=0.5\times(0)=0is2/math/660e9363a712569b.svg
• \lambda_k^i=\pi_k\mathcal N(x_i\mid\mu_k,\sigma_k^2)/\sum_l\pi_l\mathcal N(x_i\mid\mu_l,\sigma_l^2)is2/math/662cdc674fe7b2c5.svg
• \bar f_{original}=799/10=79.9is2/math/66d0b787f8c6800c.svg
• 50(0.25)+30(0.50)+20(0.30)=12.5+15+6=33.5is2/math/66d48e5543eb7f89.svg
• P^2=\begin{bmatrix}0.6625&0.3375\\0.54&0.46\end{bmatrix}is2/math/6708ca668a7d9dcd.svg
• H:\quad z_L=50+0.3(65-50)=54.5,\quad z_R=80-0.3(80-65)=75.5is2/math/6757779ccc7556ad.svg
• A_{new}=15+16+7.5=38.5is2/math/67abb9b77cf134f6.svg
• 7\times7=49is2/math/67b20e8d00098635.svg
• f(x)=2x^3-3x^2+8xis2/math/68385220447af94f.svg
• 1\times 16+0\times 8+0\times 4+1\times 2+1\times 1=19is2/math/684f11c4b4f0e966.svg
• 0.6\times10=6is2/math/6862aff895141018.svg
• child_1=\alpha P+(1-\alpha)Q,\quad child_2=(1-\alpha)P+\alpha Qis2/math/686b6d11dfecb89a.svg
• P(a\land b)=P(a\mid b)P(b)is2/math/68a6596a80d9767b.svg
• \ell(\theta)=\log\prod_{i=1}^{N}\sum_k\pi_k\mathcal N(x_i\mid\mu_k,\Sigma_k)=\sum_i\log\left[\sum_k\pi_k\mathcal N(x_i\mid\mu_k,\Sigma_k)\right]is2/math/68bab3da61241a5e.svg
• p(x)=\sum_k\pi_k\mathcal N(x\mid\mu_k,\sigma_k^2)is2/math/68bcea7fd026bedf.svg
• x^2=12\times12=144\qquad x^3=144\times12=1728is2/math/68dcfd678cb95894.svg
• \pi=\pi P,\quad\sum_i\pi_i=1is2/math/690921434c640510.svg
• v^{(1)}=0.05+0+0=0.05is2/math/6950d20a25859fe4.svg
• 0.6\times10=6\qquad0.4\times(-2)=-0.8is2/math/6978caf34fd95e0c.svg
• P_{ii}=1is2/math/698cf8e871a3b571.svg
• R_{.003}=\begin{bmatrix}1&1&0&1\\0&1&1&1\\1&1&1&0\\0&1&0&1\end{bmatrix}is2/math/69c4d4d1862cfea7.svg
• P(A\cap B)=0.6\times0.04=0.024is2/math/69ed4b855796ade5.svg
• R=\begin{bmatrix}0.2&0.8\\0.6&0.3\end{bmatrix}is2/math/6a3980f8eb0faad9.svg
• P(3,2,1)=\frac12\cdot\frac12\cdot\frac13=\frac1{12}\approx0.0833333is2/math/6a536bdcde0eb5da.svg
• \mathbf P(X\mid e)=\alpha\,\mathbf P(X,e),\;\alpha=1/\sum_xP(x,e)is2/math/6a7c981fbbb8b9db.svg
• P(j)=P(j\mid a)P(a)+P(j\mid\lnot a)P(\lnot a)=0.90(0.002516)+0.05(0.997484)=0.0521is2/math/6aa5c702fa516ce7.svg
• x^{(2)}=1+(-2)=-1is2/math/6aabd1c885e75a7f.svg
• \mu_M(t)=\begin{cases}(t-1)/49&1\le t\le50\\(100-t)/50&50<t\le100\end{cases}is2/math/6ad9728f32af11c4.svg
• \pi(\text{stay}\mid s_1)=0.5,\;\pi(\text{go}\mid s_1)=0.5is2/math/6b25a58130fefd91.svg
• \mu_L(t)=\begin{cases}(50-t)/49&1\le t\le50\\0&50<t\le100\end{cases}is2/math/6b2ac0fe5ac34f60.svg
• d_g=3.14-(5.9064)=-2.7664is2/math/6b3316f35248ffdf.svg
• \frac{7}{10}=0.7is2/math/6b3a1b0f1231fc45.svg
• z^*=\frac{10}{2}=5is2/math/6b4a65f7e66f3cfb.svg
• 20\times3=60is2/math/6b7da7f3a0b59a78.svg
• P(B_1\mid W)=\frac{3/20}{37/80}=\frac3{20}\frac{80}{37}=\frac{12}{37}\approx0.324324is2/math/6bf86459a0c31fc3.svg
• P(W)=0.18+0.08=0.26is2/math/6c276fca73f350e2.svg
• f(-0.5)=(-0.5)^2=0.25=0.25is2/math/6c453bb065df6e4a.svg
• (1,3)=\max[\min(.4,.3),\min(.5,.3),\min(.3,.1)]=0.3is2/math/6ce0ff88a64ad0ef.svg
• 19.5+42.5=62is2/math/6cfaa5029f5a6cb9.svg
• =\max\{0.2,0.3,0.1\}=0.3is2/math/6d394999310022b5.svg
• S=0.5\times(-4)=-2is2/math/6d9a3c8d94a7312e.svg
• x^{(1)}=4-1=3is2/math/6df71b72d0a31e70.svg
• P(\text{colour})=\frac{\text{balls of that colour}}{\text{all balls}}is2/math/6e0b3ead99011455.svg
• a=0.1b+0.3i+0.9ais2/math/6e1d2349f082467b.svg
• P(Yes\mid x)=\frac0{0+0.03125}=0,\quad P(No\mid x)=\frac{0.03125}{0.03125}=1is2/math/6e2299fa87aa4056.svg
• v=0.5+0.45v+2+0.09v=2.5+0.54vis2/math/6e4eca135d9eae35.svg
• -2+0.45(-2.0525)=-2-0.9236=-2.9236\;\checkmarkis2/math/6e971d442b2340c9.svg
• I=0.5\times0=0is2/math/6f4daee36ea96185.svg
• P(\text{Reputation}=\text{Good}\mid C=\text{No})=\frac{1}{4}=0.25is2/math/6f502d0eec66aa0e.svg
• \mu_{High}(55)=\frac{55-50}{75-50}=0.2is2/math/6f64cb8c41f3eca0.svg
• d_g=-2-(-2)=0is2/math/6f82367c11e33631.svg
• q_*(s,a)=\sum_{s'}P(s'\mid s,a)\,\big[\,R(s,a,s')+\gamma\max_{a'}q_*(s',a')\,\big]is2/math/6f856204425ff6bb.svg
• 0.3(0.1)+0.5(0.2)+0.1(0.7)=0.03+0.10+0.07=0.20=\pi_I\;\checkmarkis2/math/6fea0316b61a8fc1.svg
• x=8b_1+4b_2+2b_3+b_4,\qquad f(x)=x^3-4x^2+6xis2/math/7000daead38032a9.svg
• x^{(2)}=1+(-1.5)=-0.5is2/math/700c5bb73aa33b26.svg
• \max(0.7,0.4)=0.7is2/math/709d5f49fad6a33b.svg
• \frac{0.135335283}{\sqrt{2\pi}}\approx0.053990967is2/math/70e444d7db1b7149.svg
• p(x)=0.12+0.02=0.14is2/math/70eb8d3c0588a981.svg
• P(A\mid B)=\frac{P(B\mid A)P(A)}{P(B\mid A)P(A)+P(B\mid\neg A)P(\neg A)}is2/math/715ffe0029a7c804.svg
• \alpha=\min(0.3,0)is2/math/7194f00ae81bc1b1.svg
• S=0.12\times(-0.9328)=-0.111936is2/math/71d9611ef2b3e668.svg
• 0.5+1+0.5=2is2/math/723c388b267a6d46.svg
• d_p=-1-(-1)=0is2/math/726263f8c97a9f8e.svg
• L=\sum_{i=1}^{n-1}d(c_i,c_{i+1})+d(c_n,c_1),\quad F=1/Lis2/math/7283c4b77917085c.svg
• P(B)=0.024+0.004=0.028is2/math/7297bb21152d3d31.svg
• f(13)=4394-845+91=3640is2/math/729d7b573feaaf95.svg
• d_p=1-(2)=-1is2/math/72aafd3f3c32f9a7.svg
• P(X_{t+1}=S\mid X_t=C,X_{t-1}=R)=P_{CS}=0.75is2/math/72bc7a0f35b98e71.svg
• (2)^2=4is2/math/730976d327ad7f5e.svg
• P(Yellow\mid Mango)=\frac{350}{650}=\frac7{13}\approx0.538462is2/math/730d82f2aaccc80b.svg
• P(W,F,C,V)=P(W)\,P(F\mid W)\,P(C\mid W)\,P(V\mid F,C)is2/math/73135e72ac19cca4.svg
• \text{retained sum}=95+80+89+94+81+88=527is2/math/73527afb9489efc7.svg
• P(c)=is2/math/73533725519e6dc2.svg
• P(\text{Temperature}=\text{Normal}\mid C=\text{No})=\frac{3}{5}=0.6is2/math/738acde5b7f3afea.svg
• P(R,W)=0.2\times0.9=0.18is2/math/7395138baece4728.svg
• f(-2)=(-2)^2=4=4is2/math/73d4f3be5484ad52.svg
• \sum_{i=1}^3 a_i=2+5+8is2/math/73da4a922a419fa3.svg
• \log\big(\sum_k\lambda_ka_k\big)\ge\sum_k\lambda_k\log a_kis2/math/742b7c09a1ce32a4.svg
• w+(0.5w+0.5c)+c=1is2/math/748747a4be5b6069.svg
• \sum F=15+57+57+37\approx166is2/math/74f6736e95b5ff63.svg
• 2^4=2\times2\times2\times2=16is2/math/750f21b62900f171.svg
• X=\begin{bmatrix}3.461457&-3.572109\\4.22962&-2.01448\end{bmatrix}is2/math/751b3a44f8f94dbc.svg
• T_{22}=\max\{\min(0.4,0.6),\min(0.7,0.3)\}is2/math/7580af7f56a5d58b.svg
• 0+0+2+1=3is2/math/76153632459f881f.svg
• 0.2+0.6+0.4=1.2is2/math/762eaecf8eca8b36.svg
• 3.88r=1\quad r=1/3.88=25/97is2/math/76317a80ed92e6f4.svg
• |(14-10)/2|=|2|=2is2/math/76742f554dcbda62.svg
• 0+0.3+0.5=0.8is2/math/76b7c63a5ed0de17.svg
• f(3.179, 7.327)=3.179^2+7.327^2-(3.179)(7.327)is2/math/76bb4ad3f6895999.svg
• x^{(1)}=3+(-1.5)=1.5is2/math/76c197431121b5e1.svg
• z^*=10/3\approx3.333333is2/math/772602a59fae475c.svg
• \tau=\begin{bmatrix}0&15&5&4\\15&0&4&8\\5&4&0&1\\4&8&1&0\end{bmatrix}is2/math/772dd747e10284ad.svg
• \max\{\min(0.2,0.2),\min(0.4,0.2),\min(0.3,0.1)\}is2/math/7738d5099a9e3ab1.svg
• \sigma^2=\frac{1+1}{2}=1is2/math/7740370cf600b4a5.svg
• \log L=\sum_i\log\sum_k\pi_k\mathcal N(x_i\mid\mu_k,\sigma_k^2)is2/math/7852bd4ef8cbe433.svg
• 1+0.1428=1.1428is2/math/7856892795a9fcf9.svg
• d_p=5.9064-(5.9064)=0is2/math/786ba18b3b62c512.svg
• -0/8=0is2/math/7875a0b4018e0854.svg
• F=1/Lis2/math/7887e5a1e9ffa26d.svg
• C_{new}=0.08+0.07+0.125=0.275is2/math/7892e8414e63bbd0.svg
• \frac{0.60653066}{\sqrt{2\pi}}\approx0.241970725is2/math/7893b72b8b5aa648.svg
• P_i=F_i/(F_A+F_B+F_C+F_D)is2/math/791fc41b905c43f5.svg
• x^{(2)}=2+(-0.5)=1.5is2/math/7929aa22070f7f4e.svg
• F(x_A^1)=(0.12825)^2+7(0.12825)+5=5.9141980625is2/math/7930f3e2597545dd.svg
• P(W,B_1)=\frac3{10}\frac12=\frac3{20}=0.15is2/math/797903c04a87ae2d.svg
• Q/L=1/14=0.0714is2/math/799732b628eacf45.svg
• P(\text{sick}\mid+)=\frac{P(+\mid\text{sick})\,P(\text{sick})}{P(+\mid\text{sick})P(\text{sick})+P(+\mid\text{healthy})P(\text{healthy})}=\frac{0.99\times0.001}{0.99\times0.001+0.02\times0.999}=0.047is2/math/7999e36c046cbb22.svg
• (a+2b)v=(1+30.8)\times0.6=31.8\times0.6=19.08is2/math/79c1733b7155a919.svg
• z^*=\frac{\sum_i z_i\mu_i}{\sum_i\mu_i}\qquad z^*=\frac{\int z\mu(z)\,dz}{\int\mu(z)\,dz}is2/math/79ddc9ad485491a0.svg
• 0.3\times0.8=0.24\quad;\quad0.24\times0.1=0.024is2/math/79f6ccced168cd06.svg
• 2x^3=5488\quad 5x^2=980\quad 7x=98is2/math/7a11853924c667dd.svg
• x=b_1 2^3+b_2 2^2+b_3 2^1+b_4 2^0is2/math/7a27d9903aa92206.svg
• jis2/math/7a5268d94266eaaf.svg
• 2x^3=18522\quad 3x^2=1323\quad 8x=168is2/math/7a74ad5911e6f0d8.svg
• x=5is2/math/7a95cf78643aae43.svg
• T_{ik}=\max_j\min(R_{ij},S_{jk})is2/math/7ad9b78ea1d9260b.svg
• \overline f_{old}=\frac{1+36+81+196}{4}=\frac{314}{4}=78.5is2/math/7aeb7f13264a06b2.svg
• s_{Yes}=\frac12\times\frac24\times\frac14\times\frac04=0is2/math/7af34167018bca92.svg
• w_B=1\times0.5=0.5\qquad w_C=1\times0.25=0.25is2/math/7b0589337b7254cb.svg
• P(A\mid R)=\frac{3/14}{55/126}=\frac{27}{55}\approx0.490909is2/math/7b275c539000ed1c.svg
• P(a\land b)=P(a\mid b)P(b)=P(b\mid a)P(a)is2/math/7b86112b191ef76c.svg
• \neg X\cap\neg Y:\min(0.7,0.1)=0.1is2/math/7b8935d4bd0523c1.svg
• v_\pi(s)=\sum_{a}\pi(a\mid s)\sum_{s'}P(s'\mid s,a)\,\big[\,R(s,a,s')+\gamma\,v_\pi(s')\,\big]is2/math/7b8a3aea9af35f5f.svg
• \alpha=0is2/math/7b95185b7bc8e8a2.svg
• v(L)=-1+0.9\,[\,0.4\,v(C)+0.6\,(0)\,]=-1+0.36\,v(C)is2/math/7b9d5ae6e490a014.svg
• v(s)=R(s)+\gamma\sum_{s'}P_{ss'}v(s')is2/math/7c2858b3388b1fca.svg
• f(3)=3^2is2/math/7cb566d00780f2b1.svg
• (3,1)=\max[\min(.2,.2),\min(.4,.2),\min(.3,.1)]=0.2,\;(3,2)=\max[.2,.3,.1]=0.3,\;(3,3)=\max[.2,.3,.1]=0.3is2/math/7cbca4576287a20a.svg
• d_g=1-(4)=-3is2/math/7cc359443926f7d3.svg
• \tau_{ij}\leftarrow(1-\rho)\,\tau_{ij}+\sum_{k=1}^{m}\Delta\tau^{k}_{ij},\qquad \Delta\tau^{k}_{ij}=\begin{cases}Q/L_k&\text{if ant }k\text{ used edge }(i,j)\\0&\text{otherwise}\end{cases}is2/math/7cdb88481f227e3c.svg
• \alpha=1/0.2=5is2/math/7ce6fe6ef36e3167.svg
• x_{new}=4+(-1)=3is2/math/7ce7c2323916220b.svg
• \propto\tau_{ij}\,(1/d_{ij})is2/math/7d200da3be6a78d4.svg
• v_{ij}^{t+1}=wv_{ij}^t+c_1r_1(pbest_{ij}-x_{ij}^t)+c_2r_2(gbest_j-x_{ij}^t)is2/math/7d9ccf789fe769f1.svg
• v_*(s)=\max_a\sum_{s'}P(s'\mid s,a)[R+\gamma v_*(s')]is2/math/7e6827fd1d3ef854.svg
• G_0=R_1+\gamma R_2+\gamma^2R_3is2/math/7e7a8fe5c95dbdc8.svg
• \piis2/math/7e7e8c545536f6a2.svg
• 0\le P(\omega)\le1is2/math/7e932392a2d209b9.svg
• P(No)=4/8=0.5is2/math/7ee037eb525f597a.svg
• R_{ij}=\min(A_i,B_j),\quad S_{ij}=\min(C_i,B_j),\quad T_{ij}=\max_k\min(R_{ik},S_{kj})is2/math/7ef66022cb4ba451.svg
• f(3)=9is2/math/7f1b6387a96fa5e5.svg
• P_{ij}=P(\text{next}=j\mid\text{now}=i)is2/math/7f4c1d9519ac27a9.svg
• P(regular\mid male,debit)=\frac{0.20}{0.05+0.20}=0.8is2/math/7f500d88c2aab86f.svg
• \sum F=0+3+1=4is2/math/7f52b89af830db1b.svg
• l=0.25p+0.5l+0.3ris2/math/809047184e208e3d.svg
• v_\pi(s_1)\approx5.43is2/math/80d83715ae3f22a7.svg
• V^\pi(s)=\sum_a\pi(a\mid s)\sum_{s',r}p(s',r\mid s,a)\left[r+\gamma V^\pi(s')\right]is2/math/80ef2d8261fa9e17.svg
• x^2=17\times17=289\qquad x^3=289\times17=4913is2/math/8106fc4d3ea404bb.svg
• 0.3+0.5=0.8is2/math/8108e5ca96fd589b.svg
• v(s)=R(s)+\gamma\sum_{s'}P(s'\mid s)v(s')is2/math/811e452127929f3e.svg
• \sigma^2=\frac{0+0+0}{3}=0is2/math/8148ccf99111a269.svg
• z^*=\frac{56}{1}=56is2/math/8184446a5afe0a48.svg
• \overline f_{new}=\frac{1+1+4+4}{4}=\frac{10}{4}=2.5is2/math/818ac9c6d2d2746f.svg
• f(4)=128-80+28=76is2/math/818af979753c0a98.svg
• \mathbb E[R]=\sum_i P_iR_iis2/math/81d37eff7c373c7f.svg
• (4)^2=16\qquad2\sigma^2=2(2^2)=8is2/math/8237f522c90ae910.svg
• v=R+\gamma Pv\Rightarrow (I-\gamma P)v=R\Rightarrow v=(I-\gamma P)^{-1}Ris2/math/8248280819c17d38.svg
• \neg X\cap\neg Y:\min(0.1,0.7)=0.1is2/math/82b3199ed4befd72.svg
• v_{12}=0.9(0.2449)+0+0.12825(-2.2150+4.0246)=0.22041+0.232081=0.452491is2/math/82dc2ed77545eeb8.svg
• q_\pi(s,a)=\mathbb E_\pi[G_t\mid S_t=s,A_t=a]is2/math/830f0f5268ecd933.svg
• d_g=7.21-(7.36)=-0.15is2/math/8346d41c793ff815.svg
• L_kis2/math/836949dff185b3ba.svg
• \overline f_{new}=\frac{196+196+196+196}{4}=\frac{784}{4}=196is2/math/83698ff516912ccd.svg
• G_t=R_{t+1}+\gamma G_{t+1}is2/math/837aecc6afff8d86.svg
• P(R)=\frac3{14}+\frac29=\frac{27+28}{126}=\frac{55}{126}is2/math/8383f96e6caef877.svg
• s(c)=P(c)\prod_j P(x_j\mid c),\qquad P(c\mid x)=\frac{s(c)}{\sum_{k}s(k)}is2/math/839a41948edad8ea.svg
• x^{(1)}=5+(-1.5)=3.5is2/math/83a647473c6fa205.svg
• 14-10=4is2/math/83b853a2ad6339ff.svg
• \mu_{out}(60)=\max(0.3,0.2)=0.3is2/math/84156b76989dd58e.svg
• T_{12}=\max\{\min(0.3,0.1),\min(0.6,0.7),\min(0.9,0.4)\}is2/math/843b8913b1abe9b7.svg
• I=0.5\times(-0.5)=-0.25is2/math/84dfd42953d2811a.svg
• F_{TSP}=1/Lis2/math/85040bacae1471d9.svg
• \mu_A(9)=0is2/math/857573d5b79c2b73.svg
• \max\{\min(0.2,0.3),\min(0.4,0.3),\min(0.3,0.1)\}is2/math/859d029599adbb59.svg
• x=(2.5,\,-3.0)is2/math/85ac77f48027edb2.svg
• 0.5(1-(-2))=1.5is2/math/85cc697d8abbcf2c.svg
• N_k=0.8+0.2=1is2/math/860f0958878c117d.svg
• V^*(s)=\max_a\sum_{s',r}p(s',r\mid s,a)\left[r+\gamma V^*(s')\right]is2/math/863310fc89ae0de3.svg
• v_{B1}^{1}=v_{B2}^{1}=0.9(0)+0.89235(1-1)+0.12825(1-1)=0is2/math/86465ec4ca89266b.svg
• a=\frac{25}{42}\div\frac32=\frac{25}{63}is2/math/8653487298ec14a0.svg
• \overline f_{new}=\frac{64+169+169+64}{4}=\frac{466}{4}=116.5is2/math/866b6bb4aa32a9a4.svg
• \gamma_1=0.12/0.15=0.8\quad\gamma_2=0.03/0.15=0.2is2/math/867693405ef3ab51.svg
• e^{0}\approx1is2/math/86787607e49b39df.svg
• \sum F=0.1+0.2+0.5+0.2\approx1is2/math/8709c94ae272ba4a.svg
• f(1)=1^2=1=1is2/math/872b6624767f4385.svg
• v^{(2)}=(-0.25)+0+(-0.25)=-0.5is2/math/8789b37bca837a83.svg
• x_{new}=4+(-3)=1is2/math/87db5516b0eef4a5.svg
• P(s'\mid s,a)is2/math/8818d06a66002156.svg
• v=0.5(1+0.9v)+0.5(4+0.18v)is2/math/886e804c14fdc761.svg
• Qis2/math/88c438d7bc523a5f.svg
• v^{(2)}=(-1)+0+(-0.5)=-1.5is2/math/88e7bce915d4dcc4.svg
• =\max\{0.4,0.4,0.1\}=0.4is2/math/891d26c6d9632710.svg
• C=0.6\times(0)=0is2/math/89df484c365b908d.svg
• \mu_{out}(z)=\max(\min(0.3,\mu_A(z)),\min(0.5,\mu_B(z)))is2/math/89f0491f916a2f27.svg
• \gammais2/math/8a226d1a655f7384.svg
• v^{t+1}=wv^t+c_1r_1(pbest^t-x^t)+c_2r_2(gbest^t-x^t)is2/math/8a9f3b6c402bae75.svg
• (E,I,M,S,A)=(1,0,1,0,1)is2/math/8adff69e2c0b00ef.svg
• b=0.3a+0.4b+0.25cis2/math/8b2ff5dead1627ab.svg
• \mu_M(32)=\frac{35-32}{35-25}=\frac3{10}=0.3is2/math/8b417cc7c6fd3b3f.svg
• P(S_{t+1}\mid S_t, S_{t-1},\dots,S_1)=P(S_{t+1}\mid S_t)is2/math/8b44c46b24e14f80.svg
• x_{t+1}=x_tPis2/math/8b6e8d7e9dedda16.svg
• x_{t+1}=x_tP,\qquad x_{t+2}=(x_tP)Pis2/math/8b9618ddde204a22.svg
• v^{(2)}=0+0+0=0is2/math/8ba6a1d6c9203afb.svg
• 1-\max(0.3,0.8)=1-0.8=0.2is2/math/8bcb03e96487eae0.svg
• x^2=15\times15=225\qquad x^3=225\times15=3375is2/math/8c2aaf117e33ac77.svg
• z^*=\frac{1924.166667}{31.500000}\approx 61.084656is2/math/8c383994ebbe02fc.svg
• p(x)=\sum_{k=1}^K\pi_k\mathcal N(x\mid\mu_k,\sigma_k^2)is2/math/8c4c98f40f54268b.svg
• f(7)=686-245+49=490is2/math/8c54966ea16cb628.svg
• q_1=0.5\frac{e^{-(1-2)^2/2}}{\sqrt{2\pi}}\approx0.120985362is2/math/8c570fbfdd0a7367.svg
• \pi(\text{stay})=\pi(\text{go})=0.5is2/math/8c95286b27a764a3.svg
• P(\neg A)=1-0.51=0.49is2/math/8cc383cc77727275.svg
• =\max\{0.1,0.6,0.4\}=0.6is2/math/8ce46b5da48e6fa8.svg
• \text{Remaining}=2-2=0is2/math/8d90e4663ec0a413.svg
• \mu(x)=\begin{cases}0,&x\le a\text{ or }x\ge d\\(x-a)/(b-a),&a<x<b\\1,&b\le x\le c\\(d-x)/(d-c),&c<x<d\end{cases}is2/math/8db3f8710aa967fd.svg
• I=0.5\times(-2)=-1is2/math/8dba29f0d505433a.svg
• P_{ij}=\dfrac{\tau_{ij}^\alpha\eta_{ij}^\beta}{\sum_{l\in\text{allowed}}\tau_{il}^\alpha\eta_{il}^\beta},\;\eta_{ij}=1/d_{ij}is2/math/8df8f5c198d8cd6b.svg
• b+i+a=1\quad b+2b+7b=1is2/math/8e00c1c8e9302c7a.svg
• \text{Number}=\text{sum of (digit}\times\text{place value)}is2/math/8e2952f36a427b0a.svg
• L=p(x_1)p(x_2)\qquad\ln L=\ln p(x_1)+\ln p(x_2)is2/math/8e3c44857ba18435.svg
• P(\text{event})=\frac{\text{favourable count}}{\text{total count}}is2/math/8e42f83b578b6254.svg
• (1,2)=\max[\min(.4,.6),\min(.5,.4),\min(.3,.1)]=\max(.4,.4,.1)=0.4is2/math/8e54167083aa61b1.svg
• x^{(1)}=1+2=3is2/math/8e886f0f9174fead.svg
• \mu_{A\cup B}(x)=\max(\mu_A(x),\mu_B(x)),\quad \mu_{A\cap B}(x)=\min(\mu_A(x),\mu_B(x)),\quad \mu_{\bar A}(x)=1-\mu_A(x)is2/math/8ef43c082b1d8a4a.svg
• P_{AB}=1-0.8=0.2,\quad P_{BA}=1-0.9=0.1is2/math/8f12974205c3ad53.svg
• P_{ij}=P(S_{t+1}=j\mid S_t=i)is2/math/8f271383e8f91295.svg
• 0.5\times(4)=2is2/math/8f4e70094d023847.svg
• p(x)is2/math/8f52f259a1ffabd6.svg
• 8-7=1is2/math/8f69aa364500c9a0.svg
• c_{next}=0.20a+0.30b+0.80cis2/math/8fa41e7634228ffe.svg
• \min(0.4,0.8)=0.4is2/math/8fd31f9127b73dc1.svg
• \alpha=\beta=1is2/math/8fe45a81f0de6694.svg
• \min(1-0.3,1-0.8)=\min(0.7,0.2)=0.2is2/math/902b79a384f1ef2a.svg
• \mu_L(32)=0is2/math/90541f01c8f3e577.svg
• x^{(2)}=1.5+(-1)=0.5is2/math/905c77866baf10ae.svg
• C=0.6\times(-0.1288)=-0.07728is2/math/9088271ed20d40af.svg
• 3+7=10is2/math/90e46c6fe6c61dca.svg
• d_g=1-(-2)=3is2/math/9102a4aae4f9ab55.svg
• P(\neg A\cap B)=0.49\times0.08=0.0392is2/math/916824e2b1c49a5f.svg
• \bar z=\frac{\sum_i w_i z_i}{\sum_i w_i}is2/math/91cd48ba0a9e464c.svg
• p(x)=\pi_1N_1+\pi_2N_2is2/math/91ef77408ecb18fc.svg
• sis2/math/92329599456283d6.svg
• A_1=4\times0.5=2is2/math/925738e6d7bc9738.svg
• (1001)_2is2/math/92700f027de744f9.svg
• x^{(1)}=(-1)+0=-1is2/math/92761bdfeb82fb87.svg
• P(B)=7/10=0.7=70\%is2/math/927c15e53970ce9d.svg
• (1011)_2=(11)_{10}is2/math/9299eb74dd4bc751.svg
• G_0=R_1+\gamma R_2+\gamma^2R_3+\gamma^3R_4is2/math/92c2d569c2ff0264.svg
• A_2=\tfrac12\times2\times0.5=0.5is2/math/931f8a69a034cfb8.svg
• d_g=3.14-(6.17)=-3.03is2/math/93416da8e366b396.svg
• P(W)=P(W,R)+P(W,\neg R)=0.4581+0.1919=0.6500is2/math/9357fadf4faa85ed.svg
• \sum_a\pi(a\mid s)\to\max_ais2/math/935b09896577bd3b.svg
• v^{(2)}=0.019+(-0.0228)+(-0.01656)=-0.02036is2/math/9363048bb26f273e.svg
• p+l+r=1\quad\Rightarrow\quad1.52r+1.36r+r=1is2/math/936703629ebeff23.svg
• e^u=e^{-2}\approx 0.135335283is2/math/93bf357c53e138fe.svg
• P(cavity\lor toothache)=0.108+0.012+0.072+0.008+0.016+0.064=0.28is2/math/93d535e63ac5c87c.svg
• \alpha\,P(bp\mid hd)P(hd)=\alpha(0.85)(0.49)=\alpha\,0.4165,\qquad\alpha\,P(bp\mid\lnot hd)P(\lnot hd)=\alpha(0.2)(0.51)=\alpha\,0.102is2/math/9429a00abcd3cccf.svg
• 1.5w=1-1.5c=1-3/7=4/7is2/math/9499078f217c4146.svg
• \mu_k=\frac{\sum\lambda_k^ix_i}{\sum\lambda_k^i},\;\sigma_k^2=\frac{\sum\lambda_k^i(x_i-\mu_k)^2}{\sum\lambda_k^i},\;\pi_k=\frac{\sum_i\lambda_k^i}{n}is2/math/94b0dd17291e3c1a.svg
• \max\{\min(0.2,0.4),\min(0.4,0.3),\min(0.3,0.1)\}is2/math/952b0167063f5e45.svg
• f(3, 2)=3^2+2^2=9+4=13is2/math/952bb71a534d7b48.svg
• (10011)_2=(19)_{10}is2/math/9557b36b09cf5595.svg
• \mathbb E[R_A]=6-0.8=5.2is2/math/95811b7af645e1fd.svg
• 2x^3=4394\quad 3x^2=507\quad 8x=104is2/math/963b6763e9667406.svg
• 2x^3=6750\quad 5x^2=1125\quad 7x=105is2/math/9652ff9e90959bb5.svg
• v_{new}=0+0+(-1)=-1is2/math/96abcf8e044d8408.svg
• 0.08+0.10+0.072+0.05+0.10+0.15+0.25+0.20=1.002is2/math/96cf20835aaf1e81.svg
• S=0.5\times(-2)=-1is2/math/96dd8f64f9d9a341.svg
• 0+4+2+0=6is2/math/972060f72bffad8b.svg
• z^*=\frac{A_1c_1+A_2c_2}{A_1+A_2}is2/math/97228eac9007fb53.svg
• \tau_{ij}^{new}=(1-\rho)\tau_{ij}^{old}+\sum_{ants\ using\ ij}Q/Lis2/math/97bee3443d137b73.svg
• G_t=R_{t+1}+\gamma R_{t+2}+\gamma^2R_{t+3}+\cdots=R_{t+1}+\gamma G_{t+1}is2/math/97caee2822bc6475.svg
• P(\lnot R)=P(C)P(\lnot R\mid C)+P(\lnot C)P(\lnot R\mid\lnot C)=0.5(0.2)+0.5(0.8)=0.5is2/math/97de213766352c7b.svg
• 0.5is2/math/97dede1883211cd7.svg
• 12-10=2is2/math/97fe26072a5dade9.svg
• \max\{\min(0.4,0.3),\min(0.4,0.3),\min(0.3,0.1)\}is2/math/98186c591a86ee1c.svg
• q_\pi(s,a)=\mathbb E_\pi[G_t\mid s,a]is2/math/99133acdbfaa7afa.svg
• d_p=8.1428-(8.1428)=0is2/math/9923de46bc28d517.svg
• \neg X:1-0.8=0.2\qquad\neg Y:1-0.5=0.5is2/math/994b18481ef2566a.svg
• v_i^{new}=wv_i+c_1r_1(pbest_i-x_i)+c_2r_2(gbest-x_i),\quad x_i^{new}=x_i+v_i^{new}is2/math/99893bd63a6f8a01.svg
• \max\{\min(0.4,0.6),\min(0.5,0.4),\min(0.3,0.1)\}is2/math/999b4f88f5256b56.svg
• P^{a}_{ss'}=P(S_{t+1}=s'\mid S_t=s,A_t=a)is2/math/9a04b612a1d5ca7f.svg
• \log L(\theta)=\sum_{i=1}^{n}\log\Big(\sum_{k=1}^{K}\pi_k\frac{1}{\sqrt{2\pi}\,\sigma_k}e^{-\frac{(x_i-\mu_k)^2}{2\sigma_k^2}}\Big)is2/math/9a0bb26e3ef13adb.svg
• v^{(1)}=1+0+0=1is2/math/9a2ea8db49561d57.svg
• v_*(s)=\max_{a}\sum_{s'}P(s'\mid s,a)\,\big[\,R(s,a,s')+\gamma\,v_*(s')\,\big]is2/math/9a59f92e2e358c6a.svg
• 35-25=10is2/math/9a5b4bf9b146b2b2.svg
• G_0=-2+0.9(-1)+0.81(-2)+0.729(0)=-2-0.9-1.62=-4.52is2/math/9a77a967b32f363f.svg
• \neg X:1-0.5=0.5\qquad\neg Y:1-0.6=0.4is2/math/9b11f9a14b9e240f.svg
• c_2=(4+4+6)/3=14/3is2/math/9b85cc1cdfb06f4d.svg
• P(toothache)=0.108+0.012+0.016+0.064=0.2,\qquad P(cavity\mid toothache)=\frac{0.12}{0.2}=0.6,\qquad P(\lnot cavity\mid toothache)=\frac{0.08}{0.2}=0.4is2/math/9b888308d7a5d7e4.svg
• \overline f_{old}=\frac{15+57+57+37}{4}=\frac{166}{4}=41.5is2/math/9b9333d937330aaa.svg
• =\max\{0.2,0\}=0.2is2/math/9ba8671f889808c7.svg
• z^*=\frac{50}{1}=50is2/math/9c1293c80ea00bab.svg
• 3\times5=15is2/math/9c14aa0e11753341.svg
• x^2=4\times4=16\qquad x^3=16\times4=64is2/math/9c1546066fc40eda.svg
• L=\sum d(c_i,c_{i+1})+d(c_n,c_1)is2/math/9c269c2e6ae2c088.svg
• v_{A1}^{1}=0.9(0)+0.89235(0-0)+0.12825(1-0)=0.12825is2/math/9c531ea472a5b2fe.svg
• 80-z_R=4.5is2/math/9c5f88f119fb8224.svg
• \tau^{new}\approx\begin{bmatrix}0&.558823529&.541666667&.600490196\\.558823529&0&.600490196&.541666667\\.541666667&.600490196&0&.558823529\\.600490196&.541666667&.558823529&0\end{bmatrix}is2/math/9cb40cf52889b49c.svg
• 80\times0.2=16is2/math/9cd289b939140103.svg
• P(A\text{ and }B)=P(A)\times P(B)\qquad P(\text{two heads})=0.5\times0.5=0.25is2/math/9d043d505b5382fc.svg
• \eta:\;C=0.25,\;D=0.2;\;\text{sum}=0.45\Rightarrow P_{BC}=0.5556,\;P_{BD}=0.4444is2/math/9d1964928f02cbff.svg
• (2-0)^2=4is2/math/9da874308fbfbb30.svg
• R=\begin{bmatrix}\min(.4,.2)&\min(.4,.7)&\min(.4,.3)\\\min(.3,.2)&\min(.3,.7)&\min(.3,.3)\\\min(.1,.2)&\min(.1,.7)&\min(.1,.3)\end{bmatrix}=\begin{bmatrix}0.2&0.4&0.3\\0.2&0.3&0.3\\0.1&0.1&0.1\end{bmatrix}is2/math/9db67d15872888f6.svg
• 2^n-1is2/math/9dc91f3c375d809f.svg
• 0.0625\times0.75=0.046875is2/math/9df28e0200ccc17b.svg
• \mu_{Low}(65)=0is2/math/9e3859fafc31a24f.svg
• z^*=\frac{\sum_{z_i\in M}z_i}{|M|}\quad\text{one plateau }[a,b]:\ z^*=(a+b)/2is2/math/9ed17a341041965a.svg
• \max\{\min(0.1,0.4),\min(0.1,0.3),\min(0.1,0.1)\}is2/math/9eda900daa88d0f0.svg
• v_\pi(s)=\sum_{a\in\mathcal A}\pi(a\mid s)\,q_\pi(s,a)is2/math/9ede9bf563f2c6c9.svg
• P^{\,n}is2/math/9ef45de9aca6f359.svg
• S=0.5\times(3)=1.5is2/math/9efd684a8627ac87.svg
• 0.6+0.4+0.2=1.2is2/math/9f1b10f7784e3b8c.svg
• score(Yes)=0\qquad score(No)=0.03is2/math/9f7a6514218ef651.svg
• r+\gamma\max_{a'}Q(s',a')=3+0.9(5)=7.5is2/math/9f91a54deb2a51d6.svg
• \text{Remaining}=1-1=0is2/math/9f9f05b8fd1b9885.svg
• e^{-0.5}\approx0.60653066is2/math/9fa1a1619e482083.svg
• X\cup Y:\max(0.6,0.4)=0.6is2/math/9fba815e535802c7.svg
• gbest^0=(2)is2/math/a00ef2dd53de0008.svg
• f(x)=x_1^2+x_2^2is2/math/a02144dad2ff9c8a.svg
• (19)_{10}=(10011)_2is2/math/a03240c4b6abd4c3.svg
• P(R\mid W)=\frac{0.4581}{0.6500}\approx0.704769231is2/math/a03cc788f3ded23d.svg
• \max\{\min(0.1,0.3),\min(0.1,0.3),\min(0.1,0.1)\}is2/math/a0554ac89675d583.svg
• C=0.5\times(-2)=-1is2/math/a08a6d09414479ce.svg
• \mu_{B^c}=1-\mu_B\qquad\mu_{A\setminus B}=\min(\mu_A,1-\mu_B)is2/math/a10c281d39aeb04c.svg
• \max(0,9-10)=0is2/math/a12fb07643a5aaa1.svg
• A=(b-a)(u+v)/2is2/math/a1493aef29965a09.svg
• P(C\text{ and }F)=P(C\cap F)=\frac{10}{100}=0.1is2/math/a166ec0150b9d8e6.svg
• \sum F=4+4+2=10is2/math/a1ef5b129218331e.svg
• \mu_{R2}(60)=\min(0.6,0.2)=0.2is2/math/a21d3bc0188016ea.svg
• \begin{bmatrix}3\\7\end{bmatrix}is2/math/a263cada62354d7d.svg
• \mu(x)=\frac{\text{distance still remaining to right zero}}{\text{total falling width}}is2/math/a265ac2e0c52f5aa.svg
• P(V\mid\lnot F)=\frac{0.2734875}{0.725}=0.3772is2/math/a27699b35a63a308.svg
• v_\pi(s)=\mathbb E_\pi[G_t\mid S_t=s]is2/math/a27ddca95f461e82.svg
• \mu(14)=e^{-2}\approx0.135335283is2/math/a282bbe0d0c6f2b0.svg
• P=\begin{bmatrix}0.60&0.25&0.15\\0.30&0.50&0.20\\0.20&0.30&0.50\end{bmatrix},\qquad x_0=[50,\;30,\;20]is2/math/a2936167c6c14fec.svg
• \pi\approx[0.392,\;0.351,\;0.258]is2/math/a2ee97608735415b.svg
• X\cup Y:\max(0.7,0.6)=0.7is2/math/a2eea0fe6bbb6b87.svg
• \bar z=\frac{62}{0.8}=77.5is2/math/a313fb00fc0904f5.svg
• z^*=\frac{\int z\mu(z)\,dz}{\int\mu(z)\,dz}is2/math/a3227ed1a9289bb5.svg
• =17.050821182+53.69430797-30.257760057=40.487369094is2/math/a32da9cd0f25ac36.svg
• P=\begin{bmatrix}0.8&0.2\\0.1&0.9\end{bmatrix}is2/math/a352d238bb02bc14.svg
• d_p=1-(1)=0is2/math/a352ebe71945e68e.svg
• P(\lnot F)=0.75(0.8)+0.25(0.5)=0.6+0.125=0.725is2/math/a396371c860e59b0.svg
• f(7)=7^2is2/math/a39be859ebcf3420.svg
• v_{12}=0.9(0.3)+0.89235(0)+0.12825(-1.5-(-3.0))=0.27+0.192375=0.462375is2/math/a3af7b24f214c46d.svg
• P_i=F_i/\sum_jF_j\quad C_i=\sum_{j=1}^iP_jis2/math/a3d3e670e29c240d.svg
• P(C\mid F)=10/30=0.33is2/math/a402991a380b1d2f.svg
• -1/(2\times1)=-0.5is2/math/a4066df8f72b8058.svg
• \text{Remaining}=11-8=3is2/math/a475ecc5627f41d9.svg
• x^{(2)}=(-1)+0=-1is2/math/a4857620c6480702.svg
• v^{(1)}=0+0+(-0.5)=-0.5is2/math/a489ee8326d49a33.svg
• \pi_2^{new}=1.5/3=0.5is2/math/a4b477c2cff182c7.svg
• 2+5=7\quad7+8=15is2/math/a4c248316325eed6.svg
• S_1, S_2, \dotsis2/math/a4c47d7eb261ab94.svg
• 2x^3=128\quad 5x^2=80\quad 7x=28is2/math/a4dd7bf5365d3c7e.svg
• z^*_{MoM}=\frac{77.5+92.5}{2}=85is2/math/a4e234f2232ff44d.svg
• \frac{z_L-70}{15}=0.5\Rightarrow z_L=77.5is2/math/a4fdc221a6cfa166.svg
• P(R,A)=\frac37\frac12=\frac3{14},\quad P(R,B)=\frac49\frac12=\frac29is2/math/a518125ef5d2bed8.svg
• v^{(2)}=1+(-1)+(-1)=-1is2/math/a51eec1438d949c1.svg
• P^2=\begin{bmatrix}3/8&5/8\\5/16&11/16\end{bmatrix}=\begin{bmatrix}0.375&0.625\\0.3125&0.6875\end{bmatrix}is2/math/a532335af3e981b5.svg
• S=0.12\times(-3.03)=-0.3636is2/math/a59d670bba6f5dac.svg
• (0)_{10}=(0000)_2is2/math/a5e664b79f15e056.svg
• v_\pi(s)=\sum_a\pi(a\mid s)q_\pi(s,a),\qquad v_*(s)=\max_a q_*(s,a)is2/math/a5e70ba1f7aaa949.svg
• f(6)=432-180+42=294is2/math/a5f219480ddd154a.svg
• P=\begin{bmatrix}0.60&0.25&0.15\\0.30&0.50&0.20\\0.20&0.30&0.50\end{bmatrix}is2/math/a6093f9acfe22fe1.svg
• =17.889685344+8.520484898+4.05812967+12.68886+(-8.05792)+5\approx40.099239912is2/math/a64c8baf9521e960.svg
• 0.5(1)+2\times\frac1{14}=0.6429is2/math/a677db1cbc02a07c.svg
• \overline f_{old}=\frac{25+9+1+25}{4}=\frac{60}{4}=15is2/math/a688edfd604798b3.svg
• 0.5(4-(4))=0is2/math/a6908257a641f5f9.svg
• R=\begin{bmatrix}0.3&0.3&0.3\\0.5&0.6&0.7\\0.5&0.6&0.8\end{bmatrix}is2/math/a6d0138a2e4b14ec.svg
• P(B)=0.08/0.20=0.4is2/math/a6f1afe6f9d94092.svg
• P(\text{Cough}=\text{Yes}\mid C=\text{Yes})=\frac{3}{3}=1is2/math/a6ffcdb19b7e5a36.svg
• U-L=5-(-2)=7\qquad2^3-1=7is2/math/a70472c6ec58ad8d.svg
• (\sigma_2^2)^{new}=\frac{1.521735533}{1.5}\approx1.014490355is2/math/a7522d8383a399aa.svg
• x^{(2)}=1+0=1is2/math/a7841f9dac4f92c7.svg
• v_{new}=0.5v+0.5(pbest-x)+0.5(gbest-x)is2/math/a86502ce620dafcc.svg
• d_p=4.15-(4.2788)=-0.1288is2/math/a86b46d750156354.svg
• P(X=1\mid B)=\frac{0.10}{0.25+0.10}=\frac{0.10}{0.35}=\frac27\approx0.285714is2/math/a88bec71b29e2cf3.svg
• x^{(1)}=3+(-1)=2is2/math/a890dd5de2855674.svg
• \alpha=\max(0.2,0.8)is2/math/a8a03390bc687846.svg
• p=0.6p+0.3l+0.2ris2/math/a8c5e214edbb94e4.svg
• =29.622243087+63.477295646-43.362885999=49.736652734is2/math/a9051d35de42b6a1.svg
• x=(3.5,\,-1.5)is2/math/a92c6e5b7c6f437f.svg
• \mathcal N(x\mid\mu,\sigma^2)=\frac1{\sqrt{2\pi}\,\sigma}\exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)is2/math/a9b2cdf84258b1c9.svg
• \begin{bmatrix}3&7\\4&2\end{bmatrix}is2/math/a9c176e9eb771437.svg
• x_{11}=2.5+0.30825=2.80825is2/math/aa0bacfeb906b753.svg
• -4/(2\times1)=-2is2/math/aa2983139384fc61.svg
• f(x)=2x^3-5x^2+7xis2/math/aacd674038ffd6be.svg
• R=\begin{bmatrix}0.6&0.3\\0.2&0.5\end{bmatrix},\quad S=\begin{bmatrix}1&0.4&0.5\\0.8&0.3&0.8\end{bmatrix}is2/math/ab04f42254fa9175.svg
• \neg X:1-0.6=0.4\qquad\neg Y:1-0.4=0.6is2/math/ab521a108b232136.svg
• P(X{=}0)=0.072+0.016+0.012+0.108=0.208,\qquad P(Y{=}1\mid X{=}0)=\frac{0.072+0.016}{0.208}=\frac{0.088}{0.208}=0.423is2/math/ab59aa2401b72e20.svg
• x^{(2)}=5.9064+(-0.463768)=5.442632is2/math/ab68ef2fa1b527a9.svg
• \neg X:1-0.3=0.7\qquad\neg Y:1-0.9=0.1is2/math/abbf5f7c84a2a614.svg
• q_1=0.5\frac{e^{-(3-2)^2/2}}{\sqrt{2\pi}}\approx0.120985362is2/math/abfe7f04db4f042d.svg
• (0.5-0.1875)l=(0.125+0.3)ris2/math/ac05d4785e692e11.svg
• P(R)=3/10=0.3=30\%is2/math/ac0dd61435e485bc.svg
• f(6.17, 8.27)=6.17^2+8.27^2-(6.17)(8.27)is2/math/ac0fc4a2897f8794.svg
• \overline f_{old}=\frac{9+25+100+144}{4}=\frac{278}{4}=69.5is2/math/ac14e9c27d115d0e.svg
• (-3)^2=(-3)\times(-3)=9is2/math/ac40c2845414cf0d.svg
• P=P(j\mid a)P(m\mid a)P(a\mid\lnot b,\lnot e)P(\lnot b)P(\lnot e)=0.90\times0.70\times0.001\times0.999\times0.998=0.000628is2/math/ac4abea66468645f.svg
• \mu_{A^c}=1-\mu_A\quad\mu_{A\setminus B}=\min(\mu_A,1-\mu_B)is2/math/ac545c5022ffd6a1.svg
• \neg X\cup\neg Y:\max(0.4,0.6)=0.6is2/math/ac5d994d1c80e0dd.svg
• \ell(\theta)=\log\mathcal L(\theta)=\sum_{i=1}^{N}\log\left[\sum_{k=1}^{K}\pi_k\mathcal N(x_i\mid\mu_k,\Sigma_k)\right]is2/math/ac7ba81fe410766e.svg
• a=0.3w+0.4a+0.3cis2/math/ac7bb8ac13980636.svg
• P_{ij}=P(\text{next state}=j\mid\text{current state}=i),\qquad\sum_jP_{ij}=1is2/math/ac80fe221bb10516.svg
• L_1=d_{AB}+d_{BC}+d_{CD}+d_{DA}=10+1+1+3=15is2/math/ac9a78a66f280841.svg
• S=\begin{bmatrix}.4&.5&.3\\.2&.2&.2\\.4&.4&.3\end{bmatrix}is2/math/acc98fbd37745e57.svg
• P(X_{t+1}\mid X_t,X_{t-1},\ldots)=P(X_{t+1}\mid X_t)is2/math/acdbeb13aace7aac.svg
• d_p=4-(4)=0is2/math/ad073051db57b851.svg
• \mu_{B^c}(u_1)=1-0.7=0.3is2/math/ad1dcd17cbaf20b9.svg
• P(Co\mid\neg C,S)=0.0756/0.27=0.28is2/math/ad2987ff1355d124.svg
• P=\begin{bmatrix}0.6&0.3&0.1\\0.2&0.5&0.3\\0&0.1&0.9\end{bmatrix}is2/math/ad531c204822e0b1.svg
• q_2=0.5\frac{e^{-(5-4)^2/2}}{\sqrt{2\pi}}\approx0.120985362is2/math/ad6c931c7f9ef8f8.svg
• P=\begin{array}{c|cc} & S & C\\\hline S & 0.80 & 0.20\\ C & 0.10 & 0.90\end{array}is2/math/ad78149edee2ba2b.svg
• d_g=3.14-(4.2788)=-1.1388is2/math/ad7db7d0feab17e6.svg
• 1/(1-0.9)=10is2/math/ad897de8c397fc5d.svg
• \gamma_1=\frac{0.120985362}{0.120985362+0.002215924}\approx0.98201379is2/math/aded404e73fc7586.svg
• S=\begin{bmatrix}0.5&0.9\\0.7&0.1\end{bmatrix}is2/math/ae4133e873ae6a3f.svg
• P(A\text{ and }B)=P(A\mid B)\,P(B)is2/math/aec0a3a12cca75ad.svg
• v(s)=2+3=5is2/math/aed5ac659044d3f1.svg
• L(ADBEC)=10+4+3+5+9=31,\quad F=1/31=0.0323is2/math/aede143ef375acd9.svg
• a_{next}=0.30a+0.10b+0.10cis2/math/aee58f980f95f13c.svg
• (P^2)_{ij}=\sum_kP_{ik}P_{kj},\qquad p^{(2)}=p^{(0)}P^2is2/math/aef8325cefd5071a.svg
• P(A\mid B)=\frac{P(A\cap B)}{P(B)},\qquad P(B)>0is2/math/af10b31123342cc5.svg
• p=0.75(1.36r)+0.5r=1.52ris2/math/af26ae261b198416.svg
• \mu_{Medium}(80)=0is2/math/af4f190a786e2c05.svg
• f(20)=16000-1200+160=14960is2/math/af62ea37a6cc0979.svg
• a=1-w-c=1-8/21-6/21=7/21=1/3is2/math/afad15a4af697ec7.svg
• Q/Lis2/math/afe42919eb73e68a.svg
• P(A\mid B)=\frac{0.00099}{0.02097}\approx0.0472103is2/math/b015ca384b1a70e6.svg
• F(3.461457275,-3.5721088)=(3.461457275)^2-(3.461457275)(-3.5721088)+(-3.5721088)^2+3(3.461457275)+4(-3.5721088)+5is2/math/b030a3c1723e90f0.svg
• \text{height}=\frac{\text{distance from left}}{\text{total rising width}}is2/math/b04bfa99f868280c.svg
• R(C)=-2,\;R(L)=-1,\;v(D)=0,\;\gamma=0.9is2/math/b06d57b864effc2d.svg
• v_\pi(s)=\mathbb E_\pi[G_t\mid s]is2/math/b08e042bdd23bc61.svg
• \neg X:1-0.7=0.3\qquad\neg Y:1-0.2=0.8is2/math/b115216f7121911a.svg
• x^2=20\times20=400\qquad x^3=400\times20=8000is2/math/b143d22c2e5129d2.svg
• \neg X\cup\neg Y:\max(0.7,0.1)=0.7is2/math/b158ec14569973da.svg
• x_{next}=x_{now}P\qquad\pi=\pi P,\quad\sum_i\pi_i=1is2/math/b179f364b70ac72c.svg
• \neg X\cup\neg Y:\max(0.1,0.7)=0.7is2/math/b188e2e037566b1e.svg
• \mu_{Low}(55)=0is2/math/b1adf4cfc2121094.svg
• 2x^3=4394\quad 5x^2=845\quad 7x=91is2/math/b223b59b645fb4b2.svg
• z_{COS}=\frac{A_1c_1+A_2c_2}{A_1+A_2}=\frac{M_1+M_2}{A_1+A_2}is2/math/b24773ebdb90677f.svg
• \mu_{agg}(z)=\begin{cases}39/49&1\le z\le11\\(50-z)/49&11<z\le43\\1/7&43<z\le650/7\\(100-z)/50&650/7<z\le100\end{cases}is2/math/b2489917bec1091e.svg
• z^*=\frac{\sum_i\alpha_i z_i}{\sum_i\alpha_i}is2/math/b26ca7a8925830d0.svg
• 0.9^2\times(2)=1.62is2/math/b276421b6d8cf042.svg
• P^{k}_{ij}=\frac{\tau_{ij}^{\,\alpha}\,\eta_{ij}^{\,\beta}}{\displaystyle\sum_{l\in\text{allowed}}\tau_{il}^{\,\alpha}\,\eta_{il}^{\,\beta}},\qquad \eta_{ij}=\frac1{d_{ij}}is2/math/b2a12d0ebbd2c2db.svg
• \text{largest unsigned value}=2^n-1is2/math/b2b8f1010a02780c.svg
• p(x_i\mid\theta)=\sum_{k=1}^K\pi_k\,\mathcal N(x_i\mid\mu_k,\Sigma_k),\quad \pi_k\ge0,\quad\sum_k\pi_k=1is2/math/b313ea1944e92313.svg
• q_1+q_2=0.12+0.03=0.15is2/math/b318c8654f7774af.svg
• S=0.12\times(-2.7664)=-0.331968is2/math/b32a5de2e3b2cefc.svg
• P(Co=T\mid C=F,S=T)=0.21+0.07=\boxed{0.28}is2/math/b36e6cc824d0d47d.svg
• a\cdot b=a_1b_1+a_2b_2is2/math/b383a742b04996eb.svg
• \sum\pi_i=1is2/math/b39286a8c89942c9.svg
• v_{k+1}(s)\leftarrow\max_a\sum_{s'}P(s'\mid s,a)\,[R+\gamma v_k(s')]is2/math/b41a10ce7769c4dd.svg
• v^{(2)}=(-1)+0+(-1)=-2is2/math/b437ff946e9d4080.svg
• P(A\mid B)=\frac{P(A\cap B)}{P(B)}is2/math/b482646983d755cb.svg
• \mu_{agg}(z)=\max\{\min(39/49,\mu_L(z)),\min(1/7,\mu_M(z))\}is2/math/b4a4496253a59f9d.svg
• \mu(35)=(40-35)/(40-30)=5/10=0.5is2/math/b4c061d09e040d15.svg
• \overline f_{new}=\frac{81+225+225+81}{4}=\frac{612}{4}=153is2/math/b4cd0bde2e1de484.svg
• \bar z=\frac{w_1z_1+w_2z_2+\cdots}{w_1+w_2+\cdots}is2/math/b4ce56a94f70502d.svg
• \begin{bmatrix}1-a&a\\b&1-b\end{bmatrix}\Rightarrow\pi=\left[\frac b{a+b},\frac a{a+b}\right]is2/math/b54df1e7f5634ea5.svg
• 0\times 2=0is2/math/b584619dcb7531a8.svg
• \tau\eta:\;A\to B=1\times\tfrac14=0.25,\quad A\to C=1\times\tfrac12=0.5,\quad A\to D=1\times\tfrac14=0.25;\quad\text{sum}=1is2/math/b5d4297d164fecda.svg
• f(1.5)=1.5^2=2.25=2.25is2/math/b61e97d213b18164.svg
• x_0=[1,\;0]is2/math/b6383902c21eeee0.svg
• L=0.2\times0.5=0.1is2/math/b651f1b57b4503e2.svg
• \eta=\tfrac19,\tfrac13\Rightarrow P_{DB}=0.25,\;P_{DC}=0.75is2/math/b66f5122ccaaac24.svg
• P(a\mid b)=P(a\land b)/P(b)is2/math/b69a34e61294d724.svg
• =\max\{0.6,0.1\}=0.6is2/math/b6bb46f767031fd8.svg
• P(Y)=\sum_zP(Y,z)is2/math/b6dbba8c2c4c1da9.svg
• P(\text{Salary}=\text{Low}\mid C=\text{Yes})=\frac{1}{4}=0.25is2/math/b6e871091f36d8f9.svg
• \overline f_{old}=\frac{9+4+1+4}{4}=\frac{18}{4}=4.5is2/math/b6ef25b43ff486a0.svg
• x^{(2)}=1.5+(-0.5)=1is2/math/b71d05464709fcf3.svg
• ris2/math/b758343af751f858.svg
• X\cup Y:\max(0.3,0.9)=0.9is2/math/b75d7a85a782c820.svg
• P(C\mid a_1,\dots,a_n)\;\propto\;P(C)\prod_{i=1}^{n}P(a_i\mid C),\qquad v_{NB}=\arg\max_{c}\,P(c)\prod_iP(a_i\mid c)is2/math/b770ed6380567e45.svg
• 0.5(1-(4))=-1.5is2/math/b774cba7be997b80.svg
• z_{COS}=\frac{13+9}{2.75+1.5}=\frac{22}{4.25}=\frac{88}{17}\approx5.176471is2/math/b790fc2742ec49ac.svg
• P(a\mid b)=\dfrac{P(a\land b)}{P(b)}is2/math/b7aba0396dc1aeb0.svg
• P(A\cap B)=0.51\times0.12=0.0612is2/math/b7d733145715c7d9.svg
• P=\begin{bmatrix}0.8&0.05&0.15\\0.75&0.2&0.05\\0.38&0.02&0.6\end{bmatrix}is2/math/b7e30adc50b915ab.svg
• C=0.6\times(-0.15)=-0.09is2/math/b80cfa7b094e421e.svg
• f(5)=250-125+35=160is2/math/b80fc8e2e02f0c86.svg
• (1,2)=\max[\min(.2,.4),\min(.3,.3),\min(.3,.1)]=\max(.2,.3,.1)=0.3is2/math/b81568f26ae092d4.svg
• P(No)=5/8=0.625is2/math/b83bf0fcbd8f3bee.svg
• tis2/math/b84a36f811c3dd68.svg
• P(X_3=F\mid X_2=N)=P_{NF}=0.65is2/math/b86a579af37c69bb.svg
• F(t,t)=t^2+7t+5\longrightarrow+\infty\quad(t\to+\infty)is2/math/b88e59abd3293f46.svg
• \begin{aligned}v(s)&=\mathbb E[G_t\mid S_t=s]\\&=\mathbb E[R_{t+1}+\gamma(R_{t+2}+\gamma R_{t+3}+\dots)\mid S_t=s]\\&=\mathbb E[R_{t+1}+\gamma G_{t+1}\mid S_t=s]\\&=R(s)+\gamma\sum_{s'\in\mathcal S}P_{ss'}\,v(s')\end{aliis2/math/b890be2892d436ac.svg
• \mu_{A\setminus B}(u_1)=\min(0.9,0.4)=0.4is2/math/b891d5b3e6b00766.svg
• \text{retained sum}=64+95+89+94+79+88=509is2/math/b892f89037dda209.svg
• \tau_{ij}^{new}=(1-\rho)\tau_{ij}+\sum_{k:\,ij\in\mathrm{tour}_k}\frac{Q}{L_k}is2/math/b8a3f5425a6d79b2.svg
• 35-32=3is2/math/b8b4baa992603f77.svg
• \max\{\min(0.1,0.2),\min(0.1,0.2),\min(0.1,0.1)\}is2/math/b8bd41528e9d1a36.svg
• 0.6(0.2)=0.12is2/math/b90648f46ba2060f.svg
• P_{AC}=0.25/0.75=1/3is2/math/b90f87b145809204.svg
• \tau^{new}\approx\begin{bmatrix}0&7.566667&2.600000&2.166667\\7.566667&0&2.166667&4.100000\\2.600000&2.166667&0&0.566667\\2.166667&4.100000&0.566667&0\end{bmatrix}is2/math/b935c574afe72660.svg
• =10.1761+54.1696-23.4784=40.8673is2/math/b9b0c1a8c0f513a8.svg
• S=0.12\times(-1.06)=-0.1272is2/math/b9fa5298c9c7cb16.svg
• v_*(s_1)is2/math/ba05a03037bd4feb.svg
• \mu_A:X\rightarrow[0,1]is2/math/ba33a453b276adb3.svg
• =17.2225+53.4361-30.3365=40.3221is2/math/ba408f391fd916c4.svg
• P(A\mid R)=\frac{P(R\mid A)P(A)}{P(R\mid A)P(A)+P(R\mid B)P(B)}is2/math/ba4aa6be7e570107.svg
• 1-\varepsilonis2/math/bad648db5f9b2803.svg
• gbest^0=(1, 1)is2/math/bae2559b2265b2a0.svg
• 4-2=2is2/math/bae97278ddb3a17b.svg
• 8+15=23is2/math/bb009310f34186b1.svg
• 0.5\pi_L=0.1875\pi_L+0.125\pi_R+0.3\pi_R\Rightarrow0.3125\pi_L=0.425\pi_R\Rightarrow\pi_L=1.36\,\pi_Ris2/math/bb6389f4842051c2.svg
• S=0.12\times(-0.15)=-0.018is2/math/bb6864a422495c83.svg
• x^{(1)}=4.15+0.1288=4.2788is2/math/bb72c3734ea8a809.svg
• d_{ij}is2/math/bb784b3ff229be34.svg
• \mathbb E[R_B]=1\times4=4is2/math/bbe9161115867c07.svg
• x^{(1)}=1+0=1is2/math/bc4c5112ff057d33.svg
• 0.5l=0.1875l+0.125r+0.3ris2/math/bc67c26ec7fc1e90.svg
• \neg X\cup\neg Y:\max(0.5,0.4)=0.5is2/math/bc8abd6badb711b9.svg
• v_{new}=0.5+0.5+1.5=2.5is2/math/bc8dd34ee10c4d4d.svg
• R(s)=\mathbb E[R_{t+1}\mid S_t=s]is2/math/bcb8b1797c06bfd8.svg
• d_p=7.31-(7.31)=0is2/math/bceb9619d4320629.svg
• v_C=-2-0.45+0.162v_C=-2.45+0.162v_Cis2/math/bd1b2a47575bc95d.svg
• Q=1is2/math/bd5a6aec418f8f8f.svg
• \pi_1=0.8\pi_1+0.3\pi_2\Rightarrow0.2\pi_1=0.3\pi_2\Rightarrow\pi_1=1.5\pi_2is2/math/bd5f35e7318e17ca.svg
• \langle \mathcal S, P\rangleis2/math/bd68aa54ee45e3af.svg
• 10-2=8is2/math/bdb801d401c03c68.svg
• \tau_{ij}is2/math/bdc505e26b5c671c.svg
• D=\begin{bmatrix}0&10&4&3\\10&0&1&2\\4&1&0&1\\3&2&1&0\end{bmatrix}is2/math/bddf9ec4c2c0baa6.svg
• P(yes)=\tfrac9{14}\approx0.643,\qquad P(no)=\tfrac5{14}\approx0.357is2/math/be7de04ac2f36060.svg
• q(s_1,\text{stay})=1+0.9\,v(s_1)is2/math/bebd3888f239dad7.svg
• \text{used edge: }(1-0.5)(1)+0.1429=0.6429\qquad\text{unused edge (AD, BC): }(1-0.5)(1)+0=0.5is2/math/beecdfda1143f6eb.svg
• X=\begin{bmatrix}2.80825&-2.537625\\3.59&-1.275\end{bmatrix}is2/math/befb5808089844a6.svg
• \pi=\pi P,\quad\pi_1+\pi_2=1is2/math/bf18c709d7b4ad37.svg
• f(3.14, 7.21)=3.14^2+7.21^2-(3.14)(7.21)is2/math/bf321f61ed66837c.svg
• 0.8\times1=0.8\qquad0.2\times5=1is2/math/bf3d6ca02caff691.svg
• 0.4p=0.3l+0.2ris2/math/bf5256a21829e5b4.svg
• Q_{new}=Q_{old}+\alpha[r+\gamma\max Q_{next}-Q_{old}]is2/math/bf626aee4ce371f4.svg
• P(\text{Fever}=\text{Yes}\mid C=\text{Yes})=\frac{3}{3}=1is2/math/bf71c2b5401bc336.svg
• 85\times0.5=42.5is2/math/bf9c01dfa8fdf718.svg
• x^{(1)}=2+0=2is2/math/bfaa1b2d391c5f4e.svg
• z^*_{MoM}=\frac{z_1+\cdots+z_m}{m}is2/math/bfe66044c82b5d4a.svg
• z^*=(19/3)/(2+0.5)=38/15\approx2.533333is2/math/c0212421c50f4e75.svg
• x_{12}=-3.0+0.462375=-2.537625is2/math/c09ac9a314e4985c.svg
• 2^nis2/math/c0a4368527e8571c.svg
• \mu=\frac{1+3+5}{3}=3is2/math/c0bd139e7dbf3b1f.svg
• f(1)=1<9is2/math/c0e5424c98d7c06b.svg
• P(a)=0.95(0.001)(0.002)+0.94(0.001)(0.998)+0.29(0.999)(0.002)+0.001(0.999)(0.998)=0.002516is2/math/c16479d9b9398167.svg
• f(-3)=2(-3)^2+1is2/math/c18467bab709f205.svg
• P(W\mid\lnot R)=\frac{0.1919}{0.5}=0.3838is2/math/c18c31810569b62d.svg
• 3-2=1is2/math/c1d7fd7ffca66c17.svg
• 1\times(2)=2is2/math/c1fb0fa1e275d956.svg
• (0110)_2=(0\times8)+(1\times4)+(1\times2)+(0\times1)=6is2/math/c20ea277129e9ff7.svg
• v^{(1)}=0+0+(-3)=-3is2/math/c239eeeeeca7f890.svg
• P(3,2,1)=\frac38\cdot\frac12\cdot\frac13=\frac1{16}=0.0625is2/math/c24f3d84f730db29.svg
• 1\times 8=8is2/math/c26b14275b7516e9.svg
• 2\times9=18is2/math/c271680245ea1d71.svg
• v(C)=-2+0.9\,[\,0.5\,v(L)+0.5\,(0)\,]=-2+0.45\,v(L)is2/math/c27d02b92c11fd77.svg
• \max(0,12-10)=2is2/math/c2860199dab61091.svg
• \mu_{max}=0.9is2/math/c286f3dd87b25655.svg
• R\circ S^T=\begin{bmatrix}0.3&0.4&0.6\\0.2&0.2&0.2\\0.3&0.4&0.6\end{bmatrix}is2/math/c371ee91de2bed68.svg
• 3+1=4is2/math/c37463b701447787.svg
• 0\times 8+1\times 4+1\times 2+0\times 1=6is2/math/c3e3fe9d6dd4079b.svg
• \mu_{Low}(35)=\frac{50-35}{50-25}=0.6is2/math/c3eeae587085fbae.svg
• A_2c_2=1\times6=6is2/math/c3ef1e042430e3fc.svg
• P(Yes)=3/8=0.375is2/math/c3f053c53b40a9b7.svg
• \alpha=1/P(e)is2/math/c3f28a2b8d2fb918.svg
• V^\pi(s)=\sum_a\pi(a\mid s)\sum_{s\prime}P(s\prime\mid s,a)\left[R(s,a,s\prime)+\gamma V^\pi(s\prime)\right]is2/math/c42f67b85b9dd072.svg
• T=\begin{bmatrix}0.7&0.2\\0.5&0.6\end{bmatrix}is2/math/c434184a79340d55.svg
• B_{new}=9+16+7.5=32.5is2/math/c43c2e3add5e9f7a.svg
• \mu_2^{new}=\frac{0.01798621(1)+0.5(3)+0.98201379(5)}{1.5}\approx4.285370107is2/math/c4600c4965dd3721.svg
• q_*(s,a)=\max_\pi q_\pi(s,a)is2/math/c4640b6ec47433ce.svg
• f(4.2788, 7.348)=4.2788^2+7.348^2-(4.2788)(7.348)is2/math/c476266715700a96.svg
• \gamma=0.9is2/math/c491a3a52610e919.svg
• 0.6(0.163)+0.2(0.255)=0.149\ne0.163is2/math/c4d46c0025cc2225.svg
• \frac{n_{ic}+1}{N_c+k}is2/math/c521050ae7174687.svg
• p_2^{(2)}=1-37/64=27/64=0.421875is2/math/c5238d8cc5680149.svg
• v^{(2)}=(-0.75)+0+(-0.25)=-1is2/math/c528696fa4ba4081.svg
• v_{11}=0.9(0.2)+0.89235(2.5-2.5)+0.12825(3.5-2.5)=0.18+0+0.12825=0.30825is2/math/c5345abf4a118373.svg
• d_g=2-(5)=-3is2/math/c53a1760c2aca6d6.svg
• P(Yes)=4/8=1/2,\quad P(No)=4/8=1/2is2/math/c54c9411a1740830.svg
• v_\pi=0.75+5.25=6is2/math/c56792e7802f455e.svg
• \neg X:1-0.7=0.3\qquad\neg Y:1-0.6=0.4is2/math/c590062072bce4a3.svg
• x_tis2/math/c5e02241cdaf66e6.svg
• v=4+0.18v\Rightarrow v=4/0.82=4.878is2/math/c5eca38ec463b77a.svg
• N_k=\sum_i\gamma_{ik},\quad \pi_k^{new}=\frac{N_k}{N},\quad\mu_k^{new}=\frac{\sum_i\gamma_{ik}x_i}{N_k}is2/math/c63266009fda4bdb.svg
• a=\pi(s)is2/math/c6437d9ecddc37a5.svg
• 32-25=7is2/math/c65340df2ae7cb64.svg
• P(A\mid B)=\frac{0.024}{0.028}\approx0.857142857is2/math/c6539c9bc8ea7676.svg
• P(Sweet\mid Banana)=\frac{300}{400}=\frac34=0.75is2/math/c65d5aa64de25ddd.svg
• 0\times 8=0is2/math/c667fae833dbf441.svg
• \max\{\min(0.2,0.4),\min(0.2,0.4),\min(0.2,0.1)\}is2/math/c683e55ca4bccac0.svg
• b=0.5a+\frac5{12}cis2/math/c6b1f0d1f25d8a7c.svg
• (0110)_2=(6)_{10}is2/math/c6cd8f6b20d7ec32.svg
• c_2r_2=0.12825is2/math/c760fda5e7728aea.svg
• x_{11}=3.1472+0.314257=3.461457is2/math/c789efe14e1349c9.svg
• I=0.5\times0.1288=0.0644is2/math/c7a3d7f19f863ddf.svg
• 2x^3=250\quad 5x^2=125\quad 7x=35is2/math/c7ab1b91c2eced73.svg
• v^{(1)}=0.15+0+0=0.15is2/math/c7ca03a89be4beea.svg
• \alpha=\min(0.7,0.5)is2/math/c7cb0e48bf98cd0f.svg
• d_p=7.31-(7.348)=-0.038is2/math/c86909b46639b9bb.svg
• v(\text{Class})\approx-2.924,\;v(\text{Lab})\approx-2.053,\;v(\text{Done})=0is2/math/c86b63e5ffd55eb7.svg
• \text{row 2 (all 0.2)}=(0.2,0.2,0.2);\quad(3,1)=\max(.4,.4,.1)=0.4,\;(3,2)=\max(.4,.4,.1)=0.4,\;(3,3)=0.3is2/math/c87a743bfa211789.svg
• x-3=5-3=2is2/math/c893e43cc49d3cae.svg
• P_{+1}(Yes\mid x)=3/13,\quad P_{+1}(No\mid x)=10/13is2/math/c94cdf7b26810ac7.svg
• =\alpha\,(0.001)\big[0.9(0.7)(0.95\times0.002+0.94\times0.998)+0.05(0.01)(0.05\times0.002+0.06\times0.998)\big]=\alpha\times0.000592is2/math/c977798f6840354f.svg
• \mu_{A\setminus B}(u_1)=\min(0.7,0.7)=0.7is2/math/c98cc0f742bee1be.svg
• x_2=x_1P=[0.75(0.75)+0.25(0.40),\;0.75(0.25)+0.25(0.60)]=[0.6625,\;0.3375]is2/math/c991023e560f0430.svg
• P_{ij}=\frac{\tau_{ij}^{\alpha}(1/d_{ij})^{\beta}}{\sum_{k\in allowed}\tau_{ik}^{\alpha}(1/d_{ik})^{\beta}}is2/math/c9ac4c197173e498.svg
• \mu_{B^c}(u_1)=1-0.6=0.4is2/math/c9c6864aea087c58.svg
• d_g=7.21-(7.21)=0is2/math/ca02c52199da4f04.svg
• \alphais2/math/ca2ff9ceb2ebf0ae.svg
• X\cup Y:\max(0.7,0.2)=0.7is2/math/ca6e5562fab61517.svg
• v_\pi(s)=\mathbb E_\pi[G_t\mid S_t=s]is2/math/ca7b3acce010f57a.svg
• R=\begin{bmatrix}0.9&0.3&0.8\\0.2&0.2&0.2\\0.6&0.3&0.6\end{bmatrix}is2/math/ca9c5503d687e49c.svg
• z^*=\frac{20+30}{2}=25is2/math/caf06fb51932ddd6.svg
• f(x)=(x-7)^2is2/math/caf5baf4d59a93e5.svg
• 1/dis2/math/cb175dc9c9dea9bf.svg
• f(-1)=(-1)^2=1=1is2/math/cb918b9a5ef7b82c.svg
• 0\times 1=0is2/math/cbd4799920019db6.svg
• \frac{80-z_R}{80-65}=0.3is2/math/cbfa4bdb9123ba28.svg
• d_p=3.14-(3.19)=-0.05is2/math/cc2af53bc1a4a60e.svg
• v_*(s)=\max_a\sum_{s'}P(s'\mid s,a)\,[R+\gamma v_*(s')],\qquad q_*(s,a)=\sum_{s'}P(s'\mid s,a)\,[R+\gamma\max_{a'}q_*(s',a')]is2/math/cc767eadfe69fc8c.svg
• x-3=(-1)-3=-4is2/math/cca2cd1e0a06e8d4.svg
• 0.625\times0.4=0.25is2/math/ccabec7597d2610a.svg
• \overline f_{new}=\frac{144+144+144+144}{4}=\frac{576}{4}=144is2/math/cd00806a61b8ba0c.svg
• v^{(2)}=0.5+(-0.5)+(-0.5)=-0.5is2/math/cd06fdfee86fe231.svg
• (0)^2=0is2/math/cd561d277fcba773.svg
• 4+30+16=50is2/math/cd5630515c6da6da.svg
• L_2=d_{AC}+d_{CB}+d_{BD}+d_{DA}=4+1+2+3=10is2/math/cd7de0bca5f15da5.svg
• v_\pi(s)=\sum_a\pi(a\mid s)\sum_{s'}P(s'\mid s,a)\,[R(s,a,s')+\gamma v_\pi(s')]is2/math/cd7f793810f9ae1e.svg
• 0.5\times0.5=0.25is2/math/cd8320ff28f21681.svg
• \gamma\to1is2/math/cd9ca536b7f46490.svg
• v_{22}=0.9(0.25)+0+0=0.225,\qquad x_{22}=-1.5+0.225=-1.275is2/math/cdc998e6ae01e141.svg
• \neg X:1-0.9=0.1\qquad\neg Y:1-0.3=0.7is2/math/ce24f9cbda57e56d.svg
• T_{11}=\max\{\min(0.2,0.5),\min(0.8,0.7)\}is2/math/ce290d4a2b6414c6.svg
• T_{12}=\max\{\min(0.2,0.9),\min(0.8,0.1)\}is2/math/ce2b2b77de4033df.svg
• \mathcal Ais2/math/ce88a93ac568728b.svg
• \max\{\min(0.4,0.6),\min(0.4,0.4),\min(0.3,0.1)\}is2/math/cedfe000344abbe9.svg
• 0\times0.25=0is2/math/cf25ab85b70dd28a.svg
• X=\begin{bmatrix}2.5&-3.0\\3.5&-1.5\end{bmatrix},\;V=\begin{bmatrix}0.2&0.3\\0.1&0.25\end{bmatrix}is2/math/cf25c61c163c59c7.svg
• T_{11}=\max\{\min(1,0.2),\min(0,0.8)\}is2/math/cf3e1892a786961d.svg
• z^*_{COA}=1329.75/17.25=1773/23\approx77.086956522\%is2/math/cf56069827f681bb.svg
• I=0.5\times(-0.2636)=-0.1318is2/math/cfc400d9e06f7358.svg
• (z-70)/15=0.3\quad\Rightarrow\quad z=74.5is2/math/d00383e24c2ca74a.svg
• S=0.5\times(0)=0is2/math/d00ab76bdfccf855.svg
• z_R=75.5is2/math/d02ea96dcfb95060.svg
• P(J,M,A,B,E)=P(J\mid A)\,P(M\mid A)\,P(A\mid B,E)\,P(B)\,P(E)is2/math/d065035a53d0c75c.svg
• \gamma=0is2/math/d06cb90e005f54d4.svg
• I_{new}=15+15+2=32is2/math/d10fe140b0e969ab.svg
• P(\text{Experience}=\text{Mid}\mid C=\text{Yes})=\frac{2}{4}=0.5is2/math/d14d2e65e85aa98c.svg
• v^{new}=wv+c_1r_1(pbest-x)+c_2r_2(gbest-x),\quad x^{new}=x+v^{new}is2/math/d17ec7429fc5876d.svg
• \mu(14)=1/(1+4)=0.2is2/math/d1a7dee6fdbdc5f8.svg
• \mu_{out}(z)=\max_r\min(\alpha_r,\mu_{C_r}(z))is2/math/d1c2a08444481d0c.svg
• 8+0+2+0=10is2/math/d1e04d751dc741ed.svg
• 0.3125l=0.425r\quad\Rightarrow\quad l=1.36ris2/math/d21c86c6a141bc54.svg
• P_{ii}is2/math/d21fe82fdd25b21f.svg
• \mu_A:U\rightarrow[0,1]is2/math/d2598c83177cc2bf.svg
• x^{(2)}=7.348+(-0.02036)=7.32764is2/math/d30e07ac98dd2bba.svg
• 0.5^2\times(8)=2is2/math/d314ab19535cf0b6.svg
• f(2, 1.5)=2^2+1.5^2=4+2.25=6.25is2/math/d342f4df799385a1.svg
• d_p=8.27-(8.27)=0is2/math/d343c09e8978bf19.svg
• d_p=3.14-(3.14)=0is2/math/d3ab4c94a5f284d2.svg
• P(\text{Cough}=\text{Yes}\mid C=\text{No})=\frac{1}{5}=0.2is2/math/d406a69469adaf9b.svg
• \overline f_{new}=\frac{169+169+169+169}{4}=\frac{676}{4}=169is2/math/d460690a1684b5f8.svg
• \text{Ant 1}=\text{Ant 2}:\;A\to C\to D\to B\to A,\qquad L=2+2+6+4=14is2/math/d48c911f3133b5dc.svg
• d_g=1-(1)=0is2/math/d4af3bfbae14d68f.svg
• \max\{\min(0.4,0.3),\min(0.5,0.3),\min(0.3,0.1)\}is2/math/d50a44d1852d902c.svg
• \tau_{ij}^{new}=(1-\rho)\tau_{ij}+\sum_{k:\,ij\in tour_k}\frac{Q}{L_k}is2/math/d5297204b4481c8e.svg
• =9.8596+51.9841-22.6394=39.2043is2/math/d54138730c346bbf.svg
• I=0.5\times(-0.1272)=-0.0636is2/math/d56f94fb044781a6.svg
• p_{ij}^{(k)}=\frac{\tau_{ij}^{\alpha}\eta_{ij}^{\beta}}{\sum_{h\in allowed_k}\tau_{ih}^{\alpha}\eta_{ih}^{\beta}}is2/math/d58589c662f1dcac.svg
• x=L+\frac d{2^n-1}(U-L)is2/math/d5a028f7f6ff09e2.svg
• \mu_{A\cup B}=\max(\mu_A,\mu_B)\qquad\mu_{A\cap B}=\min(\mu_A,\mu_B)is2/math/d5a9710049ed9d75.svg
• v_{21}=0.9(0.1)+0+0=0.09,\qquad x_{21}=3.5+0.09=3.59is2/math/d5ef6ea1fa183f88.svg
• \mu_{Medium}(35)=\frac{35-25}{50-25}=0.4is2/math/d607a558d66f8791.svg
• \lambda_k^iis2/math/d61decbc739ada16.svg
• L_k=\sum_{(i,j)\in tour_k}d_{ij},\quad\Delta\tau_{ij}^{k}=\begin{cases}Q/L_k&(i,j)\in tour_k\\0&\text{otherwise}\end{cases}is2/math/d62616a1401fbf2a.svg
• 1.5\pi_2+\pi_2=1\Rightarrow\pi_2=0.4,\quad\pi_1=0.6is2/math/d630dbfb75f1cbad.svg
• f(x)=2x^3-3x^2+8xis2/math/d678d50d2d5c4ae4.svg
• M=\frac{b-a}{6}\big[(2a+b)u+(a+2b)v\big]is2/math/d68335c9ce0df62c.svg
• T_{12}=\max\{\min(1,0.6),\min(0,0.3)\}is2/math/d6b9af12c569f647.svg
• gbest^0=(1)is2/math/d6d8eee6e10c4922.svg
• Q(s,a)\leftarrow Q(s,a)+\alpha\,\big[\,r+\gamma\max_{a'}Q(s',a')-Q(s,a)\,\big]is2/math/d6e7feae89b08cf3.svg
• x^{(1)}=4+(-3)=1is2/math/d71458dcf56b7a77.svg
• K=5:\quad n_{free}=5d+\frac{5d(d+1)}2+4is2/math/d72da4a5707e5b6a.svg
• \rho=0.5is2/math/d7320dc626d3d883.svg
• \min(0.7,0.6)=0.6is2/math/d73caed266f7eed3.svg
• \eta:\;B=1,\;C=0.5,\;D=0.3333;\;\text{sum}=1.8333\Rightarrow P=0.5455,\;0.2727,\;0.1818is2/math/d7a22769c0bad254.svg
• d_p=7.21-(7.21)=0is2/math/d7b5c8d3808f4c73.svg
• v=(I-\gamma P)^{-1}Ris2/math/d7e46a6497b0a4a6.svg
• \pi'(s)=\arg\max_a q_\pi(s,a)is2/math/d82f4ea4f553e1e7.svg
• (1001)_2=(9)_{10}is2/math/d83c62995f78037e.svg
• q_*(s,a)=\sum_{s'}P(s'\mid s,a)[R+\gamma\max_{a'}q_*(s',a')]is2/math/d84f2e508180aed8.svg
• F(2.80825,-2.537625)=15.772536+21.378856+6.439541+14.04125-5.07525+7=59.5569is2/math/d85fb6275e35f72d.svg
• P(Y\mid x)=\frac{2/45}{2/45+1/160}=\frac{64}{73}\approx0.876712is2/math/d8ec61024e51e1a4.svg
• \mu(4)=\frac{2}{4}=0.5is2/math/d97d0314d106677e.svg
• \eta_{ij}=1/d_{ij}\qquad P_{ij}=\tau_{ij}\eta_{ij}/\sum_{l\in allowed}\tau_{il}\eta_{il}is2/math/d98041105cc02612.svg
• T=\begin{bmatrix}0.6&0.4&0.5\\0.5&0.3&0.5\end{bmatrix}is2/math/d993cd17e860e89f.svg
• W=0.9,\;c_1=c_2=1.5,\;r_1=0.5949,\;r_2=0.0855is2/math/d99737a5fc79423b.svg
• (110)_2=(1\times4)+(1\times2)+(0\times1)=6is2/math/d9a164fc63da4660.svg
• \eta_{AB}=1/2=0.5\qquad\eta_{AC}=1/4=0.25is2/math/d9c5253b54fcf92e.svg
• f(21)=18522-1323+168=17367is2/math/da828d5642e53dc9.svg
• X\cup Y:\max(0.9,0.1)=0.9is2/math/da89b631e703aa27.svg
• Q(\theta\mid\theta^{old})=\sum_i\sum_k\gamma_{ik}\left[\log\pi_k+\log\mathcal N(x_i\mid\mu_k,\Sigma_k)\right]is2/math/da8bce8c31e000fa.svg
• x_{12}=-4.0246+0.452491=-3.572109is2/math/daa1f4fff8808c55.svg
• v_{i,j}^{t+1}=w v_{i,j}^{t}+c_1r_1(pbest_{i,j}^{t}-x_{i,j}^{t})+c_2r_2(gbest_j^{t}-x_{i,j}^{t})is2/math/dac83013fb52dd1c.svg
• v_{k+1}=\max(1+0.9v_k,\;4+0.18v_k)is2/math/db13b37224954848.svg
• I=0.5\times0.5=0.25is2/math/db302571bbe707b4.svg
• \sum M=1522.144is2/math/db33614c63a7fe19.svg
• \mu(10)=e^{0}\approx1is2/math/db4e90ad0923df06.svg
• x^2=21\times21=441\qquad x^3=441\times21=9261is2/math/db60547636676c53.svg
• C=0.6\times(-0.05)=-0.03is2/math/db60d8d854aa5ee6.svg
• x^{(1)}=8.27+(-0.1272)=8.1428is2/math/dba6d056546c8535.svg
• \bar z=\frac{140}{4}=35is2/math/dbdc025973fedfed.svg
• 1.5a=\frac57-\frac5{42}=\frac{25}{42}is2/math/dbf72a6e6a9bedfd.svg
• G_0=(-1)+(4.5)+(1.62)=5.12is2/math/dc498059f2b74dca.svg
• 40+16=56is2/math/dc4a907872eaaa0b.svg
• I=0.5\times(-1)=-0.5is2/math/dc7ed342b0db88bc.svg
• \sigma=\sqrt{1}\approx1is2/math/dc84a3fdebee073d.svg
• \mu(15)=(15-10)/(20-10)=5/10=0.5is2/math/dcb555d1584d4cd8.svg
• d_p=1.5-(1.5)=0is2/math/dcb7c97c6295715d.svg
• A_1=0.5+2+0.25=2.75,\quad M_1=7/6+10+11/6=13,\quad c_1=13/2.75=52/11is2/math/dcc4edef182cedf3.svg
• f(-1)=(-4)^2=(-4)\times(-4)=16is2/math/dce608d3d6bd8495.svg
• z^*=\frac{1522.144}{35.58}\approx42.780888139is2/math/dd239737d0e6ec58.svg
• X\cap Y:\min(0.5,0.6)=0.5is2/math/dd263e46a0f4e96c.svg
• \mu_{\mathrm{aggregate}}(z)=\max_r\mu_{\mathrm{rule}\ r}(z)is2/math/dd5c7064495bb995.svg
• P(N\mid x)=\frac9{73}\approx0.123288is2/math/dd660d78450f1519.svg
• X\cup Y:\max(0.2,0.8)=0.8is2/math/ddba21d8acd6b044.svg
• A_{next}=0.8A+0.1B,\quad B_{next}=0.2A+0.9Bis2/math/ddeb66c690da3b9e.svg
• d_g=-1-(1)=-2is2/math/de044b0c7b70c868.svg
• 2^{2(1)}=4is2/math/de2c89102c7ce99d.svg
• \mu_{High}(65)=\frac{65-50}{75-50}=0.6is2/math/de5666dfbbcb02fc.svg
• v^{(2)}=(-0.1318)+0+(-0.331968)=-0.463768is2/math/de7212cc169aa2f9.svg
• x^{(1)}=3.14+0.05=3.19is2/math/de8ab0140b454ff9.svg
• x_{t+1}is2/math/de9a8c96e0ee7db9.svg
• F(9)=81-100(0)=81is2/math/deb272d3158b70b9.svg
• v^{(1)}=0+0+0=0is2/math/dec9907b2180e0af.svg
• F(12)=144-100(2)=-56is2/math/decf48b28bc4a229.svg
• q_*(s_1,\text{go})=5.8is2/math/defe0522e073671d.svg
• \frac{32-25}{35-25}is2/math/df0b8ec6b8392714.svg
• v^{(2)}=0.075+(-0.09)+(-0.018)=-0.033is2/math/df1003fde626354c.svg
• \frac{3}{10}=0.3is2/math/df25a279590f4f48.svg
• F(3.59,-1.275)=25.7762+13.73175+1.625625+17.95-2.55+7=63.5336is2/math/df29fd84a6cdf039.svg
• d_g=2-(2)=0is2/math/df4d0e49facbaae8.svg
• s_{Yes}^{(+1)}=\frac12\frac37\frac27\frac16=\frac1{98},\quad s_{No}^{(+1)}=\frac12\frac27\frac27\frac56=\frac5{147}is2/math/df5992f5cc97daef.svg
• c_1r_1=1.5(0.5949)=0.89235,\quad c_2r_2=1.5(0.0855)=0.12825is2/math/df63d53634007eba.svg
• \bar f_{fitness}=\frac{527+378}{10}=90.5is2/math/df9902dbae821985.svg
• R_{ij}=\min(A_i,B_j)\quad T_{ik}=\max_j\min(R_{ij},S_{jk})is2/math/dfabdb4f1a2b77c3.svg
• P(S,C,P,Co,F)=P(S)P(C\mid S)P(P)P(Co\mid C,P)P(F\mid P)is2/math/dfc06603bb26a3be.svg
• d_p=3.5-(3.5)=0is2/math/dfcb6b54b5bd04e2.svg
• 0.5\times85=42.5is2/math/dfe0b2ebeb9b2e0c.svg
• I=0.5\times4=2is2/math/e009c778bab94586.svg
• score(C)=P(C)\prod_jP(a_j\mid C)is2/math/e0195b1e4ddb0473.svg
• v_\pi(s)=\sum_a\pi(a\mid s)\sum_{s'}P(s'\mid s,a)[R+\gamma v_\pi(s')]is2/math/e0207f466cf8054f.svg
• 7-6=1is2/math/e03d8b2e4fa9e9e0.svg
• \mu(45)=0is2/math/e052435017d8c2af.svg
• G_t=R_{t+1}+\gamma G_{t+1}is2/math/e076ba9a7dcf48af.svg
• A_{new}=5+9+18=32is2/math/e07af325efaf974e.svg
• \pi_jis2/math/e0c070505a488343.svg
• 0.024\times0.75=0.018\quad;\quad0.018\times0.1=0.0018is2/math/e0cfdfc6521b1e91.svg
• =10.106041+53.684929-23.292533=40.498437is2/math/e1463199434aa355.svg
• v^{(1)}=(-0.5)+0+(-1)=-1.5is2/math/e15db2dfd9b255cf.svg
• T_{ij}=\max_{k=1,2}\min(R_{ik},S_{kj})is2/math/e1639920f7163459.svg
• \log\sum_k\lambda_ka_k\ge\sum_k\lambda_k\log a_kis2/math/e188562c73c0cd46.svg
• c_2r_2=1.2\times0.1=0.12is2/math/e19d636861b5bba6.svg
• 0.7\times0.2\times0.5\times0.6\times0.6=0.0252is2/math/e1a6504a28d510dd.svg
• C_{new}=6+8+15=29is2/math/e1d10b6213a4f6c2.svg
• P(V\mid\neg F)=\frac{\sum_{w,c}P(w)P(\neg F\mid w)P(c\mid w)P(V\mid\neg F,c)}{\sum_wP(w)P(\neg F\mid w)}is2/math/e2104f63677bf585.svg
• f(x)=x^2is2/math/e23769019c76d37b.svg
• P(Y{=}1\mid X{=}1,Z{=}1)=\frac{0.576}{0.576+0.008}=\frac{0.576}{0.584}=0.986is2/math/e2428bb7484a2222.svg
• (0110)_2is2/math/e242d7ba8a184724.svg
• g(10)=\frac{10-2}{4}is2/math/e276417a70627f82.svg
• P(E)=\frac{\text{sum of weights for }E}{\text{sum of all weights}}is2/math/e29a33f81d36c3d3.svg
• \mu_{R1}(60)=\min(0.3,0.8)=0.3is2/math/e2cf40ea63fed461.svg
• \mathbf P(X_i\mid Parents(X_i))is2/math/e2e32da4799205fe.svg
• T_{22}=\max\{\min(0.6,0.9),\min(0.3,0.1)\}is2/math/e2fa4ca72e9eecdf.svg
• \pi=\pi P,\qquad \sum_i \pi_i=1,\qquad \pi_i\ge 0is2/math/e3293de0e0ea04b1.svg
• \mu_{X\cup Y}=\max(\mu_X,\mu_Y),\quad\mu_{X\cap Y}=\min(\mu_X,\mu_Y)is2/math/e32ad02c5ba3dae5.svg
• P=\begin{bmatrix}0.8&0.2\\0.1&0.9\end{bmatrix}is2/math/e333fb4ce294957a.svg
• \varepsilonis2/math/e34d6bfa22904706.svg
• \mathcal L(\theta)=\prod_{i=1}^{N}\left[\sum_{k=1}^{K}\pi_k\mathcal N(x_i\mid\mu_k,\Sigma_k)\right]is2/math/e3640e9995db4c3c.svg
• s_{No}=\frac12\times\frac14\times\frac14\times\frac44=\frac1{32}=0.03125is2/math/e376b189fb5efc12.svg
• z_{COA,1}=\frac{1+1.875+1.25}{.75+.75+.375}=\frac{4.125}{1.875}=2.2is2/math/e3b301b5ea6a74cd.svg
• =\max\{0.2,0.2,0.1\}=0.2is2/math/e40652bfd4dc8015.svg
• \mu_H(t)=\begin{cases}0&1\le t\le50\\(t-50)/50&50<t\le100\end{cases}is2/math/e4429c897c410965.svg
• \alpha_{AND}=\min(a,b)\qquad\alpha_{OR}=\max(a,b)is2/math/e446839a4e5e2d93.svg
• 0.3\times65=19.5is2/math/e44a237331cdc730.svg
• \neg X\cup\neg Y:\max(0.2,0.5)=0.5is2/math/e450f31b46fa1e42.svg
• \eta=\tfrac15,\tfrac16,\tfrac17=0.2,\;0.1667,\;0.1429;\;\text{sum}=0.5095\Rightarrow P=0.3925,\;0.3271,\;0.2804is2/math/e47560808b5cfa0e.svg
• P(B_1\mid W)=\frac{P(W\mid B_1)P(B_1)}{P(W\mid B_1)P(B_1)+P(W\mid B_2)P(B_2)}is2/math/e49e2acd2af82aac.svg
• I=0.5\times0.05=0.025is2/math/e4bd35a53c831c76.svg
• P(F\mid C)=\frac{\text{both}}{\text{cricket}}=\frac{10}{40}=0.25is2/math/e504a0ad42ea83d7.svg
• 0.375\times0=0is2/math/e5245fc04fe5be62.svg
• d_p=1-(3)=-2is2/math/e56ef95c25fb8bde.svg
• P_{ij}=P(\text{next}=j\mid\text{current}=i)is2/math/e570af7d131502f2.svg
• \pi=\pi Pis2/math/e595caf895fa199c.svg
• \pi_kis2/math/e60970d7047375ff.svg
• r=3is2/math/e61c4165fe299ce3.svg
• b=0.6b+0.2iis2/math/e6252d668732cace.svg
• P(Y)=\sum_z P(Y,z)is2/math/e6ca224bc18b80c0.svg
• P(a\mid e)=\dfrac{P(a\land e)}{P(e)}is2/math/e6e697dab9ccb462.svg
• 0.5\times0=0is2/math/e71e2d0a61e9d301.svg
• 0.1a=0.1b+0.3(2b)=0.7bis2/math/e730f86d7c6fb366.svg
• S=0.12\times(-0.05)=-0.006is2/math/e7739769871dabe8.svg
• 0.1+0.7=0.8is2/math/e794808140502a26.svg
• p=0.75l+0.5ris2/math/e7f829189002f918.svg
• d_p=2-(2)=0is2/math/e7f9deeb68816f5a.svg
• f(1, 1)=1^2+1^2=1+1=2is2/math/e7fb8b8ea59ba406.svg
• x_{11}is2/math/e82facc124656ec6.svg
• P(C\cup F)=P(C)+P(F)-P(C\cap F)=0.4+0.3-0.1=0.6is2/math/e851c0d30cd17e85.svg
• 0\le P_{ij}\le 1is2/math/e864eb4c19313d10.svg
• \mathcal N(x\mid\mu,\Sigma)=\frac{\exp[-\frac12(x-\mu)^T\Sigma^{-1}(x-\mu)]}{(2\pi)^{d/2}|\Sigma|^{1/2}}is2/math/e8919c0c3aee3455.svg
• f(2)=4>1is2/math/e8b8ef3456701e2b.svg
• 0.70\times0.30=0.21,\qquad0.10\times0.70=0.07is2/math/e8bf2d5761946110.svg
• \gamma_2=1-\gamma_1\approx0.01798621is2/math/e951eddaa67ec02a.svg
• P(\text{Salary}=\text{Low}\mid C=\text{No})=\frac{1}{4}=0.25is2/math/e95fc4843fd44476.svg
• P(S_{t+1}\mid S_t,\dots,S_1)=P(S_{t+1}\mid S_t)is2/math/e99a6e85c6b029e3.svg
• S=0.12\times(-0.138)=-0.01656is2/math/e9a578685c025bc9.svg
• 2x^3=686\quad 5x^2=245\quad 7x=49is2/math/e9a8667a88c96f77.svg
• \mu_{A\times B}(x,y)=\min(\mu_A(x),\mu_B(y)),\quad T_{ij}=\max_k\min(U_{ik},V_{kj})is2/math/e9a9f703d571640c.svg
• I=0.5\times0.15=0.075is2/math/e9c97812dc9ecc86.svg
• d_g=7.21-(8.27)=-1.06is2/math/ea093f799c7411f8.svg
• C_1=12+36=48is2/math/ea2f674cf5833607.svg
• 12\times12=144is2/math/ea37e28a2a0fc9ec.svg
• \neg X\cap\neg Y:\min(0.8,0.2)=0.2is2/math/ea5cfbc43203761d.svg
• v=(0.1908,\,0.2228)is2/math/ea7da7e3a3880e01.svg
• \pi_P=0.75(1.36\pi_R)+0.5\pi_R=1.52\,\pi_Ris2/math/ea975cda05879034.svg
• \pi_j=\sum_i\pi_iP_{ij}is2/math/ea9b496718656056.svg
• 0.7c=0.2\quad c=2/7is2/math/eaaea4df9a4e16b6.svg
• (n-1)!/2is2/math/eab927d7255210e1.svg
• x^2=7\times7=49\qquad x^3=49\times7=343is2/math/eac1f56ad09d694c.svg
• \mu_1^{new}=\frac{0.98201379(1)+0.5(3)+0.01798621(5)}{1.5}\approx1.714629893is2/math/eac77d4f47719aec.svg
• 0+12+36+8=56is2/math/eacf08e4fd9566bf.svg
• x^{(2)}=4.2788+(-0.149536)=4.129264is2/math/eae7d898d5a551e1.svg
• \mu_{A\cup B}=\max(\mu_A,\mu_B)\quad\mu_{A\cap B}=\min(\mu_A,\mu_B)is2/math/eaf940ee2fdf1f17.svg
• P(B)=0.0612+0.0392=0.1004is2/math/eafd0d70d15c9138.svg
• R=\begin{bmatrix}0.3&0.6&0.5\\0.2&0.2&0.2\end{bmatrix}is2/math/eb5e24dcbf5263d2.svg
• N_1=0.98201379+0.5+0.01798621=1.5is2/math/eb730836d82ad0e1.svg
• d_g=-2-(4)=-6is2/math/eb7d3c2c066600f2.svg
• I=0.5\times0.3=0.15is2/math/eb88185c1e62a59c.svg
• v(s)=R(s)+\gamma\sum_{s'}P_{ss'}v(s')is2/math/eb908cf2d66f9fa5.svg
• P(\neg A)=1-0.001=0.999is2/math/eba03740537f2993.svg
• v(C)=-2+0.45(-1+0.36\,v(C))=-2.45+0.162\,v(C)\Rightarrow0.838\,v(C)=-2.45is2/math/ebb39216a5765bf5.svg
• =\max\{0.3,0.2,0.5\}=0.5is2/math/ebeaf1e5ce8f9612.svg
• v^{(2)}=0.0644+(-0.07728)+(-0.136656)=-0.149536is2/math/ebf74e4d56d1b7b9.svg
• \mu_{High}(80)=\frac{100-80}{100-75}=0.8is2/math/ec0cdd51e84c6204.svg
• P(C\mid X=1)=\frac{0.08/1.20}{0.30/1.20}=\frac{0.08}{0.30}=\frac4{15}\approx0.266667is2/math/ec2f2af190c229ec.svg
• P(W\mid\neg R)=\frac{\sum_{c,s}P(c)P(s\mid c)P(\neg R\mid c)P(W\mid s,\neg R)}{\sum_cP(c)P(\neg R\mid c)}is2/math/ec8db3b28ef749de.svg
• X\cap Y:\min(0.7,0.6)=0.6is2/math/ecd4a7a60c39fa85.svg
• f(12)=3456-432+96=3120is2/math/ece74f225f6dc343.svg
• v_*(s)=\max_\pi v_\pi(s)is2/math/ed035ab67f88529a.svg
• A=(b-a)\frac{u+v}{2},\qquad M=aA+\frac{(b-a)^2(u+2v)}6,\qquad z^*=\frac{\sum M}{\sum A}is2/math/ed18c6126c4569a0.svg
• P=\begin{bmatrix}1/2&1/2\\1/4&3/4\end{bmatrix}is2/math/ed5dd48ea3dd8f9e.svg
• \frac{140}{160}=0.875is2/math/ed912694e6123813.svg
• P(a\land b)=P(a)P(b)is2/math/ed94fa4d4b6553f6.svg
• \neg X\cap\neg Y:\min(0.6,0.3)=0.3is2/math/edebdcb00abb9544.svg
• 0\times 8+0\times 4+0\times 2+0\times 1=0is2/math/edfc0319e8a1b583.svg
• M=\frac{b-a}{6}[(2a+b)u+(a+2b)v]is2/math/ee3fb707739e8b69.svg
• 3K-1is2/math/eec5afc3e1c07c8c.svg
• g(x)=\frac{x-2}{4}is2/math/ef554c6170bf393e.svg
• f(x)=x_1^2+x_2^2-x_1x_2is2/math/ef9d55cf7c043920.svg
• Q(s,a)=2is2/math/efb25431f54f8edc.svg
• x_{t+1}=x_t\,P,\qquad x_{t+n}=x_t\,P^{\,n}is2/math/efd1264e0038146b.svg
• (2a+b)u=(2+15.4)\times0=0is2/math/f077b4473411d432.svg
• f(16)=8192-768+128=7552is2/math/f0bfce15122fad53.svg
• \mu(x)=\max\left(0,\min\left(\frac{x-a}{b-a},\frac{c-x}{c-b}\right)\right)is2/math/f0ca7a326bd064e8.svg
• \gamma_1=\frac{0.002215924}{0.002215924+0.120985362}\approx0.01798621is2/math/f0ca8e430dd11e63.svg
• \mu_{X\cup Y}(c)=\max(\mu_X(c),\mu_Y(c)),\quad\mu_{X\cap Y}(c)=\min(\mu_X(c),\mu_Y(c))is2/math/f1157faae69b5acb.svg
• =\max\{0.3,0.3,0.1\}=0.3is2/math/f11631b96248ce3d.svg
• P(credit\mid premium)=\frac{0.08+0.10}{0.302}=\frac{0.18}{0.302}\approx0.596026is2/math/f1511f5b6ec07db0.svg
• \pi(a\mid s)is2/math/f155e46af3f71ce4.svg
• \sigma=\sqrt{2.666666667}\approx1.632993162is2/math/f193c4f2114bfb50.svg
• P(X_{t+1}=S,X_{t+2}=R\mid X_t=S)=P_{SS}P_{SR}=0.8\times0.15=0.12is2/math/f206e4e596cad948.svg
• \langle\mathcal S,P\rangleis2/math/f25474192d3bf3f0.svg
• P(A\mid B)=\frac{P(B\mid A)P(A)}{P(B)}is2/math/f27b652409444deb.svg
• P(\lnot a\mid b,\lnot e)is2/math/f2ae2e9bf6ce6704.svg
• \text{Ant 1: }A\to C\to B\to D\to A,\quad L_1=6+2+9+7=24is2/math/f2d71691d61a5d31.svg
• \mu_{Medium}(65)=\frac{75-65}{75-50}=0.4is2/math/f35f654b9795c088.svg
• A_1c_1+A_2c_2=2(2)+0.5(14/3)=19/3is2/math/f37e6894a3c89e61.svg
• 8+4+2+1=15is2/math/f3965404fe997578.svg
• \mathbf n_{t+1}=\mathbf n_tPis2/math/f39ae5af858943b8.svg
• \pi_*(s)=\arg\max_a q_*(s,a)is2/math/f3cf791d9745c384.svg
• 0.12+0.08=0.20is2/math/f3e0db112ddd0eb5.svg
• \max\{\min(0.7,0.2),\min(0.4,0.8)\}is2/math/f5116e7e9b08a405.svg
• R=\begin{bmatrix}.4&.6&.3\\.4&.4&.3\\.1&.1&.1\end{bmatrix}is2/math/f54af3b992bbd86c.svg
• x=(3.1472,\,-4.0246)is2/math/f58367ac748ee874.svg
• G_0=(2)+(2)+(2)=6is2/math/f58d97b5beb6cfcc.svg
• \neg X:1-0.9=0.1\qquad\neg Y:1-0.1=0.9is2/math/f594ae61dd359b85.svg
• \min(0.7,0.4)=0.4is2/math/f5bde40636dee1a9.svg
• 50\times0.6=30is2/math/f5c2b1ec334e1160.svg
• v_{A2}^{1}=0.9(0)+0.89235(0-0)+0.12825(1-0)=0.12825is2/math/f5e85d68eed20ee1.svg
• 1\times 8+0\times 4+1\times 2+1\times 1=11is2/math/f63d65f39c03b2d0.svg
• x_i\sim\mathcal N(\mu_k,\sigma_k^2)is2/math/f66cfb5e517110e8.svg
• P(a\lor b)=P(a)+P(b)-P(a\land b)is2/math/f67fe6f37e8347cc.svg
• x^{(1)}=7.21+0.15=7.36is2/math/f693927b5eebad50.svg
• P=\begin{bmatrix}0.60&0.40\\0.30&0.70\end{bmatrix}_{(B,C)}is2/math/f73b0238f5d9fefe.svg
• z^*=\frac{26}{1.2}=21.666666667is2/math/f7410bf10136fdc7.svg
• 0.5l=0.25(0.75l+0.5r)+0.3ris2/math/f791f1205fe97449.svg
• \sigma^2=\frac{4+0+4}{3}=2.666666667is2/math/f8055b1e2e609fe7.svg
• x=b_1 2^{n-1}+b_2 2^{n-2}+\cdots+b_n2^0is2/math/f86e986776de93a7.svg
• P(A\mid B)=\frac{P(A\cap B)}{P(B)}=\frac{\text{count satisfying A and B}}{\text{count satisfying B}}is2/math/f893878b79f9bfdd.svg
• \neg X\cap\neg Y:\min(0.4,0.6)=0.4is2/math/f8ad2bf1c74a8449.svg
• S=0.12\times(-0.1)=-0.012is2/math/f8e69de5d9161da2.svg
• P(\neg A\cap B)=0.999\times0.02=0.01998is2/math/f8f7a41ed738725f.svg
• \overline f_{new}=\frac{9+4+1+4}{4}=\frac{18}{4}=4.5is2/math/f966da034fd08628.svg
• x^{(1)}=1+1=2is2/math/f9d52a6143caaf7b.svg
• \pi=\pi P,\;\sum\pi_i=1is2/math/fa1b266acbaaacb0.svg
• (E,I,M,S,A)=(0,1,0,1,0)is2/math/fa60b6f67d870af3.svg
• P(Yes)=4/8=0.5is2/math/fa6d94e12c8c8a44.svg
• P(W,R)=0.5(0.1)(0.8)(0.99)+0.5(0.9)(0.8)(0.90)+0.5(0.5)(0.2)(0.99)+0.5(0.5)(0.2)(0.90)is2/math/fa867c388246e93b.svg
• \mu_{out}(z)=\max_r\min(\alpha_r,\mu_{B_r}(z))is2/math/fa8dd4982ac39cbc.svg
• (0-0)^2=0is2/math/faf8b496eb6d2931.svg
• f(x)=2x^3-5x^2+7xis2/math/fb065d212fc56261.svg
• v_{new}=(-1)+(-0.5)+(-1.5)=-3is2/math/fb1a87666bd393ac.svg
• fan=\frac{0.5(50)+0.5(90)}{0.5+0.5}=\frac{25+45}{1}=70\%is2/math/fb20d690460ddd1a.svg
• \max\{\min(0.2,0.4),\min(0.3,0.3),\min(0.3,0.1)\}is2/math/fb42face9e153c77.svg
• 60\times0.8=48\qquad40\times0.1=4is2/math/fb4be71da02b8086.svg
• Q\leftarrow2+0.5(7.5-2)=2+2.75=4.75is2/math/fb6236e8c0963234.svg
• f(13)=4394-507+104=3991is2/math/fb78bd652840ab88.svg
• \frac{100-z_R}{15}=0.5\Rightarrow z_R=92.5is2/math/fba2b937f7b534a8.svg
• \sum_k\lambda_k^i=1is2/math/fbe0441b97e0b9c1.svg
• \max(0.4,0.6)=0.6\ne1is2/math/fbe8846f3efa93f5.svg
• P(x_1,\ldots,x_n)=\prod_iP(x_i\mid parents_i)is2/math/fbf52d09e73db054.svg
• \betais2/math/fc31c14dd22861fd.svg
• T=\begin{bmatrix}0.5&0.6\end{bmatrix}is2/math/fc69cd14a88a24b1.svg
• (s,a,r,s')is2/math/fc6fae0a339bc5e8.svg
• z^*=\frac{3}{1}=3is2/math/fc739e8ece1c9e47.svg
• \overline f_{old}=\frac{64+169+169+64}{4}=\frac{466}{4}=116.5is2/math/fcafd07eb412a037.svg
• (R_\lambda)_{ij}=\begin{cases}1&R_{ij}\geq\lambda\\0&R_{ij}<\lambda\end{cases}is2/math/fcd0258f1fdf1909.svg
• 0.05\times0.6=0.03is2/math/fcec3b47d13c61a4.svg
• P(\text{not }C)=1-P(C)=1-0.4=0.6\qquad P(\text{neither})=1-P(C\cup F)=0.4is2/math/fd0572e2636f2a68.svg
• P(A)=0.12/0.20=0.6is2/math/fd57d6dd0af66643.svg
• z^*=(54.5+75.5)/2=65is2/math/fd9133cfe48a7467.svg
• x_1=x_0Pis2/math/fe018f0ddfd05790.svg
• d_g=7.21-(7.348)=-0.138is2/math/fe0c41420480a542.svg
• P_{ij}is2/math/fe1d46e7593684d1.svg
• vis2/math/fe332ce60af0f3c3.svg
• R=\begin{bmatrix}1&0.1&0&0.4\\0.001&0.5&0.5&0.6\\0.3&1&0.5&0\\0&0.4&0.002&0.1\end{bmatrix}is2/math/fe401cd453e2d902.svg
• x=2is2/math/feb11b3ddc232396.svg
• =\max\{0.4,0.3\}=0.4is2/math/fed46295bb39d225.svg
• \mu_A(3)=\frac{3-2}{5-2}=\frac{1}{3}=0.333333333is2/math/feefc8b7a33635af.svg
• T=S\odot R=\begin{bmatrix}0.6&0.6&0.4\\0.3&0.3&0.3\\0.5&0.5&0.4\end{bmatrix}is2/math/fef1c19f50df11d5.svg
• D=\begin{array}{c|cccc}&A&B&C&D\\\hline A&0&4&2&4\\B&4&0&3&6\\C&2&3&0&2\\D&4&6&2&0\end{array}\qquad \tau=\begin{array}{c|cccc}&A&B&C&D\\\hline A&0&1&1&1\\B&1&0&1&1\\C&1&1&0&1\\D&1&1&1&0\end{array}is2/math/ff22cd24b937efcc.svg
• s(Y)=\frac6{10}\frac26\frac26\frac46=\frac2{45}\approx0.0444444is2/math/ff3411bced348831.svg
• 0\times 4=0is2/math/ff395ff0ae2a5dd3.svg
• [0.6,0.4]P=[0.48+0.12,\;0.12+0.28]=[0.6,0.4]is2/math/ff5d9937f5d667c5.svg
• (P^2)_{FH}=.075+.100+.1125=.2875is2/math/ff6357103803b6cd.svg
• \hat\mu_k=\frac{\sum_i\lambda_k^ix_i}{\sum_i\lambda_k^i},\qquad\hat\sigma_k^2=\frac{\sum_i\lambda_k^i(x_i-\hat\mu_k)^2}{\sum_i\lambda_k^i},\qquad\hat\pi_k=\frac{\sum_i\lambda_k^i}{n}is2/math/ffbe3f9b626d2a86.svg
• \text{Child}=ABEDC,\quad L=2+3+1+8+9=23,\quad F=0.0435is2/math/ffc7b04e07bd4328.svg
• 0.8+0.2=1is2/math/ffd94fb58f716964.svg
• \mathbf P(Weather)=\langle0.6,0.1,0.29,0.01\rangleis2/math/ffe772be82ede0e4.svg
• s(\text{the})=0.02\times0.05=0.001nlp/math/0057723bfc328243.svg
• P(\text{cat}\mid N)=\frac12nlp/math/00ba54f74d066f93.svg
• B\cdot C=(0)(1)+(1)(0)+(1)(1)+(1)(0)+(0)(1)+(0)(1)=1nlp/math/00d6bb1eb81ee107.svg
• C\cap G_2=\{\mathrm{deposits}\},\qquad \mathrm{score}(2)=1nlp/math/0107901b24c804b2.svg
• \text{Number of tag sequences}=N^Tnlp/math/014ed66c3c0df2b1.svg
• P_{+1}(w_i\mid w_{i-1})=\frac{C(w_{i-1},w_i)+1}{C(w_{i-1})+V}nlp/math/0167293b8abe1cce.svg
• 827+1=828nlp/math/02a3d76972ac332b.svg
• \|A\|=\sqrt{1^2+1^2+1^2}=\sqrt3nlp/math/0319d5a65cf344a9.svg
• 1-dnlp/math/03ba8413969b1f03.svg
• M=N^Tnlp/math/0496ec54e0350270.svg
• P(\mathbf w,\mathbf t)=P(t_1\mid\mathrm{Start})\prod_iP(w_i\mid t_i)\prod_{i=2}^{n}P(t_i\mid t_{i-1})P(\mathrm{End}\mid t_n)nlp/math/054557fae74f519d.svg
• d\times Vnlp/math/058d387c1daaff14.svg
• s(H)=\frac12\times\frac18\times\frac18=\frac{1}{128}=0.0078125nlp/math/05ad87647bbe0b7d.svg
• M_{ij}=\sum_{s=1}^{5}\mathbf{1}[\text{target}_i\text{ and context}_j\text{ occur in sentence }s]nlp/math/05bf57345047ff24.svg
• P(\text{stream}\mid R)=\frac{1+1}{23}=\frac2{23}nlp/math/0612a35cf28c3394.svg
• 1/P=8nlp/math/063b3931740c7830.svg
• 3^5=3\times3\times3\times3\times3nlp/math/064d395af2cadd10.svg
• P_A=0.5\times0.5\times0.5=0.125=1/8nlp/math/072e354cb1ef7f5e.svg
• \frac{5}{20}=0.25nlp/math/07337f908c856c5a.svg
• 0.0448+0.048+0.0016+0.006=0.1004.nlp/math/073c637441080500.svg
• 0.8(0.2)(0.3)(0.1)=0.0048nlp/math/076745077c7c6109.svg
• S\to NP\ VP,\quad NP\to Det\ N,\quad VP\to V\ NPnlp/math/08274dbc564789d5.svg
• =\max(0.096,0.048)(0.4)=0.0384nlp/math/088fd26aaafea320.svg
• P(\text{path,words})=(\text{other required factors})\times b_V(can)nlp/math/0957b6ab0a1aa0cc.svg
• P(w_1,\ldots,w_n,\langle/s\rangle\mid\langle s\rangle)=\prod_{i=1}^{n+1}P(w_i\mid w_{i-1})nlp/math/09cc94e28baa5a90.svg
• 0.15\times0.8=0.12nlp/math/0a217c93f9b6444f.svg
• 0.5\times0.4=0.2nlp/math/0a2405c4d5291b52.svg
• S(b\mid h)=0.4(0.5)=0.2.nlp/math/0aa2b77f15f8a15c.svg
• D(1,2)=\min(2+1,\ 0+1,\ 1+1)=1nlp/math/0ab677a634f01ae0.svg
• A\cdot C=(1)(1)+(1)(0)+(1)(1)+(0)(0)+(0)(1)+(0)(1)=2nlp/math/0add9a13e11f4556.svg
• P=\prod_i P(w_i\mid w_{i-2},w_{i-1})nlp/math/0b4f2f8e9cf3fc17.svg
• J_{ij}=f(X)(w^T\widetilde w+b+\widetilde b-\ln X)^2nlp/math/0b9576850c99f85a.svg
• P(eat)\approx0.244728nlp/math/0be039cb18c2d44a.svg
• 0.4\times0.01=0.004nlp/math/0c266d4a40683c9b.svg
• P(w_i\mid w_{i-2},w_{i-1})=\frac{C(w_{i-2},w_{i-1},w_i)}{C(w_{i-2},w_{i-1})}nlp/math/0ca97d5e894e09b1.svg
• P(c)=\frac{s(c)}{\sum_j s(j)}nlp/math/0cbb12db950a8417.svg
• P_{\mathrm{Laplace}}(w)=\frac{C(w)+1}{N+V}nlp/math/0d44559007b7dea6.svg
• D(3,2)=\min(1+1,\ 2+1,\ 1+1)=2nlp/math/0d6484992af2f005.svg
• D(4,1)=\min(2+1,\ 4+1,\ 3+1)=3nlp/math/0dd1d63ad4a25f3e.svg
• \frac{2}{3}\times\frac{1}{2}=\frac{1}{3}nlp/math/0e0cb943fc030998.svg
• \mathrm{score}(c)=P(c)\prod_jP(x_j\mid c)nlp/math/0ec49ccf4b6243b5.svg
• J(A,B)=\frac{|A\cap B|}{|A\cup B|}nlp/math/0eeb7c5512da08e8.svg
• Vnlp/math/0ef287f11eb0a00f.svg
• \mathrm{BoW}_j(d)=C(v_j,d)nlp/math/0f69259be30dd3d0.svg
• P_L=1.6/20=0.08nlp/math/0fd3e6bc0e9d2f40.svg
• P(s\mid C)=\frac{\operatorname{score}(s)}{\sum_{s'}\operatorname{score}(s')}.nlp/math/0fd59effd2bb7926.svg
• \operatorname{PP}=2^3=8nlp/math/0ffdeccced5f3a45.svg
• \alpha_1(H)=0.8(0.4)=0.32,\quad\alpha_1(C)=0.2(0.1)=0.02.nlp/math/1058f8c72f5016fb.svg
• d_1(a,b)=\sum_i|a_i-b_i|nlp/math/105c539acb46c968.svg
• 0.6\times0.5=0.3nlp/math/115f9e8e47cda84d.svg
• \mathrm{score}(A)=0.5\times\frac1{31}\times\frac1{31}\times\frac1{31}\approx0.00001678359236nlp/math/118f04dc70642898.svg
• \frac{1}{10}\times\frac{2}{17}=\frac{1}{85}nlp/math/11b6a388c4744fa9.svg
• P(w_i\mid w_{i-2},w_{i-1})nlp/math/12c99aed7b535ad6.svg
• P(w_o\mid w_i)=\frac{\exp(u_{w_o}^{T}v_{w_i})}{\sum_{w\in V}\exp(u_w^{T}v_{w_i})}.nlp/math/12d30b8166786ac1.svg
• D(2,1)=\min(0+1,\ 2+1,\ 1+1)=1nlp/math/132b0bf994fa1fe9.svg
• c^*=(1+1)(20/25)=1.6nlp/math/13803fe6f5045ef1.svg
• \cos(A,C)=\frac2{\sqrt3\cdot2}=\frac1{\sqrt3}\approx0.577350nlp/math/13f584cbebce7216.svg
• P(F\mid\text{loan})=\frac{s(F)}{s(F)+s(R)}nlp/math/1414aff0cfb8fbd2.svg
• d=1-Jnlp/math/14ba87d96baf7247.svg
• P_{\mathrm{MLE}}(w)=\frac{C(w)}{N}\qquad P(w_1,\ldots,w_m)\approx\prod_{i=1}^{m}P(w_i)nlp/math/151df11dd9b4e4ca.svg
• \frac{1}{12}\times\frac{1}{21}=\frac{1}{252}nlp/math/15653088bc649e84.svg
• \cos(\mathbf t,\mathbf b)=\frac1{\sqrt2\sqrt2}=0.5nlp/math/15909e375167f411.svg
• =\max(0.096,0.012)(0.5)=0.048nlp/math/15d2f168f4fe9604.svg
• P=\frac23\times1\times\frac12\times1=\frac13nlp/math/15f0c8914a694355.svg
• P=\frac{6}{6+2}=0.750000nlp/math/16635e3dc332a6fb.svg
• \#\text{ n-grams}=T-n+1nlp/math/167913d5b2acc327.svg
• P(w\mid h)=\frac{C(h,w)}{\sum_{v\in V}C(h,v)}=\frac{C(h,w)}{C(h)}nlp/math/16ee67bf50a203c3.svg
• \alpha_2(C)=[0.16(0.3)+0.10(0.6)](0.1)nlp/math/16ff8201d135ae01.svg
• P(visited\mid prime,minister)=1/12nlp/math/170c0f0886c42082.svg
• \operatorname{score}(s)=P(s)\prod_j P(w_j\mid s).nlp/math/1711a8d0c3782b27.svg
• \max(0.32(0.7),0.02(0.4))(0.2)nlp/math/1716d45167a355a0.svg
• \frac1{7776}\times\frac49=\frac1{17496}\approx0.0000571559nlp/math/171a6c8b22e17d35.svg
• 4/7+2/7+1/7=1nlp/math/172a91eacd7bcf59.svg
• d_2=\sqrt{\sum|\Delta_i|^2}nlp/math/182b357a15210739.svg
• 0+5=5nlp/math/190ddc58c7516f91.svg
• N+V=4+3=7nlp/math/1918c0a9ceaec186.svg
• w_inlp/math/1990ec0f29e8d861.svg
• \delta_2(t)=P(w_2\mid t)\max_s[\delta_1(s)P(t\mid s)]nlp/math/19da4a0018c4e11f.svg
• \frac{2}{75}\times\frac{2}{3}=\frac{4}{225}nlp/math/1a1bff7a79a3effd.svg
• \#\text{bigrams}=(m+2)-1=m+1nlp/math/1b167f71a6a973b4.svg
• 16/100=0.16nlp/math/1b290111c4a2b057.svg
• P(t\mid w)=\frac{P(w\mid t)P(t)}{\sum_{u}P(w\mid u)P(u)}nlp/math/1b37c73f9aa13957.svg
• P(k\mid i)=X_{ik}/X_inlp/math/1b6f3de99376d438.svg
• P(W)\approx\prod_iP(w_i\mid w_{i-1})nlp/math/1bb8f8f2c685b8c8.svg
• z=w^Tx+b,\qquad P(y=1\mid x)=\sigma(z)=\frac{1}{1+e^{-z}}nlp/math/1bbea17565b5567a.svg
• A=\frac{4+12}{4+0+4+12}=0.800000nlp/math/1cc17a482eb72469.svg
• P(w\mid h)=\frac{C(hw)}{C(h)}nlp/math/1ce8860e624e2650.svg
• P(D\mid D)=0.8nlp/math/1cf8b238b738730b.svg
• P(D,D,R,R)=P(D)P(D\mid D)P(R\mid D)P(R\mid R)nlp/math/1d6fc24a22e2f64d.svg
• N_snlp/math/1d9c73ba4ad13811.svg
• P(t_j\mid t_i)=\frac{c(t_i,t_j)}{\sum_kc(t_i,t_k)}nlp/math/1d9ef5a8985e2f56.svg
• knlp/math/1dc6f9b1a0213681.svg
• P_L(sleep\mid I)=(0+1)/7=1/7nlp/math/1e0b14bd8e108207.svg
• P_L(w\mid h)=\frac{0+1}{0+V}nlp/math/1e1f42d7a9537f57.svg
• 0.06+0.005=0.065nlp/math/1f46aceeecb6ce54.svg
• =[0.048+0.060](0.1)=0.0108.nlp/math/1f61deb821d3eda8.svg
• r_{t+1}(j)=\sum_i r_t(i)P(i\to j)nlp/math/1f9486c13badc4b9.svg
• 0.06(0.1)=0.006nlp/math/2039ca068efac123.svg
• a_{ij}=\frac{c(t_i,t_j)}{c(t_i)},\qquad b_i(w)=\frac{c(t_i,w)}{c(t_i)}nlp/math/204793f5263e1ae9.svg
• P(\text{a cat, end})=P(\text{a}\mid\text{start})P(\text{cat}\mid\text{a})P(\text{end}\mid\text{cat})nlp/math/20d792a0426e11b9.svg
• z_j=u_j^Tvnlp/math/211ede92a411f755.svg
• P(t'\mid t)=\frac{C(t,t')}{C(t)\text{ with a following tag/end event}}nlp/math/2145c612575540f3.svg
• P(D,D)=0.6\times0.8=0.48nlp/math/21b9a2b02a067056.svg
• P(\langle/s\rangle\mid like,tea)=2/2=1nlp/math/221a8d0201199a49.svg
• 0.5(0)=0,\qquad0.3(0.2)=0.06,\qquad0.2(0.1)=0.02.nlp/math/226058093bf8c6aa.svg
• P(AAA)=0.25^3=\frac1{64}=0.015625nlp/math/227b8a572c539a9c.svg
• P(tea\mid I,like)=1/2nlp/math/22c4756e3a5a9146.svg
• f(16)=(16/100)^{0.75}nlp/math/2307be092fac7a7f.svg
• P=\frac8{5625}\approx0.0014222222nlp/math/236caebf4a4b2830.svg
• P(O)=\sum_j\alpha_T(j).nlp/math/237f4e30ab6d371f.svg
• P=\pi_M b_M(will)a_{MN}b_N(jane)a_{NV}b_V(spot)a_{VN}b_N(mary)a_{N,END}nlp/math/23c70e8086b83a3b.svg
• P(loan\mid R)=\frac1{14},\quad P(water\mid R)=\frac4{14}nlp/math/23fae5f134fd8df2.svg
• P(W)=(1/10)^5,\qquad \operatorname{PP}=P(W)^{-1/5}nlp/math/245311d6fb2b9841.svg
• h=[4/2,2/2]=[2,1]nlp/math/24cc9ba7bac967c1.svg
• d_p=\left(\sum_i|x_i-y_i|^p\right)^{1/p},\quad p\ge1\quad\text{Minkowski}.nlp/math/24cfaa0ff892eddd.svg
• P(\text{rain})=\frac{5}{20}nlp/math/24d2e511eeadd9c0.svg
• 0.16\times0.3=0.048nlp/math/24d6c09c3069c91f.svg
• D(3,3)=\min(1+1,\ 1+1,\ 0+0)=0nlp/math/24fd76b9544c3db3.svg
• C\to H:\ 0.10(0.4)=0.040nlp/math/256055f1b5c81a60.svg
• P(3,1)=0.0464+0.054=0.1004.nlp/math/260e76274f7db291.svg
• D(2,2)=\min(1+1,\ 1+1,\ 0+1)=1nlp/math/26524a35ae995b95.svg
• P(w)=\frac{C(w)}{N}nlp/math/2711f5706c9dc86b.svg
• P(y\mid x)=\frac{P(x\mid y)P(y)}{P(x)}nlp/math/279bb8404538a591.svg
• Nnlp/math/284cee6629b09719.svg
• C\to H:\ 0.006(0.4)=0.0024nlp/math/286832c2c26620dd.svg
• score(s)=|context\cap gloss(s)|nlp/math/28ab6543e5b805ca.svg
• P=\frac23\times\frac12\times1\times\frac25\times\frac35\times\frac23\times\frac23\times\frac15\times\frac25nlp/math/2931148a79074475.svg
• P(red\ green)=P(red)P(green)nlp/math/2ad862e46bb0787a.svg
• J=\sum_{i,j}f(X_{ij})(w_i^T\widetilde w_j+b_i+\widetilde b_j-\log X_{ij})^2nlp/math/2aeda5f45a0334aa.svg
• P(1,3)=0.0448+0.0048+0.016+0.006=0.0716nlp/math/2b05babdadbfdf90.svg
• \frac{8}{1125}\times\frac{2}{5}=\frac{16}{5625}nlp/math/2baf230e35df5c28.svg
• P=\frac23\times1\times\frac23\times1nlp/math/2c319bafd1386357.svg
• P(\text{cat}\mid N)=\frac35=0.6nlp/math/2c58a6fe2e2bd177.svg
• 10/50=0.2nlp/math/2d17e669222fac35.svg
• D/df=1nlp/math/2d65862ad68e4d56.svg
• 0.2(0.5)(0.4)(0.4)=0.016nlp/math/2dbad158b7e2df38.svg
• F_1=\frac{12}{12+2+4}=0.666667nlp/math/2de5ef4a0ac69e0e.svg
• \delta_1(H)=0.8(0.4)=0.32,\quad\delta_1(C)=0.2(0.1)=0.02nlp/math/2e38d10b83c569f4.svg
• J=\sum_{i,j:X_{ij}>0}f(X_{ij})\left(w_i^T\widetilde w_j+b_i+\widetilde b_j-\ln X_{ij}\right)^2.nlp/math/2e4b741639de8430.svg
• P(i\to j)nlp/math/2e580e2c56ce5fb8.svg
• P(F)=\frac36=\frac12,\qquad P(R)=\frac36=\frac12.nlp/math/2f05aa3a89468f5e.svg
• P(H,H,C\mid1,3,1)=\frac{0.00672}{0.021736}=\frac{840}{2717}\approx0.309164.nlp/math/2fa6cd41aa88e033.svg
• N_F=N_R=9nlp/math/2fe9227521f0d9d8.svg
• P(D\mid R)=0.7,\quad0.4(0.7)=0.28nlp/math/2ff067547e9b897f.svg
• \sqrt{(1)^2+(0)^2}=\sqrt{1}.nlp/math/300c6695721d0668.svg
• \mathrm{score}(c)=P(c)P(Sunny\mid c)P(Cool\mid c)P(High\mid c)P(True\mid c)nlp/math/300d051b2a893047.svg
• \approx4.086161nlp/math/308d6aae3a132ab1.svg
• \frac{1}{85}\times\frac{1}{6}=\frac{1}{510}nlp/math/3123e759352a19f6.svg
• =[0.04256+0.00432](0.2)=0.009376.nlp/math/3146e799c63b1663.svg
• 0.0024(0.2)=0.00048nlp/math/315780f838015183.svg
• 1/P(W)=1/0.0625=16nlp/math/31c8e928444ef07c.svg
• NN:0.3\times0.3\times0.1=0.009nlp/math/31db017cf8adde14.svg
• z_w=u_w^Thnlp/math/3203e6549ee39a87.svg
• TF(t,d)=\frac{C(t,d)}{\sum_w C(w,d)}nlp/math/326e2d23a0474bba.svg
• P(k\mid i)/P(k\mid j)nlp/math/330098eff375745b.svg
• [2/3,1/3,0]nlp/math/33108c032f375ccc.svg
• \mathrm{TFIDF}(t,d)=C(t,d)\ln\frac{N}{\mathrm{df}(t)}nlp/math/341f27f737ff351a.svg
• \delta_1(V)=0.4\times0.4=0.16nlp/math/3422eafaea5af482.svg
• \sqrt{(3-3)^2+(3-3)^2}=0nlp/math/3428ab21f04811a8.svg
• C(a)+1=4,\quad C(b)+1=2,\quad C(c)+1=1nlp/math/345a4b4f00828435.svg
• 4+4=8nlp/math/34a0153bb137796a.svg
• 0.112(0.4)=0.0448nlp/math/359ae9ea2534c8f2.svg
• \log_{10}5\approx0.698970nlp/math/35a02628576f1000.svg
• VN:0.16\times0.8\times0.1=0.0128nlp/math/35f92b0ccc050c81.svg
• a=\tfrac14\log_{10}(3/2)\approx0.0440228nlp/math/36558fafa06680e2.svg
• C(w,s)nlp/math/365a4595c962f2f2.svg
• 0.5\times0.2=0.1nlp/math/3666eec3c2831548.svg
• P_{+1}(\text{tea}\mid i,\text{like})=\frac{3+1}{5+4}=\frac49\approx0.444444nlp/math/36b3ccec8e82a718.svg
• P(R\mid C)=\frac{3/12167}{4/12167}=\frac34=0.75.nlp/math/371000172a7d5f5f.svg
• \sqrt{8}\approx 2.828427.nlp/math/37625b38e47e1914.svg
• \operatorname{score}(R)=2nlp/math/384740febd8dd4b0.svg
• u_wnlp/math/388d7b079b70bcfd.svg
• 0.5\times0.6=0.3nlp/math/389d6af9cfa7b052.svg
• P_L(w\mid h)=1/5=0.2,\qquad5(0.2)=1nlp/math/39038359834be22e.svg
• \frac{4}{1125}\times\frac{2}{5}=\frac{8}{5625}nlp/math/391357e72bb5bd3d.svg
• \psi_t(j)=\arg\max_i[\delta_{t-1}(i)a_{ij}]nlp/math/3935aac800940d84.svg
• P=\frac38\times\frac38\times\frac14\times\frac27=\frac9{896}\approx0.010044643nlp/math/395a3f897d526ccd.svg
• [0,1/2,1/2]nlp/math/3a45dfb5933f8388.svg
• \sqrt[3]{91}\approx 4.497941.nlp/math/3a576bee1de26274.svg
• \frac{8}{225}\times\frac{1}{5}=\frac{8}{1125}nlp/math/3a78d82833b21a72.svg
• 0.1(0.6)=0.06nlp/math/3af50c5661f75952.svg
• \alpha_2(C)=[0.32(0.3)+0.02(0.6)](0.5)=0.054.nlp/math/3af92aa3b08aa15a.svg
• 0.1\times0.1=0.01nlp/math/3b2af630e6ad1d43.svg
• \frac{1}{3}\times1=\frac{1}{3}nlp/math/3b61cadbe8860930.svg
• P_{+1}(\text{bright}\mid\text{very})=\frac1{12}\approx0.083333nlp/math/3b6878cf6c2ef959.svg
• s(R)=0.5\times0.1\times0.1=0.005nlp/math/3b8218c2f9695700.svg
• \frac{2}{25}\times\frac{1}{3}=\frac{2}{75}nlp/math/3bb8a5077b206779.svg
• P(\text{cat}\mid\text{the})=\frac{4}{10}=0.4nlp/math/3bf149513726477a.svg
• inlp/math/3c3a2b02c199adb2.svg
• \mathbf f\cdot\mathbf p=(2)(2)+(0)(0)+(0)(1)+(1)(1)=5nlp/math/3c4e8c9a3d5b0e2a.svg
• \mathrm{IDF}(energy)=\ln(3/3)=0nlp/math/3ca0cca554234bcd.svg
• \operatorname{PP}=e^{0.924196}\approx2.519842nlp/math/3d34de424885d00e.svg
• \cos(\mathbf f,\mathbf s)=\frac2{\sqrt{10}}\approx0.632456nlp/math/3d552a0f1e7f606d.svg
• M=3\times3\times3\times3nlp/math/3d7b06b7e39ae0a5.svg
• 91125/2025=45nlp/math/3daee8fcfcc1adea.svg
• P(\text{loan}\mid F)nlp/math/3e3b57b2d1341cd6.svg
• 0.5+0.3+0.2=1.nlp/math/3e929c377d815fdc.svg
• D(i,j)=\min\{D(i-1,j)+1,D(i,j-1)+1,D(i-1,j-1)+[a_i\ne b_j]\}nlp/math/3eb73cb97ea627b1.svg
• \hat w=\arg\max_w P(x\mid w)P(w)nlp/math/3ede23f6f61aee5c.svg
• P(W)=\prod_i p_i,\qquad\operatorname{PP}=P(W)^{-1/3}nlp/math/3f347a2e35262f3f.svg
• P_i=s_i/(s_1+s_2)nlp/math/3f5210a24ca75d45.svg
• P=1/20=0.05nlp/math/3fc24a506f12b888.svg
• R=\frac6{10}=0.6nlp/math/402ac41b5040dfdd.svg
• P_{\mathrm{mix}}(w\mid u,v)=\lambda_3P(w\mid u,v)+\lambda_2P(w\mid v)+\lambda_1P(w)nlp/math/402b0524fed55b5a.svg
• P=\frac{TP}{TP+FP},\quad R=\frac{TP}{TP+FN}nlp/math/409ab1514075c74b.svg
• \frac1{96}\times\frac19\times\frac14=\frac1{3456}nlp/math/40b213c06ef8b7d2.svg
• P(\text{win}\mid H)=P(\text{prize}\mid H)=\frac{0+1}{8}=\frac18nlp/math/4134fa5958e2a1ed.svg
• \sqrt{25}\approx 5.000000.nlp/math/4150d3f3df4a566b.svg
• P=\frac33\times\frac23\times\frac02\times\frac11=0nlp/math/420a50ccd335ade1.svg
• P(a\mid h)=\frac{3+0.5}{5.5}=\frac7{11}\approx0.636364.nlp/math/429115b3bdd79ba5.svg
• \approx[0,0.088046,0.088046]nlp/math/42c4ea9d39b355c1.svg
• P(w_1,\ldots,w_m)=\prod_{i=1}^{m}P(w_i\mid w_1,\ldots,w_{i-1})nlp/math/430725f0e531e2f7.svg
• 0.5(0.4)=0.20,\qquad0.3(0.1)=0.03,\qquad0.2(0.05)=0.01.nlp/math/4318d6a3e52440bd.svg
• 20/100=0.2nlp/math/4373563868df4a11.svg
• \delta_1(H)=0.8(0.2)=0.16nlp/math/44fc29f14406bbca.svg
• 27\times3=81,\quad81\times3=243nlp/math/4502edf461289ab6.svg
• 0.4\times0.5=0.2nlp/math/4528e6e7a97bb8e8.svg
• 0.2\times0.2=0.04nlp/math/45c99ba534e1771d.svg
• 0.2/0.2=1nlp/math/468ec9ecee826ded.svg
• |\cdot|nlp/math/468f3b6d2f1b95a0.svg
• P(j)=e^{z_j}/\sum_k e^{z_k}nlp/math/469b017eccd019fc.svg
• \pi_t=C(START,t)/2,\quad a_{tu}=C(t,u)/C(t),\quad b_t(w)=C(w,t)/C(t)nlp/math/46d6d50083ab5629.svg
• \mathbf f\cdot\mathbf u=(2)(1)+(0)(0)+(0)(1)+(1)(0)=2nlp/math/46e846a5d9fa6557.svg
• F_1=\frac{2PR}{P+R}nlp/math/4743421649dff6a6.svg
• \operatorname{onehot}(w)_j=\begin{cases}1,&j=\operatorname{index}(w)\\0,&\text{otherwise.}\end{cases}nlp/math/47e78a40b0a5b0bd.svg
• P(NN\mid DT)=\frac{56509}{116454}\approx0.485247nlp/math/47e9bc9d8bc16788.svg
• P(is\mid VBZ)=\frac{10073}{21627}\approx0.465760nlp/math/481557575db95324.svg
• VV:0.16\times0.2\times0.6=0.0192nlp/math/4898969d0dee0618.svg
• N=4+3+1=8nlp/math/489d9ca17e456579.svg
• \delta_3(H)=0.03136(0.2)=0.006272nlp/math/48b05bab6b13e38a.svg
• P(\text{shore}\mid F)=\frac{0+1}{23}=\frac1{23}nlp/math/48b17e3adf6f62bb.svg
• \alpha_3(C)=[0.0608(0.3)+0.0108(0.6)](0.5)nlp/math/48f64c6c6ce831b9.svg
• \sqrt[3]{16}\approx 2.519842.nlp/math/498a102c3a9a2537.svg
• r_1(B)=0.4+0.3=0.7nlp/math/4a0be788a6cd2a62.svg
• 0.6/0.05=12nlp/math/4a1735e84fe18a45.svg
• \#\text{positions}=m-n+1nlp/math/4ad44404e3f41cdd.svg
• \mathrm{score}(c)=P(c)\prod_{w\in V}\left(\frac{c(w,c)+1}{N_c+|V|}\right)^{c(w,d)}nlp/math/4ad88d5fb5558250.svg
• P(b\mid h)=\frac{1+0.5}{5.5}=\frac3{11}\approx0.272727.nlp/math/4ad9b8ae622d15d2.svg
• P(F\mid C)=\frac{1/12167}{4/12167}=\frac14=0.25nlp/math/4b082736e405b96c.svg
• \text{score}(s)=|\text{context}\cap\text{gloss}(s)|nlp/math/4b90e7419dc87033.svg
• C\to C:\ 0.10(0.6)=0.060nlp/math/4c0863275b12a91e.svg
• 1(1)+1(-1)=0nlp/math/4c1cadf015f2425f.svg
• C(v_j,d)nlp/math/4c241e4d8abb2427.svg
• (2/3)\times0=0nlp/math/4c86e2ed24bee6fb.svg
• -\ln P/3\approx2.772589/3=0.924196nlp/math/4d6cc3cf68cb5127.svg
• =\max(0.224,0.008)(0.2)=0.0448nlp/math/4db3393304c21c23.svg
• 8+4=12nlp/math/4dcd6d36e73676fe.svg
• d_E(u,v)=\sqrt{\sum_i(u_i-v_i)^2}nlp/math/4df9d01057f7efb0.svg
• P(red)=2/3nlp/math/4e4dbe2f8003f7d7.svg
• P(\text{cat})=\frac{5}{20}=0.25nlp/math/4e5251cd8736de84.svg
• z_{sleep}=0(1)+1(-1)=-1nlp/math/4e7aa48204aedcfe.svg
• P(is\mid VBZ)=C(is,VBZ)/C(VBZ)nlp/math/4ec13ac6a8a9bdae.svg
• 0.16^{0.75}=0.16^{3/4}\approx0.252982nlp/math/4ec2c364222018a7.svg
• 0.01/0.2=0.05nlp/math/4ed5ae8db66197fd.svg
• P(\text{books}\mid\text{read})=\frac6{10}=0.6nlp/math/4f1e78e98a40bd62.svg
• \operatorname{PP}=2^Hnlp/math/50941856f700d45e.svg
• c^*=(c+1)N/(N+V)nlp/math/509478366d59c2ce.svg
• \delta_t(j)=b_j(o_t)\max_i[\delta_{t-1}(i)a_{ij}]nlp/math/50951d9d0cf5dcf6.svg
• 7+5=12nlp/math/519c72927bad6998.svg
• P(I,like,tea)=P(I)P(like\mid I)P(tea\mid I,like)nlp/math/51a05236a3c51a81.svg
• 0.5+0+0-1.386294=-0.886294nlp/math/51c0d80ff79ed2cd.svg
• 3/12=0.25nlp/math/51ccdc5dee01783f.svg
• 0.12\times0.4=0.048nlp/math/52a11bdd907960f4.svg
• d_1=\sum|\Delta_i|nlp/math/52b6164a35837b22.svg
• \delta_1(N)=0.6\times0.5=0.3nlp/math/52ba9de4a33ff58d.svg
• 0.25(0.33)=0.0825nlp/math/536a06c85da5d9e1.svg
• wnlp/math/5388050eb01d765c.svg
• P_{mix}(coffee)=0.20+0.03+0.01=0.24.nlp/math/538b559c97ffd180.svg
• C(\text{cat})=3nlp/math/53df638c65412b2e.svg
• \sqrt{(1)^2+(1)^2}=\sqrt{2}.nlp/math/546bd34545f00851.svg
• \mathrm{TFIDF}(t,d)=\mathrm{TF}(t,d)\,\mathrm{IDF}(t)nlp/math/555e615abfdacbc0.svg
• 0.2\times0.5=0.1nlp/math/5574a2d8fcf3abb3.svg
• P(1,3,1)=0.009376+0.01236=0.021736.nlp/math/55ab2780a51cb820.svg
• S(w\mid h)=\begin{cases}C(h,w)/C(h),&C(h,w)>0\\ \alpha S(w\mid h'),&C(h,w)=0.\end{cases}nlp/math/55b65d9f12d95541.svg
• 0.00009075(0.5)=0.000045375nlp/math/55d45996875f92e6.svg
• \frac{4}{225}\times\frac{1}{5}=\frac{4}{1125}nlp/math/55f13fa2c61c6016.svg
• \operatorname{PP}=16^{1/4}nlp/math/560b6664723185b3.svg
• \alpha_1(H)=0.8(0.2)=0.16,\qquad\alpha_1(C)=0.2(0.5)=0.10.nlp/math/5624ccfc45b5b3b0.svg
• 2533+1446=3979nlp/math/56cd165dd5075aa7.svg
• \delta_1(H)=0.8(0.4)=0.32,\quad\delta_1(C)=0.2(0.4)=0.08nlp/math/56ea5abfdb65b387.svg
• P(C)nlp/math/57aab4348bb4d572.svg
• \alpha=0.75nlp/math/57c3b9fea49de9d2.svg
• u\cdot v=\sum_i u_iv_inlp/math/57c675782454b924.svg
• \widetilde w_jnlp/math/57e45239d1931442.svg
• X_{ij}=C(\text{target }i\text{ immediately followed by }j)nlp/math/584bbdf22ac0ea3c.svg
• P=\frac{\text{relevant retrieved}}{\text{retrieved}},\quad R=\frac{\text{relevant retrieved}}{\text{all relevant}}nlp/math/58a11f3930a42815.svg
• \frac{1}{1/16}=16nlp/math/58b85b07a845828b.svg
• P(\text{noun})=\frac{0.036}{0.084}=\frac37\approx0.428571nlp/math/58bb63e903487b21.svg
• r_{t+1}(j)=\frac{1-d}{N}+d\sum_i r_t(i)P(i\rightarrow j).nlp/math/58c328eed762e371.svg
• =[0.01824+0.00648](0.5)=0.01236.nlp/math/58d25b69e54ff73f.svg
• P(w)=e^{z_w}/\sum_j e^{z_j}nlp/math/5908df64ce67fd12.svg
• 0.14/0.20=0.7nlp/math/593a5443e39d17c3.svg
• 0.2(0.5)(0.6)(0.1)=0.006nlp/math/595d09221813b65c.svg
• P(w\mid t)=C(w,t)/C(t)nlp/math/597b97fa150cfb9b.svg
• v=(v_1+v_2+v_3)/3nlp/math/59b4e5d5faf31ee8.svg
• 0.048\times0.1=0.0048nlp/math/59bb445235ff63a6.svg
• J(A,B)=\frac{|A\cap B|}{|A\cup B|}.nlp/math/59eaa3e10117805d.svg
• P(c\mid h)=\frac{0+0.5}{5.5}=\frac1{11}\approx0.090909.nlp/math/5a6ab3e28bf5bbad.svg
• 0.8(0.2)(0.7)(0.4)(0.3)(0.5)=0.00672nlp/math/5aa4b0fd88f37247.svg
• \mathbf t\cdot\mathbf s=(1)(0)+(1)(1)+(0)(0)+(0)(1)=1nlp/math/5ac5a4d62f663029.svg
• P(green)=0/3=0nlp/math/5ae67848fc965273.svg
• 0.16\times0.25=0.04nlp/math/5bbc9c588899b389.svg
• \frac{1}{5292}\times\frac{4}{23}=\frac{1}{30429}nlp/math/5be8f2428aa87996.svg
• \frac{1}{30429}\times\frac{2}{23}=\frac{2}{699867}nlp/math/5c01b946da36062b.svg
• \ln(e^2)=2nlp/math/5c3429ffd674bcbd.svg
• P(w_1,w_2,w_3)=P(w_1)P(w_2\mid w_1)P(w_3\mid w_1,w_2)nlp/math/5c8b1d21b7b346a9.svg
• z_{sat}=1(2)+0(1)=2nlp/math/5ce861a41af2767b.svg
• 0/N_s=0nlp/math/5d29fa66aa3ef2ee.svg
• P(C\mid F)nlp/math/5d8f091329836633.svg
• [1+3,\,2+0]=[4,2]nlp/math/5e48edec2b0848fd.svg
• P_L=0.8/20=0.04nlp/math/5e8f939253d4321b.svg
• 2-2=0nlp/math/5e9eb71fd28f428c.svg
• \|\mathbf u\|=\sqrt2,\quad\cos(\mathbf f,\mathbf u)=\frac2{\sqrt{10}}\approx0.632456nlp/math/5eb8385d74c2a577.svg
• 0.7+0.1=0.8nlp/math/5f21379b0e25bf90.svg
• 0.252982\times4\approx1.011929nlp/math/5f73059a0e8e07a5.svg
• P(A,B,A)=P(A)P(B\mid A)P(A\mid B)nlp/math/5f8aae6191f91906.svg
• I=0nlp/math/5fd951e2f99cd39c.svg
• X_{ik}nlp/math/6053336a6b0040cd.svg
• H\to H:\ 0.16(0.7)=0.112nlp/math/61623e2cd024bab3.svg
• \|a\|_2=\sqrt{\sum_i a_i^2}nlp/math/61ee9e4ad4caa506.svg
• P(\text{loan}\mid F)=\frac{1+1}{23}=\frac2{23}nlp/math/622a2e615b1ad17f.svg
• P=\frac{2}{20}\times\frac{2}{17}\times\frac{3}{18}\times\frac{1}{18}\times\frac{2}{17}\times\frac{4}{19}nlp/math/62ad7b9fcca8c7b7.svg
• Dnlp/math/62ef1c41dad1fbc8.svg
• \prod_{i=1}^{3}p_i=p_1p_2p_3nlp/math/62eff7a248154cbf.svg
• P(R)=0.4nlp/math/634e8d0183974542.svg
• P(AAA)=P(ABC)=(1/3)^3=1/27nlp/math/6366e364bc9f8ccd.svg
• c^*=(c+1)N/(N+V),\quad P_L=c^*/N,\quad d=c^*/cnlp/math/63cfdcd2de177903.svg
• \sum_{i=1}^{3}p_i=p_1+p_2+p_3nlp/math/63e5285ab6522f55.svg
• P=\frac24\times\frac12\times\frac12\times\frac23\times\frac12nlp/math/6420e3d4b5982a43.svg
• d_2=\sqrt{\sum_i(x_i-y_i)^2}\quad\text{Euclidean}nlp/math/64b5f21c9b008394.svg
• P(B\mid A)=\frac{C(A,B)}{C(A)}nlp/math/650cfba7e65676b7.svg
• \text{Recall}=\frac{TP}{TP+FN}nlp/math/65330df51cf428ee.svg
• s(F)=\frac12\times\frac3{14}\times\frac1{14}=\frac3{392}\approx0.007653061nlp/math/6564e561ec421080.svg
• R=\frac{4}{4+4}=0.500000nlp/math/656d499e455af020.svg
• 3\times3=9,\quad9\times3=27,\quad27\times3=81nlp/math/6570633fef12ecb5.svg
• d=\frac{c^*}{c}\quad(c>0)nlp/math/6597274c03aa7f00.svg
• \frac{2}{\sqrt{5}\sqrt{5}}\approx 0.400000.nlp/math/659dfd20c2e54394.svg
• 25/50=0.5nlp/math/65bda32fec36f0b1.svg
• \|\mathbf p\|=\sqrt{2^2+0^2+1^2+1^2}=\sqrt6nlp/math/65e703346d258ca7.svg
• P=\frac{2}{24}\times\frac{1}{21}\times\frac{1}{21}\times\frac{4}{23}\times\frac{2}{23}\times\frac{2}{21}nlp/math/661dddaa8d668f4e.svg
• \operatorname{score}(R)=\frac12\times\frac2{23}\times\frac3{23}\times\frac1{23}=\frac3{12167}\approx0.00024657.nlp/math/665624fd363dd007.svg
• 45^2=2025nlp/math/672617aa8909800b.svg
• X_i=\sum_k X_{ik},\qquad P_{ik}=\frac{X_{ik}}{X_i}.nlp/math/678c585ed4cafa14.svg
• 0.06+0.14=0.20nlp/math/67c63d0fb7e2a504.svg
• s(c)=P(c)\prod_wP(w\mid c)nlp/math/684c27f96a1d3e08.svg
• 1\times dnlp/math/689036be40a9e9d7.svg
• \operatorname{score}(s)=|C\cap G_s|,\qquad s^*=\arg\max_s\operatorname{score}(s)nlp/math/691fb54513999f0d.svg
• \frac{2}{15}\times\frac{3}{5}=\frac{2}{25}nlp/math/69517c6e474a3048.svg
• s(c)=P(c)P(\text{loan}\mid c)P(\text{money}\mid c)nlp/math/69a207d02fe88fc1.svg
• 0.000045375(0.68)=0.000030855nlp/math/69c2e9e932b1b959.svg
• P(of\mid prime,minister)=8/12=2/3nlp/math/69d53078e2ba6e25.svg
• \alpha_1(j)=\pi_jb_j(o_1)nlp/math/69e9cae75f1bf784.svg
• 1-Jnlp/math/6a1612157fb25e7c.svg
• 3\times3=9,\quad9\times3=27nlp/math/6a2ca9dea8abf7e6.svg
• P=\frac68=0.75nlp/math/6a30a7f6694209de.svg
• F_1=\frac{2TP}{2TP+FP+FN},\quad A=\frac{TP+TN}{TP+FP+FN+TN}nlp/math/6a365832b41faf96.svg
• \|\mathbf s\|=\sqrt{0^2+1^2+0^2+1^2}=\sqrt2nlp/math/6a3ba3507695e199.svg
• 1+1+1+0=3nlp/math/6a517f05661478a7.svg
• \mathrm{score}(s)=|C\cap G_s|,\qquad \hat s=\arg\max_s\mathrm{score}(s)nlp/math/6ad61c84bee7b920.svg
• \frac{1}{9180}\times\frac{2}{17}=\frac{1}{78030}nlp/math/6b756a42abed1715.svg
• \frac24\times\frac12=\frac14,\quad \frac14\times\frac12=\frac18nlp/math/6c021a4ae77c326e.svg
• P(w)=C(w)/Nnlp/math/6ce6bece84751874.svg
• P(F\mid C)nlp/math/6d368ab85d9177c5.svg
• 1+0.2=1.2\ne1.nlp/math/6d9f4f0a09441a66.svg
• s(c)=P(c)\prod_{w\in V}P(w\mid c)^{C(w,d)}nlp/math/6dd9b21835eaff13.svg
• v_{w_i}nlp/math/6e048d845033ae63.svg
• P(F\mid C)=\frac{1/162}{1/162+1/512}=\frac{512}{674}=\frac{256}{337}\approx0.759644.nlp/math/6e05aa52a40b6b55.svg
• H\to H:\ 0.0448(0.7)=0.03136nlp/math/6e2373be2aa769a8.svg
• 2>0nlp/math/6e25b81ec200909a.svg
• 0.00192\times0.2=0.000384nlp/math/6e3b740dc0fd787f.svg
• H\to C:\ 0.16(0.3)=0.048nlp/math/6e9c32ed85546537.svg
• 0.02\times0.1=0.002nlp/math/6ea85a9a0fdfcde1.svg
• P=0.2\times0.5\times0.4nlp/math/6ed38d660fa17e1d.svg
• \delta_3(C)=0.01344(0.5)=0.00672nlp/math/709ea0da7f36d663.svg
• \frac{1}{\sqrt{2}\sqrt{1}}\approx 0.707107.nlp/math/70b251889baa0790.svg
• c^*=(c+1)\frac{N}{N+V},\qquad P_L(w)=\frac{c^*}{N}nlp/math/70d92382a6cc1e9c.svg
• \delta_1(j)=\pi_jb_j(o_1),\quad\delta_t(j)=b_j(o_t)\max_i[\delta_{t-1}(i)a_{ij}]nlp/math/712c98f9c4966f84.svg
• D(1,1)=\min(1+1,\ 1+1,\ 0+0)=0nlp/math/715c07a4ab771469.svg
• N_dnlp/math/716cbce384bea429.svg
• s(F)=0.6\times0.5=0.30nlp/math/72319d32a8913027.svg
• s(B)=0.08\times0.06=0.0048.nlp/math/723c1308f4a70743.svg
• P(\text{win}\mid S)=\frac{3+1}{8}=\frac12nlp/math/724a47033195de19.svg
• D(3,3)=\min(2+1,\ 2+1,\ 1+0)=1nlp/math/725867dfbce579ce.svg
• P=P(I\mid\langle s\rangle)P(like\mid I)P(tea\mid like)P(\langle/s\rangle\mid tea)nlp/math/7363c7dc5cab171f.svg
• P_{ik}nlp/math/737c2c9c574f2ad5.svg
• v_{sentence}=\frac1n\sum_{i=1}^n v_{w_i}nlp/math/739624bae7d9725e.svg
• P(loan\mid R)=(0+1)/8=1/8nlp/math/739f9e9752ebca3d.svg
• N+V=20+5=25nlp/math/749a0f4867a4b3db.svg
• P_{+1}(w\mid h)=\frac{C(hw)+1}{C(h)+|V|}nlp/math/749bec61cf2a8b8e.svg
• 20/80=0.25nlp/math/74c949323dee3d6f.svg
• P(F\mid x)=\frac{0.06}{0.065}=\frac{12}{13}\approx0.923077nlp/math/74cbf2f696cd1b4b.svg
• 0.5\times0.4=0.20nlp/math/7510fe35c0e59838.svg
• \max(0.32(0.7),0.08(0.4))(0.4)nlp/math/75472da37ba442a1.svg
• =[0.112+0.040](0.4)=0.0608.nlp/math/758961ee3abb481e.svg
• P_k(w\mid h)=\frac{C(h,w)+k}{C(h)+kV}nlp/math/75aafa44d0208bdf.svg
• P(\text{dog}\mid\text{the})=\frac{6}{10}=0.6nlp/math/75de664b10c4bbcd.svg
• \cos(\mathbf a,\mathbf b)=\frac{\mathbf a\cdot\mathbf b}{\|\mathbf a\|\|\mathbf b\|}nlp/math/75e740509c95134b.svg
• 0.3+0.7=1nlp/math/76cecf94aaf6078d.svg
• S=-2+3-2=-1nlp/math/76ffe09c2be2d0ce.svg
• 0.5+0.2=0.7nlp/math/7759ec5773e64d72.svg
• 0.8\times0.2=0.16nlp/math/78369e147be76567.svg
• 2^2=4nlp/math/7839fb9a78007523.svg
• \mathbf t\cdot\mathbf b=(1)(1)+(1)(0)+(0)(1)+(0)(0)=1nlp/math/786f2eeb23b20d27.svg
• \operatorname{PP}(W)=\left(\frac1{P(W)}\right)^{1/N}nlp/math/7873b84adddec337.svg
• \cos(u,v)=\frac{\sum_i u_i v_i}{\sqrt{\sum_i u_i^2}\sqrt{\sum_i v_i^2}}nlp/math/789b38cafe4954af.svg
• P(t_j\mid t_i)=\frac{C(t_i,t_j)}{\sum_k C(t_i,t_k)}nlp/math/791852c27a57a918.svg
• 0+0+0+0=0nlp/math/79d13a087a88c477.svg
• P(\text{papers}\mid\text{read})=\frac4{10}=0.4nlp/math/79d625a295e85470.svg
• s(S)=\frac12\times\frac12\times\frac38=\frac{3}{32}=0.09375nlp/math/79f02eee2a0a5ddb.svg
• \cos(a,b)=\frac{\sum_i a_ib_i}{\sqrt{\sum_i a_i^2}\sqrt{\sum_i b_i^2}}nlp/math/79fe3d8fd2840024.svg
• P_L(w\mid I)=\frac{C(I,w)+1}{C(I)+V}nlp/math/7a1c0980897f9361.svg
• P(w_i\mid w_{i-1})=C(w_{i-1},w_i)/C(w_{i-1})nlp/math/7a356598298a4b8a.svg
• \log_b x=y\quad\Longleftrightarrow\quad b^y=xnlp/math/7a41936887c316a1.svg
• jnlp/math/7a5268d94266eaaf.svg
• \pi_N=2/2=1,\quad \pi_V=0/2=0nlp/math/7a5ae0d0edad4515.svg
• P(w\mid h)=\frac{C(h,w)}{C(h)},\qquad P_L(w\mid h)=\frac{C(h,w)+1}{C(h)+12}nlp/math/7a627bcb8a0b3808.svg
• \frac14\times\frac34=\frac3{16}nlp/math/7a7d30f8cd240d72.svg
• D(2,2)=\min(1+1,\ 1+1,\ 0+0)=0nlp/math/7acd2ee646c6109c.svg
• P=\frac23\times\frac12\times1\times\frac25\times\frac35\times\frac13\times\frac23\times\frac15\times\frac25nlp/math/7b611360d98ac780.svg
• A\cdot B=(1)(0)+(1)(1)+(1)(1)+(0)(1)+(0)(0)+(0)(0)=2nlp/math/7b652d0c42b280a1.svg
• \sqrt{(0)^2+(1)^2+(2)^2}=\sqrt{5}.nlp/math/7c552c58c81e3b57.svg
• \operatorname{PP}_B=32^{1/3}\approx3.174802nlp/math/7cad7e80704e87a9.svg
• P(\text{cat}\mid V)=\frac14=0.25nlp/math/7ce20bd4ba2c5e3d.svg
• D(2,2)=\min(1+1,\ 2+1,\ 1+1)=2nlp/math/7cf9b393d47f5d6d.svg
• \operatorname{PP}=e^2\approx7.389056nlp/math/7d421123292e12a1.svg
• P(w_1\mid\langle s\rangle)=\frac{C(\langle s\rangle,w_1)}{C(\langle s\rangle)}nlp/math/7d997b6731cad785.svg
• P=\pi_H b_H(1)a_{HC}b_C(3)a_{CH}b_H(1)nlp/math/7d9c99617b5e16c0.svg
• P=0.4\times0.5\times0.8\times0.25nlp/math/7e1f0a0c2ffc974b.svg
• 1+1+1=3nlp/math/7e22dbf6b1c68d2b.svg
• 5+3=8nlp/math/7e931d91ea22e68e.svg
• \delta_1(j)=\pi_jb_j(3),\quad\delta_2(j)=b_j(1)\max_i[\delta_1(i)a_{ij}]nlp/math/801327bb8fdbd394.svg
• 0.8(0.2)=0.16nlp/math/806228a802bab106.svg
• d_p(a,b)=\left(\sum_i|a_i-b_i|^p\right)^{1/p}nlp/math/80afdfa17b2f9274.svg
• P(w)=C(w)/N,\qquad P_L(w)=(C(w)+1)/(N+V)nlp/math/80d7161647b260e8.svg
• P(w\mid h)=C(h,w)/C(h)nlp/math/8150207b365ec07a.svg
• s(c)=P(c)P(\text{win}\mid c)P(\text{prize}\mid c)nlp/math/81645a009f312045.svg
• \mathrm{TPR}=\frac{TP}{TP+FN},\qquad\mathrm{FPR}=\frac{FP}{FP+TN}nlp/math/81a0419dccf2b651.svg
• \sqrt{(1)^2+(2)^2+(0)^2}=\sqrt{5}.nlp/math/8232d196e4629829.svg
• PP=P^{-1/N}=\left(\frac1P\right)^{1/N}nlp/math/8263823864a08bf0.svg
• df(cat)=df(dog)=df(fish)=2nlp/math/8279f303c7e95ae1.svg
• x_1=\log_{10}36=1.556302501nlp/math/82be852b42963d6b.svg
• 4+(0.5)(3)=5.5.nlp/math/82f30cc20819ed96.svg
• P(D,D)=P(D)P(D\mid D)nlp/math/83080c3651cd8fc6.svg
• \#\text{positions}=4-2+1=3nlp/math/8326144ada817e2c.svg
• \frac{1}{510}\times\frac{1}{18}=\frac{1}{9180}nlp/math/8396eb2fae42ff25.svg
• \operatorname{TTR}=\frac{\text{number of distinct types}}{\text{number of tokens}}.nlp/math/83c58242dcad1838.svg
• P=\pi_H b_H(1)a_{HH}b_H(3)a_{HH}b_H(1)nlp/math/83f5cb754dc77195.svg
• \operatorname{score}(s)=P(s)P(\text{caught}\mid s)P(\text{lake}\mid s)nlp/math/8414ab183de734ec.svg
• P(w\mid s)=\frac{C(w,s)+1}{N_s+V}.nlp/math/8423df4009ba426a.svg
• P_L(like\mid I)=(3+1)/7=4/7nlp/math/848e1d743f230579.svg
• P=\frac{4}{14697207}\approx 2.72160554\times10^{-7}nlp/math/84afc176da8ea9ec.svg
• S(a\mid h)=2/2=1.nlp/math/84f0ae8328abe0b8.svg
• P(sat)\approx0.731059nlp/math/85b24a818aa9d900.svg
• 2\times2\times2\times2\times2=32nlp/math/85ec03497588b6d5.svg
• J(A,B)=\frac{|A\cap B|}{|A\cup B|},\quad d_E(u,v)=\sqrt{\sum_i(u_i-v_i)^2}nlp/math/862c39e0f08fa618.svg
• 0+2+1=3nlp/math/8658ccf788305270.svg
• c^*=(10+1)\frac{20}{25}=11(0.8)=8.8nlp/math/866c6ee174b3d21b.svg
• x\cdot y=a^2+0+0+a^2+0=2a^2nlp/math/8684c855bb57efcc.svg
• \max(0.32(0.3),0.08(0.6))(0.4)nlp/math/87b9dd83765528d1.svg
• 0.036+0.048=0.084nlp/math/884e5c8a756ca87a.svg
• P(N\mid book)=0.12/0.32=0.375nlp/math/886a6346c0609831.svg
• M_{ij}=\sum_s\mathbf1[\text{target}_i\text{ and context}_j\text{ present in }s]nlp/math/888eadfcee94dff3.svg
• P(I\mid\langle s\rangle,\langle s\rangle)=2/3nlp/math/88a613de059c9165.svg
• \alpha_2(H)=[0.16(0.7)+0.10(0.4)](0.4)nlp/math/88c29e3099046db2.svg
• \ln4\approx1.386294nlp/math/88f99752ef835c83.svg
• A=\frac{6+8}{6+2+4+8}=0.700000nlp/math/8934313942e2b516.svg
• 8/12+3/12+1/12=12/12=1nlp/math/8a19e47f93490280.svg
• P_L=\frac4{15}\times\frac3{15}\times\frac1{14}\times\frac2{13}nlp/math/8b4c9f0eec18e2b4.svg
• P=\frac{2}{741285}\approx 2.69801763\times10^{-6}nlp/math/8bf61abdfc6b7956.svg
• x_{d,j}nlp/math/8bfe4f4285f4b225.svg
• L-1nlp/math/8c8ba2cc6117f74c.svg
• h=(v_1+v_2)/2nlp/math/8d14d18bc7e6e64e.svg
• P(t_j\mid t_i)=\frac{C(t_i,t_j)}{\sum_kC(t_i,t_k)}nlp/math/8da165d58e49419c.svg
• \frac{1}{3}\times\frac{2}{5}=\frac{2}{15}nlp/math/8e3868104c41a9da.svg
• 0.2\times0.08=0.016nlp/math/8f15b3cdfefd2ca8.svg
• 1\times\frac{1}{10}=\frac{1}{10}nlp/math/8fc51cfa0d4d18ef.svg
• \ln0.5\approx-0.693147,\quad\ln0.25\approx-1.386294nlp/math/8fc9ba50109d8156.svg
• \cos(A,B)=\frac2{\sqrt3\sqrt3}=\frac23\approx0.666667nlp/math/905c4c3c1ff47cfe.svg
• 0.12+0.002=0.122nlp/math/908f94107aa11373.svg
• \text{Number of unpadded n-gram positions}=\max(0,m-n+1)nlp/math/9097807b6ad98d38.svg
• -\ln P=12nlp/math/90abe91c0eba5423.svg
• P(t_j\mid t_i)=\frac{C(t_i\to t_j)}{\sum_k C(t_i\to t_k)}nlp/math/91497b7e4448d55d.svg
• \frac1{0.0016}=625nlp/math/92106edf250f4af4.svg
• \prod_i x_inlp/math/9213d2b315baaeb2.svg
• snlp/math/92329599456283d6.svg
• P(w)=C(w)/N,\qquad P(I\ like\ tea)=P(I)P(like)P(tea)nlp/math/92c670591895f879.svg
• P(\text{sentence})=\prod P(\text{next token}\mid\text{previous token})nlp/math/92e554afbd43b835.svg
• \operatorname{df}(t)nlp/math/92f4a76a32f5c806.svg
• \lambda_1,\lambda_2,\lambda_3\ge0,\qquad\lambda_1+\lambda_2+\lambda_3=1.nlp/math/934ad91d11cebd33.svg
• \frac3{16}\times\frac14\times\frac29=\frac1{96}nlp/math/934eaa8bf4f9d42a.svg
• \delta_1(t)=P(t)P(w_1\mid t)nlp/math/9376db481e8e0e16.svg
• 0.6\times0.5=0.3nlp/math/93f751cb6f79ddc6.svg
• \operatorname{score}(F)=\frac23\times\frac13\times\frac13\times\frac1{12}=\frac1{162}\approx0.00617284.nlp/math/94333571a54ef7df.svg
• P(Fish\mid C)=0.12/0.122\approx0.983607nlp/math/957c68e55d605895.svg
• 45^3=2025\times45=91125nlp/math/96a0c5c55e4f8048.svg
• P(\text{tea}\mid\text{like})=\frac{1+1}{2+6}=\frac14nlp/math/96b2bf69d275056e.svg
• 3/3=1nlp/math/96b707212e8196a2.svg
• 0.3\times0.5=0.15nlp/math/96f131bd67f1d124.svg
• P(\text{sleeps}\mid V)=\frac12nlp/math/976665ab2b58cdd4.svg
• P(S)=\frac36=0.5,\qquad P(A)=\frac36=0.5nlp/math/979bfbb6524dbd77.svg
• P=\frac{9}{9+3}=0.750000nlp/math/97b392f1812900f2.svg
• P(a\,b\,a)=P(a)P(b)P(a)nlp/math/97d27988b1cff86a.svg
• 0.2\times0=0nlp/math/97ff1a04375e7994.svg
• e^{-1}\approx0.367879nlp/math/985133cf5ad35907.svg
• f(x)=\begin{cases}(x/x_{\max})^\alpha,&0<x<x_{\max}\\1,&x\ge x_{\max}.\end{cases}nlp/math/9889df20ad438723.svg
• 9+14=23nlp/math/989b4387cd367e7a.svg
• 0.075\times0.8=0.06nlp/math/98c43f5a93d1dcf9.svg
• 0.006(0.4)=0.0024nlp/math/99539fe7f4612c73.svg
• P(like\mid\langle s\rangle,I)=2/2=1nlp/math/99919b0a3770ff5c.svg
• 0.3\times0.4=0.12nlp/math/99bd356c706eee9f.svg
• P(F\mid\text{loan})=\frac{0.30}{0.34}=\frac{15}{17}\approx0.882353nlp/math/99c028fbba7fd3e1.svg
• P(w\mid t)=\frac{c(t,w)}{c(t)}nlp/math/9a06539e1ddc3c75.svg
• \mathbf f\cdot\mathbf s=(2)(1)+(0)(1)+(0)(0)+(1)(0)=2nlp/math/9a19db4a92763869.svg
• \log_{10}(8/8)=\log_{10}1=0nlp/math/9a61d3220cb1a861.svg
• P(c\mid x)=\frac{P(x\mid c)P(c)}{P(x)}nlp/math/9afdb53fbf7886dd.svg
• P(ABC)=0.25(0.5)(0.25)=\frac1{32}=0.03125nlp/math/9b87b238251977b6.svg
• \ln P=\sum_i\ln p_i,\quad P=e^{\ln P},\quad\operatorname{PP}=e^{-\ln P/N}nlp/math/9bced9ebe87e44c7.svg
• v_{\text{sentence}}=\frac1n\sum_{i=1}^{n}v_{w_i}.nlp/math/9bd733523385a6f0.svg
• \alpha_t(j)=b_j(o_t)\sum_i\alpha_{t-1}(i)a_{ij}nlp/math/9c7ba7157770eec8.svg
• P=\frac34\times1\times\frac13\times\frac12\times1=\frac18nlp/math/9c8a03a484dfb763.svg
• P(w\mid c)=\frac{C(w,c)+1}{N_c+|V|}nlp/math/9ccf47e73d1f48a3.svg
• 1\times\frac{1}{12}=\frac{1}{12}nlp/math/9cf9ede86d758f1a.svg
• \approx10.107338nlp/math/9d4eb33e3fa2df52.svg
• \operatorname{score}(s)=\sum_{w\in C} I(w,\operatorname{category}(s))nlp/math/9d7b8384c080650b.svg
• 12/6=2nlp/math/9d8a38c0a5d6d98a.svg
• TF=C/N_dnlp/math/9db57785102d90fc.svg
• 1+0+2=3nlp/math/9e9f60c0d5d4390a.svg
• e^2\approx7.389056nlp/math/9ebe22f13d18b3d6.svg
• P(c\mid x)=\frac{s(c)}{s(F)+s(R)}nlp/math/9ff803a2123f521d.svg
• \operatorname{score}(s)=|C\cap G_s|nlp/math/a0d33079bc4a9aaa.svg
• D(1,3)=\min(3+1,\ 1+1,\ 2+1)=2nlp/math/a0d37091a67b769a.svg
• \frac1{3456}\times1\times\frac49=\frac1{7776}nlp/math/a0e9571f3b95594c.svg
• \operatorname{PP}=(10^5)^{1/5}=10nlp/math/a14770a9a80e5984.svg
• \delta_1(C)=0.2(0.5)=0.10nlp/math/a1500f80244db39b.svg
• R=\frac{6}{6+4}=0.600000nlp/math/a1b816258848d9dc.svg
• \|\mathbf f\|=\sqrt{2^2+0^2+0^2+1^2}=\sqrt5,\quad\|\mathbf s\|=\sqrt2nlp/math/a1c307e5c2f60d4e.svg
• P(w\mid h)=\frac{C(h,w)}{C(h)}nlp/math/a1f5e1649537aa36.svg
• F_1=\frac{8}{8+0+4}=0.666667nlp/math/a2397b5600ed6ed2.svg
• 2/40=0.05nlp/math/a23f9c2192e787e1.svg
• dnlp/math/a2ff71daba472383.svg
• \frac{4}{75}\times\frac{2}{3}=\frac{8}{225}nlp/math/a35b29cd9dbe9b05.svg
• P=827/2533\approx0.326490nlp/math/a38462b5efb9f1e3.svg
• \|C\|=\sqrt{1^2+1^2+1^2+1^2}=2nlp/math/a3902ff8f18f763e.svg
• D(2,1)=\min(1+1,\ 2+1,\ 1+1)=2nlp/math/a3ee153834d01ea4.svg
• \ln 0nlp/math/a4450c8290ea350d.svg
• P=\frac12\times\frac38\times\frac12nlp/math/a46bf6ca459b44e0.svg
• \delta_1(j)=\pi_jb_j(2),\quad\delta_2(j)=b_j(2)\max_i[\delta_1(i)a_{ij}]nlp/math/a478f4924f42dec2.svg
• P(V\mid book)=0.20/0.32=0.625nlp/math/a4c885afdd1366d5.svg
• G_snlp/math/a5684e27b28daffa.svg
• 0.6\times0.8=0.48nlp/math/a5e47f480eb58213.svg
• H\to C:\ 0.0448(0.3)=0.01344nlp/math/a5f431d2a23d6fe1.svg
• D(1,2)=\min(2+1,\ 1+1,\ 1+0)=1nlp/math/a6adb2bfe71ef208.svg
• P(w\mid t)=\frac{C(t,w)}{C(t)}nlp/math/a732ea965eb9fa65.svg
• P(said\mid prime,minister)=3/12=1/4nlp/math/a742b6dcdf3bfef5.svg
• P(D\mid start)=P(N\mid D)=P(V\mid N)=P(end\mid V)=\frac22=1nlp/math/a7ee998df3e74fd3.svg
• 1/P(W)=10^5nlp/math/a7f36a9eff1f3a2c.svg
• P(t_j\mid t_i)=C(t_i\to t_j)/\sum_k C(t_i\to t_k)nlp/math/a89cf52a2261bb02.svg
• P(w\mid t)=\frac{C(w,t)}{C(t)}nlp/math/a89d9641a3681129.svg
• \delta_2(C)=\max(0.048,0.060)(0.1)=0.006nlp/math/a8ea78adb0cc4576.svg
• \mathrm{score}(S)=0.5\times\frac2{32}\times\frac2{32}\times\frac2{32}=0.0001220703125nlp/math/a90c8cf82da50b4f.svg
• D=3nlp/math/a912b37fed2477fc.svg
• P_L(love\mid I)=\frac{2+1}{3+12}=\frac3{15}nlp/math/a94897ddface7219.svg
• [(2/3)0.176091,(1/3)0.176091,0]\approx[0.117394,0.058697,0]nlp/math/a969ee9cc56b474e.svg
• \frac{2}{699867}\times\frac{2}{21}=\frac{4}{14697207}nlp/math/aa10a9e9b5ec1f5a.svg
• \log(ab)=\log a+\log bnlp/math/aa22afa6e5e6a3ea.svg
• J_{ij}=f(16)\times2^2nlp/math/aa2cbf1be1ca48e7.svg
• P=0.4\times0.5\times0.2nlp/math/aacb5800d20f6f86.svg
• 0.7\times0.4=0.28nlp/math/aaf2c572f2d6daa0.svg
• \hat e=\arg\max_e P(f\mid e)P(e).nlp/math/ab454138c207cedf.svg
• TF\!\operatorname{-}\!IDF(t,d)=TF(t,d)IDF(t)nlp/math/ab5dadabf0c331ca.svg
• 1\times0^2=0nlp/math/ab6810c2ebb5c0c0.svg
• \mathrm{score}(s)=P(s)\prod_{w\in\mathrm{context}}P(w\mid s)nlp/math/ab6d19b2902f96bb.svg
• P(W)=e^{\ln P(W)},\qquad \operatorname{PP}(W)=\exp\left(-\frac1N\sum_i\ln p_i\right)nlp/math/ab927afa9cd05bef.svg
• [2+1,2+1]=[3,3]nlp/math/ad3021d33d70757e.svg
• \operatorname{score}(y)=P(x\mid y)P(y)nlp/math/ad804659d9b93edc.svg
• s(R)=0.4\times0.1=0.04nlp/math/adf5b8b758b9a296.svg
• \|\mathbf t\|=\sqrt{1^2+1^2+0^2+0^2}=\sqrt2nlp/math/ae259d10b965e91c.svg
• P(W,T)=\pi_{t_1}b_{t_1}(w_1)\prod_{i=2}^{n}a_{t_{i-1},t_i}b_{t_i}(w_i)nlp/math/aed9bbaaf2578ce2.svg
• F_1=\frac{2(0.75)(0.6)}{0.75+0.6}=\frac{0.9}{1.35}=\frac23\approx0.666667nlp/math/aedd722bd9955777.svg
• P_{MLE}(\text{bright}\mid\text{very})=\frac07=0nlp/math/af58dca7b04e5ea2.svg
• 0.03136(0.2)=0.006272nlp/math/af971b25ad6922c0.svg
• P(\text{the}\mid D)=\frac12nlp/math/afa17db444c44e08.svg
• \alpha_3(H)=[0.0608(0.7)+0.0108(0.4)](0.2)nlp/math/b06aab93d1b8d35f.svg
• M_2=45^2,\quad M_3=45^3nlp/math/b0806abcd5d0a43e.svg
• \mathrm{TF}(t,d)=C(t,d),\qquad \mathrm{IDF}(t)=\ln\frac{N}{\mathrm{df}(t)}nlp/math/b0a5cfe1defbb82d.svg
• 4/20=0.2nlp/math/b0de7a9a9a0c728c.svg
• 0.48\times0.2=0.096nlp/math/b0e4bbd812b1ba4d.svg
• P(\text{water}\mid\text{drink})=\frac{C(\text{drink water})}{C(\text{drink})}nlp/math/b16537e98cb2aea8.svg
• I=1nlp/math/b186266d0d476166.svg
• P_{mix}(tea)=0+0.06+0.02=0.08.nlp/math/b1c5370fd614e492.svg
• 0.4\times0.05=0.02nlp/math/b1d1c4fd715bfbf9.svg
• J_{ij}=f(X)(\text{dot}+b+\widetilde b-\ln X)^2nlp/math/b1d9328b46f12df0.svg
• P(F)nlp/math/b22ff6e7eb0a8abb.svg
• 1\times\frac{2}{3}=\frac{2}{3}nlp/math/b32eeb64b6ce8d87.svg
• D(3,2)=\min(0+1,\ 2+1,\ 1+1)=1nlp/math/b374d7c3144e3404.svg
• P=\frac{4}{216}=\frac{1}{54}\approx0.018519nlp/math/b3a5496aef6f0526.svg
• P(drive)\approx0.665241nlp/math/b3ad8b21b03e9eb1.svg
• 1(0.5)+2(0)=0.5nlp/math/b3e5abe1a60dc7f4.svg
• r_1(A)=0.1+0.2=0.3nlp/math/b3f4e8d5ae51ab54.svg
• \delta_1(j)=\pi_jb_j(o_1)nlp/math/b3fa79f3d8232431.svg
• TTR=V/Nnlp/math/b4293b5e8a5a504c.svg
• 12/20=0.6nlp/math/b553cd0f20af7e2a.svg
• \|\mathbf b\|=\sqrt{1^2+0^2+1^2+0^2}=\sqrt2nlp/math/b579713956a7c270.svg
• x_{d,j}=C(v_j,d).nlp/math/b587e59a2f7e4537.svg
• F_1=\frac{18}{18+3+1}=0.818182nlp/math/b58f8dd0b3cf6f35.svg
• Cnlp/math/b5a9729b8c25c0f7.svg
• [3-1,2-0]=[2,2]nlp/math/b6233b70d329ba84.svg
• C\to C:\ 0.006(0.6)=0.0036nlp/math/b69919a26d75a560.svg
• P_{MLE}(w\mid h)=\frac{C(hw)}{C(h)}nlp/math/b6cda4dbf7b89bd7.svg
• 0.4+0.6=1nlp/math/b7387cc7f2626c50.svg
• s(\text{not good})=-2nlp/math/b751e154a1c1b2a0.svg
• [1/3,0,2/3]nlp/math/b75534a4556b2589.svg
• \max_c s(c)nlp/math/b75d6d79f6931032.svg
• IDF=\log_{10}(D/df)nlp/math/b7c5a7e09e1e5ec5.svg
• P=1\times\frac12\times1\times\frac12\times1\times\frac12\times1=\frac18=0.125nlp/math/b84bd52617fb1279.svg
• P=e^{-2.772589}\approx0.0625nlp/math/b886c03a804cf5a4.svg
• P(c)=D_c/Dnlp/math/b91d8ff6433f1da4.svg
• J(A,B)=\frac24=0.5nlp/math/b91de05a3cafb1a8.svg
• P(\text{tea}\mid i,\text{like})=\frac35=0.6nlp/math/b943d3d9935a9541.svg
• \cos(B,C)=\frac1{\sqrt3\cdot2}\approx0.288675nlp/math/b9b3f31b850daf8d.svg
• \frac7{11}+\frac3{11}+\frac1{11}=1.nlp/math/ba980a00aba773f0.svg
• \delta_i(t)=b_t(w_i)\max_s[\delta_{i-1}(s)a_{s,t}]nlp/math/bb085669dbdb35d4.svg
• |G_{P_i}\cap G_{C_j}|nlp/math/bb2259d0f8a3b048.svg
• \alpha_2(H)=[0.32(0.7)+0.02(0.4)](0.2)=0.0464.nlp/math/bb239a1f267ec561.svg
• \frac{1}{252}\times\frac{1}{21}=\frac{1}{5292}nlp/math/bc2636bde4d927d0.svg
• P=\frac13\times1\times\frac13\times1=\frac19\approx0.111111nlp/math/bccffa0a6f23966a.svg
• \cos(u,v)=\frac{u\cdot v}{\|u\|\|v\|}nlp/math/bcd9c35430c84240.svg
• P(w\mid h)=\frac{c(h,w)}{\sum_v c(h,v)}nlp/math/bd18c85b739d1cb7.svg
• \sqrt{(1)^2+(2)^2}=\sqrt{5}.nlp/math/bd2d42d1e7a24550.svg
• \max(0.32(0.3),0.02(0.6))(0.5)nlp/math/bd4ebf48e761f108.svg
• \operatorname{score}(F)=0nlp/math/bdc75dbcd361db69.svg
• C\cap G_1=\varnothing,\qquad \mathrm{score}(1)=0nlp/math/bdca68bf983394ca.svg
• 0.3\times0.25=0.075nlp/math/bdfb9291e6974dcf.svg
• d=8.8/10=0.88nlp/math/be28007774d8c2d5.svg
• P_L=\frac{24}{40950}=\frac4{6825}\approx0.000586081nlp/math/be31ee9de701fcab.svg
• PP=P(W)^{-1/N}nlp/math/be3b71a436d652a8.svg
• NV:0.3\times0.7\times0.6=0.126nlp/math/be98de9bc4ba04ab.svg
• 0.2\times0.6=0.12nlp/math/bef5ede62cecaa67.svg
• \|x\|=\|y\|=\sqrt{2a^2}nlp/math/bfb4b0e406e7642a.svg
• P(\text{loan}\mid R)=\frac{0+1}{23}=\frac1{23}nlp/math/bff44b72ce8cc8e7.svg
• d_2(a,b)=\sqrt{\sum_i(a_i-b_i)^2}nlp/math/bffa849afa7462de.svg
• s(w)=P(x\mid w)P(w)nlp/math/c03956cd222f4789.svg
• \operatorname{PP}=8^{1/5}\approx1.515717nlp/math/c0fbb6bb85c1b79f.svg
• 0.0048\times0.4=0.00192nlp/math/c1146803e8605f2e.svg
• 2^4=16\quad\Rightarrow\quad\operatorname{PP}=2nlp/math/c15dbdf7cfc85ea8.svg
• Lnlp/math/c16b65225745dbd1.svg
• P_k(w\mid h)=\frac{C(h,w)+k}{C(h)+kV},\qquad k>0.nlp/math/c1743621aeed43c4.svg
• P(tea\mid\langle s\rangle)=0/3=0nlp/math/c17c8267050aec19.svg
• P(\text{cat}\mid\text{the})nlp/math/c19161969f7b5db0.svg
• \cos(\mathbf f,\mathbf p)=\frac5{\sqrt{30}}\approx0.912871nlp/math/c1f1082cbeeb42e0.svg
• 0.016>0.004nlp/math/c236ae69fa45b6c2.svg
• PP=27^{1/3}=3nlp/math/c2e12c89ac512fd3.svg
• 0.8(0.2)(0.7)(0.4)=0.0448nlp/math/c316d3e7d10ea4e8.svg
• \operatorname{TFIDF}(t,d)=\operatorname{TF}(t,d)\operatorname{IDF}(t).nlp/math/c3205280fe205579.svg
• \text{Accuracy}=\frac{TP+TN}{TP+TN+FP+FN}nlp/math/c35bcbb3c320de01.svg
• 3+1=4nlp/math/c37463b701447787.svg
• P=\pi_C b_C(1)a_{CC}b_C(3)a_{CH}b_H(1)nlp/math/c39dbc8fa9b14a01.svg
• \frac{1}{78030}\times\frac{4}{19}=\frac{2}{741285}nlp/math/c4371974205f854f.svg
• P(R,D,R)=P(R)P(D\mid R)P(R\mid D)nlp/math/c4b2bba9bbf4c148.svg
• P(\text{water}\mid\text{drink})=\frac68=0.75nlp/math/c4bcc480ea0098b5.svg
• P=P(i\mid START)P(want\mid i)P(english\mid want)P(food\mid english)P(END\mid food)nlp/math/c573bb6be6835221.svg
• P(\text{coffee}\mid i,\text{like})=\frac25=0.4nlp/math/c5f56302282fa692.svg
• A=\frac{TP+TN}{TP+TN+FP+FN}nlp/math/c65e0cb33ac1ccf1.svg
• \max(T-n+1,0)nlp/math/c6f5691757b3dce9.svg
• P(\text{shore}\mid R)=\frac{2+1}{23}=\frac3{23}nlp/math/c71632f8b691648b.svg
• \capnlp/math/c7de95563fc530c3.svg
• \cos(x,y)=\frac{\sum_i x_i y_i}{\sqrt{\sum_i x_i^2}\sqrt{\sum_i y_i^2}}.nlp/math/c7fc299613856e1a.svg
• P(loan\mid F)=(3+1)/12=1/3nlp/math/c83ba89d49bb7c55.svg
• P(\mathbf w,\mathbf t)=P(t_1\mid\mathrm{Start})\prod_{i=1}^{n}P(w_i\mid t_i)\prod_{i=2}^{n}P(t_i\mid t_{i-1})P(\mathrm{End}\mid t_n)nlp/math/c8c4a0c66e131a8f.svg
• s(c)=P(c)P(\text{loan}\mid c)nlp/math/c8e3aa7e814574bf.svg
• P=\frac3{32}=0.09375nlp/math/c91fa325852e9b20.svg
• P(s_t\mid s_1,\ldots,s_{t-1})=P(s_t\mid s_{t-1})nlp/math/c9611ed107ec9047.svg
• P(a)=4/8=1/2,\qquad P(b)=3/8nlp/math/c986ed6d03f3c421.svg
• P_L=828/3979\approx0.208092nlp/math/ca005fa10935e238.svg
• P(R\mid D)=0.2,\quad0.28(0.2)=0.056nlp/math/ca3a09e7de1ba004.svg
• e^1\approx2.718282nlp/math/ca9caa03620bfec2.svg
• \operatorname{score}(F)=\frac12\times\frac1{23}\times\frac1{23}\times\frac2{23}=\frac1{23^3}=\frac1{12167}\approx0.00008219.nlp/math/ca9dd31ae9d5dc12.svg
• 1\times Vnlp/math/cad69b71ff59db43.svg
• P(river\mid R)=(2+1)/8=3/8nlp/math/cb1959ca98dd60bb.svg
• TTR=3/5=0.6nlp/math/cbe1c80ba427d405.svg
• \ln P(W)=\sum_{i=1}^{N}\ln p_inlp/math/cbf31ffd35c5c322.svg
• 0.0825(0.0011)=0.00009075nlp/math/cc8c5266050b1825.svg
• D(2,3)=\min(2+1,\ 2+1,\ 1+0)=1nlp/math/ccb8721250bf5a16.svg
• P(want\mid i)=827/2533,\qquad P_L(want\mid i)=\frac{827+1}{2533+1446}nlp/math/ccfa3c73248a102f.svg
• P(w\mid h)=C(h,w)/C(h),\qquad\operatorname{PP}=P(W)^{-1/5}nlp/math/cdcbde9621e0e24d.svg
• P(A)=\frac{C(A)}{N}nlp/math/cde1f475268fa5b4.svg
• 0.5\times0.8=0.4nlp/math/ce0a35fc09c4be55.svg
• \frac{10}{\sqrt{5}\sqrt{20}}\approx 1.000000.nlp/math/ce85a11409a92a5a.svg
• PP(AAA)=64^{1/3}=4nlp/math/ced7ad162e418ee6.svg
• P(D)=0.6nlp/math/cf847b7db44f5fad.svg
• P_L(\langle/s\rangle\mid Python)=\frac{1+1}{1+12}=\frac2{13}nlp/math/cfdc24dda127984e.svg
• s(e)=P(f\mid e)P(e).nlp/math/cff4fc3f67bedf64.svg
• D(4,2)=\min(1+1,\ 3+1,\ 2+1)=2nlp/math/cff6b6d394377532.svg
• r_t(i)nlp/math/d0ac5d68953e63fb.svg
• IDF(t)=\log_{10}\frac{D}{df(t)}nlp/math/d11da5cd42950e34.svg
• 0.1\times0.4=0.04nlp/math/d17f98269b12c97d.svg
• P(F\mid C)=\frac{P(C\mid F)P(F)}{P(C)}.nlp/math/d2028c752e4c5347.svg
• \hat y=\arg\max_y P(y\mid x)=\arg\max_y P(x\mid y)P(y)nlp/math/d214e55148d91eec.svg
• \hat c=\arg\max_c s(c)nlp/math/d264cbc0f9fb7e13.svg
• \hat c=\arg\max_c P(c)\prod_iP(w_i\mid c)nlp/math/d28d2ec54104fe0c.svg
• 0.0448(0.7)=0.03136nlp/math/d2be9254f3d2eee1.svg
• x_{\max}=100nlp/math/d2d81040be91ac39.svg
• C(I)+V=4+3=7nlp/math/d3c1688373da86b2.svg
• D(3,1)=\min(1+1,\ 3+1,\ 2+1)=2nlp/math/d443351dab39e649.svg
• P=\frac{16}{5625}\approx0.0028444444nlp/math/d4673af7839c15e4.svg
• \cos(\mathbf a,\mathbf b)=\frac{\sum_j a_jb_j}{\sqrt{\sum_j a_j^2}\sqrt{\sum_j b_j^2}}nlp/math/d47266edc91f0bb5.svg
• P=\frac49\approx0.444444nlp/math/d5107e2738f0177f.svg
• P(\text{loan})=0.30+0.04=0.34nlp/math/d526374c15c8e154.svg
• \arg\max_c s(c)nlp/math/d6495ab774b17309.svg
• P(w,t)=P(t_1\mid start)P(w_1\mid t_1)P(t_2\mid t_1)P(w_2\mid t_2)P(end\mid t_2)nlp/math/d74602d45a8f6006.svg
• PP=625^{1/4}=5nlp/math/d84b412c777e5d8e.svg
• P(Music\mid C)=0.002/0.122\approx0.016393.nlp/math/d8a1b99447a84030.svg
• TTR=\frac{\#\text{types}}{\#\text{tokens}}nlp/math/d8b9766d1f538431.svg
• P(w_i\mid w_{i-2},w_{i-1})=C(w_{i-2},w_{i-1},w_i)/C(w_{i-2},w_{i-1})nlp/math/d95e106d3b255b6a.svg
• \text{Precision}=\frac{TP}{TP+FP}nlp/math/d9fa10a04601a634.svg
• P(i\mid\langle s\rangle)=\frac{2+1}{2+6}=\frac38nlp/math/d9fa36d9af3de632.svg
• 0.096\times0.3=0.0288nlp/math/da540845c4cc326e.svg
• \operatorname{PP}(W)=P(W)^{-1/N}=\sqrt[N]{\frac1{P(W)}}nlp/math/dab12face5c56858.svg
• s(R)=\frac12\times\frac1{14}\times\frac4{14}=\frac4{392}\approx0.010204082nlp/math/db5c659ae2e27a2e.svg
• D(i,j)=\min(D(i-1,j)+1,D(i,j-1)+1,D(i-1,j-1)+c)nlp/math/dbb6791fed13f314.svg
• P(W)=0.1^5=0.00001=10^{-5}nlp/math/dbea309801bb43da.svg
• c^*=(c+1)N/(N+V),\qquad P_L=c^*/Nnlp/math/dc6d3ca40a034b28.svg
• \operatorname{IDF}(t)=\log_{10}\frac{D}{\operatorname{df}(t)}nlp/math/dd1eb7f26551c5f0.svg
• 0+1=1nlp/math/dd66590db88ae5fa.svg
• score(R)=2nlp/math/dd672eb60c452a19.svg
• A\in T[i,j]\quad\text{if}\quad A\to BC,\ B\in T[i,k],\ C\in T[k+1,j],\ i\le k<j.nlp/math/de5d4255c3f9d98c.svg
• \operatorname{TF}(t,d)=\frac{C(t,d)}{N_d}nlp/math/deca468dd207619e.svg
• P(\text{stream}\mid F)=\frac{0+1}{23}=\frac1{23}nlp/math/dece6048e91d14ee.svg
• A=\frac{9+7}{9+3+1+7}=0.800000nlp/math/df21b30335e9b090.svg
• P(w\mid s)=\frac{C(w,s)+1}{N_s+V}nlp/math/df7198279b2281ba.svg
• \arg\maxnlp/math/dfccac0c4f581832.svg
• 10/2=5nlp/math/e03ff475f4b9e538.svg
• c^*=(0+1)(20/25)=0.8nlp/math/e073545bb6c88eb4.svg
• M=2^5nlp/math/e0b25a2374d09c2d.svg
• D(1,1)=\min(1+1,\ 1+1,\ 0+1)=1nlp/math/e16686e02e9f18c0.svg
• \sum_i x_inlp/math/e1e73674daae58eb.svg
• s(A)=0.12\times0.03=0.0036.nlp/math/e1fe9eb8c63ff3f5.svg
• P(\text{path,words})=0nlp/math/e24e316f36777dd5.svg
• P(o_1,o_2)=\sum_{s_1,s_2}\pi_{s_1}b_{s_1}(o_1)a_{s_1s_2}b_{s_2}(o_2)nlp/math/e24e591ebf359dce.svg
• X_{ij}nlp/math/e26c85d7637ed303.svg
• 1/3\approx0.333333nlp/math/e3565e86f7464cf8.svg
• P_{\mathrm{Laplace}}(w\mid h)=\frac{C(h,w)+1}{C(h)+V}nlp/math/e38a73cc66cdd3f1.svg
• P=0.5\times0.6\times0.25\times0.8nlp/math/e3c41e56b889ac9f.svg
• P_B=0.5\times0.25\times0.25=0.03125=1/32nlp/math/e4c44123baab46cd.svg
• P(s_1,\ldots,s_T)=P(s_1)\prod_{t=2}^{T}P(s_t\mid s_{t-1})nlp/math/e50f813caaa8d825.svg
• X_inlp/math/e51016ac22dc64d1.svg
• P(we\mid\langle s\rangle)=2/8=0.25nlp/math/e61e15f5786a3aaa.svg
• \sqrt{(2)^2+(4)^2}=\sqrt{20}.nlp/math/e6c0085ce3b598a8.svg
• 1.5+0.2+0.3=2nlp/math/e84d0514f15f7e73.svg
• P=\frac{2}{6}\times\frac{2}{6}\times\frac{1}{6}nlp/math/e8823227ec8a6a1a.svg
• P(t_{1:T},w_{1:T})=\pi_{t_1}b_{t_1}(w_1)\prod_{k=2}^{T}a_{t_{k-1},t_k}b_{t_k}(w_k)nlp/math/e8849ce22bc4d03e.svg
• P(\langle/ s\rangle\mid\text{tea})=\frac{1+1}{1+6}=\frac27nlp/math/e8b1fabbba875fd8.svg
• P(w\mid prime,minister)=C(prime,minister,w)/12nlp/math/e8ba899fcd0d6112.svg
• 0.06/0.20=0.3nlp/math/e8d4386c048d353e.svg
• \frac{2}{25}\times\frac{2}{3}=\frac{4}{75}nlp/math/e9a1944302b07ad1.svg
• \frac18\times\frac23=\frac1{12},\quad \frac1{12}\times\frac12=\frac1{24}nlp/math/e9b12d6d627642db.svg
• D(2,3)=\min(2+1,\ 0+1,\ 1+1)=1nlp/math/e9bf43ed11bd7fac.svg
• P(c_1c_2c_3)=P(c_1)P(c_2)P(c_3),\qquad PP=P(W)^{-1/3}nlp/math/e9f6c28675d3dd90.svg
• 0.2(0.5)=0.1nlp/math/ea1ebd665b713576.svg
• 0.6+0.4=1nlp/math/ea2d73cc17f05950.svg
• \operatorname{PP}_A=8^{1/3}=2nlp/math/eaa36684164a8ec6.svg
• a\cdot b=\sum_i a_ib_inlp/math/eb860722f2212f6a.svg
• P(loan\mid F)=\frac3{14},\quad P(water\mid F)=\frac1{14}nlp/math/eb9dda37004ceb2f.svg
• 3/6=0.5nlp/math/ebc0c6fa20ed6a98.svg
• 0.5\times0.6=0.3nlp/math/ec4de7a9661ab386.svg
• z_{drive}=1(1)+0(-1)=1nlp/math/ec8f12c38d59cf41.svg
• 2a^2nlp/math/eca0d1bd6ebd3288.svg
• N=3+3=6nlp/math/ece835ece0dea071.svg
• K^Tnlp/math/ed1eadf4591c422c.svg
• P(river\mid F)=(0+1)/12=1/12nlp/math/ed4e2c4c1b9881ea.svg
• P_L(study\mid I)=(1+1)/7=2/7nlp/math/ed5edd3a14936914.svg
• e^0=1nlp/math/edbc4e3eb26d26ce.svg
• \delta_2(H)=\max(0.112,0.040)(0.4)=0.0448nlp/math/ee04f7fda3bf226b.svg
• \cos(\mathbf t,\mathbf s)=\frac1{\sqrt2\sqrt2}=\frac12=0.5nlp/math/eea3730ff7194b51.svg
• R=\frac{9}{9+1}=0.900000nlp/math/eec2430e01f99107.svg
• s(\text{ten})=0.01\times0.01=0.0001nlp/math/ef19fb34efc392f6.svg
• D(i,j)=\min\begin{cases}D(i-1,j)+1\\D(i,j-1)+1\\D(i-1,j-1)+[a_i\ne b_j]\end{cases}nlp/math/ef2777a0390860c8.svg
• D(2,3)=\min(2+1,\ 1+1,\ 1+1)=2nlp/math/ef96295cbb47c0f8.svg
• R=P(gas\mid ice)/P(gas\mid steam)nlp/math/efe28e7e3bde3014.svg
• P(ran)\approx0.268941nlp/math/efe53e94b12751a6.svg
• P_L(Python\mid love)=\frac{0+1}{2+12}=\frac1{14}nlp/math/f023b28c09df9a1a.svg
• v_jnlp/math/f032d29fd7d0f0b5.svg
• 0.16(0.7)=0.112nlp/math/f1106c30e00cc2ce.svg
• P=\frac{4}{4+0}=1.000000nlp/math/f133094efa30d271.svg
• PP(ABC)=32^{1/3}\approx3.174802nlp/math/f1633e91cb3db49f.svg
• \operatorname{PP}=\exp(-\ln P(W)/N)nlp/math/f166d1cce0943ce3.svg
• P_L(I\mid\langle s\rangle)=\frac{3+1}{3+12}=\frac4{15}nlp/math/f16fc85077f5670b.svg
• 6+8=14nlp/math/f1b9154da4378a4a.svg
• D(4,3)=\min(0+1,\ 2+1,\ 1+1)=1nlp/math/f23dc80f45d4021b.svg
• P(book)=0.12+0.20=0.32nlp/math/f24842cc6cea4777.svg
• V\times dnlp/math/f3392b41ba4b84ba.svg
• P=\frac{TP}{TP+FP},\qquad R=\frac{TP}{TP+FN}nlp/math/f39c3956a7771d5d.svg
• s(F)=0.5\times0.4\times0.3=0.06nlp/math/f3df697a20d8a380.svg
• \|B\|=\sqrt{1^2+1^2+1^2}=\sqrt3nlp/math/f3f38c6241a4b948.svg
• 0+0+0=0nlp/math/f4ccd7a1f1f516c9.svg
• P(NN\mid DT)=C(DT,NN)/C(DT)nlp/math/f5163e6a68004bc8.svg
• \approx[0.058697,0,0.117394]nlp/math/f530e35b75352a92.svg
• 0.5\times0.2=0.1nlp/math/f5b0bcf0ed9035b5.svg
• \cos(x,y)=(x\cdot y)/(\|x\|\|y\|)nlp/math/f5ba4ff6b359de70.svg
• P_L=8.8/20=0.44nlp/math/f5fb26f5353edaa9.svg
• P(I\mid\langle s\rangle)=6/8=0.75nlp/math/f61ebf6be6e02d19.svg
• P(w\mid h)=\frac{C(hw)+1}{C(h)+|V|}nlp/math/f6db957bd553fd41.svg
• d_3=\sqrt[3]{\sum|\Delta_i|^3}nlp/math/f6e68c062a394495.svg
• P(\text{cat})nlp/math/f74362866e81db35.svg
• P(\text{like}\mid i)=\frac38nlp/math/f7bf3c73f1ebb6dc.svg
• P=P(we\mid s,s)P(study\mid s,we)P(NLP\mid we,study)P(end\mid study,NLP)nlp/math/f7ce4e6b173900c6.svg
• R=P(k\mid ice)/P(k\mid steam)nlp/math/f8639f057ec07b30.svg
• 0.25\times0.698970\approx0.174743nlp/math/f8ed981eba287429.svg
• S=\sum_i s_inlp/math/f9356cb0249ff14a.svg
• =\max(0.224,0.032)(0.4)=0.0896nlp/math/f9a0610a246e0ed0.svg
• 1/100=0.01nlp/math/fa188fed61277d35.svg
• F_1=\frac{2PR}{P+R}=\frac{2TP}{2TP+FP+FN}nlp/math/fa461c3c89d25a66.svg
• \mathrm{IDF}(solar)=\mathrm{IDF}(wind)=\mathrm{IDF}(policy)=\ln3\approx1.098612nlp/math/faa332781751eb0a.svg
• (-0.886294)^2\approx0.785518nlp/math/faebe1abd20fb32f.svg
• P_{mix}=\lambda_3P_3+\lambda_2P_2+\lambda_1P_1nlp/math/fbe778721993c743.svg
• P(\text{prize}\mid S)=\frac{2+1}{8}=\frac38nlp/math/fc2cdb5225a070b5.svg
• z_{ran}=0(2)+1(1)=1nlp/math/fca7b5321c9a075e.svg
• \ln P=-0.693147-1.386294-0.693147\approx-2.772589nlp/math/fcdc8edcec916f1f.svg
• PP=16^{1/4}=2nlp/math/fcf353f0637dd633.svg
• 0.2\times0.8=0.16nlp/math/fd4748e690ad03d9.svg
• \mathrm{TFIDF}(w,d)=c(w,d)\ln\frac{N}{\mathrm{df}(w)}nlp/math/fd85ae95ce50c427.svg
• b_V(can)=C(can,V)/C(V)=0/4=0nlp/math/fdca0f8e7f8d838a.svg
• =\log_{10}(3/2)\approx0.176091nlp/math/fdd435a4274ddcc4.svg
• P(sleep)\approx0.090031nlp/math/fe584f91a1ab5142.svg
• d_1=\sum_i|x_i-y_i|\quad\text{Manhattan}nlp/math/fe89867dfd286879.svg
• 0.28\times0.3=0.084nlp/math/fea2bef4fcebf46d.svg
• \cos(x,y)=2a^2/(2a^2)=1nlp/math/fea488d21ebb7221.svg
• \operatorname{score}(R)=\frac13\times\frac18\times\frac18\times\frac38=\frac1{512}\approx0.001953125.nlp/math/ffa63ee974248624.svg
• J=|A\cap B|/|A\cup B|nlp/math/ffa98f175c301d1c.svg