From 0147f627672acafb284ddd3fa3e0ad31a541318b Mon Sep 17 00:00:00 2001 From: Morten Hjorth-Jensen Date: Thu, 21 Oct 2021 08:54:07 +0200 Subject: [PATCH] more updates --- doc/pub/week42/html/._week42-bs000.html | 67 +++--- doc/pub/week42/html/._week42-bs001.html | 67 +++--- doc/pub/week42/html/._week42-bs002.html | 67 +++--- doc/pub/week42/html/._week42-bs003.html | 67 +++--- doc/pub/week42/html/._week42-bs004.html | 67 +++--- doc/pub/week42/html/._week42-bs005.html | 67 +++--- doc/pub/week42/html/._week42-bs006.html | 67 +++--- doc/pub/week42/html/._week42-bs007.html | 67 +++--- doc/pub/week42/html/._week42-bs008.html | 67 +++--- doc/pub/week42/html/._week42-bs009.html | 67 +++--- doc/pub/week42/html/._week42-bs010.html | 67 +++--- doc/pub/week42/html/._week42-bs011.html | 67 +++--- doc/pub/week42/html/._week42-bs012.html | 67 +++--- doc/pub/week42/html/._week42-bs013.html | 67 +++--- doc/pub/week42/html/._week42-bs014.html | 67 +++--- doc/pub/week42/html/._week42-bs015.html | 67 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doc/pub/week42/html/._week42-bs072.html | 88 ++++---- doc/pub/week42/html/._week42-bs073.html | 121 +++++------ doc/pub/week42/html/._week42-bs074.html | 109 +++++----- doc/pub/week42/html/._week42-bs075.html | 108 +++++----- doc/pub/week42/html/week42-bs.html | 67 +++--- doc/pub/week42/html/week42-reveal.html | 149 +++++++++++++- doc/pub/week42/html/week42-solarized.html | 150 +++++++++++++- doc/pub/week42/html/week42.html | 150 +++++++++++++- doc/pub/week42/ipynb/ipynb-week42-src.tar.gz | Bin 87691 -> 87691 bytes doc/pub/week42/ipynb/week42.ipynb | 204 ++++++++++++++++++- doc/src/week42/week42.do.txt | 131 +++++++++++- 83 files changed, 4231 insertions(+), 2334 deletions(-) diff --git a/doc/pub/week42/html/._week42-bs000.html b/doc/pub/week42/html/._week42-bs000.html index cbef1a23a..1ddf0f142 100644 --- a/doc/pub/week42/html/._week42-bs000.html +++ b/doc/pub/week42/html/._week42-bs000.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -424,7 +439,7 @@ MathJax.Hub.Config({
  • 9
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  • »
    • diff --git a/doc/pub/week42/html/._week42-bs001.html b/doc/pub/week42/html/._week42-bs001.html index 114bc3739..8c42e96c8 100644 --- a/doc/pub/week42/html/._week42-bs001.html +++ b/doc/pub/week42/html/._week42-bs001.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -433,7 +448,7 @@ MathJax.Hub.Config({
  • 10
  • 11
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    • -
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  • »
    • diff --git a/doc/pub/week42/html/._week42-bs002.html b/doc/pub/week42/html/._week42-bs002.html index 541095256..3f799bd7c 100644 --- a/doc/pub/week42/html/._week42-bs002.html +++ b/doc/pub/week42/html/._week42-bs002.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -405,7 +420,7 @@ we will also study the usage of 11
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    • diff --git a/doc/pub/week42/html/._week42-bs003.html b/doc/pub/week42/html/._week42-bs003.html index dc7f4212c..f81d5af10 100644 --- a/doc/pub/week42/html/._week42-bs003.html +++ b/doc/pub/week42/html/._week42-bs003.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -422,7 +437,7 @@ and output layer to any given precision.
  • 12
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    • diff --git a/doc/pub/week42/html/._week42-bs004.html b/doc/pub/week42/html/._week42-bs004.html index 92d65523f..be2683862 100644 --- a/doc/pub/week42/html/._week42-bs004.html +++ b/doc/pub/week42/html/._week42-bs004.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -425,7 +440,7 @@ for the solution to be unique.
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    • diff --git a/doc/pub/week42/html/._week42-bs005.html b/doc/pub/week42/html/._week42-bs005.html index 3158ec573..b1f0e562f 100644 --- a/doc/pub/week42/html/._week42-bs005.html +++ b/doc/pub/week42/html/._week42-bs005.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -430,7 +445,7 @@ As described previously, an optimization method could be used to minimize the pa
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    • diff --git a/doc/pub/week42/html/._week42-bs006.html b/doc/pub/week42/html/._week42-bs006.html index 28822d027..25d9632ec 100644 --- a/doc/pub/week42/html/._week42-bs006.html +++ b/doc/pub/week42/html/._week42-bs006.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -432,7 +447,7 @@ The neural net should then find the parameters \( P \) that minimizes the cost f
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    • diff --git a/doc/pub/week42/html/._week42-bs007.html b/doc/pub/week42/html/._week42-bs007.html index d89802fd6..eb11e62e5 100644 --- a/doc/pub/week42/html/._week42-bs007.html +++ b/doc/pub/week42/html/._week42-bs007.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -414,7 +429,7 @@ Automatic differentiation is a method of finding the derivatives numerically wit
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    • diff --git a/doc/pub/week42/html/._week42-bs008.html b/doc/pub/week42/html/._week42-bs008.html index 5e5562f23..63664794b 100644 --- a/doc/pub/week42/html/._week42-bs008.html +++ b/doc/pub/week42/html/._week42-bs008.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -432,7 +447,7 @@ Having an analytical solution at hand, it is possible to use it to compare how w
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    • diff --git a/doc/pub/week42/html/._week42-bs009.html b/doc/pub/week42/html/._week42-bs009.html index 47e7f704a..32cd3dde7 100644 --- a/doc/pub/week42/html/._week42-bs009.html +++ b/doc/pub/week42/html/._week42-bs009.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -423,7 +438,7 @@ In this example, \( \gamma = 2 \) and \( g_0 = 10 \).
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    • diff --git a/doc/pub/week42/html/._week42-bs010.html b/doc/pub/week42/html/._week42-bs010.html index 0e1ed0f8e..b7a619ba5 100644 --- a/doc/pub/week42/html/._week42-bs010.html +++ b/doc/pub/week42/html/._week42-bs010.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -417,7 +432,7 @@ with \( h_1(x) \) ensuring that \( g_t(x) \) satisfies some conditions and \( h_
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    • diff --git a/doc/pub/week42/html/._week42-bs011.html b/doc/pub/week42/html/._week42-bs011.html index db4c0b075..a18ce572f 100644 --- a/doc/pub/week42/html/._week42-bs011.html +++ b/doc/pub/week42/html/._week42-bs011.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -429,7 +444,7 @@ $$
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    • diff --git a/doc/pub/week42/html/._week42-bs012.html b/doc/pub/week42/html/._week42-bs012.html index 5eebd5097..e778a69fd 100644 --- a/doc/pub/week42/html/._week42-bs012.html +++ b/doc/pub/week42/html/._week42-bs012.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -438,7 +453,7 @@ is fulfilled as best as possible.
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    • diff --git a/doc/pub/week42/html/._week42-bs013.html b/doc/pub/week42/html/._week42-bs013.html index 62de6019e..75b58766b 100644 --- a/doc/pub/week42/html/._week42-bs013.html +++ b/doc/pub/week42/html/._week42-bs013.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -434,7 +449,7 @@ for an input value \( x \).
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    • diff --git a/doc/pub/week42/html/._week42-bs014.html b/doc/pub/week42/html/._week42-bs014.html index c8cffe1e2..1cb77319e 100644 --- a/doc/pub/week42/html/._week42-bs014.html +++ b/doc/pub/week42/html/._week42-bs014.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -432,7 +447,7 @@ $$
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    • diff --git a/doc/pub/week42/html/._week42-bs015.html b/doc/pub/week42/html/._week42-bs015.html index dcad8812e..9e12143db 100644 --- a/doc/pub/week42/html/._week42-bs015.html +++ b/doc/pub/week42/html/._week42-bs015.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -417,7 +432,7 @@ The input layer will consist of \( N_{\text{input} } \) neurons, passing its ele
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    • diff --git a/doc/pub/week42/html/._week42-bs016.html b/doc/pub/week42/html/._week42-bs016.html index d56208013..24cce3e51 100644 --- a/doc/pub/week42/html/._week42-bs016.html +++ b/doc/pub/week42/html/._week42-bs016.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -426,7 +441,7 @@ $$
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    • diff --git a/doc/pub/week42/html/._week42-bs017.html b/doc/pub/week42/html/._week42-bs017.html index b770e5d47..1b0c8b3d2 100644 --- a/doc/pub/week42/html/._week42-bs017.html +++ b/doc/pub/week42/html/._week42-bs017.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -427,7 +442,7 @@ $$
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    • diff --git a/doc/pub/week42/html/._week42-bs018.html b/doc/pub/week42/html/._week42-bs018.html index b16c86f2a..053b9457b 100644 --- a/doc/pub/week42/html/._week42-bs018.html +++ b/doc/pub/week42/html/._week42-bs018.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -442,7 +457,7 @@ it is assumes that the number of neurons in the output layer is one.
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    • diff --git a/doc/pub/week42/html/._week42-bs019.html b/doc/pub/week42/html/._week42-bs019.html index 36913c414..4a0597e20 100644 --- a/doc/pub/week42/html/._week42-bs019.html +++ b/doc/pub/week42/html/._week42-bs019.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -425,7 +440,7 @@ $$
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    • diff --git a/doc/pub/week42/html/._week42-bs020.html b/doc/pub/week42/html/._week42-bs020.html index b67affdde..3e0f7b25e 100644 --- a/doc/pub/week42/html/._week42-bs020.html +++ b/doc/pub/week42/html/._week42-bs020.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -426,7 +441,7 @@ In this case we seek a continuous range of values since we are approximating a f
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    • diff --git a/doc/pub/week42/html/._week42-bs021.html b/doc/pub/week42/html/._week42-bs021.html index 145c04b4a..62cdc4222 100644 --- a/doc/pub/week42/html/._week42-bs021.html +++ b/doc/pub/week42/html/._week42-bs021.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -425,7 +440,7 @@ Here, gradient descent with a constant step size has been chosen.
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    • diff --git a/doc/pub/week42/html/._week42-bs022.html b/doc/pub/week42/html/._week42-bs022.html index 358d76e3c..4d4effcb8 100644 --- a/doc/pub/week42/html/._week42-bs022.html +++ b/doc/pub/week42/html/._week42-bs022.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -447,7 +462,7 @@ $$
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    • diff --git a/doc/pub/week42/html/._week42-bs023.html b/doc/pub/week42/html/._week42-bs023.html index 2bb8c7c0d..3d3293dd1 100644 --- a/doc/pub/week42/html/._week42-bs023.html +++ b/doc/pub/week42/html/._week42-bs023.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -557,7 +572,7 @@ MathJax.Hub.Config({
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    • diff --git a/doc/pub/week42/html/._week42-bs024.html b/doc/pub/week42/html/._week42-bs024.html index c4922abb5..c5221f2b6 100644 --- a/doc/pub/week42/html/._week42-bs024.html +++ b/doc/pub/week42/html/._week42-bs024.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -577,7 +592,7 @@ The number of neurons within each hidden layer are given as a list of integers i
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    • diff --git a/doc/pub/week42/html/._week42-bs025.html b/doc/pub/week42/html/._week42-bs025.html index bdeea7b05..8292b740f 100644 --- a/doc/pub/week42/html/._week42-bs025.html +++ b/doc/pub/week42/html/._week42-bs025.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -429,7 +444,7 @@ Here, we stay with a more simple approach and implement for comparison, the simp
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    • diff --git a/doc/pub/week42/html/._week42-bs026.html b/doc/pub/week42/html/._week42-bs026.html index ee6ea7225..da67462a5 100644 --- a/doc/pub/week42/html/._week42-bs026.html +++ b/doc/pub/week42/html/._week42-bs026.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -425,7 +440,7 @@ In this example, we let \( \alpha = 2 \), \( A = 1 \), and \( g_0 = 1.2 \).
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    • diff --git a/doc/pub/week42/html/._week42-bs027.html b/doc/pub/week42/html/._week42-bs027.html index 57ecb7d72..1af800383 100644 --- a/doc/pub/week42/html/._week42-bs027.html +++ b/doc/pub/week42/html/._week42-bs027.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -430,7 +445,7 @@ $$
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    • diff --git a/doc/pub/week42/html/._week42-bs028.html b/doc/pub/week42/html/._week42-bs028.html index c61d9aeff..e620e3e61 100644 --- a/doc/pub/week42/html/._week42-bs028.html +++ b/doc/pub/week42/html/._week42-bs028.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -579,7 +594,7 @@ The network will be the similar as for the exponential decay example, but with s
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    • diff --git a/doc/pub/week42/html/._week42-bs029.html b/doc/pub/week42/html/._week42-bs029.html index f26d78b32..f49a81b5e 100644 --- a/doc/pub/week42/html/._week42-bs029.html +++ b/doc/pub/week42/html/._week42-bs029.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -533,7 +548,7 @@ extending the program that uses the network using Autograd:
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    • diff --git a/doc/pub/week42/html/._week42-bs030.html b/doc/pub/week42/html/._week42-bs030.html index 308fbb861..374b0093c 100644 --- a/doc/pub/week42/html/._week42-bs030.html +++ b/doc/pub/week42/html/._week42-bs030.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -435,7 +450,7 @@ In addition, it could be interesting to see how a typical method for numerically
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    • diff --git a/doc/pub/week42/html/._week42-bs031.html b/doc/pub/week42/html/._week42-bs031.html index 89ec009ca..64cbf39e0 100644 --- a/doc/pub/week42/html/._week42-bs031.html +++ b/doc/pub/week42/html/._week42-bs031.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -442,7 +457,7 @@ $$
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    • diff --git a/doc/pub/week42/html/._week42-bs032.html b/doc/pub/week42/html/._week42-bs032.html index 20987bc82..c636bfde7 100644 --- a/doc/pub/week42/html/._week42-bs032.html +++ b/doc/pub/week42/html/._week42-bs032.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -563,7 +578,7 @@ MathJax.Hub.Config({
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    • diff --git a/doc/pub/week42/html/._week42-bs033.html b/doc/pub/week42/html/._week42-bs033.html index 721c309f8..6adbe0ba2 100644 --- a/doc/pub/week42/html/._week42-bs033.html +++ b/doc/pub/week42/html/._week42-bs033.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -502,7 +517,7 @@ which makes it possible to solve for the vector \( \boldsymbol{g} \).
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    • diff --git a/doc/pub/week42/html/._week42-bs034.html b/doc/pub/week42/html/._week42-bs034.html index 62ee97617..eb63ec4bd 100644 --- a/doc/pub/week42/html/._week42-bs034.html +++ b/doc/pub/week42/html/._week42-bs034.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -606,7 +621,7 @@ We can then compare the result from this numerical scheme with the output from o
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    • diff --git a/doc/pub/week42/html/._week42-bs035.html b/doc/pub/week42/html/._week42-bs035.html index 50e0d85c9..bcb7bc668 100644 --- a/doc/pub/week42/html/._week42-bs035.html +++ b/doc/pub/week42/html/._week42-bs035.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -427,7 +442,7 @@ where \( f \) is an expression involving all kinds of possible mixed derivatives
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    • diff --git a/doc/pub/week42/html/._week42-bs036.html b/doc/pub/week42/html/._week42-bs036.html index 33e9673b8..f8c3a0b5c 100644 --- a/doc/pub/week42/html/._week42-bs036.html +++ b/doc/pub/week42/html/._week42-bs036.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -427,7 +442,7 @@ The role of the function \( h_2(x_1,\dots,x_N,N(x_1,\dots,x_N,P)) \), is to ensu
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    • diff --git a/doc/pub/week42/html/._week42-bs037.html b/doc/pub/week42/html/._week42-bs037.html index 38f99f949..6065742b8 100644 --- a/doc/pub/week42/html/._week42-bs037.html +++ b/doc/pub/week42/html/._week42-bs037.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -427,7 +442,7 @@ $$
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    • diff --git a/doc/pub/week42/html/._week42-bs038.html b/doc/pub/week42/html/._week42-bs038.html index 7f325a6f7..5c896c2f1 100644 --- a/doc/pub/week42/html/._week42-bs038.html +++ b/doc/pub/week42/html/._week42-bs038.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -423,7 +438,7 @@ $$
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    • diff --git a/doc/pub/week42/html/._week42-bs039.html b/doc/pub/week42/html/._week42-bs039.html index ed3b5cac7..ab44cbb57 100644 --- a/doc/pub/week42/html/._week42-bs039.html +++ b/doc/pub/week42/html/._week42-bs039.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -429,7 +444,7 @@ with \( u(x) \) being some given function.
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    • diff --git a/doc/pub/week42/html/._week42-bs040.html b/doc/pub/week42/html/._week42-bs040.html index b9e9ae624..b781d777f 100644 --- a/doc/pub/week42/html/._week42-bs040.html +++ b/doc/pub/week42/html/._week42-bs040.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -436,7 +451,7 @@ First, we will look into how Autograd could be used in a network tailored to sol
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    • diff --git a/doc/pub/week42/html/._week42-bs041.html b/doc/pub/week42/html/._week42-bs041.html index 16ea3ed35..04689be43 100644 --- a/doc/pub/week42/html/._week42-bs041.html +++ b/doc/pub/week42/html/._week42-bs041.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -466,7 +481,7 @@ network at each possible pair \( (x,t) \), given an array for the desired
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    • diff --git a/doc/pub/week42/html/._week42-bs042.html b/doc/pub/week42/html/._week42-bs042.html index 9088ae409..4077fd39a 100644 --- a/doc/pub/week42/html/._week42-bs042.html +++ b/doc/pub/week42/html/._week42-bs042.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -433,7 +448,7 @@ since \( (0) = u(1) = 0 \) and \( u(x) = \sin(\pi x) \).
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    • diff --git a/doc/pub/week42/html/._week42-bs043.html b/doc/pub/week42/html/._week42-bs043.html index 6e30681d3..c6df28be7 100644 --- a/doc/pub/week42/html/._week42-bs043.html +++ b/doc/pub/week42/html/._week42-bs043.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -471,7 +486,7 @@ mixed derivatives of \( g(x,t) \).
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    • diff --git a/doc/pub/week42/html/._week42-bs044.html b/doc/pub/week42/html/._week42-bs044.html index 746d20cf7..bdd972a8d 100644 --- a/doc/pub/week42/html/._week42-bs044.html +++ b/doc/pub/week42/html/._week42-bs044.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -655,7 +670,7 @@ Using TensorFlow results in a much better execution time. Try it!
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    • diff --git a/doc/pub/week42/html/._week42-bs045.html b/doc/pub/week42/html/._week42-bs045.html index a01d76ba7..56631b93a 100644 --- a/doc/pub/week42/html/._week42-bs045.html +++ b/doc/pub/week42/html/._week42-bs045.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -433,7 +448,7 @@ where \( \frac{\partial g(x,t)}{\partial t} \Big |_{t = 0} \) means the derivati
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    • diff --git a/doc/pub/week42/html/._week42-bs046.html b/doc/pub/week42/html/._week42-bs046.html index 947f910bc..a078783b5 100644 --- a/doc/pub/week42/html/._week42-bs046.html +++ b/doc/pub/week42/html/._week42-bs046.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -434,7 +449,7 @@ In this example, let \( c = 1 \) and \( u(x) = \sin(\pi x) \) and \( v(x) = -\pi
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    • diff --git a/doc/pub/week42/html/._week42-bs047.html b/doc/pub/week42/html/._week42-bs047.html index fbcbf1419..050b4dae4 100644 --- a/doc/pub/week42/html/._week42-bs047.html +++ b/doc/pub/week42/html/._week42-bs047.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -428,7 +443,7 @@ Note that this trial solution satisfies the conditions only if \( u(0) = v(0) =
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    • diff --git a/doc/pub/week42/html/._week42-bs048.html b/doc/pub/week42/html/._week42-bs048.html index 06766f951..73c196992 100644 --- a/doc/pub/week42/html/._week42-bs048.html +++ b/doc/pub/week42/html/._week42-bs048.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -416,7 +431,7 @@ $$
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    • diff --git a/doc/pub/week42/html/._week42-bs049.html b/doc/pub/week42/html/._week42-bs049.html index 703f34ef3..da6d31f22 100644 --- a/doc/pub/week42/html/._week42-bs049.html +++ b/doc/pub/week42/html/._week42-bs049.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -633,7 +648,7 @@ MathJax.Hub.Config({
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    • diff --git a/doc/pub/week42/html/._week42-bs050.html b/doc/pub/week42/html/._week42-bs050.html index 2e8ae2ee9..318acba4c 100644 --- a/doc/pub/week42/html/._week42-bs050.html +++ b/doc/pub/week42/html/._week42-bs050.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -415,7 +430,7 @@ MathJax.Hub.Config({
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    • diff --git a/doc/pub/week42/html/._week42-bs051.html b/doc/pub/week42/html/._week42-bs051.html index 6adb3f2c8..13d6e6ee8 100644 --- a/doc/pub/week42/html/._week42-bs051.html +++ b/doc/pub/week42/html/._week42-bs051.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -443,7 +458,7 @@ Another good read is the article here 60
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    • diff --git a/doc/pub/week42/html/._week42-bs052.html b/doc/pub/week42/html/._week42-bs052.html index eb05abcaa..7e417a629 100644 --- a/doc/pub/week42/html/._week42-bs052.html +++ b/doc/pub/week42/html/._week42-bs052.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -418,7 +433,7 @@ before the transformation.
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    • diff --git a/doc/pub/week42/html/._week42-bs053.html b/doc/pub/week42/html/._week42-bs053.html index 21ab949db..cd1511355 100644 --- a/doc/pub/week42/html/._week42-bs053.html +++ b/doc/pub/week42/html/._week42-bs053.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -432,7 +447,7 @@ in the input).
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    • diff --git a/doc/pub/week42/html/._week42-bs054.html b/doc/pub/week42/html/._week42-bs054.html index 20dbf54f9..9b3634bbf 100644 --- a/doc/pub/week42/html/._week42-bs054.html +++ b/doc/pub/week42/html/._week42-bs054.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -432,7 +447,7 @@ would quickly lead to possible overfitting.
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    • diff --git a/doc/pub/week42/html/._week42-bs055.html b/doc/pub/week42/html/._week42-bs055.html index 660ddab75..6102b4b6d 100644 --- a/doc/pub/week42/html/._week42-bs055.html +++ b/doc/pub/week42/html/._week42-bs055.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -444,7 +459,7 @@ dimension.
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    • diff --git a/doc/pub/week42/html/._week42-bs056.html b/doc/pub/week42/html/._week42-bs056.html index a87e28613..a8248f759 100644 --- a/doc/pub/week42/html/._week42-bs056.html +++ b/doc/pub/week42/html/._week42-bs056.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -427,7 +442,7 @@ A simple CNN for image classification could have the architecture:
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    • diff --git a/doc/pub/week42/html/._week42-bs057.html b/doc/pub/week42/html/._week42-bs057.html index c42f64393..dda308b35 100644 --- a/doc/pub/week42/html/._week42-bs057.html +++ b/doc/pub/week42/html/._week42-bs057.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -423,7 +438,7 @@ are consistent with the labels in the training set for each image.
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    • diff --git a/doc/pub/week42/html/._week42-bs058.html b/doc/pub/week42/html/._week42-bs058.html index 2633fe27a..9bea3152d 100644 --- a/doc/pub/week42/html/._week42-bs058.html +++ b/doc/pub/week42/html/._week42-bs058.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -425,7 +440,7 @@ and the slides of 67
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    • diff --git a/doc/pub/week42/html/._week42-bs059.html b/doc/pub/week42/html/._week42-bs059.html index e949475d4..039e31780 100644 --- a/doc/pub/week42/html/._week42-bs059.html +++ b/doc/pub/week42/html/._week42-bs059.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -443,7 +458,7 @@ How can we use this? And what does it mean? Let us study some familiar examples
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    • diff --git a/doc/pub/week42/html/._week42-bs060.html b/doc/pub/week42/html/._week42-bs060.html index 301b1ada8..22db352ee 100644 --- a/doc/pub/week42/html/._week42-bs060.html +++ b/doc/pub/week42/html/._week42-bs060.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -384,8 +399,24 @@ MathJax.Hub.Config({

    Convolution Examples: Polynomial multiplication

    -We have already met such an example in project 1 when we tried to set up the design matrix for a two-dimensional function. -Let us remind of this and recast it in terms of the mathematical operation of convolution. +We have already met such an example in project 1 when we tried to set +up the design matrix for a two-dimensional function. This was an +example of polynomial multiplication. Let us recast such a problem in terms of the convolution operation. +Let us look a the following polynomials to second and third order, respectively: +$$ +p(t) = \alpha_0+\alpha_1 t+\alpha_2 t^2, +$$ + +and +$$ +s(t) = \beta_0+\beta_1 t+\beta_2 t^2+\beta_3 t^3. +$$ + +

    +The polynomial multiplication gives us a new polynomial of degree \( 5 \) +$$ +z(t) = \delta_0+\delta_1 t+\delta_2 t^2+\delta_3 t^3+\delta_4 t^4+\delta_5 t^5. +$$

    @@ -413,7 +444,7 @@ Let us remind of this and recast it in terms of the mathematical operation of co

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    • diff --git a/doc/pub/week42/html/._week42-bs061.html b/doc/pub/week42/html/._week42-bs061.html index 39a1511d8..b00f331b8 100644 --- a/doc/pub/week42/html/._week42-bs061.html +++ b/doc/pub/week42/html/._week42-bs061.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -381,7 +396,39 @@ MathJax.Hub.Config({ -

    Convolution Examples: Probability Theory

    +

    Efficient Polynomial Multiplication

    + +

    +Computing polynomial products can be implemented efficiently if we rewrite the the more brute force multiplications using convolution. +We note first that the new coefficients are given as + +$$ +\begin{split} +\delta_0=&\alpha_0\beta_0\\ +\delta_1=&\alpha_1\beta_0+\beta_0\alpha_1\\ +\delta_2=&\alpha_0\beta_2+\beta_1\alpha_1+\alpha_2\beta_0\\ +\delta_3=&\alpha_1\beta_2+\beta_1\alpha_2+\alpha_0\beta_3\\ +\delta_4=&\alpha_2\beta_2+\beta_3\alpha_1\\ +\delta_5=&\alpha_2\beta_3.\\ +\end{split} +$$ + +

    +We note that \( \alpha_i=0 \) except for \( i\in \left{0,1,2\right} \) and \( \beta_i=0 \) except for \( i\in\left{0,1,2,3\right} \). + +

    +We can then rewrite the coefficients \( \delta_j \) using a discrete convolution as +$$ +\delta_j = \sum_{i=-\infty}^{i=\infty}\alpha_i\beta_{j-i}=(\alpha * \beta)_j, +$$ + +or as a double sum with restriction \( l=i+j \) +$$ +\delta_l = \sum_{ij}\alpha_i\beta_{j}. +$$ + +

    +Do you see a potential drawback with these equations?

    @@ -409,7 +456,7 @@ MathJax.Hub.Config({

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    • diff --git a/doc/pub/week42/html/._week42-bs062.html b/doc/pub/week42/html/._week42-bs062.html index b826c1a4a..984df9d52 100644 --- a/doc/pub/week42/html/._week42-bs062.html +++ b/doc/pub/week42/html/._week42-bs062.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -381,27 +396,25 @@ MathJax.Hub.Config({ -

    Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)

    +

    A more efficient way of coding the above Convolution

    -For problems with so-called harmonic oscillations, given by for example the following differential equation -$$ -m\frac{d^2x(t)}(dt^2}+\eta\frac{dx}{dt}+x(t)=F(t), -$$ +Since we only have a finite number of \( \alpha \) and \( \beta \) values +which are non-zero, we can rewrite the above convolution expressions +as a matrix-vector multiplication -where \( F(t) \) is an applied external force acting on the system (often called a driving force), one can use the theory of Fourier transformations to find the solutions of this type of equations. +$$ +\boldsymbol{\delta}=\begin{bmatriax}\alpha_0 & 0 & 0 & 0 \\ + \alpha_1 & \alpha_0 & 0 & 0 \\ + \alpha_2 & \alpha_1 & \alpha_0 & 0 \\ + 0 & \alpha_2 & \alpha_1 & \alpha_0 \\ + 0 & 0 & \alpha_2 & \alpha_1 \\ + 0 & 0 & 0 & \alpha_2 + \end{bmatrix}\begin{bmatrix} \beta_0 \\ \beta_1 \\ \beta_2 \\ \beta_3\end{bmatrix} +$$

    -If one has several driving forces, \( F(t)=\sum_n F_n(t) \), one can find -the particular solution to each \( F_n \), \( x_{pn}(t) \), and the particular -solution for the entire driving force is then given by a series like - -$$ -\begin{equation} -x_p(t)=\sum_nx_{pn}(t). -\tag{21} -\end{equation} -$$ +The process is commutative and we can easily see that we can rewrite the multiplication in terms of a martrix holding \( \beta \) and a vector holding \( \alpha \).

    @@ -429,7 +442,7 @@ $$

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    • diff --git a/doc/pub/week42/html/._week42-bs063.html b/doc/pub/week42/html/._week42-bs063.html index b86b86e53..8b7f7c7f8 100644 --- a/doc/pub/week42/html/._week42-bs063.html +++ b/doc/pub/week42/html/._week42-bs063.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -381,32 +396,27 @@ MathJax.Hub.Config({ -

    Principle of Superposition

    +

    Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)

    -This is known as the principle of superposition. It only applies when -the homogenous equation is linear. If there were an anharmonic term -such as \( x^3 \) in the homogenous equation, then when one summed various -solutions, \( x=(\sum_n x_n)^2 \), one would get cross -terms. Superposition is especially useful when \( F(t) \) can be written -as a sum of sinusoidal terms, because the solutions for each -sinusoidal (sine or cosine) term is analytic. - -

    -Driving forces are often periodic, even when they are not -sinusoidal. Periodicity implies that for some time \( \tau \) - +For problems with so-called harmonic oscillations, given by for example the following differential equation $$ -\begin{eqnarray} -F(t+\tau)=F(t). -\end{eqnarray} +m\frac{d^2x}{dt^2}+\eta\frac{dx}{dt}+x(t)=F(t), $$ +where \( F(t) \) is an applied external force acting on the system (often called a driving force), one can use the theory of Fourier transformations to find the solutions of this type of equations. +

    -One example of a non-sinusoidal periodic force is a square wave. Many -components in electric circuits are non-linear, e.g. diodes, which -makes many wave forms non-sinusoidal even when the circuits are being -driven by purely sinusoidal sources. +If one has several driving forces, \( F(t)=\sum_n F_n(t) \), one can find +the particular solution to each \( F_n \), \( x_{pn}(t) \), and the particular +solution for the entire driving force is then given by a series like + +$$ +\begin{equation} +x_p(t)=\sum_nx_{pn}(t). +\tag{21} +\end{equation} +$$

    @@ -434,7 +444,7 @@ driven by purely sinusoidal sources.

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    • diff --git a/doc/pub/week42/html/._week42-bs064.html b/doc/pub/week42/html/._week42-bs064.html index 352aa8962..238a4dbef 100644 --- a/doc/pub/week42/html/._week42-bs064.html +++ b/doc/pub/week42/html/._week42-bs064.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -381,46 +396,33 @@ MathJax.Hub.Config({ -

    Simple Code Example

    +

    Principle of Superposition

    -The code here shows a typical example of such a square wave generated using the functionality included in the scipy Python package. We have used a period of \( \tau=0.2 \). +This is known as the principle of superposition. It only applies when +the homogenous equation is linear. If there were an anharmonic term +such as \( x^3 \) in the homogenous equation, then when one summed various +solutions, \( x=(\sum_n x_n)^2 \), one would get cross +terms. Superposition is especially useful when \( F(t) \) can be written +as a sum of sinusoidal terms, because the solutions for each +sinusoidal (sine or cosine) term is analytic.

    - - -

    import numpy as np
-import math
-from scipy import signal
-import matplotlib.pyplot as plt
-
-# number of points                                                                                       
-n = 500
-# start and final times                                                                                  
-t0 = 0.0
-tn = 1.0
-# Period                                                                                                 
-t = np.linspace(t0, tn, n, endpoint=False)
-SqrSignal = np.zeros(n)
-SqrSignal = 1.0+signal.square(2*np.pi*5*t)
-plt.plot(t, SqrSignal)
-plt.ylim(-0.5, 2.5)
-plt.show()
-
    -

    -For the sinusoidal example the -period is \( \tau=2\pi/\omega \). However, higher harmonics can also -satisfy the periodicity requirement. In general, any force that -satisfies the periodicity requirement can be expressed as a sum over -harmonics, +Driving forces are often periodic, even when they are not +sinusoidal. Periodicity implies that for some time \( \tau \) $$ -\begin{equation} -F(t)=\frac{f_0}{2}+\sum_{n>0} f_n\cos(2n\pi t/\tau)+g_n\sin(2n\pi t/\tau). -\tag{22} -\end{equation} +\begin{eqnarray} +F(t+\tau)=F(t). +\end{eqnarray} $$ +

    +One example of a non-sinusoidal periodic force is a square wave. Many +components in electric circuits are non-linear, e.g. diodes, which +makes many wave forms non-sinusoidal even when the circuits are being +driven by purely sinusoidal sources. +

    @@ -447,7 +449,7 @@ $$

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    • diff --git a/doc/pub/week42/html/._week42-bs065.html b/doc/pub/week42/html/._week42-bs065.html index 025064c86..46781c38f 100644 --- a/doc/pub/week42/html/._week42-bs065.html +++ b/doc/pub/week42/html/._week42-bs065.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -381,37 +396,46 @@ MathJax.Hub.Config({ -

    Wrapping up Fourier transforms

    +

    Simple Code Example

    -We can write down the answer for -\( x_{pn}(t) \), by substituting \( f_n/m \) or \( g_n/m \) for \( F_0/m \). By -writing each factor \( 2n\pi t/\tau \) as \( n\omega t \), with \( \omega\equiv -2\pi/\tau \), +The code here shows a typical example of such a square wave generated using the functionality included in the scipy Python package. We have used a period of \( \tau=0.2 \). + +

    + + +

    import numpy as np
+import math
+from scipy import signal
+import matplotlib.pyplot as plt
+
+# number of points                                                                                       
+n = 500
+# start and final times                                                                                  
+t0 = 0.0
+tn = 1.0
+# Period                                                                                                 
+t = np.linspace(t0, tn, n, endpoint=False)
+SqrSignal = np.zeros(n)
+SqrSignal = 1.0+signal.square(2*np.pi*5*t)
+plt.plot(t, SqrSignal)
+plt.ylim(-0.5, 2.5)
+plt.show()
+
    +

    +For the sinusoidal example the +period is \( \tau=2\pi/\omega \). However, higher harmonics can also +satisfy the periodicity requirement. In general, any force that +satisfies the periodicity requirement can be expressed as a sum over +harmonics, $$ \begin{equation} -\tag{23} -F(t)=\frac{f_0}{2}+\sum_{n>0}f_n\cos(n\omega t)+g_n\sin(n\omega t). +F(t)=\frac{f_0}{2}+\sum_{n>0} f_n\cos(2n\pi t/\tau)+g_n\sin(2n\pi t/\tau). +\tag{22} \end{equation} $$ -

    -The solutions for \( x(t) \) then come from replacing \( \omega \) with -\( n\omega \) for each term in the particular solution, - -$$ -\begin{eqnarray} -x_p(t)&=&\frac{f_0}{2k}+\sum_{n>0} \alpha_n\cos(n\omega t-\delta_n)+\beta_n\sin(n\omega t-\delta_n),\\ -\nonumber -\alpha_n&=&\frac{f_n/m}{\sqrt{((n\omega)^2-\omega_0^2)+4\beta^2n^2\omega^2}},\\ -\nonumber -\beta_n&=&\frac{g_n/m}{\sqrt{((n\omega)^2-\omega_0^2)+4\beta^2n^2\omega^2}},\\ -\nonumber -\delta_n&=&\tan^{-1}\left(\frac{2\beta n\omega}{\omega_0^2-n^2\omega^2}\right). -\end{eqnarray} -$$ -

    @@ -438,7 +462,7 @@ $$

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    • diff --git a/doc/pub/week42/html/._week42-bs066.html b/doc/pub/week42/html/._week42-bs066.html index 13365f352..20550e1a9 100644 --- a/doc/pub/week42/html/._week42-bs066.html +++ b/doc/pub/week42/html/._week42-bs066.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -381,73 +396,37 @@ MathJax.Hub.Config({ -

    Finding the Coefficients

    +

    Wrapping up Fourier transforms

    -Because the forces have been applied for a long time, any non-zero -damping eliminates the homogenous parts of the solution, so one need -only consider the particular solution for each \( n \). - -

    -The problem is considered solved if one can find expressions for the -coefficients \( f_n \) and \( g_n \), even though the solutions are expressed -as an infinite sum. The coefficients can be extracted from the -function \( F(t) \) by - -$$ -\begin{eqnarray} -\tag{24} -f_n&=&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~F(t)\cos(2n\pi t/\tau),\\ -\nonumber -g_n&=&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~F(t)\sin(2n\pi t/\tau). -\end{eqnarray} -$$ - -

    -To check the consistency of these expressions and to verify -Eq. (24), one can insert the expansion of \( F(t) \) in -Eq. (23) into the expression for the coefficients in -Eq. (24) and see whether - -$$ -\begin{eqnarray} -f_n&=?&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~\left\{ -\frac{f_0}{2}+\sum_{m>0}f_m\cos(m\omega t)+g_m\sin(m\omega t) -\right\}\cos(n\omega t). -\end{eqnarray} -$$ - -

    -Immediately, one can throw away all the terms with \( g_m \) because they -convolute an even and an odd function. The term with \( f_0/2 \) -disappears because \( \cos(n\omega t) \) is equally positive and negative -over the interval and will integrate to zero. For all the terms -\( f_m\cos(m\omega t) \) appearing in the sum, one can use angle addition -formulas to see that \( \cos(m\omega t)\cos(n\omega -t)=(1/2)(\cos[(m+n)\omega t]+\cos[(m-n)\omega t] \). This will integrate -to zero unless \( m=n \). In that case the \( m=n \) term gives +We can write down the answer for +\( x_{pn}(t) \), by substituting \( f_n/m \) or \( g_n/m \) for \( F_0/m \). By +writing each factor \( 2n\pi t/\tau \) as \( n\omega t \), with \( \omega\equiv +2\pi/\tau \), $$ \begin{equation} -\int_{-\tau/2}^{\tau/2}dt~\cos^2(m\omega t)=\frac{\tau}{2}, -\tag{25} +\tag{23} +F(t)=\frac{f_0}{2}+\sum_{n>0}f_n\cos(n\omega t)+g_n\sin(n\omega t). \end{equation} $$

    -and +The solutions for \( x(t) \) then come from replacing \( \omega \) with +\( n\omega \) for each term in the particular solution, $$ \begin{eqnarray} -f_n&=?&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~f_n/2\\ +x_p(t)&=&\frac{f_0}{2k}+\sum_{n>0} \alpha_n\cos(n\omega t-\delta_n)+\beta_n\sin(n\omega t-\delta_n),\\ \nonumber -&=&f_n~\checkmark. +\alpha_n&=&\frac{f_n/m}{\sqrt{((n\omega)^2-\omega_0^2)+4\beta^2n^2\omega^2}},\\ +\nonumber +\beta_n&=&\frac{g_n/m}{\sqrt{((n\omega)^2-\omega_0^2)+4\beta^2n^2\omega^2}},\\ +\nonumber +\delta_n&=&\tan^{-1}\left(\frac{2\beta n\omega}{\omega_0^2-n^2\omega^2}\right). \end{eqnarray} $$ -

    -The same method can be used to check for the consistency of \( g_n \). -

    @@ -474,7 +453,7 @@ The same method can be used to check for the consistency of \( g_n \).

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    • diff --git a/doc/pub/week42/html/._week42-bs067.html b/doc/pub/week42/html/._week42-bs067.html index 8c5861e99..07e575e7d 100644 --- a/doc/pub/week42/html/._week42-bs067.html +++ b/doc/pub/week42/html/._week42-bs067.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -381,18 +396,72 @@ MathJax.Hub.Config({ -

    CNNs in more detail, building convolutional neural networks in Tensorflow and Keras

    +

    Finding the Coefficients

    -As discussed above, CNNs are neural networks built from the assumption that the inputs -to the network are 2D images. This is important because the number of features or pixels in images -grows very fast with the image size, and an enormous number of weights and biases are needed in order to build an accurate network. +Because the forces have been applied for a long time, any non-zero +damping eliminates the homogenous parts of the solution, so one need +only consider the particular solution for each \( n \).

    -As before, we still have our input, a hidden layer and an output. What's novel about convolutional networks -are the convolutional and pooling layers stacked in pairs between the input and the hidden layer. -In addition, the data is no longer represented as a 2D feature matrix, instead each input is a number of 2D -matrices, typically 1 for each color dimension (Red, Green, Blue). +The problem is considered solved if one can find expressions for the +coefficients \( f_n \) and \( g_n \), even though the solutions are expressed +as an infinite sum. The coefficients can be extracted from the +function \( F(t) \) by + +$$ +\begin{eqnarray} +\tag{24} +f_n&=&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~F(t)\cos(2n\pi t/\tau),\\ +\nonumber +g_n&=&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~F(t)\sin(2n\pi t/\tau). +\end{eqnarray} +$$ + +

    +To check the consistency of these expressions and to verify +Eq. (24), one can insert the expansion of \( F(t) \) in +Eq. (23) into the expression for the coefficients in +Eq. (24) and see whether + +$$ +\begin{eqnarray} +f_n&=?&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~\left\{ +\frac{f_0}{2}+\sum_{m>0}f_m\cos(m\omega t)+g_m\sin(m\omega t) +\right\}\cos(n\omega t). +\end{eqnarray} +$$ + +

    +Immediately, one can throw away all the terms with \( g_m \) because they +convolute an even and an odd function. The term with \( f_0/2 \) +disappears because \( \cos(n\omega t) \) is equally positive and negative +over the interval and will integrate to zero. For all the terms +\( f_m\cos(m\omega t) \) appearing in the sum, one can use angle addition +formulas to see that \( \cos(m\omega t)\cos(n\omega +t)=(1/2)(\cos[(m+n)\omega t]+\cos[(m-n)\omega t] \). This will integrate +to zero unless \( m=n \). In that case the \( m=n \) term gives + +$$ +\begin{equation} +\int_{-\tau/2}^{\tau/2}dt~\cos^2(m\omega t)=\frac{\tau}{2}, +\tag{25} +\end{equation} +$$ + +

    +and + +$$ +\begin{eqnarray} +f_n&=?&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~f_n/2\\ +\nonumber +&=&f_n~\checkmark. +\end{eqnarray} +$$ + +

    +The same method can be used to check for the consistency of \( g_n \).

    @@ -420,7 +489,7 @@ matrices, typically 1 for each color dimension (Red, Green, Blue).

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    • diff --git a/doc/pub/week42/html/._week42-bs068.html b/doc/pub/week42/html/._week42-bs068.html index 884e01b3e..7a54db58a 100644 --- a/doc/pub/week42/html/._week42-bs068.html +++ b/doc/pub/week42/html/._week42-bs068.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -381,15 +396,50 @@ MathJax.Hub.Config({ -

    Setting it up

    +

    Final words on Fourier Transforms

    -It means that to represent the entire -dataset of images, we require a 4D matrix or tensor. This tensor has the dimensions: -$$ -(n_{inputs},\, n_{pixels, width},\, n_{pixels, height},\, depth) . -$$ +The code here uses the Fourier series applied to a +square wave signal. The code here +visualizes the various approximations given by Fourier series compared +with a square wave with period \( T=0.2 \) (dimensionless time), width \( 0.1 \) and max value of the force \( F=2 \). We +see that when we increase the number of components in the Fourier +series, the Fourier series approximation gets closer and closer to the +square wave signal. +

    + + +

    import numpy as np
+import math
+from scipy import signal
+import matplotlib.pyplot as plt
+
+# number of points                                                                                       
+n = 500
+# start and final times                                                                                  
+t0 = 0.0
+tn = 1.0
+# Period                                                                                                 
+T =0.2
+# Max value of square signal                                                                             
+Fmax= 2.0
+# Width of signal   
+Width = 0.1
+t = np.linspace(t0, tn, n, endpoint=False)
+SqrSignal = np.zeros(n)
+FourierSeriesSignal = np.zeros(n)
+SqrSignal = 1.0+signal.square(2*np.pi*5*t+np.pi*Width/T)
+a0 = Fmax*Width/T
+FourierSeriesSignal = a0
+Factor = 2.0*Fmax/np.pi
+for i in range(1,500):
+    FourierSeriesSignal += Factor/(i)*np.sin(np.pi*i*Width/T)*np.cos(i*t*2*np.pi/T)
+plt.plot(t, SqrSignal)
+plt.plot(t, FourierSeriesSignal)
+plt.ylim(-0.5, 2.5)
+plt.show()
+

    @@ -416,7 +466,7 @@ $$

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    • diff --git a/doc/pub/week42/html/._week42-bs069.html b/doc/pub/week42/html/._week42-bs069.html index ec3baf136..e455fd991 100644 --- a/doc/pub/week42/html/._week42-bs069.html +++ b/doc/pub/week42/html/._week42-bs069.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -381,20 +396,10 @@ MathJax.Hub.Config({ -

    The MNIST dataset again

    +

    Convolution Examples: Probability Theory

    -The MNIST dataset consists of grayscale images with a pixel size of -\( 28\times 28 \), meaning we require \( 28 \times 28 = 724 \) weights to each -neuron in the first hidden layer. - -

    -If we were to analyze images of size \( 128\times 128 \) we would require -\( 128 \times 128 = 16384 \) weights to each neuron. Even worse if we were -dealing with color images, as most images are, we have an image matrix -of size \( 128\times 128 \) for each color dimension (Red, Green, Blue), -meaning 3 times the number of weights \( = 49152 \) are required for every -single neuron in the first hidden layer. +More text will be added here

    @@ -422,7 +427,7 @@ single neuron in the first hidden layer.

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    • diff --git a/doc/pub/week42/html/._week42-bs070.html b/doc/pub/week42/html/._week42-bs070.html index e38a96ffa..37a010cf0 100644 --- a/doc/pub/week42/html/._week42-bs070.html +++ b/doc/pub/week42/html/._week42-bs070.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -381,20 +396,18 @@ MathJax.Hub.Config({ -

    Strong correlations

    +

    CNNs in more detail, building convolutional neural networks in Tensorflow and Keras

    -Images typically have strong local correlations, meaning that a small -part of the image varies little from its neighboring regions. If for -example we have an image of a blue car, we can roughly assume that a -small blue part of the image is surrounded by other blue regions. +As discussed above, CNNs are neural networks built from the assumption that the inputs +to the network are 2D images. This is important because the number of features or pixels in images +grows very fast with the image size, and an enormous number of weights and biases are needed in order to build an accurate network.

    -Therefore, instead of connecting every single pixel to a neuron in the -first hidden layer, as we have previously done with deep neural -networks, we can instead connect each neuron to a small part of the -image (in all 3 RGB depth dimensions). The size of each small area is -fixed, and known as a receptive. +As before, we still have our input, a hidden layer and an output. What's novel about convolutional networks +are the convolutional and pooling layers stacked in pairs between the input and the hidden layer. +In addition, the data is no longer represented as a 2D feature matrix, instead each input is a number of 2D +matrices, typically 1 for each color dimension (Red, Green, Blue).

    @@ -422,7 +435,7 @@ fixed, and known as a 79

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  • »
    • diff --git a/doc/pub/week42/html/._week42-bs071.html b/doc/pub/week42/html/._week42-bs071.html index 5eca31447..fb7d042dd 100644 --- a/doc/pub/week42/html/._week42-bs071.html +++ b/doc/pub/week42/html/._week42-bs071.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -379,26 +394,16 @@ MathJax.Hub.Config({

     

     

     

    - + -

    Layers of a CNN

    -The layers of a convolutional neural network arrange neurons in 3D: width, height and depth. -The input image is typically a square matrix of depth 3. +

    Setting it up

    -A convolution is performed on the image which outputs -a 3D volume of neurons. The weights to the input are arranged in a number of 2D matrices, known as filters. - -

    -Each filter slides along the input image, taking the dot product -between each small part of the image and the filter, in all depth -dimensions. This is then passed through a non-linear function, -typically the Rectified Linear (ReLu) function, which serves as the -activation of the neurons in the first convolutional layer. This is -further passed through a pooling layer, which reduces the size of the -convolutional layer, e.g. by taking the maximum or average across some -small regions, and this serves as input to the next convolutional -layer. +It means that to represent the entire +dataset of images, we require a 4D matrix or tensor. This tensor has the dimensions: +$$ +(n_{inputs},\, n_{pixels, width},\, n_{pixels, height},\, depth) . +$$

    @@ -426,7 +431,7 @@ layer.

  • 80
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    • -
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  • »
    • diff --git a/doc/pub/week42/html/._week42-bs072.html b/doc/pub/week42/html/._week42-bs072.html index b0eef899c..67fd04dbc 100644 --- a/doc/pub/week42/html/._week42-bs072.html +++ b/doc/pub/week42/html/._week42-bs072.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -381,17 +396,20 @@ MathJax.Hub.Config({ -

    Systematic reduction

    +

    The MNIST dataset again

    -By systematically reducing the size of the input volume, through -convolution and pooling, the network should create representations of -small parts of the input, and then from them assemble representations -of larger areas. The final pooling layer is flattened to serve as -input to a hidden layer, such that each neuron in the final pooling -layer is connected to every single neuron in the hidden layer. This -then serves as input to the output layer, e.g. a softmax output for -classification. +The MNIST dataset consists of grayscale images with a pixel size of +\( 28\times 28 \), meaning we require \( 28 \times 28 = 724 \) weights to each +neuron in the first hidden layer. + +

    +If we were to analyze images of size \( 128\times 128 \) we would require +\( 128 \times 128 = 16384 \) weights to each neuron. Even worse if we were +dealing with color images, as most images are, we have an image matrix +of size \( 128\times 128 \) for each color dimension (Red, Green, Blue), +meaning 3 times the number of weights \( = 49152 \) are required for every +single neuron in the first hidden layer.

    @@ -419,7 +437,7 @@ classification.

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    • diff --git a/doc/pub/week42/html/._week42-bs073.html b/doc/pub/week42/html/._week42-bs073.html index 7c092099a..57db5dc17 100644 --- a/doc/pub/week42/html/._week42-bs073.html +++ b/doc/pub/week42/html/._week42-bs073.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -381,51 +396,21 @@ MathJax.Hub.Config({ -

    Prerequisites: Collect and pre-process data

    +

    Strong correlations

    +

    +Images typically have strong local correlations, meaning that a small +part of the image varies little from its neighboring regions. If for +example we have an image of a blue car, we can roughly assume that a +small blue part of the image is surrounded by other blue regions. - -

    # import necessary packages
-import numpy as np
-import matplotlib.pyplot as plt
-from sklearn import datasets
+
    
+Therefore, instead of connecting every single pixel to a neuron in the
+first hidden layer, as we have previously done with deep neural
+networks, we can instead connect each neuron to a small part of the
+image (in all 3 RGB depth dimensions).  The size of each small area is
+fixed, and known as a receptive.
 
-
-# ensure the same random numbers appear every time
-np.random.seed(0)
-
-# display images in notebook
-%matplotlib inline
-plt.rcParams['figure.figsize'] = (12,12)
-
-
-# download MNIST dataset
-digits = datasets.load_digits()
-
-# define inputs and labels
-inputs = digits.images
-labels = digits.target
-
-# RGB images have a depth of 3
-# our images are grayscale so they should have a depth of 1
-inputs = inputs[:,:,:,np.newaxis]
-
-print("inputs = (n_inputs, pixel_width, pixel_height, depth) = " + str(inputs.shape))
-print("labels = (n_inputs) = " + str(labels.shape))
-
-
-# choose some random images to display
-n_inputs = len(inputs)
-indices = np.arange(n_inputs)
-random_indices = np.random.choice(indices, size=5)
-
-for i, image in enumerate(digits.images[random_indices]):
-    plt.subplot(1, 5, i+1)
-    plt.axis('off')
-    plt.imshow(image, cmap=plt.cm.gray_r, interpolation='nearest')
-    plt.title("Label: %d" % digits.target[random_indices[i]])
-plt.show()
-

    @@ -452,7 +437,7 @@ plt.show()

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    • diff --git a/doc/pub/week42/html/._week42-bs074.html b/doc/pub/week42/html/._week42-bs074.html index 6cf8283cc..e88f8e75c 100644 --- a/doc/pub/week42/html/._week42-bs074.html +++ b/doc/pub/week42/html/._week42-bs074.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -379,35 +394,27 @@ MathJax.Hub.Config({

     

     

     

    - + + +

    Layers of a CNN

    +The layers of a convolutional neural network arrange neurons in 3D: width, height and depth. +The input image is typically a square matrix of depth 3. -

    Importing Keras and Tensorflow

    +A convolution is performed on the image which outputs +a 3D volume of neurons. The weights to the input are arranged in a number of 2D matrices, known as filters. - -

    from tensorflow.keras import datasets, layers, models
-from tensorflow.keras.layers import Input
-from tensorflow.keras.models import Sequential      #This allows appending layers to existing models
-from tensorflow.keras.layers import Dense           #This allows defining the characteristics of a particular layer
-from tensorflow.keras import optimizers             #This allows using whichever optimiser we want (sgd,adam,RMSprop)
-from tensorflow.keras import regularizers           #This allows using whichever regularizer we want (l1,l2,l1_l2)
-from tensorflow.keras.utils import to_categorical   #This allows using categorical cross entropy as the cost function
-#from tensorflow.keras import Conv2D
-#from tensorflow.keras import MaxPooling2D
-#from tensorflow.keras import Flatten
+
    
+Each filter slides along the input image, taking the dot product
+between each small part of the image and the filter, in all depth
+dimensions. This is then passed through a non-linear function,
+typically the Rectified Linear (ReLu) function, which serves as the
+activation of the neurons in the first convolutional layer. This is
+further passed through a pooling layer, which reduces the size of the
+convolutional layer, e.g. by taking the maximum or average across some
+small regions, and this serves as input to the next convolutional
+layer.
 
-from sklearn.model_selection import train_test_split
-
-# representation of labels
-labels = to_categorical(labels)
-
-# split into train and test data
-# one-liner from scikit-learn library
-train_size = 0.8
-test_size = 1 - train_size
-X_train, X_test, Y_train, Y_test = train_test_split(inputs, labels, train_size=train_size,
-                                                    test_size=test_size)
-

    @@ -433,6 +440,8 @@ X_train, X_test, Y_train, Y_test = train_tes

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    • diff --git a/doc/pub/week42/html/._week42-bs075.html b/doc/pub/week42/html/._week42-bs075.html index 27b559e79..ad5f57d86 100644 --- a/doc/pub/week42/html/._week42-bs075.html +++ b/doc/pub/week42/html/._week42-bs075.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -379,40 +394,20 @@ MathJax.Hub.Config({

     

     

     

    - + -

    Running with Keras

    +

    Systematic reduction

    +By systematically reducing the size of the input volume, through +convolution and pooling, the network should create representations of +small parts of the input, and then from them assemble representations +of larger areas. The final pooling layer is flattened to serve as +input to a hidden layer, such that each neuron in the final pooling +layer is connected to every single neuron in the hidden layer. This +then serves as input to the output layer, e.g. a softmax output for +classification. - -

    def create_convolutional_neural_network_keras(input_shape, receptive_field,
-                                              n_filters, n_neurons_connected, n_categories,
-                                              eta, lmbd):
-    model = Sequential()
-    model.add(layers.Conv2D(n_filters, (receptive_field, receptive_field), input_shape=input_shape, padding='same',
-              activation='relu', kernel_regularizer=regularizers.l2(lmbd)))
-    model.add(layers.MaxPooling2D(pool_size=(2, 2)))
-    model.add(layers.Flatten())
-    model.add(layers.Dense(n_neurons_connected, activation='relu', kernel_regularizer=regularizers.l2(lmbd)))
-    model.add(layers.Dense(n_categories, activation='softmax', kernel_regularizer=regularizers.l2(lmbd)))
-    
-    sgd = optimizers.SGD(lr=eta)
-    model.compile(loss='categorical_crossentropy', optimizer=sgd, metrics=['accuracy'])
-    
-    return model
-
-epochs = 100
-batch_size = 100
-input_shape = X_train.shape[1:4]
-receptive_field = 3
-n_filters = 10
-n_neurons_connected = 50
-n_categories = 10
-
-eta_vals = np.logspace(-5, 1, 7)
-lmbd_vals = np.logspace(-5, 1, 7)
-

    @@ -437,6 +432,9 @@ lmbd_vals = np.

  • 82
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    • +
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    • +
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    • +
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  • »
    • diff --git a/doc/pub/week42/html/week42-bs.html b/doc/pub/week42/html/week42-bs.html index cbef1a23a..1ddf0f142 100644 --- a/doc/pub/week42/html/week42-bs.html +++ b/doc/pub/week42/html/week42-bs.html @@ -194,10 +194,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -213,6 +217,14 @@ Automatically generated HTML file from DocOnce source None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -343,29 +355,32 @@ MathJax.Hub.Config({
  • CNNs in brief
  • Mathematics of CNNs
  • Convolution Examples: Polynomial multiplication
    • -
  • Convolution Examples: Probability Theory
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Convolution Examples: Probability Theory
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -424,7 +439,7 @@ MathJax.Hub.Config({
  • 9
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  • ...
    • -
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  • »
    • diff --git a/doc/pub/week42/html/week42-reveal.html b/doc/pub/week42/html/week42-reveal.html index 654c56dbe..c9cd01597 100644 --- a/doc/pub/week42/html/week42-reveal.html +++ b/doc/pub/week42/html/week42-reveal.html @@ -3226,13 +3226,98 @@ How can we use this? And what does it mean? Let us study some familiar examples

    Convolution Examples: Polynomial multiplication

    -We have already met such an example in project 1 when we tried to set up the design matrix for a two-dimensional function. -Let us remind of this and recast it in terms of the mathematical operation of convolution. +We have already met such an example in project 1 when we tried to set +up the design matrix for a two-dimensional function. This was an +example of polynomial multiplication. Let us recast such a problem in terms of the convolution operation. +Let us look a the following polynomials to second and third order, respectively: +

     
    +$$ +p(t) = \alpha_0+\alpha_1 t+\alpha_2 t^2, +$$ +

     
    + +and +

     
    +$$ +s(t) = \beta_0+\beta_1 t+\beta_2 t^2+\beta_3 t^3. +$$ +

     
    + +

    +The polynomial multiplication gives us a new polynomial of degree \( 5 \) +

     
    +$$ +z(t) = \delta_0+\delta_1 t+\delta_2 t^2+\delta_3 t^3+\delta_4 t^4+\delta_5 t^5. +$$ +

     

    -

    Convolution Examples: Probability Theory

    +

    Efficient Polynomial Multiplication

    + +

    +Computing polynomial products can be implemented efficiently if we rewrite the the more brute force multiplications using convolution. +We note first that the new coefficients are given as + +

     
    +$$ +\begin{split} +\delta_0=&\alpha_0\beta_0\\ +\delta_1=&\alpha_1\beta_0+\beta_0\alpha_1\\ +\delta_2=&\alpha_0\beta_2+\beta_1\alpha_1+\alpha_2\beta_0\\ +\delta_3=&\alpha_1\beta_2+\beta_1\alpha_2+\alpha_0\beta_3\\ +\delta_4=&\alpha_2\beta_2+\beta_3\alpha_1\\ +\delta_5=&\alpha_2\beta_3.\\ +\end{split} +$$ +

     
    + +

    +We note that \( \alpha_i=0 \) except for \( i\in \left{0,1,2\right} \) and \( \beta_i=0 \) except for \( i\in\left{0,1,2,3\right} \). + +

    +We can then rewrite the coefficients \( \delta_j \) using a discrete convolution as +

     
    +$$ +\delta_j = \sum_{i=-\infty}^{i=\infty}\alpha_i\beta_{j-i}=(\alpha * \beta)_j, +$$ +

     
    + +or as a double sum with restriction \( l=i+j \) +

     
    +$$ +\delta_l = \sum_{ij}\alpha_i\beta_{j}. +$$ +

     
    + +

    +Do you see a potential drawback with these equations? +

    + + +
    +

    A more efficient way of coding the above Convolution

    + +

    +Since we only have a finite number of \( \alpha \) and \( \beta \) values +which are non-zero, we can rewrite the above convolution expressions +as a matrix-vector multiplication + +

     
    +$$ +\boldsymbol{\delta}=\begin{bmatriax}\alpha_0 & 0 & 0 & 0 \\ + \alpha_1 & \alpha_0 & 0 & 0 \\ + \alpha_2 & \alpha_1 & \alpha_0 & 0 \\ + 0 & \alpha_2 & \alpha_1 & \alpha_0 \\ + 0 & 0 & \alpha_2 & \alpha_1 \\ + 0 & 0 & 0 & \alpha_2 + \end{bmatrix}\begin{bmatrix} \beta_0 \\ \beta_1 \\ \beta_2 \\ \beta_3\end{bmatrix} +$$ +

     
    + +

    +The process is commutative and we can easily see that we can rewrite the multiplication in terms of a martrix holding \( \beta \) and a vector holding \( \alpha \).

    @@ -3243,7 +3328,7 @@ Let us remind of this and recast it in terms of the mathematical operation of co For problems with so-called harmonic oscillations, given by for example the following differential equation

     
    $$ -m\frac{d^2x(t)}(dt^2}+\eta\frac{dx}{dt}+x(t)=F(t), +m\frac{d^2x}{dt^2}+\eta\frac{dx}{dt}+x(t)=F(t), $$

     
    @@ -3458,6 +3543,62 @@ The same method can be used to check for the consistency of \( g_n \). +

    +

    Final words on Fourier Transforms

    + +

    +The code here uses the Fourier series applied to a +square wave signal. The code here +visualizes the various approximations given by Fourier series compared +with a square wave with period \( T=0.2 \) (dimensionless time), width \( 0.1 \) and max value of the force \( F=2 \). We +see that when we increase the number of components in the Fourier +series, the Fourier series approximation gets closer and closer to the +square wave signal. + +

    + + +

    import numpy as np
+import math
+from scipy import signal
+import matplotlib.pyplot as plt
+
+# number of points                                                                                       
+n = 500
+# start and final times                                                                                  
+t0 = 0.0
+tn = 1.0
+# Period                                                                                                 
+T =0.2
+# Max value of square signal                                                                             
+Fmax= 2.0
+# Width of signal   
+Width = 0.1
+t = np.linspace(t0, tn, n, endpoint=False)
+SqrSignal = np.zeros(n)
+FourierSeriesSignal = np.zeros(n)
+SqrSignal = 1.0+signal.square(2*np.pi*5*t+np.pi*Width/T)
+a0 = Fmax*Width/T
+FourierSeriesSignal = a0
+Factor = 2.0*Fmax/np.pi
+for i in range(1,500):
+    FourierSeriesSignal += Factor/(i)*np.sin(np.pi*i*Width/T)*np.cos(i*t*2*np.pi/T)
+plt.plot(t, SqrSignal)
+plt.plot(t, FourierSeriesSignal)
+plt.ylim(-0.5, 2.5)
+plt.show()
+
    +
    + + +
    +

    Convolution Examples: Probability Theory

    + +

    +More text will be added here +

    + +

    CNNs in more detail, building convolutional neural networks in Tensorflow and Keras

    diff --git a/doc/pub/week42/html/week42-solarized.html b/doc/pub/week42/html/week42-solarized.html index c1d5beee2..1591bb175 100644 --- a/doc/pub/week42/html/week42-solarized.html +++ b/doc/pub/week42/html/week42-solarized.html @@ -214,10 +214,14 @@ div { text-align: justify; text-justify: inter-word; } 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -233,6 +237,14 @@ div { text-align: justify; text-justify: inter-word; } None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -3239,13 +3251,84 @@ How can we use this? And what does it mean? Let us study some familiar examples

    Convolution Examples: Polynomial multiplication

    -We have already met such an example in project 1 when we tried to set up the design matrix for a two-dimensional function. -Let us remind of this and recast it in terms of the mathematical operation of convolution. +We have already met such an example in project 1 when we tried to set +up the design matrix for a two-dimensional function. This was an +example of polynomial multiplication. Let us recast such a problem in terms of the convolution operation. +Let us look a the following polynomials to second and third order, respectively: +$$ +p(t) = \alpha_0+\alpha_1 t+\alpha_2 t^2, +$$ + +and +$$ +s(t) = \beta_0+\beta_1 t+\beta_2 t^2+\beta_3 t^3. +$$ + +

    +The polynomial multiplication gives us a new polynomial of degree \( 5 \) +$$ +z(t) = \delta_0+\delta_1 t+\delta_2 t^2+\delta_3 t^3+\delta_4 t^4+\delta_5 t^5. +$$











    -

    Convolution Examples: Probability Theory

    +

    Efficient Polynomial Multiplication

    + +

    +Computing polynomial products can be implemented efficiently if we rewrite the the more brute force multiplications using convolution. +We note first that the new coefficients are given as + +$$ +\begin{split} +\delta_0=&\alpha_0\beta_0\\ +\delta_1=&\alpha_1\beta_0+\beta_0\alpha_1\\ +\delta_2=&\alpha_0\beta_2+\beta_1\alpha_1+\alpha_2\beta_0\\ +\delta_3=&\alpha_1\beta_2+\beta_1\alpha_2+\alpha_0\beta_3\\ +\delta_4=&\alpha_2\beta_2+\beta_3\alpha_1\\ +\delta_5=&\alpha_2\beta_3.\\ +\end{split} +$$ + +

    +We note that \( \alpha_i=0 \) except for \( i\in \left{0,1,2\right} \) and \( \beta_i=0 \) except for \( i\in\left{0,1,2,3\right} \). + +

    +We can then rewrite the coefficients \( \delta_j \) using a discrete convolution as +$$ +\delta_j = \sum_{i=-\infty}^{i=\infty}\alpha_i\beta_{j-i}=(\alpha * \beta)_j, +$$ + +or as a double sum with restriction \( l=i+j \) +$$ +\delta_l = \sum_{ij}\alpha_i\beta_{j}. +$$ + +

    +Do you see a potential drawback with these equations? + +

    +









    + +

    A more efficient way of coding the above Convolution

    + +

    +Since we only have a finite number of \( \alpha \) and \( \beta \) values +which are non-zero, we can rewrite the above convolution expressions +as a matrix-vector multiplication + +$$ +\boldsymbol{\delta}=\begin{bmatriax}\alpha_0 & 0 & 0 & 0 \\ + \alpha_1 & \alpha_0 & 0 & 0 \\ + \alpha_2 & \alpha_1 & \alpha_0 & 0 \\ + 0 & \alpha_2 & \alpha_1 & \alpha_0 \\ + 0 & 0 & \alpha_2 & \alpha_1 \\ + 0 & 0 & 0 & \alpha_2 + \end{bmatrix}\begin{bmatrix} \beta_0 \\ \beta_1 \\ \beta_2 \\ \beta_3\end{bmatrix} +$$ + +

    +The process is commutative and we can easily see that we can rewrite the multiplication in terms of a martrix holding \( \beta \) and a vector holding \( \alpha \).











    @@ -3255,7 +3338,7 @@ Let us remind of this and recast it in terms of the mathematical operation of co

    For problems with so-called harmonic oscillations, given by for example the following differential equation $$ -m\frac{d^2x(t)}(dt^2}+\eta\frac{dx}{dt}+x(t)=F(t), +m\frac{d^2x}{dt^2}+\eta\frac{dx}{dt}+x(t)=F(t), $$ where \( F(t) \) is an applied external force acting on the system (often called a driving force), one can use the theory of Fourier transformations to find the solutions of this type of equations. @@ -3452,6 +3535,61 @@ The same method can be used to check for the consistency of \( g_n \).











    +

    Final words on Fourier Transforms

    + +

    +The code here uses the Fourier series applied to a +square wave signal. The code here +visualizes the various approximations given by Fourier series compared +with a square wave with period \( T=0.2 \) (dimensionless time), width \( 0.1 \) and max value of the force \( F=2 \). We +see that when we increase the number of components in the Fourier +series, the Fourier series approximation gets closer and closer to the +square wave signal. + +

    + + +

    import numpy as np
+import math
+from scipy import signal
+import matplotlib.pyplot as plt
+
+# number of points                                                                                       
+n = 500
+# start and final times                                                                                  
+t0 = 0.0
+tn = 1.0
+# Period                                                                                                 
+T =0.2
+# Max value of square signal                                                                             
+Fmax= 2.0
+# Width of signal   
+Width = 0.1
+t = np.linspace(t0, tn, n, endpoint=False)
+SqrSignal = np.zeros(n)
+FourierSeriesSignal = np.zeros(n)
+SqrSignal = 1.0+signal.square(2*np.pi*5*t+np.pi*Width/T)
+a0 = Fmax*Width/T
+FourierSeriesSignal = a0
+Factor = 2.0*Fmax/np.pi
+for i in range(1,500):
+    FourierSeriesSignal += Factor/(i)*np.sin(np.pi*i*Width/T)*np.cos(i*t*2*np.pi/T)
+plt.plot(t, SqrSignal)
+plt.plot(t, FourierSeriesSignal)
+plt.ylim(-0.5, 2.5)
+plt.show()
+
    +

    +









    + +

    Convolution Examples: Probability Theory

    + +

    +More text will be added here + +

    +









    +

    CNNs in more detail, building convolutional neural networks in Tensorflow and Keras

    diff --git a/doc/pub/week42/html/week42.html b/doc/pub/week42/html/week42.html index b667683db..009050326 100644 --- a/doc/pub/week42/html/week42.html +++ b/doc/pub/week42/html/week42.html @@ -219,10 +219,14 @@ div { text-align: justify; text-justify: inter-word; } 2, None, 'convolution-examples-polynomial-multiplication'), - ('Convolution Examples: Probability Theory', + ('Efficient Polynomial Multiplication', 2, None, - 'convolution-examples-probability-theory'), + 'efficient-polynomial-multiplication'), + ('A more efficient way of coding the above Convolution', + 2, + None, + 'a-more-efficient-way-of-coding-the-above-convolution'), ('Convolution Examples: Principle of Superposition and Periodic ' 'Forces (Fourier Transforms)', 2, @@ -238,6 +242,14 @@ div { text-align: justify; text-justify: inter-word; } None, 'wrapping-up-fourier-transforms'), ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), + ('Final words on Fourier Transforms', + 2, + None, + 'final-words-on-fourier-transforms'), + ('Convolution Examples: Probability Theory', + 2, + None, + 'convolution-examples-probability-theory'), ('CNNs in more detail, building convolutional neural networks in ' 'Tensorflow and Keras', 2, @@ -3244,13 +3256,84 @@ How can we use this? And what does it mean? Let us study some familiar examples

    Convolution Examples: Polynomial multiplication

    -We have already met such an example in project 1 when we tried to set up the design matrix for a two-dimensional function. -Let us remind of this and recast it in terms of the mathematical operation of convolution. +We have already met such an example in project 1 when we tried to set +up the design matrix for a two-dimensional function. This was an +example of polynomial multiplication. Let us recast such a problem in terms of the convolution operation. +Let us look a the following polynomials to second and third order, respectively: +$$ +p(t) = \alpha_0+\alpha_1 t+\alpha_2 t^2, +$$ + +and +$$ +s(t) = \beta_0+\beta_1 t+\beta_2 t^2+\beta_3 t^3. +$$ + +

    +The polynomial multiplication gives us a new polynomial of degree \( 5 \) +$$ +z(t) = \delta_0+\delta_1 t+\delta_2 t^2+\delta_3 t^3+\delta_4 t^4+\delta_5 t^5. +$$











    -

    Convolution Examples: Probability Theory

    +

    Efficient Polynomial Multiplication

    + +

    +Computing polynomial products can be implemented efficiently if we rewrite the the more brute force multiplications using convolution. +We note first that the new coefficients are given as + +$$ +\begin{split} +\delta_0=&\alpha_0\beta_0\\ +\delta_1=&\alpha_1\beta_0+\beta_0\alpha_1\\ +\delta_2=&\alpha_0\beta_2+\beta_1\alpha_1+\alpha_2\beta_0\\ +\delta_3=&\alpha_1\beta_2+\beta_1\alpha_2+\alpha_0\beta_3\\ +\delta_4=&\alpha_2\beta_2+\beta_3\alpha_1\\ +\delta_5=&\alpha_2\beta_3.\\ +\end{split} +$$ + +

    +We note that \( \alpha_i=0 \) except for \( i\in \left{0,1,2\right} \) and \( \beta_i=0 \) except for \( i\in\left{0,1,2,3\right} \). + +

    +We can then rewrite the coefficients \( \delta_j \) using a discrete convolution as +$$ +\delta_j = \sum_{i=-\infty}^{i=\infty}\alpha_i\beta_{j-i}=(\alpha * \beta)_j, +$$ + +or as a double sum with restriction \( l=i+j \) +$$ +\delta_l = \sum_{ij}\alpha_i\beta_{j}. +$$ + +

    +Do you see a potential drawback with these equations? + +

    +









    + +

    A more efficient way of coding the above Convolution

    + +

    +Since we only have a finite number of \( \alpha \) and \( \beta \) values +which are non-zero, we can rewrite the above convolution expressions +as a matrix-vector multiplication + +$$ +\boldsymbol{\delta}=\begin{bmatriax}\alpha_0 & 0 & 0 & 0 \\ + \alpha_1 & \alpha_0 & 0 & 0 \\ + \alpha_2 & \alpha_1 & \alpha_0 & 0 \\ + 0 & \alpha_2 & \alpha_1 & \alpha_0 \\ + 0 & 0 & \alpha_2 & \alpha_1 \\ + 0 & 0 & 0 & \alpha_2 + \end{bmatrix}\begin{bmatrix} \beta_0 \\ \beta_1 \\ \beta_2 \\ \beta_3\end{bmatrix} +$$ + +

    +The process is commutative and we can easily see that we can rewrite the multiplication in terms of a martrix holding \( \beta \) and a vector holding \( \alpha \).











    @@ -3260,7 +3343,7 @@ Let us remind of this and recast it in terms of the mathematical operation of co

    For problems with so-called harmonic oscillations, given by for example the following differential equation $$ -m\frac{d^2x(t)}(dt^2}+\eta\frac{dx}{dt}+x(t)=F(t), +m\frac{d^2x}{dt^2}+\eta\frac{dx}{dt}+x(t)=F(t), $$ where \( F(t) \) is an applied external force acting on the system (often called a driving force), one can use the theory of Fourier transformations to find the solutions of this type of equations. @@ -3457,6 +3540,61 @@ The same method can be used to check for the consistency of \( g_n \).











    +

    Final words on Fourier Transforms

    + +

    +The code here uses the Fourier series applied to a +square wave signal. The code here +visualizes the various approximations given by Fourier series compared +with a square wave with period \( T=0.2 \) (dimensionless time), width \( 0.1 \) and max value of the force \( F=2 \). We +see that when we increase the number of components in the Fourier +series, the Fourier series approximation gets closer and closer to the +square wave signal. + +

    + + +

    import numpy as np
+import math
+from scipy import signal
+import matplotlib.pyplot as plt
+
+# number of points                                                                                       
+n = 500
+# start and final times                                                                                  
+t0 = 0.0
+tn = 1.0
+# Period                                                                                                 
+T =0.2
+# Max value of square signal                                                                             
+Fmax= 2.0
+# Width of signal   
+Width = 0.1
+t = np.linspace(t0, tn, n, endpoint=False)
+SqrSignal = np.zeros(n)
+FourierSeriesSignal = np.zeros(n)
+SqrSignal = 1.0+signal.square(2*np.pi*5*t+np.pi*Width/T)
+a0 = Fmax*Width/T
+FourierSeriesSignal = a0
+Factor = 2.0*Fmax/np.pi
+for i in range(1,500):
+    FourierSeriesSignal += Factor/(i)*np.sin(np.pi*i*Width/T)*np.cos(i*t*2*np.pi/T)
+plt.plot(t, SqrSignal)
+plt.plot(t, FourierSeriesSignal)
+plt.ylim(-0.5, 2.5)
+plt.show()
+
    +

    +









    + +

    Convolution Examples: Probability Theory

    + +

    +More text will be added here + +

    +









    +

    CNNs in more detail, building convolutional neural networks in Tensorflow and Keras

    diff --git a/doc/pub/week42/ipynb/ipynb-week42-src.tar.gz b/doc/pub/week42/ipynb/ipynb-week42-src.tar.gz index e6627596269262872d28a10044ccab3ee6b43497..1dfa21863c5a9a1c3708407ed614cd67e39a0693 100644 GIT binary patch delta 20 bcmeBP%i6t`l})~zgQ115k!>p*V`~@yLj48f delta 20 bcmeBP%i6t`l})~zgW(->BimLs#?~+ZMqdU( diff --git a/doc/pub/week42/ipynb/week42.ipynb b/doc/pub/week42/ipynb/week42.ipynb index 9f1147cb4..a75354186 100644 --- a/doc/pub/week42/ipynb/week42.ipynb +++ b/doc/pub/week42/ipynb/week42.ipynb @@ -3350,16 +3350,150 @@ "\n", "## Convolution Examples: Polynomial multiplication\n", "\n", - "We have already met such an example in project 1 when we tried to set up the design matrix for a two-dimensional function.\n", - "Let us remind of this and recast it in terms of the mathematical operation of convolution.\n", + "We have already met such an example in project 1 when we tried to set\n", + "up the design matrix for a two-dimensional function. This was an\n", + "example of polynomial multiplication. Let us recast such a problem in terms of the convolution operation.\n", + "Let us look a the following polynomials to second and third order, respectively:" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "p(t) = \\alpha_0+\\alpha_1 t+\\alpha_2 t^2,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "and" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "s(t) = \\beta_0+\\beta_1 t+\\beta_2 t^2+\\beta_3 t^3.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The polynomial multiplication gives us a new polynomial of degree $5$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "z(t) = \\delta_0+\\delta_1 t+\\delta_2 t^2+\\delta_3 t^3+\\delta_4 t^4+\\delta_5 t^5.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Efficient Polynomial Multiplication\n", + "\n", + "Computing polynomial products can be implemented efficiently if we rewrite the the more brute force multiplications using convolution.\n", + "We note first that the new coefficients are given as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\begin{split}\n", + "\\delta_0=&\\alpha_0\\beta_0\\\\\n", + "\\delta_1=&\\alpha_1\\beta_0+\\beta_0\\alpha_1\\\\\n", + "\\delta_2=&\\alpha_0\\beta_2+\\beta_1\\alpha_1+\\alpha_2\\beta_0\\\\\n", + "\\delta_3=&\\alpha_1\\beta_2+\\beta_1\\alpha_2+\\alpha_0\\beta_3\\\\\n", + "\\delta_4=&\\alpha_2\\beta_2+\\beta_3\\alpha_1\\\\\n", + "\\delta_5=&\\alpha_2\\beta_3.\\\\\n", + "\\end{split}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We note that $\\alpha_i=0$ except for $i\\in \\left{0,1,2\\right}$ and $\\beta_i=0$ except for $i\\in\\left{0,1,2,3\\right}$.\n", + "\n", + "We can then rewrite the coefficients $\\delta_j$ using a discrete convolution as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\delta_j = \\sum_{i=-\\infty}^{i=\\infty}\\alpha_i\\beta_{j-i}=(\\alpha * \\beta)_j,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "or as a double sum with restriction $l=i+j$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\delta_l = \\sum_{ij}\\alpha_i\\beta_{j}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Do you see a potential drawback with these equations?\n", + "\n", + "## A more efficient way of coding the above Convolution\n", + "\n", + "Since we only have a finite number of $\\alpha$ and $\\beta$ values\n", + "which are non-zero, we can rewrite the above convolution expressions\n", + "as a matrix-vector multiplication" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{\\delta}=\\begin{bmatriax}\\alpha_0 & 0 & 0 & 0 \\\\\n", + " \\alpha_1 & \\alpha_0 & 0 & 0 \\\\\n", + "\t\t\t \\alpha_2 & \\alpha_1 & \\alpha_0 & 0 \\\\\n", + "\t\t\t 0 & \\alpha_2 & \\alpha_1 & \\alpha_0 \\\\\n", + "\t\t\t 0 & 0 & \\alpha_2 & \\alpha_1 \\\\\n", + "\t\t\t 0 & 0 & 0 & \\alpha_2\n", + "\t\t\t \\end{bmatrix}\\begin{bmatrix} \\beta_0 \\\\ \\beta_1 \\\\ \\beta_2 \\\\ \\beta_3\\end{bmatrix}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The process is commutative and we can easily see that we can rewrite the multiplication in terms of a martrix holding $\\beta$ and a vector holding $\\alpha$.\n", "\n", "\n", "\n", "\n", - "\n", - "## Convolution Examples: Probability Theory\n", - "\n", - "\n", "## Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)\n", "\n", "For problems with so-called harmonic oscillations, given by for example the following differential equation" @@ -3370,7 +3504,7 @@ "metadata": {}, "source": [ "$$\n", - "m\\frac{d^2x(t)}(dt^2}+\\eta\\frac{dx}{dt}+x(t)=F(t),\n", + "m\\frac{d^2x}{dt^2}+\\eta\\frac{dx}{dt}+x(t)=F(t),\n", "$$" ] }, @@ -3662,8 +3796,64 @@ "\n", "\n", "\n", + "## Final words on Fourier Transforms\n", "\n", + "The code here uses the Fourier series applied to a \n", + "square wave signal. The code here\n", + "visualizes the various approximations given by Fourier series compared\n", + "with a square wave with period $T=0.2$ (dimensionless time), width $0.1$ and max value of the force $F=2$. We\n", + "see that when we increase the number of components in the Fourier\n", + "series, the Fourier series approximation gets closer and closer to the\n", + "square wave signal." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "import numpy as np\n", + "import math\n", + "from scipy import signal\n", + "import matplotlib.pyplot as plt\n", "\n", + "# number of points \n", + "n = 500\n", + "# start and final times \n", + "t0 = 0.0\n", + "tn = 1.0\n", + "# Period \n", + "T =0.2\n", + "# Max value of square signal \n", + "Fmax= 2.0\n", + "# Width of signal \n", + "Width = 0.1\n", + "t = np.linspace(t0, tn, n, endpoint=False)\n", + "SqrSignal = np.zeros(n)\n", + "FourierSeriesSignal = np.zeros(n)\n", + "SqrSignal = 1.0+signal.square(2*np.pi*5*t+np.pi*Width/T)\n", + "a0 = Fmax*Width/T\n", + "FourierSeriesSignal = a0\n", + "Factor = 2.0*Fmax/np.pi\n", + "for i in range(1,500):\n", + " FourierSeriesSignal += Factor/(i)*np.sin(np.pi*i*Width/T)*np.cos(i*t*2*np.pi/T)\n", + "plt.plot(t, SqrSignal)\n", + "plt.plot(t, FourierSeriesSignal)\n", + "plt.ylim(-0.5, 2.5)\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Convolution Examples: Probability Theory\n", + "\n", + "More text will be added here\n", "\n", "\n", "\n", diff --git a/doc/src/week42/week42.do.txt b/doc/src/week42/week42.do.txt index a431401b8..99cd9b644 100644 --- a/doc/src/week42/week42.do.txt +++ b/doc/src/week42/week42.do.txt @@ -2637,15 +2637,86 @@ How can we use this? And what does it mean? Let us study some familiar examples !split ===== Convolution Examples: Polynomial multiplication ===== -We have already met such an example in project 1 when we tried to set up the design matrix for a two-dimensional function. -Let us remind of this and recast it in terms of the mathematical operation of convolution. - - - +We have already met such an example in project 1 when we tried to set +up the design matrix for a two-dimensional function. This was an +example of polynomial multiplication. Let us recast such a problem in terms of the convolution operation. +Let us look a the following polynomials to second and third order, respectively: +!bt +\[ +p(t) = \alpha_0+\alpha_1 t+\alpha_2 t^2, +\] +!et +and +!bt +\[ +s(t) = \beta_0+\beta_1 t+\beta_2 t^2+\beta_3 t^3. +\] +!et +The polynomial multiplication gives us a new polynomial of degree $5$ +!bt +\[ +z(t) = \delta_0+\delta_1 t+\delta_2 t^2+\delta_3 t^3+\delta_4 t^4+\delta_5 t^5. +\] +!et !split -===== Convolution Examples: Probability Theory ===== +===== Efficient Polynomial Multiplication ===== + +Computing polynomial products can be implemented efficiently if we rewrite the the more brute force multiplications using convolution. +We note first that the new coefficients are given as + +!bt +\begin{split} +\delta_0=&\alpha_0\beta_0\\ +\delta_1=&\alpha_1\beta_0+\beta_0\alpha_1\\ +\delta_2=&\alpha_0\beta_2+\beta_1\alpha_1+\alpha_2\beta_0\\ +\delta_3=&\alpha_1\beta_2+\beta_1\alpha_2+\alpha_0\beta_3\\ +\delta_4=&\alpha_2\beta_2+\beta_3\alpha_1\\ +\delta_5=&\alpha_2\beta_3.\\ +\end{split} +!et + + +We note that $\alpha_i=0$ except for $i\in \left{0,1,2\right}$ and $\beta_i=0$ except for $i\in\left{0,1,2,3\right}$. + +We can then rewrite the coefficients $\delta_j$ using a discrete convolution as +!bt +\[ +\delta_j = \sum_{i=-\infty}^{i=\infty}\alpha_i\beta_{j-i}=(\alpha * \beta)_j, +\] +!et +or as a double sum with restriction $l=i+j$ +!bt +\[ +\delta_l = \sum_{ij}\alpha_i\beta_{j}. +\] +!et + +Do you see a potential drawback with these equations? + +!split +===== A more efficient way of coding the above Convolution ===== + +Since we only have a finite number of $\alpha$ and $\beta$ values +which are non-zero, we can rewrite the above convolution expressions +as a matrix-vector multiplication + +!bt +\[ +\bm{\delta}=\begin{bmatriax}\alpha_0 & 0 & 0 & 0 \\ + \alpha_1 & \alpha_0 & 0 & 0 \\ + \alpha_2 & \alpha_1 & \alpha_0 & 0 \\ + 0 & \alpha_2 & \alpha_1 & \alpha_0 \\ + 0 & 0 & \alpha_2 & \alpha_1 \\ + 0 & 0 & 0 & \alpha_2 + \end{bmatrix}\begin{bmatrix} \beta_0 \\ \beta_1 \\ \beta_2 \\ \beta_3\end{bmatrix} +\] +!et + +The process is commutative and we can easily see that we can rewrite the multiplication in terms of a martrix holding $\beta$ and a vector holding $\alpha$. + + !split @@ -2654,7 +2725,7 @@ Let us remind of this and recast it in terms of the mathematical operation of co For problems with so-called harmonic oscillations, given by for example the following differential equation !bt \[ -m\frac{d^2x(t)}(dt^2}+\eta\frac{dx}{dt}+x(t)=F(t), +m\frac{d^2x}{dt^2}+\eta\frac{dx}{dt}+x(t)=F(t), \] !et where $F(t)$ is an applied external force acting on the system (often called a driving force), one can use the theory of Fourier transformations to find the solutions of this type of equations. @@ -2825,8 +2896,54 @@ The same method can be used to check for the consistency of $g_n$. +!split +===== Final words on Fourier Transforms ===== + +The code here uses the Fourier series applied to a +square wave signal. The code here +visualizes the various approximations given by Fourier series compared +with a square wave with period $T=0.2$ (dimensionless time), width $0.1$ and max value of the force $F=2$. We +see that when we increase the number of components in the Fourier +series, the Fourier series approximation gets closer and closer to the +square wave signal. + +!bc pycod +import numpy as np +import math +from scipy import signal +import matplotlib.pyplot as plt + +# number of points +n = 500 +# start and final times +t0 = 0.0 +tn = 1.0 +# Period +T =0.2 +# Max value of square signal +Fmax= 2.0 +# Width of signal +Width = 0.1 +t = np.linspace(t0, tn, n, endpoint=False) +SqrSignal = np.zeros(n) +FourierSeriesSignal = np.zeros(n) +SqrSignal = 1.0+signal.square(2*np.pi*5*t+np.pi*Width/T) +a0 = Fmax*Width/T +FourierSeriesSignal = a0 +Factor = 2.0*Fmax/np.pi +for i in range(1,500): + FourierSeriesSignal += Factor/(i)*np.sin(np.pi*i*Width/T)*np.cos(i*t*2*np.pi/T) +plt.plot(t, SqrSignal) +plt.plot(t, FourierSeriesSignal) +plt.ylim(-0.5, 2.5) +plt.show() +!ec +!split +===== Convolution Examples: Probability Theory ===== + +More text will be added here