From a405c5a10a2508347470f9d3f1e0855fb829a995 Mon Sep 17 00:00:00 2001 From: mhjensen Date: Sat, 16 Nov 2019 22:45:50 +0100 Subject: [PATCH] added a simple svm example --- doc/pub/svm/html/._svm-bs000.html | 80 +++---- doc/pub/svm/html/._svm-bs001.html | 78 ++++--- doc/pub/svm/html/._svm-bs002.html | 78 ++++--- doc/pub/svm/html/._svm-bs003.html | 78 ++++--- doc/pub/svm/html/._svm-bs004.html | 78 ++++--- doc/pub/svm/html/._svm-bs005.html | 78 ++++--- doc/pub/svm/html/._svm-bs006.html | 78 ++++--- doc/pub/svm/html/._svm-bs007.html | 78 ++++--- doc/pub/svm/html/._svm-bs008.html | 90 ++++---- doc/pub/svm/html/._svm-bs009.html | 118 ++++------ doc/pub/svm/html/._svm-bs010.html | 128 ++++------- doc/pub/svm/html/._svm-bs011.html | 131 +++++------ doc/pub/svm/html/._svm-bs012.html | 129 ++++++----- doc/pub/svm/html/._svm-bs013.html | 121 ++++++---- doc/pub/svm/html/._svm-bs014.html | 115 +++++----- doc/pub/svm/html/._svm-bs015.html | 113 +++++----- doc/pub/svm/html/._svm-bs016.html | 124 +++++----- doc/pub/svm/html/._svm-bs017.html | 162 ++++++-------- doc/pub/svm/html/._svm-bs018.html | 131 ++++++----- doc/pub/svm/html/._svm-bs019.html | 173 ++++++++------ doc/pub/svm/html/._svm-bs020.html | 129 ++++++----- doc/pub/svm/html/._svm-bs021.html | 298 ++++++------------------- doc/pub/svm/html/._svm-bs022.html | 116 +++++----- doc/pub/svm/html/._svm-bs023.html | 286 +++++++++++++++++++----- doc/pub/svm/html/._svm-bs024.html | 145 +++++------- doc/pub/svm/html/._svm-bs025.html | 111 ++++----- doc/pub/svm/html/._svm-bs026.html | 166 +++++++++----- doc/pub/svm/html/svm-bs.html | 80 +++---- doc/pub/svm/html/svm-reveal.html | 73 ++++-- doc/pub/svm/html/svm-solarized.html | 110 +++++---- doc/pub/svm/html/svm.html | 110 +++++---- doc/pub/svm/ipynb/ipynb-svm-src.tar.gz | Bin 189 -> 189 bytes doc/pub/svm/ipynb/svm.ipynb | 32 ++- doc/pub/svm/pdf/svm-minted.pdf | Bin 265437 -> 266199 bytes doc/src/SupportVMachines/svm.do.txt | 35 ++- 35 files changed, 2033 insertions(+), 1819 deletions(-) diff --git a/doc/pub/svm/html/._svm-bs000.html b/doc/pub/svm/html/._svm-bs000.html index 6fb243975..811e86f4d 100644 --- a/doc/pub/svm/html/._svm-bs000.html +++ b/doc/pub/svm/html/._svm-bs000.html @@ -49,30 +49,32 @@ Automatically generated HTML file from DocOnce source ('Getting into the details', 2, None, '___sec5'), ('First attempt at a minimization approach', 2, None, '___sec6'), ('Solving the equations', 2, None, '___sec7'), - ('A better approach', 2, None, '___sec8'), + ('Code Example', 2, None, '___sec8'), + ('Problems with the Simpler Approach', 2, None, '___sec9'), + ('A better approach', 2, None, '___sec10'), ('A quick reminder on Lagrangian multipliers', 2, None, - '___sec9'), - ('Adding the muliplier', 2, None, '___sec10'), - ('Setting up the problem', 2, None, '___sec11'), - ('The problem to solve', 2, None, '___sec12'), - ('The last steps', 2, None, '___sec13'), - ('A soft classifier', 2, None, '___sec14'), - ('Soft optmization problem', 2, None, '___sec15'), - ('Kernels and non-linearity', 2, None, '___sec16'), - ('The equations', 2, None, '___sec17'), - ('The problem to solve', 2, None, '___sec18'), - ("Different kernels and Mercer's theorem", 2, None, '___sec19'), - ('The moons example', 2, None, '___sec20'), + '___sec11'), + ('Adding the muliplier', 2, None, '___sec12'), + ('Setting up the problem', 2, None, '___sec13'), + ('The problem to solve', 2, None, '___sec14'), + ('The last steps', 2, None, '___sec15'), + ('A soft classifier', 2, None, '___sec16'), + ('Soft optmization problem', 2, None, '___sec17'), + ('Kernels and non-linearity', 2, None, '___sec18'), + ('The equations', 2, None, '___sec19'), + ('The problem to solve', 2, None, '___sec20'), + ("Different kernels and Mercer's theorem", 2, None, '___sec21'), + ('The moons example', 2, None, '___sec22'), ('Mathematical optimization of convex functions', 2, None, - '___sec21'), - ('How do we solve these problems?', 2, None, '___sec22'), - ('A simple example', 2, None, '___sec23'), - ('Back to the more realistic cases', 2, None, '___sec24'), - ('Code Example', 2, None, '___sec25')]} + '___sec23'), + ('How do we solve these problems?', 2, None, '___sec24'), + ('A simple example', 2, None, '___sec25'), + ('Back to the more realistic cases', 2, None, '___sec26'), + ('Code Example', 2, None, '___sec27')]} end of tocinfo --> @@ -118,24 +120,26 @@ MathJax.Hub.Config({
  • Getting into the details
  • First attempt at a minimization approach
  • Solving the equations
    • -
  • A better approach
    • -
  • A quick reminder on Lagrangian multipliers
    • -
  • Adding the muliplier
    • -
  • Setting up the problem
    • -
  • The problem to solve
    • -
  • The last steps
    • -
  • A soft classifier
    • -
  • Soft optmization problem
    • -
  • Kernels and non-linearity
    • -
  • The equations
    • -
  • The problem to solve
    • -
  • Different kernels and Mercer's theorem
    • -
  • The moons example
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simple example
    • -
  • Back to the more realistic cases
    • -
  • Code Example
    • +
  • Code Example
    • +
  • Problems with the Simpler Approach
    • +
  • A better approach
    • +
  • A quick reminder on Lagrangian multipliers
    • +
  • Adding the muliplier
    • +
  • Setting up the problem
    • +
  • The problem to solve
    • +
  • The last steps
    • +
  • A soft classifier
    • +
  • Soft optmization problem
    • +
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • The problem to solve
    • +
  • Different kernels and Mercer's theorem
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simple example
    • +
  • Back to the more realistic cases
    • +
  • Code Example
    • @@ -170,7 +174,7 @@ MathJax.Hub.Config({
    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

    -

    Nov 15, 2019

    +

    Nov 16, 2019


    @@ -194,7 +198,7 @@ MathJax.Hub.Config({

  • 9
  • 10
  • ...
    • -
  • 27
    • +
  • 29
  • »
    • diff --git a/doc/pub/svm/html/._svm-bs001.html b/doc/pub/svm/html/._svm-bs001.html index edb9268bf..bfb7ac6b8 100644 --- a/doc/pub/svm/html/._svm-bs001.html +++ b/doc/pub/svm/html/._svm-bs001.html @@ -49,30 +49,32 @@ Automatically generated HTML file from DocOnce source ('Getting into the details', 2, None, '___sec5'), ('First attempt at a minimization approach', 2, None, '___sec6'), ('Solving the equations', 2, None, '___sec7'), - ('A better approach', 2, None, '___sec8'), + ('Code Example', 2, None, '___sec8'), + ('Problems with the Simpler Approach', 2, None, '___sec9'), + ('A better approach', 2, None, '___sec10'), ('A quick reminder on Lagrangian multipliers', 2, None, - '___sec9'), - ('Adding the muliplier', 2, None, '___sec10'), - ('Setting up the problem', 2, None, '___sec11'), - ('The problem to solve', 2, None, '___sec12'), - ('The last steps', 2, None, '___sec13'), - ('A soft classifier', 2, None, '___sec14'), - ('Soft optmization problem', 2, None, '___sec15'), - ('Kernels and non-linearity', 2, None, '___sec16'), - ('The equations', 2, None, '___sec17'), - ('The problem to solve', 2, None, '___sec18'), - ("Different kernels and Mercer's theorem", 2, None, '___sec19'), - ('The moons example', 2, None, '___sec20'), + '___sec11'), + ('Adding the muliplier', 2, None, '___sec12'), + ('Setting up the problem', 2, None, '___sec13'), + ('The problem to solve', 2, None, '___sec14'), + ('The last steps', 2, None, '___sec15'), + ('A soft classifier', 2, None, '___sec16'), + ('Soft optmization problem', 2, None, '___sec17'), + ('Kernels and non-linearity', 2, None, '___sec18'), + ('The equations', 2, None, '___sec19'), + ('The problem to solve', 2, None, '___sec20'), + ("Different kernels and Mercer's theorem", 2, None, '___sec21'), + ('The moons example', 2, None, '___sec22'), ('Mathematical optimization of convex functions', 2, None, - '___sec21'), - ('How do we solve these problems?', 2, None, '___sec22'), - ('A simple example', 2, None, '___sec23'), - ('Back to the more realistic cases', 2, None, '___sec24'), - ('Code Example', 2, None, '___sec25')]} + '___sec23'), + ('How do we solve these problems?', 2, None, '___sec24'), + ('A simple example', 2, None, '___sec25'), + ('Back to the more realistic cases', 2, None, '___sec26'), + ('Code Example', 2, None, '___sec27')]} end of tocinfo --> @@ -118,24 +120,26 @@ MathJax.Hub.Config({
  • Getting into the details
  • First attempt at a minimization approach
  • Solving the equations
    • -
  • A better approach
    • -
  • A quick reminder on Lagrangian multipliers
    • -
  • Adding the muliplier
    • -
  • Setting up the problem
    • -
  • The problem to solve
    • -
  • The last steps
    • -
  • A soft classifier
    • -
  • Soft optmization problem
    • -
  • Kernels and non-linearity
    • -
  • The equations
    • -
  • The problem to solve
    • -
  • Different kernels and Mercer's theorem
    • -
  • The moons example
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simple example
    • -
  • Back to the more realistic cases
    • -
  • Code Example
    • +
  • Code Example
    • +
  • Problems with the Simpler Approach
    • +
  • A better approach
    • +
  • A quick reminder on Lagrangian multipliers
    • +
  • Adding the muliplier
    • +
  • Setting up the problem
    • +
  • The problem to solve
    • +
  • The last steps
    • +
  • A soft classifier
    • +
  • Soft optmization problem
    • +
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • The problem to solve
    • +
  • Different kernels and Mercer's theorem
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simple example
    • +
  • Back to the more realistic cases
    • +
  • Code Example
    • @@ -196,7 +200,7 @@ unlikely that we can separate classes easily by say straight lines.
  • 10
  • 11
  • ...
    • -
  • 27
    • +
  • 29
  • »
    • diff --git a/doc/pub/svm/html/._svm-bs002.html b/doc/pub/svm/html/._svm-bs002.html index bb5af5309..1751f46d8 100644 --- a/doc/pub/svm/html/._svm-bs002.html +++ b/doc/pub/svm/html/._svm-bs002.html @@ -49,30 +49,32 @@ Automatically generated HTML file from DocOnce source ('Getting into the details', 2, None, '___sec5'), ('First attempt at a minimization approach', 2, None, '___sec6'), ('Solving the equations', 2, None, '___sec7'), - ('A better approach', 2, None, '___sec8'), + ('Code Example', 2, None, '___sec8'), + ('Problems with the Simpler Approach', 2, None, '___sec9'), + ('A better approach', 2, None, '___sec10'), ('A quick reminder on Lagrangian multipliers', 2, None, - '___sec9'), - ('Adding the muliplier', 2, None, '___sec10'), - ('Setting up the problem', 2, None, '___sec11'), - ('The problem to solve', 2, None, '___sec12'), - ('The last steps', 2, None, '___sec13'), - ('A soft classifier', 2, None, '___sec14'), - ('Soft optmization problem', 2, None, '___sec15'), - ('Kernels and non-linearity', 2, None, '___sec16'), - ('The equations', 2, None, '___sec17'), - ('The problem to solve', 2, None, '___sec18'), - ("Different kernels and Mercer's theorem", 2, None, '___sec19'), - ('The moons example', 2, None, '___sec20'), + '___sec11'), + ('Adding the muliplier', 2, None, '___sec12'), + ('Setting up the problem', 2, None, '___sec13'), + ('The problem to solve', 2, None, '___sec14'), + ('The last steps', 2, None, '___sec15'), + ('A soft classifier', 2, None, '___sec16'), + ('Soft optmization problem', 2, None, '___sec17'), + ('Kernels and non-linearity', 2, None, '___sec18'), + ('The equations', 2, None, '___sec19'), + ('The problem to solve', 2, None, '___sec20'), + ("Different kernels and Mercer's theorem", 2, None, '___sec21'), + ('The moons example', 2, None, '___sec22'), ('Mathematical optimization of convex functions', 2, None, - '___sec21'), - ('How do we solve these problems?', 2, None, '___sec22'), - ('A simple example', 2, None, '___sec23'), - ('Back to the more realistic cases', 2, None, '___sec24'), - ('Code Example', 2, None, '___sec25')]} + '___sec23'), + ('How do we solve these problems?', 2, None, '___sec24'), + ('A simple example', 2, None, '___sec25'), + ('Back to the more realistic cases', 2, None, '___sec26'), + ('Code Example', 2, None, '___sec27')]} end of tocinfo --> @@ -118,24 +120,26 @@ MathJax.Hub.Config({
  • Getting into the details
  • First attempt at a minimization approach
  • Solving the equations
    • -
  • A better approach
    • -
  • A quick reminder on Lagrangian multipliers
    • -
  • Adding the muliplier
    • -
  • Setting up the problem
    • -
  • The problem to solve
    • -
  • The last steps
    • -
  • A soft classifier
    • -
  • Soft optmization problem
    • -
  • Kernels and non-linearity
    • -
  • The equations
    • -
  • The problem to solve
    • -
  • Different kernels and Mercer's theorem
    • -
  • The moons example
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simple example
    • -
  • Back to the more realistic cases
    • -
  • Code Example
    • +
  • Code Example
    • +
  • Problems with the Simpler Approach
    • +
  • A better approach
    • +
  • A quick reminder on Lagrangian multipliers
    • +
  • Adding the muliplier
    • +
  • Setting up the problem
    • +
  • The problem to solve
    • +
  • The last steps
    • +
  • A soft classifier
    • +
  • Soft optmization problem
    • +
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • The problem to solve
    • +
  • Different kernels and Mercer's theorem
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simple example
    • +
  • Back to the more realistic cases
    • +
  • Code Example
    • @@ -253,7 +257,7 @@ plt.show()
  • 11
  • 12
  • ...
    • -
  • 27
    • +
  • 29
  • »
    • diff --git a/doc/pub/svm/html/._svm-bs003.html b/doc/pub/svm/html/._svm-bs003.html index 3d295f011..4283e0279 100644 --- a/doc/pub/svm/html/._svm-bs003.html +++ b/doc/pub/svm/html/._svm-bs003.html @@ -49,30 +49,32 @@ Automatically generated HTML file from DocOnce source ('Getting into the details', 2, None, '___sec5'), ('First attempt at a minimization approach', 2, None, '___sec6'), ('Solving the equations', 2, None, '___sec7'), - ('A better approach', 2, None, '___sec8'), + ('Code Example', 2, None, '___sec8'), + ('Problems with the Simpler Approach', 2, None, '___sec9'), + ('A better approach', 2, None, '___sec10'), ('A quick reminder on Lagrangian multipliers', 2, None, - '___sec9'), - ('Adding the muliplier', 2, None, '___sec10'), - ('Setting up the problem', 2, None, '___sec11'), - ('The problem to solve', 2, None, '___sec12'), - ('The last steps', 2, None, '___sec13'), - ('A soft classifier', 2, None, '___sec14'), - ('Soft optmization problem', 2, None, '___sec15'), - ('Kernels and non-linearity', 2, None, '___sec16'), - ('The equations', 2, None, '___sec17'), - ('The problem to solve', 2, None, '___sec18'), - ("Different kernels and Mercer's theorem", 2, None, '___sec19'), - ('The moons example', 2, None, '___sec20'), + '___sec11'), + ('Adding the muliplier', 2, None, '___sec12'), + ('Setting up the problem', 2, None, '___sec13'), + ('The problem to solve', 2, None, '___sec14'), + ('The last steps', 2, None, '___sec15'), + ('A soft classifier', 2, None, '___sec16'), + ('Soft optmization problem', 2, None, '___sec17'), + ('Kernels and non-linearity', 2, None, '___sec18'), + ('The equations', 2, None, '___sec19'), + ('The problem to solve', 2, None, '___sec20'), + ("Different kernels and Mercer's theorem", 2, None, '___sec21'), + ('The moons example', 2, None, '___sec22'), ('Mathematical optimization of convex functions', 2, None, - '___sec21'), - ('How do we solve these problems?', 2, None, '___sec22'), - ('A simple example', 2, None, '___sec23'), - ('Back to the more realistic cases', 2, None, '___sec24'), - ('Code Example', 2, None, '___sec25')]} + '___sec23'), + ('How do we solve these problems?', 2, None, '___sec24'), + ('A simple example', 2, None, '___sec25'), + ('Back to the more realistic cases', 2, None, '___sec26'), + ('Code Example', 2, None, '___sec27')]} end of tocinfo --> @@ -118,24 +120,26 @@ MathJax.Hub.Config({
  • Getting into the details
  • First attempt at a minimization approach
  • Solving the equations
    • -
  • A better approach
    • -
  • A quick reminder on Lagrangian multipliers
    • -
  • Adding the muliplier
    • -
  • Setting up the problem
    • -
  • The problem to solve
    • -
  • The last steps
    • -
  • A soft classifier
    • -
  • Soft optmization problem
    • -
  • Kernels and non-linearity
    • -
  • The equations
    • -
  • The problem to solve
    • -
  • Different kernels and Mercer's theorem
    • -
  • The moons example
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simple example
    • -
  • Back to the more realistic cases
    • -
  • Code Example
    • +
  • Code Example
    • +
  • Problems with the Simpler Approach
    • +
  • A better approach
    • +
  • A quick reminder on Lagrangian multipliers
    • +
  • Adding the muliplier
    • +
  • Setting up the problem
    • +
  • The problem to solve
    • +
  • The last steps
    • +
  • A soft classifier
    • +
  • Soft optmization problem
    • +
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • The problem to solve
    • +
  • Different kernels and Mercer's theorem
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simple example
    • +
  • Back to the more realistic cases
    • +
  • Code Example
    • @@ -194,7 +198,7 @@ $$
  • 12
  • 13
  • ...
    • -
  • 27
    • +
  • 29
  • »
    • diff --git a/doc/pub/svm/html/._svm-bs004.html b/doc/pub/svm/html/._svm-bs004.html index 3a87906a9..80a25a37c 100644 --- a/doc/pub/svm/html/._svm-bs004.html +++ b/doc/pub/svm/html/._svm-bs004.html @@ -49,30 +49,32 @@ Automatically generated HTML file from DocOnce source ('Getting into the details', 2, None, '___sec5'), ('First attempt at a minimization approach', 2, None, '___sec6'), ('Solving the equations', 2, None, '___sec7'), - ('A better approach', 2, None, '___sec8'), + ('Code Example', 2, None, '___sec8'), + ('Problems with the Simpler Approach', 2, None, '___sec9'), + ('A better approach', 2, None, '___sec10'), ('A quick reminder on Lagrangian multipliers', 2, None, - '___sec9'), - ('Adding the muliplier', 2, None, '___sec10'), - ('Setting up the problem', 2, None, '___sec11'), - ('The problem to solve', 2, None, '___sec12'), - ('The last steps', 2, None, '___sec13'), - ('A soft classifier', 2, None, '___sec14'), - ('Soft optmization problem', 2, None, '___sec15'), - ('Kernels and non-linearity', 2, None, '___sec16'), - ('The equations', 2, None, '___sec17'), - ('The problem to solve', 2, None, '___sec18'), - ("Different kernels and Mercer's theorem", 2, None, '___sec19'), - ('The moons example', 2, None, '___sec20'), + '___sec11'), + ('Adding the muliplier', 2, None, '___sec12'), + ('Setting up the problem', 2, None, '___sec13'), + ('The problem to solve', 2, None, '___sec14'), + ('The last steps', 2, None, '___sec15'), + ('A soft classifier', 2, None, '___sec16'), + ('Soft optmization problem', 2, None, '___sec17'), + ('Kernels and non-linearity', 2, None, '___sec18'), + ('The equations', 2, None, '___sec19'), + ('The problem to solve', 2, None, '___sec20'), + ("Different kernels and Mercer's theorem", 2, None, '___sec21'), + ('The moons example', 2, None, '___sec22'), ('Mathematical optimization of convex functions', 2, None, - '___sec21'), - ('How do we solve these problems?', 2, None, '___sec22'), - ('A simple example', 2, None, '___sec23'), - ('Back to the more realistic cases', 2, None, '___sec24'), - ('Code Example', 2, None, '___sec25')]} + '___sec23'), + ('How do we solve these problems?', 2, None, '___sec24'), + ('A simple example', 2, None, '___sec25'), + ('Back to the more realistic cases', 2, None, '___sec26'), + ('Code Example', 2, None, '___sec27')]} end of tocinfo --> @@ -118,24 +120,26 @@ MathJax.Hub.Config({
  • Getting into the details
  • First attempt at a minimization approach
  • Solving the equations
    • -
  • A better approach
    • -
  • A quick reminder on Lagrangian multipliers
    • -
  • Adding the muliplier
    • -
  • Setting up the problem
    • -
  • The problem to solve
    • -
  • The last steps
    • -
  • A soft classifier
    • -
  • Soft optmization problem
    • -
  • Kernels and non-linearity
    • -
  • The equations
    • -
  • The problem to solve
    • -
  • Different kernels and Mercer's theorem
    • -
  • The moons example
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simple example
    • -
  • Back to the more realistic cases
    • -
  • Code Example
    • +
  • Code Example
    • +
  • Problems with the Simpler Approach
    • +
  • A better approach
    • +
  • A quick reminder on Lagrangian multipliers
    • +
  • Adding the muliplier
    • +
  • Setting up the problem
    • +
  • The problem to solve
    • +
  • The last steps
    • +
  • A soft classifier
    • +
  • Soft optmization problem
    • +
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • The problem to solve
    • +
  • Different kernels and Mercer's theorem
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simple example
    • +
  • Back to the more realistic cases
    • +
  • Code Example
    • @@ -210,7 +214,7 @@ When we try to separate hyperplanes, if it exists, we can use it to construct a
  • 13
  • 14
  • ...
    • -
  • 27
    • +
  • 29
  • »
    • diff --git a/doc/pub/svm/html/._svm-bs005.html b/doc/pub/svm/html/._svm-bs005.html index d1e920a97..e173f9bc4 100644 --- a/doc/pub/svm/html/._svm-bs005.html +++ b/doc/pub/svm/html/._svm-bs005.html @@ -49,30 +49,32 @@ Automatically generated HTML file from DocOnce source ('Getting into the details', 2, None, '___sec5'), ('First attempt at a minimization approach', 2, None, '___sec6'), ('Solving the equations', 2, None, '___sec7'), - ('A better approach', 2, None, '___sec8'), + ('Code Example', 2, None, '___sec8'), + ('Problems with the Simpler Approach', 2, None, '___sec9'), + ('A better approach', 2, None, '___sec10'), ('A quick reminder on Lagrangian multipliers', 2, None, - '___sec9'), - ('Adding the muliplier', 2, None, '___sec10'), - ('Setting up the problem', 2, None, '___sec11'), - ('The problem to solve', 2, None, '___sec12'), - ('The last steps', 2, None, '___sec13'), - ('A soft classifier', 2, None, '___sec14'), - ('Soft optmization problem', 2, None, '___sec15'), - ('Kernels and non-linearity', 2, None, '___sec16'), - ('The equations', 2, None, '___sec17'), - ('The problem to solve', 2, None, '___sec18'), - ("Different kernels and Mercer's theorem", 2, None, '___sec19'), - ('The moons example', 2, None, '___sec20'), + '___sec11'), + ('Adding the muliplier', 2, None, '___sec12'), + ('Setting up the problem', 2, None, '___sec13'), + ('The problem to solve', 2, None, '___sec14'), + ('The last steps', 2, None, '___sec15'), + ('A soft classifier', 2, None, '___sec16'), + ('Soft optmization problem', 2, None, '___sec17'), + ('Kernels and non-linearity', 2, None, '___sec18'), + ('The equations', 2, None, '___sec19'), + ('The problem to solve', 2, None, '___sec20'), + ("Different kernels and Mercer's theorem", 2, None, '___sec21'), + ('The moons example', 2, None, '___sec22'), ('Mathematical optimization of convex functions', 2, None, - '___sec21'), - ('How do we solve these problems?', 2, None, '___sec22'), - ('A simple example', 2, None, '___sec23'), - ('Back to the more realistic cases', 2, None, '___sec24'), - ('Code Example', 2, None, '___sec25')]} + '___sec23'), + ('How do we solve these problems?', 2, None, '___sec24'), + ('A simple example', 2, None, '___sec25'), + ('Back to the more realistic cases', 2, None, '___sec26'), + ('Code Example', 2, None, '___sec27')]} end of tocinfo --> @@ -118,24 +120,26 @@ MathJax.Hub.Config({
  • Getting into the details
  • First attempt at a minimization approach
  • Solving the equations
    • -
  • A better approach
    • -
  • A quick reminder on Lagrangian multipliers
    • -
  • Adding the muliplier
    • -
  • Setting up the problem
    • -
  • The problem to solve
    • -
  • The last steps
    • -
  • A soft classifier
    • -
  • Soft optmization problem
    • -
  • Kernels and non-linearity
    • -
  • The equations
    • -
  • The problem to solve
    • -
  • Different kernels and Mercer's theorem
    • -
  • The moons example
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simple example
    • -
  • Back to the more realistic cases
    • -
  • Code Example
    • +
  • Code Example
    • +
  • Problems with the Simpler Approach
    • +
  • A better approach
    • +
  • A quick reminder on Lagrangian multipliers
    • +
  • Adding the muliplier
    • +
  • Setting up the problem
    • +
  • The problem to solve
    • +
  • The last steps
    • +
  • A soft classifier
    • +
  • Soft optmization problem
    • +
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • The problem to solve
    • +
  • Different kernels and Mercer's theorem
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simple example
    • +
  • Back to the more realistic cases
    • +
  • Code Example
    • @@ -196,7 +200,7 @@ for our data sample.
  • 14
  • 15
  • ...
    • -
  • 27
    • +
  • 29
  • »
    • diff --git a/doc/pub/svm/html/._svm-bs006.html b/doc/pub/svm/html/._svm-bs006.html index 436d6cc85..f32cde5f5 100644 --- a/doc/pub/svm/html/._svm-bs006.html +++ b/doc/pub/svm/html/._svm-bs006.html @@ -49,30 +49,32 @@ Automatically generated HTML file from DocOnce source ('Getting into the details', 2, None, '___sec5'), ('First attempt at a minimization approach', 2, None, '___sec6'), ('Solving the equations', 2, None, '___sec7'), - ('A better approach', 2, None, '___sec8'), + ('Code Example', 2, None, '___sec8'), + ('Problems with the Simpler Approach', 2, None, '___sec9'), + ('A better approach', 2, None, '___sec10'), ('A quick reminder on Lagrangian multipliers', 2, None, - '___sec9'), - ('Adding the muliplier', 2, None, '___sec10'), - ('Setting up the problem', 2, None, '___sec11'), - ('The problem to solve', 2, None, '___sec12'), - ('The last steps', 2, None, '___sec13'), - ('A soft classifier', 2, None, '___sec14'), - ('Soft optmization problem', 2, None, '___sec15'), - ('Kernels and non-linearity', 2, None, '___sec16'), - ('The equations', 2, None, '___sec17'), - ('The problem to solve', 2, None, '___sec18'), - ("Different kernels and Mercer's theorem", 2, None, '___sec19'), - ('The moons example', 2, None, '___sec20'), + '___sec11'), + ('Adding the muliplier', 2, None, '___sec12'), + ('Setting up the problem', 2, None, '___sec13'), + ('The problem to solve', 2, None, '___sec14'), + ('The last steps', 2, None, '___sec15'), + ('A soft classifier', 2, None, '___sec16'), + ('Soft optmization problem', 2, None, '___sec17'), + ('Kernels and non-linearity', 2, None, '___sec18'), + ('The equations', 2, None, '___sec19'), + ('The problem to solve', 2, None, '___sec20'), + ("Different kernels and Mercer's theorem", 2, None, '___sec21'), + ('The moons example', 2, None, '___sec22'), ('Mathematical optimization of convex functions', 2, None, - '___sec21'), - ('How do we solve these problems?', 2, None, '___sec22'), - ('A simple example', 2, None, '___sec23'), - ('Back to the more realistic cases', 2, None, '___sec24'), - ('Code Example', 2, None, '___sec25')]} + '___sec23'), + ('How do we solve these problems?', 2, None, '___sec24'), + ('A simple example', 2, None, '___sec25'), + ('Back to the more realistic cases', 2, None, '___sec26'), + ('Code Example', 2, None, '___sec27')]} end of tocinfo --> @@ -118,24 +120,26 @@ MathJax.Hub.Config({
  • Getting into the details
  • First attempt at a minimization approach
  • Solving the equations
    • -
  • A better approach
    • -
  • A quick reminder on Lagrangian multipliers
    • -
  • Adding the muliplier
    • -
  • Setting up the problem
    • -
  • The problem to solve
    • -
  • The last steps
    • -
  • A soft classifier
    • -
  • Soft optmization problem
    • -
  • Kernels and non-linearity
    • -
  • The equations
    • -
  • The problem to solve
    • -
  • Different kernels and Mercer's theorem
    • -
  • The moons example
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simple example
    • -
  • Back to the more realistic cases
    • -
  • Code Example
    • +
  • Code Example
    • +
  • Problems with the Simpler Approach
    • +
  • A better approach
    • +
  • A quick reminder on Lagrangian multipliers
    • +
  • Adding the muliplier
    • +
  • Setting up the problem
    • +
  • The problem to solve
    • +
  • The last steps
    • +
  • A soft classifier
    • +
  • Soft optmization problem
    • +
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • The problem to solve
    • +
  • Different kernels and Mercer's theorem
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simple example
    • +
  • Back to the more realistic cases
    • +
  • Code Example
    • @@ -192,7 +196,7 @@ $$
  • 15
  • 16
  • ...
    • -
  • 27
    • +
  • 29
  • »
    • diff --git a/doc/pub/svm/html/._svm-bs007.html b/doc/pub/svm/html/._svm-bs007.html index b3ad07f9d..b66e1dc7f 100644 --- a/doc/pub/svm/html/._svm-bs007.html +++ b/doc/pub/svm/html/._svm-bs007.html @@ -49,30 +49,32 @@ Automatically generated HTML file from DocOnce source ('Getting into the details', 2, None, '___sec5'), ('First attempt at a minimization approach', 2, None, '___sec6'), ('Solving the equations', 2, None, '___sec7'), - ('A better approach', 2, None, '___sec8'), + ('Code Example', 2, None, '___sec8'), + ('Problems with the Simpler Approach', 2, None, '___sec9'), + ('A better approach', 2, None, '___sec10'), ('A quick reminder on Lagrangian multipliers', 2, None, - '___sec9'), - ('Adding the muliplier', 2, None, '___sec10'), - ('Setting up the problem', 2, None, '___sec11'), - ('The problem to solve', 2, None, '___sec12'), - ('The last steps', 2, None, '___sec13'), - ('A soft classifier', 2, None, '___sec14'), - ('Soft optmization problem', 2, None, '___sec15'), - ('Kernels and non-linearity', 2, None, '___sec16'), - ('The equations', 2, None, '___sec17'), - ('The problem to solve', 2, None, '___sec18'), - ("Different kernels and Mercer's theorem", 2, None, '___sec19'), - ('The moons example', 2, None, '___sec20'), + '___sec11'), + ('Adding the muliplier', 2, None, '___sec12'), + ('Setting up the problem', 2, None, '___sec13'), + ('The problem to solve', 2, None, '___sec14'), + ('The last steps', 2, None, '___sec15'), + ('A soft classifier', 2, None, '___sec16'), + ('Soft optmization problem', 2, None, '___sec17'), + ('Kernels and non-linearity', 2, None, '___sec18'), + ('The equations', 2, None, '___sec19'), + ('The problem to solve', 2, None, '___sec20'), + ("Different kernels and Mercer's theorem", 2, None, '___sec21'), + ('The moons example', 2, None, '___sec22'), ('Mathematical optimization of convex functions', 2, None, - '___sec21'), - ('How do we solve these problems?', 2, None, '___sec22'), - ('A simple example', 2, None, '___sec23'), - ('Back to the more realistic cases', 2, None, '___sec24'), - ('Code Example', 2, None, '___sec25')]} + '___sec23'), + ('How do we solve these problems?', 2, None, '___sec24'), + ('A simple example', 2, None, '___sec25'), + ('Back to the more realistic cases', 2, None, '___sec26'), + ('Code Example', 2, None, '___sec27')]} end of tocinfo --> @@ -118,24 +120,26 @@ MathJax.Hub.Config({
  • Getting into the details
  • First attempt at a minimization approach
  • Solving the equations
    • -
  • A better approach
    • -
  • A quick reminder on Lagrangian multipliers
    • -
  • Adding the muliplier
    • -
  • Setting up the problem
    • -
  • The problem to solve
    • -
  • The last steps
    • -
  • A soft classifier
    • -
  • Soft optmization problem
    • -
  • Kernels and non-linearity
    • -
  • The equations
    • -
  • The problem to solve
    • -
  • Different kernels and Mercer's theorem
    • -
  • The moons example
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simple example
    • -
  • Back to the more realistic cases
    • -
  • Code Example
    • +
  • Code Example
    • +
  • Problems with the Simpler Approach
    • +
  • A better approach
    • +
  • A quick reminder on Lagrangian multipliers
    • +
  • Adding the muliplier
    • +
  • Setting up the problem
    • +
  • The problem to solve
    • +
  • The last steps
    • +
  • A soft classifier
    • +
  • Soft optmization problem
    • +
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • The problem to solve
    • +
  • Different kernels and Mercer's theorem
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simple example
    • +
  • Back to the more realistic cases
    • +
  • Code Example
    • @@ -196,7 +200,7 @@ $$
  • 16
  • 17
  • ...
    • -
  • 27
    • +
  • 29
  • »
    • diff --git a/doc/pub/svm/html/._svm-bs008.html b/doc/pub/svm/html/._svm-bs008.html index 9310989f2..4e45296b0 100644 --- a/doc/pub/svm/html/._svm-bs008.html +++ b/doc/pub/svm/html/._svm-bs008.html @@ -49,30 +49,32 @@ Automatically generated HTML file from DocOnce source ('Getting into the details', 2, None, '___sec5'), ('First attempt at a minimization approach', 2, None, '___sec6'), ('Solving the equations', 2, None, '___sec7'), - ('A better approach', 2, None, '___sec8'), + ('Code Example', 2, None, '___sec8'), + ('Problems with the Simpler Approach', 2, None, '___sec9'), + ('A better approach', 2, None, '___sec10'), ('A quick reminder on Lagrangian multipliers', 2, None, - '___sec9'), - ('Adding the muliplier', 2, None, '___sec10'), - ('Setting up the problem', 2, None, '___sec11'), - ('The problem to solve', 2, None, '___sec12'), - ('The last steps', 2, None, '___sec13'), - ('A soft classifier', 2, None, '___sec14'), - ('Soft optmization problem', 2, None, '___sec15'), - ('Kernels and non-linearity', 2, None, '___sec16'), - ('The equations', 2, None, '___sec17'), - ('The problem to solve', 2, None, '___sec18'), - ("Different kernels and Mercer's theorem", 2, None, '___sec19'), - ('The moons example', 2, None, '___sec20'), + '___sec11'), + ('Adding the muliplier', 2, None, '___sec12'), + ('Setting up the problem', 2, None, '___sec13'), + ('The problem to solve', 2, None, '___sec14'), + ('The last steps', 2, None, '___sec15'), + ('A soft classifier', 2, None, '___sec16'), + ('Soft optmization problem', 2, None, '___sec17'), + ('Kernels and non-linearity', 2, None, '___sec18'), + ('The equations', 2, None, '___sec19'), + ('The problem to solve', 2, None, '___sec20'), + ("Different kernels and Mercer's theorem", 2, None, '___sec21'), + ('The moons example', 2, None, '___sec22'), ('Mathematical optimization of convex functions', 2, None, - '___sec21'), - ('How do we solve these problems?', 2, None, '___sec22'), - ('A simple example', 2, None, '___sec23'), - ('Back to the more realistic cases', 2, None, '___sec24'), - ('Code Example', 2, None, '___sec25')]} + '___sec23'), + ('How do we solve these problems?', 2, None, '___sec24'), + ('A simple example', 2, None, '___sec25'), + ('Back to the more realistic cases', 2, None, '___sec26'), + ('Code Example', 2, None, '___sec27')]} end of tocinfo --> @@ -118,24 +120,26 @@ MathJax.Hub.Config({
  • Getting into the details
  • First attempt at a minimization approach
  • Solving the equations
    • -
  • A better approach
    • -
  • A quick reminder on Lagrangian multipliers
    • -
  • Adding the muliplier
    • -
  • Setting up the problem
    • -
  • The problem to solve
    • -
  • The last steps
    • -
  • A soft classifier
    • -
  • Soft optmization problem
    • -
  • Kernels and non-linearity
    • -
  • The equations
    • -
  • The problem to solve
    • -
  • Different kernels and Mercer's theorem
    • -
  • The moons example
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simple example
    • -
  • Back to the more realistic cases
    • -
  • Code Example
    • +
  • Code Example
    • +
  • Problems with the Simpler Approach
    • +
  • A better approach
    • +
  • A quick reminder on Lagrangian multipliers
    • +
  • Adding the muliplier
    • +
  • Setting up the problem
    • +
  • The problem to solve
    • +
  • The last steps
    • +
  • A soft classifier
    • +
  • Soft optmization problem
    • +
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • The problem to solve
    • +
  • Different kernels and Mercer's theorem
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simple example
    • +
  • Back to the more realistic cases
    • +
  • Code Example
    • @@ -154,7 +158,7 @@ MathJax.Hub.Config({

    Solving the equations

    -We can now use the Newton-Raphson method or gradient descent to solve the equations +We can now use the Newton-Raphson method or different variants of the gradient descent family (from plain gradient descent to various stochastic gradient descent approaches) to solve the equations $$ b \leftarrow b +\eta \frac{\partial C}{\partial b}, $$ @@ -166,16 +170,6 @@ $$ where \( \eta \) is our by now well-known learning rate. -

    -There are however problems with this approach, although it looks -pretty straightforward to implement. In case we separate our data into -two distinct classes, we may up with many possible lines, as indicated -in the figure and shown by running the following program. For small -gaps between the entries, we may also end up needing many iterations -before the solutions converge and if the data cannot be separated -properly into two distinct classes, we may not experience a converge -at all. -

    @@ -200,7 +194,7 @@ at all.

  • 17
  • 18
  • ...
    • -
  • 27
    • +
  • 29
  • »
    • diff --git a/doc/pub/svm/html/._svm-bs009.html b/doc/pub/svm/html/._svm-bs009.html index 403a65868..2b389e3b0 100644 --- a/doc/pub/svm/html/._svm-bs009.html +++ b/doc/pub/svm/html/._svm-bs009.html @@ -49,30 +49,32 @@ Automatically generated HTML file from DocOnce source ('Getting into the details', 2, None, '___sec5'), ('First attempt at a minimization approach', 2, None, '___sec6'), ('Solving the equations', 2, None, '___sec7'), - ('A better approach', 2, None, '___sec8'), + ('Code Example', 2, None, '___sec8'), + ('Problems with the Simpler Approach', 2, None, '___sec9'), + ('A better approach', 2, None, '___sec10'), ('A quick reminder on Lagrangian multipliers', 2, None, - '___sec9'), - ('Adding the muliplier', 2, None, '___sec10'), - ('Setting up the problem', 2, None, '___sec11'), - ('The problem to solve', 2, None, '___sec12'), - ('The last steps', 2, None, '___sec13'), - ('A soft classifier', 2, None, '___sec14'), - ('Soft optmization problem', 2, None, '___sec15'), - ('Kernels and non-linearity', 2, None, '___sec16'), - ('The equations', 2, None, '___sec17'), - ('The problem to solve', 2, None, '___sec18'), - ("Different kernels and Mercer's theorem", 2, None, '___sec19'), - ('The moons example', 2, None, '___sec20'), + '___sec11'), + ('Adding the muliplier', 2, None, '___sec12'), + ('Setting up the problem', 2, None, '___sec13'), + ('The problem to solve', 2, None, '___sec14'), + ('The last steps', 2, None, '___sec15'), + ('A soft classifier', 2, None, '___sec16'), + ('Soft optmization problem', 2, None, '___sec17'), + ('Kernels and non-linearity', 2, None, '___sec18'), + ('The equations', 2, None, '___sec19'), + ('The problem to solve', 2, None, '___sec20'), + ("Different kernels and Mercer's theorem", 2, None, '___sec21'), + ('The moons example', 2, None, '___sec22'), ('Mathematical optimization of convex functions', 2, None, - '___sec21'), - ('How do we solve these problems?', 2, None, '___sec22'), - ('A simple example', 2, None, '___sec23'), - ('Back to the more realistic cases', 2, None, '___sec24'), - ('Code Example', 2, None, '___sec25')]} + '___sec23'), + ('How do we solve these problems?', 2, None, '___sec24'), + ('A simple example', 2, None, '___sec25'), + ('Back to the more realistic cases', 2, None, '___sec26'), + ('Code Example', 2, None, '___sec27')]} end of tocinfo --> @@ -118,24 +120,26 @@ MathJax.Hub.Config({
  • Getting into the details
  • First attempt at a minimization approach
  • Solving the equations
    • -
  • A better approach
    • -
  • A quick reminder on Lagrangian multipliers
    • -
  • Adding the muliplier
    • -
  • Setting up the problem
    • -
  • The problem to solve
    • -
  • The last steps
    • -
  • A soft classifier
    • -
  • Soft optmization problem
    • -
  • Kernels and non-linearity
    • -
  • The equations
    • -
  • The problem to solve
    • -
  • Different kernels and Mercer's theorem
    • -
  • The moons example
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simple example
    • -
  • Back to the more realistic cases
    • -
  • Code Example
    • +
  • Code Example
    • +
  • Problems with the Simpler Approach
    • +
  • A better approach
    • +
  • A quick reminder on Lagrangian multipliers
    • +
  • Adding the muliplier
    • +
  • Setting up the problem
    • +
  • The problem to solve
    • +
  • The last steps
    • +
  • A soft classifier
    • +
  • Soft optmization problem
    • +
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • The problem to solve
    • +
  • Different kernels and Mercer's theorem
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simple example
    • +
  • Back to the more realistic cases
    • +
  • Code Example
    • @@ -151,42 +155,18 @@ MathJax.Hub.Config({ -

    A better approach

    +

    Code Example

    -A better approach is rather to try to define a large margin between -the two classes (if they are well separated from the beginning). - +The equations we discussed above can be coded rather easily (the +framework is rather similar to what we developed for logistic +regression). We use the standard moons example to set up our data +set. We want to find a line which separates the two classes

    -Thus, we wish to find a margin \( M \) with \( \boldsymbol{w} \) normalized to -\( \vert\vert \boldsymbol{w}\vert\vert =1 \) subject to the condition - -$$ -y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) \geq M \hspace{0.1cm}\forall i=1,2,\dots, p. -$$ - -All points are thus at a signed distance from the decision boundary defined by the line \( L \). The parameters \( b \) and \( w_1 \) and \( w_2 \) define this line. - -

    -We seek thus the largest value \( M \) defined by -$$ -\frac{1}{\vert \vert \boldsymbol{w}\vert\vert}y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) \geq M \hspace{0.1cm}\forall i=1,2,\dots, n, -$$ - -or just -$$ -y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) \geq M\vert \vert \boldsymbol{w}\vert\vert \hspace{0.1cm}\forall i. -$$ - -If we scale the equation so that \( \vert \vert \boldsymbol{w}\vert\vert = 1/M \), we have to find the minimum of -\( \boldsymbol{w}^T\boldsymbol{w}=\vert \vert \boldsymbol{w}\vert\vert \) (the norm) subject to the condition -$$ -y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) \geq 1 \hspace{0.1cm}\forall i. -$$ - -

    -We have thus defined our margin as the invers of the norm of \( \boldsymbol{w} \). We want to minimize the norm in order to have a as large as possible margin \( M \). Before we proceed, we need to remind ourselves about Lagrangian multipliers. + +

    
+

    @@ -212,7 +192,7 @@ We have thus defined our margin as the invers of the norm of \( \boldsymbol{w} \

  • 18
  • 19
  • ...
    • -
  • 27
    • +
  • 29
  • »
    • diff --git a/doc/pub/svm/html/._svm-bs010.html b/doc/pub/svm/html/._svm-bs010.html index 553790da7..432f59530 100644 --- a/doc/pub/svm/html/._svm-bs010.html +++ b/doc/pub/svm/html/._svm-bs010.html @@ -49,30 +49,32 @@ Automatically generated HTML file from DocOnce source ('Getting into the details', 2, None, '___sec5'), ('First attempt at a minimization approach', 2, None, '___sec6'), ('Solving the equations', 2, None, '___sec7'), - ('A better approach', 2, None, '___sec8'), + ('Code Example', 2, None, '___sec8'), + ('Problems with the Simpler Approach', 2, None, '___sec9'), + ('A better approach', 2, None, '___sec10'), ('A quick reminder on Lagrangian multipliers', 2, None, - '___sec9'), - ('Adding the muliplier', 2, None, '___sec10'), - ('Setting up the problem', 2, None, '___sec11'), - ('The problem to solve', 2, None, '___sec12'), - ('The last steps', 2, None, '___sec13'), - ('A soft classifier', 2, None, '___sec14'), - ('Soft optmization problem', 2, None, '___sec15'), - ('Kernels and non-linearity', 2, None, '___sec16'), - ('The equations', 2, None, '___sec17'), - ('The problem to solve', 2, None, '___sec18'), - ("Different kernels and Mercer's theorem", 2, None, '___sec19'), - ('The moons example', 2, None, '___sec20'), + '___sec11'), + ('Adding the muliplier', 2, None, '___sec12'), + ('Setting up the problem', 2, None, '___sec13'), + ('The problem to solve', 2, None, '___sec14'), + ('The last steps', 2, None, '___sec15'), + ('A soft classifier', 2, None, '___sec16'), + ('Soft optmization problem', 2, None, '___sec17'), + ('Kernels and non-linearity', 2, None, '___sec18'), + ('The equations', 2, None, '___sec19'), + ('The problem to solve', 2, None, '___sec20'), + ("Different kernels and Mercer's theorem", 2, None, '___sec21'), + ('The moons example', 2, None, '___sec22'), ('Mathematical optimization of convex functions', 2, None, - '___sec21'), - ('How do we solve these problems?', 2, None, '___sec22'), - ('A simple example', 2, None, '___sec23'), - ('Back to the more realistic cases', 2, None, '___sec24'), - ('Code Example', 2, None, '___sec25')]} + '___sec23'), + ('How do we solve these problems?', 2, None, '___sec24'), + ('A simple example', 2, None, '___sec25'), + ('Back to the more realistic cases', 2, None, '___sec26'), + ('Code Example', 2, None, '___sec27')]} end of tocinfo --> @@ -118,24 +120,26 @@ MathJax.Hub.Config({
  • Getting into the details
  • First attempt at a minimization approach
  • Solving the equations
    • -
  • A better approach
    • -
  • A quick reminder on Lagrangian multipliers
    • -
  • Adding the muliplier
    • -
  • Setting up the problem
    • -
  • The problem to solve
    • -
  • The last steps
    • -
  • A soft classifier
    • -
  • Soft optmization problem
    • -
  • Kernels and non-linearity
    • -
  • The equations
    • -
  • The problem to solve
    • -
  • Different kernels and Mercer's theorem
    • -
  • The moons example
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simple example
    • -
  • Back to the more realistic cases
    • -
  • Code Example
    • +
  • Code Example
    • +
  • Problems with the Simpler Approach
    • +
  • A better approach
    • +
  • A quick reminder on Lagrangian multipliers
    • +
  • Adding the muliplier
    • +
  • Setting up the problem
    • +
  • The problem to solve
    • +
  • The last steps
    • +
  • A soft classifier
    • +
  • Soft optmization problem
    • +
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • The problem to solve
    • +
  • Different kernels and Mercer's theorem
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simple example
    • +
  • Back to the more realistic cases
    • +
  • Code Example
    • @@ -151,52 +155,18 @@ MathJax.Hub.Config({ -

    A quick reminder on Lagrangian multipliers

    +

    Problems with the Simpler Approach

    -Consider a function of three independent variables \( f(x,y,z) \) . For the function \( f \) to be an -extreme we have -$$ -df=0. -$$ - -A necessary and sufficient condition is -$$ -\frac{\partial f}{\partial x} =\frac{\partial f}{\partial y}=\frac{\partial f}{\partial z}=0, -$$ - -due to -$$ -df = \frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy+\frac{\partial f}{\partial z}dz. -$$ - -In many problems the variables \( x,y,z \) are often subject to constraints (such as those above for the margin) -so that they are no longer all independent. It is possible at least in principle to use each -constraint to eliminate one variable -and to proceed with a new and smaller set of independent varables. +There are however problems with this approach, although it looks +pretty straightforward to implement. When running the above code, we see that we can easily end up with many diffeent lines which separate the two classes.

    -The use of so-called Lagrangian multipliers is an alternative technique when the elimination -of variables is incovenient or undesirable. Assume that we have an equation of constraint on -the variables \( x,y,z \) -$$ -\phi(x,y,z) = 0, -$$ - - resulting in -$$ -d\phi = \frac{\partial \phi}{\partial x}dx+\frac{\partial \phi}{\partial y}dy+\frac{\partial \phi}{\partial z}dz =0. -$$ - -Now we cannot set anymore -$$ -\frac{\partial f}{\partial x} =\frac{\partial f}{\partial y}=\frac{\partial f}{\partial z}=0, -$$ - -if \( df=0 \) is wanted -because there are now only two independent variables! Assume \( x \) and \( y \) are the independent -variables. -Then \( dz \) is no longer arbitrary. +For small +gaps between the entries, we may also end up needing many iterations +before the solutions converge and if the data cannot be separated +properly into two distinct classes, we may not experience a converge +at all.

    @@ -224,7 +194,7 @@ Then \( dz \) is no longer arbitrary.

  • 19
  • 20
  • ...
    • -
  • 27
    • +
  • 29
  • »
    • diff --git a/doc/pub/svm/html/._svm-bs011.html b/doc/pub/svm/html/._svm-bs011.html index 9f522eef9..316db0caa 100644 --- a/doc/pub/svm/html/._svm-bs011.html +++ b/doc/pub/svm/html/._svm-bs011.html @@ -49,30 +49,32 @@ Automatically generated HTML file from DocOnce source ('Getting into the details', 2, None, '___sec5'), ('First attempt at a minimization approach', 2, None, '___sec6'), ('Solving the equations', 2, None, '___sec7'), - ('A better approach', 2, None, '___sec8'), + ('Code Example', 2, None, '___sec8'), + ('Problems with the Simpler Approach', 2, None, '___sec9'), + ('A better approach', 2, None, '___sec10'), ('A quick reminder on Lagrangian multipliers', 2, None, - '___sec9'), - ('Adding the muliplier', 2, None, '___sec10'), - ('Setting up the problem', 2, None, '___sec11'), - ('The problem to solve', 2, None, '___sec12'), - ('The last steps', 2, None, '___sec13'), - ('A soft classifier', 2, None, '___sec14'), - ('Soft optmization problem', 2, None, '___sec15'), - ('Kernels and non-linearity', 2, None, '___sec16'), - ('The equations', 2, None, '___sec17'), - ('The problem to solve', 2, None, '___sec18'), - ("Different kernels and Mercer's theorem", 2, None, '___sec19'), - ('The moons example', 2, None, '___sec20'), + '___sec11'), + ('Adding the muliplier', 2, None, '___sec12'), + ('Setting up the problem', 2, None, '___sec13'), + ('The problem to solve', 2, None, '___sec14'), + ('The last steps', 2, None, '___sec15'), + ('A soft classifier', 2, None, '___sec16'), + ('Soft optmization problem', 2, None, '___sec17'), + ('Kernels and non-linearity', 2, None, '___sec18'), + ('The equations', 2, None, '___sec19'), + ('The problem to solve', 2, None, '___sec20'), + ("Different kernels and Mercer's theorem", 2, None, '___sec21'), + ('The moons example', 2, None, '___sec22'), ('Mathematical optimization of convex functions', 2, None, - '___sec21'), - ('How do we solve these problems?', 2, None, '___sec22'), - ('A simple example', 2, None, '___sec23'), - ('Back to the more realistic cases', 2, None, '___sec24'), - ('Code Example', 2, None, '___sec25')]} + '___sec23'), + ('How do we solve these problems?', 2, None, '___sec24'), + ('A simple example', 2, None, '___sec25'), + ('Back to the more realistic cases', 2, None, '___sec26'), + ('Code Example', 2, None, '___sec27')]} end of tocinfo --> @@ -118,24 +120,26 @@ MathJax.Hub.Config({
  • Getting into the details
  • First attempt at a minimization approach
  • Solving the equations
    • -
  • A better approach
    • -
  • A quick reminder on Lagrangian multipliers
    • -
  • Adding the muliplier
    • -
  • Setting up the problem
    • -
  • The problem to solve
    • -
  • The last steps
    • -
  • A soft classifier
    • -
  • Soft optmization problem
    • -
  • Kernels and non-linearity
    • -
  • The equations
    • -
  • The problem to solve
    • -
  • Different kernels and Mercer's theorem
    • -
  • The moons example
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simple example
    • -
  • Back to the more realistic cases
    • -
  • Code Example
    • +
  • Code Example
    • +
  • Problems with the Simpler Approach
    • +
  • A better approach
    • +
  • A quick reminder on Lagrangian multipliers
    • +
  • Adding the muliplier
    • +
  • Setting up the problem
    • +
  • The problem to solve
    • +
  • The last steps
    • +
  • A soft classifier
    • +
  • Soft optmization problem
    • +
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • The problem to solve
    • +
  • Different kernels and Mercer's theorem
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simple example
    • +
  • Back to the more realistic cases
    • +
  • Code Example
    • @@ -151,45 +155,44 @@ MathJax.Hub.Config({ -

    Adding the muliplier

    +

    A better approach

    -However, we can add to +A better approach is rather to try to define a large margin between +the two classes (if they are well separated from the beginning). + +

    +Thus, we wish to find a margin \( M \) with \( \boldsymbol{w} \) normalized to +\( \vert\vert \boldsymbol{w}\vert\vert =1 \) subject to the condition + $$ -df = \frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy+\frac{\partial f}{\partial z}dz, +y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) \geq M \hspace{0.1cm}\forall i=1,2,\dots, p. $$ -a multiplum of \( d\phi \), viz. \( \lambda d\phi \), resulting in +All points are thus at a signed distance from the decision boundary defined by the line \( L \). The parameters \( b \) and \( w_1 \) and \( w_2 \) define this line. + +

    +We seek thus the largest value \( M \) defined by $$ -df+\lambda d\phi = (\frac{\partial f}{\partial z}+\lambda -\frac{\partial \phi}{\partial x})dx+(\frac{\partial f}{\partial y}+\lambda\frac{\partial \phi}{\partial y})dy+ -(\frac{\partial f}{\partial z}+\lambda\frac{\partial \phi}{\partial z})dz =0. +\frac{1}{\vert \vert \boldsymbol{w}\vert\vert}y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) \geq M \hspace{0.1cm}\forall i=1,2,\dots, n, $$ -Our multiplier is chosen so that +or just $$ -\frac{\partial f}{\partial z}+\lambda\frac{\partial \phi}{\partial z} =0. +y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) \geq M\vert \vert \boldsymbol{w}\vert\vert \hspace{0.1cm}\forall i. +$$ + +If we scale the equation so that \( \vert \vert \boldsymbol{w}\vert\vert = 1/M \), we have to find the minimum of +\( \boldsymbol{w}^T\boldsymbol{w}=\vert \vert \boldsymbol{w}\vert\vert \) (the norm) subject to the condition +$$ +y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) \geq 1 \hspace{0.1cm}\forall i. $$

    -We need to remember that we took \( dx \) and \( dy \) to be arbitrary and thus we must have -$$ -\frac{\partial f}{\partial x}+\lambda\frac{\partial \phi}{\partial x} =0, -$$ - -and -$$ -\frac{\partial f}{\partial y}+\lambda\frac{\partial \phi}{\partial y} =0. -$$ - -When all these equations are satisfied, \( df=0 \). We have four unknowns, \( x,y,z \) and -\( \lambda \). Actually we want only \( x,y,z \), \( \lambda \) needs not to be determined, -it is therefore often called -Lagrange's undetermined multiplier. -If we have a set of constraints \( \phi_k \) we have the equations -$$ -\frac{\partial f}{\partial x_i}+\sum_k\lambda_k\frac{\partial \phi_k}{\partial x_i} =0. -$$ +We have thus defined our margin as the invers of the norm of +\( \boldsymbol{w} \). We want to minimize the norm in order to have a as large as +possible margin \( M \). Before we proceed, we need to remind ourselves +about Lagrangian multipliers.

    @@ -217,7 +220,7 @@ $$

  • 20
  • 21
  • ...
    • -
  • 27
    • +
  • 29
  • »
    • diff --git a/doc/pub/svm/html/._svm-bs012.html b/doc/pub/svm/html/._svm-bs012.html index f3c3baf16..b600ced75 100644 --- a/doc/pub/svm/html/._svm-bs012.html +++ b/doc/pub/svm/html/._svm-bs012.html @@ -49,30 +49,32 @@ Automatically generated HTML file from DocOnce source ('Getting into the details', 2, None, '___sec5'), ('First attempt at a minimization approach', 2, None, '___sec6'), ('Solving the equations', 2, None, '___sec7'), - ('A better approach', 2, None, '___sec8'), + ('Code Example', 2, None, '___sec8'), + ('Problems with the Simpler Approach', 2, None, '___sec9'), + ('A better approach', 2, None, '___sec10'), ('A quick reminder on Lagrangian multipliers', 2, None, - '___sec9'), - ('Adding the muliplier', 2, None, '___sec10'), - ('Setting up the problem', 2, None, '___sec11'), - ('The problem to solve', 2, None, '___sec12'), - ('The last steps', 2, None, '___sec13'), - ('A soft classifier', 2, None, '___sec14'), - ('Soft optmization problem', 2, None, '___sec15'), - ('Kernels and non-linearity', 2, None, '___sec16'), - ('The equations', 2, None, '___sec17'), - ('The problem to solve', 2, None, '___sec18'), - ("Different kernels and Mercer's theorem", 2, None, '___sec19'), - ('The moons example', 2, None, '___sec20'), + '___sec11'), + ('Adding the muliplier', 2, None, '___sec12'), + ('Setting up the problem', 2, None, '___sec13'), + ('The problem to solve', 2, None, '___sec14'), + ('The last steps', 2, None, '___sec15'), + ('A soft classifier', 2, None, '___sec16'), + ('Soft optmization problem', 2, None, '___sec17'), + ('Kernels and non-linearity', 2, None, '___sec18'), + ('The equations', 2, None, '___sec19'), + ('The problem to solve', 2, None, '___sec20'), + ("Different kernels and Mercer's theorem", 2, None, '___sec21'), + ('The moons example', 2, None, '___sec22'), ('Mathematical optimization of convex functions', 2, None, - '___sec21'), - ('How do we solve these problems?', 2, None, '___sec22'), - ('A simple example', 2, None, '___sec23'), - ('Back to the more realistic cases', 2, None, '___sec24'), - ('Code Example', 2, None, '___sec25')]} + '___sec23'), + ('How do we solve these problems?', 2, None, '___sec24'), + ('A simple example', 2, None, '___sec25'), + ('Back to the more realistic cases', 2, None, '___sec26'), + ('Code Example', 2, None, '___sec27')]} end of tocinfo --> @@ -118,24 +120,26 @@ MathJax.Hub.Config({
  • Getting into the details
  • First attempt at a minimization approach
  • Solving the equations
    • -
  • A better approach
    • -
  • A quick reminder on Lagrangian multipliers
    • -
  • Adding the muliplier
    • -
  • Setting up the problem
    • -
  • The problem to solve
    • -
  • The last steps
    • -
  • A soft classifier
    • -
  • Soft optmization problem
    • -
  • Kernels and non-linearity
    • -
  • The equations
    • -
  • The problem to solve
    • -
  • Different kernels and Mercer's theorem
    • -
  • The moons example
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simple example
    • -
  • Back to the more realistic cases
    • -
  • Code Example
    • +
  • Code Example
    • +
  • Problems with the Simpler Approach
    • +
  • A better approach
    • +
  • A quick reminder on Lagrangian multipliers
    • +
  • Adding the muliplier
    • +
  • Setting up the problem
    • +
  • The problem to solve
    • +
  • The last steps
    • +
  • A soft classifier
    • +
  • Soft optmization problem
    • +
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • The problem to solve
    • +
  • Different kernels and Mercer's theorem
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simple example
    • +
  • Back to the more realistic cases
    • +
  • Code Example
    • @@ -151,43 +155,52 @@ MathJax.Hub.Config({ -

    Setting up the problem

    -In order to solve the above problem, we define the following Lagrangian function to be minimized -$$ -{\cal L}(\lambda,b,\boldsymbol{w})=\frac{1}{2}\boldsymbol{w}^T\boldsymbol{w}-\sum_{i=1}^n\lambda_i\left[y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)-1\right], -$$ - -where \( \lambda_i \) is a so-called Lagrange multiplier subject to the condition \( \lambda_i \geq 0 \). +

    A quick reminder on Lagrangian multipliers

    -Taking the derivatives with respect to \( b \) and \( \boldsymbol{w} \) we obtain +Consider a function of three independent variables \( f(x,y,z) \) . For the function \( f \) to be an +extreme we have $$ -\frac{\partial {\cal L}}{\partial b} = -\sum_{i} \lambda_iy_i=0, +df=0. $$ -and +A necessary and sufficient condition is $$ -\frac{\partial {\cal L}}{\partial \boldsymbol{w}} = 0 = \boldsymbol{w}-\sum_{i} \lambda_iy_i\boldsymbol{x}_i. +\frac{\partial f}{\partial x} =\frac{\partial f}{\partial y}=\frac{\partial f}{\partial z}=0, $$ -Inserting these constraints into the equation for \( {\cal L} \) we obtain +due to $$ -{\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{x}_i^T\boldsymbol{x}_j, +df = \frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy+\frac{\partial f}{\partial z}dz. $$ -subject to the constraints \( \lambda_i\geq 0 \) and \( \sum_i\lambda_iy_i=0 \). -We must in addition satisfy the Karush-Kuhn-Tucker (KKT) condition +In many problems the variables \( x,y,z \) are often subject to constraints (such as those above for the margin) +so that they are no longer all independent. It is possible at least in principle to use each +constraint to eliminate one variable +and to proceed with a new and smaller set of independent varables. + +

    +The use of so-called Lagrangian multipliers is an alternative technique when the elimination +of variables is incovenient or undesirable. Assume that we have an equation of constraint on +the variables \( x,y,z \) $$ -\lambda_i\left[y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) -1\right] \hspace{0.1cm}\forall i. +\phi(x,y,z) = 0, $$ + resulting in +$$ +d\phi = \frac{\partial \phi}{\partial x}dx+\frac{\partial \phi}{\partial y}dy+\frac{\partial \phi}{\partial z}dz =0. +$$ -

      -
    1. If \( \lambda_i > 0 \), then \( y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)=1 \) and we say that \( x_i \) is on the boundary.
      2. -
    2. If \( y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)> 1 \), we say \( x_i \) is not on the boundary and we set \( \lambda_i=0 \).
      3. -
    +Now we cannot set anymore +$$ +\frac{\partial f}{\partial x} =\frac{\partial f}{\partial y}=\frac{\partial f}{\partial z}=0, +$$ -When \( \lambda_i > 0 \), the vectors \( \boldsymbol{x}_i \) are called support vectors. They are the vectors closest to the line (or hyperplane) and define the margin \( M \). +if \( df=0 \) is wanted +because there are now only two independent variables! Assume \( x \) and \( y \) are the independent +variables. +Then \( dz \) is no longer arbitrary.

    @@ -215,7 +228,7 @@ When \( \lambda_i > 0 \), the vectors \( \boldsymbol{x}_i \) are called support

  • 21
  • 22
  • ...
    • -
  • 27
    • +
  • 29
  • »
    • diff --git a/doc/pub/svm/html/._svm-bs013.html b/doc/pub/svm/html/._svm-bs013.html index 1ce154cf5..302ea6e65 100644 --- a/doc/pub/svm/html/._svm-bs013.html +++ b/doc/pub/svm/html/._svm-bs013.html @@ -49,30 +49,32 @@ Automatically generated HTML file from DocOnce source ('Getting into the details', 2, None, '___sec5'), ('First attempt at a minimization approach', 2, None, '___sec6'), ('Solving the equations', 2, None, '___sec7'), - ('A better approach', 2, None, '___sec8'), + ('Code Example', 2, None, '___sec8'), + ('Problems with the Simpler Approach', 2, None, '___sec9'), + ('A better approach', 2, None, '___sec10'), ('A quick reminder on Lagrangian multipliers', 2, None, - '___sec9'), - ('Adding the muliplier', 2, None, '___sec10'), - ('Setting up the problem', 2, None, '___sec11'), - ('The problem to solve', 2, None, '___sec12'), - ('The last steps', 2, None, '___sec13'), - ('A soft classifier', 2, None, '___sec14'), - ('Soft optmization problem', 2, None, '___sec15'), - ('Kernels and non-linearity', 2, None, '___sec16'), - ('The equations', 2, None, '___sec17'), - ('The problem to solve', 2, None, '___sec18'), - ("Different kernels and Mercer's theorem", 2, None, '___sec19'), - ('The moons example', 2, None, '___sec20'), + '___sec11'), + ('Adding the muliplier', 2, None, '___sec12'), + ('Setting up the problem', 2, None, '___sec13'), + ('The problem to solve', 2, None, '___sec14'), + ('The last steps', 2, None, '___sec15'), + ('A soft classifier', 2, None, '___sec16'), + ('Soft optmization problem', 2, None, '___sec17'), + ('Kernels and non-linearity', 2, None, '___sec18'), + ('The equations', 2, None, '___sec19'), + ('The problem to solve', 2, None, '___sec20'), + ("Different kernels and Mercer's theorem", 2, None, '___sec21'), + ('The moons example', 2, None, '___sec22'), ('Mathematical optimization of convex functions', 2, None, - '___sec21'), - ('How do we solve these problems?', 2, None, '___sec22'), - ('A simple example', 2, None, '___sec23'), - ('Back to the more realistic cases', 2, None, '___sec24'), - ('Code Example', 2, None, '___sec25')]} + '___sec23'), + ('How do we solve these problems?', 2, None, '___sec24'), + ('A simple example', 2, None, '___sec25'), + ('Back to the more realistic cases', 2, None, '___sec26'), + ('Code Example', 2, None, '___sec27')]} end of tocinfo --> @@ -118,24 +120,26 @@ MathJax.Hub.Config({
  • Getting into the details
  • First attempt at a minimization approach
  • Solving the equations
    • -
  • A better approach
    • -
  • A quick reminder on Lagrangian multipliers
    • -
  • Adding the muliplier
    • -
  • Setting up the problem
    • -
  • The problem to solve
    • -
  • The last steps
    • -
  • A soft classifier
    • -
  • Soft optmization problem
    • -
  • Kernels and non-linearity
    • -
  • The equations
    • -
  • The problem to solve
    • -
  • Different kernels and Mercer's theorem
    • -
  • The moons example
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simple example
    • -
  • Back to the more realistic cases
    • -
  • Code Example
    • +
  • Code Example
    • +
  • Problems with the Simpler Approach
    • +
  • A better approach
    • +
  • A quick reminder on Lagrangian multipliers
    • +
  • Adding the muliplier
    • +
  • Setting up the problem
    • +
  • The problem to solve
    • +
  • The last steps
    • +
  • A soft classifier
    • +
  • Soft optmization problem
    • +
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • The problem to solve
    • +
  • Different kernels and Mercer's theorem
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simple example
    • +
  • Back to the more realistic cases
    • +
  • Code Example
    • @@ -151,26 +155,45 @@ MathJax.Hub.Config({ -

    The problem to solve

    +

    Adding the muliplier

    -We can rewrite +However, we can add to $$ -{\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{x}_i^T\boldsymbol{x}_j, +df = \frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy+\frac{\partial f}{\partial z}dz, $$ -and its constraints in terms of a matrix-vector problem where we minimize w.r.t. \( \lambda \) the following problem +a multiplum of \( d\phi \), viz. \( \lambda d\phi \), resulting in $$ -\frac{1}{2} \boldsymbol{\lambda}^T\begin{bmatrix} y_1y_1\boldsymbol{x}_1^T\boldsymbol{x}_1 & y_1y_2\boldsymbol{x}_1^T\boldsymbol{x}_2 & \dots & \dots & y_1y_n\boldsymbol{x}_1^T\boldsymbol{x}_n \\ -y_2y_1\boldsymbol{x}_2^T\boldsymbol{x}_1 & y_2y_2\boldsymbol{x}_2^T\boldsymbol{x}_2 & \dots & \dots & y_1y_n\boldsymbol{x}_2^T\boldsymbol{x}_n \\ -\dots & \dots & \dots & \dots & \dots \\ -\dots & \dots & \dots & \dots & \dots \\ -y_ny_1\boldsymbol{x}_n^T\boldsymbol{x}_1 & y_ny_2\boldsymbol{x}_n^T\boldsymbol{x}_2 & \dots & \dots & y_ny_n\boldsymbol{x}_n^T\boldsymbol{x}_n \\ -\end{bmatrix}\boldsymbol{\lambda}-\mathbb{1}\boldsymbol{\lambda}, +df+\lambda d\phi = (\frac{\partial f}{\partial z}+\lambda +\frac{\partial \phi}{\partial x})dx+(\frac{\partial f}{\partial y}+\lambda\frac{\partial \phi}{\partial y})dy+ +(\frac{\partial f}{\partial z}+\lambda\frac{\partial \phi}{\partial z})dz =0. $$ -subject to \( \boldsymbol{y}^T\boldsymbol{\lambda}=0 \). Here we defined the vectors \( \boldsymbol{\lambda} =[\lambda_1,\lambda_2,\dots,\lambda_n] \) and -\( \boldsymbol{y}=[y_1,y_2,\dots,y_n] \). +Our multiplier is chosen so that +$$ +\frac{\partial f}{\partial z}+\lambda\frac{\partial \phi}{\partial z} =0. +$$ + +

    +We need to remember that we took \( dx \) and \( dy \) to be arbitrary and thus we must have +$$ +\frac{\partial f}{\partial x}+\lambda\frac{\partial \phi}{\partial x} =0, +$$ + +and +$$ +\frac{\partial f}{\partial y}+\lambda\frac{\partial \phi}{\partial y} =0. +$$ + +When all these equations are satisfied, \( df=0 \). We have four unknowns, \( x,y,z \) and +\( \lambda \). Actually we want only \( x,y,z \), \( \lambda \) needs not to be determined, +it is therefore often called +Lagrange's undetermined multiplier. +If we have a set of constraints \( \phi_k \) we have the equations +$$ +\frac{\partial f}{\partial x_i}+\sum_k\lambda_k\frac{\partial \phi_k}{\partial x_i} =0. +$$

    @@ -198,7 +221,7 @@ subject to \( \boldsymbol{y}^T\boldsymbol{\lambda}=0 \). Here we defined the vec

  • 22
  • 23
  • ...
    • -
  • 27
    • +
  • 29
  • »
    • diff --git a/doc/pub/svm/html/._svm-bs014.html b/doc/pub/svm/html/._svm-bs014.html index 47f389a3b..562fea828 100644 --- a/doc/pub/svm/html/._svm-bs014.html +++ b/doc/pub/svm/html/._svm-bs014.html @@ -49,30 +49,32 @@ Automatically generated HTML file from DocOnce source ('Getting into the details', 2, None, '___sec5'), ('First attempt at a minimization approach', 2, None, '___sec6'), ('Solving the equations', 2, None, '___sec7'), - ('A better approach', 2, None, '___sec8'), + ('Code Example', 2, None, '___sec8'), + ('Problems with the Simpler Approach', 2, None, '___sec9'), + ('A better approach', 2, None, '___sec10'), ('A quick reminder on Lagrangian multipliers', 2, None, - '___sec9'), - ('Adding the muliplier', 2, None, '___sec10'), - ('Setting up the problem', 2, None, '___sec11'), - ('The problem to solve', 2, None, '___sec12'), - ('The last steps', 2, None, '___sec13'), - ('A soft classifier', 2, None, '___sec14'), - ('Soft optmization problem', 2, None, '___sec15'), - ('Kernels and non-linearity', 2, None, '___sec16'), - ('The equations', 2, None, '___sec17'), - ('The problem to solve', 2, None, '___sec18'), - ("Different kernels and Mercer's theorem", 2, None, '___sec19'), - ('The moons example', 2, None, '___sec20'), + '___sec11'), + ('Adding the muliplier', 2, None, '___sec12'), + ('Setting up the problem', 2, None, '___sec13'), + ('The problem to solve', 2, None, '___sec14'), + ('The last steps', 2, None, '___sec15'), + ('A soft classifier', 2, None, '___sec16'), + ('Soft optmization problem', 2, None, '___sec17'), + ('Kernels and non-linearity', 2, None, '___sec18'), + ('The equations', 2, None, '___sec19'), + ('The problem to solve', 2, None, '___sec20'), + ("Different kernels and Mercer's theorem", 2, None, '___sec21'), + ('The moons example', 2, None, '___sec22'), ('Mathematical optimization of convex functions', 2, None, - '___sec21'), - ('How do we solve these problems?', 2, None, '___sec22'), - ('A simple example', 2, None, '___sec23'), - ('Back to the more realistic cases', 2, None, '___sec24'), - ('Code Example', 2, None, '___sec25')]} + '___sec23'), + ('How do we solve these problems?', 2, None, '___sec24'), + ('A simple example', 2, None, '___sec25'), + ('Back to the more realistic cases', 2, None, '___sec26'), + ('Code Example', 2, None, '___sec27')]} end of tocinfo --> @@ -118,24 +120,26 @@ MathJax.Hub.Config({
  • Getting into the details
  • First attempt at a minimization approach
  • Solving the equations
    • -
  • A better approach
    • -
  • A quick reminder on Lagrangian multipliers
    • -
  • Adding the muliplier
    • -
  • Setting up the problem
    • -
  • The problem to solve
    • -
  • The last steps
    • -
  • A soft classifier
    • -
  • Soft optmization problem
    • -
  • Kernels and non-linearity
    • -
  • The equations
    • -
  • The problem to solve
    • -
  • Different kernels and Mercer's theorem
    • -
  • The moons example
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simple example
    • -
  • Back to the more realistic cases
    • -
  • Code Example
    • +
  • Code Example
    • +
  • Problems with the Simpler Approach
    • +
  • A better approach
    • +
  • A quick reminder on Lagrangian multipliers
    • +
  • Adding the muliplier
    • +
  • Setting up the problem
    • +
  • The problem to solve
    • +
  • The last steps
    • +
  • A soft classifier
    • +
  • Soft optmization problem
    • +
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • The problem to solve
    • +
  • Different kernels and Mercer's theorem
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simple example
    • +
  • Back to the more realistic cases
    • +
  • Code Example
    • @@ -151,36 +155,43 @@ MathJax.Hub.Config({ -

    The last steps

    +

    Setting up the problem

    +In order to solve the above problem, we define the following Lagrangian function to be minimized +$$ +{\cal L}(\lambda,b,\boldsymbol{w})=\frac{1}{2}\boldsymbol{w}^T\boldsymbol{w}-\sum_{i=1}^n\lambda_i\left[y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)-1\right], +$$ + +where \( \lambda_i \) is a so-called Lagrange multiplier subject to the condition \( \lambda_i \geq 0 \).

    -Solving the above problem, yields the values of \( \lambda_i \). -To find the coefficients of your hyperplane we need simply to compute +Taking the derivatives with respect to \( b \) and \( \boldsymbol{w} \) we obtain $$ -\boldsymbol{w}=\sum_{i} \lambda_iy_i\boldsymbol{x}_i. +\frac{\partial {\cal L}}{\partial b} = -\sum_{i} \lambda_iy_i=0, $$ -With our vector \( \boldsymbol{w} \) we can in turn find the value of the intercept \( b \) (here in two dimensions) via +and $$ -y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)=1, +\frac{\partial {\cal L}}{\partial \boldsymbol{w}} = 0 = \boldsymbol{w}-\sum_{i} \lambda_iy_i\boldsymbol{x}_i. $$ -resulting in +Inserting these constraints into the equation for \( {\cal L} \) we obtain $$ -b = \frac{1}{y_i}-\boldsymbol{w}^T\boldsymbol{x}_i, +{\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{x}_i^T\boldsymbol{x}_j, $$ -or if we write it out in terms of the support vectors only, with \( N_s \) being their number, we have +subject to the constraints \( \lambda_i\geq 0 \) and \( \sum_i\lambda_iy_i=0 \). +We must in addition satisfy the Karush-Kuhn-Tucker (KKT) condition $$ -b = \frac{1}{N_s}\sum_{j\in N_s}\left(y_j-\sum_{i=1}^n\lambda_iy_i\boldsymbol{x}_i^T\boldsymbol{x}_j\right). +\lambda_i\left[y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) -1\right] \hspace{0.1cm}\forall i. $$ -With our hyperplane coefficients we can use our classifier to assign any observation by simply using -$$ -y_i = \mathrm{sign}(\boldsymbol{w}^T\boldsymbol{x}_i+b). -$$ -Below we discuss how to find the optimal values of \( \lambda_i \). Before we proceed however, we discuss now the so-called soft classifier. +

      +
    1. If \( \lambda_i > 0 \), then \( y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)=1 \) and we say that \( x_i \) is on the boundary.
      2. +
    2. If \( y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)> 1 \), we say \( x_i \) is not on the boundary and we set \( \lambda_i=0 \).
      3. +
    + +When \( \lambda_i > 0 \), the vectors \( \boldsymbol{x}_i \) are called support vectors. They are the vectors closest to the line (or hyperplane) and define the margin \( M \).

    @@ -208,7 +219,7 @@ Below we discuss how to find the optimal values of \( \lambda_i \). Before we pr

  • 23
  • 24
  • ...
    • -
  • 27
    • +
  • 29
  • »
    • diff --git a/doc/pub/svm/html/._svm-bs015.html b/doc/pub/svm/html/._svm-bs015.html index 49fe21701..fde86dc1a 100644 --- a/doc/pub/svm/html/._svm-bs015.html +++ b/doc/pub/svm/html/._svm-bs015.html @@ -49,30 +49,32 @@ Automatically generated HTML file from DocOnce source ('Getting into the details', 2, None, '___sec5'), ('First attempt at a minimization approach', 2, None, '___sec6'), ('Solving the equations', 2, None, '___sec7'), - ('A better approach', 2, None, '___sec8'), + ('Code Example', 2, None, '___sec8'), + ('Problems with the Simpler Approach', 2, None, '___sec9'), + ('A better approach', 2, None, '___sec10'), ('A quick reminder on Lagrangian multipliers', 2, None, - '___sec9'), - ('Adding the muliplier', 2, None, '___sec10'), - ('Setting up the problem', 2, None, '___sec11'), - ('The problem to solve', 2, None, '___sec12'), - ('The last steps', 2, None, '___sec13'), - ('A soft classifier', 2, None, '___sec14'), - ('Soft optmization problem', 2, None, '___sec15'), - ('Kernels and non-linearity', 2, None, '___sec16'), - ('The equations', 2, None, '___sec17'), - ('The problem to solve', 2, None, '___sec18'), - ("Different kernels and Mercer's theorem", 2, None, '___sec19'), - ('The moons example', 2, None, '___sec20'), + '___sec11'), + ('Adding the muliplier', 2, None, '___sec12'), + ('Setting up the problem', 2, None, '___sec13'), + ('The problem to solve', 2, None, '___sec14'), + ('The last steps', 2, None, '___sec15'), + ('A soft classifier', 2, None, '___sec16'), + ('Soft optmization problem', 2, None, '___sec17'), + ('Kernels and non-linearity', 2, None, '___sec18'), + ('The equations', 2, None, '___sec19'), + ('The problem to solve', 2, None, '___sec20'), + ("Different kernels and Mercer's theorem", 2, None, '___sec21'), + ('The moons example', 2, None, '___sec22'), ('Mathematical optimization of convex functions', 2, None, - '___sec21'), - ('How do we solve these problems?', 2, None, '___sec22'), - ('A simple example', 2, None, '___sec23'), - ('Back to the more realistic cases', 2, None, '___sec24'), - ('Code Example', 2, None, '___sec25')]} + '___sec23'), + ('How do we solve these problems?', 2, None, '___sec24'), + ('A simple example', 2, None, '___sec25'), + ('Back to the more realistic cases', 2, None, '___sec26'), + ('Code Example', 2, None, '___sec27')]} end of tocinfo --> @@ -118,24 +120,26 @@ MathJax.Hub.Config({
  • Getting into the details
  • First attempt at a minimization approach
  • Solving the equations
    • -
  • A better approach
    • -
  • A quick reminder on Lagrangian multipliers
    • -
  • Adding the muliplier
    • -
  • Setting up the problem
    • -
  • The problem to solve
    • -
  • The last steps
    • -
  • A soft classifier
    • -
  • Soft optmization problem
    • -
  • Kernels and non-linearity
    • -
  • The equations
    • -
  • The problem to solve
    • -
  • Different kernels and Mercer's theorem
    • -
  • The moons example
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simple example
    • -
  • Back to the more realistic cases
    • -
  • Code Example
    • +
  • Code Example
    • +
  • Problems with the Simpler Approach
    • +
  • A better approach
    • +
  • A quick reminder on Lagrangian multipliers
    • +
  • Adding the muliplier
    • +
  • Setting up the problem
    • +
  • The problem to solve
    • +
  • The last steps
    • +
  • A soft classifier
    • +
  • Soft optmization problem
    • +
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • The problem to solve
    • +
  • Different kernels and Mercer's theorem
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simple example
    • +
  • Back to the more realistic cases
    • +
  • Code Example
    • @@ -151,37 +155,26 @@ MathJax.Hub.Config({ -

    A soft classifier

    +

    The problem to solve

    -Till now, the margin is strictly defined by the support vectors. This defines what is called a hard classifier, that is the margins are well defined. - -

    -Suppose now that classes overlap in feature space, as shown in the -figure here. One way to deal with this problem before we define the -so-called kernel approach, is to allow a kind of slack in the sense -that we allow some points to be on the wrong side of the margin. - -

    -We introduce thus the so-called slack variables \( \boldsymbol{\xi} =[\xi_1,x_2,\dots,x_n] \) and -modify our previous equation +We can rewrite $$ -y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)=1, +{\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{x}_i^T\boldsymbol{x}_j, $$ -to +and its constraints in terms of a matrix-vector problem where we minimize w.r.t. \( \lambda \) the following problem $$ -y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)=1-\xi_i, +\frac{1}{2} \boldsymbol{\lambda}^T\begin{bmatrix} y_1y_1\boldsymbol{x}_1^T\boldsymbol{x}_1 & y_1y_2\boldsymbol{x}_1^T\boldsymbol{x}_2 & \dots & \dots & y_1y_n\boldsymbol{x}_1^T\boldsymbol{x}_n \\ +y_2y_1\boldsymbol{x}_2^T\boldsymbol{x}_1 & y_2y_2\boldsymbol{x}_2^T\boldsymbol{x}_2 & \dots & \dots & y_1y_n\boldsymbol{x}_2^T\boldsymbol{x}_n \\ +\dots & \dots & \dots & \dots & \dots \\ +\dots & \dots & \dots & \dots & \dots \\ +y_ny_1\boldsymbol{x}_n^T\boldsymbol{x}_1 & y_ny_2\boldsymbol{x}_n^T\boldsymbol{x}_2 & \dots & \dots & y_ny_n\boldsymbol{x}_n^T\boldsymbol{x}_n \\ +\end{bmatrix}\boldsymbol{\lambda}-\mathbb{1}\boldsymbol{\lambda}, $$ -with the requirement \( \xi_i\geq 0 \). The total violation is now \( \sum_i\xi \). -The value \( \xi_i \) in the constraint the last constraint corresponds to the amount by which the prediction -\( y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)=1 \) is on the wrong side of its margin. Hence by bounding the sum \( \sum_i \xi_i \), -we bound the total amount by which predictions fall on the wrong side of their margins. - -

    -Misclassifications occur when \( \xi_i > 1 \). Thus bounding the total sum by some value \( C \) bounds in turn the total number of -misclassifications. +subject to \( \boldsymbol{y}^T\boldsymbol{\lambda}=0 \). Here we defined the vectors \( \boldsymbol{\lambda} =[\lambda_1,\lambda_2,\dots,\lambda_n] \) and +\( \boldsymbol{y}=[y_1,y_2,\dots,y_n] \).

    @@ -209,7 +202,7 @@ misclassifications.

  • 24
  • 25
  • ...
    • -
  • 27
    • +
  • 29
  • »
    • diff --git a/doc/pub/svm/html/._svm-bs016.html b/doc/pub/svm/html/._svm-bs016.html index 868552403..496a4d385 100644 --- a/doc/pub/svm/html/._svm-bs016.html +++ b/doc/pub/svm/html/._svm-bs016.html @@ -49,30 +49,32 @@ Automatically generated HTML file from DocOnce source ('Getting into the details', 2, None, '___sec5'), ('First attempt at a minimization approach', 2, None, '___sec6'), ('Solving the equations', 2, None, '___sec7'), - ('A better approach', 2, None, '___sec8'), + ('Code Example', 2, None, '___sec8'), + ('Problems with the Simpler Approach', 2, None, '___sec9'), + ('A better approach', 2, None, '___sec10'), ('A quick reminder on Lagrangian multipliers', 2, None, - '___sec9'), - ('Adding the muliplier', 2, None, '___sec10'), - ('Setting up the problem', 2, None, '___sec11'), - ('The problem to solve', 2, None, '___sec12'), - ('The last steps', 2, None, '___sec13'), - ('A soft classifier', 2, None, '___sec14'), - ('Soft optmization problem', 2, None, '___sec15'), - ('Kernels and non-linearity', 2, None, '___sec16'), - ('The equations', 2, None, '___sec17'), - ('The problem to solve', 2, None, '___sec18'), - ("Different kernels and Mercer's theorem", 2, None, '___sec19'), - ('The moons example', 2, None, '___sec20'), + '___sec11'), + ('Adding the muliplier', 2, None, '___sec12'), + ('Setting up the problem', 2, None, '___sec13'), + ('The problem to solve', 2, None, '___sec14'), + ('The last steps', 2, None, '___sec15'), + ('A soft classifier', 2, None, '___sec16'), + ('Soft optmization problem', 2, None, '___sec17'), + ('Kernels and non-linearity', 2, None, '___sec18'), + ('The equations', 2, None, '___sec19'), + ('The problem to solve', 2, None, '___sec20'), + ("Different kernels and Mercer's theorem", 2, None, '___sec21'), + ('The moons example', 2, None, '___sec22'), ('Mathematical optimization of convex functions', 2, None, - '___sec21'), - ('How do we solve these problems?', 2, None, '___sec22'), - ('A simple example', 2, None, '___sec23'), - ('Back to the more realistic cases', 2, None, '___sec24'), - ('Code Example', 2, None, '___sec25')]} + '___sec23'), + ('How do we solve these problems?', 2, None, '___sec24'), + ('A simple example', 2, None, '___sec25'), + ('Back to the more realistic cases', 2, None, '___sec26'), + ('Code Example', 2, None, '___sec27')]} end of tocinfo --> @@ -118,24 +120,26 @@ MathJax.Hub.Config({
  • Getting into the details
  • First attempt at a minimization approach
  • Solving the equations
    • -
  • A better approach
    • -
  • A quick reminder on Lagrangian multipliers
    • -
  • Adding the muliplier
    • -
  • Setting up the problem
    • -
  • The problem to solve
    • -
  • The last steps
    • -
  • A soft classifier
    • -
  • Soft optmization problem
    • -
  • Kernels and non-linearity
    • -
  • The equations
    • -
  • The problem to solve
    • -
  • Different kernels and Mercer's theorem
    • -
  • The moons example
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simple example
    • -
  • Back to the more realistic cases
    • -
  • Code Example
    • +
  • Code Example
    • +
  • Problems with the Simpler Approach
    • +
  • A better approach
    • +
  • A quick reminder on Lagrangian multipliers
    • +
  • Adding the muliplier
    • +
  • Setting up the problem
    • +
  • The problem to solve
    • +
  • The last steps
    • +
  • A soft classifier
    • +
  • Soft optmization problem
    • +
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • The problem to solve
    • +
  • Different kernels and Mercer's theorem
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simple example
    • +
  • Back to the more realistic cases
    • +
  • Code Example
    • @@ -151,56 +155,36 @@ MathJax.Hub.Config({ -

    Soft optmization problem

    +

    The last steps

    -This has in turn the consequences that we change our optmization problem to finding the minimum of +Solving the above problem, yields the values of \( \lambda_i \). +To find the coefficients of your hyperplane we need simply to compute $$ -{\cal L}=\frac{1}{2}\boldsymbol{w}^T\boldsymbol{w}-\sum_{i=1}^n\lambda_i\left[y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)-(1-\xi_)\right]+C\sum_{i=1}^n\xi_i-\sum_{i=1}^n\gamma_i\xi_i, +\boldsymbol{w}=\sum_{i} \lambda_iy_i\boldsymbol{x}_i. $$ -subject to +With our vector \( \boldsymbol{w} \) we can in turn find the value of the intercept \( b \) (here in two dimensions) via $$ -y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)=1-\xi_i \hspace{0.1cm}\forall i, +y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)=1, $$ -with the requirement \( \xi_i\geq 0 \). - -

    -Taking the derivatives with respect to \( b \) and \( \boldsymbol{w} \) we obtain +resulting in $$ -\frac{\partial {\cal L}}{\partial b} = -\sum_{i} \lambda_iy_i=0, +b = \frac{1}{y_i}-\boldsymbol{w}^T\boldsymbol{x}_i, $$ -and +or if we write it out in terms of the support vectors only, with \( N_s \) being their number, we have $$ -\frac{\partial {\cal L}}{\partial \boldsymbol{w}} = 0 = \boldsymbol{w}-\sum_{i} \lambda_iy_i\boldsymbol{x}_i, +b = \frac{1}{N_s}\sum_{j\in N_s}\left(y_j-\sum_{i=1}^n\lambda_iy_i\boldsymbol{x}_i^T\boldsymbol{x}_j\right). $$ -and +With our hyperplane coefficients we can use our classifier to assign any observation by simply using $$ -\lambda_i = C-\gamma_i \hspace{0.1cm}\forall i. +y_i = \mathrm{sign}(\boldsymbol{w}^T\boldsymbol{x}_i+b). $$ -Inserting these constraints into the equation for \( {\cal L} \) we obtain the same equation as before -$$ -{\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{x}_i^T\boldsymbol{x}_j, -$$ - -but now subject to the constraints \( \lambda_i\geq 0 \), \( \sum_i\lambda_iy_i=0 \) and \( 0\leq\lambda_i \leq C \). -We must in addition satisfy the Karush-Kuhn-Tucker condition which now reads -$$ -\lambda_i\left[y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) -(1-\xi_)\right]=0 \hspace{0.1cm}\forall i, -$$ - -$$ -\gamma_i\xi_i = 0, -$$ - -and -$$ -y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) -(1-\xi_) \geq 0 \hspace{0.1cm}\forall i. -$$ +Below we discuss how to find the optimal values of \( \lambda_i \). Before we proceed however, we discuss now the so-called soft classifier.

    @@ -228,7 +212,7 @@ $$

  • 25
  • 26
  • ...
    • -
  • 27
    • +
  • 29
  • »
    • diff --git a/doc/pub/svm/html/._svm-bs017.html b/doc/pub/svm/html/._svm-bs017.html index c89ed61be..d3188a034 100644 --- a/doc/pub/svm/html/._svm-bs017.html +++ b/doc/pub/svm/html/._svm-bs017.html @@ -49,30 +49,32 @@ Automatically generated HTML file from DocOnce source ('Getting into the details', 2, None, '___sec5'), ('First attempt at a minimization approach', 2, None, '___sec6'), ('Solving the equations', 2, None, '___sec7'), - ('A better approach', 2, None, '___sec8'), + ('Code Example', 2, None, '___sec8'), + ('Problems with the Simpler Approach', 2, None, '___sec9'), + ('A better approach', 2, None, '___sec10'), ('A quick reminder on Lagrangian multipliers', 2, None, - '___sec9'), - ('Adding the muliplier', 2, None, '___sec10'), - ('Setting up the problem', 2, None, '___sec11'), - ('The problem to solve', 2, None, '___sec12'), - ('The last steps', 2, None, '___sec13'), - ('A soft classifier', 2, None, '___sec14'), - ('Soft optmization problem', 2, None, '___sec15'), - ('Kernels and non-linearity', 2, None, '___sec16'), - ('The equations', 2, None, '___sec17'), - ('The problem to solve', 2, None, '___sec18'), - ("Different kernels and Mercer's theorem", 2, None, '___sec19'), - ('The moons example', 2, None, '___sec20'), + '___sec11'), + ('Adding the muliplier', 2, None, '___sec12'), + ('Setting up the problem', 2, None, '___sec13'), + ('The problem to solve', 2, None, '___sec14'), + ('The last steps', 2, None, '___sec15'), + ('A soft classifier', 2, None, '___sec16'), + ('Soft optmization problem', 2, None, '___sec17'), + ('Kernels and non-linearity', 2, None, '___sec18'), + ('The equations', 2, None, '___sec19'), + ('The problem to solve', 2, None, '___sec20'), + ("Different kernels and Mercer's theorem", 2, None, '___sec21'), + ('The moons example', 2, None, '___sec22'), ('Mathematical optimization of convex functions', 2, None, - '___sec21'), - ('How do we solve these problems?', 2, None, '___sec22'), - ('A simple example', 2, None, '___sec23'), - ('Back to the more realistic cases', 2, None, '___sec24'), - ('Code Example', 2, None, '___sec25')]} + '___sec23'), + ('How do we solve these problems?', 2, None, '___sec24'), + ('A simple example', 2, None, '___sec25'), + ('Back to the more realistic cases', 2, None, '___sec26'), + ('Code Example', 2, None, '___sec27')]} end of tocinfo --> @@ -118,24 +120,26 @@ MathJax.Hub.Config({
  • Getting into the details
  • First attempt at a minimization approach
  • Solving the equations
    • -
  • A better approach
    • -
  • A quick reminder on Lagrangian multipliers
    • -
  • Adding the muliplier
    • -
  • Setting up the problem
    • -
  • The problem to solve
    • -
  • The last steps
    • -
  • A soft classifier
    • -
  • Soft optmization problem
    • -
  • Kernels and non-linearity
    • -
  • The equations
    • -
  • The problem to solve
    • -
  • Different kernels and Mercer's theorem
    • -
  • The moons example
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simple example
    • -
  • Back to the more realistic cases
    • -
  • Code Example
    • +
  • Code Example
    • +
  • Problems with the Simpler Approach
    • +
  • A better approach
    • +
  • A quick reminder on Lagrangian multipliers
    • +
  • Adding the muliplier
    • +
  • Setting up the problem
    • +
  • The problem to solve
    • +
  • The last steps
    • +
  • A soft classifier
    • +
  • Soft optmization problem
    • +
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • The problem to solve
    • +
  • Different kernels and Mercer's theorem
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simple example
    • +
  • Back to the more realistic cases
    • +
  • Code Example
    • @@ -151,76 +155,38 @@ MathJax.Hub.Config({ -

    Kernels and non-linearity

    +

    A soft classifier

    -The cases we have studied till were all characterized by two classes -with a close to linear separability. The classifiers we have described -so far find linear boundaries in our input feature space. It is -possible to make our procedure more flexible by exploring the feature -space using other basis expansions such higher-order polynomials, -wavelets, splines etc. +Till now, the margin is strictly defined by the support vectors. This defines what is called a hard classifier, that is the margins are well defined.

    -If our feature space is not easy to separate, as shown in the figure -here, we can achieve a better separation by introducing more complex -basis functions. The ideal would be, as shown in the next figure, to, via a specific transformation to -obtain a separation between the classes which is almost linear. +Suppose now that classes overlap in feature space, as shown in the +figure here. One way to deal with this problem before we define the +so-called kernel approach, is to allow a kind of slack in the sense +that we allow some points to be on the wrong side of the margin.

    -The change of basis, from \( x\rightarrow z=\phi(x) \) leads to the same type of equations to be solved, except that -we need to introduce for example a polynomial transformation to a two-dimensional training set. +We introduce thus the so-called slack variables \( \boldsymbol{\xi} =[\xi_1,x_2,\dots,x_n] \) and +modify our previous equation +$$ +y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)=1, +$$ + +to +$$ +y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)=1-\xi_i, +$$ + +with the requirement \( \xi_i\geq 0 \). The total violation is now \( \sum_i\xi \). +The value \( \xi_i \) in the constraint the last constraint corresponds to the amount by which the prediction +\( y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)=1 \) is on the wrong side of its margin. Hence by bounding the sum \( \sum_i \xi_i \), +we bound the total amount by which predictions fall on the wrong side of their margins.

    +Misclassifications occur when \( \xi_i > 1 \). Thus bounding the total sum by some value \( C \) bounds in turn the total number of +misclassifications. - -

    import numpy as np
-import os
-
-np.random.seed(42)
-
-# To plot pretty figures
-import matplotlib
-import matplotlib.pyplot as plt
-plt.rcParams['axes.labelsize'] = 14
-plt.rcParams['xtick.labelsize'] = 12
-plt.rcParams['ytick.labelsize'] = 12
-
-
-from sklearn.svm import SVC
-from sklearn import datasets
-
-
-
-X1D = np.linspace(-4, 4, 9).reshape(-1, 1)
-X2D = np.c_[X1D, X1D**2]
-y = np.array([0, 0, 1, 1, 1, 1, 1, 0, 0])
-
-plt.figure(figsize=(11, 4))
-
-plt.subplot(121)
-plt.grid(True, which='both')
-plt.axhline(y=0, color='k')
-plt.plot(X1D[:, 0][y==0], np.zeros(4), "bs")
-plt.plot(X1D[:, 0][y==1], np.zeros(5), "g^")
-plt.gca().get_yaxis().set_ticks([])
-plt.xlabel(r"$x_1$", fontsize=20)
-plt.axis([-4.5, 4.5, -0.2, 0.2])
-
-plt.subplot(122)
-plt.grid(True, which='both')
-plt.axhline(y=0, color='k')
-plt.axvline(x=0, color='k')
-plt.plot(X2D[:, 0][y==0], X2D[:, 1][y==0], "bs")
-plt.plot(X2D[:, 0][y==1], X2D[:, 1][y==1], "g^")
-plt.xlabel(r"$x_1$", fontsize=20)
-plt.ylabel(r"$x_2$", fontsize=20, rotation=0)
-plt.gca().get_yaxis().set_ticks([0, 4, 8, 12, 16])
-plt.plot([-4.5, 4.5], [6.5, 6.5], "r--", linewidth=3)
-plt.axis([-4.5, 4.5, -1, 17])
-plt.subplots_adjust(right=1)
-plt.show()
-

    @@ -246,6 +212,8 @@ plt.show()

  • 25
  • 26
  • 27
    • +
  • ...
    • +
  • 29
  • »
    • diff --git a/doc/pub/svm/html/._svm-bs018.html b/doc/pub/svm/html/._svm-bs018.html index e5914bda7..7f64bc0d4 100644 --- a/doc/pub/svm/html/._svm-bs018.html +++ b/doc/pub/svm/html/._svm-bs018.html @@ -49,30 +49,32 @@ Automatically generated HTML file from DocOnce source ('Getting into the details', 2, None, '___sec5'), ('First attempt at a minimization approach', 2, None, '___sec6'), ('Solving the equations', 2, None, '___sec7'), - ('A better approach', 2, None, '___sec8'), + ('Code Example', 2, None, '___sec8'), + ('Problems with the Simpler Approach', 2, None, '___sec9'), + ('A better approach', 2, None, '___sec10'), ('A quick reminder on Lagrangian multipliers', 2, None, - '___sec9'), - ('Adding the muliplier', 2, None, '___sec10'), - ('Setting up the problem', 2, None, '___sec11'), - ('The problem to solve', 2, None, '___sec12'), - ('The last steps', 2, None, '___sec13'), - ('A soft classifier', 2, None, '___sec14'), - ('Soft optmization problem', 2, None, '___sec15'), - ('Kernels and non-linearity', 2, None, '___sec16'), - ('The equations', 2, None, '___sec17'), - ('The problem to solve', 2, None, '___sec18'), - ("Different kernels and Mercer's theorem", 2, None, '___sec19'), - ('The moons example', 2, None, '___sec20'), + '___sec11'), + ('Adding the muliplier', 2, None, '___sec12'), + ('Setting up the problem', 2, None, '___sec13'), + ('The problem to solve', 2, None, '___sec14'), + ('The last steps', 2, None, '___sec15'), + ('A soft classifier', 2, None, '___sec16'), + ('Soft optmization problem', 2, None, '___sec17'), + ('Kernels and non-linearity', 2, None, '___sec18'), + ('The equations', 2, None, '___sec19'), + ('The problem to solve', 2, None, '___sec20'), + ("Different kernels and Mercer's theorem", 2, None, '___sec21'), + ('The moons example', 2, None, '___sec22'), ('Mathematical optimization of convex functions', 2, None, - '___sec21'), - ('How do we solve these problems?', 2, None, '___sec22'), - ('A simple example', 2, None, '___sec23'), - ('Back to the more realistic cases', 2, None, '___sec24'), - ('Code Example', 2, None, '___sec25')]} + '___sec23'), + ('How do we solve these problems?', 2, None, '___sec24'), + ('A simple example', 2, None, '___sec25'), + ('Back to the more realistic cases', 2, None, '___sec26'), + ('Code Example', 2, None, '___sec27')]} end of tocinfo --> @@ -118,24 +120,26 @@ MathJax.Hub.Config({
  • Getting into the details
  • First attempt at a minimization approach
  • Solving the equations
    • -
  • A better approach
    • -
  • A quick reminder on Lagrangian multipliers
    • -
  • Adding the muliplier
    • -
  • Setting up the problem
    • -
  • The problem to solve
    • -
  • The last steps
    • -
  • A soft classifier
    • -
  • Soft optmization problem
    • -
  • Kernels and non-linearity
    • -
  • The equations
    • -
  • The problem to solve
    • -
  • Different kernels and Mercer's theorem
    • -
  • The moons example
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simple example
    • -
  • Back to the more realistic cases
    • -
  • Code Example
    • +
  • Code Example
    • +
  • Problems with the Simpler Approach
    • +
  • A better approach
    • +
  • A quick reminder on Lagrangian multipliers
    • +
  • Adding the muliplier
    • +
  • Setting up the problem
    • +
  • The problem to solve
    • +
  • The last steps
    • +
  • A soft classifier
    • +
  • Soft optmization problem
    • +
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • The problem to solve
    • +
  • Different kernels and Mercer's theorem
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simple example
    • +
  • Back to the more realistic cases
    • +
  • Code Example
    • @@ -151,44 +155,56 @@ MathJax.Hub.Config({ -

    The equations

    +

    Soft optmization problem

    -Suppose we define a polynomial transformation of degree two only (we continue to live in a plane with \( x_i \) and \( y_i \) as variables) +This has in turn the consequences that we change our optmization problem to finding the minimum of $$ -z = \phi(x_i) =\left(x_i^2, y_i^2, \sqrt{2}x_iy_i\right). +{\cal L}=\frac{1}{2}\boldsymbol{w}^T\boldsymbol{w}-\sum_{i=1}^n\lambda_i\left[y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)-(1-\xi_)\right]+C\sum_{i=1}^n\xi_i-\sum_{i=1}^n\gamma_i\xi_i, $$ +subject to +$$ +y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)=1-\xi_i \hspace{0.1cm}\forall i, +$$ + +with the requirement \( \xi_i\geq 0 \). +

    -With our new basis, the equations we solved earlier are basically the same, that is we have now (without the slack option for simplicity) +Taking the derivatives with respect to \( b \) and \( \boldsymbol{w} \) we obtain $$ -{\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{x}_i^T\boldsymbol{z}_j, +\frac{\partial {\cal L}}{\partial b} = -\sum_{i} \lambda_iy_i=0, $$ -subject to the constraints \( \lambda_i\geq 0 \), \( \sum_i\lambda_iy_i=0 \), and for the support vectors +and $$ -y_i(\boldsymbol{w}^T\boldsymbol{z}_i+b)= 1 \hspace{0.1cm}\forall i, +\frac{\partial {\cal L}}{\partial \boldsymbol{w}} = 0 = \boldsymbol{w}-\sum_{i} \lambda_iy_i\boldsymbol{x}_i, $$ -from which we also find \( b \). -To compute \( \boldsymbol{z}_i^T\boldsymbol{z}_j \) we define the kerne \( K(\boldsymbol{x}_i,\boldsymbol{x}_j) \) as +and $$ -K(\boldsymbol{x}_i,\boldsymbol{x}_j)=\boldsymbol{z}_i^T\boldsymbol{z}_j= \phi(\boldsymbol{x}_i)^T\phi(\boldsymbol{x}_j). +\lambda_i = C-\gamma_i \hspace{0.1cm}\forall i. $$ -For the above example, the kernel reads +Inserting these constraints into the equation for \( {\cal L} \) we obtain the same equation as before $$ -K(\boldsymbol{x}_i,\boldsymbol{x}_j)=[x_i^2, y_i^2, \sqrt{2}x_iy_i]^T\begin{bmatrix} x_j^2 \\ y_j^2 \\ \sqrt{2}x_jy_j \end{bmatrix}=x_i^2x_j^2+2x_ix_jy_iy_j+y_i^2y_j^2. +{\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{x}_i^T\boldsymbol{x}_j, $$ -

    -We note that this is nothing but the dot product of the two original -vectors \( (\boldsymbol{x}_i^T\boldsymbol{x}_j)^2 \). Instead of thus computing the -product in the Lagrangian of \( \boldsymbol{z}_i^T\boldsymbol{z}_j \) we simply compute -the dot product \( (\boldsymbol{x}_i^T\boldsymbol{x}_j)^2 \). This leads to the so-called -kernel trick and the result leads to the same as if we went through -the trouble of performing the transformation -\( (\phi(\boldsymbol{x}_i)^T\phi(\boldsymbol{x}_j) \) during the SVM calculations. +but now subject to the constraints \( \lambda_i\geq 0 \), \( \sum_i\lambda_iy_i=0 \) and \( 0\leq\lambda_i \leq C \). +We must in addition satisfy the Karush-Kuhn-Tucker condition which now reads +$$ +\lambda_i\left[y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) -(1-\xi_)\right]=0 \hspace{0.1cm}\forall i, +$$ + +$$ +\gamma_i\xi_i = 0, +$$ + +and +$$ +y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) -(1-\xi_) \geq 0 \hspace{0.1cm}\forall i. +$$

    @@ -214,6 +230,9 @@ the trouble of performing the transformation

  • 25
  • 26
  • 27
    • +
  • 28
    • +
  • ...
    • +
  • 29
  • »
    • diff --git a/doc/pub/svm/html/._svm-bs019.html b/doc/pub/svm/html/._svm-bs019.html index 25b07df62..8c92dae33 100644 --- a/doc/pub/svm/html/._svm-bs019.html +++ b/doc/pub/svm/html/._svm-bs019.html @@ -49,30 +49,32 @@ Automatically generated HTML file from DocOnce source ('Getting into the details', 2, None, '___sec5'), ('First attempt at a minimization approach', 2, None, '___sec6'), ('Solving the equations', 2, None, '___sec7'), - ('A better approach', 2, None, '___sec8'), + ('Code Example', 2, None, '___sec8'), + ('Problems with the Simpler Approach', 2, None, '___sec9'), + ('A better approach', 2, None, '___sec10'), ('A quick reminder on Lagrangian multipliers', 2, None, - '___sec9'), - ('Adding the muliplier', 2, None, '___sec10'), - ('Setting up the problem', 2, None, '___sec11'), - ('The problem to solve', 2, None, '___sec12'), - ('The last steps', 2, None, '___sec13'), - ('A soft classifier', 2, None, '___sec14'), - ('Soft optmization problem', 2, None, '___sec15'), - ('Kernels and non-linearity', 2, None, '___sec16'), - ('The equations', 2, None, '___sec17'), - ('The problem to solve', 2, None, '___sec18'), - ("Different kernels and Mercer's theorem", 2, None, '___sec19'), - ('The moons example', 2, None, '___sec20'), + '___sec11'), + ('Adding the muliplier', 2, None, '___sec12'), + ('Setting up the problem', 2, None, '___sec13'), + ('The problem to solve', 2, None, '___sec14'), + ('The last steps', 2, None, '___sec15'), + ('A soft classifier', 2, None, '___sec16'), + ('Soft optmization problem', 2, None, '___sec17'), + ('Kernels and non-linearity', 2, None, '___sec18'), + ('The equations', 2, None, '___sec19'), + ('The problem to solve', 2, None, '___sec20'), + ("Different kernels and Mercer's theorem", 2, None, '___sec21'), + ('The moons example', 2, None, '___sec22'), ('Mathematical optimization of convex functions', 2, None, - '___sec21'), - ('How do we solve these problems?', 2, None, '___sec22'), - ('A simple example', 2, None, '___sec23'), - ('Back to the more realistic cases', 2, None, '___sec24'), - ('Code Example', 2, None, '___sec25')]} + '___sec23'), + ('How do we solve these problems?', 2, None, '___sec24'), + ('A simple example', 2, None, '___sec25'), + ('Back to the more realistic cases', 2, None, '___sec26'), + ('Code Example', 2, None, '___sec27')]} end of tocinfo --> @@ -118,24 +120,26 @@ MathJax.Hub.Config({
  • Getting into the details
  • First attempt at a minimization approach
  • Solving the equations
    • -
  • A better approach
    • -
  • A quick reminder on Lagrangian multipliers
    • -
  • Adding the muliplier
    • -
  • Setting up the problem
    • -
  • The problem to solve
    • -
  • The last steps
    • -
  • A soft classifier
    • -
  • Soft optmization problem
    • -
  • Kernels and non-linearity
    • -
  • The equations
    • -
  • The problem to solve
    • -
  • Different kernels and Mercer's theorem
    • -
  • The moons example
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simple example
    • -
  • Back to the more realistic cases
    • -
  • Code Example
    • +
  • Code Example
    • +
  • Problems with the Simpler Approach
    • +
  • A better approach
    • +
  • A quick reminder on Lagrangian multipliers
    • +
  • Adding the muliplier
    • +
  • Setting up the problem
    • +
  • The problem to solve
    • +
  • The last steps
    • +
  • A soft classifier
    • +
  • Soft optmization problem
    • +
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • The problem to solve
    • +
  • Different kernels and Mercer's theorem
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simple example
    • +
  • Back to the more realistic cases
    • +
  • Code Example
    • @@ -151,39 +155,76 @@ MathJax.Hub.Config({ -

    The problem to solve

    -Using our definition of the kernel We can rewrite again the Lagrangian -$$ -{\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{x}_i^T\boldsymbol{z}_j, -$$ - -subject to the constraints \( \lambda_i\geq 0 \), \( \sum_i\lambda_iy_i=0 \) in terms of a convex optimization problem -$$ -\frac{1}{2} \boldsymbol{\lambda}^T\begin{bmatrix} y_1y_1K(\boldsymbol{x}_1,\boldsymbol{x}_1) & y_1y_2K(\boldsymbol{x}_1,\boldsymbol{x}_2) & \dots & \dots & y_1y_nK(\boldsymbol{x}_1,\boldsymbol{x}_n) \\ -y_2y_1K(\boldsymbol{x}_2,\boldsymbol{x}_1) & y_2y_2(\boldsymbol{x}_2,\boldsymbol{x}_2) & \dots & \dots & y_1y_nK(\boldsymbol{x}_2,\boldsymbol{x}_n) \\ -\dots & \dots & \dots & \dots & \dots \\ -\dots & \dots & \dots & \dots & \dots \\ -y_ny_1K(\boldsymbol{x}_n,\boldsymbol{x}_1) & y_ny_2K(\boldsymbol{x}_n\boldsymbol{x}_2) & \dots & \dots & y_ny_nK(\boldsymbol{x}_n,\boldsymbol{x}_n) \\ -\end{bmatrix}\boldsymbol{\lambda}-\mathbb{1}\boldsymbol{\lambda}, -$$ - -subject to \( \boldsymbol{y}^T\boldsymbol{\lambda}=0 \). Here we defined the vectors \( \boldsymbol{\lambda} =[\lambda_1,\lambda_2,\dots,\lambda_n] \) and -\( \boldsymbol{y}=[y_1,y_2,\dots,y_n] \). -If we add the slack constants this leads to the additional constraint \( 0\leq \lambda_i \leq C \). +

    Kernels and non-linearity

    -We can rewrite this (see the solutions below) in terms of a convex optimization problem of the type -$$ -\begin{align*} - &\mathrm{min}_{\lambda}\hspace{0.2cm} \frac{1}{2}\boldsymbol{\lambda}^T\boldsymbol{P}\boldsymbol{\lambda}+\boldsymbol{q}^T\boldsymbol{\lambda},\\ \nonumber - &\mathrm{subject\hspace{0.1cm}to} \hspace{0.2cm} \boldsymbol{G}\boldsymbol{\lambda} \preceq \boldsymbol{h} \hspace{0.2cm} \wedge \boldsymbol{A}\boldsymbol{\lambda}=f. -\end{align*} -$$ +The cases we have studied till were all characterized by two classes +with a close to linear separability. The classifiers we have described +so far find linear boundaries in our input feature space. It is +possible to make our procedure more flexible by exploring the feature +space using other basis expansions such higher-order polynomials, +wavelets, splines etc. -Below we discuss how to solve these equations. Here we note that the matrix \( \boldsymbol{P} \) has matrix elements \( p_{ij}=y_iy_jK(\boldsymbol{x}_i,\boldsymbol{x}_j) \). -Given a kernel \( K \) and the targets \( y_i \) this matrix is easy to set up. The constraint \( \boldsymbol{y}^T\boldsymbol{\lambda}=0 \) leads to \( f=0 \) and \( \boldsymbol{A}=\boldsymbol{y} \). How to set up the matrix \( \boldsymbol{G} \) is discussed later. Here note that the inequalities \( 0\leq \lambda_i \leq C \) can be split up into -\( 0\leq \lambda_i \) and \( \lambda_i \leq C \). These two inequalities define then the matrix \( \boldsymbol{G} \) and the vector \( \boldsymbol{h} \). +

    +If our feature space is not easy to separate, as shown in the figure +here, we can achieve a better separation by introducing more complex +basis functions. The ideal would be, as shown in the next figure, to, via a specific transformation to +obtain a separation between the classes which is almost linear. +

    +The change of basis, from \( x\rightarrow z=\phi(x) \) leads to the same type of equations to be solved, except that +we need to introduce for example a polynomial transformation to a two-dimensional training set. + +

    + + +

    import numpy as np
+import os
+
+np.random.seed(42)
+
+# To plot pretty figures
+import matplotlib
+import matplotlib.pyplot as plt
+plt.rcParams['axes.labelsize'] = 14
+plt.rcParams['xtick.labelsize'] = 12
+plt.rcParams['ytick.labelsize'] = 12
+
+
+from sklearn.svm import SVC
+from sklearn import datasets
+
+
+
+X1D = np.linspace(-4, 4, 9).reshape(-1, 1)
+X2D = np.c_[X1D, X1D**2]
+y = np.array([0, 0, 1, 1, 1, 1, 1, 0, 0])
+
+plt.figure(figsize=(11, 4))
+
+plt.subplot(121)
+plt.grid(True, which='both')
+plt.axhline(y=0, color='k')
+plt.plot(X1D[:, 0][y==0], np.zeros(4), "bs")
+plt.plot(X1D[:, 0][y==1], np.zeros(5), "g^")
+plt.gca().get_yaxis().set_ticks([])
+plt.xlabel(r"$x_1$", fontsize=20)
+plt.axis([-4.5, 4.5, -0.2, 0.2])
+
+plt.subplot(122)
+plt.grid(True, which='both')
+plt.axhline(y=0, color='k')
+plt.axvline(x=0, color='k')
+plt.plot(X2D[:, 0][y==0], X2D[:, 1][y==0], "bs")
+plt.plot(X2D[:, 0][y==1], X2D[:, 1][y==1], "g^")
+plt.xlabel(r"$x_1$", fontsize=20)
+plt.ylabel(r"$x_2$", fontsize=20, rotation=0)
+plt.gca().get_yaxis().set_ticks([0, 4, 8, 12, 16])
+plt.plot([-4.5, 4.5], [6.5, 6.5], "r--", linewidth=3)
+plt.axis([-4.5, 4.5, -1, 17])
+plt.subplots_adjust(right=1)
+plt.show()
+

    @@ -207,6 +248,8 @@ Given a kernel \( K \) and the targets \( y_i \) this matrix is easy to set up.

  • 25
  • 26
  • 27
    • +
  • 28
    • +
  • 29
  • »
    • diff --git a/doc/pub/svm/html/._svm-bs020.html b/doc/pub/svm/html/._svm-bs020.html index bd289d77f..9e90df4cc 100644 --- a/doc/pub/svm/html/._svm-bs020.html +++ b/doc/pub/svm/html/._svm-bs020.html @@ -49,30 +49,32 @@ Automatically generated HTML file from DocOnce source ('Getting into the details', 2, None, '___sec5'), ('First attempt at a minimization approach', 2, None, '___sec6'), ('Solving the equations', 2, None, '___sec7'), - ('A better approach', 2, None, '___sec8'), + ('Code Example', 2, None, '___sec8'), + ('Problems with the Simpler Approach', 2, None, '___sec9'), + ('A better approach', 2, None, '___sec10'), ('A quick reminder on Lagrangian multipliers', 2, None, - '___sec9'), - ('Adding the muliplier', 2, None, '___sec10'), - ('Setting up the problem', 2, None, '___sec11'), - ('The problem to solve', 2, None, '___sec12'), - ('The last steps', 2, None, '___sec13'), - ('A soft classifier', 2, None, '___sec14'), - ('Soft optmization problem', 2, None, '___sec15'), - ('Kernels and non-linearity', 2, None, '___sec16'), - ('The equations', 2, None, '___sec17'), - ('The problem to solve', 2, None, '___sec18'), - ("Different kernels and Mercer's theorem", 2, None, '___sec19'), - ('The moons example', 2, None, '___sec20'), + '___sec11'), + ('Adding the muliplier', 2, None, '___sec12'), + ('Setting up the problem', 2, None, '___sec13'), + ('The problem to solve', 2, None, '___sec14'), + ('The last steps', 2, None, '___sec15'), + ('A soft classifier', 2, None, '___sec16'), + ('Soft optmization problem', 2, None, '___sec17'), + ('Kernels and non-linearity', 2, None, '___sec18'), + ('The equations', 2, None, '___sec19'), + ('The problem to solve', 2, None, '___sec20'), + ("Different kernels and Mercer's theorem", 2, None, '___sec21'), + ('The moons example', 2, None, '___sec22'), ('Mathematical optimization of convex functions', 2, None, - '___sec21'), - ('How do we solve these problems?', 2, None, '___sec22'), - ('A simple example', 2, None, '___sec23'), - ('Back to the more realistic cases', 2, None, '___sec24'), - ('Code Example', 2, None, '___sec25')]} + '___sec23'), + ('How do we solve these problems?', 2, None, '___sec24'), + ('A simple example', 2, None, '___sec25'), + ('Back to the more realistic cases', 2, None, '___sec26'), + ('Code Example', 2, None, '___sec27')]} end of tocinfo --> @@ -118,24 +120,26 @@ MathJax.Hub.Config({
  • Getting into the details
  • First attempt at a minimization approach
  • Solving the equations
    • -
  • A better approach
    • -
  • A quick reminder on Lagrangian multipliers
    • -
  • Adding the muliplier
    • -
  • Setting up the problem
    • -
  • The problem to solve
    • -
  • The last steps
    • -
  • A soft classifier
    • -
  • Soft optmization problem
    • -
  • Kernels and non-linearity
    • -
  • The equations
    • -
  • The problem to solve
    • -
  • Different kernels and Mercer's theorem
    • -
  • The moons example
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simple example
    • -
  • Back to the more realistic cases
    • -
  • Code Example
    • +
  • Code Example
    • +
  • Problems with the Simpler Approach
    • +
  • A better approach
    • +
  • A quick reminder on Lagrangian multipliers
    • +
  • Adding the muliplier
    • +
  • Setting up the problem
    • +
  • The problem to solve
    • +
  • The last steps
    • +
  • A soft classifier
    • +
  • Soft optmization problem
    • +
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • The problem to solve
    • +
  • Different kernels and Mercer's theorem
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simple example
    • +
  • Back to the more realistic cases
    • +
  • Code Example
    • @@ -151,33 +155,44 @@ MathJax.Hub.Config({ -

    Different kernels and Mercer's theorem

    +

    The equations

    -There are several popular kernels being used. These are - -

      -
    1. Linear: \( K(\boldsymbol{x},\boldsymbol{y})=\boldsymbol{x}^T\boldsymbol{y} \),
      2. -
    2. Polynomial: \( K(\boldsymbol{x},\boldsymbol{y})=(\boldsymbol{x}^T\boldsymbol{y}+\gamma)^d \),
      3. -
    3. Gaussian Radial Basis Function: \( K(\boldsymbol{x},\boldsymbol{y})=\exp{\left(-\gamma\vert\vert\boldsymbol{x}-\boldsymbol{y}\vert\vert^2\right)} \),
      4. -
    4. Tanh: \( K(\boldsymbol{x},\boldsymbol{y})=\tanh{(\boldsymbol{x}^T\boldsymbol{y}+\gamma)} \),
      5. -
    - -and many other ones. +Suppose we define a polynomial transformation of degree two only (we continue to live in a plane with \( x_i \) and \( y_i \) as variables) +$$ +z = \phi(x_i) =\left(x_i^2, y_i^2, \sqrt{2}x_iy_i\right). +$$

    -An important theorem for us is Mercer's theorem. -The theorem states that if a kernel function \( K \) is symmetric, continuous and leads to a positive semi-definite matrix \( \boldsymbol{P} \) then -there exists a function \( \phi \) that maps \( \boldsymbol{x}_i \) and \( \boldsymbol{x}_j \) into another space -(possibly with much higher dimensions) such that +With our new basis, the equations we solved earlier are basically the same, that is we have now (without the slack option for simplicity) $$ -K(\boldsymbol{x}_i,\boldsymbol{x}_j)=\phi(\boldsymbol{x}_i)^T\phi(\boldsymbol{x}_j). +{\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{x}_i^T\boldsymbol{z}_j, $$ -So you can use \( K \) as a kernel since you know \( \phi \) exists, even if -you don’t know what \( \phi \) is. -Note that some frequently used kernels (such as the Sigmoid kernel) don’t respect all of Mercer’s conditions, yet they generally work -well in practice. +subject to the constraints \( \lambda_i\geq 0 \), \( \sum_i\lambda_iy_i=0 \), and for the support vectors +$$ +y_i(\boldsymbol{w}^T\boldsymbol{z}_i+b)= 1 \hspace{0.1cm}\forall i, +$$ + +from which we also find \( b \). +To compute \( \boldsymbol{z}_i^T\boldsymbol{z}_j \) we define the kerne \( K(\boldsymbol{x}_i,\boldsymbol{x}_j) \) as +$$ +K(\boldsymbol{x}_i,\boldsymbol{x}_j)=\boldsymbol{z}_i^T\boldsymbol{z}_j= \phi(\boldsymbol{x}_i)^T\phi(\boldsymbol{x}_j). +$$ + +For the above example, the kernel reads +$$ +K(\boldsymbol{x}_i,\boldsymbol{x}_j)=[x_i^2, y_i^2, \sqrt{2}x_iy_i]^T\begin{bmatrix} x_j^2 \\ y_j^2 \\ \sqrt{2}x_jy_j \end{bmatrix}=x_i^2x_j^2+2x_ix_jy_iy_j+y_i^2y_j^2. +$$ + +

    +We note that this is nothing but the dot product of the two original +vectors \( (\boldsymbol{x}_i^T\boldsymbol{x}_j)^2 \). Instead of thus computing the +product in the Lagrangian of \( \boldsymbol{z}_i^T\boldsymbol{z}_j \) we simply compute +the dot product \( (\boldsymbol{x}_i^T\boldsymbol{x}_j)^2 \). This leads to the so-called +kernel trick and the result leads to the same as if we went through +the trouble of performing the transformation +\( (\phi(\boldsymbol{x}_i)^T\phi(\boldsymbol{x}_j) \) during the SVM calculations.

    @@ -201,6 +216,8 @@ well in practice.

  • 25
  • 26
  • 27
    • +
  • 28
    • +
  • 29
  • »
    • diff --git a/doc/pub/svm/html/._svm-bs021.html b/doc/pub/svm/html/._svm-bs021.html index 8af5cea84..814f24f67 100644 --- a/doc/pub/svm/html/._svm-bs021.html +++ b/doc/pub/svm/html/._svm-bs021.html @@ -49,30 +49,32 @@ Automatically generated HTML file from DocOnce source ('Getting into the details', 2, None, '___sec5'), ('First attempt at a minimization approach', 2, None, '___sec6'), ('Solving the equations', 2, None, '___sec7'), - ('A better approach', 2, None, '___sec8'), + ('Code Example', 2, None, '___sec8'), + ('Problems with the Simpler Approach', 2, None, '___sec9'), + ('A better approach', 2, None, '___sec10'), ('A quick reminder on Lagrangian multipliers', 2, None, - '___sec9'), - ('Adding the muliplier', 2, None, '___sec10'), - ('Setting up the problem', 2, None, '___sec11'), - ('The problem to solve', 2, None, '___sec12'), - ('The last steps', 2, None, '___sec13'), - ('A soft classifier', 2, None, '___sec14'), - ('Soft optmization problem', 2, None, '___sec15'), - ('Kernels and non-linearity', 2, None, '___sec16'), - ('The equations', 2, None, '___sec17'), - ('The problem to solve', 2, None, '___sec18'), - ("Different kernels and Mercer's theorem", 2, None, '___sec19'), - ('The moons example', 2, None, '___sec20'), + '___sec11'), + ('Adding the muliplier', 2, None, '___sec12'), + ('Setting up the problem', 2, None, '___sec13'), + ('The problem to solve', 2, None, '___sec14'), + ('The last steps', 2, None, '___sec15'), + ('A soft classifier', 2, None, '___sec16'), + ('Soft optmization problem', 2, None, '___sec17'), + ('Kernels and non-linearity', 2, None, '___sec18'), + ('The equations', 2, None, '___sec19'), + ('The problem to solve', 2, None, '___sec20'), + ("Different kernels and Mercer's theorem", 2, None, '___sec21'), + ('The moons example', 2, None, '___sec22'), ('Mathematical optimization of convex functions', 2, None, - '___sec21'), - ('How do we solve these problems?', 2, None, '___sec22'), - ('A simple example', 2, None, '___sec23'), - ('Back to the more realistic cases', 2, None, '___sec24'), - ('Code Example', 2, None, '___sec25')]} + '___sec23'), + ('How do we solve these problems?', 2, None, '___sec24'), + ('A simple example', 2, None, '___sec25'), + ('Back to the more realistic cases', 2, None, '___sec26'), + ('Code Example', 2, None, '___sec27')]} end of tocinfo --> @@ -118,24 +120,26 @@ MathJax.Hub.Config({
  • Getting into the details
  • First attempt at a minimization approach
  • Solving the equations
    • -
  • A better approach
    • -
  • A quick reminder on Lagrangian multipliers
    • -
  • Adding the muliplier
    • -
  • Setting up the problem
    • -
  • The problem to solve
    • -
  • The last steps
    • -
  • A soft classifier
    • -
  • Soft optmization problem
    • -
  • Kernels and non-linearity
    • -
  • The equations
    • -
  • The problem to solve
    • -
  • Different kernels and Mercer's theorem
    • -
  • The moons example
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simple example
    • -
  • Back to the more realistic cases
    • -
  • Code Example
    • +
  • Code Example
    • +
  • Problems with the Simpler Approach
    • +
  • A better approach
    • +
  • A quick reminder on Lagrangian multipliers
    • +
  • Adding the muliplier
    • +
  • Setting up the problem
    • +
  • The problem to solve
    • +
  • The last steps
    • +
  • A soft classifier
    • +
  • Soft optmization problem
    • +
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • The problem to solve
    • +
  • Different kernels and Mercer's theorem
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simple example
    • +
  • Back to the more realistic cases
    • +
  • Code Example
    • @@ -151,199 +155,39 @@ MathJax.Hub.Config({ -

    The moons example

    +

    The problem to solve

    +Using our definition of the kernel We can rewrite again the Lagrangian +$$ +{\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{x}_i^T\boldsymbol{z}_j, +$$ + +subject to the constraints \( \lambda_i\geq 0 \), \( \sum_i\lambda_iy_i=0 \) in terms of a convex optimization problem +$$ +\frac{1}{2} \boldsymbol{\lambda}^T\begin{bmatrix} y_1y_1K(\boldsymbol{x}_1,\boldsymbol{x}_1) & y_1y_2K(\boldsymbol{x}_1,\boldsymbol{x}_2) & \dots & \dots & y_1y_nK(\boldsymbol{x}_1,\boldsymbol{x}_n) \\ +y_2y_1K(\boldsymbol{x}_2,\boldsymbol{x}_1) & y_2y_2(\boldsymbol{x}_2,\boldsymbol{x}_2) & \dots & \dots & y_1y_nK(\boldsymbol{x}_2,\boldsymbol{x}_n) \\ +\dots & \dots & \dots & \dots & \dots \\ +\dots & \dots & \dots & \dots & \dots \\ +y_ny_1K(\boldsymbol{x}_n,\boldsymbol{x}_1) & y_ny_2K(\boldsymbol{x}_n\boldsymbol{x}_2) & \dots & \dots & y_ny_nK(\boldsymbol{x}_n,\boldsymbol{x}_n) \\ +\end{bmatrix}\boldsymbol{\lambda}-\mathbb{1}\boldsymbol{\lambda}, +$$ + +subject to \( \boldsymbol{y}^T\boldsymbol{\lambda}=0 \). Here we defined the vectors \( \boldsymbol{\lambda} =[\lambda_1,\lambda_2,\dots,\lambda_n] \) and +\( \boldsymbol{y}=[y_1,y_2,\dots,y_n] \). +If we add the slack constants this leads to the additional constraint \( 0\leq \lambda_i \leq C \). +

    +We can rewrite this (see the solutions below) in terms of a convex optimization problem of the type +$$ +\begin{align*} + &\mathrm{min}_{\lambda}\hspace{0.2cm} \frac{1}{2}\boldsymbol{\lambda}^T\boldsymbol{P}\boldsymbol{\lambda}+\boldsymbol{q}^T\boldsymbol{\lambda},\\ \nonumber + &\mathrm{subject\hspace{0.1cm}to} \hspace{0.2cm} \boldsymbol{G}\boldsymbol{\lambda} \preceq \boldsymbol{h} \hspace{0.2cm} \wedge \boldsymbol{A}\boldsymbol{\lambda}=f. +\end{align*} +$$ - -

    from __future__ import division, print_function, unicode_literals
+Below we discuss how to solve these equations. Here we note that the matrix \( \boldsymbol{P} \) has matrix elements \( p_{ij}=y_iy_jK(\boldsymbol{x}_i,\boldsymbol{x}_j) \).
+Given a kernel \( K \) and the targets \( y_i \) this matrix is easy to set up. The constraint \( \boldsymbol{y}^T\boldsymbol{\lambda}=0 \) leads to \( f=0 \) and \( \boldsymbol{A}=\boldsymbol{y} \). How to set up the matrix \( \boldsymbol{G} \) is discussed later. Here note that the inequalities \( 0\leq \lambda_i \leq C \) can be split up into
+\( 0\leq \lambda_i \) and \( \lambda_i \leq C \). These two inequalities define then the matrix \( \boldsymbol{G} \) and the vector \( \boldsymbol{h} \).
 
-import numpy as np
-np.random.seed(42)
-
-import matplotlib
-import matplotlib.pyplot as plt
-plt.rcParams['axes.labelsize'] = 14
-plt.rcParams['xtick.labelsize'] = 12
-plt.rcParams['ytick.labelsize'] = 12
-
-
-from sklearn.svm import SVC
-from sklearn import datasets
-
-
-
-from sklearn.pipeline import Pipeline
-from sklearn.preprocessing import StandardScaler
-from sklearn.svm import LinearSVC
-
-
-from sklearn.datasets import make_moons
-X, y = make_moons(n_samples=100, noise=0.15, random_state=42)
-
-def plot_dataset(X, y, axes):
-    plt.plot(X[:, 0][y==0], X[:, 1][y==0], "bs")
-    plt.plot(X[:, 0][y==1], X[:, 1][y==1], "g^")
-    plt.axis(axes)
-    plt.grid(True, which='both')
-    plt.xlabel(r"$x_1$", fontsize=20)
-    plt.ylabel(r"$x_2$", fontsize=20, rotation=0)
-
-plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])
-plt.show()
-
-from sklearn.datasets import make_moons
-from sklearn.pipeline import Pipeline
-from sklearn.preprocessing import PolynomialFeatures
-
-polynomial_svm_clf = Pipeline([
-        ("poly_features", PolynomialFeatures(degree=3)),
-        ("scaler", StandardScaler()),
-        ("svm_clf", LinearSVC(C=10, loss="hinge", random_state=42))
-    ])
-
-polynomial_svm_clf.fit(X, y)
-
-def plot_predictions(clf, axes):
-    x0s = np.linspace(axes[0], axes[1], 100)
-    x1s = np.linspace(axes[2], axes[3], 100)
-    x0, x1 = np.meshgrid(x0s, x1s)
-    X = np.c_[x0.ravel(), x1.ravel()]
-    y_pred = clf.predict(X).reshape(x0.shape)
-    y_decision = clf.decision_function(X).reshape(x0.shape)
-    plt.contourf(x0, x1, y_pred, cmap=plt.cm.brg, alpha=0.2)
-    plt.contourf(x0, x1, y_decision, cmap=plt.cm.brg, alpha=0.1)
-
-plot_predictions(polynomial_svm_clf, [-1.5, 2.5, -1, 1.5])
-plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])
-
-plt.show()
-
-
-from sklearn.svm import SVC
-
-poly_kernel_svm_clf = Pipeline([
-        ("scaler", StandardScaler()),
-        ("svm_clf", SVC(kernel="poly", degree=3, coef0=1, C=5))
-    ])
-poly_kernel_svm_clf.fit(X, y)
-
-poly100_kernel_svm_clf = Pipeline([
-        ("scaler", StandardScaler()),
-        ("svm_clf", SVC(kernel="poly", degree=10, coef0=100, C=5))
-    ])
-poly100_kernel_svm_clf.fit(X, y)
-
-plt.figure(figsize=(11, 4))
-
-plt.subplot(121)
-plot_predictions(poly_kernel_svm_clf, [-1.5, 2.5, -1, 1.5])
-plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])
-plt.title(r"$d=3, r=1, C=5$", fontsize=18)
-
-plt.subplot(122)
-plot_predictions(poly100_kernel_svm_clf, [-1.5, 2.5, -1, 1.5])
-plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])
-plt.title(r"$d=10, r=100, C=5$", fontsize=18)
-
-plt.show()
-
-def gaussian_rbf(x, landmark, gamma):
-    return np.exp(-gamma * np.linalg.norm(x - landmark, axis=1)**2)
-
-gamma = 0.3
-
-x1s = np.linspace(-4.5, 4.5, 200).reshape(-1, 1)
-x2s = gaussian_rbf(x1s, -2, gamma)
-x3s = gaussian_rbf(x1s, 1, gamma)
-
-XK = np.c_[gaussian_rbf(X1D, -2, gamma), gaussian_rbf(X1D, 1, gamma)]
-yk = np.array([0, 0, 1, 1, 1, 1, 1, 0, 0])
-
-plt.figure(figsize=(11, 4))
-
-plt.subplot(121)
-plt.grid(True, which='both')
-plt.axhline(y=0, color='k')
-plt.scatter(x=[-2, 1], y=[0, 0], s=150, alpha=0.5, c="red")
-plt.plot(X1D[:, 0][yk==0], np.zeros(4), "bs")
-plt.plot(X1D[:, 0][yk==1], np.zeros(5), "g^")
-plt.plot(x1s, x2s, "g--")
-plt.plot(x1s, x3s, "b:")
-plt.gca().get_yaxis().set_ticks([0, 0.25, 0.5, 0.75, 1])
-plt.xlabel(r"$x_1$", fontsize=20)
-plt.ylabel(r"Similarity", fontsize=14)
-plt.annotate(r'$\mathbf{x}$',
-             xy=(X1D[3, 0], 0),
-             xytext=(-0.5, 0.20),
-             ha="center",
-             arrowprops=dict(facecolor='black', shrink=0.1),
-             fontsize=18,
-            )
-plt.text(-2, 0.9, "$x_2$", ha="center", fontsize=20)
-plt.text(1, 0.9, "$x_3$", ha="center", fontsize=20)
-plt.axis([-4.5, 4.5, -0.1, 1.1])
-
-plt.subplot(122)
-plt.grid(True, which='both')
-plt.axhline(y=0, color='k')
-plt.axvline(x=0, color='k')
-plt.plot(XK[:, 0][yk==0], XK[:, 1][yk==0], "bs")
-plt.plot(XK[:, 0][yk==1], XK[:, 1][yk==1], "g^")
-plt.xlabel(r"$x_2$", fontsize=20)
-plt.ylabel(r"$x_3$  ", fontsize=20, rotation=0)
-plt.annotate(r'$\phi\left(\mathbf{x}\right)$',
-             xy=(XK[3, 0], XK[3, 1]),
-             xytext=(0.65, 0.50),
-             ha="center",
-             arrowprops=dict(facecolor='black', shrink=0.1),
-             fontsize=18,
-            )
-plt.plot([-0.1, 1.1], [0.57, -0.1], "r--", linewidth=3)
-plt.axis([-0.1, 1.1, -0.1, 1.1])
-    
-plt.subplots_adjust(right=1)
-
-plt.show()
-
-
-x1_example = X1D[3, 0]
-for landmark in (-2, 1):
-    k = gaussian_rbf(np.array([[x1_example]]), np.array([[landmark]]), gamma)
-    print("Phi({}, {}) = {}".format(x1_example, landmark, k))
-
-rbf_kernel_svm_clf = Pipeline([
-        ("scaler", StandardScaler()),
-        ("svm_clf", SVC(kernel="rbf", gamma=5, C=0.001))
-    ])
-rbf_kernel_svm_clf.fit(X, y)
-
-
-from sklearn.svm import SVC
-
-gamma1, gamma2 = 0.1, 5
-C1, C2 = 0.001, 1000
-hyperparams = (gamma1, C1), (gamma1, C2), (gamma2, C1), (gamma2, C2)
-
-svm_clfs = []
-for gamma, C in hyperparams:
-    rbf_kernel_svm_clf = Pipeline([
-            ("scaler", StandardScaler()),
-            ("svm_clf", SVC(kernel="rbf", gamma=gamma, C=C))
-        ])
-    rbf_kernel_svm_clf.fit(X, y)
-    svm_clfs.append(rbf_kernel_svm_clf)
-
-plt.figure(figsize=(11, 7))
-
-for i, svm_clf in enumerate(svm_clfs):
-    plt.subplot(221 + i)
-    plot_predictions(svm_clf, [-1.5, 2.5, -1, 1.5])
-    plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])
-    gamma, C = hyperparams[i]
-    plt.title(r"$\gamma = {}, C = {}$".format(gamma, C), fontsize=16)
-
-plt.show()
-

    @@ -365,6 +209,8 @@ plt.show()

  • 25
  • 26
  • 27
    • +
  • 28
    • +
  • 29
  • »
    • diff --git a/doc/pub/svm/html/._svm-bs022.html b/doc/pub/svm/html/._svm-bs022.html index 1cb7942ac..a529ff2e2 100644 --- a/doc/pub/svm/html/._svm-bs022.html +++ b/doc/pub/svm/html/._svm-bs022.html @@ -49,30 +49,32 @@ Automatically generated HTML file from DocOnce source ('Getting into the details', 2, None, '___sec5'), ('First attempt at a minimization approach', 2, None, '___sec6'), ('Solving the equations', 2, None, '___sec7'), - ('A better approach', 2, None, '___sec8'), + ('Code Example', 2, None, '___sec8'), + ('Problems with the Simpler Approach', 2, None, '___sec9'), + ('A better approach', 2, None, '___sec10'), ('A quick reminder on Lagrangian multipliers', 2, None, - '___sec9'), - ('Adding the muliplier', 2, None, '___sec10'), - ('Setting up the problem', 2, None, '___sec11'), - ('The problem to solve', 2, None, '___sec12'), - ('The last steps', 2, None, '___sec13'), - ('A soft classifier', 2, None, '___sec14'), - ('Soft optmization problem', 2, None, '___sec15'), - ('Kernels and non-linearity', 2, None, '___sec16'), - ('The equations', 2, None, '___sec17'), - ('The problem to solve', 2, None, '___sec18'), - ("Different kernels and Mercer's theorem", 2, None, '___sec19'), - ('The moons example', 2, None, '___sec20'), + '___sec11'), + ('Adding the muliplier', 2, None, '___sec12'), + ('Setting up the problem', 2, None, '___sec13'), + ('The problem to solve', 2, None, '___sec14'), + ('The last steps', 2, None, '___sec15'), + ('A soft classifier', 2, None, '___sec16'), + ('Soft optmization problem', 2, None, '___sec17'), + ('Kernels and non-linearity', 2, None, '___sec18'), + ('The equations', 2, None, '___sec19'), + ('The problem to solve', 2, None, '___sec20'), + ("Different kernels and Mercer's theorem", 2, None, '___sec21'), + ('The moons example', 2, None, '___sec22'), ('Mathematical optimization of convex functions', 2, None, - '___sec21'), - ('How do we solve these problems?', 2, None, '___sec22'), - ('A simple example', 2, None, '___sec23'), - ('Back to the more realistic cases', 2, None, '___sec24'), - ('Code Example', 2, None, '___sec25')]} + '___sec23'), + ('How do we solve these problems?', 2, None, '___sec24'), + ('A simple example', 2, None, '___sec25'), + ('Back to the more realistic cases', 2, None, '___sec26'), + ('Code Example', 2, None, '___sec27')]} end of tocinfo --> @@ -118,24 +120,26 @@ MathJax.Hub.Config({
  • Getting into the details
  • First attempt at a minimization approach
  • Solving the equations
    • -
  • A better approach
    • -
  • A quick reminder on Lagrangian multipliers
    • -
  • Adding the muliplier
    • -
  • Setting up the problem
    • -
  • The problem to solve
    • -
  • The last steps
    • -
  • A soft classifier
    • -
  • Soft optmization problem
    • -
  • Kernels and non-linearity
    • -
  • The equations
    • -
  • The problem to solve
    • -
  • Different kernels and Mercer's theorem
    • -
  • The moons example
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simple example
    • -
  • Back to the more realistic cases
    • -
  • Code Example
    • +
  • Code Example
    • +
  • Problems with the Simpler Approach
    • +
  • A better approach
    • +
  • A quick reminder on Lagrangian multipliers
    • +
  • Adding the muliplier
    • +
  • Setting up the problem
    • +
  • The problem to solve
    • +
  • The last steps
    • +
  • A soft classifier
    • +
  • Soft optmization problem
    • +
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • The problem to solve
    • +
  • Different kernels and Mercer's theorem
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simple example
    • +
  • Back to the more realistic cases
    • +
  • Code Example
    • @@ -151,27 +155,33 @@ MathJax.Hub.Config({ -

    Mathematical optimization of convex functions

    +

    Different kernels and Mercer's theorem

    -A mathematical (quadratic) optimization problem, or just optimization problem, has the form +There are several popular kernels being used. These are + +

      +
    1. Linear: \( K(\boldsymbol{x},\boldsymbol{y})=\boldsymbol{x}^T\boldsymbol{y} \),
      2. +
    2. Polynomial: \( K(\boldsymbol{x},\boldsymbol{y})=(\boldsymbol{x}^T\boldsymbol{y}+\gamma)^d \),
      3. +
    3. Gaussian Radial Basis Function: \( K(\boldsymbol{x},\boldsymbol{y})=\exp{\left(-\gamma\vert\vert\boldsymbol{x}-\boldsymbol{y}\vert\vert^2\right)} \),
      4. +
    4. Tanh: \( K(\boldsymbol{x},\boldsymbol{y})=\tanh{(\boldsymbol{x}^T\boldsymbol{y}+\gamma)} \),
      5. +
    + +and many other ones. + +

    +An important theorem for us is Mercer's theorem. +The theorem states that if a kernel function \( K \) is symmetric, continuous and leads to a positive semi-definite matrix \( \boldsymbol{P} \) then +there exists a function \( \phi \) that maps \( \boldsymbol{x}_i \) and \( \boldsymbol{x}_j \) into another space +(possibly with much higher dimensions) such that $$ -\begin{align*} - &\mathrm{min}_{\lambda}\hspace{0.2cm} \frac{1}{2}\boldsymbol{\lambda}^T\boldsymbol{P}\boldsymbol{\lambda}+\boldsymbol{q}^T\boldsymbol{\lambda},\\ \nonumber - &\mathrm{subject\hspace{0.1cm}to} \hspace{0.2cm} \boldsymbol{G}\boldsymbol{\lambda} \preceq \boldsymbol{h} \wedge \boldsymbol{A}\boldsymbol{\lambda}=f. -\end{align*} +K(\boldsymbol{x}_i,\boldsymbol{x}_j)=\phi(\boldsymbol{x}_i)^T\phi(\boldsymbol{x}_j). $$ -subject to some constraints for say a selected set \( i=1,2,\dots, n \). -In our case we are optimizing with respect to the Lagrangian multipliers \( \lambda_i \), and the -vector \( \boldsymbol{\lambda}=[\lambda_1, \lambda_2,\dots, \lambda_n] \) is the optimization variable we are dealing with. - -

    -In our case we are particularly interested in a class of optimization problems called convex optmization problems. -In our discussion on gradient descent methods we discussed at length the definition of a convex function. - -

    -Convex optimization problems play a central role in applied mathematics and we recommend strongly Boyd and Vandenberghe's text on the topics. +So you can use \( K \) as a kernel since you know \( \phi \) exists, even if +you don’t know what \( \phi \) is. +Note that some frequently used kernels (such as the Sigmoid kernel) don’t respect all of Mercer’s conditions, yet they generally work +well in practice.

    @@ -193,6 +203,8 @@ Convex optimization problems play a central role in applied mathematics and we r

  • 25
  • 26
  • 27
    • +
  • 28
    • +
  • 29
  • »
    • diff --git a/doc/pub/svm/html/._svm-bs023.html b/doc/pub/svm/html/._svm-bs023.html index c03305e33..7523745b2 100644 --- a/doc/pub/svm/html/._svm-bs023.html +++ b/doc/pub/svm/html/._svm-bs023.html @@ -49,30 +49,32 @@ Automatically generated HTML file from DocOnce source ('Getting into the details', 2, None, '___sec5'), ('First attempt at a minimization approach', 2, None, '___sec6'), ('Solving the equations', 2, None, '___sec7'), - ('A better approach', 2, None, '___sec8'), + ('Code Example', 2, None, '___sec8'), + ('Problems with the Simpler Approach', 2, None, '___sec9'), + ('A better approach', 2, None, '___sec10'), ('A quick reminder on Lagrangian multipliers', 2, None, - '___sec9'), - ('Adding the muliplier', 2, None, '___sec10'), - ('Setting up the problem', 2, None, '___sec11'), - ('The problem to solve', 2, None, '___sec12'), - ('The last steps', 2, None, '___sec13'), - ('A soft classifier', 2, None, '___sec14'), - ('Soft optmization problem', 2, None, '___sec15'), - ('Kernels and non-linearity', 2, None, '___sec16'), - ('The equations', 2, None, '___sec17'), - ('The problem to solve', 2, None, '___sec18'), - ("Different kernels and Mercer's theorem", 2, None, '___sec19'), - ('The moons example', 2, None, '___sec20'), + '___sec11'), + ('Adding the muliplier', 2, None, '___sec12'), + ('Setting up the problem', 2, None, '___sec13'), + ('The problem to solve', 2, None, '___sec14'), + ('The last steps', 2, None, '___sec15'), + ('A soft classifier', 2, None, '___sec16'), + ('Soft optmization problem', 2, None, '___sec17'), + ('Kernels and non-linearity', 2, None, '___sec18'), + ('The equations', 2, None, '___sec19'), + ('The problem to solve', 2, None, '___sec20'), + ("Different kernels and Mercer's theorem", 2, None, '___sec21'), + ('The moons example', 2, None, '___sec22'), ('Mathematical optimization of convex functions', 2, None, - '___sec21'), - ('How do we solve these problems?', 2, None, '___sec22'), - ('A simple example', 2, None, '___sec23'), - ('Back to the more realistic cases', 2, None, '___sec24'), - ('Code Example', 2, None, '___sec25')]} + '___sec23'), + ('How do we solve these problems?', 2, None, '___sec24'), + ('A simple example', 2, None, '___sec25'), + ('Back to the more realistic cases', 2, None, '___sec26'), + ('Code Example', 2, None, '___sec27')]} end of tocinfo --> @@ -118,24 +120,26 @@ MathJax.Hub.Config({
  • Getting into the details
  • First attempt at a minimization approach
  • Solving the equations
    • -
  • A better approach
    • -
  • A quick reminder on Lagrangian multipliers
    • -
  • Adding the muliplier
    • -
  • Setting up the problem
    • -
  • The problem to solve
    • -
  • The last steps
    • -
  • A soft classifier
    • -
  • Soft optmization problem
    • -
  • Kernels and non-linearity
    • -
  • The equations
    • -
  • The problem to solve
    • -
  • Different kernels and Mercer's theorem
    • -
  • The moons example
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simple example
    • -
  • Back to the more realistic cases
    • -
  • Code Example
    • +
  • Code Example
    • +
  • Problems with the Simpler Approach
    • +
  • A better approach
    • +
  • A quick reminder on Lagrangian multipliers
    • +
  • Adding the muliplier
    • +
  • Setting up the problem
    • +
  • The problem to solve
    • +
  • The last steps
    • +
  • A soft classifier
    • +
  • Soft optmization problem
    • +
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • The problem to solve
    • +
  • Different kernels and Mercer's theorem
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simple example
    • +
  • Back to the more realistic cases
    • +
  • Code Example
    • @@ -151,29 +155,199 @@ MathJax.Hub.Config({ -

    How do we solve these problems?

    - -

    -If we use Python as programming language and wish to venture beyond -scikit-learn, tensorflow and similar software which makes our -lives so much easier, we need to dive into the wonderful world of -quadratic programming. We can, if we wish, solve the minimization -problem using say standard gradient methods or conjugate gradient -methods. However, these methods tend to exhibit a rather slow -converge. So, welcome to the promised land of quadratic programming. - -

    -The functions we need are contained in the quadratic programming package CVXOPT and we need to import it together with numpy as - +

    The moons example

    -

    import numpy
-import cvxopt
-
    -

    -This will make our life much easier. You don't need t write your own optimizer. +

    from __future__ import division, print_function, unicode_literals
 
+import numpy as np
+np.random.seed(42)
+
+import matplotlib
+import matplotlib.pyplot as plt
+plt.rcParams['axes.labelsize'] = 14
+plt.rcParams['xtick.labelsize'] = 12
+plt.rcParams['ytick.labelsize'] = 12
+
+
+from sklearn.svm import SVC
+from sklearn import datasets
+
+
+
+from sklearn.pipeline import Pipeline
+from sklearn.preprocessing import StandardScaler
+from sklearn.svm import LinearSVC
+
+
+from sklearn.datasets import make_moons
+X, y = make_moons(n_samples=100, noise=0.15, random_state=42)
+
+def plot_dataset(X, y, axes):
+    plt.plot(X[:, 0][y==0], X[:, 1][y==0], "bs")
+    plt.plot(X[:, 0][y==1], X[:, 1][y==1], "g^")
+    plt.axis(axes)
+    plt.grid(True, which='both')
+    plt.xlabel(r"$x_1$", fontsize=20)
+    plt.ylabel(r"$x_2$", fontsize=20, rotation=0)
+
+plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])
+plt.show()
+
+from sklearn.datasets import make_moons
+from sklearn.pipeline import Pipeline
+from sklearn.preprocessing import PolynomialFeatures
+
+polynomial_svm_clf = Pipeline([
+        ("poly_features", PolynomialFeatures(degree=3)),
+        ("scaler", StandardScaler()),
+        ("svm_clf", LinearSVC(C=10, loss="hinge", random_state=42))
+    ])
+
+polynomial_svm_clf.fit(X, y)
+
+def plot_predictions(clf, axes):
+    x0s = np.linspace(axes[0], axes[1], 100)
+    x1s = np.linspace(axes[2], axes[3], 100)
+    x0, x1 = np.meshgrid(x0s, x1s)
+    X = np.c_[x0.ravel(), x1.ravel()]
+    y_pred = clf.predict(X).reshape(x0.shape)
+    y_decision = clf.decision_function(X).reshape(x0.shape)
+    plt.contourf(x0, x1, y_pred, cmap=plt.cm.brg, alpha=0.2)
+    plt.contourf(x0, x1, y_decision, cmap=plt.cm.brg, alpha=0.1)
+
+plot_predictions(polynomial_svm_clf, [-1.5, 2.5, -1, 1.5])
+plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])
+
+plt.show()
+
+
+from sklearn.svm import SVC
+
+poly_kernel_svm_clf = Pipeline([
+        ("scaler", StandardScaler()),
+        ("svm_clf", SVC(kernel="poly", degree=3, coef0=1, C=5))
+    ])
+poly_kernel_svm_clf.fit(X, y)
+
+poly100_kernel_svm_clf = Pipeline([
+        ("scaler", StandardScaler()),
+        ("svm_clf", SVC(kernel="poly", degree=10, coef0=100, C=5))
+    ])
+poly100_kernel_svm_clf.fit(X, y)
+
+plt.figure(figsize=(11, 4))
+
+plt.subplot(121)
+plot_predictions(poly_kernel_svm_clf, [-1.5, 2.5, -1, 1.5])
+plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])
+plt.title(r"$d=3, r=1, C=5$", fontsize=18)
+
+plt.subplot(122)
+plot_predictions(poly100_kernel_svm_clf, [-1.5, 2.5, -1, 1.5])
+plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])
+plt.title(r"$d=10, r=100, C=5$", fontsize=18)
+
+plt.show()
+
+def gaussian_rbf(x, landmark, gamma):
+    return np.exp(-gamma * np.linalg.norm(x - landmark, axis=1)**2)
+
+gamma = 0.3
+
+x1s = np.linspace(-4.5, 4.5, 200).reshape(-1, 1)
+x2s = gaussian_rbf(x1s, -2, gamma)
+x3s = gaussian_rbf(x1s, 1, gamma)
+
+XK = np.c_[gaussian_rbf(X1D, -2, gamma), gaussian_rbf(X1D, 1, gamma)]
+yk = np.array([0, 0, 1, 1, 1, 1, 1, 0, 0])
+
+plt.figure(figsize=(11, 4))
+
+plt.subplot(121)
+plt.grid(True, which='both')
+plt.axhline(y=0, color='k')
+plt.scatter(x=[-2, 1], y=[0, 0], s=150, alpha=0.5, c="red")
+plt.plot(X1D[:, 0][yk==0], np.zeros(4), "bs")
+plt.plot(X1D[:, 0][yk==1], np.zeros(5), "g^")
+plt.plot(x1s, x2s, "g--")
+plt.plot(x1s, x3s, "b:")
+plt.gca().get_yaxis().set_ticks([0, 0.25, 0.5, 0.75, 1])
+plt.xlabel(r"$x_1$", fontsize=20)
+plt.ylabel(r"Similarity", fontsize=14)
+plt.annotate(r'$\mathbf{x}$',
+             xy=(X1D[3, 0], 0),
+             xytext=(-0.5, 0.20),
+             ha="center",
+             arrowprops=dict(facecolor='black', shrink=0.1),
+             fontsize=18,
+            )
+plt.text(-2, 0.9, "$x_2$", ha="center", fontsize=20)
+plt.text(1, 0.9, "$x_3$", ha="center", fontsize=20)
+plt.axis([-4.5, 4.5, -0.1, 1.1])
+
+plt.subplot(122)
+plt.grid(True, which='both')
+plt.axhline(y=0, color='k')
+plt.axvline(x=0, color='k')
+plt.plot(XK[:, 0][yk==0], XK[:, 1][yk==0], "bs")
+plt.plot(XK[:, 0][yk==1], XK[:, 1][yk==1], "g^")
+plt.xlabel(r"$x_2$", fontsize=20)
+plt.ylabel(r"$x_3$  ", fontsize=20, rotation=0)
+plt.annotate(r'$\phi\left(\mathbf{x}\right)$',
+             xy=(XK[3, 0], XK[3, 1]),
+             xytext=(0.65, 0.50),
+             ha="center",
+             arrowprops=dict(facecolor='black', shrink=0.1),
+             fontsize=18,
+            )
+plt.plot([-0.1, 1.1], [0.57, -0.1], "r--", linewidth=3)
+plt.axis([-0.1, 1.1, -0.1, 1.1])
+    
+plt.subplots_adjust(right=1)
+
+plt.show()
+
+
+x1_example = X1D[3, 0]
+for landmark in (-2, 1):
+    k = gaussian_rbf(np.array([[x1_example]]), np.array([[landmark]]), gamma)
+    print("Phi({}, {}) = {}".format(x1_example, landmark, k))
+
+rbf_kernel_svm_clf = Pipeline([
+        ("scaler", StandardScaler()),
+        ("svm_clf", SVC(kernel="rbf", gamma=5, C=0.001))
+    ])
+rbf_kernel_svm_clf.fit(X, y)
+
+
+from sklearn.svm import SVC
+
+gamma1, gamma2 = 0.1, 5
+C1, C2 = 0.001, 1000
+hyperparams = (gamma1, C1), (gamma1, C2), (gamma2, C1), (gamma2, C2)
+
+svm_clfs = []
+for gamma, C in hyperparams:
+    rbf_kernel_svm_clf = Pipeline([
+            ("scaler", StandardScaler()),
+            ("svm_clf", SVC(kernel="rbf", gamma=gamma, C=C))
+        ])
+    rbf_kernel_svm_clf.fit(X, y)
+    svm_clfs.append(rbf_kernel_svm_clf)
+
+plt.figure(figsize=(11, 7))
+
+for i, svm_clf in enumerate(svm_clfs):
+    plt.subplot(221 + i)
+    plot_predictions(svm_clf, [-1.5, 2.5, -1, 1.5])
+    plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])
+    gamma, C = hyperparams[i]
+    plt.title(r"$\gamma = {}, C = {}$".format(gamma, C), fontsize=16)
+
+plt.show()
+

    @@ -193,6 +367,8 @@ This will make our life much easier. You don't need t write your own optimizer.

  • 25
  • 26
  • 27
    • +
  • 28
    • +
  • 29
  • »
    • diff --git a/doc/pub/svm/html/._svm-bs024.html b/doc/pub/svm/html/._svm-bs024.html index d0ae1ed08..276edbbfc 100644 --- a/doc/pub/svm/html/._svm-bs024.html +++ b/doc/pub/svm/html/._svm-bs024.html @@ -49,30 +49,32 @@ Automatically generated HTML file from DocOnce source ('Getting into the details', 2, None, '___sec5'), ('First attempt at a minimization approach', 2, None, '___sec6'), ('Solving the equations', 2, None, '___sec7'), - ('A better approach', 2, None, '___sec8'), + ('Code Example', 2, None, '___sec8'), + ('Problems with the Simpler Approach', 2, None, '___sec9'), + ('A better approach', 2, None, '___sec10'), ('A quick reminder on Lagrangian multipliers', 2, None, - '___sec9'), - ('Adding the muliplier', 2, None, '___sec10'), - ('Setting up the problem', 2, None, '___sec11'), - ('The problem to solve', 2, None, '___sec12'), - ('The last steps', 2, None, '___sec13'), - ('A soft classifier', 2, None, '___sec14'), - ('Soft optmization problem', 2, None, '___sec15'), - ('Kernels and non-linearity', 2, None, '___sec16'), - ('The equations', 2, None, '___sec17'), - ('The problem to solve', 2, None, '___sec18'), - ("Different kernels and Mercer's theorem", 2, None, '___sec19'), - ('The moons example', 2, None, '___sec20'), + '___sec11'), + ('Adding the muliplier', 2, None, '___sec12'), + ('Setting up the problem', 2, None, '___sec13'), + ('The problem to solve', 2, None, '___sec14'), + ('The last steps', 2, None, '___sec15'), + ('A soft classifier', 2, None, '___sec16'), + ('Soft optmization problem', 2, None, '___sec17'), + ('Kernels and non-linearity', 2, None, '___sec18'), + ('The equations', 2, None, '___sec19'), + ('The problem to solve', 2, None, '___sec20'), + ("Different kernels and Mercer's theorem", 2, None, '___sec21'), + ('The moons example', 2, None, '___sec22'), ('Mathematical optimization of convex functions', 2, None, - '___sec21'), - ('How do we solve these problems?', 2, None, '___sec22'), - ('A simple example', 2, None, '___sec23'), - ('Back to the more realistic cases', 2, None, '___sec24'), - ('Code Example', 2, None, '___sec25')]} + '___sec23'), + ('How do we solve these problems?', 2, None, '___sec24'), + ('A simple example', 2, None, '___sec25'), + ('Back to the more realistic cases', 2, None, '___sec26'), + ('Code Example', 2, None, '___sec27')]} end of tocinfo --> @@ -118,24 +120,26 @@ MathJax.Hub.Config({
  • Getting into the details
  • First attempt at a minimization approach
  • Solving the equations
    • -
  • A better approach
    • -
  • A quick reminder on Lagrangian multipliers
    • -
  • Adding the muliplier
    • -
  • Setting up the problem
    • -
  • The problem to solve
    • -
  • The last steps
    • -
  • A soft classifier
    • -
  • Soft optmization problem
    • -
  • Kernels and non-linearity
    • -
  • The equations
    • -
  • The problem to solve
    • -
  • Different kernels and Mercer's theorem
    • -
  • The moons example
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simple example
    • -
  • Back to the more realistic cases
    • -
  • Code Example
    • +
  • Code Example
    • +
  • Problems with the Simpler Approach
    • +
  • A better approach
    • +
  • A quick reminder on Lagrangian multipliers
    • +
  • Adding the muliplier
    • +
  • Setting up the problem
    • +
  • The problem to solve
    • +
  • The last steps
    • +
  • A soft classifier
    • +
  • Soft optmization problem
    • +
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • The problem to solve
    • +
  • Different kernels and Mercer's theorem
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simple example
    • +
  • Back to the more realistic cases
    • +
  • Code Example
    • @@ -151,71 +155,28 @@ MathJax.Hub.Config({ -

    A simple example

    +

    Mathematical optimization of convex functions

    -We remind ourselves about the general problem we want to solve +A mathematical (quadratic) optimization problem, or just optimization problem, has the form $$ \begin{align*} - &\mathrm{min}_{x}\hspace{0.2cm} \frac{1}{2}\boldsymbol{x}^T\boldsymbol{P}\boldsymbol{x}+\boldsymbol{q}^T\boldsymbol{x},\\ \nonumber - &\mathrm{subject\hspace{0.1cm} to} \hspace{0.2cm} \boldsymbol{G}\boldsymbol{x} \preceq \boldsymbol{h} \wedge \boldsymbol{A}\boldsymbol{x}=f. + &\mathrm{min}_{\lambda}\hspace{0.2cm} \frac{1}{2}\boldsymbol{\lambda}^T\boldsymbol{P}\boldsymbol{\lambda}+\boldsymbol{q}^T\boldsymbol{\lambda},\\ \nonumber + &\mathrm{subject\hspace{0.1cm}to} \hspace{0.2cm} \boldsymbol{G}\boldsymbol{\lambda} \preceq \boldsymbol{h} \wedge \boldsymbol{A}\boldsymbol{\lambda}=f. \end{align*} $$ -

    -Let us show how to perform the optmization using a simple case. Assume we want to optimize the following problem -$$ -\begin{align*} - &\mathrm{min}_{x}\hspace{0.2cm} \frac{1}{2}x^2+5x+3y \\ \nonumber - &\mathrm{subject to} \\ \nonumber - &x, y \geq 0 \\ \nonumber - &x+3y \geq 15 \\ \nonumber - &2x+5y \leq 100 \\ \nonumber - &3x+4y \leq 80. \\ \nonumber -\end{align*} -$$ - -The minimization problem can be rewritten in terms of vectors and matrices as (with \( x \) and \( y \) being the unknowns) -$$ -\frac{1}{2}\begin{bmatrix} x\\ y \end{bmatrix}^T \begin{bmatrix} 1 & 0\\ 0 & 0 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} + \begin{bmatrix}3\\ 4 \end{bmatrix}^T \begin{bmatrix}x \\ y \end{bmatrix}. -$$ - -Similarly, we can now set up the inequalities (we need to change \( \geq \) to \( \leq \) by multiplying with \( -1 \) on bot sides) as the following matrix-vector equation -$$ -\begin{bmatrix} -1 & 0 \\ 0 & -1 \\ -1 & -3 \\ 2 & 5 \\ 3 & 4\end{bmatrix}\begin{bmatrix} x \\ y\end{bmatrix} \preceq \begin{bmatrix}0 \\ 0\\ -15 \\ 100 \\ 80\end{bmatrix}. -$$ - -We have collapsed all the inequalities into a single matrix \( \boldsymbol{G} \). We see also that our matrix -$$ -\boldsymbol{P} =\begin{bmatrix} 1 & 0\\ 0 & 0 \end{bmatrix} -$$ - -is clearly positive semi-definite (all eigenvalues larger or equal zero). -Finally, the vector \( \boldsymbol{h} \) is defined as -$$ -\boldsymbol{h} = \begin{bmatrix}0 \\ 0\\ -15 \\ 100 \\ 80\end{bmatrix}. -$$ +subject to some constraints for say a selected set \( i=1,2,\dots, n \). +In our case we are optimizing with respect to the Lagrangian multipliers \( \lambda_i \), and the +vector \( \boldsymbol{\lambda}=[\lambda_1, \lambda_2,\dots, \lambda_n] \) is the optimization variable we are dealing with.

    -Since we don't have any equalities the matrix \( \boldsymbol{A} \) is set to zero -The following code solves the equations for us -

    +In our case we are particularly interested in a class of optimization problems called convex optmization problems. +In our discussion on gradient descent methods we discussed at length the definition of a convex function. + +

    +Convex optimization problems play a central role in applied mathematics and we recommend strongly Boyd and Vandenberghe's text on the topics. - -

    # Import the necessary packages
-import numpy
-from cvxopt import matrix
-from cvxopt import solvers
-P = matrix(numpy.diag([1,0]), tc=’d’)
-q = matrix(numpy.array([3,4]), tc=’d’)
-G = matrix(numpy.array([[-1,0],[0,-1],[-1,-3],[2,5],[3,4]]), tc=’d’)
-h = matrix(numpy.array([0,0,-15,100,80]), tc=’d’)
-# Construct the QP, invoke solver
-sol = solvers.qp(P,q,G,h)
-# Extract optimal value and solution
-sol[’x’] 
-sol[’primal objective’]
-

    @@ -234,6 +195,8 @@ sol[’primal objective’]

  • 25
  • 26
  • 27
    • +
  • 28
    • +
  • 29
  • »
    • diff --git a/doc/pub/svm/html/._svm-bs025.html b/doc/pub/svm/html/._svm-bs025.html index 432960660..e857d117b 100644 --- a/doc/pub/svm/html/._svm-bs025.html +++ b/doc/pub/svm/html/._svm-bs025.html @@ -49,30 +49,32 @@ Automatically generated HTML file from DocOnce source ('Getting into the details', 2, None, '___sec5'), ('First attempt at a minimization approach', 2, None, '___sec6'), ('Solving the equations', 2, None, '___sec7'), - ('A better approach', 2, None, '___sec8'), + ('Code Example', 2, None, '___sec8'), + ('Problems with the Simpler Approach', 2, None, '___sec9'), + ('A better approach', 2, None, '___sec10'), ('A quick reminder on Lagrangian multipliers', 2, None, - '___sec9'), - ('Adding the muliplier', 2, None, '___sec10'), - ('Setting up the problem', 2, None, '___sec11'), - ('The problem to solve', 2, None, '___sec12'), - ('The last steps', 2, None, '___sec13'), - ('A soft classifier', 2, None, '___sec14'), - ('Soft optmization problem', 2, None, '___sec15'), - ('Kernels and non-linearity', 2, None, '___sec16'), - ('The equations', 2, None, '___sec17'), - ('The problem to solve', 2, None, '___sec18'), - ("Different kernels and Mercer's theorem", 2, None, '___sec19'), - ('The moons example', 2, None, '___sec20'), + '___sec11'), + ('Adding the muliplier', 2, None, '___sec12'), + ('Setting up the problem', 2, None, '___sec13'), + ('The problem to solve', 2, None, '___sec14'), + ('The last steps', 2, None, '___sec15'), + ('A soft classifier', 2, None, '___sec16'), + ('Soft optmization problem', 2, None, '___sec17'), + ('Kernels and non-linearity', 2, None, '___sec18'), + ('The equations', 2, None, '___sec19'), + ('The problem to solve', 2, None, '___sec20'), + ("Different kernels and Mercer's theorem", 2, None, '___sec21'), + ('The moons example', 2, None, '___sec22'), ('Mathematical optimization of convex functions', 2, None, - '___sec21'), - ('How do we solve these problems?', 2, None, '___sec22'), - ('A simple example', 2, None, '___sec23'), - ('Back to the more realistic cases', 2, None, '___sec24'), - ('Code Example', 2, None, '___sec25')]} + '___sec23'), + ('How do we solve these problems?', 2, None, '___sec24'), + ('A simple example', 2, None, '___sec25'), + ('Back to the more realistic cases', 2, None, '___sec26'), + ('Code Example', 2, None, '___sec27')]} end of tocinfo --> @@ -118,24 +120,26 @@ MathJax.Hub.Config({
  • Getting into the details
  • First attempt at a minimization approach
  • Solving the equations
    • -
  • A better approach
    • -
  • A quick reminder on Lagrangian multipliers
    • -
  • Adding the muliplier
    • -
  • Setting up the problem
    • -
  • The problem to solve
    • -
  • The last steps
    • -
  • A soft classifier
    • -
  • Soft optmization problem
    • -
  • Kernels and non-linearity
    • -
  • The equations
    • -
  • The problem to solve
    • -
  • Different kernels and Mercer's theorem
    • -
  • The moons example
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simple example
    • -
  • Back to the more realistic cases
    • -
  • Code Example
    • +
  • Code Example
    • +
  • Problems with the Simpler Approach
    • +
  • A better approach
    • +
  • A quick reminder on Lagrangian multipliers
    • +
  • Adding the muliplier
    • +
  • Setting up the problem
    • +
  • The problem to solve
    • +
  • The last steps
    • +
  • A soft classifier
    • +
  • Soft optmization problem
    • +
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • The problem to solve
    • +
  • Different kernels and Mercer's theorem
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simple example
    • +
  • Back to the more realistic cases
    • +
  • Code Example
    • @@ -151,25 +155,28 @@ MathJax.Hub.Config({ -

    Back to the more realistic cases

    +

    How do we solve these problems?

    -We are now ready to return to our setup of the optmization problem for a more realistic case. Introducing the slack parameter \( C \) we have -$$ -\frac{1}{2} \boldsymbol{\lambda}^T\begin{bmatrix} y_1y_1K(\boldsymbol{x}_1,\boldsymbol{x}_1) & y_1y_2K(\boldsymbol{x}_1,\boldsymbol{x}_2) & \dots & \dots & y_1y_nK(\boldsymbol{x}_1,\boldsymbol{x}_n) \\ -y_2y_1K(\boldsymbol{x}_2,\boldsymbol{x}_1) & y_2y_2K(\boldsymbol{x}_2,\boldsymbol{x}_2) & \dots & \dots & y_1y_nK(\boldsymbol{x}_2,\boldsymbol{x}_n) \\ -\dots & \dots & \dots & \dots & \dots \\ -\dots & \dots & \dots & \dots & \dots \\ -y_ny_1K(\boldsymbol{x}_n,\boldsymbol{x}_1) & y_ny_2K(\boldsymbol{x}_n\boldsymbol{x}_2) & \dots & \dots & y_ny_nK(\boldsymbol{x}_n,\boldsymbol{x}_n) \\ -\end{bmatrix}\boldsymbol{\lambda}-\mathbb{I}\boldsymbol{\lambda}, -$$ - -subject to \( \boldsymbol{y}^T\boldsymbol{\lambda}=0 \). Here we defined the vectors \( \boldsymbol{\lambda} =[\lambda_1,\lambda_2,\dots,\lambda_n] \) and -\( \boldsymbol{y}=[y_1,y_2,\dots,y_n] \). -With the slack constants this leads to the additional constraint \( 0\leq \lambda_i \leq C \). +If we use Python as programming language and wish to venture beyond +scikit-learn, tensorflow and similar software which makes our +lives so much easier, we need to dive into the wonderful world of +quadratic programming. We can, if we wish, solve the minimization +problem using say standard gradient methods or conjugate gradient +methods. However, these methods tend to exhibit a rather slow +converge. So, welcome to the promised land of quadratic programming.

    -code will be added +The functions we need are contained in the quadratic programming package CVXOPT and we need to import it together with numpy as + +

    + + +

    import numpy
+import cvxopt
+
    +

    +This will make our life much easier. You don't need t write your own optimizer.

    @@ -188,6 +195,8 @@ With the slack constants this leads to the additional constraint \( 0\leq \lamb

  • 25
  • 26
  • 27
    • +
  • 28
    • +
  • 29
  • »
    • diff --git a/doc/pub/svm/html/._svm-bs026.html b/doc/pub/svm/html/._svm-bs026.html index 18b76c041..09c865519 100644 --- a/doc/pub/svm/html/._svm-bs026.html +++ b/doc/pub/svm/html/._svm-bs026.html @@ -49,30 +49,32 @@ Automatically generated HTML file from DocOnce source ('Getting into the details', 2, None, '___sec5'), ('First attempt at a minimization approach', 2, None, '___sec6'), ('Solving the equations', 2, None, '___sec7'), - ('A better approach', 2, None, '___sec8'), + ('Code Example', 2, None, '___sec8'), + ('Problems with the Simpler Approach', 2, None, '___sec9'), + ('A better approach', 2, None, '___sec10'), ('A quick reminder on Lagrangian multipliers', 2, None, - '___sec9'), - ('Adding the muliplier', 2, None, '___sec10'), - ('Setting up the problem', 2, None, '___sec11'), - ('The problem to solve', 2, None, '___sec12'), - ('The last steps', 2, None, '___sec13'), - ('A soft classifier', 2, None, '___sec14'), - ('Soft optmization problem', 2, None, '___sec15'), - ('Kernels and non-linearity', 2, None, '___sec16'), - ('The equations', 2, None, '___sec17'), - ('The problem to solve', 2, None, '___sec18'), - ("Different kernels and Mercer's theorem", 2, None, '___sec19'), - ('The moons example', 2, None, '___sec20'), + '___sec11'), + ('Adding the muliplier', 2, None, '___sec12'), + ('Setting up the problem', 2, None, '___sec13'), + ('The problem to solve', 2, None, '___sec14'), + ('The last steps', 2, None, '___sec15'), + ('A soft classifier', 2, None, '___sec16'), + ('Soft optmization problem', 2, None, '___sec17'), + ('Kernels and non-linearity', 2, None, '___sec18'), + ('The equations', 2, None, '___sec19'), + ('The problem to solve', 2, None, '___sec20'), + ("Different kernels and Mercer's theorem", 2, None, '___sec21'), + ('The moons example', 2, None, '___sec22'), ('Mathematical optimization of convex functions', 2, None, - '___sec21'), - ('How do we solve these problems?', 2, None, '___sec22'), - ('A simple example', 2, None, '___sec23'), - ('Back to the more realistic cases', 2, None, '___sec24'), - ('Code Example', 2, None, '___sec25')]} + '___sec23'), + ('How do we solve these problems?', 2, None, '___sec24'), + ('A simple example', 2, None, '___sec25'), + ('Back to the more realistic cases', 2, None, '___sec26'), + ('Code Example', 2, None, '___sec27')]} end of tocinfo --> @@ -118,24 +120,26 @@ MathJax.Hub.Config({
  • Getting into the details
  • First attempt at a minimization approach
  • Solving the equations
    • -
  • A better approach
    • -
  • A quick reminder on Lagrangian multipliers
    • -
  • Adding the muliplier
    • -
  • Setting up the problem
    • -
  • The problem to solve
    • -
  • The last steps
    • -
  • A soft classifier
    • -
  • Soft optmization problem
    • -
  • Kernels and non-linearity
    • -
  • The equations
    • -
  • The problem to solve
    • -
  • Different kernels and Mercer's theorem
    • -
  • The moons example
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simple example
    • -
  • Back to the more realistic cases
    • -
  • Code Example
    • +
  • Code Example
    • +
  • Problems with the Simpler Approach
    • +
  • A better approach
    • +
  • A quick reminder on Lagrangian multipliers
    • +
  • Adding the muliplier
    • +
  • Setting up the problem
    • +
  • The problem to solve
    • +
  • The last steps
    • +
  • A soft classifier
    • +
  • Soft optmization problem
    • +
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • The problem to solve
    • +
  • Different kernels and Mercer's theorem
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simple example
    • +
  • Back to the more realistic cases
    • +
  • Code Example
    • @@ -151,37 +155,72 @@ MathJax.Hub.Config({ -

    Code Example

    +

    A simple example

    + +

    +We remind ourselves about the general problem we want to solve +$$ +\begin{align*} + &\mathrm{min}_{x}\hspace{0.2cm} \frac{1}{2}\boldsymbol{x}^T\boldsymbol{P}\boldsymbol{x}+\boldsymbol{q}^T\boldsymbol{x},\\ \nonumber + &\mathrm{subject\hspace{0.1cm} to} \hspace{0.2cm} \boldsymbol{G}\boldsymbol{x} \preceq \boldsymbol{h} \wedge \boldsymbol{A}\boldsymbol{x}=f. +\end{align*} +$$ + +

    +Let us show how to perform the optmization using a simple case. Assume we want to optimize the following problem +$$ +\begin{align*} + &\mathrm{min}_{x}\hspace{0.2cm} \frac{1}{2}x^2+5x+3y \\ \nonumber + &\mathrm{subject to} \\ \nonumber + &x, y \geq 0 \\ \nonumber + &x+3y \geq 15 \\ \nonumber + &2x+5y \leq 100 \\ \nonumber + &3x+4y \leq 80. \\ \nonumber +\end{align*} +$$ + +The minimization problem can be rewritten in terms of vectors and matrices as (with \( x \) and \( y \) being the unknowns) +$$ +\frac{1}{2}\begin{bmatrix} x\\ y \end{bmatrix}^T \begin{bmatrix} 1 & 0\\ 0 & 0 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} + \begin{bmatrix}3\\ 4 \end{bmatrix}^T \begin{bmatrix}x \\ y \end{bmatrix}. +$$ + +Similarly, we can now set up the inequalities (we need to change \( \geq \) to \( \leq \) by multiplying with \( -1 \) on bot sides) as the following matrix-vector equation +$$ +\begin{bmatrix} -1 & 0 \\ 0 & -1 \\ -1 & -3 \\ 2 & 5 \\ 3 & 4\end{bmatrix}\begin{bmatrix} x \\ y\end{bmatrix} \preceq \begin{bmatrix}0 \\ 0\\ -15 \\ 100 \\ 80\end{bmatrix}. +$$ + +We have collapsed all the inequalities into a single matrix \( \boldsymbol{G} \). We see also that our matrix +$$ +\boldsymbol{P} =\begin{bmatrix} 1 & 0\\ 0 & 0 \end{bmatrix} +$$ + +is clearly positive semi-definite (all eigenvalues larger or equal zero). +Finally, the vector \( \boldsymbol{h} \) is defined as +$$ +\boldsymbol{h} = \begin{bmatrix}0 \\ 0\\ -15 \\ 100 \\ 80\end{bmatrix}. +$$ + +

    +Since we don't have any equalities the matrix \( \boldsymbol{A} \) is set to zero +The following code solves the equations for us

    -

    import numpy as np
-from scipy.optimize import minimize
-def rosen(x):
-    return sum(100.0*(x[1:]-x[:-1]**2.0)**2.0 + (1-x[:-1])**2.0)
-def rosen_der(x):
-    xm = x[1:-1]
-    xm_m1 = x[:-2]
-    xm_p1 = x[2:]
-    der = np.zeros_like(x)
-    der[1:-1] = 200*(xm-xm_m1**2) - 400*(xm_p1 - xm**2)*xm - 2*(1-xm)
-    der[0] = -400*x[0]*(x[1]-x[0]**2) - 2*(1-x[0])
-    der[-1] = 200*(x[-1]-x[-2]**2)
-    return der
-
-
-
-x0 = np.array([1.2, 0.1, 0.8, 1.9, 1.2])
-print(x0)
-res = minimize(rosen, x0, method='nelder-mead',options={'xtol': 1e-8, 'disp': True})
-print(x0)
-
-
-res2 = minimize(rosen, x0, method='BFGS', jac=rosen_der, options={'disp': True})
-print(x0)
+
    # Import the necessary packages
+import numpy
+from cvxopt import matrix
+from cvxopt import solvers
+P = matrix(numpy.diag([1,0]), tc=’d’)
+q = matrix(numpy.array([3,4]), tc=’d’)
+G = matrix(numpy.array([[-1,0],[0,-1],[-1,-3],[2,5],[3,4]]), tc=’d’)
+h = matrix(numpy.array([0,0,-15,100,80]), tc=’d’)
+# Construct the QP, invoke solver
+sol = solvers.qp(P,q,G,h)
+# Extract optimal value and solution
+sol[’x’] 
+sol[’primal objective’]
 
    
 
    
-
 
    
 
 
      
@@ -197,6 +236,9 @@ res2 = minimize(rosen, x0, method25
   
    • 26
      • 
   
    • 27
      • 
+  
    • 28
      • 
+  
    • 29
      • 
+  
    • »
      • 
 
    
 
 
diff --git a/doc/pub/svm/html/svm-bs.html b/doc/pub/svm/html/svm-bs.html
index 6fb243975..811e86f4d 100644
--- a/doc/pub/svm/html/svm-bs.html
+++ b/doc/pub/svm/html/svm-bs.html
@@ -49,30 +49,32 @@ Automatically generated HTML file from DocOnce source
               ('Getting into the details', 2, None, '___sec5'),
               ('First attempt at a minimization approach', 2, None, '___sec6'),
               ('Solving the equations', 2, None, '___sec7'),
-              ('A better approach', 2, None, '___sec8'),
+              ('Code Example', 2, None, '___sec8'),
+              ('Problems with the Simpler Approach', 2, None, '___sec9'),
+              ('A better approach', 2, None, '___sec10'),
               ('A quick reminder on Lagrangian multipliers',
                2,
                None,
-               '___sec9'),
-              ('Adding the muliplier', 2, None, '___sec10'),
-              ('Setting up the problem', 2, None, '___sec11'),
-              ('The problem to solve', 2, None, '___sec12'),
-              ('The last steps', 2, None, '___sec13'),
-              ('A soft classifier', 2, None, '___sec14'),
-              ('Soft optmization problem', 2, None, '___sec15'),
-              ('Kernels and non-linearity', 2, None, '___sec16'),
-              ('The equations', 2, None, '___sec17'),
-              ('The problem to solve', 2, None, '___sec18'),
-              ("Different kernels and Mercer's theorem", 2, None, '___sec19'),
-              ('The moons example', 2, None, '___sec20'),
+               '___sec11'),
+              ('Adding the muliplier', 2, None, '___sec12'),
+              ('Setting up the problem', 2, None, '___sec13'),
+              ('The problem to solve', 2, None, '___sec14'),
+              ('The last steps', 2, None, '___sec15'),
+              ('A soft classifier', 2, None, '___sec16'),
+              ('Soft optmization problem', 2, None, '___sec17'),
+              ('Kernels and non-linearity', 2, None, '___sec18'),
+              ('The equations', 2, None, '___sec19'),
+              ('The problem to solve', 2, None, '___sec20'),
+              ("Different kernels and Mercer's theorem", 2, None, '___sec21'),
+              ('The moons example', 2, None, '___sec22'),
               ('Mathematical optimization of convex functions',
                2,
                None,
-               '___sec21'),
-              ('How do we solve these problems?', 2, None, '___sec22'),
-              ('A simple example', 2, None, '___sec23'),
-              ('Back to the more realistic cases', 2, None, '___sec24'),
-              ('Code Example', 2, None, '___sec25')]}
+               '___sec23'),
+              ('How do we solve these problems?', 2, None, '___sec24'),
+              ('A simple example', 2, None, '___sec25'),
+              ('Back to the more realistic cases', 2, None, '___sec26'),
+              ('Code Example', 2, None, '___sec27')]}
 end of tocinfo -->
 
 
@@ -118,24 +120,26 @@ MathJax.Hub.Config({
       
  • Getting into the details
    • 
       
  • First attempt at a minimization approach
    • 
       
  • Solving the equations
    • 
-      
  • A better approach
    • 
-      
  • A quick reminder on Lagrangian multipliers
    • 
-      
  • Adding the muliplier
    • 
-      
  • Setting up the problem
    • 
-      
  • The problem to solve
    • 
-      
  • The last steps
    • 
-      
  • A soft classifier
    • 
-      
  • Soft optmization problem
    • 
-      
  • Kernels and non-linearity
    • 
-      
  • The equations
    • 
-      
  • The problem to solve
    • 
-      
  • Different kernels and Mercer's theorem
    • 
-      
  • The moons example
    • 
-      
  • Mathematical optimization of convex functions
    • 
-      
  • How do we solve these problems?
    • 
-      
  • A simple example
    • 
-      
  • Back to the more realistic cases
    • 
-      
  • Code Example
    • 
+      
  • Code Example
    • 
+      
  • Problems with the Simpler Approach
    • 
+      
  • A better approach
    • 
+      
  • A quick reminder on Lagrangian multipliers
    • 
+      
  • Adding the muliplier
    • 
+      
  • Setting up the problem
    • 
+      
  • The problem to solve
    • 
+      
  • The last steps
    • 
+      
  • A soft classifier
    • 
+      
  • Soft optmization problem
    • 
+      
  • Kernels and non-linearity
    • 
+      
  • The equations
    • 
+      
  • The problem to solve
    • 
+      
  • Different kernels and Mercer's theorem
    • 
+      
  • The moons example
    • 
+      
  • Mathematical optimization of convex functions
    • 
+      
  • How do we solve these problems?
    • 
+      
  • A simple example
    • 
+      
  • Back to the more realistic cases
    • 
+      
  • Code Example
    • 
 
         
       
@@ -170,7 +174,7 @@ MathJax.Hub.Config({
 
    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University
    
 
    
 
    
-
    Nov 15, 2019
     
+
    Nov 16, 2019
     
 
    
 
    
 
@@ -194,7 +198,7 @@ MathJax.Hub.Config({
   
  • 9
    • 
   
  • 10
    • 
   
  • ...
    • 
-  
  • 27
    • 
+  
  • 29
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/svm/html/svm-reveal.html b/doc/pub/svm/html/svm-reveal.html
index 0de24dc3a..941055f1b 100644
--- a/doc/pub/svm/html/svm-reveal.html
+++ b/doc/pub/svm/html/svm-reveal.html
@@ -148,7 +148,7 @@ MathJax.Hub.Config({
 
    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University
    
 
    
 
     
    
-
    Nov 15, 2019
     
+
    Nov 16, 2019
     
 
    
 
    
 
@@ -448,7 +448,7 @@ $$
 
    Solving the equations 
    
 
 
    
-We can now use the Newton-Raphson method or gradient descent to solve the equations
+We can now use the Newton-Raphson method or different variants of the gradient descent family (from plain gradient descent to various stochastic gradient descent approaches) to solve the equations
 
     
    
 $$
 b \leftarrow b +\eta \frac{\partial C}{\partial b},
@@ -463,12 +463,34 @@ $$
 
     
    
 
 where \( \eta \) is our by now well-known learning rate.
+
+
+
+
    
+
    Code Example 
    
+
+
    
+The equations we discussed above can be coded rather easily (the
+framework is rather similar to what we developed for logistic
+regression). We use the standard moons example to set up our data
+set. We want to find a line which separates the two classes
+
    
+
+
+
    
+
    
+
    
+
+
+
    
+
    Problems with the Simpler Approach 
    
 
 
    
 There are however problems with this approach, although it looks
-pretty straightforward to implement. In case we separate our data into
-two distinct classes, we may up with many possible lines, as indicated
-in the figure and shown by running the following program. For small
+pretty straightforward to implement. When running the above code, we see that we can easily end up with many diffeent lines which separate the two classes.
+
+
    
+For small
 gaps between the entries, we may also end up needing many iterations
 before the solutions converge and if the data cannot be separated
 properly into two distinct classes, we may not experience a converge
@@ -477,7 +499,7 @@ at all.
 
 
 
    
-
    A better approach 
    
+
    A better approach 
    
 
 
    
 A better approach is rather to try to define a large margin between
@@ -519,12 +541,15 @@ $$
 
     
    
 
 
    
-We have thus defined our margin as the invers of the norm of \( \boldsymbol{w} \). We want to minimize the norm in order to have a as large as possible margin \( M \). Before we proceed, we need to remind ourselves about Lagrangian multipliers.
+We have thus defined our margin as the invers of the norm of
+\( \boldsymbol{w} \). We want to minimize the norm in order to have a as large as
+possible margin \( M \). Before we proceed, we need to remind ourselves
+about Lagrangian multipliers.
 
    
 
 
 
    
-
    A quick reminder on Lagrangian multipliers 
    
+
    A quick reminder on Lagrangian multipliers 
    
 
 
    
 Consider a function of three independent variables \( f(x,y,z) \) . For the function \( f \) to be an
@@ -586,7 +611,7 @@ Then \( dz \) is no longer arbitrary.
 
 
 
    
-
    Adding the muliplier 
    
+
    Adding the muliplier 
    
 
 
    
 However, we can add to
@@ -641,7 +666,7 @@ $$
 
 
 
    
-
    Setting up the problem 
    
+
    Setting up the problem 
    
 In order to solve the above problem, we define the following Lagrangian function to be minimized 
 
     
    
 $$
@@ -693,7 +718,7 @@ When \( \lambda_i > 0 \), the vectors \( \boldsymbol{x}_i \) are called support
 
 
 
    
-
    The problem to solve 
    
+
    The problem to solve 
    
 
 
    
 We can rewrite 
@@ -721,7 +746,7 @@ subject to \( \boldsymbol{y}^T\boldsymbol{\lambda}=0 \). Here we defined the vec
 
 
 
    
-
    The last steps 
    
+
    The last steps 
    
 
 
    
 Solving the above problem, yields the values of \( \lambda_i \).
@@ -765,7 +790,7 @@ Below we discuss how to find the optimal values of \( \lambda_i \). Before we pr
 
 
 
    
-
    A soft classifier 
    
+
    A soft classifier 
    
 
 
    
 Till now, the margin is strictly defined by the support vectors. This defines what is called a hard classifier, that is the margins are well defined.
@@ -804,7 +829,7 @@ misclassifications.
 
 
 
    
-
    Soft optmization problem 
    
+
    Soft optmization problem 
    
 
 
    
 This has in turn the consequences that we change our optmization problem to finding the minimum of 
@@ -876,7 +901,7 @@ $$
 
 
 
    
-
    Kernels and non-linearity 
    
+
    Kernels and non-linearity 
    
 
 
    
 The cases we have studied till were all characterized by two classes
@@ -950,7 +975,7 @@ plt.show()
 
 
 
    
-
    The equations 
    
+
    The equations 
    
 
 
    
 Suppose we define a polynomial transformation of degree two only (we continue to live in a plane with \( x_i \) and \( y_i \) as variables)
@@ -1002,7 +1027,7 @@ the trouble of performing the transformation
 
 
 
    
-
    The problem to solve 
    
+
    The problem to solve 
    
 Using our definition of the kernel We can rewrite again the Lagrangian
 
     
    
 $$
@@ -1044,7 +1069,7 @@ Given a kernel \( K \) and the targets \( y_i \) this matrix is easy to set up.
 
 
 
    
-
    Different kernels and Mercer's theorem 
    
+
    Different kernels and Mercer's theorem 
    
 
 
    
 There are several popular kernels being used. These are
@@ -1078,7 +1103,7 @@ well in practice.
 
 
 
    
-
    The moons example 
    
+
    The moons example 
    
 
    
 
 
@@ -1275,7 +1300,7 @@ plt.show()
 
 
 
    
-
    Mathematical optimization of convex functions 
    
+
    Mathematical optimization of convex functions 
    
 
 
    
 A mathematical (quadratic) optimization problem, or just optimization problem, has the form
@@ -1302,7 +1327,7 @@ Convex optimization problems play a central role in applied mathematics and we r
 
 
 
    
-
    How do we solve these problems? 
    
+
    How do we solve these problems? 
    
 
 
    
 If we use Python as programming language and wish to venture beyond
@@ -1328,7 +1353,7 @@ This will make our life much easier. You don't need t write your own optimizer.
 
 
 
    
-
    A simple example 
    
+
    A simple example 
    
 
 
    
 We remind ourselves about the general problem we want to solve
@@ -1409,7 +1434,7 @@ sol[’primal obj
 
 
 
    
-
    Back to the more realistic cases 
    
+
    Back to the more realistic cases 
    
 
 
    
 We are now ready to return to our setup of the optmization problem for a more realistic case. Introducing the slack parameter \( C \) we have
@@ -1434,7 +1459,7 @@ With  the slack constants this leads to the additional constraint \( 0\leq \lamb
 
 
 
    
-
    Code Example 
    
+
    Code Example 
    
 
    
 
 
diff --git a/doc/pub/svm/html/svm-solarized.html b/doc/pub/svm/html/svm-solarized.html
index 6d50f6a4e..c4b108daf 100644
--- a/doc/pub/svm/html/svm-solarized.html
+++ b/doc/pub/svm/html/svm-solarized.html
@@ -43,30 +43,32 @@ div { text-align: justify; text-justify: inter-word; }
               ('Getting into the details', 2, None, '___sec5'),
               ('First attempt at a minimization approach', 2, None, '___sec6'),
               ('Solving the equations', 2, None, '___sec7'),
-              ('A better approach', 2, None, '___sec8'),
+              ('Code Example', 2, None, '___sec8'),
+              ('Problems with the Simpler Approach', 2, None, '___sec9'),
+              ('A better approach', 2, None, '___sec10'),
               ('A quick reminder on Lagrangian multipliers',
                2,
                None,
-               '___sec9'),
-              ('Adding the muliplier', 2, None, '___sec10'),
-              ('Setting up the problem', 2, None, '___sec11'),
-              ('The problem to solve', 2, None, '___sec12'),
-              ('The last steps', 2, None, '___sec13'),
-              ('A soft classifier', 2, None, '___sec14'),
-              ('Soft optmization problem', 2, None, '___sec15'),
-              ('Kernels and non-linearity', 2, None, '___sec16'),
-              ('The equations', 2, None, '___sec17'),
-              ('The problem to solve', 2, None, '___sec18'),
-              ("Different kernels and Mercer's theorem", 2, None, '___sec19'),
-              ('The moons example', 2, None, '___sec20'),
+               '___sec11'),
+              ('Adding the muliplier', 2, None, '___sec12'),
+              ('Setting up the problem', 2, None, '___sec13'),
+              ('The problem to solve', 2, None, '___sec14'),
+              ('The last steps', 2, None, '___sec15'),
+              ('A soft classifier', 2, None, '___sec16'),
+              ('Soft optmization problem', 2, None, '___sec17'),
+              ('Kernels and non-linearity', 2, None, '___sec18'),
+              ('The equations', 2, None, '___sec19'),
+              ('The problem to solve', 2, None, '___sec20'),
+              ("Different kernels and Mercer's theorem", 2, None, '___sec21'),
+              ('The moons example', 2, None, '___sec22'),
               ('Mathematical optimization of convex functions',
                2,
                None,
-               '___sec21'),
-              ('How do we solve these problems?', 2, None, '___sec22'),
-              ('A simple example', 2, None, '___sec23'),
-              ('Back to the more realistic cases', 2, None, '___sec24'),
-              ('Code Example', 2, None, '___sec25')]}
+               '___sec23'),
+              ('How do we solve these problems?', 2, None, '___sec24'),
+              ('A simple example', 2, None, '___sec25'),
+              ('Back to the more realistic cases', 2, None, '___sec26'),
+              ('Code Example', 2, None, '___sec27')]}
 end of tocinfo -->
 
 
@@ -108,7 +110,7 @@ MathJax.Hub.Config({
 
    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University
    
 
    
 
    
-
    Nov 15, 2019
     
+
    Nov 16, 2019
     
 
    
 
    
 









    
@@ -377,7 +379,7 @@ $$
 
    Solving the equations 
    
 
 
    
-We can now use the Newton-Raphson method or gradient descent to solve the equations
+We can now use the Newton-Raphson method or different variants of the gradient descent family (from plain gradient descent to various stochastic gradient descent approaches) to solve the equations
 $$
 b \leftarrow b +\eta \frac{\partial C}{\partial b},
 $$
@@ -389,11 +391,32 @@ $$
 
 where \( \eta \) is our by now well-known learning rate.
 
+
    
+









    
+
+
    Code Example 
    
+
+
    
+The equations we discussed above can be coded rather easily (the
+framework is rather similar to what we developed for logistic
+regression). We use the standard moons example to set up our data
+set. We want to find a line which separates the two classes
+
    
+
+
+
    
+
    
+
    
+









    
+
+
    Problems with the Simpler Approach 
    
+
 
    
 There are however problems with this approach, although it looks
-pretty straightforward to implement. In case we separate our data into
-two distinct classes, we may up with many possible lines, as indicated
-in the figure and shown by running the following program. For small
+pretty straightforward to implement. When running the above code, we see that we can easily end up with many diffeent lines which separate the two classes.
+
+
    
+For small
 gaps between the entries, we may also end up needing many iterations
 before the solutions converge and if the data cannot be separated
 properly into two distinct classes, we may not experience a converge
@@ -402,7 +425,7 @@ at all.
 
    
 









    
 
-
    A better approach 
    
+
    A better approach 
    
 
 
    
 A better approach is rather to try to define a large margin between
@@ -436,12 +459,15 @@ y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) \geq 1 \hspace{0.1cm}\forall i.
 $$
 
 
    
-We have thus defined our margin as the invers of the norm of \( \boldsymbol{w} \). We want to minimize the norm in order to have a as large as possible margin \( M \). Before we proceed, we need to remind ourselves about Lagrangian multipliers.
+We have thus defined our margin as the invers of the norm of
+\( \boldsymbol{w} \). We want to minimize the norm in order to have a as large as
+possible margin \( M \). Before we proceed, we need to remind ourselves
+about Lagrangian multipliers.
 
 
    
 









    
 
-
    A quick reminder on Lagrangian multipliers 
    
+
    A quick reminder on Lagrangian multipliers 
    
 
 
    
 Consider a function of three independent variables \( f(x,y,z) \) . For the function \( f \) to be an
@@ -491,7 +517,7 @@ Then \( dz \) is no longer arbitrary.
 
    
 









    
 
-
    Adding the muliplier 
    
+
    Adding the muliplier 
    
 
 
    
 However, we can add to
@@ -534,7 +560,7 @@ $$
 
    
 









    
 
-
    Setting up the problem 
    
+
    Setting up the problem 
    
 In order to solve the above problem, we define the following Lagrangian function to be minimized 
 $$
 {\cal L}(\lambda,b,\boldsymbol{w})=\frac{1}{2}\boldsymbol{w}^T\boldsymbol{w}-\sum_{i=1}^n\lambda_i\left[y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)-1\right],
@@ -575,7 +601,7 @@ When \( \lambda_i > 0 \), the vectors \( \boldsymbol{x}_i \) are called support
 
    
 









    
 
-
    The problem to solve 
    
+
    The problem to solve 
    
 
 
    
 We can rewrite 
@@ -599,7 +625,7 @@ subject to \( \boldsymbol{y}^T\boldsymbol{\lambda}=0 \). Here we defined the vec
 
    
 









    
 
-
    The last steps 
    
+
    The last steps 
    
 
 
    
 Solving the above problem, yields the values of \( \lambda_i \).
@@ -633,7 +659,7 @@ Below we discuss how to find the optimal values of \( \lambda_i \). Before we pr
 
    
 









    
 
-
    A soft classifier 
    
+
    A soft classifier 
    
 
 
    
 Till now, the margin is strictly defined by the support vectors. This defines what is called a hard classifier, that is the margins are well defined.
@@ -668,7 +694,7 @@ misclassifications.
 
    
 









    
 
-
    Soft optmization problem 
    
+
    Soft optmization problem 
    
 
 
    
 This has in turn the consequences that we change our optmization problem to finding the minimum of 
@@ -722,7 +748,7 @@ $$
 
    
 









    
 
-
    Kernels and non-linearity 
    
+
    Kernels and non-linearity 
    
 
 
    
 The cases we have studied till were all characterized by two classes
@@ -795,7 +821,7 @@ plt.show()
 
    
 









    
 
-
    The equations 
    
+
    The equations 
    
 
 
    
 Suppose we define a polynomial transformation of degree two only (we continue to live in a plane with \( x_i \) and \( y_i \) as variables)
@@ -837,7 +863,7 @@ the trouble of performing the transformation
 
    
 









    
 
-
    The problem to solve 
    
+
    The problem to solve 
    
 Using our definition of the kernel We can rewrite again the Lagrangian
 $$
 {\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{x}_i^T\boldsymbol{z}_j,
@@ -873,7 +899,7 @@ Given a kernel \( K \) and the targets \( y_i \) this matrix is easy to set up.
 
    
 









    
 
-
    Different kernels and Mercer's theorem 
    
+
    Different kernels and Mercer's theorem 
    
 
 
    
 There are several popular kernels being used. These are
@@ -904,7 +930,7 @@ well in practice.
 
    
 









    
 
-
    The moons example 
    
+
    The moons example 
    
 
    
 
 
@@ -1100,7 +1126,7 @@ plt.show()
 
    
 









    
 
-
    Mathematical optimization of convex functions 
    
+
    Mathematical optimization of convex functions 
    
 
 
    
 A mathematical (quadratic) optimization problem, or just optimization problem, has the form
@@ -1125,7 +1151,7 @@ Convex optimization problems play a central role in applied mathematics and we r
 
    
 









    
 
-
    How do we solve these problems? 
    
+
    How do we solve these problems? 
    
 
 
    
 If we use Python as programming language and wish to venture beyond
@@ -1151,7 +1177,7 @@ This will make our life much easier. You don't need t write your own optimizer.
 
    
 









    
 
-
    A simple example 
    
+
    A simple example 
    
 
 
    
 We remind ourselves about the general problem we want to solve
@@ -1219,7 +1245,7 @@ sol[’primal obj
 
    
 









    
 
-
    Back to the more realistic cases 
    
+
    Back to the more realistic cases 
    
 
 
    
 We are now ready to return to our setup of the optmization problem for a more realistic case. Introducing the slack parameter \( C \) we have
@@ -1242,7 +1268,7 @@ With  the slack constants this leads to the additional constraint \( 0\leq \lamb
 
    
 









    
 
-
    Code Example 
    
+
    Code Example 
    
 
    
 
 
diff --git a/doc/pub/svm/html/svm.html b/doc/pub/svm/html/svm.html
index 0ac067ea2..ad3d86721 100644
--- a/doc/pub/svm/html/svm.html
+++ b/doc/pub/svm/html/svm.html
@@ -48,30 +48,32 @@ div { text-align: justify; text-justify: inter-word; }
               ('Getting into the details', 2, None, '___sec5'),
               ('First attempt at a minimization approach', 2, None, '___sec6'),
               ('Solving the equations', 2, None, '___sec7'),
-              ('A better approach', 2, None, '___sec8'),
+              ('Code Example', 2, None, '___sec8'),
+              ('Problems with the Simpler Approach', 2, None, '___sec9'),
+              ('A better approach', 2, None, '___sec10'),
               ('A quick reminder on Lagrangian multipliers',
                2,
                None,
-               '___sec9'),
-              ('Adding the muliplier', 2, None, '___sec10'),
-              ('Setting up the problem', 2, None, '___sec11'),
-              ('The problem to solve', 2, None, '___sec12'),
-              ('The last steps', 2, None, '___sec13'),
-              ('A soft classifier', 2, None, '___sec14'),
-              ('Soft optmization problem', 2, None, '___sec15'),
-              ('Kernels and non-linearity', 2, None, '___sec16'),
-              ('The equations', 2, None, '___sec17'),
-              ('The problem to solve', 2, None, '___sec18'),
-              ("Different kernels and Mercer's theorem", 2, None, '___sec19'),
-              ('The moons example', 2, None, '___sec20'),
+               '___sec11'),
+              ('Adding the muliplier', 2, None, '___sec12'),
+              ('Setting up the problem', 2, None, '___sec13'),
+              ('The problem to solve', 2, None, '___sec14'),
+              ('The last steps', 2, None, '___sec15'),
+              ('A soft classifier', 2, None, '___sec16'),
+              ('Soft optmization problem', 2, None, '___sec17'),
+              ('Kernels and non-linearity', 2, None, '___sec18'),
+              ('The equations', 2, None, '___sec19'),
+              ('The problem to solve', 2, None, '___sec20'),
+              ("Different kernels and Mercer's theorem", 2, None, '___sec21'),
+              ('The moons example', 2, None, '___sec22'),
               ('Mathematical optimization of convex functions',
                2,
                None,
-               '___sec21'),
-              ('How do we solve these problems?', 2, None, '___sec22'),
-              ('A simple example', 2, None, '___sec23'),
-              ('Back to the more realistic cases', 2, None, '___sec24'),
-              ('Code Example', 2, None, '___sec25')]}
+               '___sec23'),
+              ('How do we solve these problems?', 2, None, '___sec24'),
+              ('A simple example', 2, None, '___sec25'),
+              ('Back to the more realistic cases', 2, None, '___sec26'),
+              ('Code Example', 2, None, '___sec27')]}
 end of tocinfo -->
 
 
@@ -113,7 +115,7 @@ MathJax.Hub.Config({
 
    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University
    
 
    
 
    
-
    Nov 15, 2019
     
+
    Nov 16, 2019
     
 
    
 
    
 









    
@@ -382,7 +384,7 @@ $$
 
    Solving the equations 
    
 
 
    
-We can now use the Newton-Raphson method or gradient descent to solve the equations
+We can now use the Newton-Raphson method or different variants of the gradient descent family (from plain gradient descent to various stochastic gradient descent approaches) to solve the equations
 $$
 b \leftarrow b +\eta \frac{\partial C}{\partial b},
 $$
@@ -394,11 +396,32 @@ $$
 
 where \( \eta \) is our by now well-known learning rate.
 
+
    
+









    
+
+
    Code Example 
    
+
+
    
+The equations we discussed above can be coded rather easily (the
+framework is rather similar to what we developed for logistic
+regression). We use the standard moons example to set up our data
+set. We want to find a line which separates the two classes
+
    
+
+
+
    
+
    
+
    
+









    
+
+
    Problems with the Simpler Approach 
    
+
 
    
 There are however problems with this approach, although it looks
-pretty straightforward to implement. In case we separate our data into
-two distinct classes, we may up with many possible lines, as indicated
-in the figure and shown by running the following program. For small
+pretty straightforward to implement. When running the above code, we see that we can easily end up with many diffeent lines which separate the two classes.
+
+
    
+For small
 gaps between the entries, we may also end up needing many iterations
 before the solutions converge and if the data cannot be separated
 properly into two distinct classes, we may not experience a converge
@@ -407,7 +430,7 @@ at all.
 
    
 









    
 
-
    A better approach 
    
+
    A better approach 
    
 
 
    
 A better approach is rather to try to define a large margin between
@@ -441,12 +464,15 @@ y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) \geq 1 \hspace{0.1cm}\forall i.
 $$
 
 
    
-We have thus defined our margin as the invers of the norm of \( \boldsymbol{w} \). We want to minimize the norm in order to have a as large as possible margin \( M \). Before we proceed, we need to remind ourselves about Lagrangian multipliers.
+We have thus defined our margin as the invers of the norm of
+\( \boldsymbol{w} \). We want to minimize the norm in order to have a as large as
+possible margin \( M \). Before we proceed, we need to remind ourselves
+about Lagrangian multipliers.
 
 
    
 









    
 
-
    A quick reminder on Lagrangian multipliers 
    
+
    A quick reminder on Lagrangian multipliers 
    
 
 
    
 Consider a function of three independent variables \( f(x,y,z) \) . For the function \( f \) to be an
@@ -496,7 +522,7 @@ Then \( dz \) is no longer arbitrary.
 
    
 









    
 
-
    Adding the muliplier 
    
+
    Adding the muliplier 
    
 
 
    
 However, we can add to
@@ -539,7 +565,7 @@ $$
 
    
 









    
 
-
    Setting up the problem 
    
+
    Setting up the problem 
    
 In order to solve the above problem, we define the following Lagrangian function to be minimized 
 $$
 {\cal L}(\lambda,b,\boldsymbol{w})=\frac{1}{2}\boldsymbol{w}^T\boldsymbol{w}-\sum_{i=1}^n\lambda_i\left[y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)-1\right],
@@ -580,7 +606,7 @@ When \( \lambda_i > 0 \), the vectors \( \boldsymbol{x}_i \) are called support
 
    
 









    
 
-
    The problem to solve 
    
+
    The problem to solve 
    
 
 
    
 We can rewrite 
@@ -604,7 +630,7 @@ subject to \( \boldsymbol{y}^T\boldsymbol{\lambda}=0 \). Here we defined the vec
 
    
 









    
 
-
    The last steps 
    
+
    The last steps 
    
 
 
    
 Solving the above problem, yields the values of \( \lambda_i \).
@@ -638,7 +664,7 @@ Below we discuss how to find the optimal values of \( \lambda_i \). Before we pr
 
    
 









    
 
-
    A soft classifier 
    
+
    A soft classifier 
    
 
 
    
 Till now, the margin is strictly defined by the support vectors. This defines what is called a hard classifier, that is the margins are well defined.
@@ -673,7 +699,7 @@ misclassifications.
 
    
 









    
 
-
    Soft optmization problem 
    
+
    Soft optmization problem 
    
 
 
    
 This has in turn the consequences that we change our optmization problem to finding the minimum of 
@@ -727,7 +753,7 @@ $$
 
    
 









    
 
-
    Kernels and non-linearity 
    
+
    Kernels and non-linearity 
    
 
 
    
 The cases we have studied till were all characterized by two classes
@@ -800,7 +826,7 @@ plt.show()
 
    
 









    
 
-
    The equations 
    
+
    The equations 
    
 
 
    
 Suppose we define a polynomial transformation of degree two only (we continue to live in a plane with \( x_i \) and \( y_i \) as variables)
@@ -842,7 +868,7 @@ the trouble of performing the transformation
 
    
 









    
 
-
    The problem to solve 
    
+
    The problem to solve 
    
 Using our definition of the kernel We can rewrite again the Lagrangian
 $$
 {\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{x}_i^T\boldsymbol{z}_j,
@@ -878,7 +904,7 @@ Given a kernel \( K \) and the targets \( y_i \) this matrix is easy to set up.
 
    
 









    
 
-
    Different kernels and Mercer's theorem 
    
+
    Different kernels and Mercer's theorem 
    
 
 
    
 There are several popular kernels being used. These are
@@ -909,7 +935,7 @@ well in practice.
 
    
 









    
 
-
    The moons example 
    
+
    The moons example 
    
 
    
 
 
@@ -1105,7 +1131,7 @@ plt.show()
 
    
 









    
 
-
    Mathematical optimization of convex functions 
    
+
    Mathematical optimization of convex functions 
    
 
 
    
 A mathematical (quadratic) optimization problem, or just optimization problem, has the form
@@ -1130,7 +1156,7 @@ Convex optimization problems play a central role in applied mathematics and we r
 
    
 









    
 
-
    How do we solve these problems? 
    
+
    How do we solve these problems? 
    
 
 
    
 If we use Python as programming language and wish to venture beyond
@@ -1156,7 +1182,7 @@ This will make our life much easier. You don't need t write your own optimizer.
 
    
 









    
 
-
    A simple example 
    
+
    A simple example 
    
 
 
    
 We remind ourselves about the general problem we want to solve
@@ -1224,7 +1250,7 @@ sol[’primal objective’]
 
    
 









    
 
-
    Back to the more realistic cases 
    
+
    Back to the more realistic cases 
    
 
 
    
 We are now ready to return to our setup of the optmization problem for a more realistic case. Introducing the slack parameter \( C \) we have
@@ -1247,7 +1273,7 @@ With  the slack constants this leads to the additional constraint \( 0\leq \lamb
 
    
 









    
 
-
    Code Example 
    
+
    Code Example 
    
 
    
 
 
diff --git a/doc/pub/svm/ipynb/ipynb-svm-src.tar.gz b/doc/pub/svm/ipynb/ipynb-svm-src.tar.gz
index ce0d030c05b174fd8cd119f8465da99421cfbdd9..ab001d657af4f9343f0ad75dfe58209119d861ea 100644
GIT binary patch
literal 189
zcmV;u07CyCiwFSoZO~l+1MSbv3c@f92k@Qu6nTQtu3L5%+`)q&;tO=Fb5+;QwiEaE
z?gR9s;9-c+-{nt22+2NQt+#pP?k-piAq0~I#>_CClEia8p)>}{QcMV9h71U03|RnV
zxszTx>$n+CsZT;#q2A67V^#HG&-@BJ^G_UVX<@VLU9A-;?L?vV8SJfzWZSC(N-gZz
r2P3YXGy=HlftRIpQY(H5TceMLH#Q1?{fXy!p7)IhrgZ|A00;m8eOy=X

literal 189
zcmV;u07CyCiwFR@G|pWB1MSaC3c@fD2H>uHia9~aqmQ*<7cK-5FObsIM(v~~so2}w
z2k1({MG+z2=4XasnAzv6^)`>(-35ywgrso{V`gbGC1GQf-d~lv>)c
r4@O=)Wh4mfL6oI(N~5@ht1Y<&-=y$6qpUv00;m8oG(~j

diff --git a/doc/pub/svm/ipynb/svm.ipynb b/doc/pub/svm/ipynb/svm.ipynb
index 32030fd82..5dc073bd4 100644
--- a/doc/pub/svm/ipynb/svm.ipynb
+++ b/doc/pub/svm/ipynb/svm.ipynb
@@ -10,7 +10,7 @@
     "  \n",
     "**Morten Hjorth-Jensen**, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University\n",
     "\n",
-    "Date: **Nov 15, 2019**\n",
+    "Date: **Nov 16, 2019**\n",
     "\n",
     "Copyright 1999-2019, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license\n",
     "\n",
@@ -385,7 +385,7 @@
    "source": [
     "## Solving the equations\n",
     "\n",
-    "We can now use the Newton-Raphson method or gradient descent to solve the equations"
+    "We can now use the Newton-Raphson method or different variants of the gradient descent family (from plain gradient descent to various stochastic gradient descent approaches) to solve the equations"
    ]
   },
   {
@@ -419,10 +419,27 @@
    "source": [
     "where $\\eta$ is our by now well-known learning rate. \n",
     "\n",
+    "\n",
+    "## Code Example\n",
+    "\n",
+    "The equations we discussed above can be coded rather easily (the\n",
+    "framework is rather similar to what we developed for logistic\n",
+    "regression). We use the standard moons example to set up our data\n",
+    "set. We want to find a line which separates the two classes"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Problems with the Simpler Approach\n",
+    "\n",
+    "\n",
     "There are however problems with this approach, although it looks\n",
-    "pretty straightforward to implement. In case we separate our data into\n",
-    "two distinct classes, we may up with many possible lines, as indicated\n",
-    "in the figure and shown by running the following program. For small\n",
+    "pretty straightforward to implement. When running the above code, we see that we can easily end up with many diffeent lines which separate the two classes.\n",
+    "\n",
+    "\n",
+    "For small\n",
     "gaps between the entries, we may also end up needing many iterations\n",
     "before the solutions converge and if the data cannot be separated\n",
     "properly into two distinct classes, we may not experience a converge\n",
@@ -501,7 +518,10 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "We have thus defined our margin as the invers of the norm of $\\boldsymbol{w}$. We want to minimize the norm in order to have a as large as possible margin $M$. Before we proceed, we need to remind ourselves about Lagrangian multipliers. \n",
+    "We have thus defined our margin as the invers of the norm of\n",
+    "$\\boldsymbol{w}$. We want to minimize the norm in order to have a as large as\n",
+    "possible margin $M$. Before we proceed, we need to remind ourselves\n",
+    "about Lagrangian multipliers.\n",
     "\n",
     "## A quick reminder on Lagrangian multipliers\n",
     "\n",
diff --git a/doc/pub/svm/pdf/svm-minted.pdf b/doc/pub/svm/pdf/svm-minted.pdf
index e2221619b78620a125177919b778f208d13b1f2e..7bcee85778afd529ed308365510779f623e9c428 100644
GIT binary patch
delta 75687
zcmV)aK&rppnGn~X5U?l$0fn4@76UXnFqgr*0VsdXS>KP_xDkHe
zze0~P8Z+Wgk(UPO^^g=OkVD%na0Tw}LA&OzmB^M)lH+Xu`p$4DS<=euT<@h1h9q)0
zBxgQ;Gec*|W0EC*yUF77ueUd^-}5|i%o$;l+lM4$&I;(u5}t5nSm?6}@~}
z=^Va}zFqx#`}f!Hg&nPk$JIz_SZ-&ld5QyEVwlUN#Q}<0tCKZn%Bkdb15#y_k{32A
zRnBs4rmEkr*22n^)7XnflEyHp^>l2}(#CdWdtVi83&tsN=RSX0tK3He7Hzc532oVO@?L{e0v__
z80Ts-rr4torA>dkZFF6II3QHgx-C0TedPl^qgR6%y3I}8whVKXF*N6R&Pym)zsuHA
zb2!dZ&2MRpKY;p>sJJ8Dv9HK0#l8-n4B<`I|EWio|
zU7w)^-FW7^1*3CKGczkA0v5fQG0oA=LE`K`sir~-SE7t2xqmMp4zQNxU>3od%qNhO
zVk8Fk;I4n4Pb}!sOoCgYz=tIVWGu@}!Z~wUT+i=20b^z0eAxa-=*p8O7!ME
zE#XEm4M_bK+?fA2+_IPo9su%9eBqNK1N|
zrErAV=5Rchk~n-D%_2flfH@=la&?7BCAW;5JW)9$jOGu8SqY)qfevL9WF@rk@`8gODS7c`)W15mnu&}_)7y`W36KbB4E>NV%pL=4eZS09ff`<$;*Vlhs
zAr@R=jrt~M2*EN7_?5uPz=VG!tPQ##@@8DV9M_(0d%CGm!(H*3$P{=jNB05sMz@`|
zKvxSY1u#u!=Z2c8gm^3lYnL&ztd9+y+dDk8Pt>>{DrD%O+Mue|pJVf$geqS&`TVzP&%TZK!GCv$&%wXjGhr_7G<9h5Gx}7&`jVuut$ooU>>Hh2uL?c-JrHp(%Dg
zETL{bQ&ZVq3KLR8pohvf>}OD)xwI`ILKOJHGon{$JZ+1PT29EOwbFsAz^LUKx@M>-L`C_no;0?g(!ogR6%NWe2k)HM}w>A`OklG;Gywd7uf~B
z+pZ}}7=$^@dN>tkW_&DIoe;Dez%ecF1S+P&jha?#P%c>7J^cq&TNBMjlKm_D|fafO74odef%h@w(iGm=mjl?&%O7X
z;9UIJ5fwZ=+f8qLj6ymZzwCT71PhnzNsQWNxm8?*lxM3`{EtZUUk|j>1`3AzC|Xcou!Nmh0z2c#qN>Zr~a)$!&ihX>hJ%02S?q(h`9ou0=Zw
zMfQW|5S-z?phq;><(p=VYYsBk$*dn<4vE5H?ailS0>@tdBSH$yHfKX#o(6IVFvx6a
z`?>*!EdS#-!`@%GymkuUYo;1LeOjz5wA8b@Vxp4%Ac+FKif7l~abCrteuC6`GVhhY
zlKP=gO^JVOX502i5c=c;*9_Z!4rwK!K$f$t3dm$rBF!$upiC1pR^yU8<
z&R1Nzua)#t?J6OYQu8A2`M*(6nD?dN3;#xqq%Qp%b$#XNem-9L)vJr8%MYw6^|?;e
zd;v#>88t4t_3{aGQE=Grcr{}>*@bFg#q{x8DSh=gohDh&R=admsgzA{N3dhGx@H6ZZ7e&??PkQR^efP=b8_F^WOck
zybP~hecM#)@Vu_Oec4>nd{u=$H+XJ$+wii#i}S6k724~r-Zi1&O<4ZWtjqSH@AFH+
z#W3?5w}XevU$1|FD5F4HnM?^um
zWf=FLOT+Uxs;jEs`thRyipe3D?QPZV!^`h~mr~{B=8!seigTBY_G12&5P%9&jEb09
zAD>tsCgzh-SGpjUQ^+8Ra7H+s#>L;SFWz7Dw#ryR3YjU&i4rDTZ7+WQwaC`^_Cr>{
zElqasdv4u?oWxstJwkE`+BP8Zxaj>(Q43TP3Nk7GZ
zOl|N5jz5OSb$SihiFpWOj11-xkFZYY389CoY|`nMLKcZlb7HhyTH|y%w`r_NDGzqBBDg_+co8+Tfy(~WuZ0y5i%8m;n
zsalxvY!(=pG(!)Qn8Qgl>?I{K)YpH1{uXgrn?AhEw*rH=LnM8f6xw-2F^d_qDFm^G
zjx$_QrSli=M};5V|GW&1&C7(}UNMH)gT#ghVrV?2T^k(t?E&E-v3YnqMVt?zN1e8A
zF$;c=S+|}>zt6UAah7$9v#eX3W8LC^!@3KiuE(zbT<%jU=rf8|
zgd*}z*8A`Eag+skT1aNm+py`PK18r|h=`*&ybXDKJ_^bDr_&nc%)%aK!@gcBRONR7
zW2JllebTs(I`m0=9rQY=ds{bu_11MhY(vBDCZV?_%9O$oEh27y%y+SAqWTzg!g(jX
z?o%44$n9nf!bwM%xXSY#{Ol|YF=G!G(gxtw?Aon&^nlY20l&YkRuPMN{ZEnFYS*l5
zZ{rD+0YV~NtTKqn>MW!o&OsXgkHw!Di}|AlmYPK)+kOJjcqDvM-usz<^Dm0fLK*O|
z2z~}Oz=BVk{H}B$eDredS%D6$0AB{i0qz)Dc84}RY{3)iCOnt1dkhCUcbwNp50h7I
z{J6WBbu!-IL-&)xLNI533nA{GkX&#ITU-os)RUW;0`A=wOTSy67(g7XmCG;Q|Hvi69!p*L&j*12X1C1fV@cw{$;duV8-TBS0
zs%rgpXd1GO3w%{i34?xk-d50VY=gmd)h6gFa@Ep>Mp(^ZAKrg|EN|Pgxvk444B2{O
zyV=+8HZ{zTL=Dj_kYW}j<`P(yxi}eP!z+x|GJk)lSC^dTzlGPZ7k`3y^MC@svnBDq
zbtA?l)7Dum^Pf|-iRtfeyQZt7rHEh;Jij?KtB7LI=sn1109P!jVgbeNE*M9JCwz(4
z+!J`1P+ONUn(h>ToRFGGWa8AN_ATMz(W|u7*;3`9hyK;=HsVxbp*H=y^)t~jH({A#AFrh=i7reB1l9#5k4uFOr
zn)#ZIGG=rr@87Gc>tYd-`6j&%Ez=Q4$;m;
zIc@!BQI%QLOI+;C60is$M7kgpMtzpk29xo&qluU{D
z(2J)a*;KxV^!X`4pOnv>C7+-$a`Qa$ArltDq~!b9%O{1M$@kG^%>2cfAcr&R305fI
z#0OjbsznOCK`LBsXfvR
zDI4fXLYD92lp$F=K4BIzxiVKFqdu2phS_+31B#Wa(%(LzDbPy`^QfB@kag@C3Q-m$
zH8xb`pjN!iwLc$HZNS-@avulon)ndzxdKd%+X**&M<|aO01i9jf^&isL0i4Qi*P3#
zc2nKY;1BF)3&mD#e=WD4VthVB^^Sd43smp6>z#rO3fBwP?|gzHXZWqg`_^7-cnB+h
zS=hzZ%I#zwUgu=WF3D7AtmALrsod(1C-p=UXPLBzUsO|
z0Ny&#tYDUUg2ET{J!vGe06U4A?^)@8q$2HTZRB!C*6O}B$AM6tWN}6!h}b^6JRbM5
zD};L&)n;@05J(PkNixr2W_5zO61y}
z!eN+7Kt%z*zbWnRp&z&x8jkR}8YKc=i8JA6oB2aGKU9R%O%d_KTDjq24x@4Ud?-
zCe{`FKZ(gx3GP_PXJPWLYSRCIo5`0a^3NhkB=X0Sd`TYv9Fp`r{#cSPK;sigoN`K7
zkHqVnQ)NKmN&xc#C+PfSDQO5V^n4`b5P;Lg6T38s6KpZmAt2$fmII9OYQup$*B1Cg
zwJJRdbS=ObZG0^IhsG0(`|z@BHV+OPfi486_XNa$O7Vg41%O42fQ9CNe>s9l9QH!p
zzxG8&5P??$VJKqh5TZUL>#D^F4bJ(GyocnytQ-+ZqQ?NFxu#f-Tn5-Bh
z;E3V&Zk=SJoXGKiCHsHyYvR@K2CD3dnG3A#n;O`$eYi9t_dHsokeE1J7b@rkPWo~2+b98cLVQmU!Ze(v_Y6_R3xB?TiKr{Ra0Wz~aJ!=aAGP9{j
z6AS@4v%OSl3jsB=Ct&Lh0Wz}_ZhQ>^GqaF;kqH4gvm%8Z2$QL^6th!}RwkE28UYov
zzO#g8mz#zG77{ZuG72wDWo~D5Xfhx&HZwRdmtlSa6a_LjH#Rkwu?hqyf3&w{RGeGV
zHcW7L*9L;SySuvwr*UZ9HMj+LcL@Xn!QDN$2X}V~`jIow%sDgb{rfEzG*{_WwX61h
zHwCG(8iR-#$OI?}vUg!%VPxh7$SbIV?2PSMSQ$h?wq^iUMrLMqcnS(JC!n#5706!P
z*agT7;Bv79D4DvvJ30Yaf0>zi;3)u7KzpFmyV49`;t5ayx)`f_IsjP!)W&}SWstKA
zgNd>8yBcV3VPy}bd3O;5Ie0o*Sy;OKiNVIe@F&tAZ&5~otg)#L$lck-3Sew+29RY`
zU<4?F+~0*(0BVpuzyxS%Y-)BkZ1
zQ&ZQFq6dhJD5^^UfSU9GDGfFCKY!JM_V4^H=mCoA@A^M^-VOh_D@dq|sOu<8urU3p
z0l)%q13Ecd{Ym>j+$i6f0sf};?rQD?vimCmfZEc<#etWJ$=%(Z(Zbc)g%RXr!RTQ7
z7e94ND`$W^$jJuqfBxwNv<3dv7*~6<_jbBi0{<1@&yfJ+txSRT&cHuTlAwPn?cQ7Y
z?({Bq`46%8M!5WmY5Om8fHM&IUui6jo&WNcS5}q>*cn^dy8!Kt?M>eeU5s5^odHIF
z$=-i~W>o(o2n2|^IywExq3~~&(|^+Zo4P3Iy|PcX-oD1}fB$=8#`dnxUjOLLf0u0v
zvUj#}c5(h!L?FQ2$`<&Cz4MR_4I>4|s29V>ckc#mN=u>-|r~eU(sdk
z%|QU3e~I6j{9E;Z$4~v&lG40ys2RxK))Qa`G>2zWe+0R_4+Hi8KRfn6xg=d}Z555}
zfYkrp(EqhEwzIPJ{BOJeG0_74p-TP#aJF)mwDJI&DOoB(4dCu2`|=Jy$3<=_B#v%HV88PMaeAptNk
z+Jju)T>uWQF1`SBkQ4l$gXH7@Fp2yT{R?pdm_+|ZTmUAqzY#ZpN&Iib17MQ)58`45
zFiHN6SO82?e7qxTw2{zmULnEnTGyz5Lsw(qC)ZwWj5ABmma-~KGj@1)HB0pBYH
z{sX?3YW_E5eUEQ$^$+!X4Ho}^@8K=~PWIjyOHT(&p#48A-ev#vf3L;nAMl;9?LXjq
ze>S_nAcY}h&fNI@>Ze{J*MzVBr?
z{R6&pcK!!^_xh*tdkL=pfbS)^{R6(2;Qo*Py_ewe5BOe!=Re?k30{A}|K1wqKaY#Q
zZbRn3*XaMAqkmyF7blPnP|M2f{T<>Te2b+dd%-v8Oyu;{pa8R>Hk*(iob8+
ze|U?Ef;_w#*xv`5ft8yFz{2r9%soVC)x?dNitx6vpMoC&vV_QJt|p_#=S+pgfQk+>nnT�-aL!>CBe+
z7Gi+L=#dytTj%c>)}L=^S1DwlDq4Q!L%A2vvNU2k*`j+aqT;s`EKat6aC!TU85?cf
zBeFz0V^k1|Jt_S
zRju+zEf|HL8_DXqON(vVuKs+FEA^qex$-uOT;Jfi;`TFT`TN{4k@VhxbUHg=Wc)$1
z8<8KOKP!dvstzo>oiAz4E!P}>`i@2rRC21hxIr!9K=9mnsQ&h825!I;f86lm_2VT4
z<~quG=n0@Yk<#ZP*w~d@>}I=?U)s)kKNcb`JLZ*OTvI~FTG|%N_V=7jy0UrcH*d0j
zypwI|P7a!30|w&JAo`EbxAMV`(Wk~{RRJ91#EDVvl%t%Zo$PDF?z1%N2N4t}acpOp
zEd#&j>lH(~j(E;_&53uKe;7b9s@dX08V#arJ5?62AIm_{)V|wc3I3rM327-G-?X+r
zi{@g+@p1cQG<*0g7aj9KUrX_t))C^^LOcw!6gSrQ`K)h22RNOpB5`Oi8zE6c>L`54
zd$Yp;XK95c^Omo2)AA&VP|r;m?3(mLhm^ZsIu<$fg;p30?B87ge;KNGeEUw`FCb#h
zhe(mcp~N#YSGdXJKfQxwV63J?TX=I?mfltdqTzVzI*$2lfw>}4m`ESb9rr~v4Z4ul
zrOymIh2j#D`yST&Dzq!&h~Lt@>P8OJ+im75_
zuvy-!!&)mRmS9+(e}M)C9!YA`v&sW8Qe1h?@+Hup=(OeJf5!e2n}?we
z<(MO45>i;@M_%AOt(VxP89~cQ^iouEuZQFQQuvaa50hG7f3W(5BS%cK>Y#f;$z8(|
zb;MdTi(TuT3^^3h{mfiweO2pDV5z6z_mXZVIw7U;F6n@W(weM@I-MZ|
zZ`s`=O^Y!8wq(rP)dv!Q@~^+~V_5xNZ_A1ETCo2DrHY(xiy$O)_D!JI3>v9Ea9>q4
zGtH3vIBz&icT!{V`82OaxEdBPZXTyqFDdM^3$17E~6UJALNT
zLfmdBn>+4tK&$2=i0o9bvJd2nB5YWlLir`22yQ%b*1CH5
zLVlQCc(bggvinz6qeqal5nBdD*B3s-eanw1f5_2?Z;3f__@+Q}*{@b^1YTL*$?Y^X
znTZHUpRamzGrO7;qH-WhwZV`0Y+9X;1YJ$vmzJ7`22J%~O`b!Ni4FC#>|kAqp!?
z!k+4GXnzmIlMmv7X^U%zTJ$bcR~M7Fe;loQpOWzD>*(J)sw2T_JaYLyS_a%wgvD43
z-~|j$L21a2>BCeW5hGEVE7?8SzlFuXTN+6KCp%qVqBh7MIj77TJ1BlXpcR?_gzWjL
z>#n#>9r;nqs_~Th`4N>Iet!4>j-B9k*FgK4I@T#|9*~g01Ox7@+UApjJ~8Tme_O{R
zdw=HO#$#a7JCW(w0iCfo(1-qDEo1NFj>C(%j4&o?$xogNHbIefXW4u^9Jj`j(j>Q^
za7&D%Ei_xk;*plZUFeZo;g;>PQCE<@npB7RyXDadr_y0@X_7*`SmG`m`IMa}BS-^t
ztl5+MdIP;$!LnO;;3qjM))N1De?QrT=4r__rZ=~7oKk1hnO)uVKBnh5FL(z;zeY_!
zjBvBs^oeGeky}3YnO|U#xf{8Ect765%5_e!;SQ0<1U7K{Z8FQ_?S{INs_lX$2396f
zo6IVQFhaXCxCw|6_@E8vvtmyT
zUj$B;VdJq^X{@Sk9`(X*HEzz4(20(@drpXEtMJ1+z<|lf9nEsWa{~1jL)leFQfNl8
z!#Zto2q{KC;d4m$4i@%Le-Zj0sO{h;EL*f$lXqqdcHE!3aUk8kdyf}=GDFlB!d?bA
zP$OzJd>moxrYPew-FX()i%+sZ_Vt
z(UBtpRWYM8Fzx9gt6K;=}
z$WLYdYDT}3JLgkF_RH>R1fvWr#bR-Ct
z+O^I?0%cSoQ&t*eReG#}L{)S}RFzUJp3HW9J1)q!*3Gwv9LW#^nFoMP)_4rsLp0nS
z?_kz;{Dt!IcqY2Hpn^UJUCQTH|1T1wOiLAvsU%|sW;h6tkDsjtw
zKoCYCUU{xG{fR{_=yW3PT{YLPU89yyu;9Rx!}7!@5CUqueiHAA~9
zevyXj3aMl;dSoFQ&Ws1^F3via4|7n%CeX2@yS-X8WGFpC*@KvMzdJxW+PG2tHxd>IeXDLV1MYi-wQXJ5wa=dQ
z#c2LYEQdk)e^>L8tu0^*-TXeD$6+w&+)mryG>B*FAVb>Ylxi32b#fYXwwD54+paw&
z+2#H?NHV9Z9yb&(dKE0+xW_^Fp-6hR-3xm01O;p_J@`ap<44%baozp1kZ|IdJw}s5
z@Q(2ouY#q?#aTh{@==i3YrJ4NLdGv&@K1DK3N5o;e=E4_$w#3AlrJ
z0;v3=&HnMtY1x)bwrBCdGTw5RApjZKEg
zh^k}!f6dd0_;!z&SV}yTAKVUe2{G87VD!F#{D*pD32qU=ZnA#nG0GVwEmxHyuACI?GK%q$?{y$i2z~
z_6sp6J=nA-Vw8%a@tbf&9v#|{;pN&<<^6QKKJ8?gx*p^rf2GrGZHgEF8HnhyI0D}8
zf3x-NK8LDwjL62axcL^cV@BE+^W(_prZJ;DEDPu~A77+qjLX)C(&hr!ZFapDQa*t@
z9}%=eO*x&$kL|#Co7eNhPKfI4;8ogYX{l>o-a1!b`sjx6IpibDwJKW2k}Ft
zP~HurZnfYFzEt9*Gh1y}x_}@Z5#$j0r;9+*GyjdXXOIhY(N0S0egcz%-K4lH?i7Li
zxe;Ia|e@WOb*Uqm#&Bf1Dhuxuu)r+7S$>L0ek4mpYEvOEA5lv
zu)b_USO`~xBuJ5)nby0yy*fTs>Wk#c4|A`+vJI2f2>m}FyLzKP2)rgkf50Q8uvP^m
z`25PUT8O=T0!5HP0%cVompOGbYf`$`RSp|%HfQP@9J;|{#o2w95gB>E{+Rq`5~3*(
z@N50x07N*a!@V;M{Yx@rUz;@g@k0{At{#@)u#593s|
z#e_62@iNS$vJ{uX-t1^<{6!6Icr#&pOhi*vutW_0DO`+d2YeD-t-|(PF{WTk9+48z
ztXZwOl<M&lusjjd4d{AeHE6|T%{8zzkc3%5%HWae`THBY?rgNe>)d|
zdCp-mKJ6g8Hr?oqv*ChToa7*6`6`drk66PzD?G5BWr>k5Fc+MXZMa!0Cn%aJNro3z
zirAQ;|AS_m-54I$${mUhCy&*{!ihVOa*j$6ZKh(&HJ(~D;fdgiu@opzl?u$n(96I1
zS#l-l3$DuE*t^Q4f1O6^uP^`6qR=Xzcm{g{zlTOKgq@8YPId4;=dfo}Sd3G+R!O!4
zlU?7Mmx4dVdqzcY9eaKx?SqdGT1>RXJ-%G;kf+=VG-x&OCrs7eUTvZtWqVT9Oho5^
z%xZ+>&^YFv9>8);m|a0S>nVQfa5f&F;X-uH$lbwoLLDOoe?{_M#b;tM*}s}OXetFB
z{qTg}bZ+8>ZZMweIH5$Mtci~Oy0ks#`mq(V9+r>|9Y*AwiG0g%g$K#pcXJ8sibm7J
z7dD-lbMTQumxaR7VOg)XqNa$g520msf~tdBWT-`6r3}2e?(zzr8u&Q@EUTNMQWzf@
z#n!CjQ%oxee=>`WIN#*~hqtxJ67)rU_?x9Vb!CU+kiVY7xtupz#*;D)A7&$`5}}II
z-sFPPjpEl2A(A}UxpRioy64ux2^WAp7nliF;tyxB)M({h)#u&MAHt>GCkxTEz3x(7
zvW0c4tYZydbMNrTZg-!*<@pr^P(g!js7kL9k~h_Te`(OSJa(D$DTXp(;sXy_DP7FW(%%|brBw#Bt=^0}6A;!2>WDE>
zUO+@z_P}&0*}T>Wwo5-lc8@_JK8{Q1xJx1?NzyESVHS}mVg3yBU@V5Sju1vAi?t5GX
zvV2Ay{9wSy6oP-NH8i{liYNE4nnCT49Y#}qFw*TLF!f6~AI?G!;X>ZiCwb_6_}yWV
zi-+ZSRI3~qx!b#s14h3&36D}N>u-?h)sQ{te}7mIR97~TxAcpF-?4sAVu&%}LhINt
zB=d}Xx%`u6K&}fKby6hhm*voFD3_7rCUP*h6W*pd@6_9I&C!6da5Tak!v1po&^;61
zLio8bPt4CckBot5+0{^!S>-^EM$VK+^InQNODSN>lcRSuS?oy5*Zl=J0PMcpTD^}{
ze-o@sSE=_YMRX`CiP`mC!|HmW`Jfwr#c0#y!H|0Gq{QCZ0Z
z9wANqULHfc%*dS=WdBDo2V(Fh7C5;reP-v9B>0mW{CSAMoT4q-m%cg!%v`@3-JZ7B
z&oeYL6(|`9pJ%(aDXVeZU({$hlUe$%elRON68I>t6f
z_45}e<_ww`5kvAKD05aV*=}~CzJ-QK@l24Mp(xss6&+?>&TxW)R@^=;73R&jh1dW%
zO=X=NQHGY2Gt8;8tItehqe-iXFs44kU6STzVB-xe8uJ6Ghx4wszohd)X0;0Ce|yjj
zm94HyyQzpRD86Fc#=z^*`aq7E%64UVTAVs{R-qpT2fFi;v?xq9Yd;#QqTYd$SnM@R
z6*gtRSbVpbAQ(+rD{NkH#VmTtP3`R-(>WC`chW(<*;P#MUin=4QBQ?^pAZMfk4=}v
z^GwJ6?0d-U7V@_}{sJw+?XKz2fAJ)u@+KsO>EFNE{dg|8HtHUGO|g
zT^?16*|Jb?8;1>wvC|?ZS7GNgmXnuLnBx+T8N&j&x_XOp&M)wme}Zo2f7*rPg84?C
zNC|^OG5fy<|6nY}RJ|1*aovn!$89)rqCK50mwxC*UOB6ZjX%cNZ#7(@Wnd_yZkWv;
zT6~Cz?vddW6(geErl1LnUAHtepy3+UQ=arO#s1nAoP5rbRzUevJWS})9kOr6a&@bj
zd3K_s`33RZ`Xu@*q4--uf7sQ{9=WED2C6mF<7?M=RN613SKyO|KS!J>JKAev)7)zk
z8UA>v{gmDWQ;YPJ+OBw05hQ~OXHVI(L5s@1%&H@=ve+<6N-JM8o?)ASzT9!t(6w>7{6f9()tTif5eBh)Qs
zuEDq6qc|nwzwSd9T9*q74ZaIaxyO0((aGjOSJwBVsbmVtR_XSwS!C%}7X3c^Zh(>M
zBy0s20HG$`u!n&KDM=&5OUJ>_pBuC{i9E28pBYs1;sS>t05_Z7TSFo^_X&3mA%35i
z_Qf!!Z~RgoKN}(Of3R#wE&g_3ncxn*;7R8yapGuZ*AUoAFa8UCx2wRSL&?GKXJXyv
zK?g8_9Lw?8vnH`<)Y|X}t#)GrOqAP3d;Rx>y|UcMDN#?~W0HTif7!%Lz5oM3DfK=~
z%u`Om5b|{Ccs>Hlr)bKs%2!0SX%mr3K?s(Z+J@IkZ82E?rruv*PJKYN7?$iEBkN0X
zH*jfQc6^=uW`lx|VWonrq!V25p9i-@U!$R5beg^xDJU}I?z)RR?TGe^b^kg-mZJoc
z%smxbDjwmke{tLk?05$~vk_Q6wquodCQ1gcTpC@%oxZA%lkK}Espn{KkLz_o3*hKD
za8u~5aht4qYzEuI3%M=PL};L31}S8F0m2_sIfli%8gfd1M=ZTJi_3|KS*^>nW>20p55WQ-_K_5>#9-}H@{@G;l8&4v;O
zUECp1e_)>F*(&huEKZ-dk{i4}#~w7$fBG(lEkeRJt1^2a7mhfj{~>243f@YX03{EZ
zU(Pcq^f0y6IyRB$16|LMW~gW$>CpV%cdEjG+z-63RHUS&{mq89J#Z)7`g&xR-mw`k
zbUClPHb*iUu`?HM_n6&o@*FS(i;83d%oe!Qf67{llHezBW}91)7Zi^_mU;<{AvvDN
zTw^rH@dX7eu<6pxpDva<4{8W)1Ui6Hf}8f;5INH(4tirWy(Br&=FXpVj<~nmD#t(}
zB|GX}FBXrL3FqB1lEQ5lG)h;fJe2JULFoM7Sy7?=vdpfdga?yDWsJ?U_y~LJ-1J|a
zf3Vyg1U14bV=6+`JtCsbR8vgBRa|7;xM$Nzrukf$G9#cd7CkBJvX
z({EK8kbTwa3-9)zulvOdgHj_C^L)dB$MbD%L{RDs$^b&dyLj#V%I}M%Zxxazzr#ah
zTjd4mmC#ju?cPUehLQWwZiLfD9&II~f5p(%(RRvxbE)v?5aQEAFprij
zFKr|v9h&)txI*Dz-vyNkl}ky+MJamnuoZm@5|5ljl@ekI*-?s)9GSul^)#zATkS2{@6(i7Cll4}+V=k+J9J
zZLa&(0YE-iL~D$$5}@JgvF?f~4-ZWjJn@QsJ>l?*7BM{ef4rei_L4A&z!qy(+iStp
z!)0?HJj2O;7MWK*#hCw?vY`E~f3sailDsvb84*$n!+3aING7^KIk@Ti;mTP0^H2zS
zw^JN6kabR}744(Q_nR2qm>HKwdsheA;@h)NV%Q96cfhNVU@dUK2-oXi~m_b&}f5_Zx_?ri^
zE4|XfWPRs-FVJ(2@;$-IbYOLmy@!E`nZdx#BoI{DA<0Tgc%>Jd42@K!CTWjhqVMWzr%
zNFHKOtJI0VJK`HcCSElXe~cAQJ2w87qZFj~637OZ&U4!zRxsV!3nRLx?V~bWJ>rx@
zj`>-#yQGK6m%}yW)89|WRQJIo@*9XHs-V}b+}R@)@eTn~kJm5b!5GQie>Cx{42?z3
zCEDy)Gl!Xy6~gfaGne6!4CLY{jzPs0i6yEP!0;8@Vo1Aam%=bJf0Fzom{LRu{CC|d
z^4|1kffEgS-6n(^2pup7~s7}BNAEmi5m
z-y^TyhOc=#sRRB5`V1CBXlS)lPcXMCDH!A9_G2oCXzKuWA
z;ZwbTf7N2z5nebQe>fhey_mE6RJR0wke!&gQJy3Z_XEN)qM&9{9BwpNay@EZmS;@>
z*1Ir+M_*djLpoX~4)oL-lE#?)z;*HV8=sPcH5T3gRPnV%qg*pSYei+B^SV0dz!Fh3
zjf3_aZ-6SY{>sB?+&fCp_DF4XfXkzs#n4K;FDAINmf=}+siGp?0RtOoH;eL!U
z9t_cMd`V?@f78QWA`;F-fzo<-3BMo*!Q6_Bcjjg
zF*pQXD2X=4kgIUs!NmPynY+~M=p4Tg&ibZs6@GZXf5c6th(tG1hzFn|*q4Q+9o6Us
z%lh?OdOvjMi39m8!2K&`;p!t3THa#zU1>l&-`D8-d5@)dV=Eq^U{fTO>Od%k>jSjh
z>NhOlnuK1(2anQ=N$$?8MiCuKol(p!Dj6>7paK(2L`4hC+Xb6OI$B%|^qx4YjuK@w
zsNt^>f19GNUw#*qm6bNRh^c5Q%^uJ_CFS4s8W6(!g#x%L{DcVmlzK!|s~7ik+c@*E
zClHlHkBPITT}#zEdOK4o4Sg8uqgWwRdJ~B?ynM}m2v*CER!TEKSI=r+{1Otq3)EaM
zXUB7hplqOUNEySaml9`eQte-W6Xg&1MK`HMf5a@jHy^~PckCRrC8J^O3*pLF%1w*O##(YK2;V>?s`Eiov|hUaH>!*BLX;fX%4gomgN_?`*w*8!Az44Lf5M2l72$^5z&yv)RM8Ix>tx*wnO#3S_28O`
zmKnMaK51%=37lh~Z6r0TlvuSJCg40F&vAV%Jk+Q;=fi7VhKCkk-qwZ8qBC=(WE0Gp
ztxjmsH}Nk?-9&Bsh`0!y9l*1)>szo~`LQ>Xl}An7cmCE?eSI$xY}_`Q{zN6+e~D`R
zQRZz4qgF@tDK|k6Gr|$`O@2h-EA;5Ame-GL4;xZ`aIA
zl|Ph^+sO{&BoypG2mlopyfX2*#>C9AtaZdCop9~YkRyQz&7dMJdIc+8Qm
zqwvkfm+(_@P2;as?xsY;&`^dae@4q2n^u_+Vm`YvT;@I)^0tn|mVX%b(sMqz3MscR
zk*yW(!v)pvIDr{WJaStgd6MTW_3&?wVv{Jtp(dF4EnMty!S?t)m^r(l)bIR
zgKjvG-70o4ryDDW!R~ri+cu7DPy6|61l2_7@v!lg$o7iWD^wfF$K2D1W}M#0woGN59dSEbSKdvqC*O&Z
z3wNfg_E+jHGKBn3j+T`}OfU)eBmIKW1p~Q;Nnu8g5pjGbf4D6XfAOEZ(oNuGFeln^
zM`mm4<07P{xw*fmYP0edxxpZzr-pZLn19rC_C)wwedenqK(H7JSN*cA{V7O|ci*32
zuf~s1@4HzcCl2rjhpi(07yD(ZvYjlN=;%s7SH;Hn(J3-7h9b6A)VYC%XWQRa`=fm4)v__-;F-S|Pze^v8a9r9i{sHh4K4Tj(_
zy{yeN>1YbY9?iCk!5Jom_n}HV#RWu#{}v@uHVv&z9(?`nKhNz@_eBJ3-fJS~N0-wK
z9w_}aOLu`<^u@o)Os5?MwZ_vHO@IuGZscufeug#XhnI;wcuHC2!eR+2u8LCj{2X3m
zp|}P-5nJWTf32yhSv`PM*B4CwzN9
zx0s%0)EPYq!e%k7p`er4pfp_1=v1fIIji`bxK<9xf0@^thPyx2k*6mK3e77u5)q*C
zQ?fOJ$soW7%zp5tP$&^(<<83|>m9Nos9g}-N`@|}EkP&GzK*yDVN_p07N`&+tu
z><3lS1v@PUMtb~OofZP%ZVukpuApIGz&?Z`$*jZ0tJUhJ!gonVr8FIGIjnGc8)PyJ
z6I)A0NkJ_P-W?(S;{`{WFH0yg7DMKCLyfoMx~p{KYuVvEM1&wv>2aPS}BtdKFH@NXWu1j
z3^$_N)|9kuf%>Q%7;uGYO?Hrqy57)l3K&M?
zzfPyJ7%?K9%J3&z;vcnla<#Z@iHsqvukuJAtZ`M@m*$TZ+U{}=d4RKHF`-MEiLI7O
z2Y)u&Tame%4#frdzf^Q!d?8eEHzC8%6&_QxiZo^mY-U?q&#t`%YO_H;7XVmcO|v8yUzJsQXOIpR9~fuKKP*w
zWtj88R`k7|SQA>e^Bk4)?CTfEE8kqnJ~6&@1o&EunIM0)nzh8X<&PZK$(FPCAAn^m
z-hu(FpwnjVo?EaRzMtD6#q^(Qjrl?SBmWH|182zozRyXmkbEgN3JX`~4
zjgod?%E&u@ytKv3oU@hsKYul1p%e1^@YUtkDrI3;XDPn!5?`g&-$9Kn>D&6t`RsFN
zX$qi`W`z5H2;n_XQ}Irg1e@M^_Vq(;5eMM0vOP(%z1A3Is>EJ&e-a6+4<^+a!h>zp
zC8a~moxXyJ*Dtt4mYi_x()Ec)r&#b8rlc~$^?mH^H3<_(zD{b)QGW=!ueHDFC4s#B
z$UWAb%;&McG|Q4-pC{{FEgLkzr%>(Pl;wws!Wc`ripHmT##CYSB_SM{m}Z^cDzYw<
z^vjrcGkUD>`lYrqM<@BJ6zzxC#3Rh1Q-gGwahcW7vG&o2bbIGUbZN&*jwdw5jGI8`
zP$~L=MG;+_OHS$5-TjtivdoI<`
zp7xrl*}%KOR7xoE7swTJd;!dxEsK962mKiKpwy|nLe!JZ`7dWvYv
zPp#(5UFKUz`x5xP6K`pc_1%VB9V3yI*$v&~IG>EV!A*WvB-X=5LZvy1S~{CNYbiEn
zV)DxMsOU--K7T)on|V1tncy#G?^jr8M<}HCb>!IZ{NO}{R$b@+hhxq?XHCewfppBfS&aGS!dwJsB1x
z5Tpt9)KYDq$C7;IeckaLE3qZcdIjY}-DOm-Orw|HG!o{7Vzy^#ZYMzk^U7bN>6vvAk+r2q=^T7nn$G9s4wDLI@usQyhTBCzZfV-9k!
z1%|EVcS|A>h3d_$AQj`RsNgbd011J!Gc7R>znaMyW5SA!nPk+r&E}z0xaQ!wJAdP4!#u$
z%}KbJ0pb0LV>YNXx?FL)sJ7EFC?>BKmw&mr2qbxRh`D&}HIUprr{FB&j(sEIx29Va
zNh<||!Oj
zt_)0cjgmw?XM0GxROpmqJ=G2j+)^7gESjDTb*0o9MhI))zQz8rmFf<6+Ah%eO@GB!
zV_)<(t#Klhl{Rp%j592t-~g-iM5Z>U;6_I#8WX`$*zxHp3Vf)82wukQmTUpPQX`
zf-MD}#8ueK0}O?~`x(0rRJtOW(|@N#aCD!12U}jP^iLoy$s}-sp|()DQCbk{(1Xn(
z$Vh#?-en0K9QOZClt!fCII)UbY=|6Si{A85=IcX3n(xVy(A60^z+e+aTzTu9WL12Y1O?83B28K(~9
zG8ubDu$O_6OUCLOp7Xh-dw&(6HrLZ6gbNy3+n^C5R*=Ak3Fz7_&gRofU0YaKPea(x
ziuWWh#2}7b<&_*uNE0M1_9nAcjdJnOXf=R~MR^A%;qy_qXRVoYa6d2tSGYBt269(2
zJ8$JOe2=lRduQ(8iXN47v>A?~N9w0`STuVEejI?I^!kE{Fc8KFj(=xH*Rsgeg3d_@
zL0=32Jzy0A^K;z3R;8q=IJDZ3!Y0YZ
z)rT+{ySWf2XRHm>a(}0G3Bg(YR(Y4LUfr5x`Ap?$Wc;x5-TegVDQru3opN^)!_?9Y
zFV{oTl?(^aw_~$jDyre2toxD8je@mw5kB;~)X3PY2bvb@sqc#@)u$Um*+1sFddpE{
zB=|*otGcZvs@s@A$@}r%2Gem%4aqMpmXn3skBiQ=mgQH)%YQb9+9>3W*j7ykSOh^e
zOE3{QBu3T-63T2qKjkfFh#hQ518osL~(>M3sU!OltNlai}1Ma*v!;F@-?9KmFvlHMpm
z4yrzR2KPL9*~V=90z1^?>;fU()@-}yVtI<~FdWmde{g)zM=SVp8eBWM`=w>H4%&aK
zqFV@B&q~KEt{e|UC)kUvt%wOfCzZ-p>@zFYrEOhLTD{+d1zqtaZhSb!JN5n;XZn`
zIIJb=rbX*7D33H*s1bVUaq67gSVzqOK^WXoskzf+pJ5CHCvzv8b$aTZjf-C{QNUS-
zv-#L9&wrT-_cfK2m%f^(*XQe``akKa!LsGYHz`lL?0f2AHDlyK8XuI3NY)@gu4k=f
zX~r?t%uV?~#dL?Gznl$$TWu4MNHOSOYq?vT4e_NiigHK?d1=vP(>lh!7^DZbf_cf;
zTJnAzdlX<|H|82+d9m8+80eTPylHK*g3_&3G=Ehy6r_)_);=4c^pP5f|6JrBDr;D8
z2=Uo&QlT|dS=#s2Qz)hZ_NRS34j8J!y@_#ranK8NI`AuD
    *J1NzOPXIj$rllwQ<
zQi!3A$(aWMy8={0@6VizbXsq)^9=a;@|Dz~59{(psaITP2gDpnmAtp^1eHOu{C7&bvc+r~T}&$r)t#^hxAI
z{`DL{9yi~qdFc4!YdIiKa9ZZ6igiOJ?Lx~j>j2nsO#db2+Wv4T!!|Uy%yAQ_Z_8X#
zCV`FYkn1hY_A8&)DyUuB$4&$_1zm4rGk<#kW7f?w5>hDf%0;XH+B_vHibpb?-GuoQ
zZiFA?;=O(Tg2b{9w}qP_uv01focX4;?Jo3M0z+*|?F+n1QgDqKgZ~elma?Ae6$gYt
z-oR({Qn5iy3ci@;Qq#*8w$Kfg*6zWls81aNG=^*sgUsuv>3iQJ7J)K;2B>BVcz@%m
z96Zt2SBe&@#)??hP~dH<{XaOPvj@vPh`)?f)?@!j`l_Fy%Uh9x13f!g-w+d2^adG6
zqMC@P6niDD>1b=DLvhq#D4{VY4NWA=h`>(uM@`#o6{RVZXX`?mjwgp60G*O-3=!Kmwyp8TRR@EW`m9h
zXcUX*pCUg8$XM+GJc`M*<5`fr$ft3~{3I)2`(YWNBcJf=>FH;!!yQC?;?{A%U)p+y
z*nW*#+WPb{KHCJql}g|XZMKSLpil{lW6FLC{bZl_nQUD@J>6foK1`j*yuF43e-zuA
z+oI8ddN98dyu0O7p??_XezA7CU1BSTdi0gxz@qdSW<3Eznoh*eT)VX(vNfAZUS9g&
z@w{JPnN-m(xwU>!n+Q$qk!t9*71*X(k61Y>ZiVR=tt_X`gMqQI6J-Au-DkaJqc=9;
z1F4Xaoli(GcMeGta$=D!bqwT{LYGh(Egm%~dxt4Vqx-W`wtvrvc3_+D3B`M-{|NT_
z4ke^8j&VdrBCDKuWLhsiSL^IHm`RFV4j!qW@LEpYAF`IPArR?$<(2;J!l7#MGX)_`CZfC7I5=v1(S`l3_v<4{iwdHugtjU>J|*YwZ&Yb?T}Ov~>G}?o
zxCW{oq_9=bL4WT_3Okz>RNY4*R>cQXSGl`K$p$ify@6-=t-+jqwYYMadM@$Cv@P`f
z2$P6ks&3&_gZCI%r1I{L&kdeQN1^pFm(R*O5}T7%tK68|1P%Mb-bUnHwst$6TlHGX
z8p>|S_e&H6cTIz?y8a9<-BLoweV1VmDG0fBd=*$m4S(D43o)&q?l0TN{Aj}&y3z5~
z6S?Hrw9bjc6*4wP0CNe>J1A_thS){FGYk4!1z1GLHJcATTGqe&Js?{PcYK_rebf9N
zgI~}0HOq*1ZMSZ58}%i$s`EU`>eKBH(zkGJ*KdOUc>yDaijK=5bV&6RHA|5eS<)H(
z#4D45%zuvjILe2P-M1T9{qLT`BrKN1y><#@cPkI9!@oE4G)nvf7kO!IrmYH;!bsQr
zQ5Up{amcDkCSSj(bSQDk@v}*D*c8T`uB3++9mlL0zVR@3ds>F#6=4I93$dmSED+w#
z#)fUv&w2vSF2%4WP;wnbX_82RB?p_91;J1*9e=k&E^I2ll9elQY15ponaY)hb;d9^
zzi9A@rar}|f%-6e~yU{tHE{v4OYVs}+lZgo#rz>a=wEGKXz05t8g++xk~
z7Js8gZ%TwP%xwZwfq`R-s-Wr(k)$??8+Sj;g#IUi3F3Lx0mR
z#F)DNc8PZ=sC!~Iu9Ns*nhmwzGY=eeZlx{YyAouW}6)x{EJK_V_6Si3Oy
z$3BWg=rA>l&q7GV-Uz`Ey?FizX#ZFOc3v@63;3O=)K*WyAsKuD|S%Y-S2mE2mJgiFFR*ww9Cr`Nd9>L@6F_L3ld2Mf3UgS5$c+jDLhG
zWF;@9AKXzMXj~>wn}DAnG-@s|~MK(UVUEwzwD(;aut;?hVVCL2n@3
z3I`~YoN+o)M}gL7l98qsnL2QArPc6o;}j_TTZ{ZPcB5WYx_^L{_sG#uevm|L<@c@4
z(}z8+^chIxg@xxzvgZ6E1!8~JSoVkNL^)=fEceZa-!w+JUYLs>bNdiuLw~ijAl3Vk
zG$rS7F=*Sf&RVG0bL{^M*(xU0JK|!o(mK7vC#MfCS=ETBY<(H-UOqG$17nyLo>ZF=
zCywr@Ax;AvX}aq9EXIRVR`zax2|-0`vYPj6}yDRY4Z!cK_6l0!MMe-
zmKKl2<#lprK?4hu?h*A3{cu9?i~^~}Np9Nxheky!d&L=#6ZUl^gn#uWefGP-ilf&r
z=Ykzg5}wVqcBQ(M@KohdtSw@)D~L)3G9TeMn;IOUZJkm7ZUxJF;R2DgCak@yABAEO
z(!E8E@q%bpw|7uvv-hh)TrgKPTeAfbS66U%ZKmdM3wNlCz{)}Ju1tLAXy6ZFI72jE
zYd@adGkCCY>mpW(_Aitk$(*Hll_C8y#an_oF}wX
zDN?%KM%Iq&;{RCi*ZTV71AweB19tCOn8yeI#r2-03`2}|^4*9~JjgnIDe$`4l>8YRpdius<-S+PV{a*Gqeqy9#+o|igp@|*5@)!jgToF~}WpPMc>
z8NLmcFxK(-ox(un-5y5J>`GL4`3nhgLf0`iMT%%>SbuspPF>_f1ZDbX&^uQMm&c_n
zd#Tv`q4jU={rMzA^b?cIQx~i0bI8%<3WB^y^TN>TQxWfVXHXu~C_9q5Ge=Y+=q+;<
zHX%xNoIx&iuwx>(DoIDVnG0mo&Kd7tyB0pwSqmghxjpvxzg(X1`wdmQ4AFXknJr!4ok*w5R)`z3VxwEl#3Iu|(~o^bJ3_a}v@noi^3xw{zcse7Sb-xL?t7EU
zN~DnCc&HA?H-CP;CF6NXpSANRge!ap?cdC0Tg6-`a9R;W@N#1b(U`Z;S3}Wv5xC(v
zEif1EMuu6Bhg_U^F0+~tT5-ZG^MA`g#5pwomggi`?1&S<6fbh!R7x}wL51^$
zI)9)Lg>=RHc)kvX73$4JXE}t{3~R#3_z|pM@THJPKAGRCeUKH2#W99AC4@bdS~a%`
z0P59#2_lY?I+3#XdIVxWoL#<9hwG6d-Xj^B$16!6_(x#w>kf_*l6-P*ao$b_x)%r=e8YSAMpY!vW!WR)3cWwsV6Fpe0;P>yX`o1_fnjA3j`LnF{wN
zigLZ9Ysb0@qs30%#jrpC%+Ehcnaw=}VatZNKJ7fdgyGH!yI>HcnBHIc@nq|+1p3s`
zqnrP*2a*|EA6A64!XugfQovK={frGJ@k
zN*WrOpzek~W*wsqI@BIecr*0hVx=nxluwaG?KoNh+*@`2B_e~ZfP$@B-l-G8fbL_<4y=;`pQiC
zoy?*h0B6tRK|dlp3QQDxcPC&zW`E~-R00M2(0x`zX7rW7kKzm7qG>YW^&Y78?bI?>
zF0@qoVB3QzxXvyyDgA>;{xsiZtZ?mgELM
zvO5eDG$S0WD>2Y&#dC;v(1fpecY6dQ|AS}en8SxbA6;D3C_EyicJ
zWm{VF6UHN<-xj(t9FjIrA14>xt5S74HgHx|tL59fsw4=S7T9XYywixy;iT3z1m^I%75{g&a{!b)l&TosS?+rE@hCoGb;P~o{op=?;_FWhn`!I
zoU5A4hqJGLYQhRv-qJ?Dis#Lpu0x7mI&?H#RDT@AZfr=_fMBp~br3lL9l1wB&#cPk
zfVY8$*YlChk+vc(h`_x6W*~?Znb+q~JW)HblQgLTB%w$ZKFxk|Q-6lx(S}GMxN2sT
z%q+J3nO&8!{?a;?Mi({Bn#?>zwRK(D2OX{+Wggi}OxsPxAry%Q-{2y`!tiV)DWh=k
zaHM=hxfs*ZMfjuDCqau^W{vPFiX7z53?p=Tw>qkq%y9JadPwIaXJ3+(mgMe3h}dH;;K<>vwLeUUC
zE#dRGi93d~e;#_0kT4wVrVMc`{YSETSlX)T=H7uTeWz}mMt=iMmrxCacuvu6PxIfn
z!Wp>*ruj=f@1TxF2!Y3P1e*5kbzBN?Wk%|a7C4R-A4hmZj#;~pxFRMoJg74SLt7mU
zStPILP*Vky(E`xsvwnN58nSl_XaftR@ZmQ@2iP1Na_Ym(n5lE>vHP9NKX3nzwf?e0
z+{K!N-R0H)nSWhEpBP0B6zU^M-7ms7#FCx*#!)sAsqPDeHd59HX(#E`7{?VEao!Gr
z(b-^wF5a<5aI1I$XV>_{V|%)4^8hN|JJ6iWvxFbqK)_u%@EF5uHZ)vNF~269$hee2
zHk*%n;$(Qw(rDw$E3%<7F1X$GX_RTxxUJ#0_YH`4F@HvxQ9`Kn3a+tFw{~0&wc3J%
zA!ex{EN9A^^qbYFKth7ouV^QfS96v#+g!V*<($xEH98H1BAg;7MqRxpDH?jM6zN7l
z&AsYU$01Y1jm9fOUM*llRG07L2(O@EYEWhNN&bQ6v$5#V?7|WTyMV9=nxp#xU62V*
zx0~@OKp!w;TjVouJl#_;#eUx%P^T%{iGT1fER6WPaZ6AZJe0XQOprQd=*Xbq
z4}>cbr-}7l-AmCZEdT@2N%f&2mR$UMEVd4BkHDKvx*&Osx(fm%_F
zlj$0XYoag{N{ywjCsF
zTF(ZF5%18?vS=`WOwq+_FcXEWO@-6lYJblUAAbWU@UHtzyXKb_u}a8iaCXX0OAqx?
ztGW^VmxbG$(#(jL#use;9s{z4ufy1<$Qy(3*-+Zj^XhvYCC^dQ%`uM#TAh{r{?V*7
zJTcz>#Ad{{s&;l0M$$%0XRfI;+L))^ZM?}y;BCxOM*`VI6tVs1Iml)tWK{VG5sis13`M0GpruPvye2rvDB*9<%_lws@YZ#TNdOG*#O^PPD4as2H1(EH8gKxyM8J=su77^JubNly3o^hwVT^}3V2FRLz@v8*L;D=B$;aTmRNDnsUumOx9
zt3&%tAzYPZoqyNYAbOdWV#k%QRGDE!_3?%hAA2}d{uoW9e93Z?HhlxzG|0p-DTNeD
z4M`U+35|eA%dxYWPTgi|n+UJ#J57+Jb(Gh$=Bo3f$71wQ$7ya5Z$vL
zdrM1YDbFtY&dEzB>^ryA(Z>wlbCn7$zk!_G^QWyaWk?wbgdo8ukHL2hcwvedboM4&
z6w-ptr-Fx#%@B69DUbH$v0`Ypa_Nn2I+RC~l$f(G;OpzX`yx4MX(31_+Sxd)(%Kyo
zBFsJ`n15UWSO1$eOv~z}U>{YpHNw?MecRk8+^TbVos$U1G)~WS-YtMO*!W7O#zWgv
zQTGop=kYk{j1uS0KXOoDfLQO4ti?)@Wu4crX|no=;IdY@isp9x!_Am6zIU=zmmxCc
zd1Jhdg8Z7VS&WFz@}XU86S-EB!bW+}`^!ALR)4<*&Zos73tA~FLoE(p7S1dzJu|R7
z%^;VeI8#KUisaehFFNrr^+k|~&vkpnIq9|!j&8R+7cLEDk0Jww$b>J^FQ7&e>MvSs
zihqeo1vW-8B;WU}z2o3NaAS#WJ6DS)?w(s;8Y7bGNl?>~NGj>_uRz#fzGX18s}HQR
zLw}2F`Q5Dytfn^I$mvQaYai-vthLsk0k!ITe`KVy|Ln!Q67L;B(E)O+d`LD(tpBClSIuZ+HqKpR8fIYf4RPiKyxC6b*bdu
zxXpj7LJ5&o=P-MoQAvVj5ucqeKbqgqKVn;lGGsld_a`&A^R!P$=Y<*J6sWz)JwOWc
z-a!MF(RtcT9)DbWTs!C%$WbHNE{_e}c{gi8Aq_2!AIFjI
z{GIwVZ_afWA^Ish;$qS4GV8XY|KoZ8K-Y*StWtIEW?{tMasg2l*wNAqHVe{z{dT=649~XrJSpD#%Nc^QEi#s9kFns4rlu7
zU(W!1QE5M%5Mq*kHlpOF^GE4e230Ys2)99;QbtnMelEFDRIS5tn;1Zs21olI%^;L6ko;fhJ=L)R^y!I+;g6dFb&+z1tne2Z&x0`}%W3
znFmd-76v1I$LB^FYir+k7UJ-NRMcgMr4DlikJP_ivOa-rMY<;?~$ejQS
z;^1|N^M}H&qv!tjfT3gTKN|Ljy8BM18_L7yT)Hx)H^iD!VQRX)za`@^D
za3F%ec|opWdw1A|vu+(xbAUCizt1*hmi6P@OYATcYm2%lP7quH!e|4VW+B`-=jA(GF}vc7Lw`rdZi(WIElU6FP)7k;
zC;e?Bg^R7s4teD6?J9SO^iSd!0@Ai|CcK`06)V~tP4gdX*g7-9ab+=c_W+P?bcACD
zFPProI8jFMmY@*Xgc;Kbup&T{^?%7bmm_u%!Bnm(val(9D_NKxzbn1b
zQV=fJLYE15Zv@2Esv^A{U16D+q0?{;&%7%)TbtvC#t-C28x_lx5}
z4+jmp3;m5o8NLP2i;*Z@a(`++e15)8`{~PtjDH4>&u)QJS2#W8+G7wJl!
zK3vD!`Yqp;v6S$|pr~Rrur4f_&h-ocv=Yi}~9e@=!}CJgGht{DY%!Er!kjlfw5~B4W~=6<@w-&qN6Xbbpta
zE)@h+^_98N!1n$2{h3>?HBSn-LZ4%Wq=|rM8GC|0*Qjf~mLMkPARRO*5HYb-kcs?u
zp*xDRLlvd@b3`E(BU;MBX96(Bc!Sp4^?g)wmw}{>4(d6gRyE}emIKWpMdO3jHF0PP
zm0>|}75g(xU@ffu<4N5TN!e9K+kX-IGBumKr}zYGQp-!?$ENY6)AR9OceDQk%9+S#
z(#exL|0?#jl6RZ}=Il2sFvJ-2=!^x?M~pl{ICRbG%Sbd)cji%FN!z
z9Ff}ZhE;$tmf>V;KgSmUAk_ZsBL}wH%ZJPK_R+M30U;(Gdce~7{B~slZGWKIXa5+p
z(Wk1?F)Anz3Mg(!IAe(ThO?!Q0%}>dvffI|<6*F#Y;c
zq@dmJn3=^Muj8#_ieT>-8>%0{j!U+ug8|an*Sc7Rka%1+GufORC*^FxS5^w2>%ZnF
z-~4Wnxx~p*C*r6Pzg+dqbblV?=`NaN#h~CMsld*10S?%Ur3Ex)-AF$+t7;aRQK)#*
z|HPx9ZQ?J%{ESZqR7dr$H(rpd9l4j``f``67qrqaf1-n*&KYf6**on+K5k!W`hwCT
zW>YVR%#FVH?!8i1X5%WLLhQ(z4fU{D-)jFOK}I={?z~KXwBg}G=6@*n(8}^@H?;aX
zBXI%i5lVCB4ca+IJS#@Fr``n++~Zb?d>RUw+w
zpI>~Cqux4{7H0&@A%9`9_Qt>GK9Y;bi5wn-|MSnMft*)bC(S5Oi&X*vJiwS;%q~kD
zV;=yl3qzJbh{@&3<$5v_X|)zv-f*1JeK<>wbZ>;_it(Pb=wIeIrrKv{^z=ASSdofp
z)U0L;5${Q>>M@=PM4aRi`qEeOggG0!nGi)wD;otD6KLfQ(u`SAsPi!
z5pjGvKzR)2K!0Iy#l;zmU)Yj~>7NJQ%3E@uINSK7*3vKpMaeU%OkDLkdS?*IKxcP{
zcVbmhhi5^?BUGbY>#OU-=a(%&P4FY-of2@mZMZ1pj>?c2u`a2A8k6zT>RUSjlzi1h
zc*3ZTb#p8iaEnTU9c-&?nD^#hS={&Ii}^YgI{<>%dPq??e`o{zv%-(-@U1_#{Fippd)ro>P>_scd#x$`_ULrp|7&9~W07
z);zzv@7D$r^s?-M0z2eS@J&TuAzpH)T5;n%gMSP{b#u^aBw4UO8N6$c+LuR|2Y%ir
z{S;R8>RFuh#Ii4C5KnCmMkt$|{_%Alen3B8k@*qb&(6}-ST7|^B+VM8Ldv}}OJB}}
zLG^Ag%s%Snk+(hO_&|x~;H7$M)g`q1Mx9UJhwRkHQ$2*Q@`kGbh5&&_;%*Z@0VfCH
zU4Om1V5ci$~$c5haHLeF6-T5D(DN2aS
z$|QlpIyBqjAdtCys`VEYNp=pH>h%UbH#!JR)`FeEjRrOtu@~hxo+k>{Vcf^hAH3Q}
zhV{0mDex0PHe}d~D)mOIcxgOi^1dW6MSsKcF{g&z<37+oDxIMl{duEQe?kW*kWMOx
z4&HokJJHT$9D@MGk`xZ~8Gj6H1)ZmiXu{Au!ONATzBW*_FP3q%j4=65#_m~C)2bjf&V`rTKEq|Z4
zc2oapfML+c8H8YusyqIm`Pfg5*)5b69;L3L^Z|s(D
zr!+TjFh$=9<)EIC^WCOZz!%Kfi&JL^4Jbp@N;Nzt=~P*{xHQS5ng{c+&$s)74nr>1
zgsA=HsyS}GSZMiOyq7k3o7#1l9Epd!&TwQgtSyPO{G+MWCr$m<7u%L_
zp8ZPjCeqo?OcP?u3FU;aCVo;!NtokYmO)}G?^ROUs#BAy&&(%p<5)ZhG<(ZD2*o0`
zqeaD>JJz}KyGJ{i~)pq(FbKK
z0o{2hPw%!`p~s~>rMxWblCBwzy#^pyxY|tsWQOG-Le5kIj&&@Ew)5G^NY^XxZob!9
zOgy&`M45xh&D`>ye`_SIhDQxwCtCV{a!$!}Dh&-?oM93JJp4rGVSiT{kGH{sW^`Uu
zi6dk!LL;O`nuLp3=Ba>rx?;+#3Cd@AqdvdC6we%pJ@utV_^;;^AvGKVLL)Dwy$Jt%
zI-7=dD)LG&<_6kaUaYOby{@_6d$e4f5_R6x{zaOCd~X2|#7Z)C*5ckH-aWV3tPZ47
z%b?v&VrCA{;3CMiNq^2952M?g0PQd(Yfwj@*0xLVI!58i+KY7Un8N}N+RmEU4C|z9
zuh=1uIEXIp33q_*_fb+@iss0T0b1#~Jw-kS@mTmnvl|!8aqpklxPj$DJdcWvd<_xI
zGI-T`o2S~Ilq((++9T{z!Gp)!^=p;^v}ksQ+X5?@@@!v19)I-bugbch=W(*zr*oU?
z;XFFik`Z&-&N8>oV*!+wEY%@2!!qySpn+rIoofdO@Jf
z-Xz#+nW>&THh(?tk(Tws5>+He@%almfqL}F)VG4E#!+v_n_s%}S8B(nSUNONAKWZ?
zHHU1l(gg1Y+XQuOz*ULD{jyXzDD(y57E3vFS?+TlxnIEJv@GI7nTqOeShy?RbyaVz>gmuyFXE%az=Bb`SXSmTY{T5P?z
zWD4$Q&t5<+d}#gZj5c7j|I8xZ@KTT)o-bbjJb_wACHdI|p+MKUr(J?T#7*XT?<~qB
zfWz`9Ie$jlA*%-6`M!<@fVb{F&hO`!@|q&0D3T)xz_wcu_pz+mm`dv*Y_>Rdwu9!rQ2OBihuS+nhW7AH|9jLgU(A@LGyX0PC4K#
zn9m!#c?`FGyl5F}0y2>rK5!8h@=PI|vG+l#^{NiYBy?E{Zu5KM6CA=4Jhx
zz(i;H;>n;mRONLsK7qY6Aikqkhl2kHc2zo73?-9GiCX%oPahs=v7E?tYSEZ{>lo`(
ziA|8mh<}rb2LlPJEJ@I8LOs&JBIOhp!}7`E#kkj8JefoWh7sBOuPUXBEOVUS;(z9a
zR;qayyR@c{f`Kykg@|5??(}9tdZ7gEiLe+1lF_(}(@sG|*mtdb$3lv2ZgpG~>hnJ4Hx&=Q9
z>3MCAr%rQ+@8|}52_AntkpF4qH8r)^wu?wS^uV#ZFJs~=vfGH0li|R^>I6tl)}oa2
zoCcl)rAbGL(FUxj39q_ZOuj^T(09Vw+1|yXjNI*d^R4I~>C^W$=6aw1unyUKK+M;kX6f5a?=W~hZX&Hx1fEKWbV*iJG+!!pzkawdC)pcWsd@0NQiE1xr3
z1VEvsCBN%DP1*-r9Or?FeoyE-e&D!ezpcC=wzzSGJ(SOJ2PX_DRt9bBH
zaX6@M2QXxF^L7i_JSeRG7#tI<;EuE%kIf6e860ns5jOlN4Q|x9^fXZ6?igrTs}1Pf
zM=eTP`8?_0E*NmBi4wdYb6q+Xkodv41AE7$1D9$iwR)L+0*^24_RVtW8;K$X8h%{m7a)2YGK$OKp2H}4MIKYE3HQIUW8+Sy^&;mhrJ
zm)Na>xgcHpaAY_J{al#4;cO;`J%m1{PP5}57J7_|1`Gtat@dKa$V_*C{av9TTo4e=L>uLes6-7HPgR_4%e%fL@tdv?7h;y
z&m5l5q3(5-)TmZ;S)qRcpz*sgZU1h^qJDkg^dISAtr>q5#2B3XJ|0c@LtbReoAKEZ
z*6STJ86YM?>e7@m-_zG*!wogD1J^O%Os$(Fi4SKK>vl2|>VsQDu_=2dgQ>uy#^8`2
zSzRQy-t-4=>qnPKP`!?=+bR!77Xe#iOoTM9Ovk49b5#sRV^zeS@NSWUvA9YBMSO0FmE
zwOP-Bi;7<~FUtXjNaNFaKyW=q3(#5o>8UcO5oT=Uw;Gp{gfVQF*CznlVh1hZ+8h}5
z=?z-JAD#4|X#0OhAynvq!{gkdWW5O|tufRHaZZkW0d9TND0#ott^S4xhtmliZr{#=
z*x2I=d2gI$l3-dOVCSrTn;&i$6D2IzBSJ5>+DUOX?@kCzL#D4XF3huzQ*X9>Rl^p4
z)NTv|H}yz}FtC}t+wO!9{z@{K&AgeD^WW9543VxgYaLBWgpQ7TO1xk!m*5u7akHu<
zSLka_qzLNp_k{AsGHcj}?}Dt$rdwNzLH~JMt(Zc$Up`4fW3Lvzu4P?Nvv+&QLhq4Y
zSYl2)f-zFVmyy>26qj6?0T#Eih5@J&msH0AC=W3?3NK7$ZfA68ATv2NF_*!+0VsdX
zTWfDzx3&GQUva(!Xi?D9$V>9T%4bBX4FxCudFwqSr3|6Bd4%X+v54M;Ao!*tfx*>qt
z`XMC0XUh;x2JsN19h@67`dt_bIJkc}l#mApmRh4T6IUgH;&_%fJe%z&XLYpe$RGKjHIlfgG|?KbU&ge99YU
zT)^77setX+fCXa{uAtuySO%W)o3QMN{)9mhd@|M@0Rs5%Merg2r~!xbLo$DZ(Og&;XDOmJcxj)ui`~4itYj1Vh}gdQjg*d|>KM+BrNQAzYH&^Sj^>!
z1UUnmk_%g?Xin72u?B^JqEY!ci{olE@Hfy?EHHCK<}%2Q_*h3v@Ronmic)e(0BrAz
zF9!8;P+!eo%?I^lc}{1n4K@KhCsSK
z^-m2C%l~ncV}fQT
z9Y)CL{QB(d%`QqOlrev%(Ey$aq(0;^*(sD(m(fBiHb|dh-vhCF3eo0<5rkWoRchL@
zPW@?h(Sc>1@@aM0lJTuhxusvjQl^1gqkEQ7u!%Y08+mg%xATYOmxBU7lm;@Ag}4#yt5T@LKQ*u|UJg?mQx
z5GG@{N!nsYGQiGFqIHJs&4e($6Y|~-b#QK|=q~tI;<_j1@s{t2(|k{?n5#$U=X3CR
z&77a>xG|k{@2UOy?esi)|TwMK+QnT~%m(zEr7u>}-
zzkI9sq`o=0yuLd5>`VJu)}Iyk&GpIo>fzP((Yx8@UytVJ$ERd^^;_&@E&Mp$x5ll$
z6>%_?X|#VXk1p6rv^bB9VIr+XEcV(=@d>*UnO-S3Y`;_J4F&D;q&~#b$R47%>6YH6
zJL&Bjs5D)!X`8HU8TA&qZlkq`ufJYs+ZDoiUfwFkT{-BMlf-?cQf^5l-BT)gx>WKO
zuM7|FwVgz33Z!E4UB$91vzTNpeUyD*)XcYzz}bJ@5A*CNW4=kc%m@QKQTdO0mu)!y
z3f`sM@-F3`Q8Cvwf(9nHO9o-1T*HGey!;Ee1^SVerD6vUl%8{
z`Q_|=7Xt6PwjosYqYy*!@Ov{mt*8D?o@@kLHp?g#a7~g)3SHr
zt|yA_5^%?XVwae9B?%9{gfhC^mcg`yz|?;$$_=dY9fYtY&U?gpAJa5q2Lt5o_Fc7q
zSANz*+*`=@TihL~!qm9-Iwn2B6~9i_j+QS4Ml;lUN@>hnN~5*jUG$V8gfTX5G2&3O
zzFMXrBF
zN;qKl*0ST-mnrCtKcl@L{qiRj!#zvckBT_7l`Un~T-)sukx+oIKUolKWeW1wu>541
zv&=D}Z1j!yY7UvPmRYbgMjjgM0&HtCq5gd8Ny~r&q(Wm+x$Y{BCn*&IfciI8b!pcL
zOXl4%Vlq4;GeE{g(aild2*0xQ@zRIG~08Pp4c
z36#1?23cVVTJWs4=*T8!>RA%Vilw%omzqb)*et$~v@30!kbF>nLR9^$NS^BMNVbxRAI{T*n*^Y9W7t+$JXE4WNhPJ{a5FG*Z#?_rk9uSWNnv=
z?6%x*LbfN&!~!?!P8`&lq5)W2?V3mKo$rdeQaEvtt_Z&>kgm6GKNsFWd+}
zpDra@EhRU7*aW7x;^4Y2tRvYpEHoEnYd5ONa%O(1~?QdFn9ltgJQCp}T6YaF*r5RGZ;~sn=
zL|$C93$6vCWeyUxI7xqg8Y0voX{PM>C8TV#{G5Y^Q9`wA?j*1`S}UBR^s*&eah4eG
zB5{g1Cflt!*e&C|rw`j^V8Kzc)q!LQq?Apz48jS42`b93#@p2FJXO|i4~DVmH-*5*UW#!x|eJ%*QnI2k+A|Y
z!QmDj*$GF{QXj>sWvEPgCAecF2i9p~XemZC;ddDEmn-;W3rW4I~
zCy{0K=tdHfWJG%?d(O;=WHcaPeq*5w2I@rd)_M(X4=F33T4pP`L{|pQ4c2-pK?Kae
znh=wHOEu{+0JDE&0yK-zD4hTwe4uclzz|eqZ%K^&m8NkP4y~S1G~0Ijk1B1xs{ePbo7X8PM&l9n`IT(jrdAZ3JI>8od_tZRfPol7;5HZq!S+=`>MOFng6
zKVuNzfpLv{rIQk;J2^>rO;MQlFtmCVrqFDF1jfE5B2r_tBUam6=Icz3ze_n16ByPa^`9U9V5ais%uUu`EgQgW*
z;bDz=4Aq$jPnpxPWSUfX8f#-2A!!4Q^TxVMAQ#2d!D(-aZTF$dX&Rv
z4n2U!$N;(M;`)r_LtcJSQE{U(3jmh;dWMsABb$_WWgj^ZY?
z9NUr?=ag6HPNS0-{=_y)6lFJL8?D6Bb-9V0uS}{8w8+^o?lYN9lsWV;iBC!`@62)-
zt>-qLH`z{4$FErpH>M3oj`irj(JQf?)9cFzQM-BxS$W`g$>VdKodzO|@*PC{U<
z-}OFZSyq0186>T2Efp`h!^v{1ldZ1~o*GF{%&RJ}=(@tNh#
zPc`<6M?DVc)Sjy>+PD=h`|FL1mz$L;nRQ=9i8cID_CZY3h8n8ncv8zC8?Q0TOTBNe
z^(;G-q_!S|I*N6lTh(gML(0Zc%BBPByD~}7KnlVQ&be!1mKRVQEEn`lrRfjbiRV#d1
z_jz$=U@C`DTSFjh$U8|(MHTG|h1y!&m|IJgii}&))_xzVwK@qhX``PK~=&JNxm9v`Tm)i3IU`fz`$KB|BG
zwyx~yDfscT)$V=KzuW4Q@BaAm$>ZOB`~1cH{lU4>pYPAlj<@gF=ZGyAC;|-)i*@~;
z%f_DhxO>lfy<@*~$B)%B^#}D_eUE*=Qh&zY57ePLQpakhemYZc0maG3i<8;8I#qvD
z@6?%kug=w6U8qZSr55TM0DXVlM5f#YnLj=M?#ZM56*Bfy$Rr|@VO=2!GO0r*-_Qm>
z4H;cNp$q#!=ZX4KJ%v_0Q{Sj>fzuD_rQr0x4X2~|+58;k`}YSGGxZbxoo)ix!}hcK
z`FeIGUmg5>JzvZ&Irj?8zl*-zRJQuz@>ED4fB(m4kADQD&kq(SUw(gB%+9Yc8Wn3@kv`|TQBuX}8Dhiue(2ePpT)|npI!^?xCceBOW?5D-y>DwER
zo6n2>Y4)LiT3jrj1i_Be^7%%?V&iemOdi4Qf^UM!c^|C$hwkZc*(HPaZ$9;pB)z(Z
zT2R!-r&kwe2OrlBn12khDdN@p=YIh!NH?{Yk=FqfmxH?j6t{4h0TUvV!YLBBmAe7I
z3zt0M0T;Kr*8zkHm+bce6t`000h$(12{P`mtlSa6a_OhG&nYwu?hqy
zf3)~zR9xHEHH;#`J-7sl1b2tv?(Xh|Q@FdkyE{RG1_%zp-Q9u)cS-QCviEaN_CD|Z
zdm9akKCF+~=a_RXASYE+p%*f>GX{#=ft=}?8JM^MvU19HwniXkCVFL{xr>dFBY=g0
ziHQx7oLs~aXyj~Z2NE@M266*9oh<+ge*dfCLZ(bObw10mdExIiRzV
zs)s$089-(97f`fwa;7&nass=7AahF)kQ$sKVrTE+XlZWY{JRD#J^k-Wzq5rI0MbS#
z)^=`A)|LPxkSRc#L5=|+Z|4R!S^}u-KmcQ)g^`UJz|IVy3e*Issfa1703?(Zf7BFJ
zs2RY8Rb1@t?HvEF7ZDXzH3>R^sF1v>7yzhF2ar%xQT_c>6$k?NZ%zl0R|Wfj^MC_?
zr^|_{3aM%-iZL_(UITy`;0knfviwc^KfO_aI|KY_4V-G`XlMHu0f5TF+1Z|(kPoA)J5LwD~(6-~91T_MMYVFt&t_j83-~0
znScYGjhtPa0EU0rz~4Yq%D)=~0z_OK9e;Dk{m13_Uz-24E^G&0nXZkOe~*#d|L&L(
z$i>O?AKLtT*(P=%Crc-1r@t!#0cMsqz~9|F{oXT6&|fw=A$cirF%?yMS@7V2=;iFd
z>VO!W-JSo6{#{QTPk0uQMe$W+A6))w5g6XI|AL@mKOIommUF#f-T
zZ4I(>19|=baF|(wOwE344{z#X&!`5nbZ`MmiT+;o{fb42{Tp&aZ0Mxo!PzYzz3QTPwy1Tc#HK_3B(V*f>)OaMmlKZqH?
zDDele02n3zAXWgQ)E@-qllgLJ`3D5I`;WyigPZ$P
znH`)80-v*g_`#Cc{b>fAZ1*oWcro^W+~8vNM&L)$zc!SO`Twl{-cIo0X0(3?I$GNO
z*&sHs5cW3Uhuoj~V2i__{rQbtfKGp15C4t%UC6=3&Kdac`pnE=9si*+Gg!1gmDztI
zpzA+WWe0~lS-Srt0bKV_IdWGNwQG-~YV}{e@MW9qp`v8kVNu
z_rQOI$Qgl8m%9!V_(8-Bwu8U^^PAqk2+03DL;sO2ENtiQMb8GFY;qq=w4SW!EK(}{5EKq7*8=SN0VVJ-onOwvWfam(zL!%cw>pG1XBD0b>}
z2`H4B5~0iuI>K+nm|lAL$a?LqC8)X@m?n0z+H;U*O~##JE8%~soE2)`Ch*mr56O)u
zg+>ezpH*-yUH>4*HJ(Lx>+);!x&A=gR%;aTWgGri$r;nYP2Zy`HWE-
zQH(nk#eW!5_>00O;ZH_$bbV+HIlOmPALF65IW9k+iys7+;V}V(BV>eObLPW!(rP7U
z6}lY7Hh#i4?m>U-lYB0{)#Ca#yvYH1iQduzY9U4?(wtAOrEwz(_7}HXbLJ8eN8*0P
zGw1LzRm9!?G;EF5R*sjGp;rk*T6ID8(9UW>yBZ3Et*?iNxy>V;9>1i#N;63BB61qT
zHng1??h<>^$RUeu#Yf6B9&Z>y$EmtGw0E*JuC0lhuTFn3LLC2D&yD)~r@G0p;h1LX
zZ}msyX!4D+-|M2(mm{BM6K__eUh=Y^k}eny0@nmSKlt_O**u#hOs
zXDc;gs~{Z_d~x=;Cd>sT?@1scUB7{TZhY4n?E`Liau)mv(!!Xg99XtK43h(0U#!aNaro{k+nVGGuj2wJEa-p
zg=X!=7JC=usf8;=)YVGzUh#V?L#hIFTl!M|61bk}Y7;>gDO&pv$#T>8&&b<9Z_JB{
z7mI(75-?CRHSulj?5U$%Csu?>5w`V3~T!uf>HFloghCo~-ZX}fTt&P@?QFu${jv|t3vFUEg~r{``wN%MKk)E_(9F?^b>%U8e&v?HVl
zIS<@SjN&rxW)G8i4*KFm`dfKz9_W!;mJ#Zz-G~NE
zCBGn7#nMlq1P2o89voaZClV=9!f14W+a-b`v}xmhmOsr)J+FUTH|dg!
z7xFyeaTt~!ZM5VGZO}Wf)>H$%GdY4_-I16*_^u~suUj1VISPNFo28>Khf!BkME+OgXIR(Iac)ypD6Nz4h6&@p5!eFX
z_IMF*JXi7ao3qirVHr++`r*XGNqB9@#;&uyzyFd&%C>Xf`OUhd7Qx3c-pxsMT96a7
zKZm<>JL-XWGr+N~=g^d*+`~;`ZhLQ^oih^sk@v+?*E`cKrf!{9)!=`oB`JpC5Grc{
zNHJQ71pR?RzfNE%wcLrfBQLD*agfZI_Kul1kcH@vvk)K>wV2_?Vn&E2T37N{EZT`;rd*Ur#^a*kFuv!=mn}sk+2l@7u
zx=qVRHCi$$(zU|M!nXz-!kp@;BUaog_W!`5j8mMiG
z%L~^n<=Z~nkFS3w*_D00)UmLjXgrhCF~pdAi6Bgf>1N}Ggolyzf3N=dRs$;n%}FO;
zJVI5uW`B_{P1}P$HgR|dSoP@a`S?^f02{nO6md1{F@+VE=$o6BS9dcW~YjIh%-66
zTE#%Ng28_Z>GXE10I!dfBkvc*JnHyNh%7Vmah2TUX&uaV=kZR^lUZWj2^K@B&AtIi
zNAeT_{d;0Q(<#XbHg%*f$UdX(;x=yYOc}Cnsk}NPOn?-DSiN7A_h!O1#WP9eEvnSR
z9EE{Ex6YlP1|f+(kD9v-g{o8pe2;i{6W>zRRITVh4=>0D6ptw|ZKmfH=&~j9O8rW~
zlKX~Z1&z!x>e9e{=%8}sxYi(yU07idZtZ^uic)(q9VHKg$PyF=rb+k&%_^Z&d^1G<
zhFFnOo&!f&paW~h5HpuT0YxUJ0g#&_m@b{gjSl-b@W7nyHJc*A>vH@%+Rz)CE%!1Z^uH9rM&{ZrKXufUkQu;EWG%SXJ9=F;l;AD@32
zr%j%GSP*QieC6Ta8LfK?EOQEE`pK+3&Fge@o_tV)HK=(xu!;J_Fn2g%CWbBl?F$lf
zgt?pDObJ5J=%y7S>1qajM*snWm{&K7H-*KP87p=J)ASGu{l{OiVRLGNDTU;bW85yN
z$&>RvGnEVMD}FDy9gfOD6Og>Q2-$z!^M-B1pJ)`n@oix&ZhAyQ#rT;iKz*&t_U26h*G3CHst
zY%?*FRsPSo+~FYUnCvW0+bor=)5qPejB9Ucw$dDRO0>3dRyZt&&zRZq6Mla&aNN-n
zbM7y=;We&thcDdDYftvj&YsQ!gRQk@&c=3?=U@cMtc8?Bi3eB$ex
zj7heCAOI8|oXdqpBTVWY?p>4BfngL7#@D15OU{bs_JI3D{^e_ciQa6W;?E_J_3A8R)G3gX|U#=w&$j}Dpb
zq60U&Ae7GNWR5?aVorbS-f62&aP-l>|Ll6QZvm2zInI34x9d5nuso_iw{p9*ZaecK
zWU`=SN65xi$mbW7MOFshb@x65DSTkbOpWC-S>5EC$9b5fb%2Q;PjekQkl@(}nt~0nr?lX+R+w&YXq=5l%FKRx&@i)ujaEXj
z&DCxhOmdsdtP;;Rw0t}FRyad(1PnwLmwQb@-WfVR=2Y4-4XOp5P~%LzVU9|{tMBqw
zvlYeou4Iir92S4uEUt(#OrYaXSOvv(r8+8Ex7)eS!oto6AAS{2qx_0?itV@%QjbNM
zmUmWP>?dkg=i(!%yZ~g`qsz@Z-G^S4`fGB|H+%XDB+*){ZB}&BPZ*4JsVKdpcu5}s
z;r3-)^*0|6^*Fq3a{b~52!BXp9~SMFNyzeQDn*%Fe-eL{C^WZAC&i9Y52v);sqwNw
ziwm);)8~BeAU-9f+jz%B=XDp+o{8BCs8m=+5NV8cEhxqxeOT(-%3G<7$JKmwx6`h_7MCqr`I4EZ?x~PiR;z)q~y)`Ol(QC
z%5mo}@4j;MboJAwDU~U0-Ar(#^xEN8(s@PihKhgS61i3lt64QiEVuQnNBt7TtlfQ8
z^KHTz$Dm+(@P+d9s_?pv`b?TW>gBaTp?=o0g$ZdY^R-Y+TQ1+~sUC%8ZeLb#3}dFX
z`;X5?%9=k+hG*0sPx8HfJS5&zzgm`!POIx)H(R|?YkB*Q(4*FQF+waa6fLdQxV8Pj
z=K6oFefSIa+P4Ez>>mxSDo
zzKuj2DSbbR^sGCUQ1t4>7{sAcWl(`&viugX#3oJz3*Mn!IaKK8(ik)3V|mgGDZ>Uc
zY$&5@(X?boxUh!A@si9PN3yh4Nh2jf*<*jF{&yE%2Q;cg$fEuoOXrrO7nzD40r=m
zrx~i20FES5^P0(AZ{!Wiz6+Qab=ZLzccysXGKR5g^w{B2mXKfC3+Ij(gu-E&p23}^
zd|R88mhPvz|?T?(h@MCsxi?mc2W&)~|FBwBut3@+IJf&?;!Lz;IEjt*)6dcCNy*;-kI
z$b2}%^JGamJw!^(%>=Pea50o-V)^c^ztK1QFH5Tg*q|ITW5C;T+%UujFD-u;-2vg6
z@by{CcA(VK6nW}KGB2#B^47LgETy8Bqbuj9k9nUrX>tTGa}9J
z{yeB_YjiXUWwpNj-q{8D?2dnm5QL8S75hQfx6L6+*apx@6*Do^%5}dAn;{4StKZ6yT=CiA^u6NTMLX!Y)l!V1FI*%ASRIEhK;d6h~mY}z~n&0wH
za04ee7}b3Z#jn4ONr%OKS17)k3j6WPys&NH9-_rWn%}(kr_nI2;De2_YiF`JRA-nj
zY79?Sv+N0ZX{!*lC`;pxA#@1qC_JT>GINDvEJr7~_=$mFKCLQ5kb%N^%Dn(j{Ez&a
zAa1(8wsI!dk#?6HK6`(f5N0Ch<09=I7Kb(sQVJ)1NB)K*sO(7KUfH=;rD;Z_STnh3
znl*{`TV8@_1!~oK>o~HOocYTpE?oL{OrNbvwU@#Qh5G1u+iE;ugAM8C-pnYc>PP?)
zg?URD^?AL2ncgWpqfOIM3kwhFynl>|)tY;{NI41VfExn^u8Du+RsoUbymv}xnXQc>
zBjwz*ll=@HUrO@5D^zJ#$Sf_D=CVquo6YEnnVTs>ASq
z!gC~NUKlX_HRcMz`w>;v%r-#%q3D)r6+`t}UxamMxS(z1ZQZ>M2T3S*ef^l9_0yX0
z&WH~W$Ayg4<&uAWtPY8r+gz56q*{RY`$RZ7)enIRiIH!j24)w+lF4Qj=nj$>@N!FG
z2~6MzP#)gpt@6a5SBoCKO1bzd0mOrJoOANNJ=kg2Es~W#@VW~_e~H6o4H~hDP8^5U
zf=IWva?ohO&tS_;U@SP?DpSI!4chW7xk6KP&*70gAX|TOfwH-QGl7B7@=VQlJRijp
zIF64Wdq}=YrnbB=+3&lbSr;q^aLcy-=(RS?S0^@E;==U@(ckGO<1mLs
z;oeE+@<
z#pY|xU2R$}wKwEF^0Ar(x4o#oiBvhey++m;#y|3XJpvqRGRj5fJ6o%cxF#x6fj5Vh
z5o5LqO1{Nko_DC-N0%cqnc-F>jh<>vWyNtcfGB^8qwIr{4LU2@kGQ``r_5iG3`=7v
zPVB#Ag$JB6Z#yhsm&3-ed|s8Xd$ad?G&=07>y<5};H+_MCiC7n-BGIF@KapatOts=
z_!P}p*Ab*76@L|jG0B2r{yPyan5Md=zCH<|682@_ZXBBQG!`7l<2fCBR1qL*&71m~
zo$-H$)O3Kg<<+FUNra+~%|#>jlpS))GI~o<#rmq@Xu~H3gs)VB{L#RkV=R+NVX9wk
zDcprgr(!-+b|&|$;m`$!9R6G@q>+lJ41=0ob*tB{!ZJJ%P4#s0!jEK4g1a*gO9kDk
zgQGryi9Y+Mj1eRk8@YAR#-?70eTf}J5W0W4SOI}mE&D=N-(ZZ=#iy-=m8$8S3`skPSVgq|YoDS(3qNIYyX)%hp?O<$V`bLfH?!VUUjI1yg-LPEL#4fr5KB
zBY*j@L-
zZ<*RYUlL7BXzTfqIv-s-iMnTs61soeyf_zgoj4r5MTc3HSxbwI+Fb%^HR2|nZ`K^e
z5otGMcK1L$a+U5F1B?g^6?`)U^A^zRhM*{yn^>Q9Rjeoovj<8HonEaFxe8NnOy6F(
z&qM>;-<3H;ewBB9HI;JqM-Pvxl(>OA2qGoG?Qf7Wgp{;L_k2U?9cBM6pbLLN>3N`s
z3+U-XNL5*8*@NUnQ%N47Vpr6_pBRiZlWtFGG-NRb{cw;!M5*2)3OM?1qqEd%)IyJt
zE%;hvMAuw6GJ#|kKHdRiZsI#NkYVYY)SenC;67xiP|J;YoxGKas(z<0i)mO(i;q1i
zIWFl6Hyy}8Ca1U!P51fSo^w
z>FoRTY9H7Q2&B%z2p5<4u@|evniJbCBd}54vGhQUS}k}_G|$5j&mVs&7twD^Ku)wr
zS>uHhSGVx9NE!ZbHk+q)wfP*j4ucTC6_3>zNcqN8?m%A=p*JD1clnG?UfzeaTEtlW
zin*+rBti4gWV62MYi@SC3qjVyR1}C{pqYB(^sTF;{EfrcIfvekW;>-^t^7fZ6u*=-
zgM}@UDL9drjZ|f=q7#2jiP;}rVd*|OvURJIE=10ez7uE)Ya;@L0r$gXj={ruI#jP=
zj*H}sHKk6eVed6ya>c9kS(MT)q8JudE}Wg8zMB^yS?sK#@Wt6jo4tO2H?+cVr$B_c
zSc0%qT3@R%YqFn>1_%o*;qeI%tCrtNEOk#;M_odT^Z797?`D6O#JWXTi^$LO{tVj!
zHI06-uAwGbe{#0Zv;wOr=`OXf)rS7WsGU*$(eV6Qs)j%o^OI65{&;T@YPd*06T&Wb^;iz6MuG{+zxQ5rl9v!`)0QDX
zk+^xtCX=wk30>?vrwD^C^0^pV-jh^po;2&5mIDxB%{NyeK556M;nY5P2gblXdc@Wj>J{zMHs+Oo93Z%Yz
zWbK#_7{whP1iCNO!872Q$$wxiZA;AZp$zlJ&c`*Xv53w9ksG;K$0c&_$ON{MbU{Vg
zZzW4+kxZHHVUNdFw$Dk%#@*#Uqf{6aB~u`>rS|pe8&seEDEdyO?feNKmz!7S7$Y(FnbF4|Ha}G(5|J0nA0Q*wXQb^DB?JHM-`-vc&vFjO&koajT!`39w$W(J9FLme
zN(^h~H^Q&cXD5&-J-ft$-`nK8wu+Y^>bZZe{)l^fUz8w$(ECJyOU|PSRY~TXxA1n*
zl_As$BpWnUMb*^sn$VSNbG6o%hA++;yjImC5$L`v%i_jmcM>Z%rx9parecl7%irUMDz4i#{gvD
zce@FyRcD@IT83U%PPAFI2lzqxOL>2yhD?nIc39H|D0KHT;;xQP?X%qbxGNUf(zCPu
zhKKN0Gh?LMIADI%PfISm*h~!UkGl=O*pIaA)(%RR$ffy@mfWsmGR4ZBT)e*7^uxS?
zW+tC0jU=IsrzT);EI2BOj>Xi~>%>1<3_)ifL2vzdT{IKq81CX`0l5#*&{2OFZ=(_`
zI+(=bxS5}Lad*)8HRjn5cYj8q5GN2}P7^1)4F8U~LOs7CniNi@QSsxbrv@)eiq0_V
zcFr;9`mI~nC&EY|5j4tO%zpJJ`@Z#Q{Y@@TB8^1rDtEU-f9TkV4nR)r*m|$<7@tUx
zI<+O+6d$ZL6OKbi_(K1<8oGbQq^PceKX$Ed=aWsu=WoP;YJ7Y*C0$n#!93?Yd|&$B
zbxI)w>!;}nJ~-WGZqp8yZzK;bE=;@UGI_8*Kg#D4+MeGgIj#*S`>_;Oy=z*43G00&
zu)&~WfoSFq&TSAJbIx^dDtz{vNSQ?JRI|n-KCdt;6kH^242UFxvGsrdVszKDH0(`J
z|7Ou`;ax+CO{gFh45fs0N_O?r7<>y@Y+ETT(ytSEU+bFd^OHl9h?juA
zVG!FOBBiW*7J_!92p>R6xc`k4mn@dU$EzXBn~5{}3A-e)aBzP(c@ML>03K;6Xd@;q
zeHJly!;<4j*-xjT`c2Maj-;pg(Yws|CGcluQVL#kI#C%3Q!uhzR1iIp*CPRM@nEC5
zpEmcaLz(I&#L?e)
zA8N~t=tqc#Ry?`yp9`|Ppql$XJ_0TRHRjMF&<1z(1G8lS>r&6vE>9b;0`S|P1fbrG
zE{vcV2VT?{Sn>2k@8{o*5~8+xC_5#f
z;aJ5#5^!a*QF@h<*^NEQ>h78GHiUC00N{w}*Fv#2`=Cjyqq}a_7TA
zc;kIAhjDtZQd#BXUN&OLl)w3A75A0G2aU8>3od5ic~~9hHK|-%yw6PZ^IbaGnmf$)
zLgY;Ni8_Q{z=T=5b$4{kdZSRI15q+u`m!ElR!pxkPXQDk2r`jioVWR6jzl8f8{hQi
z<3WGH)Hwn8+95TfHvNKV7A%87nCNf#IFa)^+;l
z){e%F^Vgg2Z{6Vw=+_GsiOkf52;{E51LK5e)owottw+v5zk@H=LZ{}syy12>9Gqr@
ztbusTn3)`>OlU`1bFvAO)~kdW4gpoJYiWO@
z$QUJKDO_JfjVEF*o|M-t%Vkj5;!G`&j=gb+w)O~?n4NSo4Opz+fx{`W<9>>4B4Xd$
z`dWB-pU*q)PjEPh?i22Yb1!D%>hILt_aRQ{+YesK?EN}#%Q%2?`GC+H_E$a(1@ohE
z5ldvrQ|Mx;@8W9yw7nFH=vSfNLk)i^WuiV2pZ^Yt=dbVoC9@mNKD52ziw%GKB!jeY
z9oArfO)CS6{*G=BZab@4rqC-s0LJ;aK}orcMo&z2bD2b49=wKmh{^Yj)Upu2Un
z^+LtEn2v@dj9wpy*=y;{ttP%iY0-yD8@vToVaY3%dspJBQtF=M{_=W1T2Uh%$6;pl
zTUMujR4kzHjH7tpM`L0qNa7vm-fd_>_!wY7~k!#8O$=C70+tX
zXHo%tT`0}oYJ50=o{?D(;+Ooi9Tb>Ey+tfvXAGNh!|~Ba40)8pz#D&COxg7mZCpXn
zOD&QuVAB3LOKQNTN`qIUfxd*)Rm|ir*GHZL#`XPxni+a*ul1_?E3-Aa9=@%eoqTRjoPTT~os9o)(2)q3q)
znK|X@7g}UoJQnAV8CGnaNN73_8Ovb@63%xG#;A|ep7OnZ70fse?K6bb6rHj_q=l*J
zn&Ppx>`>vy?htVz5?%NkattR3f+0_hT8NV)odSakqhC%diGY7s-qMh$rw%isdGwYpZEpFnfY=P>g5<#!qA5~g%yFS(2Dv4>$d9w)C&J`hY(VuGH
zh75C+K@ZMOExR5cRIAK?)~vTBTC1KT3$?_cvoCKiFDwJiI6Pj7x5lwdYj`24YMH3o
zw&|;im45-!PW6^*Q66moBD@M8xGvx$^@W&;vjSn=gHDIq*T0Wih4n$G-;v8($?=23>rqqwT6Y-s
zZGO(Cn&t6#z|L$9QWQ76zpHY`<~`reICIvN`1K@oDF3d!sV*%=GLzESmDbOY8Iy(}
zfU@6=52^?vCPnxHiB(uoe(SXOi}hzGsJliANe0}L2i^$g0t`u*??KtM9gTgrM(f_!ZiIi7)YQz}0b2;Xkl6zp%P
zX(skBFhXlSk0e3crvrqsIW$E)MHE}UGZCJ8BdXjE>)%|
zzTs&&u4Vi-r?9?2lY&02%mU(ipC89H&cgnnX2hZm|WN;sz>VX)CNeM)~Vueav3-|sLhZifEa
za0(;4L=4sU#$}AJV1h3iecoq&d;UdCrK-e#S^MCOByt?wU{c^O{Sxd2;9cjlgkSH>
zl$2-O)x5U`O8T+BHO9^dmYbOzqR5;GnXPFW+|#>2RLl7>KL7e4QJk%8+`A=F1Gy^F
zj|!J}xDR9B&ngi)FJRsWR^z(w_{fSDUNgDgjH`E>#@<9jN`Ks9|4o*ha20J2~yO8qZ;PNt`$MsGTr>IJs^ww-Oz;7
z534BL?fr}G`tmPs(5?;-HSi|fn7z&Pn=3w}f*zr^G}wRV>E_Ji&=
zfu%_C6_3Z_!Ud@%g6wyWttnL??~sOn<~Y9S?c370HT)9OncEgJ%XZVTqLPx5ZX=eR
zd&W$k?rH$G=$a?*ePDeqy0_cv>dWp!3YDbv%TMn*NW~JGFSAo^4HQk`nb$B5PAoiD
zxq7MtDo<<0>q$k>+H1ytIGV`H
ztzg@?_+M*T!giv3_=K6-iipM`mwo@bgn;yE_Oi~_Ij)3RjLvICp3YI2`^>RJ_r@f2
zAx+1A2LfuM7E6I%Bgv)qqfFE?hj9AKDNlAC!!tFa=3N#07qOPO9jwuhI5NO|8#_bB>%vlo6mu@yDtM^Y
z(UB^H^N2l1I;^i7i-#2lI%rGXP3v@r4^Ej3rXn5dkk33wM;R#!C
z0ftG^aMRp8*;e~;u!HyFZy@Ncj)!)CWoVaR=lZ`|92{kC|I$A@MrJvGi)0ksK{_}g
z$P6yZ58n8h*#m%mU%*qwf@h);i#^zwYrPCOwln<{e^4eU4<*)IU=gw-j}%N{ylS(Y
z^Pm4QJizIDAL!%{uxl5o?ry~Ct?{%9O*HU}AEq(1nT?Q;NywvC;d84^yP5s)znT+s
z2_l=LkYRs7{8ApU)*?TDdBwq79F;41C#W6{TE4_B>Q2a)(dTM}Ll_|~gj-rbU2NKs3!S^VX>sbLj
zGHZ`CMK6sis>XZ7#ZyHD)KQnUO)okWE_;Udge@cNTJc{uXsrDlW;^@
zZL1$ap9l>JnhG0dSBu1r;(ZE2Gt4SK3@0IK8U@?KuO`8PIvi
z)uBWfT`d`ZBr^P=ZJHw`7*R$Yi!w?-CNqEawEaXaV?eGyx{<{aNv7qHF+>b}+^E5#hgNb_R?H2*NHfqcJ
z6}=BisUV+q-u-HJ36_Vh;$bp{wj`oHcPrsP^Q)V4IgBGRkt&
zgizvvD!NT82y}Su*HB5<2GuOs)&ws_zP
z`JKLTnc<-K_uk-K%`5ksupGl3E;ATYWfr0e2!8vAp0VS}XiON>E*h*1FNtpYVvDVR
zIzd(O<&c(rZS*J)&UBPMS>*(UH>f32KMT))+oil{pr$dsxxZqMo(SH(F$5V$`PV(bvwr7!
z&_ZhW17)-28wt<%gPt#%@=y8;h<0TVu_FeN+GN(
zZeOHynbuTY06sK@5|(ez+|W6H%@g`FH@3=H2;WDhv?r^GK~kyURegH3(xjSyDvPJX
z)g6u{x+qpY2tC&*6v|KfYT1U+q5Ta>>?^yaQ_o}NbGfyU;R+iqRwq5S%Cd2-QYX!n
zO}PDZ%>W(x1b&_InE2)j!q1d--g)>2n1K$sG2g?k?R30fr0`913!+`n19P@&Dg_+`
z&t%SQ)HROBE1|v=8i7ITc7OOYPPv
zc!8UCl~H-CK)XsBYBi~M1R8aUC0{~+o{XU$?BJvnakN_2T#h4ZAbgI0e&TP6m^
zAaCHElVfhbfZULM&oRp@>N_PplGDoO@A_*KqV(ZdDC+hJ0odh5DG@ld0B;%j=1%QC
z(1LzytttOn{C2&zLo-t~;v@6y>l+$;7DDfP{Y#S%4{kQJs8iR
zFe_*)U(zWCUeK7&avR|L|m(i3slQ&vGSffu3RW>hO
zOfEOA-I!i4dE9w(Y4)Y=HuGQUHRtmWgD^D>e#R1~QcSROBu8$4YrWNLpBS9NRwIO~
zSQ^qbN?oIZv$^Rj=#$5*YHs}X95i(Cu9Dl@k?jYj?KMk{p_BqCtO;e}`-r;Rwnv($
zMgBCk1u+sLlvV1CE6`~UORtlW0E2t9`Z2o+xAC=2+W76qE^+Y@I)
z7nu}DPy5}1(=aZ7@G{*2fh1q`V%G*aqEK@ua7`fQ>scvQnX|AFnsNy-+W2XKi(xE%
zXe2UbSobXV;hgt*_dRxe#ODW$v;DNu)q%PqzAt*?hQe137mHdR^Ur3D0LIVC^OFsmb&q
zs9%xi@bu8hKl%c}3buo5Tzc`Hz
zR`>uFtrbUInd3dPqKgg~CYY-O1W37mAc30m`gBo7W5hU|SOpXHMxX8~ZZ?frJ)H6!
zRq2%X;CmJxZa$2@lu*?qJBnaizSKf1KWix{qw&;#b_@5&`k-Z%SsEO%S9)+4Esmow
zo7Seq{Tk36Q4h*1p+OQKj@Z&66!nE
zN@&z5t)z8fLxt$*OQ=FoVDlFbB6Aejz80})72)TPp4AAnTDitY?e!~s?F}dln#3JH
z(rQyuTukELF4QCSHM8LG&Oh
z#iT>ynAQ^xXJW_fhl)XCuO)#;4_;;{LhKQFcQ#J2bBL%H^k#x{b`kzj7T+(8fm&!d
zQ3y?GC89}WIE+czRETS-SMlPg8>XDf@G;xfbOJ?ue8NdFtnn?$u0IuEHCJ}G-rF^Q
z=U6iMFc4E{Ro#n+1%4%L{hIUVSG&8-ypL2$mWvV+qd8erckOdvRb
zB?rhh<1&#&JQLhU*;=Nns*@)2)K#p1^O!+Nztpn|d)R?uk4I_vNOdjxvLAaNg&MG#
zwRmd_#eOPR*_7S6qWL4x6LpNaxuXJkyFV(B&rp&3wjbsan%Sl*QVc<74tDONPwJm=A5R3`|b8&bGXHHF!+}-&1X+vL!ym>1#bm4=4u~U9I
zLbaQnk=Vxvo|91lZZS&yz!Q$5(1^x;wc7Zvz`#E1mq0D~5^I_~jxhr)37iskvAha&Qw7y2}kDsT63v~Rhp#s+H@y(B$AA~K3bWG7n7dyt(G*dHwzJwo(*T*c>
zm<#2+22y9B>+1;Yqulg+*_hwce9>DC`YdURAyZp8ka9{!J~fCB(P-nJ#8Ih
z?IM*8WB#ZbFAI)BpRa*;@Vj>3%UilbHh+||yPN-yt#jBFhGClRwr$(CZQI7PZQHhO
z+qP}nwypo0yvZ4yJE%dWQk_osB7#uA+B5`KZ~fOJtHFV1*%^(0?`gBzkX?aHvA^NY
zAn!Iz1NjY0#y2VpAPw;MIUNAd4}r}r1wq0qR+2gOh9{*;`c1UVx^?RBFt65C)10s^
zRvpJ1`v_aO^q~iMJ9WJu)%@*s!0i~~a9FS?4lqDo(j<#2E59>vv_>i4_tFwH$f?tT
zyk!bS^e`fQkr9%AgTjdz!E;Nf`4|?e0ddol$kTk5+RV+hp^J+&V5fh(0x){~K8(u7
zBZN{aWW(!dYRRQWio0yx#S3je3<7U`Z*H=!v-;OmZ$pmu8n7eZNB9$7!Q{;bSsErj
z@t3O-kZp-s+f@E>kk3*@J?_t;{lzmvd~GX8uY+YjK*fZAoFwAM8`6r?I78?hs4eSZ
z2RNHkW9aiv?IExv0N71m_W}+{5|(34C;=Zcz2f>pz4vn=t7CU*h_U=;&{jqqK#6;#
zQf=C>Z|=zR+s+v17$s_Z;JcN8-74g_Cd-EHx5ls_q=*7smF6^KZ8vAe)&8?6Yf;4dCB&Os@R3>;u%8cVD;
z7Ffp}Bhvc&th}?>hK5!87PZ(r=AnIA?Z|lfK7(;T3W|7QRAJ0XsBtsnD=_Kd3JuaU
z2gG6QL)|g1#UbbHVKs3(absHDTdU%BRd{&4)0*giY<}h3$*MWm?Yt=@JY|J{glvff
zl00eFDSh*~lgNvC|T6)EQr$Ip$ABxSzZZwO|M4
z;dGucm_bibqZkl&-j+%2%S4H$fvm{D<8BfI1I_p}!o&iN@6N#GXudGR4P+~{yINAJ
zHA!}VvlRxz3%^ejyIrBkXpIFeRK(Zl{sMdhF&wXlHtk5IHS&W(kL4lzXII=bl8Elz
z1@HzUo0Z^~b@-PcTPp%*mP_7_A&(o30nGJ%A^R2hCf`-nSJ$Bh3VK;0U)y~KOI9^r7*zFS8yXW0RJcB|m5Pj0h5(R8i08fw|32e%?f)iF;
zr+C_FP(HeWLMJ+0v#
z1!c_XB>1$V2u6^obK_*KVsHsIY+D9@)!LOZrjUo8ICd9c@|WZSGuKcMr>ZC!vn(8(4P2yJCsk;s|zcC`t}PwQ&fMbR#sE%fk+
zBjj0ccNdK$4{kKssZ1s4;Jk9NI0UNUJq>pXeqQ
zvBXS{iW4QcE)4~U@SIh`8;-Arqtr#8ef^Kl<&Q4zwAJ>l2^{%)odpdvTUz9_Y!4XSC-T~hOk^+SO8b9TGFsvHqr>_d+?V}C(O;J
z&L3izn1fc-YF%}>j$W5Gmd}XLO&iBdt3rI>^@;1*XP@lwa-a`HmrJp~F$+NhCI2+#
z5IjxBPWR#h$1$$a9EiNn5@(|G_o!(BHS(nX8n0H&5}&yBb}fd_YW#^~j=${@W(+8M
z@O%-bn(;(`%wseWFyz|&ur3NJcS&CgWT;8q?Wd>-2_#q$hz%xcFcM!>bBriGEcwrR
zBO_HDhWH&6y@YL8_^x})G+#n%LRz99)p2|r@19(yWx-XxmJz%%{qb^SCT+KlMKwTw
zbY}ngWxx$4vyDPU=gg||J+n%FD)Q#=@u_BN%h0cX&4R-Ve4{O_YpqH%*O-V@IAYo-cN%KH1_(B&&In
zSB4GKR5NLyhd)w1SIYT4LV!yJH}0C@r$^B`ud%8*%e13zLx
z_PvR{v`*I%W>A_G^m;ASB<~MhoUEI8PyLyHx0!vg)6ruo7@RsX(K~=ByB-E5X1Nyr
zKu@z1Olr){#r}CFbkbKz;t5#(ro|BDcD_Hndrt7~BU^+J_-@}01n5f@pZoMO%Gwv7Y**xpcN0%H?Y!^4%IJ4_
zN65{V*}Mk?V$4=rJ-$Mi^`auPBRR5v_@#Bl7$u$ksh0>hl@B!2;$6A{13Y=352|zG
zKRd}_ve|#KU3|ElBH=ATEKgA;3@yB7@gA)Xt&#tC<&aur9$;Jd0g{K$&c#)=mS!ne
z>rwDz%q#0b~H$`Q%?#T*E+7?LLP5{$d5gl|yAKE6_B?Vp
zlz2PM5jWHO{nR&@(-?jR^PdiX$H2c$2Z!^U=%3;@w(2_zng9;hB7(=LaTPfEhjtqj
zp51^C_pCPB&L%c)tV)cu`OIvak?f$;I()SJaL)5_-NFK$VyC-|cN$Xx2}3h`1HnDf
zK|0y)0&VzwNdF*B-xlk|128kvG0KX>Kj8;e9b`YT&t-x^PNK%=8pdoa}grYz`3M$#bF|?te}u7Y~BdJj5iNr2G}l@;RP}Warc#dT
zt(6@tUFt~&US72s@oQawfIa|GVOMMpncmqU2>~_?M!8aZE}*1ezTD7xbyrR5h?t=+
zRBmGIVMvQf>a-%3eh^1U#N4dt&RiiEf@}!loUuPFZ7vViLRZ8ga9jw3wUF>ZKBimP
z41x_@2)ggX$wi^1xI?t9X?v?30?$OgycLl^Jv9G(g~OdRwehomL6}Y&GhC5Um)IOd
z5V8yq)25M6yg7{jB;dEOhT&)5;Pf;qccf$JjiX4Saf_;E
zDFp`B-YVXv<0Um|I)JjwX5%R#xsouKpZr!_ihebiR_SgLeKZKwa%UA!Y_9wm#3ABO
z0`f&QZdJU8bN|7A+R?mroyE>jm9Fqj)f_IwHAt;^qOAZHs{Co{85DBhBTNawfqxjV#G*i7{BJ4JAuMV8gk
z!6RQkl-Y8BA$|6xhlsFx;D?PGLJDs4M!G^%72YT?3)Y6na7A?8R<{l~QjcfB5^6~(
zZ)4X}$fLsMaY3ZMi(lr@`kqnD&Bi;bjYQj5B>
zyKi7%hQ`An>{Y08@m5Fg$!#Fvy0gmX0*1Rk`ZeG
zUx*Zbm$wTUpjP@RfXd~^r)ALe%ed0wyYIKCo+UyVOqt!*WZPIV{RAyhwiR
zqgE7UIxSginJ-h^G;g);aqr3&o#ZCf0Bulx)71MGK0#S@k{7y&pl^-ShjUs}tmaAZ
z3Bu&D;m5T8nk*VNB1`-ix^cdy1BXPSpG|&?t&r{!S~az1`#z76Y)YlGA$8
z>Xpn{ZRq`G)+*n59)V%prp!ZQ=lJy`lA6#v8-k|G;z+Z}Qz7(maGQinRng$mSNL9#cErcK9?)>fW9iR*y;M(=x
z_;Xa_u^wS(z?1C0rn5U#RKI4$dyT+q&4l+{+XJC=rcS-NrXoq~2*JyLx#sMnOWwRR
zV!4~|-H0-J0Bq>3DHt?j`88y&`4c3@HuUpaut{fq)EW9b+m!sz!pX0}l|w-sk6Pl~
ze;-+`9nmr45uVjFE|rDscXPcEvJE@L{?n-~s5&m`t80iT3x)?w!#KcQ7WRcZPEx8N
z$znDK=St3{EYbxSp6lp;<2Up--^Y9%bYb0oF3GO@d|)#09Pwrdg`)`UchVnV1hZvZ
zvA(lS=>!+fi(wlKP>1SAyT;!>R8qOh*39)dSzAtAy4?M}swWbS2Cbt^>ld03CqvMt
z)Fe*#u<+ns_S_=!#68F`B`Ccn*5_t+YcLlt6XKrAU8l6nJFSv`R5P;(Xiz&2RnNg%
zABo`akmpLREy~#QHsk9;h$22B_NLB@zsz$$hsyWB>RH0mzt11Rw%N#oDMZd;Lw0)4
zb&X4xa146KvBo58X{f8ZW@)Wgl0Bc~r?3)$XZl!6o%}M)_5bSGf-7CwjPjF!=9@v1
z8GJ1^t*_?MJz9W&!=C`R(kpQ6SSr=I_rwD;q2~wwwATIWD$HnJMDabA0yPvZIT`D((it~t6uitlciG#
ze5w=8wsqL+V9385$NWjPB3I^oZLA+<|)_^1HblJD#@~*H1^?HAn
znn2Vuv&3B9K*i(UPdlA&bLr$q2#X|o11@F}$=VXqD_BJm1#IAiY(hsBNcN&VH
zngFK1f~47Z`-3whd;?H*xuA-csQAW4RNnp7)1Co
zuO$J|Dr7JNZ}*HvS16h@H7IuQX!3-ACr8Sho6u#+>|{F3Zlor7=~XLf*5i9N?v6z*
z9t(KRUX@ova9>(@<`C5_wR;{`R0WzTlxw{t4nh~CKOTZ-W&@316iG-lgzo%6O$P8@
z`Ep~nc^bo|go;AIYsU8fSmX6DsWXm3JQf31P$-3o0uCiHf3o4(0Hb!Zec+aVEPFAF
zx&}~egI_904_MBR*KGMd3!iiAcHT0!>C7F!nfwQj(=+1c*<19v#i6=RB$RWUPlff6
zb++Zk8B}-@g<>*bpJb{ErBVDiY>=JuV4=0HE>u|H2{*vuHAMA5mt-|(rwK%k&Dg5~
z7eWd58a2r~0Prj~t3o@isO2bso<>zo_dSam`(wbL%Vl&oB8CGkJPUl7h!Gl6*WA?`
zAki&`)~C(SD9oyha2ZYSLA9UOkWbiVO;zsFL8(3sf}AySDl-a_s3g(w5h%drw$OuV
zQaXVVRAZ@by;a!V{Of;u(d60)V6Kd`s1tPb+)H-5-QRI&%G|8iWJ-~Li}Y%#3Y>aL
zx3vVkk>}0MrV*s=ZT_U0y>}#kgq5-ZVuq+j9
zkI$^i1uDaZvk;8Kbnne_Q_rpUl$x_c{QhzBaniT$v#_%Z9H+b=;mSZ$cMiepE<@x%
z$RaNYJz`s@Zx8byJt4?{)X_S*yB^#Q(2U;rQ?{=bM=4edNj)yWN9aG)SY^~^j
z%8U2|L*d&q#xkPCF_e2ULmsu7S{S*~B9OUf{C#JR91)x&rr{=k$_>tM2Zku&HEVS-
zt;&BLiE9l5HSG?Cj3n_O1_w^E6?L(AZ^F=VRW;Ao8ilCwhE>JG6CwPVJ4AE#(A2D+
z4QajgF}~V>jFF26K2&#?yJzmVdz7FYH5;y+
zG&$5pR`SNTJ}`lQ!@d_EiU&4h1A?_-QE--4Dnf?u<1_qi^S|wgCK3U=dk@mc#y77ZFZ}q&)Sv-q#LYM?Te~WRcjCCoCy0M1gGmM!M418+
z>jNbjqB)z@9Wx_KJrpTCQ)k`6jg>@R9T%nCO#Vy=w
z+NL82C?!J?53bG;HE*G3mrCyLFj|U|LN5ACYwy2~j9y8@8SHtIFDSpXd|vNgw`j27
z%RGw2GLc+=VR<%M=*>w@*S$Yh9@jUu=N)T6{sP;>DnVd*Nuw(yC;ZRO?Ohu2A07?-
zHP+Jkl&hY)&C3dP2`REn`U|xm0HrGCJ^akj^eCLP=@E2rH-Ro6H=vor>RLxz4Pz%!
zJmXXM(6}398=J$t`AsXdt{b4v;Bi7S{t7cPUkD_BuSLsR8g_ucWQ~^0#!_q+JpMEJOwasTKf81GH>l?nqYsp?*QgbBnNF?BTd5obvP)w~?`^5JvU_lir%C
zNwhHQk^|nZ3sNVbeq>0lgosrH)TQ6
zCut*p_}TQrd2lnHL#VfxievTVJ20Swfn3){W`p=80p|t=`LJ(CVZXu#DY{k?6tWUV
zgI3skD;G7lW0V*_TjNHEG!O5)ULqw?#&sv@LpKFFT~fYTB)e$*zyO3468GyZDiLuf
zL+JE6`5H+F${P95f8mia{j6BgT&R}CMYcJA^gkx8rdp5~%6qV4rTvbFk3@)Dz{~id
zc1p66ET@$?63RbuA;Y4qUBXj&e06~=Yaz@@Q>&bbWuroH`OiCUAgV7kIf=*yH|jEM
zF#}ZR#pcJm&f7fs3N`Ueu#f!%BYC)gY@Jd_HZzWsHx7Pd+kid13~_c|DUj-L`$*d@
zfXIdd*1*Kg#ju*)5TghtPyZ1FKY(8h{3Uv(XO>y2GV1P~6Z>LVE&vCY)dqJum8unf>j59@a%NoU=9Y
zOnQV0+mSF;zU`3i`1tVDNxN7kI?L7uZuyQy9bJtXj_6VAOm%zbhVDmaZ
zh??TGi2ZdS+RzKVMVD}Yl+m^ZAX0&$2cCqNfMucqI=|SjH=FMt?Z59D`Ist~J2LN2
zTA0i_FxGd?)gKc`k50eqZQz1Jgve5B)tl
zFm5|7bjYw#&djvCDDJR=kQwQc9yb`bz5K?H))<(k=2==dTz-DPeAa7z?vH>n$!vYe
zCQ3Rq{xO$~jGFC#)>ti+duqqnSaF~1>*_x0C(gK?Ekbrc(lE^W%`S#`ToNd-a)@Ek
zuE(?5^x?j&@*INPIFzV9}w$0pWQMcBz1X0xS;?}p)mw6{6
znzGtK%P>EOOBYl5k>ujibIOh3PJb{LzVQ;K5kRNgNHQ=EK~l6eG4vCL{GOtfnvnNx
z2BJa9=?lnzvc?Lw>lmXG83*V9K?04Gz{lZ_>9!!OAkL=C@W@!Ge1&@D@1@7@_+-)F
zXGIn@yVWw%gxoDvc)hwC|Cd9!J$8oh$_q{0{f~^zpS+{6u#oI5SaGmwQq?rp((spE_}+#kQq7Mt2#25Qmk<-p8C(|9H)RdQ#RIn++ZlnR8Yy@$q1-bZ(B2
z`snk2_pAvom)>}#l4x@2xS7D40f~g7$i8M=YxV1tz;FX0cKNjxFF_|GRTR1EkH%49Al}BR=E8vcJ+Vv=|Sn#Y(OX@
z_+I3$+c9qi?NBiWuQIa%fR0s7N&F7XH~_N^DH6-L>cdxbBqTHtwsJ*!`H6e(s5kQT
ztOFeXy!Dk4oycelw@Up#NT8#Q_7>GnhY;0&BZh@edw*(L7s~|iS>4Vmy_%H(S@ZLA
zc9F0&V8;`M*pThxQXN8%-%hq4Ob^W9=`K1L6DO$Z*?jVyd%C_6cM`7ih^3B1X|8w)
zq-{Lxc?vJ{j*jG=q@iBqFu
zKB58$3Lv>{2{|aPDEC{UIZBZ6Ktd_zn$gC+8<4J}X=ctNsGsgBH`@y}@arvB7@fG6<7p+Jfzv5
z6}Ug6i6`__WRreFT(ywQPC~3+chFVo&%_)!=6OIZ*90F>S_Wt6jOTpdphGIHN{wK
zA*K0v+5~0fGq1%ppwUe!BQ&UgSrfiQtb4Xy^<)NSznyM@nB-+tQn3q1nn;gN*>=_>
zDXjcU21%To^G5oU%E?_+xtdwfuw^zlw6s0}m5EY+3bw8>A)C(}U6(WPehd+0F;LiK
zp`Qs_tPl;GX+!iLpt>N|G*Y&Z@x~`VBA#9iMj#NXir+>nesUTghEs+(H#a7{+*ujdyTfZyg(&
zk+FmrlEnVVTPDChJde07(82PC%V91m&+@>;4QsndHSRDj#47xMNdnyb&WAu9Yrt0R
z`fMZm0uSF8dZw9-&ylo3KyHp0#IiDMxwTHaXNWScC`U5)wX^cqE&$UCXa_|8EJngU
zl_=vInD|OUC$SYeMkA^NxHvez?t2glvLZf?rZ@l#m}Bih=|g#>CQul;npIKvIN`6x
za{UyR_VW0&Z5z&iUKBk;K?0uOe@c#BENP`vs0gy9R7hl@nLY_!b$V$(CKKUsa>;+K
zb7X^&H8*%BGHvvO={~S9M|37FciW>h59N<5ETcavE{xdtP5XB?Kt5lq0L!WLh#^6n
ztS$DsmGCk6TJq+BYxSVf^~!ygf^5)m=&W^sG!F%#95)nyCMsZSeg-g@C)hpB?Qjz1
ze{z%C>})&Mvrs+x$$Bf$YU}`Hg{{e_#cto>3-Yx(VBJH7dny(@gOz6NOFhB1@tPi+
zM&f#~quC)tvT&IEAW&0?^&Q8i#l6>(X^CweZw2|Vqa~M7zD&i*2obqJLO(d9YYgoD
zrBrG(wQ**Dy*0qwY?99|{ZA|Tm0pv0!aYOYrd`ojJ1Nd4@TS2r+1r>aX5PGEZmgti
z{^`$v4pG^mHcc8Lye``qsT!NBW-T|X+jBo3XM_@u3Ac&OU+Q`p4}Yv0ebZ^0@!qL%
zT~GEo%0SLusl-GHBDbVRo?%BfTCm9lqOBC~Xu8UO!sv}YWc}wi_YVjauk9bmKNp?7
z-$b~H_EBQ=jP?V@>oE1Yfn3p{2{Gz$yUJsPff?zgH|QmwO0w-#-I3BSI4u%@<9cDV
zmC9yMqdNV(4!9C6zR6;x&tsdJx&SzCVRWmywmm3R2gNrdJ!48+COsN{&H_eE^l=5N
z3A5kh4X8CctwRObQAj%F5AvY>I-?YCeUEn7s_)dO4c-ZKop#(ak*%WTSw*a83)K1h
z0t5lyJ}}I?`rA6zMh)N7w3lLow{T<1Q11DEVKgO=@=xKHttrp!2ru>!KOaT!0qI*N
z;h;M>QZj8IoMy|`praz&)~!j!K4pG+(-YoNy7#tS!Ch<_(omuAaC*OFzwG|RSw9JX
zoTNAE1*Kr_KuN>Y%mPL~l#o`t;~mx6?ZAGOhcqG+s)oG_6L94qos3he-Ce2N8jrev
z4gTv2xhHi@jsq_}DWfYS@_Zzg@Dsz7dI7^T-oPwrsOE}4a{s=3A2;+kgl}J3H-GaTe~+ay59?{L>Ag
zU8fTlt7_93Im|hQhKf2u^{@#=>#;(AVv+TBH`v+DD>zQ+u@_#8*J*!-SrRoD1dtOQ
z#r@1y7a-S5*rpQ~X*i7`sNq9qR|$Z!@FJ5n{*px~GQYjvYs*M&kX>Mlb)wI-$E%2?
zVujZZy$3w8d%~KC^E(`D_7yqgG4Kn2gmOe82qIHoBSF1Yr6q1A*!Nnh`uwGT)^Ti)
zpP~xyc!H}Ze*WRu>~9a4UuzO=B=4GI=&DnR)qBF^-nGCDEBrVlrP9u>u;U(rkc|y&
zH9(2ob^6Y^s$^~=`(j4E9QgNXCL(*to829s(T8-MTha6-ijdCwkrp
zMFMZp&9seNd>I_IldgqbRZtvYlLQubcXtmgZi_>3cXtWy78Z948r(fda19>ZHMmQ#
z#T_np_t(AN+c#ZPH80;o*VJ@(qJHo{{Q=|VF7u~YeUX$J%imF;%>7;_Z;c_8CT0up
z%LQi>1-k>%wA$u$1`(r?&m>Qk|DiE3@N+WkbVMP^87@kRmg*$Z*40M!-%mZ0WXuDV
zT}S}5pRuHmH~|%yzArOEfb-v&0~6`{G^T~DE{N9QrGz*X{P#Z8i-fr{RUIcb<(Kt0F)^~o)P4}t9W|Y46Sl%a4^
zUkLb6Y~2+{!hJOje*V0!iG+D}mtIK;Or#}ZFVnSNH>>ADDpX8x#PV%GtpFqL+a8P5
z*Xr{$AI5lo9Giur)|=96OB>41UyVrF_i=W#u!U#Bg!UCx{Pje)y2+Ks#9QOTQ|E
z;UDn+=^#k9*_SWQBGsIkOm<+$UJIJO?}|e75@|#1uFjK2n@l9TYDT4?h5p%WlJ0k<
zs~E~bMVq?}DV`;S=YLwHpKRPy6GZRb2_Wi3dMe155DicB+mG9w;u!pp{Q0sxuO~WN
z>L11=jRax|xE@(O6i&C^?Ngb`$MQf+tJDujeCD_`nyJ{f^ObLy>MiyE|-
zXz6Fqdl%tV{t%*@gje8+VWX>O-TQ@f_WmJpy_$cSpS2$FrLe{k?Qeg1u|IRRc76_`
zR}4IU-IrmF6=G|VAfPMFg8WiU7|EDQZ;J=XJ-+ie@l&L73ux2VCyJD{^sy(BL*waZ
zMz+ypf^8*PPuwqtnX(=*mfs{rZ{7{8Rj);Ts1rO*8WT2JbbuloYviA4$GR=bDy=5G
zuosRulCd%6XZGB=AtQTc%>>!{rt;Z13M~qgyqagTi^C1RO4zSQ
zbO!K|I=``ZGli_jlnk-2ZHhyujyl#w1C%XfLMkx7$;77m^UAO?yEr)MF&u
zcX67CUjNo9t}K!RSMJzuu286nm=o?gh@}esq?sHQT`~G49>O`HUHeZi^irM5_X(F!
zKog~u)XcgCcYSXy4AjT)$
zgC`B2aezlhwLe?V-N#W=ne!GW^t6_V5G;0B8|pVAI0*ItOYs!b{SHl5`LXNcCRBQN
z4dZYJt5yhHeF;%^5?{;7RcAta;EZBtPns3_u@0g~@!xG4qQCEcJ2;~~4c-M}@rySc%smd#=Wm{ox~
z`etmvGes5P1EHx01?l9?pd*Cy;Yv$H?=T<`4!12wWJjW#+
z=Yj9gK}3Q5o5FcV2RDkLG?yszQ!`^+blvBTAzn8_f^0A>!mhbV1+q8wQqJ6l0-X7Y
znvormqZo3!=T22Aw)rK=iQN%cb38S|zGqkrS*c#&WnN1%1zH+BkI?aE-b|rEv$2Gm
zHn>)S^CL<>zZ!k;?JXT$|!Vx?1J3_7Wn1h%vsMvR76F*m7!Ub!qnoNmJ
zIk#LdW@lyDJk)__4ng2=x3Av0evfw;uTSLxjN$L(}iKXw!ma^IVgvutaI
z8+T(RvNpU*Nwdr9uT<#7>u%-({jpVF?>RQNO_zW|DVHjQ!AGr<`lcnu%>L@u(|A_*
zi9VrtUToWnyrrr|adv8X5j5wWxD~LTA*U=6V(Qj4n>nDPSI@2g`xvpPS)g{XHW_PaAk
z5BXSUQoNldg!KF)B^b~D&RRSyCE-Jk@uP~Mu=C!
zy>`BysH~@Zj|v^tD-%H*e>WV*Fjmu3?;?1RA9}p;>2)3rd8&+0<>TJMok0O#neHfr
zwtbyJ8mQP9@r3>k91(j+9CVM|t2xsmjV=;4Of!?wUHV03FEJz%HOxRDo~Do|7fQk$
zD!N?Xc(hvtQ;vqn!fr1$4w;jshLxpVot)Vjj7?a{=7&m>tBHwV_@#^^EC#*Tn*Mi5
z4x>pUKB`tNn_K+wD<))ze)AEGidiM%M*9P>2@fArcQktAey9Av`{`hmU~GY~K+i(B
z&!6V<#sY+4UkTPJm%sS4oT1cPd-V&=J*=fVhnX8FGimk-Yfz~A_050cXl%PRuN~JEnuZ`
zs^WcmL@-65U0+Cyn^}ahXb)?WPfS(+YTDpuw&IsD>ClgrFdt-0R~Mz?@RWo1Oge$<
zZ0~7kq7Ri+eL9qqdm((XBeY%c!QX{cI^cax63-Dzf{#ktup_YD{UGT%+SjHY;xeeeFoO1+FY`%3C7&IUm_`W7n|I|0#sOQ15
zxSzLPZfV-4=UTAhJ=_p^aocX>KSZJrZHafjoCr{#7DI@&-gYjkKRUK1GiWm%Dzg$6rTLgTe~TimnE$ZhursB{
zMVIq}QSxwoFVhjhAD1(n3BxYShwKbD)ow;sLhQ2J;q;^N4xr4+MPFN-;#~#!x8|hi
zj;Z||wpCXhkO%{~d&SX|_^fk6!EaC+Pl5hw2T`7yI0tQ?-D_b@kCe_eAjKh
z2{XZ_mNG(7LLyq@?a=p?C$xFp9xH|v8_2sk^6Hu5_=;oAC4@59e>-96a=r{zVF7cc
z3)D4ex3&b+)T1<;ZW?k7URC4Bb=($&-@BJIN
zcU)#X(BvDvMYtpN2B;#I^52=r;nJ}!bvp+Bup^w^ahg}2;<`M+LD4Eg8>j3>`CE@=
zF5n~U^nqfLJ+-jVj@h})wDnDwbj9%bN*`c}DjmQ>5Z@v^s>~o2m@DN}0%jAtLrL@V
zSS5wFbK(+&D+FE&2u$rJ9ubRF3KN=$&
z>GIoKRSVh6`yn|y|zJ|PY?o2?~hhitIEloda)@wXP;Q4Vn$6Q{}`fi0^n9Il-`
zZ0l!@U<@qq_|gDIZ#o~38yOb0JkAF32K%>Eu;YI1wZDO+Xzk&eAA;aBh>MyO!Nb&e
zP;*iQsfbNI-cL^ZN2_`*dm;wpK*mF{$~W@;w8J>lTa>)#=0TkXwGLy=Jb~{B6CG@-
zO)s4gRddFXDN!KcHi7Yx=!(yc>GC}g!{oo35?L;w~@V_(c|^pV66;
za*>6QN%jz&KJNnUa_wAt81ToPla|cu)Ab~9>gT*>L1y_S
z0^uS{RM5qlz`o2^u){gsMwK_-#w#B~H6%
zf;Vc9bbk+J!
zN+zZ^D?v6l-AC|=+eYBz8u(iI_XFdpTtV7AyG!NLkp{m@B=W(A$0BXE&Y25EriIJ<
z@lZDIpA=>t6=Ha^p#22H0;Q2r!_W1*hoYv%d_xpp$ZE@O#^`_v88K~J=>VU5aK3-T
zDo^E=pA7QxNYPBIPp1x%QVQb)plP;DART8G!5$|~D+ei8BoX`;1rDWqfI4HWi>SFK
zU;FhvzwoD4L3}#G^&b{Et3C36=DPZZmkb(XAst(!8R
zDIl&?bCTZ-$DQqWBo;xYyx1lu9O}pqDr`K(&>gtAg?Em=ZhmrxF7w08J0+V?DhWAd
z(o2aB(t?ob2#jW;n%8)x;2r9A~`hA1(ZdZh?J?Yv$=w6-0&SY)9_MChtjAWR;ra
zk(FrqTYxmW(fOG|}9
z!;?IQu__4)tPg>jSA!LFxh8WuX}^zoVBT(*x)gzDa(Por0x60~5e0{`EvlP#B$&QW
zAQ(COL}0|%ev7+(L(eBe)(_o(sNz50=755dt77~ok=0NrtT8Y-bnj3;eL74grf~(i
zS#VSm&PwRCPi+`+bZEjUUiJn&xx~v;q-|Gl`j+l0&-0>Iu}r|_pz=_A
zaJKuCQM8UJfR}eFSprsi&3~rv!J+!@vt5zhzZlduH)&6+0>Hy;nXjbPgJ0(KCEu>#
zdhc)6;b~Y}^$n9_UB${AAB)o?n8~}1sT=^ffsV_GVb~#G>FZlIN+-hSB4cdX^Y_GM
zWS}=z6j4a9ckGF8lu95tM^-;2se9EzVWt|!MstDamRn3NE1us#=O-1Bomi+eDikSItru~Zx=m;_nbV)Fcmu{}kq#C^~S=1e_mAqEH
zy^-=Ke$URT`@Pwh2rm)*xqtja*EpAC$A+8ln~ceh@^%dc9ELpxuo}+%Fcl
zL<5w5W?Lx&q2Oqtor1lXYWL~bZs=gHH7LN%0GF!#&g}b_h3uAI35nnKDAYK$SI&!W
zBZsFkMIL4!!olbhX`qQ~*8Yn%>g22v>;&B`nKD40dfz-o!h`h`WM}%RZn9PwfqN$HUdt`3b;ry7ZBwooqzb$md
zg47_mMYKRbeRYp{Eq*hbD6T_lAIn@h;}M`0V!hlGsu_E0B`x#*N?Fp-`|BZgSv)lc4QT_E3=0G4D{bM{hv<~~`-tSaC
zJlu*Rw4gSLg1~706XRIqb@T1(i}DPn$DdSsyCF?&tkKuzh6Lg8mLP)p7G+p>eLcga
zhLz;ej?Vb+ufoBjtMWx+C)i8TECx~O7If=Q)&7qMIPWq=e9>lR@>ICkiRw8O<;Rl@
z5{oWTKq-0Zalfv$M3PDNe%WlnNIVY`IEv%7W14?`M5&^LdEu&@N&i2O}kfIq#
z6VY;n)x4oW!7yEJxcWEF7o(7xXP7&UM*2_Icl+j|i`3Fr?CnE*|57dFxY?WT?KKu{
zpPC?>oq=%=F_)|qq4L=b$O;yj|FRDG?(w;AjwRtyLeJx4hNXevfKe*7_h2!yinV0H
z^dNgOX$-ewbyubEH&AM)(0=@0dEf+K^#81t8;vVG+kl5UU$S*aB?
zZoVkfHf?5?;>E<>itT!4wt57Cw8Nrpwuc(e+=))QlHlE?Xj-=^hn4X4{5$dpIQoEs
zM(yki%*z749YHPllFi8gUN7lKe50A{Fc9o;+8FsQmyeEl@3e)5>{tzsh+)cTg_t47
zN0KvTQPC7&P;@M5$wjo%b~ujgoB7FFHJ?D~dl)byKj;|CBwn{A6pqlI)~%n5|BEE_
z;F=u+Jso;?MZS}w8!f*tPY_|w<7SlWK0dRQ!>POo6wfZW)c;hxbP<)EnXBF>pE*G1
zpD}1H1qve%OxRRA)o0?_rrYW7GqfdK(XaiUrU={L
zt?*wd93a$9kb`}LH*8aKrR?{`HKw*9>TztBn?i)U$CvJv_{YI97=fSrYWc}_evD*L
zU<#+b@%k8WF3AMr>j6w?9W`Dh$$N4uL=JMTUz6-Q-vtwf;PA;BIlb>9*!Pt*wEo*)
z0;t6THL9z%wO$)2ow>W=O!*3^)F`b*vwH!Wijqanl&<1d#@DEd4G)hZ>J!I-h4N+(
zX%-2I=H|1aN9q(0rGWmo%fa_H-D!MB7{h+rTvB*OkDW7@?pGt-u|G04+7?FJXq#Ui
z&9ru0L~=gIzxxj?Ux)3)_B?9DXM9_R#>psbgTZfA353xQ%wU88;I_83!MMt0IM@LQ
zT0t;?Pc|9D)Bn(5JAjOJhWjo6uXTMIkb#}CcM8C7jot<5BW0N0{X-v50X9f5d>pN2
zcL2qIs>h7lcR(ZpFA&Jh1LVvQ{s55vUt#hANPynm?
z+5N(jo=49)Ey7IXeeG9EG}sN3`>Fqn8`_$T~j)kdC0bq
z$|;)$zKPVpwbEMd4SH51symnWH0IMtgiFAxnPJ(Gs>yxBg~=ne8`@^VqFYe1ZAFrq
zqpqry&{UJEAz?}&bf=-iN(n)&n2RDp*~d*Pjcdb+az}#A6gLD5j^U=Gl9q}YsRB~}
z#piUgKt-w6qx*-O!RW#AH3o|(W9#C72HFMDi+RrQCpWYq%As{;O3e|5^NYtI#y5$@
zVD-v<+s&dZOG?f`ZIhPQ3vx^X_VNWS!SZtYsH2|&OEN7&ppXgxK~2>4(;{r^C|
zbNEfmvAo21CUn8O1xS`r7CH)+72yN=R09H0N?&08H<&`>=A0U(e}C2m!s_*#{Z_R^
z8Hu18k0w_g*nz812kb&IQ893`;1az)9YL9hOi6|%ChG+fS{HL?9mxqY{x0uZs4=8K
z+am7<5ut~^z8jGb&PF9~#JV&@0a`iH>4~rpqTk;3q2IvW<3y`g*xa&V;`+(XfUb=A
zem(TpfvHf_KvJ0$e?nqV?}dJAi^7Jv@4%)5b|eP?C&QQj4R)1q`_m#*9}{Lf#@slE
zY#jvpJxB$DnZIiwEMGPcm3YM;cLNm(xkOii;p3AVpv2H1-QoYfnE)V-)Qhn3NjMVE
zk`)(DxBy>3gcXY|d890o^_5TR223>3-+^|7Ymj-N_P=gdo^DFY7%r{tRm<>1gnh(SQymPq>t5onm;
z5j)H+UUX;r5gq{(+iBkPR^|Kfj*)Rsc>*(e@Y$~AJ#8N*LF*3Wg*wXhkpsp|;<3$L
z6pY0l7#Q}*{@Z($dI%>V@si@`Vdn?FNM6Xg%`EF6lJd4UmR`S}tLv4vu|ec_pKm*t
zk1x~{*NCbCCbXD&F=R}~>?mNU16IFYfvL-Rj=`sC*iVo|YHD%*{*MT|InDYNsXo5)
zd=U6aokh21sjg)1grFf7J3R2f`E=P=44fF43kh1IcelqmoKl(I$}?J=sBzVYe!}|h
z4yT}Iz;U%UaT2)4R#RP{SgORo;dR%pMo1p`4B6QWoNUk=07f-h>aDPyXjJ39`#}Xh
z|ANfM_A|)l&Est@KuT*F(%AQ2sq0<~XxN(7Y0t#H$*UuUWvTPBgM@j98q^lHgr3_T?uy2IG7{(4Xs4Ej0h8DlDEQ#I
zYRiIVz7!5YF?iO%tVIOqgq8E>^gq$Zk_>H%RFM42XAtCz{vtQNz+gI&%PtUvCS%>h6fa;hxTZ@z2BVRbFj#YU
zNx0^hdi$jBd&>zTXW!KZ(lh?bj8Asu&h8DLxe7X%-o|~-J}H`uj)6I14@u}`4#sQ^
zbiFoI3Jt3W(ZCy0m=1yOikkP7s6)=A-X>^HWpN&pDF4wFlVpNxHRC9nCNtxzL$_6^
zbKCte4zkoOzJ3XDPFmDj0p9`+8&uuJZ#oHhs~i5V#fC|zo43W&=h3c@u<&=Ab)#Ch
z2zPDqqtXn1J`ITMg>_vjn4#trZxE-+=VN4z9D_Cdd)ZYl0!uBM==vXFHUm$GU3$Tes2Jx5Mt*nkP)EIqS8LL;MfeJb8&WhQqQfb0j
zdZkmmpV@FwZ-2!z?l=8X)QuNj`Yf${GhHfLe#PuUQ<)x(90Lpp`2lgGg#0jUVfCp2
zu8lw9lng)k_N@roc!PVZOl8NZ3HAOOkTm=|*ik`015ukJT8Awwqh7Xn?W!A1MN>&)
zzd~wAdZ>cfoo>vJ>8Xg$5cH0<3p9Md_lJk)r=dHLjX!#KF^nW{Y4$k%;k26K^j&QaHydNm<)YJBn9kSDgDdjVOa1|Ni1
zcrilO*Lmic9}I_=4KRENZ%3Q)f>K|q7p0)ufbZwYk6x=)=!CKY6d1ykxRF3D2b?sxYp+CGycVJs5
znp52r$X@t$%HLQ^FLK3xhaimP@E2r(AlJWR;D?9(-P=f8RB3-#%rq?0G*?7+b_zIZ
zP5Ya*I?PY8geBFoJ0&xAco+uC68X&u!sj$9(R%*rqK@qCOt~$OHyz-GuRnv8>d;eW
zFRakNf_KrGS>d|P-$dv$#+kSvUhqg0CQQd=*oygebf
zH^|Bv4gNd`hsfOTNoSPv--FQ7UvPpluy>}R%x&$~lK!6N@p?kxp@=0s-3Ak8Y!a>z
zybIQw9G=>v@_jy!=iwM!)BT(Mmu*^-vw-n~8I00EdBBe%dr>_A&grUfm=2x^vsdX?
zE?i3q5^u1X?K_D7N)|OPjron+ag4Hj?)*&-GyN|VAi8qO6*Kz+asXSjYD>aToUJ3~
zRaC-uJ(aY&{k>6iDGydHtA&w>w6#d&Bxm@ZrbTFS@+~UU#d>oIl;YIF#!$BI5Q%rb
z4lWvKa&2@(hxW?s$_qcQfAoJ*8<1Jzg$bHiTY=(mM#0^_Zqb9%olWN)Q5`YS%Iq=;u~q@R>8Q1NJydsPoulJ1_e5(pdYyK=iD9XpUu-N&3_*2sNe$SYd%{tt77
z;pfYML58J9HZ)mH~1~bMo*?^Kr?D1G%L5ffC#ToRT2Q|G(wmxc?AUZhj!;{}NTYxpoyzHQeE=
zn@;B-Xc@IZI9DlGI08d4LuhZf2RK}5I0GCZ5{nX53oCa89GovGEX?Mw0Mu}=7Wx($
zi@2Y2Pwx((rwxIJZtsg6609pe>5?R45PEWfF&3LKJ)`|TuP)4~DV|54o9@^z?1i6(
z@VWJQTUmwTYVSK4>4o5L_ClIwY)$I6)HPvvtOSzMv%Kdv1iccr!O~|2O4oGCKtl^Zs^7r%2={$~{K)=z4<`QO
z(ENyzrkD|?gy;iU{BXm#iB*AT{5e4>F8InUdEK6#urfQSvfEvM0Q?9?dcYWu;QcGZ
zFa?*)E#coRE_lC9b3zdO&}s)@*MqV-kXpA}c89wB+l#-yeRwO&kx}H9l^$K@UdTJs
z%L!=8b2Vp+d<=pWZ-Yu1?)^=L~x^x?|>YG(uzicQm!H?D{L!
zhL-)x>G8J6`FW}z-{vb}eqsGWUC&tu{DM&w<$9Zd8}07h<_eug1-Y}i<;xS-<`!SG
zyhGoP8W4@76UakFqbgn0Vsd%T1$`H$PvEJ
zui%kFqISO@b{5Dch=VNR0J}Rt0<8DIk?0X^iqeXd$CF>5s_rI5QmxTAvp{m#gF&)Y
zUEN*v^;eHE&F(Xr{q=^%=bvtGUcF@|bHr(`v)d0DCC+mAjb->+sqvSiW0un?Ho_54%HnDZdokeT@@@%h%<5Dr6I<7j+$8
zJ};DkuVdLRe!l(g)mv^yJ7RG+LTF;_Y&VZ_g3B~((Y9cm3QvDtMr&0~G+@z2%RHd%
zN?glpH@yplAV%9vOGUscUiqd#+0Vm-S8t_E*o!GmB-(2<^&w#{OO%FV5Y2S3)@?$m
zCfw+3Ig-?dd&hTp0W;TgT^#n5eT;!K$7;PFU%$4m<^q7xR7)1sx(kCJ9G4LLGn0mT(TmgKht~SSp?W8KrOj
za(na3O}`2^9HhW802R(Jb%yM)o13M&&UA_9`$q(m{avzIvePspi|#GOd3g519nhy$!q17_ij2s444
z;3F}x2XlY@dOV>=5dm(A0`F1|poCJLF-9DX`}w*XFjfZ6hvScg-XyTNqv2U)l=}0L}7q2#4&XZ9vY-vri>wiiYL(_i3Eu!&=FPP!wld!r)ofjWg*H!9F3Ac
z?&{Bh`izRy2QTF>M{w6|5g7gFOlA;Ok&5*r8lsfvf
z1YW|D@bZuHq1!dfABz2^-8ErsTXvfrM&EoLboiVPez-pr>#A%}VHZZ$WxJXTbh|JG
zQzL&k;{ue7mQ3-uLi6;&VP=6T3?u`hMYn(*T_o_k1jG$od2vFxq$y<_qH!;iVqr|O
z22`eU7knG7!hI~nR0;kWd4)d&vxI4rNl1JwH!lh=0$|`Eo=Xb^M)_HIK@HULaX?9O
zC36t!4B{0QWHYLqrlJB&%rPT`z}7?hAw_>SKn>iSyJD(s?206@LFQ!CHDB@miLn;~NmRM@%hMK9g
zMDxMgRm`Y%N8|t@0a9?;AH(2@Ti}O~6W8_fQygxBP+b;>CZ>5JKNMX_0xAJQvx0wA
za-dWjQ@UG!hNZB;W&I1#5)jIRXO)of;jS#t_lMmbFiu@=+c5A@b(=7!+xRn}nmvdW
zE1b9aJq$So#VJtT@ji-HU6@?^qSxpCpW@v8q3k+b_wj7=k;f!}T}Co)yF*cdR*>(Y
zV)mvf4#=j828x7dh4u3m*%PE7D^`D0<$r6!Z?S-cx1rR4_qN;@Xu$9lTmmoaqAQ}x
zK*e!^ZZK9HSXXUVHLGqYM`uS6zyQ)j`}+d7Y|&6upc;qU;7;b-B4PkPj$o^OcnvlW
z5)H~nk-m&-tGha?sH!5P@4R8aoZ;Pdil=)i>wLT`oPwh6e
zJK*)c*tg-$*RFZlI{+M5HnUY?)I3zd4lr^mD!DgtkF}JejDnI}#>%yB5v+jbGT21t
z$9))R%5v=wfgj!}(QgW{ULJp9sP$9sypu{mYZ(Nh-F4pX>f<1&^3|@1eF^}?;XVpQ
zcPL*XPo{fx_UBOZgJ_6uG`eP0G|jFHRbbps_~>!J4xiwS0IBj&KgPxjpzwr!km&+5
zA?{*X*<;^-jlg|s()jlQ-1sqUr~KkA3PLOgmo=*>T_jT)tjK*_^rwFs>}8|9kS&WY
z@F9SQq_RfpOBM@(#MAJ%AwdMI2nM+7Jnz9kPiZi^NhghCni4J(6w+--C;Zk>D?qtN
zYtRY<15uiM6GqU!ZFmJoY!vN;7sHn4!doTkGEFzu!Oy1V-&6y
z=x5~Yn#Op>ErbNE`*?+oKLM>N^x4iks`&q3s#
ztorUkho+z7AR~}$I>+<)6E9J0ZIdd%(m9-4#0n%)xXR2aP}vfR!#zZ
z^;E;hkIB4B2{p^BCMrW?Gz#=8&e~paT?xg8f%CCq6F+k{J>8WTq
z_Bo`rgP;{5J>U<>WL+T{r}9OpJ)p{j9Llk0RmtjdfALh>syN$hzS
zS`x;@)h_f%>Qa9f`ufh%{cODRtF>FI`oo-Bo$_SG5;)RC%W=`6(yfb8x8+?5-~zlh
zNZ5pq381TFsXA-}k!fs?%;{DX*kq{1-Xsq#xt(`v>_i~^eI#mtLfqu9@u9=R^p7`T
zaR2&3BSTMv5{FQyb!-8A8`zljX=BW3+b|fJ8`Jtgr*VI{3wcu!KHKWP2`tWh;4hG^
zSABkmF+4E)kdJ{_iCvSbt$Y^*2Eskv9h-IGqnnorQElBz_)H;rd%@aT6-nsH5EJ-`5&UNNcxfOC0m&@!
z&s^G#yfJHHLmw~kM-UTu4io7hy
z>1PHxbuGyKdNEG_7UTZMF^+vjjBCu~8E3t$CV|vLCNBzHB}WGat$czt{I{QZ^nmMtHPceUSl=3!
zlNlTk5B>va+H?{MWo~41baG{3Z3<;>WN%_>3N$#AAbk`9GcuPj;{hmt#adf$xzBgL
z+xf;Wviq#a{&rKu-|ug4zIsP>W{FiyWw&=3B?ZG!knVbf#3T`ZYqIgLwA;<}s)!nv5Q
zlPdp5SoIRJoLM62Y~h!4U{(IGpn2)qpGTehh~=*Z7tK20JHWnwt($u5CLgw;V|SN8
zkIpqLB!Ptf8WBupGexZCrUw(ZCW@+n=xt(CMWDWHIhu7$!A=RsQBgDBri`HgHfOr~
z{o0=>!w_?t@oOQAfFS32vun5LprO?r-op+|{-Ij!Lyw1_i+otH+BNIioBFw>DCk!J
z6h?`qCY<(^09X)z1G8UJBCH6r>I8P2Z~!x!(H}1wBci0
zw)b@tzRNg0MgyH=AOsJSS8W{J-AyJLZ}4IItFb~bXMQswju#{soWd4oi^yUrxS0TO
zbXP3>ZGC>o3SzCCMJZ`45ypm1f*ZzwS3i4S?BgK1qfqGb;Gk#MuB$fAifaehO&*t(
zVS)z|%?-eRt)^WceK^>4UHyI&hoTBz&J!L5HWMI~BF0h+!HCp$y2wE%Z2rBogu7i^
zd923KT+mEv4eJ!Wf_mY2{;}Qp%de_x{blGHl8p;|=pKhSp0^dyjeRhVVi^c1>4DQxU-+cz$C*RiWjs7p$Yg7e=Bv`wc!XBy6ayOIS_!8l?aukZ_dAmG08?Wsrj_
zDBjM07M*fK`XHnxuEliC<2Q>XRT?R32~+nCl|NrP-7=6Eb#^+vfIhb{Y{94Nyn(Cy
z-NL}++F7O@wD7W6gG#9-9DHkx-{dA1$%OV1yf^re>*Hx<)OPi_hw)W|!weH1%2RYr
zT}c38q?M&N_ehD>3x{t~p+Hga6ET8Mm^`U}K?!B+eqt*3EU6I$QFlo-v7DXU%6s{$
z>bkOh>V;efDZ2wK7FkBE>S}DbCwAkTb<%j5fg1~)XOTE%IPcGkC5W(>^yaU(H=k|>
zlCLPyNHIc5tjbo~n_qq_vNgtB*yq+{M?Z7x;)J#il1=vW%}=3NnH=o4a_L1#1#&%q
zIruRJu_OxX6){{)_~%p_oJuKXX>mysgl2}R*N_BTNhMQ~Jf$TOuj>={g%nP`4d(jyX?~P+uxHdF@`un^=6J0=wkV9*8<_+%VW&-s(=_Q;Lb_hgY
zq~)X1zR9i%R5t87;_*TfiUJ@Z-cv80a@F`qzNfVQl%N;oGjEblP@)QR9r-}`3Sm<6
zJ@@iSVJGrEyL6d<{YIXH>)=ZLLrzR@TSvQ%Px)wpBZ8r9yXYh^6}dlC1Mw>oCj9agUEA+o$K
z0DS)T#f<_(j=als|d_&1k*5
zuJ;Kt0Ip|sKbHkcxrq2}@Tnd(e1wxM>|(icH(7_~w4Ac@aw>Gz@wcB;ZuV6RT
z*^OtH!5n;19)1r1xvM>Yj-y0PXw4ysqnI3p25Y@b|S4tv=W(cZmxO`ZlF|4u%iSw4BJOq|SpsJwXYY-gk`
zb9!_OAEaa~tNngpIOY#R4@(C_ZZ9#+^b;-QGd+ilaxJQ8Lf>#Rq(NPkm20UxB
z5^@MI>f*vJjaxEX>PV1qbjxwZ`1cLR?_6IfA*xmBiJ&aO=pu5+p+*p${SIG5M@ooY+C6
z?sFws!u9Ze@|67JtUQsDSn%=FGw}4GeazP;uk4;k3dgsT55@>MV)zr@I$4Wyu=c@7
z>#qO%aEPQp8$j9DW-hR{?`zP<_Gw{6?n{B;pDZ$11OqBdL+pv%fSA3IQ;)%|2@j0Wh;UN)rqLH?vJvX$t`}v$tXE
z4FNE-qi=i-0Wq^6eUS+PH?y*a9S8w6m!bIq6SL!vRwk3rjuf*^wS{JrIJFiMGcY&`
zFHB`_XLM*XATl;GG&Pf8wG{+6HZeArG2;Ozf3&w{RGeGZHHy2tQ$TQccPF?@a0+*K
z_u%dt90Gyh?(Po3-CcrSa{BE)r+a*V@2F8Sm(I2J+Iv4mPNJwnFKl9G1eCC|b*5)#
zVB!JD$|>8~7}_$k(2Lkvn*dlCn3&k$$jL<=fric&cD7=M&Ojajr?WXg!Pxo3(GkGH
zf5gNMM-Gq#+5#OvlqLWp4}cud*-+KP9>@%!GW-iD+BrGX8yPx%sDZX-7Pdg@4;N88
zdk;qoGjr!ZF<9y8|3v!ZEy4hhF*LTab91t?02tbu0Av{C7y$BiZXZGm0F|9BzzAq=
zXl)9xGXs!jNo55!MHOm>e~-W_F820zj{nC+R7F)yk`5pyEUzjK0IJgg
zB-K<@|9q+fZ9n*%(E;RDKlFd{d>H<5mlIbNR@G7zXJ-6U1ArOe3UqX`_>=a3xKVsC
z1N=?x!`0N$&gQQK04j55XL}w-MmIM%1~V5YX9hb*GX{I>zxb(|TQ~vS>>RBCe;?nD
zKx^P%jd8Ix`DmxJIq+Wr{wxVV*1{NQ>jeDcBw_b2rOihxKb$_~&i^6y(Fo^1F|Ge)
z4sZej|0|8Tq0?W!vWkkb02@OKTW6rHp{?M
zVej;3%`9yHlF13nOG}8WsM5=R44y5$oZUxvY#E&0o&U1_6Hi!7mIuJa!~tODW&<#N
z45_%SiKv~8%?E2IxIgU^v-s$fvz?;{?Y;ER3B$hWulr!2K0n+Sb$#!2K`r2a|uR{%`rI
z{@SF}A3M~<&eqxkU;;FSe`A!lbN(0xs{eoX*#G2`aIvt!#l#P5=&$zf{1F9{dmAkH-ID3t$wLe-oEgR-pZ#arw(g
z+}7C6#KP7Lz{1V}Fm!Y@^nhdfm;n}cc7PZ2$0(Zs-TxXA03(B~o%4qaz~05#2ViRF
z2=`|pIoJV=!hb~nLL2}_k-rh=N9Ml~7l2XhZ^R8?6#oz6WCAcs{Ee6ajFNvN767Bv
z-{>RX=fBZMKH0y~e@8yK{~#__0Hey^hz-D~`ZxOUQ~w)%_-XtHaepAgztIP4qrcGy
zYvcbQ_79!0o%P51{9D4t_D5o4^S3`U(?@A0|9~I00sjF%iZlHivV6ohwfKknqXx5o
zz>o0ee<%BBg}H~lIneeW79X;I`hV17^$++#*!myvBb&|Ne~|fu=|4g-e=xQC2mD}a
z{}1@V%;E3V{DWM8PJa#Se{9$|{z&bdf&bdSfBSxv;rI{u!P)5_@Wbn$#vdiP`~!ZJ
z;Q9~vQG(k)`u9GH1^;`WDE_%1{yP7d{@ytMcXR%QRh%8|tbiI8
zCLd1?|1gm=e{^=VaMxk_IJ}rYJ}PO6%`HBl1U
zG4z%5#YYks9?ex}Pz%hITke{$7jYM;+#lXu4iegK7-?{`d-?(R~?iv|LH%&J<&Pa4p26PwAZxyws!nl664
zPpkEz+PSh;@|@q{IO7g7W%&BsFc5X#fwWq?Af$XjbDNQ$paK=bd6WlMTrZY2=2vP?
zzWhMN4=O%WU)rP+x5s~JJW_rCG7CH40j7WXfA;B$9Ag9dBJ>ncok-z*8EojnC3?G4
z$tPuFc@PU8mmU3vKdvsWWhrHiY5jX%I$hDUD
zhp00{lPZ7qaiYYpZWN;&qn&K)!)|lbs)rHer*W+37%c<87wY9hx{kRocua|Qo9OML
zf0eVvhSVBF)_1GSU_O=FK~eeageCZeUM8fae0tZ|2^7i2h~wq*&1iP_Uio&y3w0yO
zV_ZjoZ4Lf7%v{u1-{-xt4dw56p^V6`#$ih!gyvrO%R)A+iX-U6j-fn*I}-g5sA~U%tF}(e;i9_
z(y_<_(Nkn#q`(UFNKjx89`#WxC=GV^6ajP?uV=n9d^iq
z_NP1B(dzR+rZmYsl7g#-`Rg%D%^X&(S2Dyx`9L
zVMHhihuVfpzE8aup2BedfT)}0&gLU=ob9U|Ybsh#Kh
zjmpkITj~jXOv;4!z&UpG+0BdI^m5?PiDgD%*1FeaDk;S2hP1lp8V9thFWHfv2~_sk
zxgZPaSEo?K_TgSe-G7JcY5W9=?ft*Pw(Rn_PoUx0cRxnU?SwxZML>jUAn)tXXN2{VISCFFddZ5oX2x%%qDcaE2fo(5(PypnO&xCT
ziB|ScTMR$8<9S`J3tD=g)V{Q=T#w#27GS+3N
zn7o`w_AL)aU<+^NlRbB&5-kgQc!M1WLAf(KD@zeKM!&t*YIBHJaeK_%mSYcm8mZF>
zN#V0Oqo6s{JtB3#!DrAATry2r4hInvWSO+k=>{*XEDn3FyQTR(6i+sY3!*8e8EV$M
zLRDQv)^fbze|1L8tE;7Z=b(xRqxQt<^JMOSM;;b!$&c$lI1Q;LGo}k&c}#>zX{uoJ
zZ2KM-4QFm34xH+AdHuRc_QWx5(%3=%`w{h<=>$a2gtnXf4prnQ4U5Jzrk5uaGPs4|
zLs&NayFES48>(2xv;{yy0wXk-lX9DP3fknTJx(3Bf6T+Vy(_n#S?^?~LkCpG{y-nv
zqouU1w;MJO!V3JDggGBsD##>x*1dW2-EiDGb4t_agM>RGY)!$rQfBwG6s|(|)N
z{KETjA6IX3di8e+-6yetJMU9j?(er$m6UCl%+WB?iJGJqIm~ycdhV(Pj1tLsO4s5P
z54i9+b?Cn|%NeST@uVttl1!uQ2#&s2J*%VIe?4>chW|DcZ3%E*A-XyArwKIt)u@{7Lox+5txqsV@PrYM92y`SJEq!MdF*shRDdV?os!r{z#knWC4KsmW8{7CIk)>m#N*9t#Q^*<6Pu{*WNY`P)FKc>rz)2A^kxQI
zmQImD_S1t;)i!^Iy`I!Pya);;G(<^>447_#QnK!A*kE%o4;0`1TXf@#$gGs
z#1VRPHVxOo%Ya!?6G{V6DaRK>aTC6IiZYqgm1jgB9aHg;4FX~trn##oALkaFrM*=Y
zuLL&Wz_V!Yz)tjlX?b=^a;oms^n#Ex-ZPT&iJ~E=}1aIqXe=@$^Jvx>G*XSpg
z{d__+78p)JYU^xA0R!4mAFdBF7kgI{&4QHQmrTHLQEEI?NLk13XsP+Ea#c5+x@6!6
z@4TDlD~6tHwtRS}=yaUoist)*eB+NBA-=grEu!W!O
z+mPTsx1-4VYIl9v%`$d5%tfl8Rc~#I7n=w`a9L2hlMr
z<%98QB(Q1BAP>_FD$Uymu^IiU^|7S6z-5O`r-g)<|K3{|^+^4*R^z93V7%4a#Zf1C
zb$0L?O|z8b4G&M9iw|8?L-;(>vH5xx6nZfo6G#^)EW=cpU~uT`V)sNsB(JB@mj30Q
zK(O#oOv%|Qe^%uh1*{-t6{WRrM~*61=!zc2w~^!CRLu3RN*0R86sV;O?QyjQFUZXM
z!yBX>tz#3cWrA&@WtfG>&5dww`&sFgkOyfn%f_YAVcx7C46oE2-3b4e_O&b^r0%rZ__>p4LTTN=y&bD
zU}~^lL4>q$P8(+$S_af^1xfeodwfndqWc|kHDNDyq!7xpY0#|_T+W+Hlyq*b=|bxt
zq$P|LBKv$9Aad@vx&C734E1d{CG{YIQO;&c%mrr}U-rTPsUBUF+vv_U0TmLZ37=3a
zvZ`1_e>E9vzi(eq-M}O$myWTPoVqYT<&a5C|J@XSY{b<&6e0_8Z8=1n_qV~9RK@wj
zsdGkqCy*-_go1&sQ#+7RYoiwBX}p^0~$gNDvJ?&mC
z?<&44q!2(EC5ROc
zE%lm|?hU1*Mysvax(54huvjrR?-c|Fo{FDS-;F}l`Tc)w93I*c%xiJ&4nzHt2szLs
ziFyJ}f`8g~5sH2p-1klbGH=|cSWM2Tjg=RTUb695W$c0*STwaMXh&@tgKp-aynCln
zf3P^On2)>QIYMe7>Ltey1RbpZs4UlHJZIQzJOdr4tSus-c8-^3B9S4#67ph0RpI;A
z(1tr3w$Dg7T?Io(@0Y^KpmNA7&e>5cJbXPgx$p<*9;{jrN2#*?b(DaaBJe@fnyOUZuKOhMBraA-%Te8%&Wue5{llqU(r;-yWr
zY&RwCIX6$O5cMzwtZ2}}7mQ@vzN_4braoKCAlKCD?mjT-OdNwxpEKcjtN
z8Jutl*ms7WWFZ1QkEKE_>#Dx!egO@aa+@kd)%3hibP6vBzLuS@*c`S`oTEEgBk;kZ^YAabww02_pEYYGmz
zNXAbu)3YIa(C?@qf2gi>AaD5>J)cAUzW5MB!llNEen{py*-F_Lb^lyvRH~#%l9-jy
z8%XDolO{3{*HfORd9T#FarIIEv2av^9D@Eb-Ozm_pF+6#Fb|ACttWn
z(DcdGG#MG-hqQYA^GC$?(?DC-Oo1{xi(itm%czV*0=J+#UN5)4O=jfoE0W)ns67!_
z6Ep1RZCxg(;v~4!8oUMY!JKc~G_QSidKkIBHQGIGZ-KMav*pMc@PTt(I~3K}Zm%jd
z9LdamH<9W)f4H0v1YAUN6xRxfmvbTz+sdD@@jJ%0Ms@R-Cg=5<7!X48A}Dg!%vo=D
zzkUx5ljNQxK1Y_fA^mogbv4Uj7qsdMx?Gqy>l$JO;4qePbU+?jNzO2(%C0^)j*TLz
zB1E4KguNok&A`GPSTf`TQVr)_@5H3@LS(fH=DSl5f0eGSNx3SCF3P{5-$lde(0D_P
z8OwBKc$l3zc2=Ps1qZnC5Vy!pH)}rWE2G@oB{AEom&k3&#F+gso5UYYS}$x~biw%c
zoSWL)J*IUgROYCKa=Ry=-n|-F`AJ8K?SKFq+m}_F*yCKw?fgf`+&0qpeZB$>f}O6J
z(D5X~f3hY-xtZU;*?hS#IXCMbHH`AYv4Qn-=d@{?-Lr7lw+(L7>lHz_S(lu1y=n8T
zS4oi^mwpT;40h-Gv)QAM?_9A@_w(ODO6LtO8|m^-=A56DideHy?iz>nim=imrq*EQ
z)mD;MQkdcrP8h=cIlFql6iu1wh7&bJH2e+Bc7Jd+Rvhhp^q2>!`XgrR&VG~%-L
zl?|ui*pcRJu1xB&8)@~tDmMNE{h(EUm4=?al&WDaduZt~BC1E4S45PMW`~?QEOx_O
zUyqt|SVwWn+Zd~&D>(UrIjw+VLM%-1$_=7#)_iTdnQ3maqxlu#!tyk#fkOScnESHR$XyNaP%{L{f7<%@-UwBTiA(TZ_b7Jp_^*f1#nzRALcJe?
z({6DdytFboP?h!ls7jfFGF94r>t4|gpi;X1w|u6H#nlU*G~VFBe;T8qCJrajacAF+&DUEEk%yJd4GkpB>E1n!^g}d6
z8Yv^;N&#@@=-P(2N=;E1zoy<_AdY=Nm1ySdT?5N2F;_4t9yUC!{ARs^kYR;_>!ed0
zu!+Mv!ip$JXsxCg137snoIN)&$6b+r(e7WzNS`Ty#PiQZ=JLll>+BEwf4g2mFRb|H
zPwkjxorw~`t5*hBuxD?o^JNvb)TJL5WCQ2f|h_FUvT>s&@_?pwjOaDuMO)DddP
z7(sH`o`CSDRQ6%9u10NDQf559hJ$8aCs~4${n}^06=9&9w2Q;*p{71O8-|6r#E+71
zKPXA|aSQn>>gB_TG6n#&f5+N0eTPPS)CTBBfz8Urh74pg3hYc&&^t_iX;XePY6#n9
zh70|=^H=Y*iRra7TX!(`k@IiVoUW8-JagiE5SoNCfubFI%_ahu+Q
z^aEy%KppKH^np*RY8fMP6g>gS`L}&zM!ZZlu5+P8L6`UN#{
zFR_OWbYFgmVhIzo&MD0uehx<%(gn?#{R(FxgpZtu#P``FDD)_`)iO4b5R|rONIg^}
zk7Q_J{|9BEe=aD`8zl(|Nq@7xbr0+*m#z+}xmRq)D{ao(p4G8*M(ph6`vXR|t1LS-
z{*pW?Ka&~GjG~6Te+1YmtjX4P)5>s)o;95LPOe+AUSDWc0mRox?^Oq5fM
z!IYe(UAgAciD!778P#`t{mXJ3C(M+oMdP-IWyiz{qv*CP^++o;`og>2>FR#*KqJ>k
zN59;%<8psr9}$o|htvZX_9|Mxxb}@P_o+fu=d*u|Y^%H^xfZ;RuigJ7ML+To+6{l!
z$gQbhuoSvBfBFD(wu-NNtXf6^ejyni6+(1&1mfP3<*A8iphdl~7*{A1?6asesdOd5
zup~)G7PhKOPV8LFzK83|M{79qz&?f2I4h
ziRU766lyTlPv=Eq6+uMi}f2wzMrDPjlU`4gk`HJZfW96+bm+k7ZYM
zS$Jr=z^P~K+bO$el(7Ek|KkaDs+X8v7^Xk4HkzB>7^;Utud&fzJFs9sjh$oyOQf4e)93!TE!f%IzLd+n%>`TFx5q0jUx
z?Trrpj3Dt{I<5?a!Xw~*EHtu}Zm+gmd}X@XytDpjH%4tV+yiPw$zV}&yg
ze~rICQwY#`@@Iod<+&aV%Ng(PhY>!|^ik@s9dmp}ik>LmTh>9~&EXvK?(e5%tOGTQ
z{BFnmwV>Ce%*j0!;T|4ChsQVL(GbziZ#1z&n%XSq3U#i+#D2DTm0*0)#Cdom1F7gM
z`=I=)_%h`xVECGKDWqMbOKzA6QT7Q$e<7k6?uT|2S#SCa|EU_Cb`$(9xR$FIZKExh
zbtLL#+F28FWZuzx2}|oyHu1`oZWM4NE$=(
zBj@G&Z#)WimRLAFyYh+_wKDbitW~80j+^SBLvsX?GFl3heIQ#
zqE-#}D_m3u15i2Kb`<8xK=9~giW_!JYI5S^)zUrUD1C)>MzavF*=QXUqD=M)YnfP_
ztVm@t5yXu^k}^P@kfm(M1)kj$69wvKEZ{RT!+jZ|-RYy=d6P=-XNEn6f5n{$1Eh3t
z6NvU8F8CtjrHD7}`_=ishGs&~{=%%ago61ukQP!zH{((282_aohkKPgRQbD9KnoNn~9y;z9^Zb=g
zN9XvZP}X<3>+qw8WiCp2e?;1mLRZOC+HjX^3NdyIv6QXP>mlBnZ-p*7CeQ)~u
zuPhKLy@^C>o<1f&1*&C6E2Ze6s^>H>e+i1*2dHn9vEkZBP&AO+r;K6OONucxDfchJ
zitzdWqMgzpWD?q6e+XjGIdKZwmR7U$0e9go;&F#1>z_+GeYJsVE~*|H>a&`>Ved6L
zu+S2=@^FatR@sqN)Ch;Gqr1`)nKcE+#186R!Y`$%F;zNuL_KOCX%E5cUuzDvF~%Nl
zw}%E_Sbp$8l#-`e~Up6Sg?iWi=#!;V`xnAVf3
zAsJtg!if1*f1!rlfINrPR1r|U4bpD<%&v(}9oS~V75eVOFX|d&{1@n`n@P<|#TM=Q
z3E0m_^PClhM`|?}ytu6^a8P0^JK7Lgv?dM|tO7Z6)d?-SMt;SqTPSUx5SE~_{kd27
zd$0=O*Z2L^xo)%Z|uZ
zK#jg>c>c_GH=xiEhCeOrhIzX0IgS;T9-dqIe#5j}`BU+vo%AS9T+YT09-zdGTPn6+
z`m7bZ#}5u#ytXIkO4)z%Y4I7ao6110hg?X3+Z6HUE1t>tGF~c|FNST?DOs!BKj$Qq(BZ$G|6PFpH2U*T?
z58u`(7O^5MN`jGh-eirAF5Zu$2QOq9?G?*PnY&tCsD?wC?IL?q+OaZdtgd(0$SiM*
z?q5G)>i6d>dc)wkt0(8bfTi
z7v}iH6%r$oN0C=#0jyvAadvB%Ro}3Hu_EQ=gfD%=?*!%sGgjZ0CP{8Hb7MUHPIkOU
zfAQ%lTbI>V0mQ5@KK<5yNBjgWVPuo8XaZe*zz&HX)Ho0Yf31p)yz
zJ-m0z^s}b3C&JGnkhc;a-fSpbIc7!kOOOiBfgk>UjW2=D50gR;Y~W9JYk9gD+ZD>v
z-7M;;?5{^SqTX;Xu0T+myk)cJ{=_m~ZGoOy*qGVazR`gW#Vdt(F-5gq2vxHef89d$
zjq9|_?amyD(S2z_hsO8(v2Om~XLY8of%n4S{8r2qT}M>3N+c>Zz{Qs-^2h7I$V=RP
zUFF2?ir3kzmIpPgq)v^0mO~xi$f}iXcY9r;#jJN@?ZdQZWgRl0x>vJZ@v5m>L`*Yvt;lv9ZKve?c)EqI?dD
z5#v}vzXU@WEkJkX*+$iOc3ulaYYwV!@qQGT4)5a-Yl-!nb?sLKQ5H~`Jv5;p><(Hc
z3baKHCmuWE!PRSl6Jx<1_j00#;CP6GJFnuDcYk7=@p)F2!2>^R4&4$GDv1?R&E=d{
zd1iy7ir0~I^^lBdqiMMNe@h)%dXj+PfJe>a=s1LbZu>cEKUOs8>
zkQILIqUd%qRB>%F8d>&D#Dg7rWj5O`we}-#zJXlC!D4TL5&`0RV|{DMrw_yFcb$e^?LtKr3xJhiZDU1pT>x^bQrzALf$IWvD{3jbbq{4dXk
zFjIxePs-p!5^%5~$&DSl`^w0ZWEzbZCiun&1=$?*KeXK+DM#5gZx@O?YJ?uH5gY;y
z8gJA~Jprj<`v&UvfAW*x;RhTA)Zvj=p4kqoqzZOh^bB~S>|R|e{$u%p8krZw3>$ZLB+z008+07yW$zl{Gno6cfDk8~`>n{0`H(%j9};It+*
z1h>4-BYCvMQD$3SIFW0+&pF}-&W%NfE~_WDnkyVyX>Ld6s(;%T72y3+(t`E@SH{_b
z2)~eP4XPV-W;q8C3lH&<2(-2#Sx0hU2cyK|tO|@-L|QtF_I#nnfB2P4dK4P~8DBF@
zY1jr>-hVt}isS2C#?<^g9A1HY1ov{Sw&v~sNjE1NJ$H^{co@$A9P#wpK;uMqdi{#x
z)V?k=+@CXHL4VAJ#MuMjdtdQxQ@>=aKc>k9s0?DDBP*6pr(X7>H}1
zT!}tW-gJ1lTC>?8Kb4yG#I}`B>^I5ga}S`v(p4`3e-^ve5ubRqgfbj$txWyUD~?(v
z?MXS6kQTIP@iy-i3|`5WAP1+^JEMYM%GsWI=@S{927kdhh~LtPOLY7B2X}JjH*|F%
za$$=cQ-q2sI`w2-7Mn%vgfp1ia|F&2ZtJNO2)tj01-MYb5vX?^>N2S^eZ$L6LMn@2
z?ryjjzwWN+vJ4dxj9Cz2YF(;$f8yNgCOn-YGUUVn%DN6qT{FD{lmX
zGK&@jMiN$=o}YDZMqqepP#QPV*ibq>61nSg2!H&*W#qt>eVjw5rIXy%@()k;Nd{Ri
zB*iTczYtA|<7SFHjToUH+jiFoE+Bp&7XV0;XVt(1*pNkEGVbqvV}3+yUw%HO8r1Go
z;e=O{2<4=691T8}6;qks9%|ek`;d=?N!Pf$_|q69?LwE5b^Ls7iIvPOHC%99!14_WSI8z?G%Wk4lme?gtvebCIUxl`H`=v;E@Z
zi_#(nz-3{5mSBCWG00Skz3l!X99ADpqBVpI)2K~Ci;z2W4IQsraD^l>>CmO^9gjx7
z=qE%$X@KMN)Z1$mCWdsA)R-d|^iXSi+kZ<8arKF7tUH<4{a|^HIln$n#;IB+XnRX*c3z$sLc&i@I~28p(h1%=6l_I`WWet&ua
zgxB7L7{gF|T-ErygYp}rYq{!
zTPRNLDe+gonmISts%@){qOoC~1%DP0d+@W#@z;pT*f3>$-$#+**v$={1GMP9rP{7j
zgew8dK_7vI2~iPZ6@P{cgV@%23?99nv<(z0SXJ&vDGcjW5Fs+Y{*@vgk(CuYX}x}=
zVYIjoFM3(|b-T03Y+{`#!GAAfUi~USLC#GSM#(HOv^X$dz5zC6h3(X&Z6be7H=rPS
z{8$YxIF?n18x*nSs+JG>Z1-5_l%`bq(?d^&88J9XLOqpaTi{rdkF1Xyoj-vsByolSsneniomlxkfrI9-%G>FDHB
z*Gns0ocIzvT7N{GJhp0xZXVOHW^pGz5%JqIt@0$5{Jjcg<3=wrH;i>YiC$9wH>r|W
z&jK3qixaUeldj}CC0Wii0|K^H
zM)kkV%!Rs8Xbr=MweQ?v{oGDN&94X*bxM|5;3lt;}Cv=!dVN+d!oYqA6WU
z1bg@C4}XxAwMxGPlHyE!M`$WDrCWtX!44gm9Q=&bx0^lYfWcwEAB1UyY7UcYI7RwM
z{?=$skEK4|#3cD1%n4nckpmnLZhEgC@*_em6eUsnJoc@t#Kl^-7Ry?MqNR~4~WNV{iDOO`4r;R7&x(9
z&=)V=i9(WcfU(Yl0nkY|y@1<&u{1;OvPD>D>al{t+=1n7GyCW?>`F;){jtQu)9D)X
zgRl4Uw2q9Em;E6~qH!T#-|d;;k!2QVgiATJAXZ4(GJ-wz44gC8-f^8S%-yO0wYeTf
zA%C1uNSb<$;IRVuR*XQGZZTHxR;t>p(O(#nS;+!85(Y9cWZa7>hxv49RT~UtWWTN!!eaI67f&qLewdiwn%@wtvbx
zZ})1~B+F(h&LH81mF*oQNKIpzyJ?lV5$mUxWO%w9k*sFe+kHPV>7}F`4$68M+1e~v
zPZ#Dzt4ocHy?&%_p_=}&gj{{LDVY6pfwQ*^Sz4S=xVNg?QoOp2(JuKQ-b-&LZn+`(
zwZ(j@aOY{ssn)#gx@g7fNE4Z?5r4~~=@1h?sAd^D0-M;tQcqlw73izD?F7Dy1)--Y
z%&T~c^kh5TEvZ_Q
zB>1G7-5Nr=*PtpB`+ekxSBmS1h~m<&+#{fUbLRlYdjkWzZgSkAF(G!bPk(9h@H!As
zNr*i~4c=He>AaGk>q`*woA^0r+$u*fm?@<;i;;q=PhY@1PG5I0+G1danw*@$rP`XU
z_npnpu!;?OybHwprrjsPP-+kL|vsn
zXUYp$(h0tTsdtX&G4P@`m4Dc>O-s04i+pXZUkX|BsBfwUXyl@N98))Qqgc$DD+zY^
z+;~u0Xj8S_tSTWiW_R4w(7HIMwPzqsoOiIFJX`G76SdQ#bQcvzn#@!PJassj_NdPBm-w)H@j#yurZR}IO9gpqP-oLP#J=jK2eg8C%GR3md>VV=XJj+v9Akd9*zOqUm@d3+
zZLxsVu9Y`d(HEeLwtv(-AE5A-9EcD6<`*iXU#|}yXfq|(nyDz|^X4HK-2gLT8;=cw
zBKKfq7+)0h3Y8A5AgCN+%m5Wjk{>P|)85
zbNWFSh;rJa!BRA9OU4Us;9hAH{WUp*Y#!c;97w-j{K?`LI@OOHUVY35Wbw~RJ(MwT
zDW#lg*yrp4yAJ8UBwgAckEB_L23Oc`19YvKic7_@knD54q*#CD^H>D6OL^M}qokne
zY;I)_pwGFQM}I;HCSE&h^xv4KeErHTktbHh)ok#hc5_RUg=?kbS{)
z+uC*?dLxdmGOZE==bRK=V?yuu6T79fr+U>Mevl{N1+7GM5QCgIy1B&os)aRlQ>nFk
z@cHYP4t{EV*2h7njkEOq9}!DHX)AB}WCKrVdtXwl)b=SKK5G@?UH-uet*W=kb9|_*1GgPK#SjpSQ_$Odu^rR^w
zESpL$D1U=7JZH~rT3WK)11jfmcdQPlPs4qmIPBv9YhP=e*k&?7@-#@Yb-LzuzL~d(l
z^XJ{;lkSRuKW&1=!H%2$WWWl7dTYny^?zK@F+R0?(ZX|Npue=mKES<*R5P9#(UWWj
zXUtck9Ht+J9xCz~ubz%>&NAFy*gI|m8|<~McZl`ZsJXRwAH$24KWvFOp5RuiNCq;c
zfEb3%m(VY^d4Z%Gy6NeD+Vx?o+@|d{ZGRG5
z*;S*i1qPO+&N1ro>C?0#CUR}ogGkq{DtUP5e#i5~z%VMKUU6ysq%snm-X~GhX)CZ!
zvmCK-klzl|{kFQ2x&Q*g%!Z%+TjYS{j+M^Ph}TYul=Na!oT+n2ihu)?bh%?7uLP=?
z(qQShNzp4zP72MBg`#~{qyx)zUw<&(EB$A%=MP9hxpDMkQeqj!#AD-nvH4mj*TGB@
ztTM1j-GsL?s{WAmgiZcPmut`bz_=8L6z!qlOx|!EpK|-Er9g55=u8AR&v7u6_HUQA
zcV2H#$Sq0;%Ho>JYP&
zqCDr?V7#OWq|f-pb1X-;=Ao0;OIU=Y;DkP{v$TY)3Q%mytvSO7=+3`fz{NtYn$Fl|
zbNA8BTg5>zO7bYOd@KQ8K7SjGZH*{-`auN~VFT)#I6qLclP%@;T=SJtG+DhIK_$$`
zSe1Nf!}j6Jl`W_(x@6tO0oxNkiEr<@^Dsd3K?!xd;i>ZFlH@rqQ4n_(v89p1zwb&2E)yb$1%D?8ueGz^`R-WT
z6tZDqfHp1|NCMMSMZ&nPab+eOO!q)=d=L`X;
za>69l7an0T^u|8;5N3RSYlsjy)p;qV{3y6N`tSLw#VE$m_B(`P?|>Y}Qi}=rm&n8X
z=C{fGoRW-wIv^q-5r2?^sAA1t4u;OV4gyk4ixf!C_{$%at_`K&^j!y>;fHwUcos{E
zuTO_6)?1Vx&jEIM-^5SC2$Q*&`rS1pe9LTY1P{l@T*viKLKUun9Qw8QVpx@ZoOe~T
z?z**W3-!tdGp-~mtv+yqYx!yp>hOXNLfN4Q_)hLrf=>7V!hfPEaZXTf-U(1sNR+*V
zezS`}@tl;E#q^b1S&7Zy<63g^Vo*xzZy-kCZIKd##l?tu*Jzh9@FsW9J3HXnMr-@8
z(MR|DmyXgA9uvgm0u;Z$j2&rgsgE;L!6-@NYuSM=Mh=FNpcrQ!LB%;S=pjJVx7=g6
zVjg?FT?QeF1b;&9Iv`pYzu*^i2|oNVapHL%My8RC=HunKaDl*bGnI=V(v+GQRgSsi
z(4*2f(xiWmX^=;=%J7XtgaSc5=HVYjVc^IDA2!*8ZQC6=&3B;yi(9IBE?)Op6Ey)W
z0+tf!c304rR6D7DPQ>^=>-h3FbM>9I$4pZ1Prq||YJbmqV^W0eM2J)4Qv!7HEthA0
z2S%#?fW>R;LOh^7oxF=xxne&XMwXo_u0(%9sWuUXz3S-CXnokyL*a_Q1VhTXqj?Z*
zt{||bxQK?RdA?`)%@{_Zva1Z^#|_F*z=@%}%|tw~Zg7JY>w;d1ezPqq-|q)xrp`?M
zG;{{d>VGW8^8~=wtT+2y^@5(|;z(=B)wmXwFQKsMHu==V(0*MH+xrE_Uo2P6;n~Y@
zpZW{M*><@KB(S0leVoWE85CbJN<^NrbNY;hrO(rb*b){t?qCm<^Z9R~;%#X20L+PA%O^AjTfCy5tk;Jn7@W1ElcAh0EInnV
z!5P~gY*Vop->7zuDkN%q(tJA$1O=W08&sZgX<
zR61OnFwQzoW1YdE9!}pd*a<}2o|VPF+9dp|XWCY9TCy`paeY8wTUf*mS!n}7+P5*;
zm}``H!-;M{%u?WJeSu9wfvqB{Iw!!Vd~;S>eIueI)9J%
z1N}yhGgZc0Hi)73&m1^z8X3uH{-r`-QAc~5A8M8eGRCVbB8zkj5?xS*N$XJ$!{QXO
zgsihNk0$BP)3iCw9D%^s953*1dy*tMP~*pSv_ccqpqb;!<@K+rVJr2CPV7NTjn!n`
zey{ejJ&5Zzye?^)5$qb`jt*}tIDa=1&qbU561PwUAU>QYPcaht=$90gb(5?OkbfBz
zs^ZP*V;+8hm4sDOS;7P@Aey_O|Fz|E-tHKtw(GB-1{48zYWDQ~xe)z8+`0g&JXqF#Wn>!F?SF2dL5pwE`
zw(W?V)9TWYTV%_lpe0CRQp~4cxY8V2pKxV+&shA&`RQS1#(KFmViKS=1GJB1QN6dR
zq%pi}eH@(peqxjo)_=pEKAyd+*+b%X
zA|?0D+iJMo2NT^%kTZP}+M@uQx2-}_Y0|orHKmggWMPp4lK0a2Yqw8cKLN!3jogjv
zBZ?(dppnlO-3i>l18c7r*XAYryRXR4IZ)5d7=uZJzeo
zAIz9sffp?pI+H>u%9)i$*4=~|=@CCASOXq|Y;m5$|9av*;c|1u2n9}sU~^jT?m?OM
zU5xKBNE^^;>wQyXP#DpmZKP{(pev@z?I6kkpnolCvPhVNwvo}W@Ae@1DZ&Y}
z3+tcli@2GLa!bzEyo1h|YqDnjvBSfd>ga8QIsrFZ=AkM56>fX2(%c=#oxs=WDPl3q
z+!Z?6*XVufB}SrCl>Ob+M0VXoYw92$aYkg$gVGAto@LJ*vd(m{tL22wfXV}UelLIa
zN~gRFIe+AF9Kq{;QC)u0v7-t-8%%{LW%L*AG5#F^>$LPeCm&lDd24Q;&36;;y7{UV
zh56h5Kx^ovMzTfpPLxd8MvG}C*zeGEVsaQbixVfduWIc2XOJxcQ0gWJRm>J;utkpC
zGiXc;2AUtz1`;3d+!&XW~%)_vIMO}=_G7l6<)8NH`9^d51
zyW&VF*i!k0A2GQsRNW!T+&%vW^OkM*rO4uA_}OiDn`J>z1RXb(1lMsDjU6(8{S>2-
zg7zbLj-*E@N5h1Bo24ty_IM$s=
z*JO6TP!Q$fQKTi1?G3<2u!q~nY*ch)?fZfEo?UvH@!gmSIa^q)
z=YKB1FZ`iyQMd|Gog}g1qIh>5*|}accYpE9h(_-t!;bQ+mtO>?%U^*BC%%_Nkn+Nz
z?eTH7rT6V5vpp<6?~pte`G0~7$<}xq@^z-qwi02ow7lN;x^FEi7DCua9`)=l$w+UK
z`a0NU(i&Wj30xBA1Q15@fOhx|f_kz?VhNL`yVwEB3j-kCIdJ?K1kyp_Oz+Pax5&6>HoLH%
z>0u4|HDFF_HAp&xZ+}8uWVCLb`iut%CtE0Pq&3}ca|YRfCp>Q$uIOTxD#ZasnHnA{
z`Cb53VtFdg11QJN17cbg)k@~~cC8Y0J<4hLk78G}uU=z({{dvS5pwqDW3wZZgZ>J}
z3nM<-+L=IItBvP<)zkmYD_DH~hu4&A$sg>-znPu8Hc%@Rwo#0u+Bw@&
z1K`w)3m;!dHXtg`bVr38ur6B7y7!yC#b1*_Q7}3OGr4~Vu(;8sB}n3(P4S<(+zZcS
zwhPR1f0B%y3fk&3au*-cZOO!<-^dl?W{3A{w}4gT>JHTd$F*J`>50z^wO+_f0?zR*
zdh8on>iKKm<4tdm1*FadAz%tPt=!sjPO6p7FS*7x_lPNrLNu)e${TmxC-KwOBW|gA
zg236y0>gjRvJox_mWSV*X!1?x0+nWITLfflKYr!|Dz|X74maKR6F9vf&yzzTVEoZe
zz8?++LF2)#YOn1M!ffasE)7d$IyR(05-~JiU5JSwQ^oP?D+sM)q?r%;fh`0!q~5|4
z!lGH41dz`~2t4nq8nmu%ybr6+D{EU8^|u%_(!Xx{!_34DKJ&8T2oDs5Mx*Ks
zw+5Y_J&ON#;%yv6AuZ;5mdc;#TbgHUHbenVSDUQ;33d9WV<=Ew*kX0j`%Q^X!woE_
zii&>{=SNKczyv?BLalH%^^#&emUOYNd7%mRx0|7+_YT7U$@{3c`PkXnh4TESDx$*b
zR7ffo9mGU`_OG3#^BqeH(-Qj8@1zqwpL!126Y%YLW;fhQhcRSyoz@cC;vW$%Lwg7_
zXAFKNT1t9DmTuO&fvc4#sJR?CAajavtQLQlSJ~4-N(cTEz>PSPb=Lvapt9*sd4$jp
z1DIeDykK;c=8iNu9wkDMGgW1a(s<@}mJ-5f8w4XEP+YLi>HB^HUM&FMx*d<+5dBpc
z@>P7;M{q}x>+B>!!KA+OQitBlj8Z8BI7EbNvdV)w|B@Uju|I(iTot;w#EBhfr(u8I
zkdytq?@RKH*-V)eVwH4DW-@+FbyGM(*rWLwdSitRXhkI7$0;W;&_opT+YZT7!>4dW
zJKW97LR$lPS*JEGbJO=-`w}1%YL>$ko`ACQ5vESJijc7BN{iuh55zbSj=ks!e7O=b
zcpQ-!Ei1~KDQ>fzg(j@-;E$@GX?D~*C9SuCka3|QKz@TbKyv_C2)^`K~wD5NM
z(p?V-1+eM_?GW?SLRUIgWQ>rr%yrY7JWqc{#ZEY3)C8Q~pyomHM{h??TG{;Di&j(I
zzAcykPuw#6>{7m)J}WEN^x)2`{u!i8TSFb#Nd#I~)F)|^FXxeLY0wqv`YY<N)HHdEt1m+Gci3h
zkU~+wK~phYrF!M!eVWEJ02hCtsJUl&Zz6%#op`O8l2lo;Yfvr8yqn|_MIuMA*v+jn)|e(Y0*ZaISq?wGVMchs
zaOKF;7|GB~n7(uv5FWSC?E56WxJX_(%$Un^cG&-rE&6r5M%F-l`$d0iGY6dTy+3?h
zQ?>Fu^BQ4Z^LTPtMqp}1{`V)>XXav
zNDH5T^)*hkp=$bLm14(vq#F9@A%yL#LIuL2t_Sm!gzr09h_S;
zX>rf`aaG|+Q~1b}t3v|2CoT8U*GUvR7#J+5b`~y&t}@m1HS_-q0rp#MUrRq*+3uGL
zFQM7faD9XzYDhZ9IJAYikD<(7WaBE7#bR4RjQW2xd16<992FbbDQK{{a`k__!^id>5Vz#X)OpS9st-cvAHuD7`kPFkTPpxk!&x8_JfGHF^5zrlI{EI+^+IVmp^6gLyV>4vaM|?{>AbZ-
zFEd@~l&=<#f)T?oC3wq5`UHpE9I!oG>6$NeQBjXbAsT-K^eAMajgw$OC=`7Z!wQ)`
zc5V04B{I)@NnVuV;z7XQRiZSzQ5_V<^`ZW{_Qj?*8`2Jc&MfMeV?!}qDPVLs$MSpU
zD-YZ#IwWxy4HGbmY5EN{(S)u$O#6)nqTmH=m
z^69W=j~st>v^1Ht>lwH3ffTL|R-@Ap)|HB)G>#oGNGoIJMbyfuvuYu+8sJ}{Z7fEb
zbh(&F*?{|I=+MDUdDJ5Cj<`Z>16lB?fibIXw^NPZ&Xb=NMTBGM#f5;{k{rlQ%A8H&
zduSPVB9_+mZyVx0n$w70Eb^bohHDRk@mRVJh%A3{3@3v{aKQ`slJ}c1G!CtWGGuO%
zSE3iadJjq4$|XBmNAyQnL{|hgq~OE_U@|Z}Vbk}HaVbdZjR?OdI@J@zMiTVXn{8jk
z;Y-`=tC?N+RltKMBN?GdxDVhpo!e{N(vt22AhEWuP{7lN+0QSurY7n>RA(2iZX=iz
zI5K~E7!y{EGy~Uu!u(_R
zzHXA6C?)5|$(TjujgC?~_C2RiuB%`rklz4^h(8g8zP(5%e2RGe5W)H<=U34QH;
zzAxdjfwu?FKm96-&SOT73bA4g^%v=m=tF;hN7AA53V5WU*{Y^r1Zn;14C@{wy9svA
zx81a2sIrrzFJ}Va|2wayLW5N|jG+X~hr0)OWUi;#5%~L}DY%76)ZF9cOR4S~x^&#_
z{G@{3=ajE1r+n=KxEs`BY5~H6{
z9-&N;oM{~;N1VL@q)^}XhvPuP(2ZV1`$lMF53+H(k$IceQYC`FsGGz6zpPh^Qd{`B
zV6rX$!bP^?c{7UM}*&IK5^uTR4DX#~QApMM3to=_>a!%TC?
zr~P6X{6x(TFwbAKx=v$=wm1~0s`lIGwgiyagEYv?*D&G008py
z5%Q!3B^WCUtctUNWsUI&6xhO`_c{axlQC`O2xprSWd@g!9beC1t;3J})fJ1B;UNEu
z3bI|5J+8hW28C~y$ca0qa}@p<2~F=%hdRI-XUH_gvd$;2`@Y!!`Cgzu6rG8_m2~yl
z)pCSdaZ$J$%XE3F>8GzMM=gJJ&(9~G+;rc_g$4BD0fFr6{U)baSDkkQnc@I;A7TWk
z9v2{a?;-a(yb>Lav^gRFKmcWz-dp|TjQy|*AD>5qgzQZIQ%Ah6JO8aaVY)F7WD70E
z0>@UhcoH;!(QOb16NKtYFM`9Fwj!qkm1kEL>eZ@aVXj&^!AlSl5~+XE;{}#lpeu@J
zuCPs!U9G{mN>%zLef_wqcN(GmD2Nu0>X?P&yBY8t9v6`1ULnpZk`
z%+;}Y@zwtJsk{xg`Q35oHFaQM4nu~wm^7RKy1my
zBU9?L$D69}*H2TYDL8y?NbkdB8Xd)&fozjM^)Z!7;{GybOo>r2iDvaee|3=@TafxT
zA0q&6pJ=gt0|hrSaiqX$GA%;LlcNp;w@}{7^lN1aN3@XLmS=w&wMmcnpb?!SD58}a
zK^?D(0*5eX4`Za~?H4v!0}d{^!a>}R>_D>U%BK%B`3)S7f>0
zs+a0TOMj*1a+molLzwz59c+7;}$`H
zy9|ml6h(_*!(o3vvKTY94zzFTc%j>0m`t1K_*%NLhE0X4?VdETCy`nL)7V=KBX()r
z7ycSZ(hjJ`hSuaqtC)^aN68bi;w34v5mykapJcV__+v`gP?76|K*fQB1{=&+8TTAa
zun?{npJPHdd$IRNl=qr+L3TqqkqI+ZpV%i;7c
zbDLy^M>|1J%%s)95B44_FQ{qrN7wEuVMt&+i(A_Ht0*zJ^-9s*;N?zz3ZuyQWdNzBHss-o5zQhJqKSU}+yEnH?l7&1oi
zH*iSAim=dLlAu4$Ex&YlhCRBh{p4VfVu97VTFlXi?Xr|9E~kXoWqrsfM2HmY6`4>N
zTE01LuehA-8H2|%5iD-hkiL)1ye#WYD^q_T1IvzOS?)xBnFH*u$vs}3J{a_%a&!M7
zd5#=yjT>#V4QD{umYboALr6y5n)CdWiyY1aU;*7*q^ie@X}oW_)-^NpTQZZ1lM-
zo&9;2YTyQiV!gAM$Moo%Bj;kHfb958Exln*_M02qMMd=Gtd0-qNUDE;4y
zRST-pWii=VZr=D;db9kV{;Xz=)ZAudkPr95km~!c(
zDEPnSTx}idV;N`GV^&?#2}?9wJTk~}HlpTp91uacrJS!fE_Ch-vLuIAso~UWJ>iSG)-Fir>VOb`HiXC##V(ov})>L_Ck9~Hu
zk89Y-y<^xy+e%ARNWggOi}Vu8Fp47rn>8T!7uf>n)Y(HmA0UpfXv1*)0|5wfaqfM|
z=)j(JlifBM6NW0Gw#3L2+;Z35(@6#HtQ6MBonYJI;doH~?WZh?h%EAML?|X{kKP!@
zcql9izka`}h4b}$6gq#SYr@Yd_d%J~4i4VJzn(pD(`jR!93TOH!rKQ#%W4{YJL@l6@s2w?sHc&V*qOE
z(0ID$XPTwZLpp?WX$Obj0cJA@uAVyI=JZ?DeX(i%HE77-M23G|=P-_s$w{E|bWUhL
zhy@Cq$7y%VOnq+I8`hSi}3V6v*)WLm6#%ma6Oh
z`R?Dl1W?5sY*2)IgOVXQHrbkZ8d1IAt%
z`C@53+V;uyxpf;(XuOJ~i8X=e1f+sNsy3?@;`WJV0|zSGbeG_ZF|SQou!CD3bItl(
zB;nQX2*`hYU*b3m=-Nf6STJN<7sJS%lI3gK#|`o=krrk<&ni|g(Q$BwMH-Fj)$J@M
zaX{vy8HXNFiJaH&JJHzoa|au{FxsI(jo|5Pa~nL$1#6?75m951
ze@S`SsVj;9z-DKmLXytpfR@LCD(7~o-S$UB_)vdjNof_oZ-g3*@V>f^Z_E!c
zKbG;Q+GG$INVQI`X^{I5R!uBY55URX4SDo0y?Z_B$2vQhJ&^1;#O(4p4_Xq
z5mbL0;N_X_qjc4C-06M>Y7OLnOSJWivPTMST4Rvx#!iLHnK{~eyK5YJ5D;)kR$UUY
zb+_>yZlJc6F3_{6-2%vFCwae;bGP*J@Tovf8}b_DSsXKKb}~+5$5wau457UNjKads
z(+qJlZ9@6$i}hvC#%Aem9$Xu(nmf7Bn>BwM;&QGWf6=Jp%H;7!$9J|s6YaZ^%k*jq
zy5+AUl}C7hgl_=Z;G=q5-F*j{e~gsK1Uks;uk^+h7I?4Pt=r3ospMhU>4Su^__W7>
zmal0j6n@X*!ht67F{G;>AKNn7;e#~#>zovptch9@V8R@OEAu_GW)r0C<#FO|BME;X
zxNo4@$guTsJ$L9)jWj@Z>4Y&N8P1SrHPFb-fT>(5e9QzE
z?7Gvkqqq|PX)m4I0I8L$L|us4Wt)Ma+aJY?2eu^J!Jh4GieQ3;U9SMV(oc!TF=vBN
zoYWd{N6{y(QE)LhMHLDLBb6J4@S=aSqn!f?CfIf*Wc=c@2fm#&T}XQg+rz_!wuXj%
z>|SQL!lLJN`u*tnkDa@QCyRdwG8y=@t||LEmG($T?*V7hbu9uRtsfq0opNp=Ty~Yh
zLVu9erm??;Kaqf>BWvMfaMIdvwUM;)cPkl{+q8t>o1QBhoBOSVvHtbu)<=Mr_5{($-lCVBOC0@Cki;$hdfYh^Ks9Pzl_ZCnlao8rlb%Myb=Ls@bK9FVGU8#zLxD>bSe%p@QO(%!yMARh^)nr6>|BN*pB&2V
zE|4r_h1alb#k=;g$uob+TX5-rR$|DZn+
z?Cq@m3diOntzj1=hl~GuaD)#$kJ5uDeBD8jU1O2wzb9{mqA7p;c^q!9wzf3r&h@|P
zqLl5E96aUf|bv
zYle;p&^wJKIi!E>lUTd2K5~K$JCwsCRE`_%D1MLP^pVjFirv(9o@iZVX0^zQ?}3Bz
zsKiwC<+2=x9%8nN6m|s#mqFPBK>k$-T*b!0y^89A4w~mJ@~rZw>7qZ^I;N!Ozm~b)
z(?VMc0)db@wP!yWFusANyfmGdXEghQI`{hVIKftYErWl)wxdojO_r28*}OCBN6n5E
zL>2H?r*^dA+eUo^?Z5e&)NO7Zz27n9A^0usJX`vqk=Jtyw3^?I6R*%hj`-un@FuM|
zXE1cUW`>{}v}pH}1I)AXw|~NWG@#3@-Zs-H`uLWy>pt_3AEXBN#;2c+s&Pv_3}p-Z
z*8v$RnIeCsbuVZbJSlLf1PQP~Rl39LLv2$=S@((_BhN8KRJ!Xet5z*l)K9e)h9PyU
z1RBFmUN5Z-0t|(hpBtdm#NSDgEzaDT4&Cp4_lWLry^>kWap$MW<}B#0T9COI*!3ff
z#QmLU@0)`?ZCuH(dUGngdqv_cnu``02l;{;0mOeN>45=)v#9Z3T0?#?y7L|aj3RCm
zIa3HiQs^4I?}i&>-6d(iOR&p4*Z*cW(eyOAww~FzS>74(rORgN+|edIT8C!#kRWzy
z|LH@Arxm*B7XD{(d_8+M@On@iE%fT5bw9+zfvOT?TNdxk9-Xv9i897;|0rypt}WM4
zX_0>#c9`<)hy02cxe`RR|HBL_xFt#~i+4PT5JT@4D3_E#ra{Qf0l5a9V-QDYfp^S_
zraE|TD-gvauiFK6!Ps{aykDFuK^ah32|#g0*T2?0uI4WYE@yU`&66dI@2}nD=3e^w
zi!144QVd<3uGW2+*NnP-(_SfKXt$f+137;qHNs^VeT4SmH~BguARJtk^38HXr9P)$
zli^P46d&y6F{x0%2;8ah1}lnLMi3uZ8=)$x-|u3QR}La~a~k$(vH$Um77bK!>|lO-
zV=$VP+f$@!YU<)k!*Z*J&$Hc|dv19#ms%)i#0I{?H}_kZNdi_OQcw5Yksb+JDyM(@
zKo6+_1HIi7XF|TiO@qPYtGQAkZdtjBWwu}5oAmwUSd4PM_Hi?xkPe7k;*YDn8R#=3
z!+CFfIs5DkL{pI2;mK)G0g@N{Wf9l8rcpE}_mv_-tVvpuWwk{ntobM%?GYE0tOP@rIus`i%~Qi0e~@%rcv+!owY3!
zLT;SUfoAE9>2%pfLAPtt`#qz->o>YV+rxyI1ClvITx&@;tkaJGGv1m?bfe^=|DwVL*mlpX_pKAVGVQX=;|!vA@$k#sHf9Tlw-y60Dg7+g2Re=;J?
z+FUmhG@jxeZ&oM>h$l|Nog05?G3zb=!P(8U=&PZ48#1v(cjk>gD5ykCXKT*lan+s2
zyv(t`R(TNxJ=zFGerZTQj|7)^md=DT;1>;g+Mhj%n^NH>yU7NyY=yhTMC#C%k>G6C
z?jop-njanyJsVf}?pU0QA(okr3I1t|`>8#b*1A$&IjbA(ywPm5qJ;*;;f|6$g+@US^gz^M1{b?E8O(`HER^p!w#L7;r9a*m(m*n7Pr=j0jUy~
zzlZ@Cw{@!l5h9mgs{s_Zn#}>d3zvVJ0VoeLF$ynCWo~D5XdpB&HZ+$o;{hmt&0AY<
z8@UyJ*RPnD1zHq4yx)pNfy9Y7d-3izN!x9Mppb3FR-;H>N$w_n=x^_L4k-=C9wlQt
zEegaO4taP;e&HMoTpKM%&ofMkm#v)0)(1(>PFT+qhsS
zIyGKJ5I5fI#+W9e-?%1$gYiv&a1cq0
z0hc03y(4}ODj}%_Y$CR)Fw(_HEr3}<
zrV6^^r3n*-g?i^n=b$={IBP(rNZ8Pz;39Ql^IB00t(t~wz!+)mz&UB+jMfyf0cB|Y
z+JZH-6E%oev8qLn9%v?397)A2tt-(8o}FlzqxP7IlukQRH6V_ES!fxon1=RDDu;@m
zeDzf$UNqwC)yq{Qo;0rvjOckIj_2Fg%^&`N?umH$;cC{1pQabHi8xvS7=
zi08A->Uw=P+rYHA;be9`pFUo_Z(d0>c+#GvUr#W6jd9R#CM%q~`F{TE-)NIA=NoH3k>6JR^f-QB@*RZQYx|uyD7C)_5XD?>k<`v+7a@>fQv-jI(zotRCXnj*K
zTHa4GT8$P>pR~wErPf3{iqIggOA+oZ!Y{lNwR_u6Nu><<=oNl!qMQTiFDYkzm_p?5XG>u{=~qy}X5a9EG_Lne5SWxdNY8l&sS_ZcNAB6Y+SpKA)|b
zlYA|niEqRaUXb^_rm4@cWF&fRqHI*RmJc(gO^S|x0=6}Q$X;BZZoyJ~KVQBTkG}ef
zClHU$w)53eybynS{tf;8=5o8e+WcOK*|Pn0{&s#vyJ%PI3qfz^7`!UZ2kAM)$)8!OgF#9PAcK2@j8`;Tp+?#NaJWtY>m^SXmN08-AG!E
zkZGlVx~m_k6|`2$p6yo(-BZvZe+o7*GWQU@4Y%~x-$`#fp!|5b`XO2A)XD+54$*37
zTJJwsXgd_bkn06KB*sIj;y}WGU#WzfQVH&UQVD*vRQjaFE5n0ZsYlVO1gWTaSFs$*
zEJj&NeVTe;)QsjP=26`b^QcE--X|@XCkIb|l>VpQWeCU5;9b%!?-K7B6=U9J3=ay9
zIs#&3^VX_XB9SvLwLZ|HU$rrY&lMd{X4vcc3mrCp-(JpE>)E?11m1PYF18N-L@D1I
zBxu&z*3ms;XqD1;TQL^=>+H1MY^TdNn7EytUyFaAu0EWLvwyxnU9H}VeyEP)U3wgU
za_NLN9M>pXi;$|uk!lY_!Z=zveDM@n4BxdL$P~tgwO2mK
z7M}Q3(7M$C7uwh4dI+J710^(39xiH1=UnRxH}7$#*>SE~%8b=06JwibMLx9=Pr9u(
zO(_+PDP=wB&nm)m@ueFf@Y-(DhoP)5)F7`QDwXE1lrITOUj`XNw{AiU1@D8U2u2$tH;J<
z3x}@eu_C;C?uSTFLh)0ROk@R+B#t6B5?I2%p<-D~MxmY%OrX?7GDr%OlZPCPAtIbD-mG
z<2>!eL9U6vUxA~vggQ&sOE}oZ*vQ&qXoZ14LRwlS)(6H`v!zC2$y>I{u?nCyYcW~@
zCC(Zy$zuQH@7T*^Ujwz-`K@eW*;C}Y$U>Z-TeS%-8~Z
zb;Ay-qbLC8R^9Q4we#ImX9~j((j?}W1yc3q4RD=;!Yyz+;kg^(=i?Mf&F;N+wS%OZzQZ%+0S8ZcKtYkqOu<{NLU6sv5{OfOWOHF}N{0~iD)L8J
z7K#lmk-f&O3tVP55j?y@BPOYQ_3%}XH7Pccu$
zk5PY@((h91xhG0+xbBh@K-2`(%??5lK~|P3)f~hgk|?5pLd*
z^3a|vCHL&f9MeGyG<$r@VSSg0C6+|w0O_*VjD&S9*<5yisN}4XVg>#5CXq)r%#pR!
zdUk4&R%$~iax^(R;Z%U=;NV~&m^nm_K;XfC+DXHDM?sn{TIfeSlui`i9axsxqbp8G
zf@19<+0)Dvk#M-soZo0DgQ8y-flaRG(Dslr^T}n@oJ&-tleEEH56Q8BDX<2_WLr{A
z@)&@bGXXMx3s)!&1MjRO;X;8SDEfp5;Zo$!G=(#BDD_BrRu0K|mNGAwNH|ol!L!!X
zV3c}}aZJ;LDkGZ{)2eRRlhiZ??A-1p$^ufPt(2UhqR%uIS`T?6b)(YkW+6{eIo&#Y
z?QF{+F^W}duJ!Vd|8j0}J8u1-=O*0-%0MUo-*S_GQtr+uI+8sYOB$c+8Kj4lnNKc5
zN^CP-X(J`Rr6ovC@YyL_Iu!)^DI@pgQ&W^VwiBHRIfYg{njE-h7pr%C^0<0N$6h0-
z9rsMfCC+txhOU}CGp}JN^(;)L>46mJ$B>+YaE;<5{a&%_E$~vCrE@`kkNahHl)0b2
z?@-)-MwCu`{Mh&84dgnPbg=Al2|i{9c6^ra4LD-J?&$}2ba%ifCa*bJisiWDBOTKy
zgCTXMBPY*ksH6mme$6S%Sc;8g0~Fzv;%k7xhcV;A
z`NT#iO=>4kN53{zw2>e=$z#XpvRB}H0!of73!0J(if}%-tdjd!>uo;C)S@
z4x{OGY$9Qk>$&eCW#L(sn2A)Hfk}M(kxRahpi_G!V7s-}T#wvGzGVBW_xrA=drIr%_pZ3_$=g!P@cSDo)CUCB&
zc6UD6*fSpaARt3|&$1}v_O#TW?^-P1sbtA?eHkUH;g4kR*fh1vp_&gSxpcho9J4If
z>-JoayhBcB^8u)$==#jQmU|vj7LHsN9q4Y!I5FdQn@Uuq3sR68TuSr!rYYpC(=!?p0D7z9@(r3w7?gP(OagYiIqYK$kkl>81}1n9U56H
zZ0h~elEk1WP!`&bWPthej^@fLmy~?Gq?)XSN6W@vx@gw&AQv4ZvpNdm)CeFo7yjh3~Kymru>TApZ~Z=Mt=;MK*&T`mkW$csE~;_w84%;My8MG
z!egNGL_8J8(28f`JMlemdci3DrJ!`STCA2R-@Th=#Kar?oA+U>LHkAgay{Gdj}3mg
zUTtSDL{!R@TVM3Rf@X0bG3zt5G$qr|aq2+u3$8d$T>AU)*@@zi;ce+57r!d$oIG
z1S?Lv_Ztn{-fPED~1&R86zPVaVKXeV4
zf7=tWRd*5b@t=-=Prg1bBF^}KO1?;+vmccu4I*es@-%R>&?n!7;w
z!^w-2Z=d!d)SrM*?I6_tGePJMo5@vwbNguXA5^Tyb_!*eY_CC+|_q({D8CA1tRE=44
zt_9?ziYoL%rgp|aaXXMRJu?FnH$YZS+0NDo#LPsm3^aGKF>(a3FfcK(A(E4eI0B8F
zE$u*}M$SNP0H?DBK!3r+85|2V8T|zm
z?VOzHjg6eZZXn3q5(K0MhltqOdpKH}TR8vD!Aej6JJauIVFrM-k%_gPo0GLAzzAdt
zkYNu0ct8@$|?W}Wq$=VMHOlWaAFk~dwV;_
z|EonrMO96L4j?KduPO!rs?z}^)KpY||5OEn!1bHc0pwM|{@*;{g}=k)#8ib;wG_pe
z8GrWxU7F+nxVg7+47B-#KmmUJh^q0{=y0VdV5ztgND;NDCHiIjd>ihvRe}zW2mNp*$
zUiv>vG=RUWrINREv^BE%AD^X@xTQPLRMFDe#Nuyh|F%mx8-eFv2xM*p1aHejR$i&Xn5@ZfwVdnrCe>plDc_1=@XMu&C9pJ?b9&A&f`(MKXU}ONXcl>Z4Z?!)3w
zS77NaJnSuipnt3Y+bsV9!Hrn`1A;}j{s#mLW%CaRF83eJFoTQxlbIbH3Id*OVGBa4TKbhHoBcSU)RAmP*cd~T<
zM*ukQpLFctE}ShKf&Z8vaJSBGcK-+ftLO3$2o~P;9}uja+dswz9Q_Y67I1oxe?YLV
zo_}-!XYvF(B>qkH@9R$S_apzWXBpF<`|tnWh5o`S&W?80Kn+V%@O$7tmY2N$0UCc1
zEi7#3?nTcAo@{!SkKk*Q4ct0-7C3$W6>IYMYsp^^LGVTWAN>2B000EK15FT@7wk-U
zg00e;Ld$){il!=H$Uicie8%C|lumfFoH^A_h$ouA{T?V#9MJ5aMNw=gE6J_rn-Tyj
z)g%wbwt3lTNv@i_F;x=WG4hr3B}9J}6B^A^XHW~wl3Vhx=zLFoD4kcR^@(jJt2LpO
z7@#(KB=WJn>pq5M;uYg6^}TLI>skSfTge>rS1d=Hx9&^m1RaD+Qyq}bulHDS(MCN&
z%QUlwg`qg3(y;d5x-fSkqe}#yqNsMWxPM`aQQbRitf(Er=|nhnAQ3^l^CN#FsxX%T
zP$uaj;<#n@%HgI!hfktHCKNk$x&#zTO^HzE1|8uyVoWbRd}O`$))G`*4NMa|S?xJU
zvnJ!tu$Ays&I&be6Zq=RhvdeSLL&x<&nh^Uu78l@8qcD;b@{dVTz{Z#t2K)FvJL;M
zaoG?)GMN~Zb=<`OHChz)}+8mdk
z&&3ad%kY>0!VxmUusQSLI%%~MvkF~~VjDl<8}}ggNj{g}YH@uV-sFJ1L~m&UwGg8c
zY0f9t(zuZX`-|JHIdh4KBXPgtnREDlp?tz6mzOH4P(jIe8|tMFkG)C+9((5k
z*!&a*OM((IX3b0^+6ZF01vQ;@WUs8iszBM*EkxYfSZ)a_x*u3b2xfD~fulOmasz96
zHk}}%3>>7on=%)%-G$i`U#2eQUq3U$oN?~5+VqRzR8SO;UQ&O%th%q<%8E2x@i=xf
zu1HJQ4?iP9S52KKFxP`bSXf9D=ChR=u~m?c2);ObTodMklJ_JKk*?oBKR3SXjP`-y
zBQ*FKTQ&lDJpm$))ez7>=m}&Yl={k-!=K&P3!!%`TT~brMISZ(S?Z?x!2uFRA26QY
z01%Tyr1O^f$l8A%@EPrb=$+Dx@j|oqVvD^C^3=kWBI;@-d9V1rl_6CDx-ESve+gXA
zbhU{fixjQ>hh(|w`)B0spEu^k#EZpfN1<)=NTxbvzI<8fMF>!=POmOfxFV*am_4UC
z@RG$3tn`}nQLooi)BKye2~a273uO5vXFnlev}EbV=5N`nlPnlQ
zbO_gN4%LhugWEF`GA_fP=^8uJKdyhY-4mJ)fV5q>Q0JxyA(-FUL|QNcW`i57(UI`+7VKOoCj_uMsXQ;vjQ**J={QW}wkk7nbk17xEn@HgIpYBnWYOSid<4@v1rAO!O;f~(nM>~vLG
zR?C-nbdh7wm$O89h+3!>mSib(lBD4^s&?bCytSK;hn>b_K8o+5sA%kYFa2bX*n~Xa
zZdP&jE}e`=VIiaTGeR!9t~buzcKn{gRAexKrObcQ>oKpAB0$Q-S2XTWu4rHWB2qOn
zVZ@jD-xBB_v7Y3spkrs`^B~xvmlHYmIZ3#
z24yp_QM94Pf%3>!C+;@iNh#e|h5RUnI;6%c59^4|TM5A|;Hv{&A|**c)bms1FNEy&
z_3?k#E&Z*$HV^bjEz1aX)ow%srjlQft72)aoOqnvJaMT}Sb_rybq@}%n-htYC}A`@
z!0i%25!$qIKg*xyrJmQVn{-LV3wfUKI1EdVHd^w8Ht3yKYpQ|XnH)i|?nulYe7%~H
zxTxBIdL(aOFXwsB4=grI+pO@y0d-d3rQS)-%KTIW2&qp`{lUG
zV^VjHAXj-uU|QUMzzz;2`sco(m-ropL$_|P19l#Y(bp}G`y7SP&C=1A!>Fq%BL6G$
zGpy_9IJc=Pl-9|2!-VnQ2yB6Gd%TD@o~!uz&Dm(*unea@{cz&pB)m3cW7pZ<-+zC}
zB4yh-@BC(6Qj6f@81Lq!IxWbF*`LGRxgGUDycytF*K=q}QSRX;F}J<9&(0Z%{>b}c
zsq3BT7E`xQt7>r5k`%*m2$i(}q!=wkg8o3EUnek>TJFT#kr!6@I7ntp@-17ySV%aa
zy^*)rD!^!-qh|_|fNn)*agjg?XA6HxXVQg^BYTtQi?fmE3>1V(%+DRIJmXp_tDu)q
zp99tvF1w^EMlwR(vPPY&m<}9Aq`E%3%(PBTRXVW?4avHK0@f3s*D-Q^m}8?K5bVh|
zASNhNyKzXT$j>cR%I#AN6FK|CQ=bOtZHQk5Kdn*jcE1?8?4g>R4D%G@i-n7-G!5L=Yy#
zbhB|o!ox`VzgK^JtAQ1P=A@G^9-*pSv%kofrtQI_EaVoMAUR?sIhcQTEt@zxcL2@o
z)5ADByhK9sJhqODXk&@!QUH9+K~CH-cHVDcBji>sGg8C0#U`ncSkRtlhjJZ%<35J#
z`JyqcLqz85H^4Lg8Yb=HpfF4JyksUMKYy`K_bP5%kr0#v9lJQ%X_?gKtTxIiE6(&N
z#5r%^3LlQ|Pt$#}As&Ap(%xH|ESLGidVPTP9I*>XIIE4|WIZyHPq}$P-M`4ma3%Es
zW&TYNGnb`kh>;+WP_t9TJj9utU9DmuTftz3bb7m0fY(RLk@t&Y9(8;sM3x!(xJqvF
zv<_yw^LQud$t6GLIn>tb#WS@W0c5xfGccu(kw^Ux8
z5hg&2KyqUS+Tp<(sstk8rA9@zglcao_Bq5XLO{Kq$N=^lb^9Wu*7}8elRRIQ5yj7f
z@Z#Yk>=g!vMlH{9(cmck#)3pGOkU;*B19%eI=`Y%E_g0KP3&3^wo6jziV=as!u|
zmzH}EdZvSa0Bv4uX$UL^qdh$eq1v?HU^}KRGTpy0@+9Fr3AyRrT!jx3DtV0uQg!9l
zXuy+btllrmdo$sh;+drK7FFtDj>15oTj$PCgOJ3YN6mj-hC)>;0=`GQyNPe9YN}Rr
zpobS^1B%BKm^Rb%3Ut{Ld8K}(V99;Mv4TeC7z8Ru_L##+C&w-;X(1A5$h?z^FfFcvq0LV=dOqb5$Mu&YI
zcwo-6T_DO_63PK!raYnrUW5qbklzdk#seKz9WDDLCmWg#hb!n%ZwGf
zfoXaOh5qBO*swV@!IVPs$T4mg)a1$eo|(!8_7%Su+zv(YR{pIh3+Mf~~P#fyN$
z5UGDKFfQ>;v22jDGXuK|zJ%lX4z`(?$twS6T<&m?bWC;@r)`!>*6HJJSH`usG+Swo
zIwe}$I4c~M!)MIw_zAxlIPPePIrkUb@EX^+!xwI6wmku=ZXsW!nJ%q9M}bzxZv{2w
zEOnCF*kK$``rM%6Q)xL)9eUx_>8~iB`p|z3PwB@h*G}hTWejVl6N)*B*7G}kcRza0
zZfV68eyAwlg52k$E;kbyJ7yQ#EubhaXG4kKd8ASWScTcxwO0J1M1{r*+2Zdb6e3dL
z)*aYxRR6*usm_gJ^!CGWSWgaNR``Hc*YX(8eLta~yD$POTSjuM$ktGIBd3i42Yi3T
z2m1{;ZrmPADX8h!hF;pFFSfl3ulhwQ5A9#J;jaE#1KJj%;#w6Q65C94f&gH_Q5hnEx_pZt6z%U93<7?83C1*u*
zd%%4n|ME4!L~lXy=1nVB@8Rrxl52mu2G7HQOgD1(t=umtS*kk%n)ANrUMFd;-E>Rb
zUYELGrH?h6duPBEu-@3d7XIQnSce|A0Dw*bk<
z9A`f2+x47OSRU1%Te)3Yx1ISAGFecvBV^+$$&k8(O}ddn=rwI06PDi_5(xA@2+w
zA9E`0msmG#B%R8$t_7gR$bMX;WUI4P}(dFiy?n8gCO8qrC=bJr!
z1(Im3)ix_S=_d?Ex>S_jQM{xNfN=Y=t@@jfhk6{|Ho1QB1B5@Mu@8%O%OqrZHI<^w
ztv`uM6q?(mlVZoHhf`Ya)OgvT#f4ba>2tn!5TBCLZM9x`&;?S&aL5+WXg&q)5kXxj-j2?-_k-aDAPCY+nsAYqwM+^Jqs^xf5PT@(l
zE7L|9=0LYIZaFZu0pC>o?njOA6`KrD0Z>g!#qW
z9q&-_sUR1T_Wqf?@ajXQ9x?j&6{$Wfz1N<}qr~*WKNITYf-Zl*r)Zd2%XLD+MSBQ-
znA2;Jj5pf%*TnT|d{Xjed?vOeTIIO&mv>(|db;{))0E1Twr(akQhM!hE9tzVcSA*R
ziCn9O)vTH$mfL#Pqkf5E*6zNl`8MHU~~P}KKzAy?b`t<_K${E@@WA*e4R(@%tfH
zK$C*Tu3CSd!+P6aiDjl$`)b;zmz2#k?8?=xaxKYkMI)ZEd$cF7nK*!4S5Eitpp{MI
zg$yy>^IpS1r(?z?MC*vLsY})aN$o8wDp!UB=7|Omr|wU^yh(1WT#
zVaTO+Qf=K&!02Dz&2yRfodNx&mv8*$r}FlcE`?KcqIB^M_Z~5xXK>|75-q<+1{dsq
zK>~l7!XeE&2SgEwKX~#g|b@Te(&spe0E1g2tr5viv1w#+vX4@
zYy)VdikxEeu<(yXW-f`CazwzW9XzJGvB&2v;DcjN_mSSl=6A(U(EWjp#ToK(K-zz>
zoW3SzMZp6By%j_|k0$f^#d}5J#dPhb770#QKXLwQmY=~2P4n4RS=YO14xvc^H%h`{
z7@bE99V%8L=kPgdOVC?g&2M=pxPcQKjOxCI;@98Cq{HIAD->T%h5h(tUf4Ep57A;G
z&2L`&(`cAh@WDpewKG{9sxwR%HHLpDt6BDhytGvaT9l=6#}GP%brhabOPRUCF_xo~
zT>Qj9FrQWxBFI4DJmp@1C;mr%O%OL-Ut2ko>qxsx4xc?u2s4rMaglZpi$j|RDTR~1
zBY(pYRCXkAuk2i_(ljGdteIRi&6-5}EiXZ|0=4SAbsSks&irK)7cPA}rq6#?rP@nj
zg+hJwylpieu)&6Ob8lvpQ*|VOh{C)jjQYIZzfA8Gp3$c1sD*`xblyM4#A?kwU8J0Z
zbij>)0@uWGtAI#z-aDnU%+|(`k#cU@$$kcpFD3cj6{<8VT=0GR)@sRZ7xejQZ2yyeIlHk>W4su
z#K<>M1G5WZ$z-z%bO*@`c)6vp1SaqUC=c)QR(ayjt3{7qrCfZK0OEf^I?g$H-yZC=
z>lVq%A9&q`p})l8vIdRVL?@0zYeA%2TRCX7;AgO9CNLHpZj~uv)CO&NmRzAJy65mn
z9*`}$K-t{DnZQ73d8Xz&o{wS)9LL9xJtSWxQ(NAbslAToSHHNf4JD-Z)P?an>Snan
z`+>GnF3b2IfI)~!Ux$C+0A6Kc2}fN#a2I!v*cu1R_GL0~kZR)E{OtGL&#ViU1Gr^d
zfAm@#=BpE%EOFubgXr({lW~~CqVRBcw&A-T?Cx-q9xkCPx^-oC^o^?-0?=Qqkz5L#
zllwXtuaR&mV^mZux<+6`zJFhbV)M1;t~M=~+8gp7`B+VY+g^WE-$bgM-CiSW4C5d9
zz8(P%H5uh1^PR0#M_dyXslc1V%7`)B1SQ|%FV8#F?xV{Qnapr2l15Lprn2HV8bB1q
zQT9Q}2Avh{N8DeeQ|7NohNZC-C-z^m!UIm3w;h(R%VA?!KCepHz1e#`8Xfl4^~x4f
zaMn0BlX-8P?kIoNZ}=%LY}NxsTYQRUtm_C;l8V2I!I)%0G5?(i7fe&#QeU5hPzn1o
za5oN3dKwE3bVtv(c
zwBeHi!dEIm{%By&F_y`sFx9WN6z;;LQ!$?@JCpm>aOi&mLk@qg71Bt>Q-(p!uDaFh
zR$&<)h^BfvdErO0Cc)hqhoypU)xlApz(k+@Q^p9Ai;dhmXk$~a#J7EXm-s93#xYW$UfC^1h2I
zq3nm=Fi3w#^Ma|qA19~9?Lfgjn~}eK*dhEUqlW2GC_Zhh*lp(d^ycN!|FAN{Q@sY5
z<^|5-Ai_~Xpkv`c^_FT-3a+B~ZV__}*OXW1pi;09r+OK=v-I53sVo+rKk
zjyw^!DX6E+vjnSCV6l_tE$f6v!Phd+-IVJ?eEGQBEQPJzM4uo`=f_PRZ85z9R!gQ;Py92
z8A3|hqkF!g^p3KB7tn>E^gPhR1@!bGq^c~l>_Kv(sU(k3u`6oePYgzyNw=po8nPIJ
zemKYe0NY`u
zMy(dSCz|JBi06-#i|Dr{ASc?RtntE$t6TV4qzwNzo6Xa@+I$XMhe3$nipOdUq5^XtG(~^ffoT-Gw0QVJZqlFwjiBar)L(
zQvSx_>zqSxN3)$$u2%jaMv7lbn!&;r$rPN(%SNiQR?&&3#O#l*uymgs*}Bz97b53K
z-w8B@wGjcrfcs%G$Kc^S9jezb$3=3+no_6Ku=g4;x#Cs&EJ|q?Q49+!7tVjqPv6Z8
zkSuoAQ265Pqs?BwzZ+U%xKkj)Tr5G@DXp*7m^InYMgxQemhkw5hgHjOC6>CUtD`QV
z#rb>~^mnsMV%;LFMdasse}-*=nnpiZ*HDwJKRMfHT7gxRbeCG#YD0fw)Xu2>Xn1}t
z)mR|{dJeO!I8fx>CI6;K-@AV$t(Ek-j^B+9hgoZ97^0X568oq1s>|WSo#<9rHfO=U
z#+zb>yT@C-NHn|U9aNq-$UkP7p;A`9alO2iGV3#;^=^^P`AMl2f4sK{HC&{h31Jtz
zdMt-jBf*5^-+M1Q$xDc}Y0D6xNZdSRlS$a&gf8}-Q-nbm`CJSw?@50uHcy)MP0Im@
zu;!bq5TCST({O5^yaQw4vKq9GEX0CGJ(f^En^lri6WaVsaywh~k`_luf@7I2x_bK3
z-3S6=-7X?&vcr8bv-77k$bMGKbDDE}vSUivi5VJ|lM|7<8)^oMjcGLHW%@F}OX7qE
z>1d$@>F#&mJFbM~SowcKdHE;JZH;F+J<|6z_>aKL-9(`g6rMax$We=2E@o8z53x%V
zMljd0s);vIEQo|49@JIZ;ucSY=`-Xno(|kQYGw>|vqQp@y*K@~5{#3L!@{>pepUDg
zvvJ2fRo>0Ai(_snWtaQ-P6#VgY?-NAI_EYa*;kYddeX9h01Mwk2lyP=~}pHV6dijpZ1*;4y@
z^$n^|e-wQu({}y@kju?0bBvK0`GnR;*&x_Ga{4W7%9Vf8%;@6}o1Zgol_eENY84(3
zLA-Jv2bU5ZtM3Nj!u8)iJ~)YFAYu=?QPAt-!xknXiO7G8od~!iIRc;_HS>m
zgl9R2<7bEIJuXD-CEI8;W{yYAa3zMd^BdvU=(7_@l%8E;!S8KyUR%XW5cOPFf5g4L
zFG`R==zSu&i_7`&(C=ZN-1|y-UjjCBCiP*-Q}o;=O-${0`HcGk0xJ9qxY5(hjqRU^8uc
zA!Zuxs<``NRg4zXL!xDk$S!$WwhnK4pr956rXrzIC&Y$gWw$K8fs>_=L5YX>Du
zGZ2Do|Kr@riltz-!##0lpHx?X~M8{(4>UH9uEQX-7kD#}H
zye^svatwEIvw++OXy_=6w^4}|9ZX_z+{{nBxI1Y48uRRjyFa5)h!cn~r-_qYhJVLg
zp`Kq6O$w*dsQ7WzQ-hZ!MQ0dwJLi~l{nmf2>l0xlkO&&(E@r>_lYQU%wEiX+Cy_>?
zb(Oo@p+9tNLlz!PxqLF}mwn8uq5AaobFI)0iKZl7Posa{08}IvK>Bqj?>?Mp;M5OqZ&8
zF_}T|D;#k7!85$daVdqLyhWJqtlQQiKnnB;5bTiAxsC;p5ei<;}zy
z{e)c-SU5PGyocFb0FSg3v=Nh*K8u*UVaai%?5ERE{U&EIN7B>$=w0Uf68N(+DFv@N
zov4h2DHvHUDu|xQ>yd!Bc(8xb+)tbP)uBxF664uzf*ke*SQ>kZb$iG9@~{D+G-2NU
zuLtNCu2XR6y-*E%2UUU#{k0dz2Bei#fp^@vSJ&b94hE6bo{4H|t0lBu88Z~OfjODp
z43JM0$cGh_!SEcp8%Y^^b(9O@54B}R^dm$=E1ul<&js0CP|f`x9|3XxQjqM?c
z125_etay5&_w(;Y2~mGrJ(Qgi&~U8c9|^cJ*(kkA$?V1+Wp($=cpJjG69LybkfL=_
z0W0UI7<5C1#q$av?)i3?1wX!KQbsIZN|s!KD^##|RhNJ&LUU9N4#iYHb?;
zx!VJf+rv9SVvwh6$DJ=~x%1&5yzxGm!#KTHsjPBxFB>sr%HMpmiu+38gGSn`1sAjM
zJgg4$npCbW-e-R%`uQ%MY|R~Jdm(Zr{6rl>FJQu~-MTwEX1!6U(Sax#E`3>#F)OCm
zn5O`W4+NP=FwWb2F-Ia1?~QMI^YNfy>YM<4?T{K#n|?tw3zos4aaCyeYALhV3coOd
z8i9O6*}a@}Pb-rC!0^sy>pJ~&Ye(b8`RmR1x9;!-^y`0xibQ5=LIiSG-+^($vud}W
zgw`YHpx?olYoSwfUEXjz8xBshLDoRLWz0;DQzo<{tvT6*N$XX@j0cc5h8VH!WA0}r
zIL#*+>56KFpor3+BQ#}Cxr6s|9##uG6YPs(eSi5wY)WeJ#Aa&*vTYCpdo`ME41I!?_nTarJjP
zW%hoZw`CkaxqLwA4f`t}hJyLgxQHdPF)
zXG@In1IilbTAOL}d3uWv(A_%PdZA)nOh(_!#o?`&q6pgQPY3A
z=S%xzCw3zW7>w_B*9>ME&x&WY=rgGRzAltzZ#6y~K+njm2k}dO+71d#qTV8wuQP_t
zxZ(KdBZfT6Vc?A|rtEr(Hm)G(r54E+Flm3BB{g7ErNOJwKwm=YDrRz*>myGA|
z*MdyIw;G+|o7HBeK+6hYx1x$)s>qy-Wq=z)hHj}MLq(COlp8boy!uRc;PR)CJ+
zuh48t3^QVulV!k$yqfAmMO%MH^o<`*TW$-Kne|Sy7B_KKwm@}LiJ;f*k18#>U7u=h
zmBh5>yjcWm=ZX-y=ufq8Lx#D^pa*BCmR*kzs#Rud*4q-TRnL)yT4K=Im$#P}mVss*
z9#^i{>mzW`~cdP}t^k2U}iUWE@_7x0n#Ld<`}S%I+bL8n9Q
z>)%JM!ulZ8@5tq?l?JW;+qbr|x
zwMw6jRS%E#s40J~I}H0aKW9_T^7uPoXSN0@iksfwRk>sHo^NNIIcrM%dJ;O6e^=gA
zmzE-#Nonj#>u1P}Nke}SK-q7`2UP?SlOlY9#40Q(zja#t#rm@o)LkQmB!g_C#(2Lv
zvz%@kJ*!y2sr{YSFiOT21Paoliq;#t#c?)vGyIx*2IXJ=_N^dlnYOT;*iAAe-Jqg|2_LxvhYwpFchI2X^dR
z6!1%AxmSLAJ-si4V{b^x73pIArfPPub~LBUgxEr7c#2`Hh!5>#o0H}NnnHB
zU!^tByXznoXdzRv;L14JKjHg+TiZO=#g~0w^XiV0{P}Bw(uX>cWriB8=Fc9ybX?wj
z&lL*6{eHG@0-b-yXqZIQL?d#u2Bly8_xq4%N~Zu%K(N2d^sOjrNX2z=mNd=|{9)U_
z2D|LGB200LEt*uR5TrPbeLVdv6dCn%yt94Iw-n*C&S{u%-`yx|3-jcbY_9YhYGQvr
zEC$Z~&!txN9-J{E3=QTHUGuJi)>zx~dXYj>8W(ZLCpuzdVdq(Yr_>WwioMWCpltdC
zdI%RmF(go)MV${c3@3|2aknK4z3@;{5sGs0r<2x@)@fpOSl0Kr5G9LwQ)0B$Yh@bd
z(*eR*a&E2lr--5wn#kPdM{_o73B1F}oCF*D6Fkh7XoL{ojA3o_r0<60Mi6_{>bM`!m2lo_1Nf4#_?H+kAO+@!;Njq1Sx94
zQ4RBB*NUKRnQs2q9+1WTZfL^khgB5r_Ws3oefbxEH)vOfhZ=YjZp>cjqb(-^=0=Vg-TNT<)?QY
zq+$tw&6nA!wg!r(@XTu%2PYODt6V+ZY$LL`8}WPRi+5#BHa`ZoB}GK4P37)8JR1V|
zm0W}?50$62;`O8=Xzev)98F~9Raz^R9~hi&#tC4%X;L
z92sD~jh!Lmbzvz(iaD2U6+Bey=tz~pdBmP09oE;4#lwmN9kiwHrggf*2d7L1Q<08;
z^~mNfN!S4_;g``q{YM@y%X}D=_o@}fAIM~5^@i!3kR>woTzcRE-uyg%i
zEe?({w}0uM9V4@xMKTKRARU|#WCj=I2XFk$>;b^OFW@O-!86f_#U5?t16I2feEnngm
zbtmM@=yNr~A&ig~!YwTzumZ*TWhR;Uo>PhDVf;kr=?D!*Ce2pyZ<#|M?FdbOYAnmd
zlytZj*M-hq-LyD$((yo}5E{FfEP?|(B%+oUI+|_@$nlmq?dUCYN$&#I(&$P)fk4?w
z)CAo>8Y$W5*PNtxKI@;EX)Y!n8)Fi`d;Ij+e9M3M8FgJsFl_bo`moPD)>3(gG+Wb%
zam-Fcjs>dKZ1%z+fT}!mjLDIIXqmM~nxdCR6;_DqKak}W+l;jGAm!Fj0Dn(8?a
zVT#V~oXSSo1~LUhgZ3M)GEsY{w$+cIPlSd9O@)oKt3~2Q@jeBi8D^CqhLaFAje_mr
zSCimC9geiF`8wh2q>|Tve0R4-IEV;G&o(oGzcQGg%P${P0j^h3618Gqzy9Pc%^_db
zzk*^YSbe*J+c~XoSmHKMI%vvRq1T@>uOOn#^8r&
zry%>j$k*6a({23;fI2(i)I1L9In~xXufRI#TOXyp$Ba7V+WammKY-rBHA&Dh;Nxz)
zn&DOXJw-`~o+#uaiSf8icKL~mD;d-_#L?W{O~l#)VZeB^=9`jp817*ic*uxR$DuGJ
zF3HE*Za+q|`@92x8?J6neMU}8#sN2gP@5A?w+$DGS3t7*&mH`*EWHlD>u2f1K^V#s
zJ?_V&^Ld(}i@`)a^!AH@T^qIK{R&=u8Ln70q?JQlinqEf&7HNg=-vg@n@(&7w1FhS
zwKOMi(G$mpGPl?1f)rN#BUjzy);_VEKC4z@$t9Mf5Ya_{s1a@fx$WIMoSI7+D6t2G
z)zDS;HO`v2BUF3xU$D(f5gBDUX+kLRKo#Al6$CoG_G_r5YlCVQZ0iDM=YyCEsyz43
zp+a+0vk*g677o%^4L)!}wlqr3+{f4|Tg4xxV#cyX(~o8wt16qB20}1PzE^xHjAS!p
z%{;GB7}Gp|`e@4Bp^(yHU|T%!g#1q5xXf_S`+IM2uI80{O<0cM4wo5>sWJ;u1q8qS
zL(kapWHcs>X%`JvhL=P)eX+$>ouI1ta!AX*HhPo?XF5usta5_F8`P4_woet8&tnu@
zF_YVuR;zee@%B*KZ=W_q&17yntREu%rzv+sg=ji|(Kd%#t+8;|%S6cgI&kR;e!k#f
z<}0m+XkPswYT14r&>2x~ioeMFhISzDS(K#)pMzN+;Sl_NO0x7(-JgZ$?NVMeP}7*+
z++VRrPXzDY7=nzW{OcazS-LC+UW`6vAa#JhS|2mag8JWZPW
zo*65D>sAw^Ssi4UAF)Kp0HqMt6}K-^x=d@TE&v~zLJ7;aXKv`6zvc=3nHyVWEQIeP
zQ`(bN#2~3u@Txw&T4_>EmBrKH>JG;eT@))Hgq~{@3gst#wQNJ^(Ef%b_Lbezspqls
zx!hXFaD|N)tCJpEW!bn^sgq{PCft6yW`GWVeFDGEcuahA1>tAPI`2Gu1I$1N+?elS
z*LFJIFH-oXxdqWK=z%#~HI;%6f@d;kHtHJ3*S$_6DWR>T
z#!KDo8t&_^h-F!F-nG3%O8U|CdbT(={Swn06{;5zSr
zAixMzoFGFg6tlNTz{ON(-lcZy6uiJqyUM7%RiIrZ4YiuoI|7Y5#gZ?fKTpO`4|Z@;
zia1&=Yc9tTH4r{WKk>I_B8xRdkT>wo$uYNIKyJvs=a}Uc^_`L)$!TTtcm1^qQTlK!
z6m|QA0POOjln5MJfVYf%bEkG6XhA=Jwbqn>Eq=RR+o74M8u5{N_Vo=7J`17uz5b=i
zhldi2AkgGo8RjB#_!XVR>K=?|P?#08l`rWOgK2vAsWJRY%!*$A=4SHS}kS%oyq+u&J}{%p&;xxA7;Cxv}~y1tlz
zWZAFJ`palaoXHz4Ags}+hANwvE+&_o)^1F%mptygxitGycboaI^qTYehe4Q{20vqo
zQz<6cIg%r{wchHrPYh0Bs}aIgEDh-zrLIxI+1&IM^vUB@H8=ix4jQ_6SIKSd$o2!%
z_L`-}P)dOm)`T+geMH@D+at|?(;|PG+JYDf5y~oc#uex^ho#raNPxjTTKyQ|LSD7!
zv#ayk{q>s~E|dj)Pm2Q0_U(zYp^Hokq^JFE!D$#5c$w~iK$5R|v1@}IQK-2SxF!(u
z^{f=D%vsn7O}T^^ZTz&r#W0pWG!hvztb3OGaL)U@`yM+!;`4*bFEj0bh2uI!<0f4~
zY-Y_fcUzI~3hVEFDM1O>B=8$}L*pAR&e&DMQ>zxz68C;gtfhr3X3nc;(;DP9U;-Z2
zosei_Vb--UBI{i@dsVlmr1A0tw$%<~KD7`8cDz0tvWM@Ub=>{nBvK%>CtLl+Y(8AK
zft))8y{_)Rgy*vpu=Wyv_|#x|Roi!6U^X6XF5>^-#{WRT>C
z3YB@=e)r1^0tUC6nj}~e*
z4>T&dI+k!q%1~k2x){5vU!2AUD|~>8){3L9%<-OC(M1Og6U^0r0Rp7lKafDpd40Ml
zqcLI}POO57dZSNw6*rqktR7Bzj;eIZd+uo1UrMNIk{v~`EnjM(m7lc~l+k!<
zyM=pXebBPXEDes>D?PZ27ROPTO>5KQevNa;niDz{-uD+9eF=27h1_V4G?NZLqOFd~
zBSacHV=ZBi$cvqS=VqSLyMxc90q4b4i<
zBK+t!LiAS#d0jY?sr!xd2iO(O@0)ZCDA9Tw>3H6<7B{XW3R?@n#p+a=@B~+n*D6sj92a!1nY+s95w2JWa
zN6%^mTCH5;qxSk0zV-$b22J9QA8EBIDJ~{^Okcn~7RV-}>mC`8FCg4*=^~#w@Uq$|
zbV(F5Znhm`ma18D6R*IlAbJp$V$z{;OzR1UGqGd#L&c!6*OI`a2QM=eA@+#8I~yn1
zIYiV8dNaX)IlBn|D2wlx#y~AJoG65*v=Y%IG91RFY%0XH)T?-L)D2TkW%!uwYC3@;
zK0e{37}offWY?bxu$n8oTkq|fb1WHr7>Fsfs_wmQU7hli7KKCL7
zp=}mn^7BJ%7&(DNMSMatCDI3MsE+RQyLaxxT10$*)19%=m2vAqR>f~N#&zGa;+zij
zz2;U5{2(~rSJ}bo4h^vH6ebWHz>))Gn{k=QBAyBEqiikHRns-YGfF+7)=|K%jCt1$4Jn$^w#lG+x(eh5Z|fVnt4gfpim
zNbYX@`?R62L*BfV8M^Sn*eSmpq1w&PNbKVS&&j9&w-_aU;0Z@jXhh?_T5bGSU|^s1
zOQ4o~i8W0g$Cv>Yh65WUWR(~Bj5sEy89xGlCYz>tZ-IWlb20Lrb{#EauW%;T;EUkt
zNZ)tWaZ!z+^lVDvCf;h+x?OmePkzkzD04W{)^4l^BtK#%MH5tfo1cOpG=(6--BH6{
zA*j;>uB2`)0&1V~_qeK;##%^f;cpK4AIwO5E}$ek#q#{mLlODT3w;_%mACsi+P7SP
zRbzv-irz2YeoXN7+#C5N(}+JQAfN_ed_^={YYfG{S;TRub4P~&=10v
zLprADq>CM6YMQATU&4>Y>thyb%!P7Z1F196^>qaHQEvJ@^84N#T-_+sprC$w7w&=z
zTEDxFWO?GVw-;sVLChmsk~#H;
zE2%>AO}Namb?Wahui919oUko_S{=s|`v_CG^q~uQJ9WJu)%@*sz~va?a9FS?1~5QY
z(jJ<_tFwH$f4bVv}FoK_%I@Mkr9%E%z+TWeM_M67#6AyanqB?-F%kX
z%*DB(gM&C=r+2#oFnatxjKaz-h+HaY!{cab$*D?$vuxeP18qMH0%v`HZ*H=!z53Tw
zZ$pOq8n7eRNAMF~!RXBjSsErb@t3O{kZp-k+f@E>kk4F2HSW)@^~F6xbZsj@r;TYp
zK*@-dB<#l%(u&Cc}#5w?@Ap
zsDKPymF6^KZ88U
zLPY$H@6N#GsJ{P(8%S4Zb~PncYLe_`EA)pKexD|GyF!sr8w*+}iLTN7`FRIo*k2EA
z+7U}@9;PizzE5R@8@Ge2NR`}0lJ#m4;l9
zr+oK20qU!O0>)Sw!Ei~Cv*-%oUvid9b$+=xoomu4=IhZnmzjV$`6
zio(t%8okA!6_?0A-9I8KWp?YPBFZRdxD
z1=j_(Y~?b4u|T;z*T%sVY_zzYo%TLy{|f!z!#zlUK?BU%?Jo$6q?R)Cqyq_nDKq?f
z_7Qfbjv=F_lXZoMs|giO;bR=lQ=_k;-vL>}Zdfy_osj^ry{d|_+QqYX&$|h^2L)Xq
z`lOV^3s|QBo*+9CSe0xACagG5akbK*2-W`1SOL_3cFLuS(d=n`&q?_YGlJ;!K&AXn
zooPZ-bLVqQpclD#Y`&!NdRoIf3d)$$iScMe;0+;B=Eg}|MPUmBu;9FQs)`zM%CD3hjtwgLONOMEIR;wW%__;wFTeWtd30DXAV
zgwVx2_b@eUuarr|!&^K`z=W#yFHdB`s<>CCe2Kl&E&1Q690WgX%4jLZ-#_dA^
zHW7ArGCKS#mfBGF>@C{LP`FLr*3)b#MflQ|+)7(rDFtdE0bRqnM`yNjlx6@`N^!Y*zW5B)iuJ(pxA
zB-g=1=Cb*TyGZ$z0fVne+M!675S-=Jyzmw|aonJeH*gS34=N`hTZG9Kj+}Y}h=&vo
zUfG}1XN8h^M+z8VIBw!8v;a
zm~?tk3PMJ+k$-C$gV{JWCG&=vC*W;{F_|X-*A+3EaZ#iSvKbQQd0;gJqnh6cv8ikU
z+z!|)RJAwtf?_m($t?Q(iDqIEOT_4?Fj0cz(olc^$5AD;;rMDWN>%jP*Z=5$TrQPw
zkda;eJZRKd7(?RgUv}zG?aOLSc>fTZ$nQ<7M_<>jC&w#ce%2xGwyxD?{Z-
zTbuCp(}&8Xu#{j{!E5l7B>>8Q)9N?2p=0eQ<$cDUt;&NykQwd=7{Ym%0l{iYg^=9(
z=0KCdus#8LzyyrjvKMDbI^NtYFRZjq`oN0|J@uE}1Qy9QYZO+YcnPNbml%#@<@=t>
zHohU5uqeT7_tK-sz2stumgjx;_vpEYU9;i1FHi0e&yOR@$x4l}ZWg{2_XYF=$1l+Es_+=yhph`HTSFv~kS1D##07pSYfV
z_Q?h(3;IBKxfJ^wvk)|YP-3hh3-4(%cDffAIF5de>OkmymN*lgzehy_sGcYJ*LbyJ
zmiWY_yK6CgR^v|;bNp=&Kci34gX;@F)r>1_9-{$|F5BjZc~MZgOY%}6O-15vKSf1=
zFU|~4q(4!Ep7^4YV@Tm)$#>Qp8L8|r#OI*kC1k_Qd)-^6@e*2p6Vej>sD|z1c=zNo
zEd#FnwG8i->5rQuJ!!jjETRtjqdjZvmjOGN%sL7coinS#`^+Noslb!N%d3*9B~7l&W|;6{_4Bu}PpZRwnroB|f8(cUG*5JI#NS;b&CQ8L2S
zG&L%Q6?wXNzSy0A`echOo~-IgRv9)-UCpSD7XC>2Tq*1KkT1UKaC(45>M#T;^^#cW
z(Eotafb|nNR|$AvoiV`Cw|a`l(#83+v~+<+wIvUE?X<7Un4+jHO+QfXWfP2%eO4iO
z*e@3Z4*2H`kM;w4!3RM^l^{O`b;rLB(CS3Bq$OopV{>JHo1`7`9P**zJ!#_47>Nq^F$39kBdOgD%45e1CZNoZ#I@
zx(LDl-M$@v2+)@*Huvdeklo2-@&;dlszdbI^Wcf0mCf9Jj5-Q7JJ%}|ye|b~chN&v
z>h8}b4Jy^=$8z6V>TnvYtXi&eN5vmD-_<+`H0M-$iEddmIGvMuY`c}Cy|OhM=Q(-?MY5FS
z!2`87*sj1C?x>qSXwM|@=QOXG?@N;3OX
zFCK0x7igx*vvdOnc=A3UROiHZc9Owpv;Snf_;5Kz%u|9;o}xq$T6oRuJz5=FBWHZ&
zkXmGa9$;Jd0g{Ku#>rW=mS!ni>rwDz#3T)TB|yhTLyx_&^GhKUefe2M0k0Wb*^Yn9
z;1=J&XpSfDe<1@3TUX*R&VKO
zS3|5-QQg<(EM`&Mff~4tAE_d#6oHt{LpY*;diZ)uTs_!~b%Wy=`fE(VrPl|=76c}e
z4MZnE_IQZzyhZNs-G>cldmcF)O0=Ekh?D94e(D>{VFWjWVXVzQV7%$zaDEg0Q~btS
zeP=-(!0uXv{}?r{3@i80ZiCFd8}Q+t)kf3V#L9(PiJmr}nQb$Y9dufUhngSGaXzkp
zQ&^x~>~xp$PHoCBZeV7wFR&*vNGsD_paqu?=^v!w+hVL?7KrH_y{jdbG8ZuBP(w}m+jOA`Tosdqzm;=*9c`*&c$
z_`3kBt6sSy6+>qfMI4P&R5eR~!9TF}R`E6+FQGx*0hDDn8&3hrnS`x?N*kV+C(^sO9v7nXBBXd$Xk%dgXf$6wHL?QPcFAUf(w6VZ5)XC5_%Q
z1jve9b-Eu*?ZT(OE4Orir$yCs#BY%5Y9B9YD%YJASE5mt0doOQkOXd*rwa+7R_ZB$(&fjeWzh8GRV`Xkhf^9!Wqs;%
z%Re0xz2!&!lrj(E23z^{0vEHOcIh?D;4ztHOi*jMGaW-Yl}P*_Fid@<47WbCH`X3F
zf|Fuv4ngSqLkfRH?e_g#jH*b_wdrd9(T9t!1;AkkDh?-qe{`1lz^{f3<Zuskh*+IUorquDaxY5MF777V{Zg8@KK9h}cS5d;W{5$@?
za4Pfy)UY`Dw^wERdpg-xP*#ABHHrn|TIyBcnEI$ODB&N$YhH+2`e3>_E?8NMhMIh7
z=pOGtDby2x47!7}(L2ZpRLb%g27f2Ci5Gdv(Ggl}vp>v_Ru`GJNx#`s>3vC)6Rl5*
z+Tm$G+#T)acLHQ(m=vH3JAd_B5#;H#WX)yXOfl2E)w;*MD_b<;n^b+&LA6a&?_0P8
zC6P%U=py{SH4Yz+X${euCxIsjlgEZ1)B0=DXqbq9EU{nc7XGMg;{iXm6ispMsQG1}
zQDoWtomLimyU8U?dSF{cr}d)ME9tY^(EH8E%i&h$9Wo4>fmq8H1_1b@$AXc`_|>CvLrsU#
z)}SFDSBrw?ZO`Kc15oC;-J|gesms|L_J~&?jjpn-mXdlvL7&jbYt~f`PPJ+raA!q-
zsz6q2!v7F;=WmDa0A*+c*RJ=*pQ9L!^$0lwo@Dnmo!z0J_%$oss|QwVB)sR^9tfs0
zcIwVG6-i)42wcuJXCGbih>jVL@T{hGsVro>o9l&;Y1kq1pH6K-(RN8+
zT|+=#FgRcw#s==PurJ(kl2i#v7PUDzS9C6AmMTE^Tt^$fp}YA$=Ix*j>-KX=cHQR%
zlZIoDH-j%6g=f2y`T)b9E!&Fqoo!0Tzi?g*+n|R!R6E)=`u3re%vG{xs>jZM+H&I5
z;p*>IIT3HvZyjY^ztDg<8G<&YB6hlmfdlul=Mt79>OuNfg4}CjeQsvA_V40lLd;XC
z>y(CRr&WS-W)>b5YR94KIau=}5&RwUT(Px932WYFd|ePh*hkpj)OqojX%6U6=^j`u
zOKAG{`6Jji8)-0w&^c_#PWQQgu5sxSmR|Qb)`)m54P{lwEUooQqUV$B6h<8IOb>IZ
zlTVtd-ngDMxYCu?Fh2=sz8NH$-q&K&`f47{qXjtp32-aD0^5$cQjKd*EHD##e&A1Y
z-M?N=TDg)7QGlX2eyOL&AxjRauHFfC>gM$Ta-XpeD#lilAE@Mxf=!u!qa5wlXMl;c
zy5zn*O)XiFe)xnba&1|rMl~sgao%Qq6L2T4
zvJB2E7SQayXN7gE<#(A1^25<^#i8_aHy6P;1CVH|?TALLPh3
zOFC);15C=Z`mTJO5EC1x$2uK&X9b$8Oy
z?YoRO7>#YCB1xJ!@tRumZBZn4#T_l#tC7v$*}n|?>>(NpTqJ^jHL|rxy_%MY0*R!+
z$A+XbdT!g3c46A~Uf1i$)yEu^E0bxinF2xYbbgS!(re-ls+yc}3Q|p1&o0*AY3KVG
zzGiNd^mvzU_v>BdvImbeot*zujc~TD!(JO*?&UbEdQcVO
zJ!7{997(&&z8!~uXN5Va*ZZ^71frgaIp*>PDjw&4+Ub0oQ#GH}j7V^(8c&`R2
zCv!Zsw{g)YcI}ISZWtxM(?IOh1Tg&-B+b6tS4>-VvvZ-_hWpbG!I|-!-tL)~3aWs|
zdS{>+5i?iaTr;-b7VZp=P4PGbbFR2R(@Aue`Q8nyUaz*XW*pWS$Mnf#wgucCUTrC2
zLwpknq;61u%%cOj)yb9o#QJH%U_{TxTXW9>M;etqsEZ;w&@y?}%}UmUCNc%>@1>QIu?P;iTosRcUSr(w2ZZ_9_w
z12J3YQM=hbuuGP`7)4zJ$hN^R6(k4D=f`Wdyq|^7xpg~l8QZkxj^B*_gU9I^
zar0~~dR$^qT_@s7InJj-x=7mFvg7p1+=+sJF&QvV(p800$bRfLNKSb$(3)2l%FJ*C
z8{ly2BD$bUG8(hf_`=6#Y*m2^p#*!48e|;+xR#q$p`BJ#vgA*rDyI9MMUDM2;LqjK
zIvWwgffk+xK8!^04XJDHY7LNR7DMaP=4a$))kQcAruU#)Pisggth1)dcj=&%pZY<6
z&gwap83jp{5~z6aHm6~dA|wiLD>Lk*uQg%-
zva#hxv}4;HB$moZMXxZU?AGZ~Q6RSBs+*
zss$w<7vLgxF%x|(XC!iB{M;n4=+%Z#496EcFz%RjSaGd>>z^GyNc$T$)D0hh!{z=8
zVlP=1(opy_9vVV*)`>Q@R`fpQg#Ce`@a!348BpWs%RQMOk6KMF3|(p9N!>I4zB5OT
z@Xry_a1vz)=eGkx6mgq1yBJsHzK+B+hk=@QheAe@xDkQ_Cs~WSn7udu(Q;Nb&(|7;
zsPcqW#lsQ8|Cl>ObM#Qxtey>jX}HKBFF%G6k
zn`hNaB9?`98k}6AjyJ`BE!=9@ro;0qCPNVouFerQZ=q$EO6=}1Sc;H9F8WJp?Z1wU
zUP-~~?|G6fD800NUhiMGs59e9KMKb(5?^6@Hd^S;Nlw?jKUN;sH?`*-t3&<*+rubA
zV0uZR$tNfL$Ik6t>hT{Q4SY4$Qu!3Co;uCT@^uL*GK_i)wI2X~rOM_#d`!@E$Q(53
z5wx&3fi536pqayJnnzm=V<%DE<5Tz0I2)uJo5MW$O)E668=%hMae~tR@-xz3@Wii0
z%bMzTfWM@TmP|&HtQn!saPPP?21@H}{prkv`U9yJ0Eh!LtYGelR{f!VK}d6pFkfup
zILI7wbQQOeu_zFKhW7lE-WsV%H2>Bm2E3_;?ia(M2bKk&&Q-jyYX5cpid3JG4fU>;
z>5*-bQZ(+TCl&Ug-?++He-)tB%5104+Hyf!o0Cn
z*VoU7c{>XG71B@9u@Wbj5jPyP!rEK8sJR`ZK>yhqH$7~w8pz~1fyN8)J3p&48;4NYSDZxw#`ki&&~RRA3Uruf
zHM|i|?Ot$ybiPFoAr}^Ou{Sh5#sBJ)I)|o{u={rd_w{3K;14|o+w{t0e2ShqX7Wu9
zXXSS9
z)6}872Qya6?|Aq~n5YH3j1Ow3BrC~sT9G}W{1XR%GAzp4B|MeeR|m+l7Q&n)waS@D
zCMpDn@4Vv%qWVIEgOGG^qb|c1BS2+dbbh?+yv>uhPy^Ql>)1aqlAF`kDTR15<2ZTa
z;5W7n*u%>Jd*_uLu@0w?q}>9DbSPjAOw3#qv)K(H3V-s{7(e&{{8A|cJ@U6&SWHb=
zhOC=^>TfRy^=H4Vx@o<+R9m
za$$dFFXLv|MuGp{3m5Hb8ra_^2XvhIxzo9SE5HQK(}Gdm^O|42{<;>*%c4mqiZU7B
zpNis701DtG5pRq>ZNAwj0c2NuO&{n~AiQ>@a-zPv?q{f
z!U-nf^P--p+28JAT_pWEYZLdRN2rh;F=OT14#|#>4|koEi)Es-Ol{zn?^rahF~Dzs
z=_;VmG>^)|o@JeP4#`HYz_{N72oflNN?}aj*+)62x$-Uf4;G!0*{AIdJ#G$^ZiO&C
ztcO&cyM{`%N6_@UI5rkNtJnEK)D(wB?5_jihHmIBnz*C1mNfvO@;^G@NjPyBMrxq*
zi~V}D`To)V`>v6XsdCvP^ZulT$*cn-J=a{lG5+*u`5fls^;8+Lw3<`?BAh*cxN*?=
z1gn$&a{JGuQAAT*#A+wlfAV#MZ$TO|`+f#e^b@wd5Ac)iiII!%Ut5-rM2G;N54oB0
z`%r*kdVo
zahBE%o1fn=m-U*T`@^q9JX>F~iJT6Ncg!g*t!le9Rtx2x+A%g(+$Zz8x{vaSJ#J?U
zpB<1i{BQkc7hNna2^3f<#Gq)`<5_k3a9>7ZxM`}!#Qzy*(6n#d+VmQKy2b68z0$}K
z1B-1<#{&aw$u&l&*@d{Pg+`AHykf1{kM5o35DXA#ff}_}DBs;O=XMHc>}%?lV}UgL
zB_&}$NsO8QlhE}p`>keq`Exhl4!u~?*Xy2chE8$0=vxh?brGPJ(O&!Ot$=nI<>?2@
zX7048TXR{QFzR@5>s#@E%e<2TRY~=rWtflMrHirrNMdp6IpxM+r$3kz&u9t55TMg-
zBpDbRKPlRp2>JJ>AL?vbZ|Jj5-LcQqc`xb3Gbdb`pxbGGCkPpGzqI|DM_;L4_6e;HxTz`y^J}f~f
zsXAQYenhk#b?v+wFKf>W5SaHJt7B+ea2zxEpMa8)9JoY(;!WfVI7IEp9)z0&mz9Zx
z%u)qASho>hRc_Ykj#Z6h&6(+#G8iR}35<;txeHwIoC!97PhChhG6$*G4R$1~A4vW(
z!|3SWze;VSF7M+1ls`seP(gTt#8UX99&`f{)yzp&p6rqCah6X7dK6$yo7(3GJQ7gD(%6^
z44bkr$h%G+##iXv2Aymy%OV+|;N&26#d(X4yzvLFC0xDu_Ms=}xuz@UKEzsC
zQ{gGOMs3#h^=$U>fx_5$1kOlK6lZ`qEM+3$z@oQ*J?qA)HM|m8=*d5y3$YyJS~IQl
zVdB*v-NA&0VRa?m5GKH#Lt9m@*E~_rrpACv;2H>f)PBxGP@4gySEMD&Xd-Pw^?{}M{A(yzmJM`2M7~bE$aJg=TJ~D5I7(vj(EO0f
z0G1?wR#6*X;5BqkD-lwnJEA7X2G!kz!;h0=m8R+1#WH>6u4}tMCzIsb{Lbq|`9&N(
zzm2r*Q1JPPdX0hkh%E9~m$#AXr#U$_SA1&>O9kJ$73aZZ46RC}^z?&fZ^
z>l)z|51KD{(i14~7~+Zx0iH;+1fzkV{X#o4LC{432_D3rY?36b&3QQuyM3
zTn8UR-lrkJ>0vLo`pKZpXnaq#k61M2Yo>ZcrYbxfT$c`;;8h>d8nuPu`97ZZCUK2E
z3QLGQc7*6tD8@~~R;@pncF7+BJ$iE1q{3Hfw}SYKo{
zDSdk=pV;84qr)Efkf4t6wavx&s%avczl$nX0`~zW|77P(;@y)5sW9Lbb&{M9y2<#@ReV#g#tZDI`|{%8V`!dMe{715lFL{_Evk*v1^`|AUiwVE!?GFd``mT8||I!DatG$F#zCdW0rUGVH_1XuS&9{@%1W
zjib~r7sZKUi3*k2G#O%kLu-aRh`Z~`v86OJHFotNx6_Wd2hm2i5cXiA)A
zc7e|&AeMM73pBrK4ir$5Cp@w_vL_kcJH!RjOPxFoVC_%-1H#vpIG(b$6!
z;~l9J57dxO=YdFs&CD%lT{x;&*0t
z1HrG2b=HdrRWMFvu1o}^9>y7t@cVr
zW>$F4;^}*Krp1+2`s4gq2w~-Jj__lO|4f4io(|D;E=C^q?b+?3oXHj%iqYI|K&tJ5
z7^z}&?3WCh*2nge-ma!qVQYtxkGs>P>4f#hC>3ByQXWPj8tP#z@Z75n2DxvUnKtQ(
zC{7M2WN7}e1DR`ol1c#u&y3&NJ2fLf>0=3(RN~EX*aww)0rSr<^Z$YM@GC8}OeDMX
z)B{=5joPmfnl5|Ktm4qil=_d`--d3U-uV+N=visz6Dh_4dD2(Y8e~s1jGzb1mE|TF
z>1`fIRR!6wo=>p8cz@}J3=S?u20|)ssy%cn8?v`TyqI-==~c9zB?G6a0`OTXQJOb8
zm0=NvzlMtSdbx@vXQ14fmI@Xnk^hZ8xROr(0LO;YfsXByK_flOnP{$l^QUO?o<^hI
zG2oN!EP+Mu^Mz8?W7Dfz$L~SLS8vv}=;U&R^sbRf;)XfrSz&>;&d7`PY1vtBPPMU0
z1dpNM8>aPtry->?ersrIrZPi%@;8D@z!kw1zg@GsXnp8T0}j$W>fb<;C+*})aaILM
zV_LhxPs$)beK7_Q`}-Kge7aJ`Z@rgeuG2uJ%=$wOMhGsH=q!U3;rZS?-%}hD=5e+!
z+d+a+s`ms`Tr!FfK|vq@S%4lg@9Sms#({R
zw`p8|00@V^paHa9GHL(ZPz#W&2vW}6pQKaHwyOEx`V%!FCD0!0aen*W1WaA)4Lyb1
z>FGLkgjN0r=g`F=T*EIXD`EJ{lHc|Ovq;jBI4G#CQD9+|5*BW{-{*K%Qs|9^U}<<1
z7OSn|)cSi~0HROm=8~de@AmSR8xmct^~wZ)MDAQtjTwoPKn{G~`4&EAWbjyw(opE6
zMOAw7tonm(S(^I$T)jEaLEV~T*hPZ399z7UozAzXd31F&Z;hQoAOnAF?F%?`haT(;
z5Cuw#^kfo16nH>^v=F+gPHJaqT<-K|3SN@?`A2zw
zsA`e#_)B%(&$sYVI;69NYjw4#u}G{JNF4QTW=Ms~P$hnfU#}BQ$Qjl#fL@V#aO5%j
z6>vAX?~IP#Wn1fLL659mtU+V@pe62~8$bG~+Gp-{AOBliUVLw;o_}3zGBq#{OW5XO
z_NAnx+QCn;kcWFflyq!6FYFD`I3Wsu`Tr%%e3a$QtQ<47RdFnkR@
zfAE>%)|qBs2|M9CIl~_1>$1NFvO1=M(w)07S9*w}4x>krnN+Uj3e6{ti7z-?~{hzWVCP
z;7^(L5Ay7Aer5*h%@X31VGwqIH+bs0IEUt%y?p9hTu*lFB0DHa2+iL|*t5o(ce-kI
z{X@~({m4&8Z^JR;=#V@)3~L@mN&J@bqO)<;)y$WmBEemV^Tw?BKx(4hin=PH+eoTn`%D{Ux}&yGw9)cXtU8
z+})jUQ!`Wd>wN28wSGYFs%P!av&98TRXhjLGILQFPV!$BT=z{X?27+nr>>$*0)G4e
zIT-&eCz4ZTX=@OYf2_aN#mSda?R^n1y_MK87y9sy{{vc1(L6Q!M*h5}@PW3@*#-_q{Ivo0H}j7Mnq6MCJ%@M6l8Y{{Mrzu4!CrwIBI;)Lt)ow
z>L85b0x2>?1f;D$JvXbHFjTbBA6mE9nSQ|*XT<*iz&-X4qP$(5v7C6-dsC5fKLaGkAw1Lc_q{L%Aq
zGBvn6Wlf{XHEv~w=7`-q@rw1;Wuc;|9{A%ErYQoo5ZD#IdSsZ-HXqrSz&!7KSs4?=
z$?yn&?Ng1scY`|T)l1338TDs66Lk7wi$PCI?K~>sT(4OgZrK`6^tRiSr(A8w0%Rvv
ztPtf#iRs^nExr4n6zdp53#N`rlI%O=w8
zQGZG$qoZ)UEX{!_(GCxqdnwEj!h0V|83^c8hz;rZo6yU`f<%a)-_sTj*o
zq?^m`V#el`T5`rLHwn!OA=UDKAJh$-CdEYdkY%?|5+!jua!VviPxb-c3L);XYOebshh)Nzp>ij%ZUDJA=ck)^#pG@(5{?S?1
zu`KdC$^zKYbN{9CwJ5XhoqRBq`gOxsZ0~v&_X+Bsl*PcM^W?2=eV99
zYuf9Co$xFM&zFbv;_PrXlp_Uph`V#)p_~DbgLTY%PtdmL_g8OTXsl1;L^cQ!pX~M<
zKWS6=6j*Ta(rhD|>WPGHQ9-RQCe8`^8u*Q{zb0lOC|nKM1)0KI(>2n}`bFRaU!zds
zgEX{9w+tP#y8Ea)T^oke)S^-Op=ddX$4(FS$VDgjZBCF!kfqf=9yJT0vqP;p12DiB
z^+DJQn0Jo4Mw-$CpXH4FDVJ>9tkpZn4-l0l^7{4K0z)$QfFBs$G%0vX1q_>s_cPau
zz2BPyj_~vDVKKBX#pB+mYfpB|AHJ0-5rxMHx~>Ya$iTJGv25OIUH{jh&kabXo+=AE
zHv3HxN|+{?=Q5FFcxf!YG)=+*n>;{`riuBz_CRxwq#6s;X06EbV+Wn`p#cXZtb3UnOmk59IrrKX~^yP#^UBB_l0$W@-j&V$o-6E*a>~b2bPZ
zIg^?)K;ety`Lb)2-&Haz>a9`l^A|QFhpPYjr*{*Tpl+WF%M>034Nc7@R+=PPrMo^U
zO*?AU*}JN8B`42@%-PqcC~3F0^7&m&2lcJ2-Hrgr6OCmufT5Ubw
z;Whz-*GQ>2yf`U+nLOYzyTb1IM87wYMgt<-HKR)@=|ZJnTC0`=k7_MabcDI`*}Xwz
zC@DKV0yTBSw+5o6Rm0%-&8P$?TPa1U9OA)VP}J{eTVtGCvYnlNC$s%&;i|o%L%)4|
zTxi#tB^3kPqLbFde@Y>f8hC7z#*J^bI>Z?1J=ps$&JuWtNC(WYYGU3NuG#t39_P(3
zdfiSQp=@boLFsA}=4MQ32bed%rWtY<>(&dj0Oxsc<5FAV{qR}|NzYj-FBT;lBk-}V
zEv1w*66cyo)?UZh3+ul+qjsLKt6CDJhH}Sy`^lX)Y*$K^Zc^x(-75k)zC1};6DR=a
zLdc*Rb;^U+&%nrjM<13+2|Q&y^tq1crVwa4)BskC(TF6`(UXi?#OIq4
ze5-9*v&_?ITtYC-c(if}o9dK*V$8fLjZrwKWN_97onG$NlJ%O;uFES
zb@eShC#H#=Y5bKOpV=#iou$dO*rsAY
z{|(RWnYF?>PYe2#bYlXU3tvdgK9~ijS0H>r3t(CrYx_>^z;U02M02usS=a|r7*w;d
z#8^NvbMSM8%WkmiQbop`Y1XV(bdfp;N{A4@IM_j{&nAN+SR&0uIbM!vo|21C_WBxw
zIrfMsr7*`7%564^W<&j8n9|mCDS!LMxG{7cZd$ZJ7o^_$c5vKu(H&&ufqALU+g-z!
z1AJGH?P^L@w)pOmw=NyHA-hEjI%9ONY`nwp(Ix@e+ezI8G_(Goc>~)yzVTrbi^?^L
z_xt0^Cx`0Ixx1NM`c}7?mP#ha=7)s$bAMvr#x6Yk_3&%7q}DVs6<~4pcONqA&xY@P
zY4Da-+`>DJbvh1k-k8Y_;aBrapS0+}1je6oj2)J=W+3j|dK?kI+zPMc)HjAa5b`B9
zz+c^EH!*Slc0!+%3gMDMC6Qo7(s{PW=|_~O#3RIJx_G41ta`oubi7u6c1(0FnAUvL
z_)Y$IN8_s8t8pOi!XnC?!Qdj|bzK=yGE*LhvF9j6!T8vmsD6e|DBu%tOnhUm4?Iyh
zD2d}q%4`-@>gS(c+y3!3da?p}HeZE@3COidBnkXB2P9t)+kTF^gzFIEy{B^5o<1VF2jU5`v{AXui+cUMT8?pYl`Iy+8nbWeLdqva%9a{v
z@ohFEzB3H$vCbZZ1N|d$kzq4!%<$OrTX{GQ?*_*L*RjJm4x74aGVRl(e9^GQXPtVw
zElbw#f>Gf;Vl_mq*;SX9xeR@$#`lnc*Q1Ryq5E6nTo*R-KPL-YOy8Db0E*-vE?+FT
zs~AQQsH}RU;>v#EEI^l_c1C>&3Xw
z+|EGjuMp&9={)POABT5;{&8;!Q6a}cYWQ?vcDcmHp0fz~@&f)x!O;>27`I)gC!d*#^B6zCKju&zT1a|P1(!5B-
z0_tT@wcL%ZnxPt;0avyhLXFKRyL116n0qr^1j$rBBv=Nfao{WQ@`vTYFWC9%I%5^U
zbGj5oakD6y9P&Tt_AQxGG@H4HA(x^Rq7p3AmU!4_HF#daR(5&|?1CrDTAPd3gwb~;
z_qJodUP;vZ=#KzoMN41-nTer5SmyXblu_~*
zXRWvh<9ku@EWAt$%T9CM@*`m3XjDlMorY~3m}+Who00U|1Tj355-;8|MBpg5=nohj
z-#%fVd2Z=6q0Ep6hL(*{_e&&tQe}4NjSDu_Ivg#{?tJa&Gb;ne2w&+7^|qrdhPx+Izt<8x5_!n?xD678^_{`}&8K9q%h4kv9*j>=kdsS}C!U_|yVT8Q*{QineBb4pGsB#zJ9zHV-c!@4hW?y~@2)P4WK2L2&
zyQ)~YZ__1
zEbw*E%Rs7yk}%wrtvzk{E7HYK)ENp8ymq~#tgpkrHgd|f9VU(E7OvLZ7NT+YQ)V(6VevBsg>
zCisU3i4x>3^Vffhs+II>F>fj9!o?ug`gQT>M`6uD>EVyDvYT-r9m!IU$l276h&cwh
z;6A2=LOrc``VV7p3VvjF(<)8c@->)`jIgEp7Z;>cNg(xP)NwoDeyrL)z92~o_m@Arq
ziBp4`j%2#@IA&ESh@c1dFC^`HRoQLJj`Qu{KJNo+Yqep${(jY%bLuY%#KLK#qiRID
zy)vU2DZl4?jDZKcw8(#3?%*{(t^epN9$uRAD!K=y1>%Vh)tSSXQNB#4Re$n3E9Vg4GV=Ros+u82WjQqCa^
z3qxMGMav6{feoNil&Vbfe}mE6hYMcdCAa<~p9Up{K6J&?{F}(RSM=9D9m1yyKJxVA
zF6aUn?N{7m&vULce?jc`%2%2Hg|8`?Ae(*j(;#(srz~tw4V-v>@Qb>ujvQu9W7g{(lwIlF&9LP
zUk{?6=?VEGa|Y+=CQ?ZoKjHV|E{Wb$7-hoII=W=OXZ5rdr)jSY^){luB3JBfxM)#L
zzIX;QmhXM25^!2cm&^%>b@})Hl3b@?rj``ajv5ix{oLUp!eActeRao{eIYzOEAY@?Zosur0Q;9;x#l}GTqr=!Y1oVtx-)|w^M2@3*wB=}hpT*NXqT2KPuD^u6
zTgU4-uct_V*WGiZw$0b3JVsPi(N$lR2m>tXhcSi><5V6jr1{Emb%1H0;DfAY+)^kf$*ryiR-eH(PI
zpHGLTou!KF!+`Nwt#x|KeQ0mOZb}AX=0V;;il06_`-M&sUB0e0q#xr5+Hjx(1;B%Jeo*hBL
zbwKUF^}z5tPD}<-@uz(SxH2Td0OQaXK{F%%vL~@K?|nwEc<_Sofu2M;znCU*`bdxH
zL*F^W8udAa)Xc3mbYCi0^!MZ+oCFvW{;by=J`~fY{cfr)aE?V9KVH$0eu31!lq_RE
z7ySLY3q{pNVgvgq1kBhf;bX`CK;Qp$qi6Rrx0O9$aBrGK=vK8-u{su9dabfdf*WjN
z&!d6^ZuJi?Wh!&rQxvJP8k#Y1OZv(}fHTy5k2@Z|Z_|}$HH?8K)7ob&{=i?5Bp5bn
z3ax%7x4fZKhaktof4B>yGpbNhwga~$3c?7F@W4m__2m`u0oXabHzYkDMi?QmU5g()
zPAPM;qtPQ#T{{&sI0DT$T?;jMK}UE8>oi~M7dROn)QMq?Ckeu#yif_yvo)9Noud!n
zXmulx>ttY>@=X`yZrpX;I|j2vL!2d*i^3Qc$YyRYG}c(8-KF6RYG;F3MdiTaBFvJ~
z*Hq>T9b&0fur8<~E3U~~4=|X?*-uc|O*_8Os=UclzR=iBa&6Fd(#f1{|Ktn^hQ%zo
z0|bN9gl7dKZkh~j1%uUOHw%;UF0tLf!FmtkSvhgI(K!FJVYi^-HQBXX|zUfe(2sl{4Op+E3j!vZPZ2zR3iH(_)
zIVmeUCwDUQ4iq)ep{4DZ-iqV%P&IAr$opHA3&iGmFP9-3&pA0DJy+!UENr!H$H^Jh
zV_gfpJ%PA!&`+E1GE#A3Vv;(Y^^t<5qqzfUB)MScEFM2La-xYE
zf2m52tI&ky@K|Lx2B?r4R0JtBcxI7=K9K;7fqI@8YBDcEu>=P~XwgyUpO_Z{j0ot|
z8<-$joi9+U6qR-m_#E9MnO+Y2M6@U#`v${avPv}GajKPXALBu51T@UkNuoR8#ujzQp*!PRbBjLl-?zDC_uA3lq%(4FT^DpYJ(y_?g7gbSr%OW
zxUt(MvW5b`;xiE(D!(FMrqKK8^vQvR)ZsVQ5!7BM=I|m-)|1_xIs>cqHDXsqw2kb0
z50QJjI9PjjrM5Ra_Gp+oMpQ|f*nuD$E_ZT(4+12AniqWzioGCeO-AjNhLbk<*ly#`&SmfR`Ha_O}x(>k#PU-1a!-=x()(G}taFy;6-X%-X4a9f${v@81V&^RWu=wMMhY3ZurM-N1*4-~7XGdETpWQB~b_{wm7QAE)6D&`)2>gw7w97qx5}g6;lSx1O
zC0i4(XqO{Ffr78MFakwXx0nJGf%Xba=Gl58)Y(TUZ=$vq!WV~s={;Do4Ww@7%pbYYj0gl#NaZ{fFMtb*wD%W`*_V{{f4780Gw{!s{N4yuNG)?L
z0fM1l$mGraP7)OSHMvI=C$huXinr)xf}9AZRGy_s4h+*Id@s2bYpXR_{I_Q>;i-#f
zFel9KnUTcO-sBE5{^kfBVR{$6&jNj#$_tDe;%!Z4T>70py5>u|6*_^-h(
z`?x9b$9R`Si=^9T<7mcOKudUCINHGy(*B}bxp)bTBSI=I+K$d&PRG_i4oGvOXT}zo
zl1=j7t?AWkZBMsHwQUB!wfk7jap`3N+2{Flu-TvBc%N~i`fR@4?(62YblT4O+k#$`
z2z^K)347t5HRSE<5H9XpVErq{ED(nT6U!_nHTGg{@iZx8L8P&D3eGJjtK)waFD&>N
zkS5CcMUek{|F#X*^+OcnL+9O?lj|LSr|hpW2Dva6PF
z-Df2FcU?N27rM@#;}+8CND?3)HpczO2O#X|f#$Qm|@PWi!zGg_$A^D$R%~r^B-F00Ragy%*cCT+i5Q53
zs+3A%@p5$=EFg2>z&&Q694&OF;C+)vRd;+rjH%osLJv;M9&WMl<_tW^BX-oRjV~f!
z4ylLdH1W)!w3G?9fkDmRo#>25m=WlHJ|WZJnPhx>N9F8_EeH$H!gHb*SER_o{oF!~
zr^V;ILD7McIRXVs{SSvkE7rlN_ItO)-_@2u%?
z`*pwIv3Dj?L+U%_G!psMr6nY9stSejuUPFc_ZTs;!&p6J056L@;mgxBtJE7U)5fMN
zs!W^>Tpr_-j6Waf{E6to47S-;S3;O*_wTp6eItmP!vZmPe(Q7F=Q8f}567T-Kd(n@
zAmrS%(UYV}Q~KM)1nr~n21XwQH<7GgJa|z3;g9&fEr6xA3-Z94m>mb{ESm>s!R`6l
zvmfZ^8Q#aF0XzCrBxRFA+->c=%S~ygHXqUSe{OLYJYwTnA&P|&+
z_;SG1?S{mB#O`~<`GnAdVn8P-J6}yR)4kttJh=+jc>kpwb)rfvel}>Z9#-BKM$2!*#5j6~fKd^~Na0g)5p*IklkS
zHS107B_L(->S#fMp8l!Sdb5lx!I^&|!7%y5N^jrpJ6DBF@x(2UEhWM|QcKyLwlh(h
z77fK}Q9vCEt=Q+TQ0>59@2Sl`|
z{F)LmAE?T_r&OnH5=2A}bmW}gx*TnV>)(?N8BTN_jfr37M(v%=6GRL>8gvPff4P%3
zxX$2R0Y{B)W5&hsVl^`K%B|7WGgsWEIt!0kR
z;v$%^(`OF8=1SS)?&&7Es;&P~GdxjipjV)#sJLIJF8(f@#FJ++UFBDeKw4*%$?;T{
zD5K#oYw0i!3
zq9rspNAe>wh#cV51v9Bwc$ko~aC3s0zLRRRkg}1o{2MCU+B*FkvXL^ADubD%ZA@+d
zIrjf^;-uRAl42|@lA=7^VjQg8EF4^t5~7^q%&gpElI-HF%&cPUf~5cdiZ053QzbhW
zH|hVT%FnYra;o332Cr`ZaCJxIA>>h)_#+`4H~S)iDYnC2IBiNoH1IJoe|*G5I-}Ol
zO$-eTEQ~}<46LcIA5^AAGCtsbpW08pm~6S|@I1%}3GoP%BJ-|*m7XpS;bg*upj@Mx
z2Tx3+bX3Es*HwLxr{^=Iu-ab_Gl$}u{O~1n!TNZ
zH4v@4F`wY#xLm169oG+F>?8W0J`Tz2hV@KUUO(3pH_L7BlzwH=v0u3OhX7()1?x#T
z&#P^i(^>foDr|bcQR2Gv_!?bzWaAZVR-O4~R-@5|t@t&a1_@N-qISm^6-_M|E~5n-
z0H80MIolR(O@u)d%wS`szF(vJ92G8
z>p+gwb;ujy82hlBM{Un%IGRy8nnijr?J;e7teas}>~PO}ya&7%%u?08Uf*2>V_K;)
zObfo-W)RC%6=#?h2I4A27hGIOxl@-h>2sKR*^kRg13tVQgKtkEDn++P6)GZ{$F7#j
z_m80}b+xwo8CnT%Zu_dLjrE7B%J-ftBXuXCm(0vOPaoVshdv4qj&PJ!A0nqLycBjE
m2gym4?;pfUlzt5fP2_0JCs(3@h>_U2nAyOeKT9Y|g8u{KlR)eM

diff --git a/doc/src/SupportVMachines/svm.do.txt b/doc/src/SupportVMachines/svm.do.txt
index 2467a8efe..fda01db6e 100644
--- a/doc/src/SupportVMachines/svm.do.txt
+++ b/doc/src/SupportVMachines/svm.do.txt
@@ -110,11 +110,6 @@ plt.show()
 
 
 
-
-
-
-
-
 !ec
 
 
@@ -260,7 +255,7 @@ and
 !split
 ===== Solving the equations =====
 
-We can now use the Newton-Raphson method or gradient descent to solve the equations
+We can now use the Newton-Raphson method or different variants of the gradient descent family (from plain gradient descent to various stochastic gradient descent approaches) to solve the equations
 !bt
 \[
 b \leftarrow b +\eta \frac{\partial C}{\partial b},
@@ -274,10 +269,27 @@ and
 !et
 where $\eta$ is our by now well-known learning rate. 
 
+
+!split
+===== Code Example =====
+
+The equations we discussed above can be coded rather easily (the
+framework is rather similar to what we developed for logistic
+regression). We use the standard moons example to set up our data
+set. We want to find a line which separates the two classes
+!bc pycod
+
+!ec
+
+!split
+===== Problems with the Simpler Approach =====
+
+
 There are however problems with this approach, although it looks
-pretty straightforward to implement. In case we separate our data into
-two distinct classes, we may up with many possible lines, as indicated
-in the figure and shown by running the following program. For small
+pretty straightforward to implement. When running the above code, we see that we can easily end up with many diffeent lines which separate the two classes.
+
+
+For small
 gaps between the entries, we may also end up needing many iterations
 before the solutions converge and if the data cannot be separated
 properly into two distinct classes, we may not experience a converge
@@ -319,7 +331,10 @@ y_i(\bm{w}^T\bm{x}_i+b) \geq 1 \hspace{0.1cm}\forall i.
 \]
 !et
 
-We have thus defined our margin as the invers of the norm of $\bm{w}$. We want to minimize the norm in order to have a as large as possible margin $M$. Before we proceed, we need to remind ourselves about Lagrangian multipliers. 
+We have thus defined our margin as the invers of the norm of
+$\bm{w}$. We want to minimize the norm in order to have a as large as
+possible margin $M$. Before we proceed, we need to remind ourselves
+about Lagrangian multipliers.
 
 !split
 ===== A quick reminder on Lagrangian multipliers =====