From 6de08dee25cbf3493b876f3988cce98475bd7f1b Mon Sep 17 00:00:00 2001 From: Morten Hjorth-Jensen Date: Wed, 27 Sep 2023 10:49:57 +0200 Subject: [PATCH] correcting typos --- .../ProjectWriting/projectwriting.ipynb | 76 +-- doc/pub/week39/html/week39-reveal.html | 28 +- doc/pub/week39/html/week39-solarized.html | 23 +- doc/pub/week39/html/week39.html | 23 +- doc/pub/week39/ipynb/ipynb-week39-src.tar.gz | Bin 193 -> 193 bytes doc/pub/week39/ipynb/week39.ipynb | 589 +++++++++--------- doc/src/week39/week39.do.txt | 19 +- 7 files changed, 407 insertions(+), 351 deletions(-) diff --git a/doc/Projects/ProjectWriting/projectwriting.ipynb b/doc/Projects/ProjectWriting/projectwriting.ipynb index 92c215f4e..531e0ccde 100644 --- a/doc/Projects/ProjectWriting/projectwriting.ipynb +++ b/doc/Projects/ProjectWriting/projectwriting.ipynb @@ -3,9 +3,7 @@ { "cell_type": "markdown", "id": "8b587bcb", - "metadata": { - "editable": true - }, + "metadata": {}, "source": [ "\n", @@ -15,9 +13,7 @@ { "cell_type": "markdown", "id": "ad025527", - "metadata": { - "editable": true - }, + "metadata": {}, "source": [ "# How to write a scientific project\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", @@ -30,9 +26,7 @@ { "cell_type": "markdown", "id": "28e54722", - "metadata": { - "editable": true - }, + "metadata": {}, "source": [ "## The report: how to write a good scienfitic/technical report\n", "**What should it contain? A typical structure.**\n", @@ -59,9 +53,7 @@ { "cell_type": "markdown", "id": "450156e7", - "metadata": { - "editable": true - }, + "metadata": {}, "source": [ "## The report, the abstract\n", "The abstract gives the reader a quick overview of what has been done and the most important results. Here is a typical example\n", @@ -75,9 +67,7 @@ { "cell_type": "markdown", "id": "a5a7f13f", - "metadata": { - "editable": true - }, + "metadata": {}, "source": [ "## The report, the introduction\n", "**What should I focus on? Introduction.**\n", @@ -93,9 +83,7 @@ { "cell_type": "markdown", "id": "a80f8a9f", - "metadata": { - "editable": true - }, + "metadata": {}, "source": [ "## The report, discussion of methods and codes\n", "**What should I focus on? Methods sections.**\n", @@ -112,9 +100,7 @@ { "cell_type": "markdown", "id": "57974a1d", - "metadata": { - "editable": true - }, + "metadata": {}, "source": [ "## The report, code part\n", "\n", @@ -134,9 +120,7 @@ { "cell_type": "markdown", "id": "448f9828", - "metadata": { - "editable": true - }, + "metadata": {}, "source": [ "## The report, results part\n", "**What should I focus on? Results.**\n", @@ -155,9 +139,7 @@ { "cell_type": "markdown", "id": "fa3190a7", - "metadata": { - "editable": true - }, + "metadata": {}, "source": [ "## The report, conclusions and perspectives\n", "**What should I focus on? Conclusions.**\n", @@ -172,9 +154,7 @@ { "cell_type": "markdown", "id": "4b15f2a0", - "metadata": { - "editable": true - }, + "metadata": {}, "source": [ "## The report, appendices\n", "**What should I focus on? additional material.**\n", @@ -191,9 +171,7 @@ { "cell_type": "markdown", "id": "e982ec47", - "metadata": { - "editable": true - }, + "metadata": {}, "source": [ "## The report, references\n", "**What should I focus on? References.**\n", @@ -208,9 +186,7 @@ { "cell_type": "markdown", "id": "ae212bd2", - "metadata": { - "editable": true - }, + "metadata": {}, "source": [ "## Where do I find scientific articles, books etc and examples of reports\n", " * With a UiO IP number you can access freely all books and scientific journals available at our [University library](http://www.ub.uio.no/)\n", @@ -221,9 +197,7 @@ { "cell_type": "markdown", "id": "7d783b6f", - "metadata": { - "editable": true - }, + "metadata": {}, "source": [ "## Other resources\n", "\n", @@ -234,9 +208,7 @@ { "cell_type": "markdown", "id": "47b07b08", - "metadata": { - "editable": true - }, + "metadata": {}, "source": [ "## Procrastination... the curse of all?\n", "\n", @@ -251,7 +223,25 @@ ] } ], - "metadata": {}, + "metadata": { + "kernelspec": { + "display_name": "Python 3 (ipykernel)", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.9.10" + } + }, "nbformat": 4, "nbformat_minor": 5 } diff --git a/doc/pub/week39/html/week39-reveal.html b/doc/pub/week39/html/week39-reveal.html index 3d840a792..41138aa98 100644 --- a/doc/pub/week39/html/week39-reveal.html +++ b/doc/pub/week39/html/week39-reveal.html @@ -244,6 +244,9 @@ MathJax.Hub.Config({

+ + +

@@ -587,9 +590,17 @@ regular polygons (triangles, rectangles, pentagons, etc...).

Convex function

-

Convex function: Let \( X \subset \mathbb{R}^n \) be a convex set. Assume that the function \( f: X \rightarrow \mathbb{R} \) is continuous, then \( f \) is said to be convex if

 
-$$f(tx_1 + (1-t)x_2) \leq tf(x_1) + (1-t)f(x_2) $$ -

 
for all \( x_1, x_2 \in X \) and for all \( t \in [0,1] \). If \( \leq \) is replaced with a strict inequaltiy in the definition, we demand \( x_1 \neq x_2 \) and \( t\in(0,1) \) then \( f \) is said to be strictly convex. For a single variable function, convexity means that if you draw a straight line connecting \( f(x_1) \) and \( f(x_2) \), the value of the function on the interval \( [x_1,x_2] \) is always below the line as illustrated below.

+

Convex function: Let \( X \subset \mathbb{R}^n \) be a convex +set. Assume that the function \( f: X \rightarrow \mathbb{R} \) is +continuous, then \( f \) is said to be convex if \( f(tx_1 + (1-t)x_2) \leq tf(x_1) + (1-t)f(x_2) \) +for all \( x_1, x_2 \in X \) and for all \( t \in [0,1] \). +If \( \leq \) is replaced with a strict inequaltiy in the +definition, we demand \( x_1 \neq x_2 \) and \( t\in(0,1) \) then \( f \) is said +to be strictly convex. For a single variable function, convexity means +that if you draw a straight line connecting \( f(x_1) \) and \( f(x_2) \), the +value of the function on the interval \( [x_1,x_2] \) is always below the +line as illustrated below. +

@@ -598,7 +609,7 @@ $$f(tx_1 + (1-t)x_2) \leq tf(x_1) + (1-t)f(x_2) $$

In the following we state first and second-order conditions which ensures convexity of a function \( f \). We write \( D_f \) to denote the domain of \( f \), i.e the subset of \( R^n \) where \( f \) is defined. For more -details and proofs we refer to: S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press. +details and proofs we refer to: S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press.

@@ -606,10 +617,11 @@ details and proofs we refer to: Optimization, the central part of any Machine Learning algortithm @@ -673,7 +675,17 @@ regular polygons (triangles, rectangles, pentagons, etc...).









Convex function

-

Convex function: Let \( X \subset \mathbb{R}^n \) be a convex set. Assume that the function \( f: X \rightarrow \mathbb{R} \) is continuous, then \( f \) is said to be convex if $$f(tx_1 + (1-t)x_2) \leq tf(x_1) + (1-t)f(x_2) $$ for all \( x_1, x_2 \in X \) and for all \( t \in [0,1] \). If \( \leq \) is replaced with a strict inequaltiy in the definition, we demand \( x_1 \neq x_2 \) and \( t\in(0,1) \) then \( f \) is said to be strictly convex. For a single variable function, convexity means that if you draw a straight line connecting \( f(x_1) \) and \( f(x_2) \), the value of the function on the interval \( [x_1,x_2] \) is always below the line as illustrated below.

+

Convex function: Let \( X \subset \mathbb{R}^n \) be a convex +set. Assume that the function \( f: X \rightarrow \mathbb{R} \) is +continuous, then \( f \) is said to be convex if \( f(tx_1 + (1-t)x_2) \leq tf(x_1) + (1-t)f(x_2) \) +for all \( x_1, x_2 \in X \) and for all \( t \in [0,1] \). +If \( \leq \) is replaced with a strict inequaltiy in the +definition, we demand \( x_1 \neq x_2 \) and \( t\in(0,1) \) then \( f \) is said +to be strictly convex. For a single variable function, convexity means +that if you draw a straight line connecting \( f(x_1) \) and \( f(x_2) \), the +value of the function on the interval \( [x_1,x_2] \) is always below the +line as illustrated below. +











Conditions on convex functions

@@ -681,7 +693,7 @@ regular polygons (triangles, rectangles, pentagons, etc...).

In the following we state first and second-order conditions which ensures convexity of a function \( f \). We write \( D_f \) to denote the domain of \( f \), i.e the subset of \( R^n \) where \( f \) is defined. For more -details and proofs we refer to: S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press. +details and proofs we refer to: S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press.

@@ -689,8 +701,11 @@ details and proofs we refer to: Optimization, the central part of any Machine Learning algortithm @@ -750,7 +752,17 @@ regular polygons (triangles, rectangles, pentagons, etc...).









Convex function

-

Convex function: Let \( X \subset \mathbb{R}^n \) be a convex set. Assume that the function \( f: X \rightarrow \mathbb{R} \) is continuous, then \( f \) is said to be convex if $$f(tx_1 + (1-t)x_2) \leq tf(x_1) + (1-t)f(x_2) $$ for all \( x_1, x_2 \in X \) and for all \( t \in [0,1] \). If \( \leq \) is replaced with a strict inequaltiy in the definition, we demand \( x_1 \neq x_2 \) and \( t\in(0,1) \) then \( f \) is said to be strictly convex. For a single variable function, convexity means that if you draw a straight line connecting \( f(x_1) \) and \( f(x_2) \), the value of the function on the interval \( [x_1,x_2] \) is always below the line as illustrated below.

+

Convex function: Let \( X \subset \mathbb{R}^n \) be a convex +set. Assume that the function \( f: X \rightarrow \mathbb{R} \) is +continuous, then \( f \) is said to be convex if \( f(tx_1 + (1-t)x_2) \leq tf(x_1) + (1-t)f(x_2) \) +for all \( x_1, x_2 \in X \) and for all \( t \in [0,1] \). +If \( \leq \) is replaced with a strict inequaltiy in the +definition, we demand \( x_1 \neq x_2 \) and \( t\in(0,1) \) then \( f \) is said +to be strictly convex. For a single variable function, convexity means +that if you draw a straight line connecting \( f(x_1) \) and \( f(x_2) \), the +value of the function on the interval \( [x_1,x_2] \) is always below the +line as illustrated below. +











Conditions on convex functions

@@ -758,7 +770,7 @@ regular polygons (triangles, rectangles, pentagons, etc...).

In the following we state first and second-order conditions which ensures convexity of a function \( f \). We write \( D_f \) to denote the domain of \( f \), i.e the subset of \( R^n \) where \( f \) is defined. For more -details and proofs we refer to: S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press. +details and proofs we refer to: S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press.

@@ -766,8 +778,11 @@ details and proofs we refer to: " ] }, { "cell_type": "markdown", - "id": "529184e5", + "id": "ad3725f3", "metadata": { "editable": true }, @@ -91,7 +93,7 @@ }, { "cell_type": "markdown", - "id": "1a65465a", + "id": "a750ab4f", "metadata": { "editable": true }, @@ -108,7 +110,7 @@ }, { "cell_type": "markdown", - "id": "b67231b3", + "id": "237464b0", "metadata": { "editable": true }, @@ -123,7 +125,7 @@ }, { "cell_type": "markdown", - "id": "3f5e03db", + "id": "0d365922", "metadata": { "editable": true }, @@ -133,7 +135,7 @@ }, { "cell_type": "markdown", - "id": "e83141ae", + "id": "17dcc22b", "metadata": { "editable": true }, @@ -149,7 +151,7 @@ }, { "cell_type": "markdown", - "id": "cef5864b", + "id": "23a1111d", "metadata": { "editable": true }, @@ -161,7 +163,7 @@ }, { "cell_type": "markdown", - "id": "2f59bbb1", + "id": "f69e6301", "metadata": { "editable": true }, @@ -172,7 +174,7 @@ }, { "cell_type": "markdown", - "id": "3869b3c6", + "id": "a98f91ae", "metadata": { "editable": true }, @@ -184,7 +186,7 @@ }, { "cell_type": "markdown", - "id": "e4656e92", + "id": "b4a058a7", "metadata": { "editable": true }, @@ -194,7 +196,7 @@ }, { "cell_type": "markdown", - "id": "b73ae554", + "id": "4bef2a5f", "metadata": { "editable": true }, @@ -208,7 +210,7 @@ }, { "cell_type": "markdown", - "id": "70a2df05", + "id": "05001ecb", "metadata": { "editable": true }, @@ -220,7 +222,7 @@ }, { "cell_type": "markdown", - "id": "2e36f976", + "id": "82d01fbd", "metadata": { "editable": true }, @@ -230,7 +232,7 @@ }, { "cell_type": "markdown", - "id": "7c4959c9", + "id": "36962ee1", "metadata": { "editable": true }, @@ -242,7 +244,7 @@ }, { "cell_type": "markdown", - "id": "c553379e", + "id": "c47c224b", "metadata": { "editable": true }, @@ -254,7 +256,7 @@ }, { "cell_type": "markdown", - "id": "145f9699", + "id": "ec3ede39", "metadata": { "editable": true }, @@ -274,7 +276,7 @@ }, { "cell_type": "markdown", - "id": "9a68f686", + "id": "b869cb18", "metadata": { "editable": true }, @@ -290,7 +292,7 @@ }, { "cell_type": "markdown", - "id": "d523ab61", + "id": "fd447d3e", "metadata": { "editable": true }, @@ -306,7 +308,7 @@ }, { "cell_type": "markdown", - "id": "8205fd3a", + "id": "1d894c75", "metadata": { "editable": true }, @@ -317,7 +319,7 @@ }, { "cell_type": "markdown", - "id": "98d52e46", + "id": "0d508e5c", "metadata": { "editable": true }, @@ -329,7 +331,7 @@ }, { "cell_type": "markdown", - "id": "203f65e8", + "id": "7ed0ad95", "metadata": { "editable": true }, @@ -339,7 +341,7 @@ }, { "cell_type": "markdown", - "id": "dd40195b", + "id": "cd997af7", "metadata": { "editable": true }, @@ -351,7 +353,7 @@ }, { "cell_type": "markdown", - "id": "e13a7340", + "id": "7050afc6", "metadata": { "editable": true }, @@ -361,7 +363,7 @@ }, { "cell_type": "markdown", - "id": "30c258c9", + "id": "a198702b", "metadata": { "editable": true }, @@ -373,7 +375,7 @@ }, { "cell_type": "markdown", - "id": "84f880c9", + "id": "1514d0ff", "metadata": { "editable": true }, @@ -395,7 +397,7 @@ }, { "cell_type": "markdown", - "id": "b237f214", + "id": "42154f58", "metadata": { "editable": true }, @@ -408,7 +410,7 @@ }, { "cell_type": "markdown", - "id": "b4ea1db2", + "id": "d1a8b99e", "metadata": { "editable": true }, @@ -421,7 +423,7 @@ }, { "cell_type": "markdown", - "id": "8b2b6606", + "id": "34b84a08", "metadata": { "editable": true }, @@ -431,7 +433,7 @@ }, { "cell_type": "markdown", - "id": "e9d217a4", + "id": "8ee3bd57", "metadata": { "editable": true }, @@ -449,7 +451,7 @@ }, { "cell_type": "markdown", - "id": "54f87f7c", + "id": "f2fc5f78", "metadata": { "editable": true }, @@ -459,7 +461,7 @@ }, { "cell_type": "markdown", - "id": "c2711fc8", + "id": "a5cd2cbc", "metadata": { "editable": true }, @@ -474,7 +476,7 @@ }, { "cell_type": "markdown", - "id": "e0409f1c", + "id": "c081ba3b", "metadata": { "editable": true }, @@ -484,7 +486,7 @@ }, { "cell_type": "markdown", - "id": "53651890", + "id": "d1c5c335", "metadata": { "editable": true }, @@ -498,7 +500,7 @@ }, { "cell_type": "markdown", - "id": "4e70e7ac", + "id": "a6e178cc", "metadata": { "editable": true }, @@ -508,7 +510,7 @@ }, { "cell_type": "markdown", - "id": "9336db1d", + "id": "622ad12d", "metadata": { "editable": true }, @@ -522,7 +524,7 @@ }, { "cell_type": "markdown", - "id": "2c2ae028", + "id": "58497924", "metadata": { "editable": true }, @@ -537,7 +539,7 @@ }, { "cell_type": "markdown", - "id": "cdf99885", + "id": "fdcbb7b0", "metadata": { "editable": true }, @@ -554,7 +556,7 @@ }, { "cell_type": "markdown", - "id": "11bb1b41", + "id": "ef9f3f1b", "metadata": { "editable": true }, @@ -566,7 +568,7 @@ }, { "cell_type": "markdown", - "id": "5d957768", + "id": "041dd3e6", "metadata": { "editable": true }, @@ -580,7 +582,7 @@ }, { "cell_type": "markdown", - "id": "455b420a", + "id": "b0be361c", "metadata": { "editable": true }, @@ -595,7 +597,7 @@ }, { "cell_type": "markdown", - "id": "aacb8b05", + "id": "96ac077e", "metadata": { "editable": true }, @@ -607,7 +609,7 @@ }, { "cell_type": "markdown", - "id": "2bb3385b", + "id": "1174ae08", "metadata": { "editable": true }, @@ -618,7 +620,7 @@ }, { "cell_type": "markdown", - "id": "21326ef0", + "id": "599532b8", "metadata": { "editable": true }, @@ -646,7 +648,7 @@ }, { "cell_type": "markdown", - "id": "a41b3cc6", + "id": "cd2b65ad", "metadata": { "editable": true }, @@ -668,7 +670,7 @@ }, { "cell_type": "markdown", - "id": "a01f11a5", + "id": "9e39246b", "metadata": { "editable": true }, @@ -690,19 +692,28 @@ }, { "cell_type": "markdown", - "id": "076a32fa", + "id": "74adc3b0", "metadata": { "editable": true }, "source": [ "## Convex function\n", "\n", - "**Convex function**: Let $X \\subset \\mathbb{R}^n$ be a convex set. Assume that the function $f: X \\rightarrow \\mathbb{R}$ is continuous, then $f$ is said to be convex if $$f(tx_1 + (1-t)x_2) \\leq tf(x_1) + (1-t)f(x_2) $$ for all $x_1, x_2 \\in X$ and for all $t \\in [0,1]$. If $\\leq$ is replaced with a strict inequaltiy in the definition, we demand $x_1 \\neq x_2$ and $t\\in(0,1)$ then $f$ is said to be strictly convex. For a single variable function, convexity means that if you draw a straight line connecting $f(x_1)$ and $f(x_2)$, the value of the function on the interval $[x_1,x_2]$ is always below the line as illustrated below." + "**Convex function**: Let $X \\subset \\mathbb{R}^n$ be a convex\n", + "set. Assume that the function $f: X \\rightarrow \\mathbb{R}$ is\n", + "continuous, then $f$ is said to be convex if $f(tx_1 + (1-t)x_2) \\leq tf(x_1) + (1-t)f(x_2)$\n", + "for all $x_1, x_2 \\in X$ and for all $t \\in [0,1]$.\n", + "If $\\leq$ is replaced with a strict inequaltiy in the\n", + "definition, we demand $x_1 \\neq x_2$ and $t\\in(0,1)$ then $f$ is said\n", + "to be strictly convex. For a single variable function, convexity means\n", + "that if you draw a straight line connecting $f(x_1)$ and $f(x_2)$, the\n", + "value of the function on the interval $[x_1,x_2]$ is always below the\n", + "line as illustrated below." ] }, { "cell_type": "markdown", - "id": "73adde3c", + "id": "e4b22b9e", "metadata": { "editable": true }, @@ -712,14 +723,16 @@ "In the following we state first and second-order conditions which\n", "ensures convexity of a function $f$. We write $D_f$ to denote the\n", "domain of $f$, i.e the subset of $R^n$ where $f$ is defined. For more\n", - "details and proofs we refer to: [S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press](http://stanford.edu/boyd/cvxbook/, 2004).\n", + "details and proofs we refer to: [S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press](http://stanford.edu/boyd/cvxbook/).\n", "\n", "**First order condition.**\n", "\n", "Suppose $f$ is differentiable (i.e $\\nabla f(x)$ is well defined for\n", "all $x$ in the domain of $f$). Then $f$ is convex if and only if $D_f$\n", - "is a convex set and $$f(y) \\geq f(x) + \\nabla f(x)^T (y-x) $$ holds\n", - "for all $x,y \\in D_f$. This condition means that for a convex function\n", + "is a convex set and $f(y) \\geq f(x) + \\nabla f(x)^T (y-x)$ holds\n", + "for all $x,y \\in D_f$.\n", + "\n", + "This condition means that for a convex function\n", "the first order Taylor expansion (right hand side above) at any point\n", "a global under estimator of the function. To convince yourself you can\n", "make a drawing of $f(x) = x^2+1$ and draw the tangent line to $f(x)$ and\n", @@ -739,7 +752,7 @@ }, { "cell_type": "markdown", - "id": "9f9ab5ff", + "id": "a917f696", "metadata": { "editable": true }, @@ -767,7 +780,7 @@ }, { "cell_type": "markdown", - "id": "0b2a482b", + "id": "c7c449a5", "metadata": { "editable": true }, @@ -797,7 +810,7 @@ }, { "cell_type": "markdown", - "id": "6566ee55", + "id": "39b3fcae", "metadata": { "editable": true }, @@ -817,7 +830,7 @@ }, { "cell_type": "markdown", - "id": "c2e30cc1", + "id": "786a9bb9", "metadata": { "editable": true }, @@ -829,7 +842,7 @@ }, { "cell_type": "markdown", - "id": "5012b398", + "id": "9ae071ef", "metadata": { "editable": true }, @@ -839,7 +852,7 @@ }, { "cell_type": "markdown", - "id": "ca65d9a9", + "id": "412b4e29", "metadata": { "editable": true }, @@ -851,7 +864,7 @@ }, { "cell_type": "markdown", - "id": "ec9323dd", + "id": "c174973c", "metadata": { "editable": true }, @@ -863,7 +876,7 @@ }, { "cell_type": "markdown", - "id": "5caf0f7f", + "id": "0e6e7ad4", "metadata": { "editable": true }, @@ -875,7 +888,7 @@ }, { "cell_type": "markdown", - "id": "07734ce6", + "id": "19f4e18b", "metadata": { "editable": true }, @@ -887,7 +900,7 @@ }, { "cell_type": "markdown", - "id": "468dcb52", + "id": "51481f28", "metadata": { "editable": true }, @@ -898,7 +911,7 @@ }, { "cell_type": "markdown", - "id": "b63a89ae", + "id": "d71266e3", "metadata": { "editable": true }, @@ -911,7 +924,7 @@ }, { "cell_type": "markdown", - "id": "56a122a3", + "id": "86d8e1e4", "metadata": { "editable": true }, @@ -923,7 +936,7 @@ }, { "cell_type": "markdown", - "id": "3256eb29", + "id": "307d9b7c", "metadata": { "editable": true }, @@ -933,7 +946,7 @@ }, { "cell_type": "markdown", - "id": "b6bab858", + "id": "71b258ad", "metadata": { "editable": true }, @@ -945,7 +958,7 @@ }, { "cell_type": "markdown", - "id": "7ff39e96", + "id": "9efc8377", "metadata": { "editable": true }, @@ -955,7 +968,7 @@ }, { "cell_type": "markdown", - "id": "6678fce9", + "id": "3ea91d96", "metadata": { "editable": true }, @@ -966,7 +979,7 @@ }, { "cell_type": "markdown", - "id": "853eb11f", + "id": "ad24211f", "metadata": { "editable": true }, @@ -978,7 +991,7 @@ }, { "cell_type": "markdown", - "id": "7c229917", + "id": "b8c8925d", "metadata": { "editable": true }, @@ -990,7 +1003,7 @@ }, { "cell_type": "markdown", - "id": "5c8f310a", + "id": "aa5c7f87", "metadata": { "editable": true }, @@ -1002,7 +1015,7 @@ }, { "cell_type": "markdown", - "id": "f8a8c317", + "id": "388a1a92", "metadata": { "editable": true }, @@ -1013,7 +1026,7 @@ }, { "cell_type": "markdown", - "id": "49b64ed0", + "id": "a69a174e", "metadata": { "editable": true }, @@ -1024,7 +1037,7 @@ }, { "cell_type": "markdown", - "id": "857ee939", + "id": "413fd43b", "metadata": { "editable": true }, @@ -1036,7 +1049,7 @@ }, { "cell_type": "markdown", - "id": "7e5611fa", + "id": "f83101cc", "metadata": { "editable": true }, @@ -1046,7 +1059,7 @@ }, { "cell_type": "markdown", - "id": "384d5aa2", + "id": "e25c728c", "metadata": { "editable": true }, @@ -1058,7 +1071,7 @@ }, { "cell_type": "markdown", - "id": "0ce677be", + "id": "24a2f936", "metadata": { "editable": true }, @@ -1068,7 +1081,7 @@ }, { "cell_type": "markdown", - "id": "e97f9044", + "id": "e629e086", "metadata": { "editable": true }, @@ -1080,7 +1093,7 @@ }, { "cell_type": "markdown", - "id": "d7fbeb68", + "id": "4ed3cc84", "metadata": { "editable": true }, @@ -1090,7 +1103,7 @@ }, { "cell_type": "markdown", - "id": "293c09b1", + "id": "1e28bd59", "metadata": { "editable": true }, @@ -1102,7 +1115,7 @@ }, { "cell_type": "markdown", - "id": "af88e065", + "id": "b63b7d78", "metadata": { "editable": true }, @@ -1112,7 +1125,7 @@ }, { "cell_type": "markdown", - "id": "2757e302", + "id": "850bb9a9", "metadata": { "editable": true }, @@ -1124,7 +1137,7 @@ }, { "cell_type": "markdown", - "id": "87aab66b", + "id": "68749604", "metadata": { "editable": true }, @@ -1135,7 +1148,7 @@ { "cell_type": "code", "execution_count": 1, - "id": "a0e20ff7", + "id": "bf454b1d", "metadata": { "collapsed": false, "editable": true @@ -1168,7 +1181,7 @@ }, { "cell_type": "markdown", - "id": "c01b471a", + "id": "3569007f", "metadata": { "editable": true }, @@ -1179,7 +1192,7 @@ { "cell_type": "code", "execution_count": 2, - "id": "5b835c85", + "id": "0df89031", "metadata": { "collapsed": false, "editable": true @@ -1193,7 +1206,7 @@ }, { "cell_type": "markdown", - "id": "d6a3c121", + "id": "d6ac010a", "metadata": { "editable": true }, @@ -1204,7 +1217,7 @@ { "cell_type": "code", "execution_count": 3, - "id": "19e1d73c", + "id": "d17a90d2", "metadata": { "collapsed": false, "editable": true @@ -1217,7 +1230,7 @@ }, { "cell_type": "markdown", - "id": "9f7b2dfc", + "id": "d6ccbd2b", "metadata": { "editable": true }, @@ -1228,7 +1241,7 @@ { "cell_type": "code", "execution_count": 4, - "id": "7d8247e6", + "id": "90cfd264", "metadata": { "collapsed": false, "editable": true @@ -1246,7 +1259,7 @@ }, { "cell_type": "markdown", - "id": "c44006da", + "id": "222f1949", "metadata": { "editable": true }, @@ -1257,7 +1270,7 @@ { "cell_type": "code", "execution_count": 5, - "id": "bb8a0fd8", + "id": "4c97a7eb", "metadata": { "collapsed": false, "editable": true @@ -1272,7 +1285,7 @@ }, { "cell_type": "markdown", - "id": "3d3c98f0", + "id": "1a84e70d", "metadata": { "editable": true }, @@ -1282,7 +1295,7 @@ }, { "cell_type": "markdown", - "id": "29e5e792", + "id": "023458ec", "metadata": { "editable": true }, @@ -1296,7 +1309,7 @@ }, { "cell_type": "markdown", - "id": "2b0e0db3", + "id": "d87ce0b4", "metadata": { "editable": true }, @@ -1308,7 +1321,7 @@ }, { "cell_type": "markdown", - "id": "401dd643", + "id": "2462039a", "metadata": { "editable": true }, @@ -1319,7 +1332,7 @@ }, { "cell_type": "markdown", - "id": "bc29d596", + "id": "b724b10a", "metadata": { "editable": true }, @@ -1331,7 +1344,7 @@ }, { "cell_type": "markdown", - "id": "c1b7adb4", + "id": "33b615af", "metadata": { "editable": true }, @@ -1342,7 +1355,7 @@ }, { "cell_type": "markdown", - "id": "18924232", + "id": "cda299f2", "metadata": { "editable": true }, @@ -1353,7 +1366,7 @@ }, { "cell_type": "markdown", - "id": "1764ac31", + "id": "b8c35ed9", "metadata": { "editable": true }, @@ -1365,7 +1378,7 @@ }, { "cell_type": "markdown", - "id": "379d5862", + "id": "0c48b439", "metadata": { "editable": true }, @@ -1375,7 +1388,7 @@ }, { "cell_type": "markdown", - "id": "9587d8bf", + "id": "144b8744", "metadata": { "editable": true }, @@ -1387,7 +1400,7 @@ }, { "cell_type": "markdown", - "id": "4c3d0bfb", + "id": "2a0d3fd0", "metadata": { "editable": true }, @@ -1399,7 +1412,7 @@ }, { "cell_type": "markdown", - "id": "4079ca1a", + "id": "6d9dfebd", "metadata": { "editable": true }, @@ -1411,7 +1424,7 @@ }, { "cell_type": "markdown", - "id": "e5b487a5", + "id": "27e9b21c", "metadata": { "editable": true }, @@ -1423,7 +1436,7 @@ }, { "cell_type": "markdown", - "id": "b623d7f7", + "id": "f4527105", "metadata": { "editable": true }, @@ -1434,7 +1447,7 @@ }, { "cell_type": "markdown", - "id": "8520c560", + "id": "edcf087b", "metadata": { "editable": true }, @@ -1446,7 +1459,7 @@ }, { "cell_type": "markdown", - "id": "52575016", + "id": "ae00b0eb", "metadata": { "editable": true }, @@ -1456,7 +1469,7 @@ }, { "cell_type": "markdown", - "id": "1b8a85bd", + "id": "46364db3", "metadata": { "editable": true }, @@ -1468,7 +1481,7 @@ }, { "cell_type": "markdown", - "id": "e53b0f45", + "id": "0c24c960", "metadata": { "editable": true }, @@ -1478,7 +1491,7 @@ }, { "cell_type": "markdown", - "id": "2238e15f", + "id": "9300d6df", "metadata": { "editable": true }, @@ -1490,7 +1503,7 @@ }, { "cell_type": "markdown", - "id": "f00e8864", + "id": "0c156c48", "metadata": { "editable": true }, @@ -1510,7 +1523,7 @@ }, { "cell_type": "markdown", - "id": "7a17895d", + "id": "3edd95c8", "metadata": { "editable": true }, @@ -1522,7 +1535,7 @@ }, { "cell_type": "markdown", - "id": "d4bafcb3", + "id": "ae2529bb", "metadata": { "editable": true }, @@ -1532,7 +1545,7 @@ }, { "cell_type": "markdown", - "id": "78a7d2c3", + "id": "11afa7f3", "metadata": { "editable": true }, @@ -1544,7 +1557,7 @@ }, { "cell_type": "markdown", - "id": "e3192cbf", + "id": "2dccd575", "metadata": { "editable": true }, @@ -1554,7 +1567,7 @@ }, { "cell_type": "markdown", - "id": "0d99ed55", + "id": "8bf55c33", "metadata": { "editable": true }, @@ -1565,7 +1578,7 @@ }, { "cell_type": "markdown", - "id": "b9653ede", + "id": "a5314db5", "metadata": { "editable": true }, @@ -1577,7 +1590,7 @@ }, { "cell_type": "markdown", - "id": "78441105", + "id": "3311a809", "metadata": { "editable": true }, @@ -1589,7 +1602,7 @@ }, { "cell_type": "markdown", - "id": "317355d2", + "id": "78866ce2", "metadata": { "editable": true }, @@ -1601,7 +1614,7 @@ }, { "cell_type": "markdown", - "id": "9bb15157", + "id": "53ce67cb", "metadata": { "editable": true }, @@ -1614,7 +1627,7 @@ }, { "cell_type": "markdown", - "id": "ac584971", + "id": "8edba574", "metadata": { "editable": true }, @@ -1625,7 +1638,7 @@ }, { "cell_type": "markdown", - "id": "911f1dfa", + "id": "225c2ea7", "metadata": { "editable": true }, @@ -1637,7 +1650,7 @@ }, { "cell_type": "markdown", - "id": "d1472568", + "id": "9204037b", "metadata": { "editable": true }, @@ -1653,7 +1666,7 @@ }, { "cell_type": "markdown", - "id": "c79708e8", + "id": "4bd89a25", "metadata": { "editable": true }, @@ -1665,7 +1678,7 @@ }, { "cell_type": "markdown", - "id": "535d3e73", + "id": "1f12325e", "metadata": { "editable": true }, @@ -1676,7 +1689,7 @@ }, { "cell_type": "markdown", - "id": "ad718f62", + "id": "95c7ffea", "metadata": { "editable": true }, @@ -1688,7 +1701,7 @@ }, { "cell_type": "markdown", - "id": "d0a90fa5", + "id": "b6712b84", "metadata": { "editable": true }, @@ -1698,7 +1711,7 @@ }, { "cell_type": "markdown", - "id": "860e9217", + "id": "5df49546", "metadata": { "editable": true }, @@ -1710,7 +1723,7 @@ }, { "cell_type": "markdown", - "id": "6d8a72c8", + "id": "b8da7313", "metadata": { "editable": true }, @@ -1720,7 +1733,7 @@ }, { "cell_type": "markdown", - "id": "746e6fc0", + "id": "2ea83ddd", "metadata": { "editable": true }, @@ -1732,7 +1745,7 @@ }, { "cell_type": "markdown", - "id": "f76c0e69", + "id": "d2ab6c8b", "metadata": { "editable": true }, @@ -1742,7 +1755,7 @@ }, { "cell_type": "markdown", - "id": "9aee35ca", + "id": "7e63917b", "metadata": { "editable": true }, @@ -1754,7 +1767,7 @@ }, { "cell_type": "markdown", - "id": "ce9ce258", + "id": "4686c2e0", "metadata": { "editable": true }, @@ -1778,7 +1791,7 @@ { "cell_type": "code", "execution_count": 6, - "id": "f902a0f2", + "id": "92e23259", "metadata": { "collapsed": false, "editable": true @@ -1791,7 +1804,7 @@ }, { "cell_type": "markdown", - "id": "36d883b2", + "id": "33335dce", "metadata": { "editable": true }, @@ -1802,7 +1815,7 @@ }, { "cell_type": "markdown", - "id": "cde21ef1", + "id": "70160dcd", "metadata": { "editable": true }, @@ -1814,7 +1827,7 @@ }, { "cell_type": "markdown", - "id": "f2a021d3", + "id": "86dc4224", "metadata": { "editable": true }, @@ -1824,7 +1837,7 @@ }, { "cell_type": "markdown", - "id": "ea0a91e4", + "id": "e375ce7a", "metadata": { "editable": true }, @@ -1836,7 +1849,7 @@ }, { "cell_type": "markdown", - "id": "854f3ebb", + "id": "34a9856c", "metadata": { "editable": true }, @@ -1850,7 +1863,7 @@ }, { "cell_type": "markdown", - "id": "dd282d2d", + "id": "8f429f87", "metadata": { "editable": true }, @@ -1866,7 +1879,7 @@ }, { "cell_type": "markdown", - "id": "fd562029", + "id": "1601c60d", "metadata": { "editable": true }, @@ -1876,7 +1889,7 @@ }, { "cell_type": "markdown", - "id": "25369bc3", + "id": "c4095285", "metadata": { "editable": true }, @@ -1888,7 +1901,7 @@ }, { "cell_type": "markdown", - "id": "a222f3ea", + "id": "f44787d2", "metadata": { "editable": true }, @@ -1898,7 +1911,7 @@ }, { "cell_type": "markdown", - "id": "1d5fb6f1", + "id": "a0866b70", "metadata": { "editable": true }, @@ -1910,7 +1923,7 @@ }, { "cell_type": "markdown", - "id": "eab2df73", + "id": "8994423a", "metadata": { "editable": true }, @@ -1924,7 +1937,7 @@ }, { "cell_type": "markdown", - "id": "daee1165", + "id": "0bea6f79", "metadata": { "editable": true }, @@ -1934,7 +1947,7 @@ }, { "cell_type": "markdown", - "id": "f2c6f5cc", + "id": "2a978bc1", "metadata": { "editable": true }, @@ -1945,7 +1958,7 @@ }, { "cell_type": "markdown", - "id": "ecce0d08", + "id": "a4faf2b7", "metadata": { "editable": true }, @@ -1960,7 +1973,7 @@ }, { "cell_type": "markdown", - "id": "c4308e5f", + "id": "7e31eb15", "metadata": { "editable": true }, @@ -1970,7 +1983,7 @@ }, { "cell_type": "markdown", - "id": "4ee64b17", + "id": "e392b73b", "metadata": { "editable": true }, @@ -1982,7 +1995,7 @@ }, { "cell_type": "markdown", - "id": "57e8db33", + "id": "6d263b11", "metadata": { "editable": true }, @@ -1994,7 +2007,7 @@ }, { "cell_type": "markdown", - "id": "c42e4032", + "id": "9f413570", "metadata": { "editable": true }, @@ -2009,7 +2022,7 @@ }, { "cell_type": "markdown", - "id": "4c430cd3", + "id": "672cfbaf", "metadata": { "editable": true }, @@ -2022,7 +2035,7 @@ { "cell_type": "code", "execution_count": 7, - "id": "9ac6096f", + "id": "d9cb4a24", "metadata": { "collapsed": false, "editable": true @@ -2079,7 +2092,7 @@ }, { "cell_type": "markdown", - "id": "df783e1d", + "id": "41aa5c4e", "metadata": { "editable": true }, @@ -2090,7 +2103,7 @@ { "cell_type": "code", "execution_count": 8, - "id": "98f08f24", + "id": "b7df293d", "metadata": { "collapsed": false, "editable": true @@ -2117,7 +2130,7 @@ }, { "cell_type": "markdown", - "id": "50a5ab0d", + "id": "985bedf1", "metadata": { "editable": true }, @@ -2129,7 +2142,7 @@ }, { "cell_type": "markdown", - "id": "b35293d4", + "id": "4dc58942", "metadata": { "editable": true }, @@ -2141,7 +2154,7 @@ }, { "cell_type": "markdown", - "id": "fed491ee", + "id": "d7a01457", "metadata": { "editable": true }, @@ -2151,7 +2164,7 @@ }, { "cell_type": "markdown", - "id": "a0b4c94e", + "id": "ccf1c302", "metadata": { "editable": true }, @@ -2165,7 +2178,7 @@ }, { "cell_type": "markdown", - "id": "e4f02dce", + "id": "c7ca4b03", "metadata": { "editable": true }, @@ -2175,7 +2188,7 @@ }, { "cell_type": "markdown", - "id": "b9f297ff", + "id": "213732f8", "metadata": { "editable": true }, @@ -2187,7 +2200,7 @@ }, { "cell_type": "markdown", - "id": "4e1caf44", + "id": "e1647f85", "metadata": { "editable": true }, @@ -2198,7 +2211,7 @@ }, { "cell_type": "markdown", - "id": "a046daea", + "id": "08068516", "metadata": { "editable": true }, @@ -2213,7 +2226,7 @@ }, { "cell_type": "markdown", - "id": "02f05574", + "id": "c00bdabb", "metadata": { "editable": true }, @@ -2227,7 +2240,7 @@ }, { "cell_type": "markdown", - "id": "45484749", + "id": "622e2efe", "metadata": { "editable": true }, @@ -2238,7 +2251,7 @@ { "cell_type": "code", "execution_count": 9, - "id": "9973cd20", + "id": "039dc017", "metadata": { "collapsed": false, "editable": true @@ -2299,7 +2312,7 @@ }, { "cell_type": "markdown", - "id": "1836d4ef", + "id": "e855579a", "metadata": { "editable": true }, @@ -2321,7 +2334,7 @@ }, { "cell_type": "markdown", - "id": "88975d3d", + "id": "5f1882ae", "metadata": { "editable": true }, @@ -2334,7 +2347,7 @@ { "cell_type": "code", "execution_count": 10, - "id": "56f415e0", + "id": "283b5e04", "metadata": { "collapsed": false, "editable": true @@ -2400,7 +2413,7 @@ }, { "cell_type": "markdown", - "id": "d3343584", + "id": "0ea14406", "metadata": { "editable": true }, @@ -2411,7 +2424,7 @@ { "cell_type": "code", "execution_count": 11, - "id": "ff1e3778", + "id": "290a1521", "metadata": { "collapsed": false, "editable": true @@ -2485,7 +2498,7 @@ }, { "cell_type": "markdown", - "id": "c1d70995", + "id": "b5a1292a", "metadata": { "editable": true }, @@ -2497,7 +2510,7 @@ }, { "cell_type": "markdown", - "id": "930a5be8", + "id": "14069d67", "metadata": { "editable": true }, @@ -2518,7 +2531,7 @@ }, { "cell_type": "markdown", - "id": "0a7fb7ef", + "id": "3e43cde8", "metadata": { "editable": true }, @@ -2550,7 +2563,7 @@ }, { "cell_type": "markdown", - "id": "dbff87b0", + "id": "af983213", "metadata": { "editable": true }, @@ -2567,7 +2580,7 @@ }, { "cell_type": "markdown", - "id": "cd292df5", + "id": "46c24cc8", "metadata": { "editable": true }, @@ -2580,7 +2593,7 @@ }, { "cell_type": "markdown", - "id": "1b2ffa4e", + "id": "aafecee1", "metadata": { "editable": true }, @@ -2593,7 +2606,7 @@ }, { "cell_type": "markdown", - "id": "d0abe4b0", + "id": "bd361b47", "metadata": { "editable": true }, @@ -2606,7 +2619,7 @@ }, { "cell_type": "markdown", - "id": "65c15c60", + "id": "3d2961b9", "metadata": { "editable": true }, @@ -2620,7 +2633,7 @@ }, { "cell_type": "markdown", - "id": "460354c0", + "id": "eddbed61", "metadata": { "editable": true }, @@ -2642,7 +2655,7 @@ }, { "cell_type": "markdown", - "id": "c2a5dfcd", + "id": "5f5c72ee", "metadata": { "editable": true }, @@ -2657,7 +2670,7 @@ }, { "cell_type": "markdown", - "id": "eeeb0fe8", + "id": "0ac15259", "metadata": { "editable": true }, @@ -2669,7 +2682,7 @@ }, { "cell_type": "markdown", - "id": "2b49c741", + "id": "5654c70b", "metadata": { "editable": true }, @@ -2682,7 +2695,7 @@ }, { "cell_type": "markdown", - "id": "8ba7b9be", + "id": "01e1c95c", "metadata": { "editable": true }, @@ -2696,7 +2709,7 @@ }, { "cell_type": "markdown", - "id": "50da33c0", + "id": "2a58e959", "metadata": { "editable": true }, @@ -2707,7 +2720,7 @@ { "cell_type": "code", "execution_count": 12, - "id": "31bd6a24", + "id": "d0323605", "metadata": { "collapsed": false, "editable": true @@ -2732,7 +2745,7 @@ }, { "cell_type": "markdown", - "id": "0deb8111", + "id": "9c9cd9f4", "metadata": { "editable": true }, @@ -2748,7 +2761,7 @@ }, { "cell_type": "markdown", - "id": "16d54f02", + "id": "0b0a1725", "metadata": { "editable": true }, @@ -2769,7 +2782,7 @@ }, { "cell_type": "markdown", - "id": "b300d06b", + "id": "668ea878", "metadata": { "editable": true }, @@ -2789,7 +2802,7 @@ }, { "cell_type": "markdown", - "id": "6bc7778d", + "id": "84debf59", "metadata": { "editable": true }, @@ -2808,7 +2821,7 @@ { "cell_type": "code", "execution_count": 13, - "id": "a60fe5bd", + "id": "656bc90a", "metadata": { "collapsed": false, "editable": true @@ -2843,7 +2856,7 @@ }, { "cell_type": "markdown", - "id": "2192721f", + "id": "7530e62f", "metadata": { "editable": true }, @@ -2856,7 +2869,7 @@ { "cell_type": "code", "execution_count": 14, - "id": "e404f2c5", + "id": "af0103e5", "metadata": { "collapsed": false, "editable": true @@ -2933,7 +2946,7 @@ }, { "cell_type": "markdown", - "id": "fffdbb91", + "id": "5f5ace3c", "metadata": { "editable": true }, @@ -2948,7 +2961,7 @@ }, { "cell_type": "markdown", - "id": "8cce7a0e", + "id": "7aff62fe", "metadata": { "editable": true }, @@ -2963,7 +2976,7 @@ }, { "cell_type": "markdown", - "id": "3154c365", + "id": "dab0d9f2", "metadata": { "editable": true }, @@ -2975,7 +2988,7 @@ }, { "cell_type": "markdown", - "id": "a2a9ceca", + "id": "63d35ba1", "metadata": { "editable": true }, @@ -2993,7 +3006,7 @@ }, { "cell_type": "markdown", - "id": "3374c700", + "id": "c596f29b", "metadata": { "editable": true }, @@ -3012,7 +3025,7 @@ }, { "cell_type": "markdown", - "id": "893d86fe", + "id": "a3fafffc", "metadata": { "editable": true }, @@ -3024,7 +3037,7 @@ }, { "cell_type": "markdown", - "id": "ca2449e8", + "id": "09a1d555", "metadata": { "editable": true }, @@ -3034,7 +3047,7 @@ }, { "cell_type": "markdown", - "id": "3cbd4adb", + "id": "b41ec406", "metadata": { "editable": true }, @@ -3050,7 +3063,7 @@ }, { "cell_type": "markdown", - "id": "e3f07cbc", + "id": "05c1b2c1", "metadata": { "editable": true }, @@ -3062,7 +3075,7 @@ }, { "cell_type": "markdown", - "id": "99f2ac0f", + "id": "6cf8dccf", "metadata": { "editable": true }, @@ -3072,7 +3085,7 @@ }, { "cell_type": "markdown", - "id": "83336244", + "id": "25fc7e7d", "metadata": { "editable": true }, @@ -3084,7 +3097,7 @@ }, { "cell_type": "markdown", - "id": "efd3e708", + "id": "94d1ac1d", "metadata": { "editable": true }, @@ -3094,7 +3107,7 @@ }, { "cell_type": "markdown", - "id": "6c24d65c", + "id": "08c8a098", "metadata": { "editable": true }, @@ -3106,7 +3119,7 @@ }, { "cell_type": "markdown", - "id": "853d885b", + "id": "0a8aca26", "metadata": { "editable": true }, @@ -3122,7 +3135,7 @@ }, { "cell_type": "markdown", - "id": "5ab54645", + "id": "4c1d7175", "metadata": { "editable": true }, @@ -3134,7 +3147,7 @@ }, { "cell_type": "markdown", - "id": "90f1503b", + "id": "58b8859e", "metadata": { "editable": true }, @@ -3167,7 +3180,7 @@ }, { "cell_type": "markdown", - "id": "d496d988", + "id": "c0baa634", "metadata": { "editable": true }, @@ -3179,7 +3192,7 @@ }, { "cell_type": "markdown", - "id": "258ca1e6", + "id": "7e095cc7", "metadata": { "editable": true }, @@ -3197,7 +3210,7 @@ }, { "cell_type": "markdown", - "id": "84278058", + "id": "e6574650", "metadata": { "editable": true }, @@ -3207,7 +3220,7 @@ }, { "cell_type": "markdown", - "id": "feaee2f4", + "id": "5838caca", "metadata": { "editable": true }, @@ -3238,7 +3251,7 @@ }, { "cell_type": "markdown", - "id": "218edcf1", + "id": "1696cb1c", "metadata": { "editable": true }, @@ -3253,7 +3266,7 @@ }, { "cell_type": "markdown", - "id": "18dbc91c", + "id": "41db75b1", "metadata": { "editable": true }, @@ -3271,7 +3284,7 @@ }, { "cell_type": "markdown", - "id": "0bfcf74a", + "id": "5d71f114", "metadata": { "editable": true }, @@ -3283,7 +3296,7 @@ }, { "cell_type": "markdown", - "id": "5fedd6f0", + "id": "adf7201e", "metadata": { "editable": true }, @@ -3295,7 +3308,7 @@ }, { "cell_type": "markdown", - "id": "b210b2a4", + "id": "266384e4", "metadata": { "editable": true }, @@ -3313,7 +3326,7 @@ }, { "cell_type": "markdown", - "id": "34ffacbb", + "id": "9f0d4429", "metadata": { "editable": true }, @@ -3342,7 +3355,7 @@ }, { "cell_type": "markdown", - "id": "cd03375d", + "id": "eac0bb77", "metadata": { "editable": true }, @@ -3360,7 +3373,7 @@ }, { "cell_type": "markdown", - "id": "0db5d6e0", + "id": "ffa6229f", "metadata": { "editable": true }, @@ -3372,7 +3385,7 @@ }, { "cell_type": "markdown", - "id": "84c709d9", + "id": "35df09ae", "metadata": { "editable": true }, @@ -3384,7 +3397,7 @@ }, { "cell_type": "markdown", - "id": "e4e47496", + "id": "c2b911fd", "metadata": { "editable": true }, @@ -3396,7 +3409,7 @@ }, { "cell_type": "markdown", - "id": "164f27df", + "id": "761d3704", "metadata": { "editable": true }, @@ -3408,7 +3421,7 @@ }, { "cell_type": "markdown", - "id": "591f4833", + "id": "49a8bed5", "metadata": { "editable": true }, @@ -3420,7 +3433,7 @@ }, { "cell_type": "markdown", - "id": "e2127e8a", + "id": "445813d3", "metadata": { "editable": true }, @@ -3437,7 +3450,7 @@ }, { "cell_type": "markdown", - "id": "5cef8b84", + "id": "0419e1d5", "metadata": { "editable": true }, @@ -3456,7 +3469,7 @@ }, { "cell_type": "markdown", - "id": "505c8905", + "id": "3e3d0a9f", "metadata": { "editable": true }, @@ -3468,7 +3481,7 @@ }, { "cell_type": "markdown", - "id": "ca96ddec", + "id": "e8caeab7", "metadata": { "editable": true }, @@ -3482,7 +3495,7 @@ }, { "cell_type": "markdown", - "id": "fa011176", + "id": "51df4797", "metadata": { "editable": true }, @@ -3502,7 +3515,7 @@ }, { "cell_type": "markdown", - "id": "b91c4543", + "id": "ffbf4496", "metadata": { "editable": true }, @@ -3540,7 +3553,7 @@ }, { "cell_type": "markdown", - "id": "f13065e5", + "id": "2d87427e", "metadata": { "editable": true }, @@ -3552,7 +3565,7 @@ }, { "cell_type": "markdown", - "id": "22937f5e", + "id": "f5424c8f", "metadata": { "editable": true }, @@ -3562,7 +3575,7 @@ }, { "cell_type": "markdown", - "id": "e1459fe1", + "id": "47f14695", "metadata": { "editable": true }, @@ -3574,7 +3587,7 @@ }, { "cell_type": "markdown", - "id": "441a109a", + "id": "e2c6cff7", "metadata": { "editable": true }, @@ -3585,7 +3598,7 @@ { "cell_type": "code", "execution_count": 15, - "id": "9043abae", + "id": "28e44462", "metadata": { "collapsed": false, "editable": true @@ -3630,7 +3643,7 @@ }, { "cell_type": "markdown", - "id": "787d5d78", + "id": "09458bca", "metadata": { "editable": true }, @@ -3647,7 +3660,7 @@ { "cell_type": "code", "execution_count": 16, - "id": "6a677479", + "id": "ef273753", "metadata": { "collapsed": false, "editable": true @@ -3675,7 +3688,7 @@ }, { "cell_type": "markdown", - "id": "f94a1d32", + "id": "ae06dfcf", "metadata": { "editable": true }, @@ -3690,7 +3703,7 @@ { "cell_type": "code", "execution_count": 17, - "id": "c0eb89fd", + "id": "7d050e00", "metadata": { "collapsed": false, "editable": true @@ -3734,7 +3747,7 @@ }, { "cell_type": "markdown", - "id": "05d7497d", + "id": "7d19a1a4", "metadata": { "editable": true }, @@ -3744,7 +3757,7 @@ }, { "cell_type": "markdown", - "id": "24e3ca02", + "id": "e22e77ea", "metadata": { "editable": true }, @@ -3755,7 +3768,7 @@ { "cell_type": "code", "execution_count": 18, - "id": "a616e696", + "id": "340b92f2", "metadata": { "collapsed": false, "editable": true @@ -3783,7 +3796,7 @@ }, { "cell_type": "markdown", - "id": "f695da56", + "id": "9114dbaa", "metadata": { "editable": true }, @@ -3798,7 +3811,7 @@ }, { "cell_type": "markdown", - "id": "5ac073ee", + "id": "8bfebd09", "metadata": { "editable": true }, @@ -3809,7 +3822,7 @@ { "cell_type": "code", "execution_count": 19, - "id": "efc8906e", + "id": "e0ac03cb", "metadata": { "collapsed": false, "editable": true @@ -3837,7 +3850,7 @@ }, { "cell_type": "markdown", - "id": "7c587e8f", + "id": "3eb22ded", "metadata": { "editable": true }, @@ -3848,7 +3861,7 @@ { "cell_type": "code", "execution_count": 20, - "id": "6428be1f", + "id": "22977dd4", "metadata": { "collapsed": false, "editable": true @@ -3873,7 +3886,7 @@ }, { "cell_type": "markdown", - "id": "8e1de777", + "id": "21eb7d2b", "metadata": { "editable": true }, @@ -3884,7 +3897,7 @@ { "cell_type": "code", "execution_count": 21, - "id": "770ff6aa", + "id": "52e57a88", "metadata": { "collapsed": false, "editable": true @@ -3920,7 +3933,7 @@ { "cell_type": "code", "execution_count": 22, - "id": "b924cc5d", + "id": "cbd8dcdc", "metadata": { "collapsed": false, "editable": true @@ -3940,7 +3953,7 @@ }, { "cell_type": "markdown", - "id": "f0b0e1e9", + "id": "039e394f", "metadata": { "editable": true }, @@ -3951,7 +3964,7 @@ { "cell_type": "code", "execution_count": 23, - "id": "1585ab28", + "id": "ee0186c2", "metadata": { "collapsed": false, "editable": true @@ -3989,7 +4002,7 @@ }, { "cell_type": "markdown", - "id": "2718df1a", + "id": "73478ba8", "metadata": { "editable": true }, @@ -3999,7 +4012,7 @@ }, { "cell_type": "markdown", - "id": "d8fa5235", + "id": "d9b9f60a", "metadata": { "editable": true }, @@ -4013,7 +4026,7 @@ { "cell_type": "code", "execution_count": 24, - "id": "196a52d6", + "id": "64f20453", "metadata": { "collapsed": false, "editable": true @@ -4035,7 +4048,7 @@ }, { "cell_type": "markdown", - "id": "9127a2c5", + "id": "dfa62def", "metadata": { "editable": true }, @@ -4045,7 +4058,7 @@ }, { "cell_type": "markdown", - "id": "2b12ed61", + "id": "9b5930f8", "metadata": { "editable": true }, @@ -4056,7 +4069,7 @@ { "cell_type": "code", "execution_count": 25, - "id": "8ced55c8", + "id": "5a6e27d7", "metadata": { "collapsed": false, "editable": true @@ -4078,7 +4091,7 @@ }, { "cell_type": "markdown", - "id": "92ebdc2b", + "id": "2c15d379", "metadata": { "editable": true }, @@ -4091,7 +4104,7 @@ { "cell_type": "code", "execution_count": 26, - "id": "276f763e", + "id": "4e8bbe52", "metadata": { "collapsed": false, "editable": true @@ -4116,7 +4129,7 @@ }, { "cell_type": "markdown", - "id": "7841ad0b", + "id": "4815cbb5", "metadata": { "editable": true }, @@ -4128,7 +4141,7 @@ { "cell_type": "code", "execution_count": 27, - "id": "10107989", + "id": "48a68d92", "metadata": { "collapsed": false, "editable": true @@ -4143,7 +4156,7 @@ }, { "cell_type": "markdown", - "id": "4c1139c0", + "id": "6b9621b8", "metadata": { "editable": true }, @@ -4158,7 +4171,7 @@ { "cell_type": "code", "execution_count": 28, - "id": "3022af88", + "id": "7031c5bf", "metadata": { "collapsed": false, "editable": true @@ -4218,7 +4231,7 @@ }, { "cell_type": "markdown", - "id": "04d09021", + "id": "151dbb95", "metadata": { "editable": true }, @@ -4229,7 +4242,7 @@ { "cell_type": "code", "execution_count": 29, - "id": "71bf4b6d", + "id": "55da74a7", "metadata": { "collapsed": false, "editable": true @@ -4293,7 +4306,7 @@ }, { "cell_type": "markdown", - "id": "ad417bba", + "id": "6a59ec6f", "metadata": { "editable": true }, @@ -4304,7 +4317,7 @@ { "cell_type": "code", "execution_count": 30, - "id": "c394bcef", + "id": "ecd93c24", "metadata": { "collapsed": false, "editable": true @@ -4353,7 +4366,7 @@ }, { "cell_type": "markdown", - "id": "2e4cf4b5", + "id": "5c094787", "metadata": { "editable": true }, @@ -4365,7 +4378,7 @@ { "cell_type": "code", "execution_count": 31, - "id": "47411bcf", + "id": "8209bb0a", "metadata": { "collapsed": false, "editable": true @@ -4449,7 +4462,7 @@ }, { "cell_type": "markdown", - "id": "f8e30af2", + "id": "c8761b25", "metadata": { "editable": true }, @@ -4460,7 +4473,7 @@ { "cell_type": "code", "execution_count": 32, - "id": "dd594924", + "id": "88782543", "metadata": { "collapsed": false, "editable": true @@ -4538,7 +4551,7 @@ }, { "cell_type": "markdown", - "id": "75c4c29d", + "id": "2c44e1d1", "metadata": { "editable": true }, @@ -4549,7 +4562,7 @@ { "cell_type": "code", "execution_count": 33, - "id": "4dd14fc5", + "id": "4f6a2867", "metadata": { "collapsed": false, "editable": true @@ -4608,7 +4621,7 @@ }, { "cell_type": "markdown", - "id": "be4ce0cd", + "id": "639027f1", "metadata": { "editable": true }, @@ -4618,7 +4631,7 @@ }, { "cell_type": "markdown", - "id": "0b739495", + "id": "2c2e20f4", "metadata": { "editable": true }, @@ -4629,7 +4642,7 @@ { "cell_type": "code", "execution_count": 34, - "id": "ae87789c", + "id": "21f48a1f", "metadata": { "collapsed": false, "editable": true @@ -4694,7 +4707,7 @@ }, { "cell_type": "markdown", - "id": "76c8872b", + "id": "bc8cc44b", "metadata": { "editable": true }, @@ -4705,7 +4718,7 @@ { "cell_type": "code", "execution_count": 35, - "id": "e99dbaa4", + "id": "84b02c51", "metadata": { "collapsed": false, "editable": true @@ -4775,7 +4788,7 @@ }, { "cell_type": "markdown", - "id": "596df570", + "id": "0bb635d0", "metadata": { "editable": true }, @@ -4786,7 +4799,7 @@ { "cell_type": "code", "execution_count": 36, - "id": "6693f042", + "id": "af9f10b1", "metadata": { "collapsed": false, "editable": true @@ -4830,7 +4843,7 @@ }, { "cell_type": "markdown", - "id": "a40ed853", + "id": "f8a20f65", "metadata": { "editable": true }, @@ -4849,7 +4862,7 @@ { "cell_type": "code", "execution_count": 37, - "id": "02f88360", + "id": "f02783dc", "metadata": { "collapsed": false, "editable": true diff --git a/doc/src/week39/week39.do.txt b/doc/src/week39/week39.do.txt index 57722437e..008a97359 100644 --- a/doc/src/week39/week39.do.txt +++ b/doc/src/week39/week39.do.txt @@ -338,7 +338,16 @@ regular polygons (triangles, rectangles, pentagons, etc...). !split ===== Convex function ===== -_Convex function_: Let $X \subset \mathbb{R}^n$ be a convex set. Assume that the function $f: X \rightarrow \mathbb{R}$ is continuous, then $f$ is said to be convex if $$f(tx_1 + (1-t)x_2) \leq tf(x_1) + (1-t)f(x_2) $$ for all $x_1, x_2 \in X$ and for all $t \in [0,1]$. If $\leq$ is replaced with a strict inequaltiy in the definition, we demand $x_1 \neq x_2$ and $t\in(0,1)$ then $f$ is said to be strictly convex. For a single variable function, convexity means that if you draw a straight line connecting $f(x_1)$ and $f(x_2)$, the value of the function on the interval $[x_1,x_2]$ is always below the line as illustrated below. +_Convex function_: Let $X \subset \mathbb{R}^n$ be a convex +set. Assume that the function $f: X \rightarrow \mathbb{R}$ is +continuous, then $f$ is said to be convex if $f(tx_1 + (1-t)x_2) \leq tf(x_1) + (1-t)f(x_2)$ +for all $x_1, x_2 \in X$ and for all $t \in [0,1]$. +If $\leq$ is replaced with a strict inequaltiy in the +definition, we demand $x_1 \neq x_2$ and $t\in(0,1)$ then $f$ is said +to be strictly convex. For a single variable function, convexity means +that if you draw a straight line connecting $f(x_1)$ and $f(x_2)$, the +value of the function on the interval $[x_1,x_2]$ is always below the +line as illustrated below. !split ===== Conditions on convex functions ===== @@ -346,13 +355,15 @@ _Convex function_: Let $X \subset \mathbb{R}^n$ be a convex set. Assume that the In the following we state first and second-order conditions which ensures convexity of a function $f$. We write $D_f$ to denote the domain of $f$, i.e the subset of $R^n$ where $f$ is defined. For more -details and proofs we refer to: "S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press":"http://stanford.edu/boyd/cvxbook/, 2004". +details and proofs we refer to: "S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press":"http://stanford.edu/boyd/cvxbook/". !bblock First order condition Suppose $f$ is differentiable (i.e $\nabla f(x)$ is well defined for all $x$ in the domain of $f$). Then $f$ is convex if and only if $D_f$ -is a convex set and $$f(y) \geq f(x) + \nabla f(x)^T (y-x) $$ holds -for all $x,y \in D_f$. This condition means that for a convex function +is a convex set and $f(y) \geq f(x) + \nabla f(x)^T (y-x)$ holds +for all $x,y \in D_f$. + +This condition means that for a convex function the first order Taylor expansion (right hand side above) at any point a global under estimator of the function. To convince yourself you can make a drawing of $f(x) = x^2+1$ and draw the tangent line to $f(x)$ and