From 6452c70a7daebbe26501bfb6fa782add2100678f Mon Sep 17 00:00:00 2001
From: burak0206 <burak0206@gmail.com>
Date: Sat, 25 Feb 2017 22:47:01 +0300
Subject: [PATCH 01/17] hw2 complete

---
 .ipynb_checkpoints/hw2-checkpoint.ipynb | 179 ++++++++++++++++++++++++
 hw2.ipynb                               | 179 ++++++++++++++++++++++++
 2 files changed, 358 insertions(+)
 create mode 100644 .ipynb_checkpoints/hw2-checkpoint.ipynb
 create mode 100644 hw2.ipynb

diff --git a/.ipynb_checkpoints/hw2-checkpoint.ipynb b/.ipynb_checkpoints/hw2-checkpoint.ipynb
new file mode 100644
index 0000000..c1edcc6
--- /dev/null
+++ b/.ipynb_checkpoints/hw2-checkpoint.ipynb
@@ -0,0 +1,179 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Homework 2\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Exercise 3.3"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "#### (a) $\\|x\\|_\\infty \\leq \\|x\\|_{2}$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$\\|x\\|_\\infty = ((\\max\\limits_{i} |x_{i}|)^2)^{1/2}\\leq (\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|)^{1/2} = \\|x\\|_{2}$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$ x_{m*1} =  [2,1,\\dots,0]^T \\hspace{5mm}, then \\hspace{3mm} \\|x\\|_\\infty = 2 \\hspace{3mm} and \\hspace{3mm} \\|x\\|_{2} = \\sqrt{5}$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "#### (b) $\\|x\\|_{2} \\leq \\sqrt{m}\\|x\\|_\\infty$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$\\|x\\|_{2} = (\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|)^{1/2}\\leq (m.(\\max\\limits_{i} |x_{i}|)^2)^{1/2} = \\sqrt{5}.\\|x\\|_\\infty$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$ x_{m*1} =  [1,1,1,1,0]^T \\hspace{5mm}, then \\hspace{3mm} m = 5 \\hspace{3mm} \\|x\\|_\\infty = 1 \\hspace{3mm} and \\hspace{3mm} \\|x\\|_{2} = \\sqrt{4}=2$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "#### (c) $\\|A\\|_\\infty \\leq \\sqrt{n}\\|A\\|_{2}$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$ \\|A\\|_{\\infty}= \\sup\\limits_{x \\neq 0}\\frac{ \\|Ax\\|_{\\infty}}  {\\|x\\|_{\\infty}} \\leq \\sup\\limits_{x \\neq 0}\\frac{ \\|Ax\\|_{2}}  {\\|x\\|_{2}} \\leq \\sup\\limits_{x \\neq 0}\\frac{ \\|Ax\\|_{2}}  {\\|x\\|_{2}/ \\sqrt{n}} = \\sqrt{n}\\|A\\|_{2}$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$ A_{mxn} = \\begin{bmatrix}\n",
+    "   1 & 1 & 1 & 1 \\\\\\\\\n",
+    "   0 & 0 & 0 & 0 \\\\\\\\\n",
+    "   0 & 0 & 0 & 0\n",
+    " \\end{bmatrix}_{mxn}  A ^T = \\begin{bmatrix}\n",
+    "   1 & 0 & 0  \\\\\\\\\n",
+    "   1 & 0 & 0  \\\\\\\\\n",
+    "   1 & 0 & 0 \\\\\\\\\n",
+    "   1 & 0 & 0 \\\\\\\\\n",
+    " \\end{bmatrix}_{nxm}  A ^T A = \\begin{bmatrix}\n",
+    "    1 & 1 & 1 & 1  \\\\\\\\\n",
+    "    1 & 1 & 1 & 1  \\\\\\\\\n",
+    "    1 & 1 & 1 & 1  \\\\\\\\\n",
+    "    1 & 1 & 1 & 1  \n",
+    " \\end{bmatrix}_{nxn}$\n",
+    " "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$\\|A\\|_\\infty = 4 = n$\n",
+    "\n",
+    "$\\|A\\|_{2} =\\sqrt{tr(A^TA)}= \\sqrt{4} = \\sqrt{n} = 2$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "#### (d) $\\|A\\|_{2} \\leq \\sqrt{m}\\|A\\|_\\infty$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$ \\|A\\|_{{2}}= \\sup\\limits_{x \\neq 0}\\frac{ \\|Ax\\|_{{2}}}  {\\|x\\|_{{2}}} \\leq \\sup\\limits_{x \\neq 0}\\frac{ \\|Ax\\|\\infty}  {\\|x\\|\\infty} \\leq \\sup\\limits_{x \\neq 0}\\frac{\\sqrt{m} \\|Ax\\|\\infty}  {\\|x\\|\\infty} = \\sqrt{m}\\|A\\|\\infty $"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$ A_{mxn} = \\begin{bmatrix}\n",
+    "   1 & 0 & 0  \\\\\\\\\n",
+    "   1 & 0 & 0 \\\\\\\\\n",
+    "   1 & 0 & 0 \\\\\\\\\n",
+    "   1 & 0 & 0 \n",
+    " \\end{bmatrix}_{mxn}  A ^T = \\begin{bmatrix}\n",
+    "   1 & 1 & 1 & 1   \\\\\\\\\n",
+    "   0 & 0 & 0 & 0   \\\\\\\\\n",
+    "   0 & 0 & 0 & 0\n",
+    " \\end{bmatrix}_{nxm}  A ^T A = \\begin{bmatrix}\n",
+    "    4 & 0 & 0  \\\\\\\\\n",
+    "    0 & 0 & 0   \\\\\\\\\n",
+    "    0 & 0 & 0  \n",
+    " \\end{bmatrix}_{nxn}$\n",
+    " "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$\\|A\\|_\\infty = 1$\n",
+    "\n",
+    "$\\|A\\|_{2} =\\sqrt{tr(A^TA)}= \\sqrt{4}  = 2$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "collapsed": true
+   },
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 2",
+   "language": "python",
+   "name": "python2"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 2
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython2",
+   "version": "2.7.13"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}
diff --git a/hw2.ipynb b/hw2.ipynb
new file mode 100644
index 0000000..c1edcc6
--- /dev/null
+++ b/hw2.ipynb
@@ -0,0 +1,179 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Homework 2\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Exercise 3.3"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "#### (a) $\\|x\\|_\\infty \\leq \\|x\\|_{2}$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$\\|x\\|_\\infty = ((\\max\\limits_{i} |x_{i}|)^2)^{1/2}\\leq (\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|)^{1/2} = \\|x\\|_{2}$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$ x_{m*1} =  [2,1,\\dots,0]^T \\hspace{5mm}, then \\hspace{3mm} \\|x\\|_\\infty = 2 \\hspace{3mm} and \\hspace{3mm} \\|x\\|_{2} = \\sqrt{5}$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "#### (b) $\\|x\\|_{2} \\leq \\sqrt{m}\\|x\\|_\\infty$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$\\|x\\|_{2} = (\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|)^{1/2}\\leq (m.(\\max\\limits_{i} |x_{i}|)^2)^{1/2} = \\sqrt{5}.\\|x\\|_\\infty$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$ x_{m*1} =  [1,1,1,1,0]^T \\hspace{5mm}, then \\hspace{3mm} m = 5 \\hspace{3mm} \\|x\\|_\\infty = 1 \\hspace{3mm} and \\hspace{3mm} \\|x\\|_{2} = \\sqrt{4}=2$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "#### (c) $\\|A\\|_\\infty \\leq \\sqrt{n}\\|A\\|_{2}$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$ \\|A\\|_{\\infty}= \\sup\\limits_{x \\neq 0}\\frac{ \\|Ax\\|_{\\infty}}  {\\|x\\|_{\\infty}} \\leq \\sup\\limits_{x \\neq 0}\\frac{ \\|Ax\\|_{2}}  {\\|x\\|_{2}} \\leq \\sup\\limits_{x \\neq 0}\\frac{ \\|Ax\\|_{2}}  {\\|x\\|_{2}/ \\sqrt{n}} = \\sqrt{n}\\|A\\|_{2}$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$ A_{mxn} = \\begin{bmatrix}\n",
+    "   1 & 1 & 1 & 1 \\\\\\\\\n",
+    "   0 & 0 & 0 & 0 \\\\\\\\\n",
+    "   0 & 0 & 0 & 0\n",
+    " \\end{bmatrix}_{mxn}  A ^T = \\begin{bmatrix}\n",
+    "   1 & 0 & 0  \\\\\\\\\n",
+    "   1 & 0 & 0  \\\\\\\\\n",
+    "   1 & 0 & 0 \\\\\\\\\n",
+    "   1 & 0 & 0 \\\\\\\\\n",
+    " \\end{bmatrix}_{nxm}  A ^T A = \\begin{bmatrix}\n",
+    "    1 & 1 & 1 & 1  \\\\\\\\\n",
+    "    1 & 1 & 1 & 1  \\\\\\\\\n",
+    "    1 & 1 & 1 & 1  \\\\\\\\\n",
+    "    1 & 1 & 1 & 1  \n",
+    " \\end{bmatrix}_{nxn}$\n",
+    " "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$\\|A\\|_\\infty = 4 = n$\n",
+    "\n",
+    "$\\|A\\|_{2} =\\sqrt{tr(A^TA)}= \\sqrt{4} = \\sqrt{n} = 2$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "#### (d) $\\|A\\|_{2} \\leq \\sqrt{m}\\|A\\|_\\infty$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$ \\|A\\|_{{2}}= \\sup\\limits_{x \\neq 0}\\frac{ \\|Ax\\|_{{2}}}  {\\|x\\|_{{2}}} \\leq \\sup\\limits_{x \\neq 0}\\frac{ \\|Ax\\|\\infty}  {\\|x\\|\\infty} \\leq \\sup\\limits_{x \\neq 0}\\frac{\\sqrt{m} \\|Ax\\|\\infty}  {\\|x\\|\\infty} = \\sqrt{m}\\|A\\|\\infty $"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$ A_{mxn} = \\begin{bmatrix}\n",
+    "   1 & 0 & 0  \\\\\\\\\n",
+    "   1 & 0 & 0 \\\\\\\\\n",
+    "   1 & 0 & 0 \\\\\\\\\n",
+    "   1 & 0 & 0 \n",
+    " \\end{bmatrix}_{mxn}  A ^T = \\begin{bmatrix}\n",
+    "   1 & 1 & 1 & 1   \\\\\\\\\n",
+    "   0 & 0 & 0 & 0   \\\\\\\\\n",
+    "   0 & 0 & 0 & 0\n",
+    " \\end{bmatrix}_{nxm}  A ^T A = \\begin{bmatrix}\n",
+    "    4 & 0 & 0  \\\\\\\\\n",
+    "    0 & 0 & 0   \\\\\\\\\n",
+    "    0 & 0 & 0  \n",
+    " \\end{bmatrix}_{nxn}$\n",
+    " "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$\\|A\\|_\\infty = 1$\n",
+    "\n",
+    "$\\|A\\|_{2} =\\sqrt{tr(A^TA)}= \\sqrt{4}  = 2$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "collapsed": true
+   },
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 2",
+   "language": "python",
+   "name": "python2"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 2
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython2",
+   "version": "2.7.13"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}

From 3004c673fe0e1ac04ed5ce981f55d149bbb4bfd5 Mon Sep 17 00:00:00 2001
From: burak0206 <burak0206@gmail.com>
Date: Sat, 25 Feb 2017 22:53:53 +0300
Subject: [PATCH 02/17] fixed

---
 hw2.ipynb | 7 +++++--
 1 file changed, 5 insertions(+), 2 deletions(-)

diff --git a/hw2.ipynb b/hw2.ipynb
index c1edcc6..8575b59 100644
--- a/hw2.ipynb
+++ b/hw2.ipynb
@@ -25,7 +25,9 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "$\\|x\\|_\\infty = ((\\max\\limits_{i} |x_{i}|)^2)^{1/2}\\leq (\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|)^{1/2} = \\|x\\|_{2}$"
+    "$\\|x\\|_\\infty = ((\\max\\limits_{i} |x_{i}|)^2)^{1/2}\\leq\n",
+    "(\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|)^{1/2} \n",
+    "= \\|x\\|_{2}$"
    ]
   },
   {
@@ -46,7 +48,8 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "$\\|x\\|_{2} = (\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|)^{1/2}\\leq (m.(\\max\\limits_{i} |x_{i}|)^2)^{1/2} = \\sqrt{5}.\\|x\\|_\\infty$"
+    "$\\|x\\|_{2} = (\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|)^{1/2}\\leq \n",
+    "(m.(\\max\\limits_{i} |x_{i}|)^2)^{1/2} = \\sqrt{5}.\\|x\\|_\\infty$"
    ]
   },
   {

From 7c863500aee41b1862a2ea9a5af1a117d7750ab6 Mon Sep 17 00:00:00 2001
From: burak0206 <burak0206@gmail.com>
Date: Sat, 25 Feb 2017 22:56:28 +0300
Subject: [PATCH 03/17] done

---
 .ipynb_checkpoints/hw2-checkpoint.ipynb |  8 ++++++--
 hw2.ipynb                               | 11 ++++++-----
 2 files changed, 12 insertions(+), 7 deletions(-)

diff --git a/.ipynb_checkpoints/hw2-checkpoint.ipynb b/.ipynb_checkpoints/hw2-checkpoint.ipynb
index c1edcc6..6e9c599 100644
--- a/.ipynb_checkpoints/hw2-checkpoint.ipynb
+++ b/.ipynb_checkpoints/hw2-checkpoint.ipynb
@@ -25,7 +25,9 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "$\\|x\\|_\\infty = ((\\max\\limits_{i} |x_{i}|)^2)^{1/2}\\leq (\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|)^{1/2} = \\|x\\|_{2}$"
+    "$\\|x\\|_\\infty = ((\\max\\limits_{i} |x_{i}|)^2)^{1/2}$\n",
+    "\n",
+    "$\\leq (\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|)^{1/2} = \\|x\\|_{2}$"
    ]
   },
   {
@@ -46,7 +48,9 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "$\\|x\\|_{2} = (\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|)^{1/2}\\leq (m.(\\max\\limits_{i} |x_{i}|)^2)^{1/2} = \\sqrt{5}.\\|x\\|_\\infty$"
+    "$\\|x\\|_{2} = (\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|)^{1/2}$\n",
+    "\n",
+    "$\\leq (m.(\\max\\limits_{i} |x_{i}|)^2)^{1/2} = \\sqrt{5}.\\|x\\|_\\infty$"
    ]
   },
   {
diff --git a/hw2.ipynb b/hw2.ipynb
index 8575b59..6e9c599 100644
--- a/hw2.ipynb
+++ b/hw2.ipynb
@@ -25,9 +25,9 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "$\\|x\\|_\\infty = ((\\max\\limits_{i} |x_{i}|)^2)^{1/2}\\leq\n",
-    "(\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|)^{1/2} \n",
-    "= \\|x\\|_{2}$"
+    "$\\|x\\|_\\infty = ((\\max\\limits_{i} |x_{i}|)^2)^{1/2}$\n",
+    "\n",
+    "$\\leq (\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|)^{1/2} = \\|x\\|_{2}$"
    ]
   },
   {
@@ -48,8 +48,9 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "$\\|x\\|_{2} = (\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|)^{1/2}\\leq \n",
-    "(m.(\\max\\limits_{i} |x_{i}|)^2)^{1/2} = \\sqrt{5}.\\|x\\|_\\infty$"
+    "$\\|x\\|_{2} = (\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|)^{1/2}$\n",
+    "\n",
+    "$\\leq (m.(\\max\\limits_{i} |x_{i}|)^2)^{1/2} = \\sqrt{5}.\\|x\\|_\\infty$"
    ]
   },
   {

From 7b473df0d70b1f9659dca426bf17264c36291d48 Mon Sep 17 00:00:00 2001
From: burak0206 <burak0206@gmail.com>
Date: Sat, 25 Feb 2017 22:58:28 +0300
Subject: [PATCH 04/17] errors are fixed

---
 .ipynb_checkpoints/hw2-checkpoint.ipynb | 18 ++++++++++++++----
 hw2.ipynb                               | 18 ++++++++++++++----
 2 files changed, 28 insertions(+), 8 deletions(-)

diff --git a/.ipynb_checkpoints/hw2-checkpoint.ipynb b/.ipynb_checkpoints/hw2-checkpoint.ipynb
index 6e9c599..10d814c 100644
--- a/.ipynb_checkpoints/hw2-checkpoint.ipynb
+++ b/.ipynb_checkpoints/hw2-checkpoint.ipynb
@@ -25,8 +25,13 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "$\\|x\\|_\\infty = ((\\max\\limits_{i} |x_{i}|)^2)^{1/2}$\n",
-    "\n",
+    "$\\|x\\|_\\infty = ((\\max\\limits_{i} |x_{i}|)^2)^{1/2}$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
     "$\\leq (\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|)^{1/2} = \\|x\\|_{2}$"
    ]
   },
@@ -48,8 +53,13 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "$\\|x\\|_{2} = (\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|)^{1/2}$\n",
-    "\n",
+    "$\\|x\\|_{2} = (\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|)^{1/2}$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
     "$\\leq (m.(\\max\\limits_{i} |x_{i}|)^2)^{1/2} = \\sqrt{5}.\\|x\\|_\\infty$"
    ]
   },
diff --git a/hw2.ipynb b/hw2.ipynb
index 6e9c599..10d814c 100644
--- a/hw2.ipynb
+++ b/hw2.ipynb
@@ -25,8 +25,13 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "$\\|x\\|_\\infty = ((\\max\\limits_{i} |x_{i}|)^2)^{1/2}$\n",
-    "\n",
+    "$\\|x\\|_\\infty = ((\\max\\limits_{i} |x_{i}|)^2)^{1/2}$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
     "$\\leq (\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|)^{1/2} = \\|x\\|_{2}$"
    ]
   },
@@ -48,8 +53,13 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "$\\|x\\|_{2} = (\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|)^{1/2}$\n",
-    "\n",
+    "$\\|x\\|_{2} = (\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|)^{1/2}$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
     "$\\leq (m.(\\max\\limits_{i} |x_{i}|)^2)^{1/2} = \\sqrt{5}.\\|x\\|_\\infty$"
    ]
   },

From 18c45ebdabfdb8dd6c3e4cb787b127f771f60f66 Mon Sep 17 00:00:00 2001
From: burak0206 <burak0206@gmail.com>
Date: Sat, 25 Feb 2017 23:00:23 +0300
Subject: [PATCH 05/17] fixing go on

---
 .ipynb_checkpoints/hw2-checkpoint.ipynb | 4 ++--
 hw2.ipynb                               | 4 ++--
 2 files changed, 4 insertions(+), 4 deletions(-)

diff --git a/.ipynb_checkpoints/hw2-checkpoint.ipynb b/.ipynb_checkpoints/hw2-checkpoint.ipynb
index 10d814c..49ffe32 100644
--- a/.ipynb_checkpoints/hw2-checkpoint.ipynb
+++ b/.ipynb_checkpoints/hw2-checkpoint.ipynb
@@ -32,7 +32,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "$\\leq (\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|)^{1/2} = \\|x\\|_{2}$"
+    "$ \\leq (\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|)^{1/2} = \\|x\\|_{2} $"
    ]
   },
   {
@@ -60,7 +60,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "$\\leq (m.(\\max\\limits_{i} |x_{i}|)^2)^{1/2} = \\sqrt{5}.\\|x\\|_\\infty$"
+    "$ \\leq (m.(\\max\\limits_{i} |x_{i}|)^2)^{1/2} = \\sqrt{5}.\\|x\\|_\\infty $"
    ]
   },
   {
diff --git a/hw2.ipynb b/hw2.ipynb
index 10d814c..49ffe32 100644
--- a/hw2.ipynb
+++ b/hw2.ipynb
@@ -32,7 +32,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "$\\leq (\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|)^{1/2} = \\|x\\|_{2}$"
+    "$ \\leq (\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|)^{1/2} = \\|x\\|_{2} $"
    ]
   },
   {
@@ -60,7 +60,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "$\\leq (m.(\\max\\limits_{i} |x_{i}|)^2)^{1/2} = \\sqrt{5}.\\|x\\|_\\infty$"
+    "$ \\leq (m.(\\max\\limits_{i} |x_{i}|)^2)^{1/2} = \\sqrt{5}.\\|x\\|_\\infty $"
    ]
   },
   {

From cc8c353671a73b78db3238582a3bd134b1a3363d Mon Sep 17 00:00:00 2001
From: burak0206 <burak0206@gmail.com>
Date: Sat, 25 Feb 2017 23:03:32 +0300
Subject: [PATCH 06/17] continue to fix

---
 .ipynb_checkpoints/hw2-checkpoint.ipynb | 4 ++--
 hw2.ipynb                               | 4 ++--
 2 files changed, 4 insertions(+), 4 deletions(-)

diff --git a/.ipynb_checkpoints/hw2-checkpoint.ipynb b/.ipynb_checkpoints/hw2-checkpoint.ipynb
index 49ffe32..59b0c0c 100644
--- a/.ipynb_checkpoints/hw2-checkpoint.ipynb
+++ b/.ipynb_checkpoints/hw2-checkpoint.ipynb
@@ -32,7 +32,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "$ \\leq (\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|)^{1/2} = \\|x\\|_{2} $"
+    "$ \\leq \\sqrt{\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|} = \\|x\\|_{2} $"
    ]
   },
   {
@@ -60,7 +60,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "$ \\leq (m.(\\max\\limits_{i} |x_{i}|)^2)^{1/2} = \\sqrt{5}.\\|x\\|_\\infty $"
+    "$ \\leq \\sqrt{m.(\\max\\limits_{i} |x_{i}|)^2} = \\sqrt{5}.\\|x\\|_\\infty $"
    ]
   },
   {
diff --git a/hw2.ipynb b/hw2.ipynb
index 49ffe32..59b0c0c 100644
--- a/hw2.ipynb
+++ b/hw2.ipynb
@@ -32,7 +32,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "$ \\leq (\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|)^{1/2} = \\|x\\|_{2} $"
+    "$ \\leq \\sqrt{\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|} = \\|x\\|_{2} $"
    ]
   },
   {
@@ -60,7 +60,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "$ \\leq (m.(\\max\\limits_{i} |x_{i}|)^2)^{1/2} = \\sqrt{5}.\\|x\\|_\\infty $"
+    "$ \\leq \\sqrt{m.(\\max\\limits_{i} |x_{i}|)^2} = \\sqrt{5}.\\|x\\|_\\infty $"
    ]
   },
   {

From 4d0181cefb8c4605d47ae91bece81fbde8823fa0 Mon Sep 17 00:00:00 2001
From: burak0206 <burak0206@gmail.com>
Date: Sat, 25 Feb 2017 23:05:33 +0300
Subject: [PATCH 07/17] merged

---
 .ipynb_checkpoints/hw2-checkpoint.ipynb | 36 ++++++++++++++-----------
 hw2.ipynb                               | 36 ++++++++++++++-----------
 2 files changed, 40 insertions(+), 32 deletions(-)

diff --git a/.ipynb_checkpoints/hw2-checkpoint.ipynb b/.ipynb_checkpoints/hw2-checkpoint.ipynb
index 59b0c0c..cdb533f 100644
--- a/.ipynb_checkpoints/hw2-checkpoint.ipynb
+++ b/.ipynb_checkpoints/hw2-checkpoint.ipynb
@@ -25,14 +25,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "$\\|x\\|_\\infty = ((\\max\\limits_{i} |x_{i}|)^2)^{1/2}$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "$ \\leq \\sqrt{\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|} = \\|x\\|_{2} $"
+    "$\\|x\\|_\\infty = ((\\max\\limits_{i} |x_{i}|)^2)^{1/2} \\leq \\sqrt{\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|} = \\|x\\|_{2} $"
    ]
   },
   {
@@ -53,14 +46,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "$\\|x\\|_{2} = (\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|)^{1/2}$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "$ \\leq \\sqrt{m.(\\max\\limits_{i} |x_{i}|)^2} = \\sqrt{5}.\\|x\\|_\\infty $"
+    "$\\|x\\|_{2} = (\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|)^{1/2}  \\leq \\sqrt{m.(\\max\\limits_{i} |x_{i}|)^2} = \\sqrt{5}.\\|x\\|_\\infty $"
    ]
   },
   {
@@ -159,6 +145,24 @@
     "$\\|A\\|_{2} =\\sqrt{tr(A^TA)}= \\sqrt{4}  = 2$"
    ]
   },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "collapsed": true
+   },
+   "outputs": [],
+   "source": []
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "collapsed": true
+   },
+   "outputs": [],
+   "source": []
+  },
   {
    "cell_type": "code",
    "execution_count": null,
diff --git a/hw2.ipynb b/hw2.ipynb
index 59b0c0c..cdb533f 100644
--- a/hw2.ipynb
+++ b/hw2.ipynb
@@ -25,14 +25,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "$\\|x\\|_\\infty = ((\\max\\limits_{i} |x_{i}|)^2)^{1/2}$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "$ \\leq \\sqrt{\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|} = \\|x\\|_{2} $"
+    "$\\|x\\|_\\infty = ((\\max\\limits_{i} |x_{i}|)^2)^{1/2} \\leq \\sqrt{\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|} = \\|x\\|_{2} $"
    ]
   },
   {
@@ -53,14 +46,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "$\\|x\\|_{2} = (\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|)^{1/2}$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "$ \\leq \\sqrt{m.(\\max\\limits_{i} |x_{i}|)^2} = \\sqrt{5}.\\|x\\|_\\infty $"
+    "$\\|x\\|_{2} = (\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|)^{1/2}  \\leq \\sqrt{m.(\\max\\limits_{i} |x_{i}|)^2} = \\sqrt{5}.\\|x\\|_\\infty $"
    ]
   },
   {
@@ -159,6 +145,24 @@
     "$\\|A\\|_{2} =\\sqrt{tr(A^TA)}= \\sqrt{4}  = 2$"
    ]
   },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "collapsed": true
+   },
+   "outputs": [],
+   "source": []
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "collapsed": true
+   },
+   "outputs": [],
+   "source": []
+  },
   {
    "cell_type": "code",
    "execution_count": null,

From 3764de170eb58f1e760ad4c2b46579536a627e80 Mon Sep 17 00:00:00 2001
From: burak0206 <burak0206@gmail.com>
Date: Sun, 26 Feb 2017 17:32:36 +0300
Subject: [PATCH 08/17] parse error

---
 .ipynb_checkpoints/hw2-checkpoint.ipynb | 31 ++-----------------------
 hw2.ipynb                               | 31 ++-----------------------
 2 files changed, 4 insertions(+), 58 deletions(-)

diff --git a/.ipynb_checkpoints/hw2-checkpoint.ipynb b/.ipynb_checkpoints/hw2-checkpoint.ipynb
index cdb533f..97978db 100644
--- a/.ipynb_checkpoints/hw2-checkpoint.ipynb
+++ b/.ipynb_checkpoints/hw2-checkpoint.ipynb
@@ -25,7 +25,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "$\\|x\\|_\\infty = ((\\max\\limits_{i} |x_{i}|)^2)^{1/2} \\leq \\sqrt{\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|} = \\|x\\|_{2} $"
+    "$ \\|x\\|_\\infty = ((\\max\\limits_{i} |x_{i}|)^2)^{1/2} \\leq \\sqrt{\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|} = \\|x\\|_{2} $"
    ]
   },
   {
@@ -46,7 +46,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "$\\|x\\|_{2} = (\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|)^{1/2}  \\leq \\sqrt{m.(\\max\\limits_{i} |x_{i}|)^2} = \\sqrt{5}.\\|x\\|_\\infty $"
+    "$ \\|x\\|_{2} = (\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|)^{1/2}  \\leq \\sqrt{m.(\\max\\limits_{i} |x_{i}|)^2} = \\sqrt{5}.\\|x\\|_\\infty $"
    ]
   },
   {
@@ -144,33 +144,6 @@
     "\n",
     "$\\|A\\|_{2} =\\sqrt{tr(A^TA)}= \\sqrt{4}  = 2$"
    ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {
-    "collapsed": true
-   },
-   "outputs": [],
-   "source": []
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {
-    "collapsed": true
-   },
-   "outputs": [],
-   "source": []
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {
-    "collapsed": true
-   },
-   "outputs": [],
-   "source": []
   }
  ],
  "metadata": {
diff --git a/hw2.ipynb b/hw2.ipynb
index cdb533f..97978db 100644
--- a/hw2.ipynb
+++ b/hw2.ipynb
@@ -25,7 +25,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "$\\|x\\|_\\infty = ((\\max\\limits_{i} |x_{i}|)^2)^{1/2} \\leq \\sqrt{\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|} = \\|x\\|_{2} $"
+    "$ \\|x\\|_\\infty = ((\\max\\limits_{i} |x_{i}|)^2)^{1/2} \\leq \\sqrt{\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|} = \\|x\\|_{2} $"
    ]
   },
   {
@@ -46,7 +46,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "$\\|x\\|_{2} = (\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|)^{1/2}  \\leq \\sqrt{m.(\\max\\limits_{i} |x_{i}|)^2} = \\sqrt{5}.\\|x\\|_\\infty $"
+    "$ \\|x\\|_{2} = (\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|)^{1/2}  \\leq \\sqrt{m.(\\max\\limits_{i} |x_{i}|)^2} = \\sqrt{5}.\\|x\\|_\\infty $"
    ]
   },
   {
@@ -144,33 +144,6 @@
     "\n",
     "$\\|A\\|_{2} =\\sqrt{tr(A^TA)}= \\sqrt{4}  = 2$"
    ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {
-    "collapsed": true
-   },
-   "outputs": [],
-   "source": []
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {
-    "collapsed": true
-   },
-   "outputs": [],
-   "source": []
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {
-    "collapsed": true
-   },
-   "outputs": [],
-   "source": []
   }
  ],
  "metadata": {

From 19cfc540ae3120d5217e63bf54f0cfe6454da64a Mon Sep 17 00:00:00 2001
From: burak0206 <burak0206@gmail.com>
Date: Sun, 26 Feb 2017 17:37:24 +0300
Subject: [PATCH 09/17] sqrt script is cleared

---
 .ipynb_checkpoints/hw2-checkpoint.ipynb | 2 +-
 hw2.ipynb                               | 2 +-
 2 files changed, 2 insertions(+), 2 deletions(-)

diff --git a/.ipynb_checkpoints/hw2-checkpoint.ipynb b/.ipynb_checkpoints/hw2-checkpoint.ipynb
index 97978db..0c63b07 100644
--- a/.ipynb_checkpoints/hw2-checkpoint.ipynb
+++ b/.ipynb_checkpoints/hw2-checkpoint.ipynb
@@ -25,7 +25,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "$ \\|x\\|_\\infty = ((\\max\\limits_{i} |x_{i}|)^2)^{1/2} \\leq \\sqrt{\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|} = \\|x\\|_{2} $"
+    "$ \\|x\\|_\\infty = ((\\max\\limits_{i} |x_{i}|)^2)^{1/2} \\leq (\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|)^{1/2} = \\|x\\|_{2} $"
    ]
   },
   {
diff --git a/hw2.ipynb b/hw2.ipynb
index 97978db..0c63b07 100644
--- a/hw2.ipynb
+++ b/hw2.ipynb
@@ -25,7 +25,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "$ \\|x\\|_\\infty = ((\\max\\limits_{i} |x_{i}|)^2)^{1/2} \\leq \\sqrt{\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|} = \\|x\\|_{2} $"
+    "$ \\|x\\|_\\infty = ((\\max\\limits_{i} |x_{i}|)^2)^{1/2} \\leq (\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|)^{1/2} = \\|x\\|_{2} $"
    ]
   },
   {

From ab1080b3b6e77f0a35301c4c83d9d136042637af Mon Sep 17 00:00:00 2001
From: burak0206 <burak0206@gmail.com>
Date: Sun, 26 Feb 2017 17:39:09 +0300
Subject: [PATCH 10/17] sort function can not be rendered

---
 .ipynb_checkpoints/hw2-checkpoint.ipynb | 9 ++++++++-
 hw2.ipynb                               | 9 ++++++++-
 2 files changed, 16 insertions(+), 2 deletions(-)

diff --git a/.ipynb_checkpoints/hw2-checkpoint.ipynb b/.ipynb_checkpoints/hw2-checkpoint.ipynb
index 0c63b07..37b846e 100644
--- a/.ipynb_checkpoints/hw2-checkpoint.ipynb
+++ b/.ipynb_checkpoints/hw2-checkpoint.ipynb
@@ -25,7 +25,14 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "$ \\|x\\|_\\infty = ((\\max\\limits_{i} |x_{i}|)^2)^{1/2} \\leq (\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|)^{1/2} = \\|x\\|_{2} $"
+    "$ \\|x\\|_\\infty = ((\\max\\limits_{i} |x_{i}|)^2)^{1/2} $ "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$ \\leq (\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|)^{1/2} = \\|x\\|_{2} $"
    ]
   },
   {
diff --git a/hw2.ipynb b/hw2.ipynb
index 0c63b07..37b846e 100644
--- a/hw2.ipynb
+++ b/hw2.ipynb
@@ -25,7 +25,14 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "$ \\|x\\|_\\infty = ((\\max\\limits_{i} |x_{i}|)^2)^{1/2} \\leq (\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|)^{1/2} = \\|x\\|_{2} $"
+    "$ \\|x\\|_\\infty = ((\\max\\limits_{i} |x_{i}|)^2)^{1/2} $ "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$ \\leq (\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|)^{1/2} = \\|x\\|_{2} $"
    ]
   },
   {

From f1d186cb907599f9d5477618b9bdf13f3d1b091a Mon Sep 17 00:00:00 2001
From: burak0206 <burak0206@gmail.com>
Date: Sun, 26 Feb 2017 17:41:29 +0300
Subject: [PATCH 11/17] try to solve render problem

---
 .ipynb_checkpoints/hw2-checkpoint.ipynb | 2 +-
 hw2.ipynb                               | 2 +-
 2 files changed, 2 insertions(+), 2 deletions(-)

diff --git a/.ipynb_checkpoints/hw2-checkpoint.ipynb b/.ipynb_checkpoints/hw2-checkpoint.ipynb
index 37b846e..06ea3ea 100644
--- a/.ipynb_checkpoints/hw2-checkpoint.ipynb
+++ b/.ipynb_checkpoints/hw2-checkpoint.ipynb
@@ -32,7 +32,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "$ \\leq (\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|)^{1/2} = \\|x\\|_{2} $"
+    "$ \\leq (\\sum \\limits_{i=1}^{m} \\max \\limits_{i} |x_{i}^2|)^{1/2} = \\|x\\|_{2} $"
    ]
   },
   {
diff --git a/hw2.ipynb b/hw2.ipynb
index 37b846e..06ea3ea 100644
--- a/hw2.ipynb
+++ b/hw2.ipynb
@@ -32,7 +32,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "$ \\leq (\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|)^{1/2} = \\|x\\|_{2} $"
+    "$ \\leq (\\sum \\limits_{i=1}^{m} \\max \\limits_{i} |x_{i}^2|)^{1/2} = \\|x\\|_{2} $"
    ]
   },
   {

From 79d20745424667ad06794f02e153d4d45ed34c94 Mon Sep 17 00:00:00 2001
From: burak0206 <burak0206@gmail.com>
Date: Sun, 26 Feb 2017 17:44:28 +0300
Subject: [PATCH 12/17] attemp to problem

---
 .ipynb_checkpoints/hw2-checkpoint.ipynb | 2 +-
 hw2.ipynb                               | 2 +-
 2 files changed, 2 insertions(+), 2 deletions(-)

diff --git a/.ipynb_checkpoints/hw2-checkpoint.ipynb b/.ipynb_checkpoints/hw2-checkpoint.ipynb
index 06ea3ea..dbf01a7 100644
--- a/.ipynb_checkpoints/hw2-checkpoint.ipynb
+++ b/.ipynb_checkpoints/hw2-checkpoint.ipynb
@@ -32,7 +32,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "$ \\leq (\\sum \\limits_{i=1}^{m} \\max \\limits_{i} |x_{i}^2|)^{1/2} = \\|x\\|_{2} $"
+    "$ \\leq (\\sum \\limits_{i=1}^{m} (\\max \\limits_{i} |x_{i}^2|))^{1/2} = \\|x\\|_{2} $"
    ]
   },
   {
diff --git a/hw2.ipynb b/hw2.ipynb
index 06ea3ea..dbf01a7 100644
--- a/hw2.ipynb
+++ b/hw2.ipynb
@@ -32,7 +32,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "$ \\leq (\\sum \\limits_{i=1}^{m} \\max \\limits_{i} |x_{i}^2|)^{1/2} = \\|x\\|_{2} $"
+    "$ \\leq (\\sum \\limits_{i=1}^{m} (\\max \\limits_{i} |x_{i}^2|))^{1/2} = \\|x\\|_{2} $"
    ]
   },
   {

From d245a7574e1d580ddaa2bfe512a2242077b3f395 Mon Sep 17 00:00:00 2001
From: burak0206 <burak0206@gmail.com>
Date: Sun, 26 Feb 2017 17:53:02 +0300
Subject: [PATCH 13/17] split a and b

---
 .ipynb_checkpoints/hw2-checkpoint.ipynb | 11 +++++++++--
 hw2.ipynb                               | 11 +++++++++--
 2 files changed, 18 insertions(+), 4 deletions(-)

diff --git a/.ipynb_checkpoints/hw2-checkpoint.ipynb b/.ipynb_checkpoints/hw2-checkpoint.ipynb
index dbf01a7..f39c160 100644
--- a/.ipynb_checkpoints/hw2-checkpoint.ipynb
+++ b/.ipynb_checkpoints/hw2-checkpoint.ipynb
@@ -32,7 +32,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "$ \\leq (\\sum \\limits_{i=1}^{m} (\\max \\limits_{i} |x_{i}^2|))^{1/2} = \\|x\\|_{2} $"
+    "$ \\leq \\sum\\limits_{i=1}^{m} (\\max\\limits_{i} |x_{i}^2|))^{1/2} = \\|x\\|_{2} $"
    ]
   },
   {
@@ -53,7 +53,14 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "$ \\|x\\|_{2} = (\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|)^{1/2}  \\leq \\sqrt{m.(\\max\\limits_{i} |x_{i}|)^2} = \\sqrt{5}.\\|x\\|_\\infty $"
+    "$ \\|x\\|_{2} = (\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|)^{1/2}  $"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$ \\leq \\sqrt{m.(\\max\\limits_{i} |x_{i}|)^2} = \\sqrt{5}.\\|x\\|_\\infty $"
    ]
   },
   {
diff --git a/hw2.ipynb b/hw2.ipynb
index dbf01a7..f39c160 100644
--- a/hw2.ipynb
+++ b/hw2.ipynb
@@ -32,7 +32,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "$ \\leq (\\sum \\limits_{i=1}^{m} (\\max \\limits_{i} |x_{i}^2|))^{1/2} = \\|x\\|_{2} $"
+    "$ \\leq \\sum\\limits_{i=1}^{m} (\\max\\limits_{i} |x_{i}^2|))^{1/2} = \\|x\\|_{2} $"
    ]
   },
   {
@@ -53,7 +53,14 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "$ \\|x\\|_{2} = (\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|)^{1/2}  \\leq \\sqrt{m.(\\max\\limits_{i} |x_{i}|)^2} = \\sqrt{5}.\\|x\\|_\\infty $"
+    "$ \\|x\\|_{2} = (\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|)^{1/2}  $"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$ \\leq \\sqrt{m.(\\max\\limits_{i} |x_{i}|)^2} = \\sqrt{5}.\\|x\\|_\\infty $"
    ]
   },
   {

From 1d2de1334a89db6cff6db569be949c22dddc038b Mon Sep 17 00:00:00 2001
From: burak0206 <burak0206@gmail.com>
Date: Sun, 26 Feb 2017 18:09:13 +0300
Subject: [PATCH 14/17] render problems are fixed

---
 .ipynb_checkpoints/hw2-checkpoint.ipynb | 18 ++----------------
 hw2.ipynb                               | 18 ++----------------
 2 files changed, 4 insertions(+), 32 deletions(-)

diff --git a/.ipynb_checkpoints/hw2-checkpoint.ipynb b/.ipynb_checkpoints/hw2-checkpoint.ipynb
index f39c160..7ba9a57 100644
--- a/.ipynb_checkpoints/hw2-checkpoint.ipynb
+++ b/.ipynb_checkpoints/hw2-checkpoint.ipynb
@@ -25,14 +25,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "$ \\|x\\|_\\infty = ((\\max\\limits_{i} |x_{i}|)^2)^{1/2} $ "
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "$ \\leq \\sum\\limits_{i=1}^{m} (\\max\\limits_{i} |x_{i}^2|))^{1/2} = \\|x\\|_{2} $"
+    "$ \\|x\\|_\\infty =  \\max\\limits_{i} |x_{i}| =  \\sqrt{(\\max\\limits_{i} |x_{i}|)^2}  \\leq \\sqrt{ \\sum\\limits_{i=1}^{m}  |x_{i}|^2} = \\|x\\|_{2} $"
    ]
   },
   {
@@ -53,14 +46,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "$ \\|x\\|_{2} = (\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|)^{1/2}  $"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "$ \\leq \\sqrt{m.(\\max\\limits_{i} |x_{i}|)^2} = \\sqrt{5}.\\|x\\|_\\infty $"
+    "$ \\|x\\|_{2} = \\sqrt {\\sum\\limits_{i=1}^{m} |x_{i}^2|}  \\leq \\sqrt{m.(\\max\\limits_{i} |x_{i}|)^2} = \\sqrt{m}.\\|x\\|_\\infty $"
    ]
   },
   {
diff --git a/hw2.ipynb b/hw2.ipynb
index f39c160..7ba9a57 100644
--- a/hw2.ipynb
+++ b/hw2.ipynb
@@ -25,14 +25,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "$ \\|x\\|_\\infty = ((\\max\\limits_{i} |x_{i}|)^2)^{1/2} $ "
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "$ \\leq \\sum\\limits_{i=1}^{m} (\\max\\limits_{i} |x_{i}^2|))^{1/2} = \\|x\\|_{2} $"
+    "$ \\|x\\|_\\infty =  \\max\\limits_{i} |x_{i}| =  \\sqrt{(\\max\\limits_{i} |x_{i}|)^2}  \\leq \\sqrt{ \\sum\\limits_{i=1}^{m}  |x_{i}|^2} = \\|x\\|_{2} $"
    ]
   },
   {
@@ -53,14 +46,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "$ \\|x\\|_{2} = (\\sum\\limits_{i=1}^{m} \\max\\limits_{i} |x_{i}^2|)^{1/2}  $"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "$ \\leq \\sqrt{m.(\\max\\limits_{i} |x_{i}|)^2} = \\sqrt{5}.\\|x\\|_\\infty $"
+    "$ \\|x\\|_{2} = \\sqrt {\\sum\\limits_{i=1}^{m} |x_{i}^2|}  \\leq \\sqrt{m.(\\max\\limits_{i} |x_{i}|)^2} = \\sqrt{m}.\\|x\\|_\\infty $"
    ]
   },
   {

From ddf4524a51f5b997a55df1bf3274208193cecc6c Mon Sep 17 00:00:00 2001
From: burak0206 <burak0206@gmail.com>
Date: Sun, 26 Feb 2017 18:12:17 +0300
Subject: [PATCH 15/17] render problem can not be solved

max and sum functions can not be seen in GitHub
---
 .ipynb_checkpoints/hw2-checkpoint.ipynb | 18 ++++++++++++++++--
 hw2.ipynb                               | 18 ++++++++++++++++--
 2 files changed, 32 insertions(+), 4 deletions(-)

diff --git a/.ipynb_checkpoints/hw2-checkpoint.ipynb b/.ipynb_checkpoints/hw2-checkpoint.ipynb
index 7ba9a57..1be6798 100644
--- a/.ipynb_checkpoints/hw2-checkpoint.ipynb
+++ b/.ipynb_checkpoints/hw2-checkpoint.ipynb
@@ -25,7 +25,14 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "$ \\|x\\|_\\infty =  \\max\\limits_{i} |x_{i}| =  \\sqrt{(\\max\\limits_{i} |x_{i}|)^2}  \\leq \\sqrt{ \\sum\\limits_{i=1}^{m}  |x_{i}|^2} = \\|x\\|_{2} $"
+    "$ \\|x\\|_\\infty =  \\max\\limits_{i} |x_{i}| =  \\sqrt{(\\max\\limits_{i} |x_{i}|)^2} $"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$\\leq \\sqrt{ \\sum\\limits_{i=1}^{m}  |x_{i}|^2} = \\|x\\|_{2} $"
    ]
   },
   {
@@ -46,7 +53,14 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "$ \\|x\\|_{2} = \\sqrt {\\sum\\limits_{i=1}^{m} |x_{i}^2|}  \\leq \\sqrt{m.(\\max\\limits_{i} |x_{i}|)^2} = \\sqrt{m}.\\|x\\|_\\infty $"
+    "$ \\|x\\|_{2} = \\sqrt {\\sum\\limits_{i=1}^{m} |x_{i}^2|} $  "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$ \\leq \\sqrt{m.(\\max\\limits_{i} |x_{i}|)^2} = \\sqrt{m}.\\|x\\|_\\infty $"
    ]
   },
   {
diff --git a/hw2.ipynb b/hw2.ipynb
index 7ba9a57..1be6798 100644
--- a/hw2.ipynb
+++ b/hw2.ipynb
@@ -25,7 +25,14 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "$ \\|x\\|_\\infty =  \\max\\limits_{i} |x_{i}| =  \\sqrt{(\\max\\limits_{i} |x_{i}|)^2}  \\leq \\sqrt{ \\sum\\limits_{i=1}^{m}  |x_{i}|^2} = \\|x\\|_{2} $"
+    "$ \\|x\\|_\\infty =  \\max\\limits_{i} |x_{i}| =  \\sqrt{(\\max\\limits_{i} |x_{i}|)^2} $"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$\\leq \\sqrt{ \\sum\\limits_{i=1}^{m}  |x_{i}|^2} = \\|x\\|_{2} $"
    ]
   },
   {
@@ -46,7 +53,14 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "$ \\|x\\|_{2} = \\sqrt {\\sum\\limits_{i=1}^{m} |x_{i}^2|}  \\leq \\sqrt{m.(\\max\\limits_{i} |x_{i}|)^2} = \\sqrt{m}.\\|x\\|_\\infty $"
+    "$ \\|x\\|_{2} = \\sqrt {\\sum\\limits_{i=1}^{m} |x_{i}^2|} $  "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$ \\leq \\sqrt{m.(\\max\\limits_{i} |x_{i}|)^2} = \\sqrt{m}.\\|x\\|_\\infty $"
    ]
   },
   {

From 8c25ad50783eefcf42092359215b34b6682b006e Mon Sep 17 00:00:00 2001
From: burak0206 <burak0206@gmail.com>
Date: Sun, 26 Feb 2017 18:18:57 +0300
Subject: [PATCH 16/17] \sum\limit is changed by \sum

---
 hw2.ipynb | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/hw2.ipynb b/hw2.ipynb
index 1be6798..c2029f2 100644
--- a/hw2.ipynb
+++ b/hw2.ipynb
@@ -32,7 +32,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "$\\leq \\sqrt{ \\sum\\limits_{i=1}^{m}  |x_{i}|^2} = \\|x\\|_{2} $"
+    "$\\leq \\sqrt{ \\sum_{i=1}^{m}  |x_{i}|^2} = \\|x\\|_{2} $"
    ]
   },
   {
@@ -53,7 +53,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "$ \\|x\\|_{2} = \\sqrt {\\sum\\limits_{i=1}^{m} |x_{i}^2|} $  "
+    "$ \\|x\\|_{2} = \\sqrt {\\sum\\limits_{i=1}^{m} |x_{i}|^2} $  "
    ]
   },
   {

From 477850f77e96d7219d024abd31decfa5eccd400b Mon Sep 17 00:00:00 2001
From: burak0206 <burak0206@gmail.com>
Date: Sun, 26 Feb 2017 18:20:46 +0300
Subject: [PATCH 17/17] all \sum_\limits are changed by \sum function

---
 .ipynb_checkpoints/hw2-checkpoint.ipynb | 18 ++----------------
 hw2.ipynb                               | 18 ++----------------
 2 files changed, 4 insertions(+), 32 deletions(-)

diff --git a/.ipynb_checkpoints/hw2-checkpoint.ipynb b/.ipynb_checkpoints/hw2-checkpoint.ipynb
index 1be6798..eae0b3b 100644
--- a/.ipynb_checkpoints/hw2-checkpoint.ipynb
+++ b/.ipynb_checkpoints/hw2-checkpoint.ipynb
@@ -25,14 +25,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "$ \\|x\\|_\\infty =  \\max\\limits_{i} |x_{i}| =  \\sqrt{(\\max\\limits_{i} |x_{i}|)^2} $"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "$\\leq \\sqrt{ \\sum\\limits_{i=1}^{m}  |x_{i}|^2} = \\|x\\|_{2} $"
+    "$ \\|x\\|_\\infty =  \\max\\limits_{i} |x_{i}| =  \\sqrt{(\\max\\limits_{i} |x_{i}|)^2} \\leq \\sqrt{ \\sum_{i=1}^{m}  |x_{i}|^2} = \\|x\\|_{2} $"
    ]
   },
   {
@@ -53,14 +46,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "$ \\|x\\|_{2} = \\sqrt {\\sum\\limits_{i=1}^{m} |x_{i}^2|} $  "
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "$ \\leq \\sqrt{m.(\\max\\limits_{i} |x_{i}|)^2} = \\sqrt{m}.\\|x\\|_\\infty $"
+    "$ \\|x\\|_{2} = \\sqrt {\\sum_{i=1}^{m} |x_{i}|^2} \\leq \\sqrt{m.(\\max\\limits_{i} |x_{i}|)^2} = \\sqrt{m}.\\|x\\|_\\infty $"
    ]
   },
   {
diff --git a/hw2.ipynb b/hw2.ipynb
index c2029f2..eae0b3b 100644
--- a/hw2.ipynb
+++ b/hw2.ipynb
@@ -25,14 +25,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "$ \\|x\\|_\\infty =  \\max\\limits_{i} |x_{i}| =  \\sqrt{(\\max\\limits_{i} |x_{i}|)^2} $"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "$\\leq \\sqrt{ \\sum_{i=1}^{m}  |x_{i}|^2} = \\|x\\|_{2} $"
+    "$ \\|x\\|_\\infty =  \\max\\limits_{i} |x_{i}| =  \\sqrt{(\\max\\limits_{i} |x_{i}|)^2} \\leq \\sqrt{ \\sum_{i=1}^{m}  |x_{i}|^2} = \\|x\\|_{2} $"
    ]
   },
   {
@@ -53,14 +46,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "$ \\|x\\|_{2} = \\sqrt {\\sum\\limits_{i=1}^{m} |x_{i}|^2} $  "
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "$ \\leq \\sqrt{m.(\\max\\limits_{i} |x_{i}|)^2} = \\sqrt{m}.\\|x\\|_\\infty $"
+    "$ \\|x\\|_{2} = \\sqrt {\\sum_{i=1}^{m} |x_{i}|^2} \\leq \\sqrt{m.(\\max\\limits_{i} |x_{i}|)^2} = \\sqrt{m}.\\|x\\|_\\infty $"
    ]
   },
   {