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math.html
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math.html
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<!DOCTYPE html><html><head><meta charset="utf-8"><meta name="layout" content="post">
<meta name="permalink" content="/math/">
<meta name="mathjax" content="true"><style>body {
width: 45em;
border: 1px solid #ddd;
outline: 1300px solid #fff;
margin: 16px auto;
}
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background: #ff0;
color: #000;
font-style: italic;
font-weight: bold;
}
.markdown-body sub,
.markdown-body sup {
font-size: 75%;
line-height: 0;
position: relative;
vertical-align: baseline;
}
.markdown-body sup {
top: -0.5em;
}
.markdown-body sub {
bottom: -0.25em;
}
.markdown-body h1 {
font-size: 2em;
margin: 0.67em 0;
}
.markdown-body img {
border: 0;
}
.markdown-body hr {
-moz-box-sizing: content-box;
box-sizing: content-box;
height: 0;
}
.markdown-body pre {
overflow: auto;
}
.markdown-body code,
.markdown-body kbd,
.markdown-body pre,
.markdown-body samp {
font-family: monospace, monospace;
font-size: 1em;
}
.markdown-body input {
color: inherit;
font: inherit;
margin: 0;
}
.markdown-body html input[disabled] {
cursor: default;
}
.markdown-body input {
line-height: normal;
}
.markdown-body input[type="checkbox"] {
box-sizing: border-box;
padding: 0;
}
.markdown-body table {
border-collapse: collapse;
border-spacing: 0;
}
.markdown-body td,
.markdown-body th {
padding: 0;
}
.markdown-body .codehilitetable {
border: 0;
border-spacing: 0;
}
.markdown-body .codehilitetable tr {
border: 0;
}
.markdown-body .codehilitetable pre,
.markdown-body .codehilitetable div.codehilite {
margin: 0;
}
.markdown-body .linenos,
.markdown-body .code,
.markdown-body .codehilitetable td {
border: 0;
padding: 0;
}
.markdown-body td:not(.linenos) .linenodiv {
padding: 0 !important;
}
.markdown-body .code {
width: 100%;
}
.markdown-body .linenos div pre,
.markdown-body .linenodiv pre,
.markdown-body .linenodiv {
border: 0;
-webkit-border-radius: 0;
-moz-border-radius: 0;
border-radius: 0;
-webkit-border-top-left-radius: 3px;
-webkit-border-bottom-left-radius: 3px;
-moz-border-radius-topleft: 3px;
-moz-border-radius-bottomleft: 3px;
border-top-left-radius: 3px;
border-bottom-left-radius: 3px;
}
.markdown-body .code div pre,
.markdown-body .code div {
border: 0;
-webkit-border-radius: 0;
-moz-border-radius: 0;
border-radius: 0;
-webkit-border-top-right-radius: 3px;
-webkit-border-bottom-right-radius: 3px;
-moz-border-radius-topright: 3px;
-moz-border-radius-bottomright: 3px;
border-top-right-radius: 3px;
border-bottom-right-radius: 3px;
}
.markdown-body * {
-moz-box-sizing: border-box;
box-sizing: border-box;
}
.markdown-body input {
font: 13px Helvetica, arial, freesans, clean, sans-serif, "Segoe UI Emoji", "Segoe UI Symbol";
line-height: 1.4;
}
.markdown-body a {
color: #4183c4;
text-decoration: none;
}
.markdown-body a:hover,
.markdown-body a:focus,
.markdown-body a:active {
text-decoration: underline;
}
.markdown-body hr {
height: 0;
margin: 15px 0;
overflow: hidden;
background: transparent;
border: 0;
border-bottom: 1px solid #ddd;
}
.markdown-body hr:before,
.markdown-body hr:after {
display: table;
content: " ";
}
.markdown-body hr:after {
clear: both;
}
.markdown-body h1,
.markdown-body h2,
.markdown-body h3,
.markdown-body h4,
.markdown-body h5,
.markdown-body h6 {
margin-top: 15px;
margin-bottom: 15px;
line-height: 1.1;
}
.markdown-body h1 {
font-size: 30px;
}
.markdown-body h2 {
font-size: 21px;
}
.markdown-body h3 {
font-size: 16px;
}
.markdown-body h4 {
font-size: 14px;
}
.markdown-body h5 {
font-size: 12px;
}
.markdown-body h6 {
font-size: 11px;
}
.markdown-body blockquote {
margin: 0;
}
.markdown-body ul,
.markdown-body ol {
padding: 0;
margin-top: 0;
margin-bottom: 0;
}
.markdown-body ol ol,
.markdown-body ul ol {
list-style-type: lower-roman;
}
.markdown-body ul ul ol,
.markdown-body ul ol ol,
.markdown-body ol ul ol,
.markdown-body ol ol ol {
list-style-type: lower-alpha;
}
.markdown-body dd {
margin-left: 0;
}
.markdown-body code,
.markdown-body pre,
.markdown-body samp {
font-family: Consolas, "Liberation Mono", Menlo, Courier, monospace;
font-size: 12px;
}
.markdown-body pre {
margin-top: 0;
margin-bottom: 0;
}
.markdown-body kbd {
background-color: #e7e7e7;
background-image: -moz-linear-gradient(#fefefe, #e7e7e7);
background-image: -webkit-linear-gradient(#fefefe, #e7e7e7);
background-image: linear-gradient(#fefefe, #e7e7e7);
background-repeat: repeat-x;
border-radius: 2px;
border: 1px solid #cfcfcf;
color: #000;
padding: 3px 5px;
line-height: 10px;
font: 11px Consolas, "Liberation Mono", Menlo, Courier, monospace;
display: inline-block;
}
.markdown-body>*:first-child {
margin-top: 0 !important;
}
.markdown-body>*:last-child {
margin-bottom: 0 !important;
}
.markdown-body .headeranchor-link {
position: absolute;
top: 0;
bottom: 0;
left: 0;
display: block;
padding-right: 6px;
padding-left: 30px;
margin-left: -30px;
}
.markdown-body .headeranchor-link:focus {
outline: none;
}
.markdown-body h1,
.markdown-body h2,
.markdown-body h3,
.markdown-body h4,
.markdown-body h5,
.markdown-body h6 {
position: relative;
margin-top: 1em;
margin-bottom: 16px;
font-weight: bold;
line-height: 1.4;
}
.markdown-body h1 .headeranchor,
.markdown-body h2 .headeranchor,
.markdown-body h3 .headeranchor,
.markdown-body h4 .headeranchor,
.markdown-body h5 .headeranchor,
.markdown-body h6 .headeranchor {
display: none;
color: #000;
vertical-align: middle;
}
.markdown-body h1:hover .headeranchor-link,
.markdown-body h2:hover .headeranchor-link,
.markdown-body h3:hover .headeranchor-link,
.markdown-body h4:hover .headeranchor-link,
.markdown-body h5:hover .headeranchor-link,
.markdown-body h6:hover .headeranchor-link {
height: 1em;
padding-left: 8px;
margin-left: -30px;
line-height: 1;
text-decoration: none;
}
.markdown-body h1:hover .headeranchor-link .headeranchor,
.markdown-body h2:hover .headeranchor-link .headeranchor,
.markdown-body h3:hover .headeranchor-link .headeranchor,
.markdown-body h4:hover .headeranchor-link .headeranchor,
.markdown-body h5:hover .headeranchor-link .headeranchor,
.markdown-body h6:hover .headeranchor-link .headeranchor {
display: inline-block;
}
.markdown-body h1 {
padding-bottom: 0.3em;
font-size: 2.25em;
line-height: 1.2;
border-bottom: 1px solid #eee;
}
.markdown-body h2 {
padding-bottom: 0.3em;
font-size: 1.75em;
line-height: 1.225;
border-bottom: 1px solid #eee;
}
.markdown-body h3 {
font-size: 1.5em;
line-height: 1.43;
}
.markdown-body h4 {
font-size: 1.25em;
}
.markdown-body h5 {
font-size: 1em;
}
.markdown-body h6 {
font-size: 1em;
color: #777;
}
.markdown-body p,
.markdown-body blockquote,
.markdown-body ul,
.markdown-body ol,
.markdown-body dl,
.markdown-body table,
.markdown-body pre,
.markdown-body .admonition {
margin-top: 0;
margin-bottom: 16px;
}
.markdown-body hr {
height: 4px;
padding: 0;
margin: 16px 0;
background-color: #e7e7e7;
border: 0 none;
}
.markdown-body ul,
.markdown-body ol {
padding-left: 2em;
}
.markdown-body ul ul,
.markdown-body ul ol,
.markdown-body ol ol,
.markdown-body ol ul {
margin-top: 0;
margin-bottom: 0;
}
.markdown-body li>p {
margin-top: 16px;
}
.markdown-body dl {
padding: 0;
}
.markdown-body dl dt {
padding: 0;
margin-top: 16px;
font-size: 1em;
font-style: italic;
font-weight: bold;
}
.markdown-body dl dd {
padding: 0 16px;
margin-bottom: 16px;
}
.markdown-body blockquote {
padding: 0 15px;
color: #777;
border-left: 4px solid #ddd;
}
.markdown-body blockquote>:first-child {
margin-top: 0;
}
.markdown-body blockquote>:last-child {
margin-bottom: 0;
}
.markdown-body table {
display: block;
width: 100%;
overflow: auto;
word-break: normal;
word-break: keep-all;
}
.markdown-body table th {
font-weight: bold;
}
.markdown-body table th,
.markdown-body table td {
padding: 6px 13px;
border: 1px solid #ddd;
}
.markdown-body table tr {
background-color: #fff;
border-top: 1px solid #ccc;
}
.markdown-body table tr:nth-child(2n) {
background-color: #f8f8f8;
}
.markdown-body img {
max-width: 100%;
-moz-box-sizing: border-box;
box-sizing: border-box;
}
.markdown-body code,
.markdown-body samp {
padding: 0;
padding-top: 0.2em;
padding-bottom: 0.2em;
margin: 0;
font-size: 85%;
background-color: rgba(0,0,0,0.04);
border-radius: 3px;
}
.markdown-body code:before,
.markdown-body code:after {
letter-spacing: -0.2em;
content: "\00a0";
}
.markdown-body pre>code {
padding: 0;
margin: 0;
font-size: 100%;
word-break: normal;
white-space: pre;
background: transparent;
border: 0;
}
.markdown-body .codehilite {
margin-bottom: 16px;
}
.markdown-body .codehilite pre,
.markdown-body pre {
padding: 16px;
overflow: auto;
font-size: 85%;
line-height: 1.45;
background-color: #f7f7f7;
border-radius: 3px;
}
.markdown-body .codehilite pre {
margin-bottom: 0;
word-break: normal;
}
.markdown-body pre {
word-wrap: normal;
}
.markdown-body pre code {
display: inline;
max-width: initial;
padding: 0;
margin: 0;
overflow: initial;
line-height: inherit;
word-wrap: normal;
background-color: transparent;
border: 0;
}
.markdown-body pre code:before,
.markdown-body pre code:after {
content: normal;
}
/* Admonition */
.markdown-body .admonition {
-webkit-border-radius: 3px;
-moz-border-radius: 3px;
position: relative;
border-radius: 3px;
border: 1px solid #e0e0e0;
border-left: 6px solid #333;
padding: 10px 10px 10px 30px;
}
.markdown-body .admonition table {
color: #333;
}
.markdown-body .admonition p {
padding: 0;
}
.markdown-body .admonition-title {
font-weight: bold;
margin: 0;
}
.markdown-body .admonition>.admonition-title {
color: #333;
}
.markdown-body .attention>.admonition-title {
color: #a6d796;
}
.markdown-body .caution>.admonition-title {
color: #d7a796;
}
.markdown-body .hint>.admonition-title {
color: #96c6d7;
}
.markdown-body .danger>.admonition-title {
color: #c25f77;
}
.markdown-body .question>.admonition-title {
color: #96a6d7;
}
.markdown-body .note>.admonition-title {
color: #d7c896;
}
.markdown-body .admonition:before,
.markdown-body .attention:before,
.markdown-body .caution:before,
.markdown-body .hint:before,
.markdown-body .danger:before,
.markdown-body .question:before,
.markdown-body .note:before {
font: normal normal 16px fontawesome-mini;
-moz-osx-font-smoothing: grayscale;
-webkit-user-select: none;
-moz-user-select: none;
-ms-user-select: none;
user-select: none;
line-height: 1.5;
color: #333;
position: absolute;
left: 0;
top: 0;
padding-top: 10px;
padding-left: 10px;
}
.markdown-body .admonition:before {
content: "\f056\00a0";
color: 333;
}
.markdown-body .attention:before {
content: "\f058\00a0";
color: #a6d796;
}
.markdown-body .caution:before {
content: "\f06a\00a0";
color: #d7a796;
}
.markdown-body .hint:before {
content: "\f05a\00a0";
color: #96c6d7;
}
.markdown-body .danger:before {
content: "\f057\00a0";
color: #c25f77;
}
.markdown-body .question:before {
content: "\f059\00a0";
color: #96a6d7;
}
.markdown-body .note:before {
content: "\f040\00a0";
color: #d7c896;
}
.markdown-body .admonition::after {
content: normal;
}
.markdown-body .attention {
border-left: 6px solid #a6d796;
}
.markdown-body .caution {
border-left: 6px solid #d7a796;
}
.markdown-body .hint {
border-left: 6px solid #96c6d7;
}
.markdown-body .danger {
border-left: 6px solid #c25f77;
}
.markdown-body .question {
border-left: 6px solid #96a6d7;
}
.markdown-body .note {
border-left: 6px solid #d7c896;
}
.markdown-body .admonition>*:first-child {
margin-top: 0 !important;
}
.markdown-body .admonition>*:last-child {
margin-bottom: 0 !important;
}
/* progress bar*/
.markdown-body .progress {
display: block;
width: 300px;
margin: 10px 0;
height: 24px;
-webkit-border-radius: 3px;
-moz-border-radius: 3px;
border-radius: 3px;
background-color: #ededed;
position: relative;
box-shadow: inset -1px 1px 3px rgba(0, 0, 0, .1);
}
.markdown-body .progress-label {
position: absolute;
text-align: center;
font-weight: bold;
width: 100%; margin: 0;
line-height: 24px;
color: #333;
text-shadow: 1px 1px 0 #fefefe, -1px -1px 0 #fefefe, -1px 1px 0 #fefefe, 1px -1px 0 #fefefe, 0 1px 0 #fefefe, 0 -1px 0 #fefefe, 1px 0 0 #fefefe, -1px 0 0 #fefefe, 1px 1px 2px #000;
-webkit-font-smoothing: antialiased !important;
white-space: nowrap;
overflow: hidden;
}
.markdown-body .progress-bar {
height: 24px;
float: left;
-webkit-border-radius: 3px;
-moz-border-radius: 3px;
border-radius: 3px;
background-color: #96c6d7;
box-shadow: inset 0 1px 0 rgba(255, 255, 255, .5), inset 0 -1px 0 rgba(0, 0, 0, .1);
background-size: 30px 30px;
background-image: -webkit-linear-gradient(
135deg, rgba(255, 255, 255, .4) 27%,
transparent 27%,
transparent 52%, rgba(255, 255, 255, .4) 52%,
rgba(255, 255, 255, .4) 77%,
transparent 77%, transparent
);
background-image: -moz-linear-gradient(
135deg,
rgba(255, 255, 255, .4) 27%, transparent 27%,
transparent 52%, rgba(255, 255, 255, .4) 52%,
rgba(255, 255, 255, .4) 77%, transparent 77%,
transparent
);
background-image: -ms-linear-gradient(
135deg,
rgba(255, 255, 255, .4) 27%, transparent 27%,
transparent 52%, rgba(255, 255, 255, .4) 52%,
rgba(255, 255, 255, .4) 77%, transparent 77%,
transparent
);
background-image: -o-linear-gradient(
135deg,
rgba(255, 255, 255, .4) 27%, transparent 27%,
transparent 52%, rgba(255, 255, 255, .4) 52%,
rgba(255, 255, 255, .4) 77%, transparent 77%,
transparent
);
background-image: linear-gradient(
135deg,
rgba(255, 255, 255, .4) 27%, transparent 27%,
transparent 52%, rgba(255, 255, 255, .4) 52%,
rgba(255, 255, 255, .4) 77%, transparent 77%,
transparent
);
}
.markdown-body .progress-100plus .progress-bar {
background-color: #a6d796;
}
.markdown-body .progress-80plus .progress-bar {
background-color: #c6d796;
}
.markdown-body .progress-60plus .progress-bar {
background-color: #d7c896;
}
.markdown-body .progress-40plus .progress-bar {
background-color: #d7a796;
}
.markdown-body .progress-20plus .progress-bar {
background-color: #d796a6;
}
.markdown-body .progress-0plus .progress-bar {
background-color: #c25f77;
}
.markdown-body .candystripe-animate .progress-bar{
-webkit-animation: animate-stripes 3s linear infinite;
-moz-animation: animate-stripes 3s linear infinite;
animation: animate-stripes 3s linear infinite;
}
@-webkit-keyframes animate-stripes {
0% {
background-position: 0 0;
}
100% {
background-position: 60px 0;
}
}
@-moz-keyframes animate-stripes {
0% {
background-position: 0 0;
}
100% {
background-position: 60px 0;
}
}
@keyframes animate-stripes {
0% {
background-position: 0 0;
}
100% {
background-position: 60px 0;
}
}
.markdown-body .gloss .progress-bar {
box-shadow:
inset 0 4px 12px rgba(255, 255, 255, .7),
inset 0 -12px 0 rgba(0, 0, 0, .05);
}
/* Multimarkdown Critic Blocks */
.markdown-body .critic_mark {
background: #ff0;
}
.markdown-body .critic_delete {
color: #c82829;
text-decoration: line-through;
}
.markdown-body .critic_insert {
color: #718c00 ;
text-decoration: underline;
}
.markdown-body .critic_comment {
color: #8e908c;
font-style: italic;
}
.markdown-body .headeranchor {
font: normal normal 16px octicons-anchor;
line-height: 1;
display: inline-block;
text-decoration: none;
-webkit-font-smoothing: antialiased;
-moz-osx-font-smoothing: grayscale;
-webkit-user-select: none;
-moz-user-select: none;
-ms-user-select: none;
user-select: none;
}
.headeranchor:before {
content: '\f05c';
}
.markdown-body .task-list-item {
list-style-type: none;
}
.markdown-body .task-list-item+.task-list-item {
margin-top: 3px;
}
.markdown-body .task-list-item input {
margin: 0 4px 0.25em -20px;
vertical-align: middle;
}
/* Media */
@media only screen and (min-width: 480px) {
.markdown-body {
font-size:14px;
}
}
@media only screen and (min-width: 768px) {
.markdown-body {
font-size:16px;
}
}
@media print {
.markdown-body * {
background: transparent !important;
color: black !important;
filter:none !important;
-ms-filter: none !important;
}
.markdown-body {
font-size:12pt;
max-width:100%;
outline:none;
border: 0;
}
.markdown-body a,
.markdown-body a:visited {
text-decoration: underline;
}
.markdown-body .headeranchor-link {
display: none;
}
.markdown-body a[href]:after {
content: " (" attr(href) ")";
}
.markdown-body abbr[title]:after {
content: " (" attr(title) ")";
}
.markdown-body .ir a:after,
.markdown-body a[href^="javascript:"]:after,
.markdown-body a[href^="#"]:after {
content: "";
}
.markdown-body pre {
white-space: pre;
white-space: pre-wrap;
word-wrap: break-word;
}
.markdown-body pre,
.markdown-body blockquote {
border: 1px solid #999;
padding-right: 1em;
page-break-inside: avoid;
}
.markdown-body .progress,
.markdown-body .progress-bar {
-moz-box-shadow: none;
-webkit-box-shadow: none;
box-shadow: none;
}
.markdown-body .progress {
border: 1px solid #ddd;
}
.markdown-body .progress-bar {
height: 22px;
border-right: 1px solid #ddd;
}
.markdown-body tr,
.markdown-body img {
page-break-inside: avoid;