{"id":2407,"date":"2011-10-24T04:18:46","date_gmt":"2011-10-23T20:18:46","guid":{"rendered":"http:\/\/1.vanabel.sinaapp.com\/?p=6"},"modified":"2011-10-24T04:18:46","modified_gmt":"2011-10-23T20:18:46","slug":"test-mathjax","status":"publish","type":"post","link":"https:\/\/lttt.vanabel.cn\/?p=2407","title":{"rendered":"Hello MathJax!!"},"content":{"rendered":"

The following equations are represented in the HTML source code as LaTeX expressions.<\/span>
\nThe Cauchy-Schwarz Inequality<\/span>
\n\\[\\left(
\n\\sum_{k=1}^n a_k b_k
\n\\right)^2
\n\\leq
\n\\left(
\n\\sum_{k=1}^n a_k^2
\n\\right)
\n\\left(
\n\\sum_{k=1}^n b_k^2
\n\\right)\\]
\nAuto Numbering and Ref? <\/span>
\nSuppose that \\(f\\) is a function with period $\\pi$, then we have<\/span>
\n$
\n\\newcommand{\\rd}{\\rm d}
\n$
\n\\begin{gather}\\label{eq:1}
\n\\int_t^{t+\\pi} f(x)\\rd x=\\int_0^\\pi f(x)\\rd x.\\tag{1}
\n\\end{gather}
\nFrom \\eqref{eq:1} we know that $\\dots$.<\/span>
\nCan we use \\newcommand to define some macro?<\/span>
\n$
\n\\newcommand{\\Re}{\\mathrm{Re}\\,}
\n\\newcommand{\\pFq}[5]{{}_{#1}\\mathrm{F}_{#2} \\left( \\genfrac{}{}{0pt}{}{#3}{#4} \\bigg| {#5} \\right)}
\n$
\ndefine newcommand as:<\/span><\/p>\n

$\\newcommand{\\Re}{\\mathrm{Re}\\,}\n\\newcommand{\\pFq}[5]{{}_{#1}\\mathrm{F}_{#2}\n  \\left(\n    \\genfrac{}{}{0pt}{}{#3}{#4} \\bigg| {#5}\n  \\right)}$<\/pre>\n
We consider, for various values of $s$, the $n$-dimensional integral
\n\\begin{align}\\label{def:Wns}
\nW_n (s):=
\n\\int_{[0, 1]^n}
\n\\left| \\sum_{k = 1}^n \\mathrm{e}^{2 \\pi \\mathrm{i} \\, x_k} \\right|^s \\mathrm{d}\\boldsymbol{x}
\n\\end{align}
\nwhich occurs in the theory of uniform random walk integrals in the plane,\u00a0where at each step a unit-step is taken in a random direction. As such, the integral \\eqref{def:Wns} expresses the $s$-th moment of the distance\u00a0to the origin after $n$ steps.<\/span>
\n
\nBy experimentation and some sketchy arguments we quickly conjectured and\u00a0strongly believed that, for $k$ a nonnegative integer<\/span>
\n\\begin{align}\\label{eq:W3k}
\nW_3(k)= \\Re \\, \\pFq32{\\frac12, -\\frac k2, -\\frac k2}{1, 1}{4}.
\n\\end{align}
\nAppropriately defined, \\eqref{eq:W3k} also holds for negative odd integers.\u00a0The reason for \\eqref{eq:W3k} was long a mystery, but it will be explained\u00a0at the end of the paper.<\/span>
\nHow about MathML?<\/span>
\nDefinition of Christoffel Symbols<\/p>\n
\n\n  \n    \n      \n        (<\/mo>\n        \n          \n            ∇<\/mo>\n          <\/mrow>\n          \n            X<\/mi>\n          <\/mrow>\n        <\/msub>\n        Y<\/mi>\n        )<\/mo>\n      <\/mrow>\n      \n        k<\/mi>\n      <\/mrow>\n    <\/msup>\n    =<\/mo>\n    \n      \n        X<\/mi>\n      <\/mrow>\n      \n        i<\/mi>\n      <\/mrow>\n    <\/msup>\n    \n      \n        (<\/mo>\n        \n          \n            ∇<\/mo>\n          <\/mrow>\n          \n            i<\/mi>\n          <\/mrow>\n        <\/msub>\n        Y<\/mi>\n        )<\/mo>\n      <\/mrow>\n      \n        k<\/mi>\n      <\/mrow>\n    <\/msup>\n    =<\/mo>\n    \n      \n        X<\/mi>\n      <\/mrow>\n      \n        i<\/mi>\n      <\/mrow>\n    <\/msup>\n    <\/mrow>\n      (<\/mo>\n      \n        \n          ∂<\/mo>\n          \n            \n              Y<\/mi>\n            <\/mrow>\n            \n              k<\/mi>\n            <\/mrow>\n          <\/msup>\n        <\/mrow>\n        \n          ∂<\/mo>\n          \n            \n              x<\/mi>\n            <\/mrow>\n            \n              i<\/mi>\n            <\/mrow>\n          <\/msup>\n        <\/mrow>\n      <\/mfrac>\n      +<\/mo>\n      \n        \n          Γ<\/mi>\n        <\/mrow>\n        \n          i<\/mi>\n          m<\/mi>\n        <\/mrow>\n        \n          k<\/mi>\n        <\/mrow>\n      <\/msubsup>\n      \n        \n          Y<\/mi>\n        <\/mrow>\n        \n          m<\/mi>\n        <\/mrow>\n      <\/msup>\n      )<\/mo>\n    <\/mrow>\n  \n<\/math>\n<\/pre>\n
(<\/mo>\u2207<\/mo><\/mrow>X<\/mi><\/mrow><\/msub>Y<\/mi>)<\/mo><\/mrow>k<\/mi><\/mrow><\/msup>=<\/mo>X<\/mi><\/mrow>i<\/mi><\/mrow><\/msup>(<\/mo>\u2207<\/mo><\/mrow>i<\/mi><\/mrow><\/msub>Y<\/mi>)<\/mo><\/mrow>k<\/mi><\/mrow><\/msup>=<\/mo>X<\/mi><\/mrow>i<\/mi><\/mrow><\/msup><\/mrow>(<\/mo>\u2202<\/mo>Y<\/mi><\/mrow>k<\/mi><\/mrow><\/msup><\/mrow>\u2202<\/mo>x<\/mi><\/mrow>i<\/mi><\/mrow><\/msup><\/mrow><\/mfrac>+<\/mo>\u0393<\/mi><\/mrow>i<\/mi>m<\/mi><\/mrow>k<\/mi><\/mrow><\/msubsup>Y<\/mi><\/mrow>m<\/mi><\/mrow><\/msup>)<\/mo><\/mrow><\/math><\/p>\n","protected":false},"excerpt":{"rendered":"
The following equations are represented  …<\/p>\n
  Hello MathJax!!<\/span> \u9605\u8bfb\u66f4\u591a »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5],"tags":[532,549,645],"class_list":["post-2407","post","type-post","status-publish","format-standard","hentry","category-admin","tag-latex-2","tag-mathjax","tag-tex-in-web"],"_links":{"self":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/2407","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=2407"}],"version-history":[{"count":0,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/2407\/revisions"}],"wp:attachment":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2407"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2407"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2407"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}