(************** Content-type: application/mathematica **************
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Cell[BoxData[{
\(Clear[k, x, y, Bnd, Cff, Poly, Polys, PrmeCalc,
ConstCalc]\[IndentingNewLine]\[IndentingNewLine] (*\
Bnd : \ Input\ a\ list\ of\ variables\ {\ a_
a_ {k}\ }, \
and\ an\ integer\ tot . \
Gives\ a\ list\ of\ ranges\ for\ summing\ the\ variables\ in\ the\
\ list\ subject\ to\ each\ variable\ being\
\[GreaterEqual]
1\ and\ the\ sum\ of\ the\ variables\ being\
tot\ *) \ \), "\[IndentingNewLine]",
\(\(Bnd[lst_, tot_] :=
Module[{i1, i2, i3, i4, i5},
Table[{lst[\([i1]\)], 1,
Sum[lst[\([i2]\)], {i2, 1, i1 - 1}] - \((Length[lst] -
i1)\)}, {i1, 1, Length[lst]}]\[IndentingNewLine] (*\
Since\ each\ variable\ is\
\[GreaterEqual] 1, \
the\ lower\ bound\ is\ 1. \ If\ we\ alread\ know\ the\ values\ of\
a_ {m - 1}\ then\ since\ sum_ {i = 1}^{k}\ a_i\
\[LessEqual] \
tot - 1\ we\ have\[IndentingNewLine]a_m\
\[LessEqual] \
sum_ {i = 1}^k\ a_i\
- \ sum_ {i = 1}^{m - 1}\ a_i\
sum_ {i = m + 1}^k\ a_i\ \[IndentingNewLine] \[LessEqual] \
- \ sum_ {i = 1}^k\ a_i\
sum_ {i = m + 1}^k\ 1\[IndentingNewLine] \[LessEqual] \
- \ sum_ {i = 1}^ka_i\
- \ \((k - m)\), \
which\ is\ our\ upper\ bound*) \[IndentingNewLine]];\)\
\[IndentingNewLine]\[IndentingNewLine] (*\
Cff : \ Input\ a\ list\ of\ variables\ {\ a_
a_ {k}\ }, \
and\ an\ integer\ tot . \ Gives\ the\ summand\ \(tot!\)\ *\
prod_ {i = 1}^{k + 1}\ \(\((2
a_i)\)!\)/\(a_i!\)\ with\ a_ {k +
sum_ {i = 1}^{k}\ a_i\ *) \), "\[IndentingNewLine]",
\(\(Cff[lst_, tot_] :=
Module[{i1, i2, i3},
Product[Factorial[2*lst[\([i1]\)]]/Factorial[lst[\([i1]\)]], {i1,
1, Length[lst]}]*Factorial[tot]*
Factorial[2*tot - 2*Sum[lst[\([i2]\)], {i2, 1, Length[lst]}]]/
Factorial[
tot - Sum[
lst[\([i3]\)], {i3, 1,
Length[lst]}]]];\)\[IndentingNewLine]\[IndentingNewLine]\
(*\ Poly : \
Input\ an\ integer\ n <
40\ and\ a\ variable\ k . \ Gives\ the\ n^{th}\ polynomial\ G_ {n,
2} \((k)\)\ in\ the\ variable\ k\ *) \), "\[IndentingNewLine]", \
\(\(Poly[n_, k_] :=
Module[{A = {a1, a2, a3, a4, a5, a6, a7, a8, a9, a10, a11, a12, a13,
a14, a15, a16, a17, a18, a19, a20, a21, a22, a23, a24, a25,
a26, a27, a28, a29, a30, a31, a32, a33, a34, a35, a36, a37,
a38, a39, a40}, X = k*Factorial[2*n], i,
Lst}, \[IndentingNewLine]If[n \[Equal] 0, X = 1,
For[i = 1, i \[LessEqual] n - 1, \(i++\), Lst = Take[A, i];
X = X + Binomial[k, i + 1]*
Apply[Sum,
Join[{Cff[Lst, n]},
Bnd[Lst, n]]];]; \[IndentingNewLine] (*\
Use\ our\ formula\ to\ write\ the\ polynomial\ as\ a\ sum\ of\ \
binomial\ coefficients\ in\ k\ multiplied\ by\ some\ constants, \
which\ we\ calculate\ by\ summing\ Cff\ over\ the\ range\ Bnd\ \
*) \[IndentingNewLine]Expand[
X]]];\)\[IndentingNewLine]\[IndentingNewLine] (*\
It\ is\ convenient\ to\ precompute\ the\ polynomials\ Poly[n,
k]\ we\ will\ use\ to\ speed\ up\ calculation\ *) \), "\
\[IndentingNewLine]",
\(\(Polys[k_] =
Table[Poly[n, k], {n, 0,
11}];\)\[IndentingNewLine]\[IndentingNewLine] (*\
ConstCalc : \ Input\ a\ polynomial\ expression\ Expr\ in\ x\ and\ y, \
and\ an\ integer\ k . \ x\ represents\ 1 -
1 = \(1 - \(\(\\\)\(sum_\)\) {i = 1}^
k\ t_i\ and\ y\ represents\ P_
2 = \(\(\\\)\(sum_\)\) {i = 1}^k\ t_i^2\[IndentingNewLine]
Output\ is\ the\ result\ of\ integrating\ Expr^2\ with\ respect\ to\ \
1\), \(\(\\\)\(dots\)\)\ t_k\ over\ the\ region\ \
\[IndentingNewLine]\\mathcal {R} _k\
= \ \(\(\\\)\({\ \((t_
1, \(\(\\\)\(dots\)\),
t_k)\)\ \\in\ [0,
: \ \(\(\\\)\(sum_\)\) {i = 1}^
< \(\(1\)\(\\\)\)\)} . \
Assumes\)\)\ that\ the\ relevent\ Polynomials\ G_ {c,
2}\ have\ been\ precomputed\ in\ the\ list\ Polys\ *) \), "\
\[IndentingNewLine]",
\(ConstCalc[Expr_, k_] :=
Module[{Sum = 0, IntExpr = Expand[Expr^2], xdeg, ydeg, b, c, tmp,
Coeff}, \[IndentingNewLine] (*we\ are\ integrating\ IntExpr, \
which\ is\ Expr\ squared*) \[IndentingNewLine]xdeg =
Exponent[IntExpr,
x]; \[IndentingNewLine] (*we\ let\ xdeg\ be\ the\ maximum\ degree\
\ of\ IntExpr\ with\ respect\ to\ x*) \[IndentingNewLine] (*we\ now\ split\ \
our\ expression\ IntExpr\ into\ monomial\ terms\ x^b*
y^c*) \[IndentingNewLine]For[b = 0, b \[LessEqual] xdeg, \(b++\),
tmp = Coefficient[IntExpr, x, b]; ydeg = Exponent[tmp, y];
For[c = 0, c \[LessEqual] ydeg, \(c++\),
Coeff = Coefficient[tmp, y,
c]; \[IndentingNewLine] (*Coeff\ is\ the\ coefficient\ of\ x^
b*y^c\ in\ IntExpr*) Sum =
Sum + Coeff*
Factorial[b]/
Factorial[k + b + 2*c]*\(Polys[
k]\)[\([c +
1]\)];];]; \[IndentingNewLine] (*we\ use\ our\ \
formula\ from\ Lemma\ 7.1\ to\ count\ the\ contribution\ from\ this\ term\
*) \[IndentingNewLine]Sum]\[IndentingNewLine]\[IndentingNewLine] (*PrimeCalc \
: \ Input\ a\ polynomial\ expression\ Expr\ in\ x\ and\ y\ and\ an\ integer\ \
k . \ \[IndentingNewLine]x\ represents\ 1 -
1 = \(1 - \(\(\\\)\(sum_\)\) {i = 1}^
k\ t_i\ and\ y\ represents\ P_
2 = \(\(\\\)\(sum_\)\) {i = 1}^k\ t_i^2\ \[IndentingNewLine]
Output\ is\ a\ polynomial\ expression\ which\ is\ the\ result\ of\ \
integrating\ Expr\ with\ resepect\ to\ t1\ between\ 0\ and\ 1 - \
\(\(\\\)\(sum_\)\) {i = 2}^k\ t_i . \
The\ polynomial\ expression\ outputted\ is\ given\ in\ terms\ \
of\ x\ and\ y\), \
where\ x\ represents\ 1 - P_
1'\ and\ y\ represents\ P_
where\ P_j' = \(\(\\\)\(sum_\)\) {i = 2}^\(kt_i^j\) . \
Calls\ constcalc, \
and\ so\ assumes\ the\ relevent\ polynomials\ have\ been\ precomputed\
\[IndentingNewLine]*) \), "\[IndentingNewLine]",
\(\(PrmeCalc[Expr_, k_] :=
Module[{NewExpr = 0, FirstExpr = Expand[Expr], n1, n2, b, c, cp, tmp,
Coeff}, \[IndentingNewLine]n1 =
Exponent[Expr, x]; \[IndentingNewLine] (*\
We\ let\ n1\ be\ the\ maximum\ degree\ of\ the\ polynomial\ with\ \
respect\ to\ x\ *) \[IndentingNewLine] (*\
We\ now\ wish\ to\ split\ our\ polynomial\ into\ terms\ of\ the\ \
form\ x^b\ y^c\ *) \[IndentingNewLine]For[b = 0, b \[LessEqual] n1, \(b++\),
tmp = Coefficient[FirstExpr, x, b]; \[IndentingNewLine] (*\
We\ extract\ the\ part\ of\ the\ polynomial\ which\ has\ degree\
\ exactly\ b\ with\ respect\ to\ x\ *) \[IndentingNewLine]n2 =
Exponent[tmp, y]; \[IndentingNewLine] (*\
We\ let\ n2\ be\ the\ maximum\ degree\ of\ this\ part\ of\ the\ \
polynomial\ with\ respect\ to\ y\ *) \[IndentingNewLine]For[c = 0,
c \[LessEqual] n2, \(c++\), \[IndentingNewLine]Coeff =
Coefficient[tmp, y, c]; \[IndentingNewLine] (*\
Coeff\ is\ the\ coefficient\ of\ x^b\ y^
c\ in\ Expr\ *) \[IndentingNewLine]NewExpr =
Coeff*Sum[
x^\((b + 2
cp + 1)\)*y^\((cp)\)*Binomial[c, cp]*
Factorial[b]*
Factorial[2*c - 2*cp]/
Factorial[b + 2*c - 2*cp + 1], {cp, 0,
c}]; \[IndentingNewLine] (*\
We\ use\ our\ formula\ from\ equation\ 7.8\ to\ produce\ the\ \
polynomial\ which\ we\ get\ after\ integrating\ with\ respect\ to\ t1\ \
*) \[IndentingNewLine] (*\
NewExpr\ is\ the\ sum\ of\ all\ of\ these\ polynomials, \
and\ so\ the\ result\ of\ integrating\ Expr\ with\ respect\ \
to\ t1 . \ Here\ x\ now\ refers\ to\ 1 -
1'\ and\ y\ to\ P_
2'\ *) \[IndentingNewLine]];\[IndentingNewLine]]; \
\[IndentingNewLine]ConstCalc[NewExpr, k - 1]\[IndentingNewLine] (*\
We\ now\ intgrate\ our\ polynomial\ squared\ over\ $\\mathcal {R} \
_ {k - 1} $\ using\ our\ function\ defined\ above\ *) \[IndentingNewLine]];\)\
\[IndentingNewLine]\[IndentingNewLine] (*\ p[n] : \ Input\ an\ integer\ n, \
output\ a\ polynomial\ which\ is\ a\ linear\ combination\ of\ all\ \
monomials\ x^b\ *\ y^c\ with\ b + 2
c \[LessEqual]
n + 1. \ The\ coefficients\ are\ given\ by\ A[\([1]\)],
... \ *) \
\), "\[IndentingNewLine]",
\(\(yExponents[n_] :=
Module[{S = {}, i, tmp},
For[i = 0, i \[LessEqual] n, \(i++\),
tmp = Append[Reverse[Range[i]], 0]; \[IndentingNewLine]S =
Join[S, tmp,
tmp]; ]; \[IndentingNewLine]S];\)\), "\[IndentingNewLine]",
\(\(xExponents[n_] :=
Module[{S = {}, i, tmp},
For[i = 1, i \[LessEqual] n + 1, \(i++\),
tmp = 2*Range[i]; \[IndentingNewLine]S =
Join[S, tmp - 2,
tmp - 1]; ]; \[IndentingNewLine]S];\)\), \
"\[IndentingNewLine]",
\(\(p[n_] :=
Module[{X = xExponents[n], Y = yExponents[n], i},
Sum[A[\([\)\(i\)\(]\)]*x^X[\([\)\(i\)\(]\)]*
y^Y[\([\)\(i\)\(]\)], {i, 1,
Length[X]}]];\)\[IndentingNewLine]\[IndentingNewLine] (*\
We\ now\ perform\ the\ actual\ computation\ for\ k =
105\ to\ calculate\ a\ lower\ bound\ for\ M_ {105}\ *) \), "\
\[IndentingNewLine]",
\(\(k = 105;\)\), "\[IndentingNewLine]",
\(poly = p[5]\), "\[IndentingNewLine]",
\(\(vars =
DeleteCases[DeleteCases[Variables[poly], x],
y];\)\[IndentingNewLine] (*\
take\ our\ polynomial\ to\ be\ all\ monomial\ combination\ \((1 -
1)\)^b\ *\ P_
2^c\ with\ b + 2
c \[LessEqual]
11. \ Let\ vars\ be\ a\ vector\ of\ the\ coefficients\ \((A[\([1]\)]\ \
A[\([42]\)])\)\ in\ some\ order\ *) \), "\[IndentingNewLine]",
\(\(M1 = \(CoefficientArrays[ConstCalc[poly, k], vars,
Symmetric \[Rule] True]\)[\([3]\)];\)\), "\[IndentingNewLine]",
\(\(M2 = \(CoefficientArrays[PrmeCalc[poly, k], vars,
Symmetric \[Rule] True]\)[\([3]\)];\)\[IndentingNewLine] (*\
Calculate\ the\ positive\ definite\ real\ symmetric\ matrices\ M1\ and\ \
M2\ corresponding\ to\ the\ I_k\ and\ J_k\ integrals\ *) \), "\
\[IndentingNewLine]",
\(\(M3 = Inverse[M1] . M2;\)\), "\[IndentingNewLine]",
\(\(RatVec =
Rationalize[\(Eigenvectors[N[M3, 150], 1]\)[\([1]\)],
10^\((\(-40\))\)];\)\[IndentingNewLine] (*\
Using\ inexact\ arithmetic, \
calculate\ the\ eigenvector\ which\ maximizes\ the\ Rayleigh\ quotient \
. \ Then\ let\ RatVec\ be\ a\ close\ rational\ approximation\ to\ this\ *) \
\), "\[IndentingNewLine]",
\(\(Ratio =
k*\((RatVec . M2 . RatVec)\)/\((RatVec . M1 .
RatVec)\);\)\[IndentingNewLine] (*\
Calculate\ the\ value\ of\ the\ ratio\ using\ exact\ arithmetic\ for\ \
this\ choice\ of\ rational\ coefficients\ *) \), "\[IndentingNewLine]",
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End of Mathematica Notebook file.
*******************************************************************)Mathematica是一个世界知名的主流数学软件。Wolfram公司的调查显示，它的用户主要分布在工程、物理学、数学、计算机科学、医药化学、航空航天等领域.。 许多人都知道，它不仅能做高精度的数值计算，还有独树一帜的符号计算功能（在四大主流数学软件中，其余三家的符号计算功能都源出于Maple一家）。能为各种数学表达式绘制数学图形，还有独特的帧帧动画基本功能。是一个完美的科技工作平台。 但是，许多人也许还不了解，Mathematica的Notebook界面同时还是一个十分完善的科技文献写作环境。
它能方便地写出传统的2D数学表达式，具有足够的文本编辑排版功能，可以生成 “多媒并茂”的科技文档（把文本、公式、图形、动画、声音集于一身）。特别可爱的是，它能十分流畅的几乎是完美无缺的处理简体中文和繁体中文，无需经过“汉化”。它甚至可以进行任何一种语言的文字处理，只要配上相应的录入手段。Mathematica的Notebook还能把它所生成的文档转换格式，输送到其它软件环境中去使用。 仅就文字处理功能而言，它并不逊色于大名鼎鼎的Mord。如果再注意到它主营业务的那些看家本事（数值计算、符号计算、绘制图形、制播动画），你该想象得出来，它可以生成多么美妙的科技文献。你肯定知道，这样的科技文献会有多么大的使用范围：讲课使用的演示课件；学术报告的讲演稿；展览会上的展播解说；呈交上司的项目总结；合作者之间交流学术见解的通信文书，??。
对于科技人员来说，与一般的通用文字处理软件相比，它的优点真是太多了。由此有人给他送了一个外号：“Math-Word”或“Sciense-Word”！
如果你对Mathematica已不陌生，用它做过计算或绘图，现在你可以通过本书掌握它的科技文档处理功能；如果你还没有熟悉它的计算绘图功能，也可以先读这本书掌握它的文字处理功能，这也并不妨碍你回头再去熟悉它的计算绘图功能。 本书内容是，作者多年跟踪使用Mathematica的心得总结，内容独特新鲜。
第1章 Notebook的基本文字处理功能
1-1. Mathematica的界面
高版本的Mathematica有三种可做人机交互的前端界面。
1-1-1.Notebook界面
Notebook最常用，所以被称为主界面。它的调用命令是[开始|程序|Mathematica5| Mathematica5]或[C：\Program Files\ Wolfram Research \Mathematica \5.0\Mathmatica]。 软件安装之后，此界面在目录中的对应文件名和图标是
。执行此文件所调出的窗口界面如图1-1。
图1- 1 高版本Mathematica的前端界面(Notebook)
1-1-2.Text based界面
这种界面的调用是命令：[开始|程序|Mathematica5| Mathematica5 Kernel]或[C：\Program Files\Wolfram Research\Mathematica\ 5.0\ Math Kernel]。
在目录中的文件名和图标是：
。所调出的窗口界面如图1-2。
Mathematica的Text based界面
1-1-3.DOS界面
调用命令：[C：\Program Files\Wolfram Research\Mathematica\5.0\Math]。
在目录中的文件名和图标是：
。窗口界面如图1-3。
Mathematica的DOS界面
三种界面之中，NB（即NoteBook）最新，也最便于使用，曾有“科学计算草纸”之美称。它具有的完善的科技文档处理功能，又被称作“Math-Word”或 “Sciense-Word”。本书主要内容就是介绍这种Notebook界面的科技文档处理功能。以后，如无特殊说明，一律使用这种NB（Notebook）。
NoteBook的结构和本质
从属性来说，NoteBook既是一个计算、绘图、编程的工作平台，又是一个完美的文档处理器，能制作出图文式并茂的科技文档，保存起来就是一个文件。
NoteBook将文字处理、数学计算、图形绘制、动画制作多种功能集于一个环境，显然它的任务相当复杂而繁重。为使软件系统顺利辨识不同的任务，设计者采用的处理策略是使用一种叫做Cells的结构设计。
于是，Notebook就成了若干Cell的集合。也就是说，NB文档的所有内容都分置于不同的Cell当中，而NB也就没有不属于任何Cell的内容。
在Notebook当中，每个Cell负有不同的使命，可分为性质迥然有别的两大类： 一类，是程序类的Cell，负责接受并执行计算、推演、绘图任务,而且要返回结果； 另一类，是文本类的Cells，负责处理文字（接受录入信息，并作编辑排版然后保存起来）。文本类的Cell，还可以按着排版格式再加以细分，成为各种各样的文本类Cell。这就涉及到后面要讲的“Cell的样式”（即CellStyle）。CellStyle里面，含有两种信息:：
(1) 表明Cell的类属的信息（是属于文本类还是程序类）；
(2) 本Cell中使用了哪些排版命令。
从直观上说，Notebook就是文档，Cell就是段落。但从本质上说，Notebook和Cell都是由命令生成的“窗口区域图形”，因而必有其相应的生成命令（语句表达式）。 有关Notebook表达式和Cell表达式的知识，一般的初级使用者可以不必深究。高中级用户可以在本讲义的第2章中得到一些比较系统的知识。
2. 对Cells的几种常用操作
本段所讲各种操作，对于各种各类的Cell大都适用。
在对Cell进行种种组合操作之前，要保证将要被操作的Cell处于“可手工组合”的状态。保证这一状态的菜单命令是 [Cell | Cell Grouping | Manual Grouping ]。
2-1. 认识Cell的记号(标记符，Bracket)
在每个Cell右端，通常都会出现一条竖直的线段，上下两端带着不同形状的小钩。它就是这个Cell的标记符（Bracket）。
Cell标记符的主要作用是：1.标记本Cell的范围，2.在标记符上表示出Cell的类别。 在默认状态下，Cell标记符的形状显示为“后半个方括号(Bracket)”。 最常见也是使用最方便的形状是：文本类Cell是一种统一标记符，上头是双横线；程序类Cell标记符的标准（Standard）形状，是上头有空心的三角形，像是一面小旗。
在非默认状态下，程序类Cell标记符会产生形状变化，但这变化是由相关属性设置产生的，无需人工干预。