这是该问题的 MWE。
\documentclass[a4paper, 12pt]{article}
\usepackage{amsfonts, amssymb, amsmath, amsthm}
\usepackage[]{fontspec}
\setmainfont{ModernMT-Extended.otf}[
FakeBold=1,
SmallCapsFont=MrsEavesSmallCaps_Regular.ttf,
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ItalicFont=ModernMT-ExtendedItalic.otf,
BoldItalicFont=ModernMTStd-BoldItalic.otf,
Ligatures=TeX,
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\begin{document}
Let $\mathbf{x}$ and $\mathbf{y}$ be two vectors of $n$ real numbers. We say that $\mathbf{x}$ \textsc{dominates} or \textit{majorizes} $y$ if
\begin{align*}
x_{1}
& \geq x_{2}\geq \ldots\geq x_{n}\\
y_{1}
& \geq y_{2}\geq \ldots\geq y_{n}\\
x_{1}+\ldots+x_{n}
& = y_{1}+\ldots+y_{n}\\
x_{1}+\ldots+x_{k}
& \geq y_{1}+\ldots+y_{k}
\end{align*}
for $1\leq k\leq n-1$. If $\mathbf{x}$ dominates $\mathbf{y}$ (resp. $\mathbf{y}$ is \textit{dominated by} $\mathbf{x}$), then we write $\mathbf{x}\succ\mathbf{y}$ (resp. $\mathbf{y}\prec\mathbf{x}$). For example, $(4,0,0)\succ(3,1,0)\succ(2,2,0)$. The vectors $\mathbf{x}$ and $\mathbf{y}$ need not be monotonic because we can just sort them into monotonic vectors.
We will also introduce the cyclic and symmetric polynomials and notations juxtaposed with them. The expression $x^{2}+y^{2}+z^{2}$ is \textit{symmetric} whereas $x^{2}y+y^{2}z+z^{2}x$ is \textit{cyclic} but not symmetric because $y^{2}x+z^{2}y+x^{2}z\neq x^{2}y+y^{2}z+z^{2}x$. A symmetric polynomial in the variables $x_{1},\ldots,x_{n}$ should remain same regardless of the order in which the variables are used. So $f(x_{1},\ldots,x_{n})$ is symmetric if $f$ remains \textit{invariant} for all permutations of $x_{1},\ldots,x_{n}$ in the expression unlike the cyclic example we just saw. For example, $xy+yz+zx$ is symmetric and so is $xyz$. But $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}$ is cyclic but not symmetric. We can use the cyclic and symmetric notations to represent the expressions in a short form. Here are some demonstrations.
\begin{align*}
a^{2}+b^{2}+c^{2}
& = \sum_{cyc}a^{2}\\
a^{2}b+b^{2}c+c^{2}a
& = \sum_{cyc}a^{2}b\\
xy+yz+zx
& = \sum_{cyc}xy
\end{align*}
Note that even though the expression $xy+yz+zx$ and $a^{2}+b^{2}+c^{2}$ are symmetric, we do not consider them symmetric polynomial sums in this notation. A symmetric polynomial sum should have all $n!$ terms in the sum since it is symmetric on all $n!$ permutations of the variables in it. Even if there can be duplicates, the total number of terms should still remain $n!$. For this reason, this sum is often denoted by $\sum{!}$. Here are some examples.
\end{document}
指定的字体放在同一目录中。
答案1
您需要使用匹配的 OpenType 数学字体并指定相同的粗体，我没有您使用的字体，但之前和之后都是使用 Latin Modern
\documentclass[a4paper, 12pt]{article}
\usepackage{amsfonts, amssymb, amsmath, amsthm}
\usepackage[]{fontspec}
\setmainfont{lmroman10-regular.otf}[
% FakeBold=3,
ItalicFont=lmroman10-italic.otf,
BoldFont=lmroman10-bold.otf,
Ligatures=TeX,
]
\usepackage{unicode-math}
\setmathfont{latinmodern-math.otf}[
%FakeBold=3
]
\begin{document}
Let $\mathbf{x}$ and $\mathbf{y}$ be two vectors of $n$ real numbers. We say that $\mathbf{x}$ \textsc{dominates} or \textit{majorizes} $y$ if
\begin{align*}
x_{1}
& \geq x_{2}\geq \ldots\geq x_{n}\\
y_{1}
& \geq y_{2}\geq \ldots\geq y_{n}\\
x_{1}+\ldots+x_{n}
& = y_{1}+\ldots+y_{n}\\
x_{1}+\ldots+x_{k}
& \geq y_{1}+\ldots+y_{k}
\end{align*}
for $1\leq k\leq n-1$. If $\mathbf{x}$ dominates $\mathbf{y}$ (resp. $\mathbf{y}$ is \textit{dominated by} $\mathbf{x}$), then we write $\mathbf{x}\succ\mathbf{y}$ (resp. $\mathbf{y}\prec\mathbf{x}$). For example, $(4,0,0)\succ(3,1,0)\succ(2,2,0)$. The vectors $\mathbf{x}$ and $\mathbf{y}$ need not be monotonic because we can just sort them into monotonic vectors.
We will also introduce the cyclic and symmetric polynomials and notations juxtaposed with them. The expression $x^{2}+y^{2}+z^{2}$ is \textit{symmetric} whereas $x^{2}y+y^{2}z+z^{2}x$ is \textit{cyclic} but not symmetric because $y^{2}x+z^{2}y+x^{2}z\neq x^{2}y+y^{2}z+z^{2}x$. A symmetric polynomial in the variables $x_{1},\ldots,x_{n}$ should remain same regardless of the order in which the variables are used. So $f(x_{1},\ldots,x_{n})$ is symmetric if $f$ remains \textit{invariant} for all permutations of $x_{1},\ldots,x_{n}$ in the expression unlike the cyclic example we just saw. For example, $xy+yz+zx$ is symmetric and so is $xyz$. But $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}$ is cyclic but not symmetric. We can use the cyclic and symmetric notations to represent the expressions in a short form. Here are some demonstrations.
\begin{align*}
a^{2}+b^{2}+c^{2}
& = \sum_{cyc}a^{2}\\
a^{2}b+b^{2}c+c^{2}a
& = \sum_{cyc}a^{2}b\\
xy+yz+zx
& = \sum_{cyc}xy
\end{align*}
Note that even though the expression $xy+yz+zx$ and $a^{2}+b^{2}+c^{2}$ are symmetric, we do not consider them symmetric polynomial sums in this notation. A symmetric polynomial sum should have all $n!$ terms in the sum since it is symmetric on all $n!$ permutations of the variables in it. Even if there can be duplicates, the total number of terms should still remain $n!$. For this reason, this sum is often denoted by $\sum{!}$. Here are some examples.
\end{document}