$$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Self-defined math definitions
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Math symbol commands
\newcommand{\intd}{\,{\rm d}} % Symbol 'd' used in integration, such as 'dx'
\newcommand{\diff}{{\rm d}} % Symbol 'd' used in differentiation
\newcommand{\Diff}{{\rm D}} % Symbol 'D' used in differentiation
\newcommand{\pdiff}{\partial} % Partial derivative
\newcommand{DD}[2]{\frac{\diff}{\diff #2}\left( #1 \right)}
\newcommand{Dd}[2]{\frac{\diff #1}{\diff #2}}
\newcommand{PD}[2]{\frac{\pdiff}{\pdiff #2}\left( #1 \right)}
\newcommand{Pd}[2]{\frac{\pdiff #1}{\pdiff #2}}
\newcommand{\rme}{{\rm e}} % Exponential e
\newcommand{\rmi}{{\rm i}} % Imaginary unit i
\newcommand{\rmj}{{\rm j}} % Imaginary unit j
\newcommand{\vect}[1]{\boldsymbol{#1}} % Vector typeset in bold and italic
\newcommand{\phs}[1]{\dot{#1}} % Scalar phasor
\newcommand{\phsvect}[1]{\boldsymbol{\dot{#1}}} % Vector phasor
\newcommand{\normvect}{\vect{n}} % Normal vector: n
\newcommand{\dform}[1]{\overset{\rightharpoonup}{\boldsymbol{#1}}} % Vector for differential form
\newcommand{\cochain}[1]{\overset{\rightharpoonup}{#1}} % Vector for cochain
\newcommand{\bigabs}[1]{\bigg\lvert#1\bigg\rvert} % Absolute value (single big vertical bar)
\newcommand{\Abs}[1]{\big\lvert#1\big\rvert} % Absolute value (single big vertical bar)
\newcommand{\abs}[1]{\lvert#1\rvert} % Absolute value (single vertical bar)
\newcommand{\bignorm}[1]{\bigg\lVert#1\bigg\rVert} % Norm (double big vertical bar)
\newcommand{\Norm}[1]{\big\lVert#1\big\rVert} % Norm (double big vertical bar)
\newcommand{\norm}[1]{\lVert#1\rVert} % Norm (double vertical bar)
\newcommand{\ouset}[3]{\overset{#3}{\underset{#2}{#1}}} % over and under set
% Super/subscript for column index of a matrix, which is used in tensor analysis.
\newcommand{\cscript}[1]{\;\; #1}
% Star symbol used as prefix in front of a paragraph with no indent
\newcommand{\prefstar}{\noindent$\ast$ }
% Big vertical line restricting the function.
% Example: $u(x)\restrict_{\Omega_0}$
\newcommand{\restrict}{\big\vert}
% Math operators which are typeset in Roman font
\DeclareMathOperator{\sgn}{sgn} % Sign function
\DeclareMathOperator{\erf}{erf} % Error function
\DeclareMathOperator{\Bd}{Bd} % Boundary of a set, used in topology
\DeclareMathOperator{\Int}{Int} % Interior of a set, used in topology
\DeclareMathOperator{\rank}{rank} % Rank of a matrix
\DeclareMathOperator{\divergence}{div} % Curl
\DeclareMathOperator{\curl}{curl} % Curl
\DeclareMathOperator{\grad}{grad} % Gradient
\DeclareMathOperator{\tr}{tr} % Trace
\DeclareMathOperator{\span}{span} % Span
$$
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Let \(M\) be regular (i.e. invertible, as defined in Hackbusch's book [Hie]) and \(M = \left( \begin{array}{cc} M_{11} & M_{12}\\ M_{21} & M_{22} \end 阅读全文
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Profiling is an indispensable measure for analyzing and optimizing the performance of your program. A typical profiler like GNU gprof will complete th 阅读全文
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Theorem (Baire) Let $X$ be a complete metric space and $\{O_k\}_{k \geq 1}$ a countable collection of dense subsets of $X$. Then $\bigcap_{k=1}^{\inft 阅读全文
