$$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Self-defined math definitions %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Math symbol commands \newcommand{\intd}{\,\mathrm{d}} % Symbol 'd' used in integration, such as 'dx' \newcommand{\diff}{\,\mathrm{d}} % Symbol 'd' used in differentiation \newcommand{\grad}{\nabla} % Gradient operator \newcommand{\laplacian}{\nabla^2} % Laplacian operator \newcommand{\partiald}[2]{\frac{\partial #1}{\partial #2}} % Partial derivative \newcommand{\dd}[2]{\frac{\mathrm{d}#1}{\mathrm{d}#2}} % Total derivative % Common functions \newcommand{\expf}{\mathrm{e}^{}} % Exponential function (e.g., expf(x)) \newcommand{\logf}{\ln} % Natural logarithm \newcommand{\sinc}{\mathrm{sinc}} % Sinc function % Neural network notation \newcommand{\h}{\mathbf{h}} % Hidden state vector \newcommand{\x}{\mathbf{x}} % Input vector \newcommand{\W}{\mathbf{W}} % Weight matrix \newcommand{\U}{\mathbf{U}} % Input weight matrix \newcommand{\b}{\mathbf{b}} % Bias vector % Other useful macros \newcommand{\eqdef}{\triangleq} % Define equality \newcommand{\approxdef}{\overset{\text{def}}{=}} % Approximate definition \newcommand{\trans}{^\top} % Transpose \newcommand{\inv}{^{-1}} % Inverse \newcommand{\norm}[1]{\left\lVert#1\right\rVert} % Norm \newcommand{\abs}[1]{\left|#1\right|} % Absolute value \newcommand{\floor}[1]{\left\lfloor#1\right\rfloor} % Floor \newcommand{\ceil}[1]{\left\lceil#1\right\rceil} % Ceiling %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % End of self-defined math definitions %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% $$

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