A common fraction is in lowest terms if the numerator and denominator have no common factors other than $1$. A fraction in lowest terms may also be said to be in simplest form.
critical point: 临界点
定义: (càdlàg process)càdlàg process A càdlàg process $X$ is a stochastic process for which the paths $t\mapsto X_t$ are right-continuous with left limits everywhere, with probability one. The word càdlàg is an acronym from the French for “continu à droite, limites à gauche”.
定义: (extreme points)extreme points Let $X$ be a vector space over $\mathbb{R}$, $K$ be a convex subset of $X$. We say that $a$ is an extrem point of $K$ iff, whenever $a=tx+(1-t)y$ for $t\in (0,1)$, we have $x=y=a$.
定义: (scatter plot) a graph in which the values of two variables are plotted along two axes, the pattern of the resulting points revealing any correlation present.
定义: (Hessian Matrix)Hessian Matrix The Hessian matrix of a multivariable function $f(x,y,z,\cdots)$ organizes all second partial derivatives into a matrix: \[ \mathbf{H} f=\left[\begin{array}{cccc} \frac{\partial^{2} f}{\partial x^{2}} & \frac{\partial^{2} f}{\partial x \partial y} & \frac{\partial^{2} f}{\partial x \partial z} & \cdots \\ \frac{\partial^{2} f}{\partial y \partial x} & \frac{\partial^{2} f}{\partial y^{2}} & \frac{\partial^{2} f}{\partial y \partial z} & \cdots \\ \frac{\partial^{2} f}{\partial z \partial x} & \frac{\partial^{2} f}{\partial z \partial y} & \frac{\partial^{2} f}{\partial z^{2}} & \cdots \\ \vdots & \vdots & \vdots & \ddots \end{array}\right] \] other notations are $\mathbf{H}(f)$, $\mathbf{H}f$ or $\mathbf{H}_f$
定义: (simple group)simple group A group $G$ is simple iff it has only $G$ and the trivial group as normal subgroups.
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