# on a point set

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##### 1: 2.1 Definitions and Elementary Properties
βΊLet $\mathbf{X}$ be a point set with a limit point $c$. … βΊ
2.1.16 $f(x)\sim a_{0}+a_{1}(x-c)+a_{2}(x-c)^{2}+\cdots,$ $x\to c$ in $\mathbf{X}$,
βΊIf the set $\mathbf{X}$ in §2.1(iii) is a closed sector $\alpha\leq\operatorname{ph}x\leq\beta$, then by definition the asymptotic property (2.1.13) holds uniformly with respect to $\operatorname{ph}x\in[\alpha,\beta]$ as $|x|\to\infty$. …Suppose $u$ is a parameter (or set of parameters) ranging over a point set (or sets) $\mathbf{U}$, and for each nonnegative integer $n$
##### 2: 1.5 Calculus of Two or More Variables
βΊA function is continuous on a point set $D$ if it is continuous at all points of $D$. … … βΊFor $f(x,y)$ defined on a point set $D$ contained in a rectangle $R$, let … βΊwhere $D$ is the image of $D^{*}$ under a mapping $(u,v)\to(x(u,v),y(u,v))$ which is one-to-one except perhaps for a set of points of area zero. …Again the mapping is one-to-one except perhaps for a set of points of volume zero. …
##### 3: 1.6 Vectors and Vector-Valued Functions
βΊand $S$ be the closed and bounded point set in the $(x,y)$ plane having a simple closed curve $C$ as boundary. … βΊSuppose $S$ is a piecewise smooth surface which forms the complete boundary of a bounded closed point set $V$, and $S$ is oriented by its normal being outwards from $V$. …
##### 4: 1.9 Calculus of a Complex Variable
βΊAn open set in $\mathbb{C}$ is one in which each point has a neighborhood that is contained in the set. βΊA point $z_{0}$ is a limit point (limiting point or accumulation point) of a set of points $S$ in $\mathbb{C}$ (or $\mathbb{C}\cup\infty$) if every neighborhood of $z_{0}$ contains a point of $S$ distinct from $z_{0}$. … βΊA domain $D$, say, is an open set in $\mathbb{C}$ that is connected, that is, any two points can be joined by a polygonal arc (a finite chain of straight-line segments) lying in the set. … βΊConversely, if at a given point $(x,y)$ the partial derivatives $\ifrac{\partial u}{\partial x}$, $\ifrac{\partial u}{\partial y}$, $\ifrac{\partial v}{\partial x}$, and $\ifrac{\partial v}{\partial y}$ exist, are continuous, and satisfy (1.9.25), then $f(z)$ is differentiable at $z=x+\mathrm{i}y$. … βΊ
1.9.49 $R=\liminf_{n\to\infty}{\left|a_{n}\right|}^{-1/n}.$
##### 5: 2.8 Differential Equations with a Parameter
βΊin which $u$ is a real or complex parameter, and asymptotic solutions are needed for large $|u|$ that are uniform with respect to $z$ in a point set $\mathbf{D}$ in $\mathbb{R}$ or $\mathbb{C}$. … βΊCorresponding to each positive integer $n$ there are solutions $W_{n,j}(u,\xi)$, $j=1,2$, that depend on arbitrarily chosen reference points $\alpha_{j}$, are $C^{\infty}$ or analytic on $\mathbf{\Delta}$, and as $u\to\infty$
##### 6: 21.3 Symmetry and Quasi-Periodicity
βΊThe set of points $\mathbf{m}_{1}+\boldsymbol{{\Omega}}\mathbf{m}_{2}$ form a $g$-dimensional lattice, the period lattice of the Riemann theta function. …
##### 7: 18.2 General Orthogonal Polynomials
βΊLet $X$ be a finite set of distinct points on $\mathbb{R}$, or a countable infinite set of distinct points on $\mathbb{R}$, and $w_{x}$, $x\in X$, be a set of positive constants. …when $X$ is a finite set of $N+1$ distinct points. … βΊIf the polynomials $p_{n}(x)$ ($n=0,1,\ldots,N$) are orthogonal on a finite set $X$ of $N+1$ distinct points as in (18.2.3), then the polynomial $p_{N+1}(x)$ of degree $N+1$, up to a constant factor defined by (18.2.8) or (18.2.10), vanishes on $X$. …
##### 8: 26.13 Permutations: Cycle Notation
βΊAn element of $\mathfrak{S}_{n}$ with $a_{1}$ fixed points, $a_{2}$ cycles of length $2,\ldots,a_{n}$ cycles of length $n$, where $n=a_{1}+2a_{2}+\cdots+na_{n}$, is said to have cycle type ${\left(a_{1},a_{2},\ldots,a_{n}\right)}$. …
##### 9: 22.19 Physical Applications
βΊAs $a\to\sqrt{1/\beta}$ from below the period diverges since $a=\pm\sqrt{1/\beta}$ are points of unstable equilibrium. … βΊFor an initial displacement with $\sqrt{1/\beta}\leq|a|<\sqrt{2/\beta}$, bounded oscillations take place near one of the two points of stable equilibrium $x=\pm\sqrt{1/\beta}$. …As $a\to\sqrt{2/\beta}$ from below the period diverges since $x=0$ is a point of unstable equlilibrium. …
##### 10: 3.11 Approximation Techniques
βΊHere $x_{j}$, $j=1,2,\dots,J$, is a given set of distinct real points and $J\geq n+1$. …