# on a point set

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## 1—10 of 109 matching pages

##### 1: 2.1 Definitions and Elementary Properties

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►Let $\mathbf{X}$ be a point set with a limit point
$c$.
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2.1.16
$$f(x)\sim {a}_{0}+{a}_{1}(x-c)+{a}_{2}{(x-c)}^{2}+\mathrm{\cdots},$$
$x\to c$ in $\mathbf{X}$,

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►If the set
$\mathbf{X}$ in §2.1(iii) is a closed sector $\alpha \le \mathrm{ph}x\le \beta $, then by definition the asymptotic property (2.1.13) holds uniformly with respect to $\mathrm{ph}x\in [\alpha ,\beta ]$ as $|x|\to \mathrm{\infty}$.
…Suppose $u$ is a parameter (or set of parameters) ranging over a point set (or sets) $\mathbf{U}$, and for each nonnegative integer $n$
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##### 2: 1.5 Calculus of Two or More Variables

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►A function is

*continuous on a point set*$D$ if it is continuous at all points of $D$. … … ►For $f(x,y)$ defined on a point set $D$ contained in a rectangle $R$, let … ►where $D$ 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: 1.9 Calculus of a Complex Variable

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►An

*open set*in $\u2102$ is one in which each point has a neighborhood that is contained in the set. ►A point ${z}_{0}$ is a*limit point*(*limiting point*or*accumulation point*) of a set of points $S$ in $\u2102$ (or $\u2102\cup \mathrm{\infty}$) if every neighborhood of ${z}_{0}$ contains a point of $S$ distinct from ${z}_{0}$. … ►A*domain*$D$, say, is an open set in $\u2102$ 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. … ►Conversely, if at a given point $(x,y)$ the partial derivatives $\partial u/\partial x$, $\partial u/\partial y$, $\partial v/\partial x$, and $\partial v/\partial y$ exist, are continuous, and satisfy (1.9.25), then $f(z)$ is differentiable at $z=x+\mathrm{i}y$. … ►
1.9.49
$$R=\underset{n\to \mathrm{\infty}}{lim\; inf}{|{a}_{n}|}^{-1/n}.$$

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##### 4: 18.2 General Orthogonal Polynomials

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►Let $X$ 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.
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##### 5: 2.8 Differential Equations with a Parameter

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►in 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 $\u2102$.
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►Corresponding 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 ${C}^{\mathrm{\infty}}$ or analytic on $\mathbf{\Delta}$, and as $u\to \mathrm{\infty}$
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##### 6: 21.3 Symmetry and Quasi-Periodicity

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►The set of points
${\mathbf{m}}_{1}+\mathbf{\Omega}{\mathbf{m}}_{2}$ form a
$g$-dimensional lattice, the

*period lattice*of the Riemann theta function. …##### 7: 1.6 Vectors and Vector-Valued Functions

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►and $S$ be the closed and bounded point set in the $(x,y)$ plane having a simple closed curve $C$ as boundary.
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►Suppose $S$ 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 $V$.
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##### 8: 4.13 Lambert $W$-Function

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##### 9: 26.13 Permutations: Cycle Notation

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►An element of ${\U0001d516}_{n}$ with ${a}_{1}$ fixed points, ${a}_{2}$ cycles of length $2,\mathrm{\dots},{a}_{n}$ cycles of length $n$, where $n={a}_{1}+2{a}_{2}+\mathrm{\cdots}+n{a}_{n}$, is said to have

*cycle type*$({a}_{1},{a}_{2},\mathrm{\dots},{a}_{n})$. …##### 10: 22.19 Physical Applications

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►As $a\to \sqrt{1/\beta}$ from below the period diverges since $a=\pm \sqrt{1/\beta}$ are points of unstable equilibrium.
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►For an initial displacement with $$, 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.
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