About the Project

on finite point sets

AdvancedHelp

(0.003 seconds)

1—10 of 26 matching pages

1: 18.2 General Orthogonal Polynomials
Orthogonality on Finite Point Sets
Let X be a finite set of distinct points on , or a countable infinite set of distinct points on , and w x , x X , be a set of positive constants. …when X is a finite set of N + 1 distinct points. …
2: 23.20 Mathematical Applications
Let T denote the set of points on C that are of finite order (that is, those points P for which there exists a positive integer n with n P = o ), and let I , K be the sets of points with integer and rational coordinates, respectively. …
3: 1.9 Calculus of a Complex Variable
A domain D , say, is an open set in 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. …
4: 36.12 Uniform Approximation of Integrals
§36.12 Uniform Approximation of Integrals
§36.12(i) General Theory for Cuspoids
Also, f is real analytic, and K + 2 f / u K + 2 > 0 for all 𝐲 such that all K + 1 critical points coincide. … In (36.12.10), both second derivatives vanish when critical points coalesce, but their ratio remains finite. … Also, Δ 1 / 4 / f + ′′ and Δ 1 / 4 / f ′′ are chosen to be positive real when y is such that both critical points are real, and by analytic continuation otherwise. …
5: 2.1 Definitions and Elementary Properties
Let 𝐗 be a point set with a limit point c . … If c is a finite limit point of 𝐗 , then … Suppose u is a parameter (or set of parameters) ranging over a point set (or sets) 𝐔 , and for each nonnegative integer n Similarly for finite limit point c in place of . … where c is a finite, or infinite, limit point of 𝐗 . …
6: 21.7 Riemann Surfaces
Consider the set of points in 2 that satisfy the equation …This compact curve may have singular points, that is, points at which the gradient of P ~ vanishes. … On this surface, we choose 2 g cycles (that is, closed oriented curves, each with at most a finite number of singular points) a j , b j , j = 1 , 2 , , g , such that their intersection indices satisfy … The zeros λ j , j = 1 , 2 , , 2 g + 1 of Q ( λ ) specify the finite branch points P j , that is, points at which μ j = 0 , on the Riemann surface. Denote the set of all branch points by B = { P 1 , P 2 , , P 2 g + 1 , P } . …
7: 1.10 Functions of a Complex Variable
If the poles are infinite in number, then the point at infinity is called an essential singularity: it is the limit point of the poles. … A cut domain is one from which the points on finitely many nonintersecting simple contours (§1.9(iii)) have been removed. … Alternatively, take z 0 to be any point in D and set F ( z 0 ) = e α ln ( 1 z 0 ) e β ln ( 1 + z 0 ) where the logarithms assume their principal values. … (The integer k may be greater than one to allow for a finite number of zero factors.) …
8: 1.4 Calculus of One Variable
where the supremum is over all sets of points x 0 < x 1 < < x n in the closure of ( a , b ) , that is, ( a , b ) with a , b added when they are finite. …
9: 36.15 Methods of Computation
Far from the bifurcation set, the leading-order asymptotic formulas of §36.11 reproduce accurately the form of the function, including the geometry of the zeros described in §36.7. Close to the bifurcation set but far from 𝐱 = 𝟎 , the uniform asymptotic approximations of §36.12 can be used. … Direct numerical evaluation can be carried out along a contour that runs along the segment of the real t -axis containing all real critical points of Φ and is deformed outside this range so as to reach infinity along the asymptotic valleys of exp ( i Φ ) . …
§36.15(iv) Integration along Finite Contour
This can be carried out by direct numerical evaluation of canonical integrals along a finite segment of the real axis including all real critical points of Φ , with contributions from the contour outside this range approximated by the first terms of an asymptotic series associated with the endpoints. …
10: 1.5 Calculus of Two or More Variables
A function is continuous on a point set D if it is continuous at all points of D . … …
Finite Integrals
Moreover, if a , b , c , d are finite or infinite constants and f ( x , y ) is piecewise continuous on the set ( a , b ) × ( c , d ) , then … Again the mapping is one-to-one except perhaps for a set of points of volume zero. …