Symmetric Forms

1970 ◽  
Vol 13 (1) ◽  
pp. 83-87 ◽  
Author(s):  
K. V. Menon
Keyword(s):  
Euclidean Space ◽  
Dimensional Euclidean Space ◽  
Image Position ◽  
Class Of Functions

Let Rm denote a m dimensional Euclidean space. When x ∊ Rm will write x = (x1, x2,..., xm). Let R+m ={x: x ∊ Rm, xi < 0 for all i} and R-m ={x: x ∊ Rm, xi < 0 for all i}. In this paper we consider a class of functions which consists of mappings, Er(K) and Hr(K) of Rm into R which are indexed by K ∊ R+m and K ∊ R-m respectively, and defined at any point α ∊ Rm by1.1

Properties of Harmonic Functions of Three Real Variables Given by Bergman-Whittaker Operators

1963 ◽  
Vol 15 ◽  
pp. 157-168 ◽  
Author(s):  
Josephine Mitchell
Keyword(s):  
Euclidean Space ◽  
Harmonic Functions ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
Rectifiable Curve ◽  
Image Position

Let be a closed rectifiable curve, not going through the origin, which bounds a domain Ω in the complex ζ-plane. Let X = (x, y, z) be a point in three-dimensional euclidean space E3 and setThe Bergman-Whittaker operator defined by

Multiple points for symmetric Lévy processes

1978 ◽  
Vol 83 (1) ◽  
pp. 83-90 ◽  
Author(s):  
John Hawkes
Keyword(s):  
Markov Process ◽  
Euclidean Space ◽  
Lévy Process ◽  
Lévy Processes ◽  
Transition Function ◽  
Levy Processes ◽  
Levy Process ◽  
Dimensional Euclidean Space ◽  
Multiple Points ◽  
Image Position

Let Xt be a Lévy process in Rd, d-dimensional euclidean space. That is X is a Markov process whose transition function satisfies

Semi-E-Preinvex Functions

2012 ◽  
pp. 677-683
Author(s):  
Yu-Ru Syau ◽  
E. Stanley Lee
Keyword(s):  
Nonlinear Programming ◽  
Euclidean Space ◽  
Convex Functions ◽  
Sufficient Conditions ◽  
Dimensional Euclidean Space ◽  
Quasiconvex Functions ◽  
Invex Set ◽  
Preinvex Functions ◽  
The Relationship ◽  
Class Of Functions

A class of functions called semi-E-preinvex functions is defined as a generalization of semi-E-convex functions. Similarly, the concept of semi-E-quasiconvex functions is also generalized to semi-E-prequasiinvex functions. Properties of these proposed classes are studied, and sufficient conditions for a nonempty subset of the n-dimensional Euclidean space to be an E-convex or E-invex set are given. The relationship between semi-E-preinvex and E-preinvex functions are discussed along with results for the corresponding nonlinear programming problems.

The number of polygons on a lattice

1961 ◽  
Vol 57 (3) ◽  
pp. 516-523 ◽  
Author(s):  
J. M. Hammersley
Keyword(s):  
Euclidean Space ◽  
Dimensional Euclidean Space ◽  
End Points ◽  
Unit Distance ◽  
Connective Constant ◽  
Image Position ◽  
Self Avoiding Walk ◽  
Self Avoiding Walks

In this paper an n-stepped self-avoiding walk is defined to be an ordered sequence of n + 1 mutually distinct points, each with (positive, negative, or zero) integer coordinates in d-dimensional Euclidean space (where d is fixed and d ≥ 2), such that any two successive points in the sequence are neighbours, i.e. are unit distance apart. If further the first and last points of such a sequence are neighbours, the sequence is called an (n + 1)-sided self-avoiding polygon. Clearly, under this definition a polygon must have an even number of sides. Let f(n) and g(n) denote the numbers of n-stepped self-avoiding walks and of n-sided self-avoiding polygons having a prescribed first point. In a previous paper (3), I proved that there exists a connective constant K such thatHere I shall prove the truth of the long-standing conjecture thatI shall also show that (2) is a particular case of an expression for the number of n-stepped self-avoiding walks with prescribed end-points, a distance o(n) apart, this being another old and popular conjecture.

Semi-E-Preinvex Functions

2010 ◽  
Vol 1 (3) ◽  
pp. 31-39
Author(s):  
Yu-Ru Syau ◽  
E. Stanley Lee
Keyword(s):  
Nonlinear Programming ◽  
Euclidean Space ◽  
Nonempty Subset ◽  
Sufficient Conditions ◽  
Dimensional Euclidean Space ◽  
Quasiconvex Functions ◽  
Invex Set ◽  
Preinvex Functions ◽  
The Relationship ◽  
Class Of Functions

A class of functions called semi--preinvex functions is defined as a generalization of semi--convex functions. Similarly, the concept of semi--quasiconvex functions is also generalized to semi--prequasiinvex functions. Properties of these proposed classes are studied, and sufficient conditions for a nonempty subset of the -dimensional Euclidean space to be an -convex or -invex set are given. The relationship between semi--preinvex and -preinvex functions are discussed along with results for the corresponding nonlinear programming problems.

Some Imbedding Theorems for Sobolev Spaces

1971 ◽  
Vol 23 (3) ◽  
pp. 517-530 ◽  
Author(s):  
R. A. Adams ◽  
John Fournier
Keyword(s):  
Banach Space ◽  
Positive Integer ◽  
Euclidean Space ◽  
Sobolev Space ◽  
Dimensional Euclidean Space ◽  
Dense Subspace ◽  
Bounded Subset ◽  
Partial Derivatives ◽  
Image Position ◽  
Derivatives Of

We shall be concerned throughout this paper with the Sobolev space Wm,p(G) and the existence and compactness (or lack of it) of its imbeddings (i.e. continuous inclusions) into various LP spaces over G, where G is an open, not necessarily bounded subset of n-dimensional Euclidean space En. For each positive integer m and each real p ≧ 1 the space Wm,p(G) consists of all u in LP(G) whose distributional partial derivatives of all orders up to and including m are also in LP(G). With respect to the norm1.1Wm,p(G) is a Banach space. It has been shown by Meyers and Serrin [9] that the set of functions in Cm(G) which, together with their partial derivatives of orders up to and including m, are in LP(G) forms a dense subspace of Wm,p(G).

Universal inequalities for eigenvalues of a system of elliptic equations

2009 ◽  
Vol 139 (2) ◽  
pp. 273-285 ◽  
Author(s):  
Qing-Ming Cheng ◽  
Hong-Cang Yang
Keyword(s):  
Euclidean Space ◽  
Eigenvalue Problem ◽  
Bounded Domain ◽  
Lower Order ◽  
Elliptic Equations ◽  
Elastic Vibration ◽  
Dimensional Euclidean Space ◽  
Explicit Method ◽  
Universal Inequalities ◽  
Image Position

Let D be a bounded domain in an n-dimensional Euclidean space ℝn. Assume thatare eigenvalues of an eigenvalue problem of a system of n elliptic equations:In particular, when n=3, the eigenvalue problem describes the behaviour of the elastic vibration. We obtain universal inequalities for eigenvalues of the above eigenvalue problem by making use of a direct and explicit method; our results are sharper than one of Hook. Furthermore, a universal inequality for lower-order eigenvalues of the above eigenvalue problem is also derived.

Continuity of solutions of Schrödinger equations

1991 ◽  
Vol 110 (3) ◽  
pp. 581-597
Author(s):  
Mitsuru Nakai
Keyword(s):  
Euclidean Space ◽  
Total Variation ◽  
Open Subset ◽  
Radon Measure ◽  
Dimensional Euclidean Space ◽  
Schrödinger Equations ◽  
Kato Class ◽  
Continuity Of Solutions ◽  
Image Position ◽  
Newtonian Kernel

We denote by N(x, y) the Newtonian kernel on the d-dimensional Euclidean space (where d ≥ 2) so that N(x, y) = log|x–y|-1 for d = 2 and N(x, y) = |x−y|2−d for d ≥ 3. A signed Radon measure μ on an open subset Ω in d is said to be of Kato class iffor every y in Ω. where |μ| is the total variation measure of μ.

scholarly journals Continuity and differentiability properties of the Nemitskii operator in Hölder spaces

1988 ◽  
Vol 30 (1) ◽  
pp. 59-65 ◽  
Author(s):  
Rita Nugari
Keyword(s):  
Banach Space ◽  
Euclidean Space ◽  
Dimensional Euclidean Space ◽  
Bounded Subset ◽  
Holder Space ◽  
Usual Norm ◽  
Image Position ◽  
Holder Spaces ◽  
Open Bounded Subset ◽  
Differentiability Properties

Let ℝn be the n-dimensional Euclidean space with the usual norm denoted by |·| In what follows 蒆 will denote an open bounded subset of ℝn, and its closure.For α ∊(0,1], is the space of all functions such that: is called the Holder space with exponent a and is a Banach space when endowed with the norm:where ‖u‖∞ is, as usual, defined by:

scholarly journals Semilinear Second Order Elliptic Oscillation

1979 ◽  
Vol 22 (2) ◽  
pp. 139-157 ◽  
Author(s):  
C. A. Swanson
Keyword(s):  
Partial Differential Equations ◽  
Euclidean Space ◽  
Recent Progress ◽  
Unbounded Domains ◽  
Elliptic Partial Differential Equations ◽  
Second Order ◽  
Dimensional Euclidean Space ◽  
Semilinear Elliptic ◽  
Oscillation Problem ◽  
Image Position

These pages summarize recent progress on the oscillation problem for semilinear elliptic partial differential equations of the form(1)in unbounded domains Ω in n-dimensional Euclidean space Rn. Our attention is restricted to the second order symmetric equation (1), and completeness is not attempted; the emphasis is on results obtained in the last five years.

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