A Note on the Dirichlet Condition for Second-Order Differential Expressions

1976 ◽  
Vol 28 (2) ◽  
pp. 312-320 ◽  
Author(s):  
W. N. Everitt
Keyword(s):  
Differential Expression ◽  
Real Line ◽  
Open Interval ◽  
Dirichlet Condition ◽  
Second Order ◽  
The Real ◽  
Half Open Interval ◽  
Image Position ◽  
Complex Valued

Let M denote the formally symmetric, second-order differential expression given by, for suitably differentiable complex-valued functions ƒ,The coefficients p and q are real-valued, Lebesgue measurable on the halfclosed, half-open interval [a, b) of the real line, with - ∞ < a < b ≦ ∞, and satisfy the basic conditions:

Some applications of renewal theory on the whole line

1970 ◽  
Vol 7 (03) ◽  
pp. 734-746
Author(s):  
Kenny S. Crump ◽  
David G. Hoel
Keyword(s):  
Distribution Function ◽  
Real Line ◽  
Renewal Theory ◽  
Open Interval ◽  
One Dimensional ◽  
The Real ◽  
Half Open Interval ◽  
Negative Function ◽  
Dimensional Distribution ◽  
Image Position

Suppose F is a one-dimensional distribution function, that is, a function from the real line to the real line that is right-continuous and non-decreasing. For any such function F we shall write F{I} = F(b)– F(a) where I is the half-open interval (a, b]. Denote the k-fold convolution of F with itself by Fk* and let Now if z is a non-negative function we may form the convolution although Z may be infinite for some (and possibly all) points.

Some applications of renewal theory on the whole line

1970 ◽  
Vol 7 (3) ◽  
pp. 734-746
Author(s):  
Kenny S. Crump ◽  
David G. Hoel
Keyword(s):  
Distribution Function ◽  
Real Line ◽  
Renewal Theory ◽  
Open Interval ◽  
One Dimensional ◽  
The Real ◽  
Half Open Interval ◽  
Negative Function ◽  
Dimensional Distribution ◽  
Image Position

Suppose F is a one-dimensional distribution function, that is, a function from the real line to the real line that is right-continuous and non-decreasing. For any such function F we shall write F{I} = F(b)– F(a) where I is the half-open interval (a, b]. Denote the k-fold convolution of F with itself by Fk* and let Now if z is a non-negative function we may form the convolution although Z may be infinite for some (and possibly all) points.

On a fourth-order singular integral inequality

1978 ◽  
Vol 80 (3-4) ◽  
pp. 249-260 ◽  
Author(s):  
A. Russell
Keyword(s):  
Differential Expression ◽  
Fourth Order ◽  
Singular Integral ◽  
Integral Inequality ◽  
Second Order ◽  
New Order ◽  
Alternative Account ◽  
The Real ◽  
Image Position ◽  
Complete Solutions

SynopsisThe inequality considered in this paper iswhereNis the real-valued symmetric differential expression defined byGeneral properties of this inequality are considered which result in giving an alternative account of a previously considered inequalityto which (*) reduces in the casep=q= 0,r= 1.Inequality (*) is also an extension of the inequalityas given by Hardy and Littlewood in 1932. This last inequality has been extended by Everitt to second-order differential expressions and the methods in this paper extend it to fourth-order differential expressions. As with many studies of symmetric differential expressions the jump from the second-order to the fourth-order introduces difficulties beyond the extension of technicalities: problems of a new order appear for which complete solutions are not available.

Functions with a Finite Number of Negative Squares

1972 ◽  
Vol 15 (3) ◽  
pp. 399-410 ◽  
Author(s):  
James Stewart
Keyword(s):  
Finite Number ◽  
Real Line ◽  
Nonnegative Integer ◽  
Hermitian Form ◽  
The Real ◽  
Image Position ◽  
Negative Squares ◽  
Complex Valued

Let f be a complex-valued function defined on the real line R with the property that for every x∊R. If k is a nonnegative integer,f is said to have k negative squares, or to be indefinite of order k, if the Hermitian form

The Existence of Continuable Solutions of a Second Order Differential Equation

1977 ◽  
Vol 29 (3) ◽  
pp. 472-479 ◽  
Author(s):  
G. J. Butler
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Real Line ◽  
Order Differential Equation ◽  
Nonlinear Ordinary Differential Equation ◽  
Second Order ◽  
The Real ◽  
Second Order Differential Equation ◽  
Image Position ◽  
Sign Restriction

A much-studied equation in recent years has been the second order nonlinear ordinary differential equationwhere q and f are continuous on the real line and, in addition, f is monotone increasing with yf(y) > 0 for y ≠ 0. Although the original interest in (1) lay largely with the case that q﹛t) ≧ 0 for all large values of t, a number of papers have recently appeared in which this sign restriction is removed.

A property of the asymptotic series for a class of Titchmarsh–Weyl m-functions

1986 ◽  
Vol 102 (3-4) ◽  
pp. 253-257 ◽  
Author(s):  
B. J. Harris
Keyword(s):  
Differential Equation ◽  
Asymptotic Expansion ◽  
Linear Differential Equation ◽  
Asymptotic Series ◽  
Second Order ◽  
Order Linear Differential Equation ◽  
Order Linear ◽  
The Real ◽  
Linear Differential ◽  
Image Position

SynopsisIn an earlier paper [6] we showed that if q ϵ CN[0, ε) for some ε > 0, then the Titchmarsh–Weyl m(λ) function associated with the second order linear differential equationhas the asymptotic expansionas |A| →∞ in a sector of the form 0 < δ < arg λ < π – δ.We show that if the real valued function q admits the expansionin a neighbourhood of 0, then

scholarly journals Distribution of rational points on the real line

1973 ◽  
Vol 15 (2) ◽  
pp. 243-256 ◽  
Author(s):  
T. K. Sheng
Keyword(s):  
Algebraic Number ◽  
Rational Number ◽  
Real Line ◽  
Rational Points ◽  
The Other ◽  
Liouville Numbers ◽  
The Real ◽  
Other Hand ◽  
Image Position

It is well known that no rational number is approximable to order higher than 1. Roth [3] showed that an algebraic number is not approximable to order greater than 2. On the other hand it is easy to construct numbers, the Liouville numbers, which are approximable to any order (see [2], p. 162). We are led to the question, “Let Nn(α, β) denote the number of distinct rational points with denominators ≦ n contained in an interval (α, β). What is the behaviour of Nn(α, + 1/n) as α varies on the real line?” We shall prove that and that there are “compressions” and “rarefactions” of rational points on the real line.

Dependent thinning of point processes

1980 ◽  
Vol 17 (04) ◽  
pp. 987-995 ◽  
Author(s):  
Valerie Isham
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Real Line ◽  
Point Processes ◽  
Compound Poisson Process ◽  
Second Order ◽  
Binary Variables ◽  
Compound Poisson ◽  
The Real ◽  
Markov Sequence

A point process, N, on the real line, is thinned using a k -dependent Markov sequence of binary variables, and is rescaled. Second-order properties of the thinned process are described when k = 1. For general k, convergence to a compound Poisson process is demonstrated.

On the spectra of non-self-adjoint realisations of second-order elliptic operators

1981 ◽  
Vol 90 (1-2) ◽  
pp. 71-105 ◽  
Author(s):  
W. D. Evans
Keyword(s):  
Spherical Shell ◽  
Essential Spectrum ◽  
Weighted Space ◽  
Elliptic Operators ◽  
Second Order ◽  
Sectorial Operator ◽  
Image Position ◽  
Second Order Elliptic Operators ◽  
Complex Valued

SynopsisLet τ denote the second-order elliptic expressionwhere the coefficients bj and q are complex-valued, and let Ω be a spherical shell Ω = {x:x ∈ ℝn, l <|x|<m} with l≧0, m≦∞. Under the conditions assumed on the coefficients of τ and with either Dirichlet or Neumann conditions on the boundary of Ω, τ generates a quasi-m-sectorial operator T in the weighted space L2(Ω;w). The main objective is to locate the spectrum and essential spectrum of T. Best possible results are obtained.

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