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

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:

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.

A one dimensional random space-filling problem

1968 ◽  
Vol 5 (02) ◽  
pp. 427-435 ◽  
Author(s):  
John P. Mullooly
Keyword(s):  
Asymptotic Behavior ◽  
Finite Number ◽  
Real Line ◽  
Random Variables ◽  
Random Interval ◽  
Space Filling ◽  
One Dimensional ◽  
The Real ◽  
Fixed Constant ◽  
Filling Problem

Consider an interval of the real line (0, x), x &gt; 0; and place in it a random subinterval S(x) defined by the random variables Xx and Yx , the position of the center of S(x) and the length of S(x). The set (0, x)– S(x) consists of two intervals of length δ and η. Let a &gt; 0 be a fixed constant. If δ ≦ a, then a random interval S(δ) defined by Xδ, Yδ is placed in the interval of length δ. If δ &lt; a, the placement of the second interval is not made. The same is done for the interval of length η. Continue to place non-intersecting random subintervals in (0, x), and require that the lengths of all the random subintervals be ≦ a. The process terminates after a finite number of steps when all the segments of (0, x) uncovered by random subintervals are of length &lt; a. At this stage, we say that (0, x) is saturated. Define N(a, x) as the number of random subintervals that have been placed when the process terminates. We are interested in the asymptotic behavior of the moments of N(a, x), for large x.

Kernels with only a finite number of characteristic values

1971 ◽  
Vol 70 (2) ◽  
pp. 257-262
Author(s):  
Dale W. Swann
Keyword(s):  
Finite Number ◽  
Higher Order ◽  
Complex Numbers ◽  
Finite Characteristic ◽  
Characteristic Values ◽  
Image Position ◽  
Complex Valued

Let K(s, t) be a complex-valued L2 kernel on the square ⋜ s, t ⋜ by which we meanand let {λν}, perhaps empty, be the set of finite characteristic values (f.c.v.) of K(s, t), i.e. complex numbers with which there are associated non-trivial L2 functions øν(s) satisfyingFor such kernels, the iterated kernels,are well-defined (1), as are the higher order tracesCarleman(2) showed that the f.c.v. of K are the zeros of the modified Fredhoim determinantthe latter expression holding only for |λ| sufficiently small (3). The δn in (3) may be calculated, at least in theory, by well-known formulae involving the higher order traces (1). For our later analysis of this case, we define and , respectively, as the minimum and maximum moduli of the zeros of , the nth section of D*(K, λ).

Probability measures with trivial Stam groups

1982 ◽  
Vol 91 (3) ◽  
pp. 477-484
Author(s):  
Gavin Brown ◽  
William Mohan
Keyword(s):  
Probability Measure ◽  
Real Number ◽  
Real Line ◽  
Probability Measures ◽  
The Real ◽  
Interesting Observation ◽  
Image Position ◽  
Positive Integers

Let μ be a probability measure on the real line ℝ, x a real number and δ(x) the probability atom concentrated at x. Stam made the interesting observation that eitheror else(ii) δ(x)* μn, are mutually singular for all positive integers n.

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.

On Fourier-Stieltjes Transforms

1955 ◽  
Vol 7 ◽  
pp. 453-461 ◽  
Author(s):  
A. P. Calderón ◽  
A. Devinatz
Keyword(s):  
Real Line ◽  
The Real ◽  
One To One ◽  
Image Position ◽  
Decreasing Functions ◽  
Stieltjes Transforms

Let be the class of bounded non-decreasing functions defined on the real line which are normalized by the conditions ϕ(− ∞) = 0 , ϕ(t + 0) = ϕ(t).Let be the class of Fourier-Stieltjes transforms of elements of i.e. the elements of and are connected by the relationwhere ϕ ∊ and Φ ∊ .It is well known, and easy to verify that this mapping from to is one to one (1, p. 67, Satz 18).

Remarks on Quasi-Hermite-Fejér Interpolation

1964 ◽  
Vol 7 (1) ◽  
pp. 101-119 ◽  
Author(s):  
A. Sharma
Keyword(s):  
Real Line ◽  
Real Numbers ◽  
The Real ◽  
The Moment ◽  
Image Position

Let1be n+2 distinct points on the real line and let us denote the corresponding real numbers, which are at the moment arbitrary, by2The problem of Hermite-Fejér interpolation is to construct the polynomials which take the values (2) at the abscissas (1) and have preassigned derivatives at these points. This idea has recently been exploited in a very interesting manner by P. Szasz [1] who has termed qua si-Hermite-Fejér interpolation to be that process wherein the derivatives are only prescribed at the points x1, x2, …, xn and the points -1, +1 are left out, while the values are prescribed at all the abscissas (1).

On the Non-Existence of Conjugate Points

1970 ◽  
Vol 13 (1) ◽  
pp. 31-37
Author(s):  
G. J. Butler ◽  
L. H. Erbe ◽  
R. M. Mathsen
Keyword(s):  
Differential Equation ◽  
Real Line ◽  
Conjugate Points ◽  
Multiple Zeros ◽  
The Real ◽  
Image Position

In this paper we consider the types of pairs of multiple zeros which a solution to the differential equationcan possess on an interval I of the real line. The results obtained generalize those in [2] and (for n = 3) in [3].I. Let f satisfy the condition1.1for all t ∊ I, u0 ≠ 0, and all u1, … un-1.

scholarly journals An inequality for probability density functions arising from a distinguishability problem

1998 ◽  
Vol 39 (3) ◽  
pp. 350-354
Author(s):  
Boris Guljaš ◽  
C. E. M. Pearce ◽  
Josip Pečarić
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Real Line ◽  
Integral Inequality ◽  
Probability Density Functions ◽  
Density Functions ◽  
The Real ◽  
Image Position

AbstractAn integral inequality is established involving a probability density function on the real line and its first two derivatives. This generalizes an earlier result of Sato and Watari. If f denotes the probability density function concerned, the inequality we prove is thatunder the conditions β > α 1 and 1/(β+1) < γ ≤ 1.

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