Two problems involving schur functions

This paper gives a number of theorems extending or related to theorems obtained by Fan in an earlier paper on strictly dissipative matrices. The new results involve the eigenvalues of A^-1A¹ and a determinantal inequality involving A and its Hermitian and skew Hermitian parts, when A is strictly dissipative. Some of the results extend to arbitrary (i.e., not necessarily dissipative) matrices.     

A note on polyharmonic functions

In this note we prove the following theorem:Suppose 0<p<∞ and α>−1. Then there is a constant C=C(p,m,n,α) such that

     

A note on maximum-type functions

The paper studies the differential properties of functions of the form

$g(x) = \mathop {\max }\limits_{y \in Y} f(x,y),$

where x ∈ X (X is an open convex set from ? ^m ) and y ∈ Y (Y is a compact from ? ⁿ ). Apart from the conventional smoothness conditions imposed on f(x, y), the condition of the concavity of g(x) on X is also imposed.

The differentiability of function g(x) on X is proved.The results of the study facilitate the derivation of the conditions ensuring the sufficiency of Pontryagin’s maximum principle.     

A note on W-I-continuous functions

In [1], Aç?kgöz et al. introduced and investigated the notions of w-I-continuous and w*-I-continuous functions in ideal topological spaces. In this paper, we investigate their relationships with continuous and θ-continuous functions.     

A note on pair-formation functions

In this paper, we take a closer look at two harmonic mean functions [1, 2] and two minimum functions (moving dominance function [3] and group-specific minimum function), in two sex multigroup populations. Comparisons between these functions are focused on proportionate mixing. We show that under some special conditions, the two harmonic mean functions are identical; and under the mixing framework of Castillo-Chavez and Busenberg [4], the two minimum functions are also identical (for both proportionate and nonproportionate mixing). Simulations of a simple demographic model with the four functions are also performed to confirm the above mentioned identity and to illustrate the behavior of these functions.     

A note on canonical functions

We construct a generic extension in which the ℵ₂nd canonical function on ℵ₁ exists. Supported by NSF and by a Fulbright grant. Publ. 378. Partially supported by the B.S.F.     

A note on feebly continuous functions

A function f from to is said to be feebly continuous at a point (x,y) if there exist sequences x_nx and y_ny with lim_n→∞lim_m→∞f(x_n,y_m)=f(x,y). Dales asked if every function has a point of feeble continuity. Our aim in this short note is to show that (assuming the Continuum Hypothesis) the answer is ‘no’. Dales also asked what happens for functions taking only two values: we show that in this case the answer is ‘yes’.     

A note on locally Lipschitzian functions

An oversight in a paper of Correa and Lemaréchal (this journal, 1993) is noted; a counterexample is given.     

A note on smoothed estimating functions

The kernel estimate of regression function in likelihood based models has been studied in Staniswalis (1989,J. Amer. Statist. Assoc.,84, 276–283). The notion of optimal estimation for the nonparametric kernel estimation of semimartingale intensity (t) is proposed. The goal is to arrive at a nonparametric estimate of ₀=(t ₀) for a fixed pointt ₀ [0, 1]. We consider the estimator that is a solution of the smoothed optimal estimating equation is the optimal estimating function as in Thavaneswaran and Thompson (1986,J. Appl. Probab.,23, 409–417).     

A note onK-convex functions

In this article we concern ourselves with a generalized version of convex functions widely used in the analysis of optimization problems, especially those involving inventory control and/or production-planning. We give some characterizations of these socalledK-convex functions and derive a few of their properties concerning measurability and continuity.

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Zusammenfassung In der vorliegenden Arbeit untersuchen wir eine Klasse von verallgemeinert konvexen Funktionen, die in der Modellierung von Lagerhaltungs- und/oder Produktionsproblemen verbreitete Anwendung gefunden haben. Wir leiten einige Charakterisierungen dieser sogenanntenK-konvexen Funktionen her und beweisen einige ihrer Eigenschaften, die Meßbarkeits- und Stetigkeitsfragen betreffen.

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Dedicated To K.F. Hinderer On His Fiftieth Anniversary     

A note on strongly Θ-precontinuous functions

This paper gives some characterizations of strongly Θ-precontinuous functions.     

A note on means of entire functions

A result of Singh and Sreenivasulu is proved under less restrictive hypothesis. It is also shown that if a condition on the coefficients is not satisfied, the theorem will not hold.     

A note on hyperharmonic and polyharmonic functions

We show that the spaces of harmonic functions with respect to the Poincaré metric in the unit ball B^N in have many different properties depending upon whether N is even or odd.