A mean value theorem for generalized Riemann derivatives

Functional differences that lead to generalized Riemann derivatives were studied by Ash and Jones in (1987). They gave a partial answer as to when these differences satisfy an analog of the Mean Value Theorem. Here we give a complete classification.

     

On Lusin's condition for the inverse function

Sia ? una funzione invertibile, continua e quasi ovunque differenziabile, da un aperto di IR ⁿ in IR ⁿ . Allora ?^?1soddisfa alla condizione di Lusin (N) se e solo se lo Jacobiano di ? è quasi ovunque diverso da zero.     

Homomorphisms with respect to a function

Let X be a realcompact space and ${H:C(X)\rightarrow\mathbb{R}}$ be an identity and order preserving group homomorphism. It is shown that H is an evaluation at some point of X if and only if there is ${\varphi\in C(\mathbb{R})}$ with ${\varphi(r)>\varphi(0)}$ for all ${r\in\mathbb{R}-\{0\}}$ for which ${H\circ\varphi=\varphi\circ H}$ . This extends (and unifies) classical results by Hewitt and Shirota.     

Homomorphisms with respect to a function

Let X be a realcompact space and H:C(X)?\mathbbR{H:C(X)\rightarrow\mathbb{R}} be an identity and order preserving group homomorphism. It is shown that H is an evaluation at some point of X if and only if there is j ? C(\mathbbR){\varphi\in C(\mathbb{R})} with ${\varphi(r)>\varphi(0)}${\varphi(r)>\varphi(0)} for all r ? \mathbbR-{0}{r\in\mathbb{R}-\{0\}} for which H°j = j°H{H\circ\varphi=\varphi\circ H} . This extends (and unifies) classical results by Hewitt and Shirota.     

Separately continuous functions with respect to a variable frame

We show that the set D(f) of discontinuity points of a function f : R ² R continuous at every point p with respect to two variable linearly independent directions e ₁(p) and e ₂(p) is a set of the first category. Furthermore, if f is differentiable along one of directions, then D(f) is a nowhere dense set.__________Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 9, pp. 1281–1286, September, 2004.     

Interpolation theorem for a continuous function on orientations of a simple graph

Let G be a simple graph. A function f from the set of orientations of G to the set of non-negative integers is called a continuous function on orientations of G if, for any two orientations O ₁ and O ₂ of G, |f(O ₁)-f(O ₂)| 1 whenever O ₁ and O ₂ differ in the orientation of exactly one edge of G. We show that any continuous function on orientations of a simple graph G has the interpolation property as follows:If there are two orientations O ₁ and O ₂ of G with f(O ₁) = p and f(O ₂) = q, where p<q, then for any integer k such that p < k < q, there are at least m orientations O of G satisfying f(O) = k, where m equals the number of edges of G.It follows that some useful invariants of digraphs including the connectivity, the arc-connectivity and the absorption number, etc., have the above interpolation property on the set of all orientations of G.     

A central limit theorem for integrals with respect to random measures

Integrals with respect to stationary random measures are considered. A central limit theorem for such integrals is proved. The results are applied to obtain a functional central limit theorem for transformed solutions of the Burgers equation with random initial data.     

Generalized Newton algorithm to minimize a function with many variables using computer-evaluated exact higher-order derivatives

This paper describes an optimization algorithm which uses up torth-order derivatives to find the optimum of anr-times continuously differentiable function of many variables. The algorithm, developed by Kalaba and Tishler (Ref. 1), obtains the exact values of the derivatives required for the optimization from the table algorithm presented in Kalabaet al. (Ref. 2) and Kalaba and Tishler (Ref. 3). The optimization algorithm described here reduces to the well-known Newton-Raphson algorithm when only first-order and second-order derivatives are used.on leave from the Faculty of Management, Tel Aviv University, Tel Aviv, Israel.     

A class of degenerate elliptic equations and a Dido's problem with respect to a measure

In this paper we consider the following class of linear elliptic problems

     

A universality theorem for the Riemann zeta-function with respect to Beatty sets

We prove that, for given positive numbers α and h, the Riemann zeta-function ζ can approximate any nonvanishing analytic function on a simply connected compact subset of the right open half of the critical strip by shifts of type ζ(s?+?ih?αn?).

     

Bochner-Riesz means with respect to a rough distance function

The generalized Bochner-Riesz operator may be defined as

where is an appropriate distance function and is the inverse Fourier transform. The behavior of on is described for , a rough distance function. We conjecture that this operator is bounded on when and , and unbounded when . This conjecture is verified for large ranges of .

     

Integral representation with respect to stopped continuous local martingales

We consider a Markovian jump process θ, with finite state space, feeding the parameters of a nonlinear diffusion process X. We observe θ and X in white noise, and—given a function f—we want to construct a finite filter for the f(X _t )-process. An algorithm is investigated which will produce a finite filter if it halts after a finite number of steps, and we give necessary and. sufficient conditions for this to happen.