On certain double integrals

We evaluate certain double integrals involving H-functions of two variables. These integrals are quite general and are capable of yielding a number of integrals as particular cases.     

On the calculation of certain singular integrals

A method for the approximate calculation of Cauchy-type integrals with logarithmic singularities is proposed. It is based on the expansion of a functionf(t) in a series in the Chebyshev polynomials.Translated from Ukrainskii Matematicheskii Zhumal, Vol. 45, No. 8, pp. 1174–1176, August, 1993.     

On a certain property of the BurkillSCP-integral

We establish a necessary condition for theSCP-integrability of a function and use it to construct an example of a function possessing a continuousSCP-majorant andSCP-minorant on a closed interval, but nonintegrable over this interval in the sense of theSCP-integral. Translated fromMatematicheskie Zametki, Vol. 65, No. 4, pp. 599–606, April, 1999.     

On certain generalized families of unified elliptic-type integrals pertaining to Euler integrals and generating functions

Elliptic-type integrals have their importance and potential in certain problems in radiation physics and nuclear technology. A number of earlier works on the subject contains several interesting unifications and generalizations of some significant families of elliptic-type integrals. The present paper is intended to obtain certain new theorems on generating functions. The results obtained in this paper are of manifold generality and basic in nature. Beside deriving various known and new elliptic-type integrals and their generalizations these theorems can be used to evaluate various Euler-type integrals involving a number of generating functions.      

On the minimum average distance of binary constant weight codes

Let β(n,M,w) denote the minimum average Hamming distance of a binary constant weight code with length n, size M and weight w. In this paper, we study the problem of determining β(n,M,w). Using the methods from coding theory and linear programming, we derive several lower bounds on the average Hamming distance of a binary constant weight code. These lower bounds enable us to determine the exact value for β(n,M,w) in several cases.     

On the homogeneity property for certain quantifier logics

Summary The local homogeneity property is defined as in [Mak]. We show thatL (Q₁) and some related logics do not have the local homogeneity property, whereas cofinality logicL (Q ^cf) has the homogeneity property. Both proofs use forcing and absoluteness arguments.     

On the martingale property of certain local martingales

The stochastic exponential ${Z_t= {\rm exp}\{M_t-M_0-(1/2)\langle M,M\rangle_t\}}$ of a continuous local martingale M is itself a continuous local martingale. We give a necessary and sufficient condition for the process Z to be a true martingale in the case where ${M_t=\int_0^t b(Y_u)\,dW_u}$ and Y is a one-dimensional diffusion driven by a Brownian motion W. Furthermore, we provide a necessary and sufficient condition for Z to be a uniformly integrable martingale in the same setting. These conditions are deterministic and expressed only in terms of the function b and the drift and diffusion coefficients of Y. As an application we provide a deterministic criterion for the absence of bubbles in a one-dimensional setting.     

Bounds for certain integrals

A certain class of definite integrals is considered in which the integrand consists of a one-signed function together with another function which has a one-signed derivative in a certain interval. By examining the Cauchy form of the remainder, sets of bounds are developed which have a certain optimum property. The integrals may be multi-dimensional. The case in which the derivative component is not one-signed is briefly considered.     

On the minimal cover property and certain notions of finite

In set theory without the axiom of choice, we investigate the deductive strength of the principle “every topological space with the minimal cover property is compact”, and its relationship with certain notions of finite as well as with properties of linearly ordered sets and partially ordered sets.     

The average distance and the independence number

We prove that in every connected graph the independence number is at least as large as the average distance between vertices.