Change of variables in a multiple Lebesgue integral

Necessary and sufficient conditions are found for a change of variables in a multiple Lebesgue integral for a class of quasibijective (one-to-one everywhere except on a set whose image has measure zero) mappings in the Euclidean space Rⁿ.Translated from Matematicheskie Zametki, Vol. 14, No. 1, pp. 39–48, July, 1973.     

Some multiple integral transformations involving the H-function of several variables

The object of the present paper is to give two general multiple integral transformations of the H-function of several complex variables, which was defined and studied elsewhere by the authors (cf., e.g., [10], [11] and [12]). It is shown how the main formulas (1,8) and (1,14) below, involving Fox's H-function ([4], p, 408) and the Laguerre polynomials, respectively, are related to each other and, of course, to a number of results given recently in the literature (see [1] and [5] through [9]). Several possible applications of the operational techniques provided by these results (and their various special cases) when viewed as multidimensional integral transformations are also indicated briefly.     

Optimal fields for integral equations

The theory of optimal fields is developed for optimal control problems in which the state variables are determined by integral equations. The Hilbert integral is considered and the Hamilton-Jacobi equations are derived. The results obtained contain two maximum principles as special cases; one reflecting the special character of field theory and one corresponding to the results previously obtained by use of variations.     

An integral basis for algebraic fields

Let A be a principal ideal domain, K be the quotient field of A, and let L be a cubic extension of K. In this paper we establish the existence of a special type of integral basis of the field L over K which is a generalization of the integral basis of Voronoi for cubic extensions of the field Q of rational numbers.Translated from Matematicheskie Zametki, Vol. 13, No. 2, pp. 229–234, February, 1973.     

On a nonlinear volterra-hammerstein integral equation in two variables

Using a fixed point theorem of Krasnosel'skii type, this article proves the existence of asymptotically stable solutions for a Volterra-Hammerstein integral equation in two variables.     

Separation of variables for some systems with a fourth-order integral of motion

We construct separation variables for Yehia’s integrable deformations of the Kovalevskaya top and the Chaplygin system on a sphere. In the general case, the corresponding quadratures are given by the Abel-Jacobi map on a two-dimensional submanifold of the Jacobian of a genus-three algebraic curve, which is not hyperelliptic.     

Power integral bases for Selmer-like number fields

The Selmer trinomials are the trinomials f(X)∈{Xn−X−1,Xn+X+1|n>1 is an integer} over Z. For these trinomials we show that the ideal C=(f(X),f^′(X))Z[X] has height two and contains the linear polynomial (n−1)X+n. We then give several necessary and sufficient conditions for D[X]/(f(X)D[X]) to be a regular ring, where f(X) is an arbitrary polynomial over a Dedekind domain D such that its ideal C has height two and contains a product of primitive linear polynomials. We next specialize to the Selmer-like trinomials bXn+cX+d and bXn+cXn−1+d over D and give several more such necessary and sufficient conditions (among them is that C is a radical ideal). We then specialize to the Selmer trinomials over Z and give quite a few more such conditions (among them is that the discriminant Disc(Xn−X−1)=±(nn−(1−n)ⁿ⁻¹) of Xn−X−1 is square-free (respectively Disc(Xn+X+1)=±(nn+(1−n)ⁿ⁻¹) of Xn+X+1 is square-free)). Finally, we show that nn+(1−n)ⁿ⁻¹ is never square-free when n≡2 (mod 3) and n>2, but, otherwise, both are very often (but not always) square-free.     

Solution of a certain multiple integral equation

An explicit solution is derived formally for a certain multiple integral equation involving a multidimensional fractional integral of Riemann-Liouville type. The main inversion theorem proved here provides a generalization of a result due to W. L. Wainwright [in “Fractional Calculus and Its Applications” (B. Ross, Ed.), pp. 298–305, Springer-Verlag, Berlin/Heidelberg/New York, 1975]. A simple illustration of the theorem, involving the classical Laguerre function, is also presented.     

Bellman-Bihari integral inequalities in several independent variables

In this paper we unify certain types of integral inequalities which are established by Bellman, Bihari, Deo and Dhongade.     

Existence of asymptotically stable solutions for a mixed functional nonlinear integral equation in N variables

Motivated by recent known results about the solvability of nonlinear functional integral equations in one, two or N variables, this paper proves the existence of asymptotically stable solutions for a mixed functional integral equation in N variables with values in a general Banach space via a fixed point theorem of Krasnosels'ki?  type. In order to illustrate the results obtained here, an example is given.     

A hyperfinite flat integral for generalized random fields

In this paper, a nonstandard construction of generalized white noise is established. This provides a (hyperfinite) flat integral representation of probability measures for generalized random fields derived as image probability measures of generalized white noise under certain measurable transformations, including Euclidean random fields obtained as convolution from generalized white noise with Euclidean kernels.     

On a p-stable multiple integral. II

Summary We construct a multiple strictly p-stable stochastic integral as a Dunford type integral with respect to a L ^q -valued vector measure, 1qp2.     

Spectral trace formula for local fields

In the paper, the space of square-integrable functions on a local field and an operator acting as a sum of an operator of multiplication by a function and a convolution with a distribution (generalized function) are considered. The main result is a trace formula for such operators.