A generalization of an inequality by N. V. Krylov

In Krylov (Journal of the Juliusz Schauder Center 4 (1994), 355–364), a parabolic Littlewood–Paley inequality and its application to an L ^p -estimate of the gradient of the heat kernel are proved. These estimates are crucial tools in the development of a theory of parabolic stochastic partial differential equations (Krylov, Mathematical Surveys and Monographs vol. 64 (1999), 185–242). We generalize these inequalities so that they can be applied to stochastic integrodifferential equations.      

A generalization of Gruenhage's example

We construct, assuming the continuum hypothesis, an example of nonmetrizable n-dimensional Cantor manifold Xn(n∈N) with the following properties: 1) is hereditarily separable for all k∈N; 2) is perfectly normal for every k∈N; 3) the space F(Xn) is hereditarily normal for every seminormal functor F that preserves weights and one-to-one points and such that sp(F)={1,k}; in particular, and λ₃Xn are hereditarily normal. This example is a generalization of famous Gruenhage's example given in Gruenhage and Nyikos (1993) [4].     

A generalization of Andô's theorem and Parrott's example

Andô's theorem states that any pair of commuting contractions on a Hilbert space can be dilated to a pair of commuting unitaries. Parrott presented an example showing that an analogous result does not hold for a triple of pairwise commuting contractions. We generalize both of these results as follows. Any -tuple of contractions that commute according to a graph without a cycle can be dilated to an -tuple of unitaries that commute according to that graph. Conversely, if the graph contains a cycle, we construct a counterexample.

     

S. V. Kovalevskaya system, its generalization and discretization

We consider an integrable three-dimensional system of ordinary differential equations introduced by S. V. Kovalevskaya in a letter to G. Mittag-Leffler. We prove its isomorphism with the three-dimensional Euler top, and propose two integrable discretizations for it. Then we present an integrable generalization of the Kovalevskaya system, and study the problem of integrable discretization for this generalized system.     

A generalization of a theorem by F. Riesz

В работе получено нео бходимое и достаточн ое условие представимости функ ции в виде неопределенного инт еграла от функции огр аниченнойp-вариации в смысле Ви нера. Этот результат является о бобщением одной теор емы Ф. Рисса.     

A generalization of a theorem by C. Miranda

Summary In this paper we study the well-posedness of the Dirichlet problem for an elliptic non divergence form second order equation. The coefficients are not assumed to be continuous but their derivatives are supposed to belong to a suitable Morrey space hence generalizing a classical result by C. Miranda.This work was partly financially supported by a national project of the Italian Ministero della Pubblica Istruzione and by G.N.A.F.A.-C.N.R.     

A generalization of an integral of Ramanujan

This paper considers a generalization of an integral introduced by S. Ramanujan in his third notebook. Ramanujan’s integral is itself a version of the dilogarithm,

A subclass of the distributions of A. Ya. Khinchin

For systems of type M¦G¦1 necessary and sufficient conditions are found for the distribution function of the time of liberation of the system from demands entering up to time v, has the form F(x/e^v), where F(x) is the distribution function of the remaining time of service of demands found in the system at the initial time; in particular, it is noted that the set of all F forms a proper subclass of the class of distributions of A. Ya. Khinchin.Translated from Staticheskie Metody, pp. 23–27, 1978.     

A generalization of an extended stochastic integral

We propose a generalization of an extended stochastic integral to the case of integration with respect to a broad class of random processes. In particular, we obtain conditions for the coincidence of the considered integral with the classical Itô stochastic integral.     

A note on an example by van Mill

Adapting an earlier example by J. van Mill, we prove that there exists a zero-dimensional compact space of countable π-weight and uncountable character that is homogeneous under , but not under .     

A generalization of A. Connes' trace formula

A new Bessel type function is found for each field of p-adic numbers, which has remarkable properties. By using this Bessel function, a Hankel type transformation is defined for every field of p-adic numbers. It is the analogue of the classical Hankel transformation of order zero (Li, 2007 [8]). A global Hankel transformation is defined on the adele group. It is closely related to the Euler product formula of the Riemann zeta function. Local and S-local trace formulas are obtained for the Hankel transformation. They are generalizations of A. Connes' corresponding trace formulas in Connes (1999) [5].     

A generalization of Ruckle's results for an ambush game

This article considers a two-person zero-sum game in which a traveller, player I chooses a point in the closed unit interval [0,1] so as to minimise the probability of being ambushed by two obstacles placed there by his adversary, player II. The payoff to player I equals 1 if the point chosen by him is not in either of the intervals chosen by player II and 0 otherwise. This paper complements the results obtained by Ruckle, Baston and Bostock, and Lee.     

A generalization of a theorem of A. Grothendieck

In this article we characterize the quasi‐barrelledness of the projective tensor product of a coechelon space of type one k ¹(A) with a Fréchet space, including homological conditions as exactness properties of the corresponding tensor product functor k ¹(A) ·: ? → ??, acting from the category of Fréchet spaces to the category of linear spaces, resp. the vanishing of its first right derivative Tor¹_π (k ¹(A),.). This generalizes and extends a classical theorem of A. Grothendieck ([13, Chap. II, §4, No. 3, Theorem 15]). Further we present an analogous theorem for complete coechelon spaces of type zero and the injective tensor product and results concerning the stronger property of barrelledness. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)