Four counterexamples to the Fubini theorem

In this paper we study signed measures. Our main results are as follows: the Fubini theorem is not true in the general case; the Jordan parts of a transition measure are not necessarily transition measures; the operation of taking the Jordan parts does not necessarily commute with multiplying by the initial measure; the product of σ-bounded measures need not be a σ-bounded measure. Translated fromMatematicheskie Zametki, Vol. 62, No. 1, pp. 124–127, July, 1997. Translated by S. S. Anisov     

Construction of non-Gaussian surface measures by Malliavin’s method

We generalize the Airault-Malliavin theorem on the existence of surface measures on infinite-dimensional spaces with Gaussian measures on surfaces. We prove that the sets of capacities generated by Sobolev classes on infinite-dimensional spaces are dense. Translated fromMatematicheskie Zametki, Vol. 65, No. 3, pp. 377–388, March, 1999.     

Generalized differentiable product measures

A class of measures on ℝ^∞ determined by sequences of functions of finitely many variables is considered. An existence theorem for such measures is proved, and their properties are examined. Examples are presented. Translated fromMatematicheskie Zametki, Vol. 63, No. 1, pp. 37–55, January, 1998. The author is greatly indebted to O. V. Zimina for many stimulating discussions. This research was supported by the Russian Foundation for Basic Research under grant No. 96-01 01701.     

On the order of growth of generalized eigenfunctions of the Sturm-Liouville operator. The Shnol’ theorem

For the generalized eigenfunctions corresponding to a Sturm-Liouville operator on a semiaxis, growth estimates are obtained. In the paper, in contrast to the known Shnol’ theorem, the dependence of the growth order on the behavior at infinity of the potential is studied. Translated fromMatematicheskie Zametki, Vol. 67, No. 1, pp. 46–51, January, 2000.     

Convergence of a sequence of weakly regular set functions

The present paper is devoted to generalizations of the Dieudonné theorem claiming that the convergence of sequences of regular Borelian measures is preserved under the passage from a system of open subsets of a compact metric space to the class of all Borelian subsets of this space. The Dieudonné theorem is proved in the case for which the set functions are weakly regular, nonadditive, defined on an algebra of sets that contains the class of open subsets of an arbitrary σ-topological space, and take values in a uniform space. Translated fromMatematicheskie Zametki, Vol. 62, No. 1, pp. 103–110, July, 1997. Translated by O. V. Sipacheva     

Value-distribution of the Lerch zeta-function with algebraic irrational parameter. I

We prove a limit theorem in the sense of weak convergence of probability measures on the complex plane for the Lerch zeta-function with algebraic irrational parameter. The limit measure in this theorem is explicitly given. Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 2, pp. 163–176, April–June, 2007.     

Simple proof of the nonrationality of three-dimensional quartics

The rigidity theorem for smooth three-dimensional quartics [1] is reproved following an idea of Corti [2] and using Shokurov’s connectedness theorem. Translated fromMatematicheskie Zametki, Vol. 65, No. 5, pp. 667–673, May, 1999.     

Hausdorff dimension and the dynamics of diffeomorphisms

A proof of the Hilbert-Smith conjecture for a free Lipschitz action is given. The proof is elementary in the sense that it does not rely on Yang’s theorem about the cohomology dimension of the orbit space of thep-acid action. The result turns out to be true for the class of spaces of finite Hausdorff volume, which is considerably wider than Riemannian manifolds. As a corollary to the Lipschitz version of the Hilbert-Smith conjecture, the theorem asserting that the diffeomorphism group of a finite-dimensional manifold has no small subgroups is obtained. Translated fromMatermaticheskie Zametki, Vol. 65, No. 3, pp. 457–463, March, 1999.     

Piecewise-smooth<Superscript>+</Superscript> version of the implicit-function theorem

We introduce a class of piecewise-smooth⁺ mappings and prove the implicit-function theorem for this class. The proof is based on the theorem on global homeomorphism, which follows from the well-known Chernavskii theorem. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 5, pp. 619–625, May, 2008.     

Some limit theorems for a Dirichlet series related to the Euler function

Using some conjectures on the mean-value of generalized numbers, limit theorems in the sense of the weak convergence of probability measures for the Dirichlet series related to the Euler function are proved. Also, the limit measure is studied. Partially supported by the Lithuanian State Science and Studies Foundation. Vilnius University, Naugarduko 24, 2006 Vilnius, Lithuania: Šiauliai University, P. Višinskio 25, 5400 Šiauliai, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 39, No. 3, pp. 331–342, July–September, 1999. Translated by A. Laurinčikas     

The Perron–Frobenius theorem – a proof with the use of Markov chains

The Perron–Frobenius theorem for an irreducible nonnegative matrix is proved using the matrix graph and the ergodic theorem of the theory of Markov chains. Bibliography: 7 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 359, 2008, pp. 5–16.     

Two analogs of the Rudin-Carleson theorem

In this article, a new asymptotic estimate for embeddings of singular measures into H^1,∞ is proved and an application to multidimensional analogs of the theorem of Rudin-Carleson is given. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 247, 1997, pp. 200–209. Translated by S. V. Kislyakov.     

Pharpening of a feller theorem on the law of iterated logarithms

In the paper, a sharpened version of a theorem of W. Feller on the law of iterated logarithms is proved. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 216, 1994, pp. 117–123. Translated by the author.     

Variational problem of the two-phase medium elasticity theory for the zero surface tension coefficient

In this article, we give a proof of the existence theorem for an equilibrium state for the surface tension coefficient σ=0 and investigate the behavior of the equilibrium state for small σ. Bibliography: 4 titles. Dedicated to N. N. Uraltseva on her jubilee Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 221, 1900, pp. 208–225. Translated by S. Yu. Pilyugin.     

On the Alexandroff decomposition theorem

We prove an Alexandroff decomposition type theorem, which extends a decomposition theorem proved in [de LUCIA, P.—MORALES, P.: Decomposition of group-valued measures in orthoalgebras, Fund. Math. 158 (1998), 109–124].      

On the theorem on asymptotic equidistribution of the convolution powers of symmetric measures on a unimodular group

The theorem on the asymptotic equidistribution of the convolution powers of a symmetric probability measure given on a unimodular group [1] deals with a measure whose convolution powers, starting from one of them, load an arbitrary prescribed nonempty subset of the group. In the present note, we indicate the modifications that arise under the replacement of the above condition by the requirement that the smallest closed subgroup (of the group considered) generated by the support of this measure coincide with the group itself. Translated fromMatematicheskie Zametki, Vol. 60, No. 1, pp. 120–126, July, 1996. This research was partially supported by the International Science Foundation under grant No. 3J000.     

On the asymptotic independence of Dirichlet series

The joint limit distribution of functions given by Dirichlet series is studied. The necessary and sufficient condition when this distribution is a product of marginal distributions is found. An example of such Dirichlet series with linear independent systems of exponents is presented. Partially supported by the Lithuanian State Science and Studies Foundation. Vilnius University, Naugarduko 24, 2006 Vilnius; Šiauliai University, P. Višinskio 25, 5419 Šiauliai, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 39, No. 1, pp. 65–73, January–March, 1999. Translated by A. Laurinčikas     

A generalization of weyl’s inequalities with implications

This paper suggests a generalization of the additive Weyl inequalities to the case of two square matrices of different orders. As a consequence of the generalized Weyl inequalities, a theorem describing the location of eigenvalues of a Hermitian matrix in terms of the eigenvalues of an arbitrary Hermitian matrix of smaller order is derived. It is demonstrated that the latter theorem provides a generalization of Kahan’s theorem on clustered eigenvalues. It is also shown that the theorem on extended interlacing intervals is another consequence of the generalized additive Weyl inequalities suggested. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 248, 1998, pp. 49–59. Translated by L. Yu. Kolotilina.     

On the law of the iterated logarithm for increments of sums of independent random variables

We obtain the law of the iterated logarithm for increments of sums of independent random variables. Our results generalize the Kolmogorov theorem and the Hartman—Wintner theorem on the law of the iterated logarithm. Bibliography: 17 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 320, 2004, pp. 174–186.     

Nonpolyhedral proof of the Michael finite-dimensional selection theorem

We suggest a new method of proving the Michael finite-dimensional selection theorem. Using it, we prove a new selection theorem. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 4, pp. 3–22, 2005.