Predicate logics of expressively strong theories

It is proved that the logicL(T) coincides with the predicate calculus for the theory of rings, the theory of fields, group theory, and all arithmetically correct theoreies. Translated fromMatematicheskie Zametki, Vol. 66, No. 5, pp. 777–788, November, 1999.     

Upper bound on the occupation time in the simple exclusion process

We consider the occupation time variance in the asymmetric exclusion process. In the case where the mean m ≠ 0 (m is the mean hopping rate) and ρ = 1/2 (ρ is the filling probability for a state), we find that the variance is bounded above by O(t ^3/2 ). __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 1, pp. 147–158, July, 2008.     

Remark on the dissipation of the magnetic helicity integral

We prove that the current helicity integral for a thin magnetic tube is directly proportional to the twist Tw of field lines, which appears in the Călugărianu formula as a summand. Using this approach, we analyze the law of the variation of magnetic helicity in a conducting medium with a nonzero magnetic diffusion coefficient and also the equation of the phase transition of the magnetic field under early universe conditions. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 158, No. 1, pp. 150–160, January, 2009.     

A short proof of the twelve-point theorem

We present a short elementary proof of the following twelve-point theorem. Let M be a convex polygon with vertices at lattice points, containing a single lattice point in its interior. Denote by m (respectively, m*) the number of lattice points in the boundary of M (respectively, in the boundary of the dual polygon). Then m + m* = 12.Translated from Matematicheskie Zametki, vol. 77, no. 1, 2005, pp. 117–120.Original Russian Text Copyright © 2005 by D. Repov, M. Skopenkov, M. Cencelj.This revised version was published online in April 2005 with a corrected issue number.     

On Estimates of Factorial Quotients

We consider the factorial quotients (2n − 1)!!/(2n)!! in connection with the Wallis formula n ⁻¹(2n)!!²/(2n − 1)!!² → π. We improve the Wallis inequalities (n + 1/2)⁻¹(2n)!!²/(2n − 1)!!² < π < n ⁻¹(2n)!!²/(2n − 1)!!² for π and obtain new estimates of factorial quotients with error order not worse than 1/n ². __________ Translated from Lietuvos Matematikos Rinkinys, Vol. 45, No. 3, pp. 349–358, July–September, 2005.     

Polynomials with high multiplicity at unity and Tarry’s problem

The existence of a polynomial with integer coefficients of relatively small length and sufficiently large multiplicity at unity is established. The proof of the corresponding statement is based on an estimate of the number of solutions of Tarry’s system. Translated fromMatematicheskie Zametki, Vol. 65, No. 6, pp. 810–815, June, 1999.     

On the slow growth of the main characteristics of entire functions

We obtain a necessary and sufficient condition for the slow growth at infinity in terms of the coefficients in the series expansion of an entire function. Translated fromMatematicheskie Zametki, Vol. 65, No. 2, pp. 206–214, February, 1999.     

Cauchy-type problem for an abstract differential equation with fractional derivatives

The uniform well-posedness of a Cauchy-type problem with two fractional derivatives and bounded operator A is proved. For an unbounded operator A we present a test for the uniform well-posedness of the problem under consideration consistent with the test for the uniform well-posedness of the Cauchy problem for an equation of second order.Translated from Matematicheskie Zametki, vol. 77, no. 1, 2005, pp. 28–41.Original Russian Text Copyright © 2005 by A. V. Glushak.This revised version was published online in April 2005 with a corrected issue number.     

Extremum problems for entire functions of exponential spherical type

We consider extremum problems for entire functions of exponential spherical type related to important extremum problems on the optimal point (the Chernykh point) in the sharp jackson inequality in the spaceL ₂(ℝ ⁿ ) and the connection between codes and designs on the torusT ⁿ . Translated fromMatematicheskie Zametki, Vol. 68, No. 2, pp. 179–187, August, 2000.     

The leading term of the spectral asymptotics for the Kohn-Laplace operator in a bounded domain

We prove the Weyl asymptotic formula for the number of eigenvalues of the Kohn-Laplace operator on a Heisenberg group and write out the leading term of asymptotics. The method of study is based on estimates of the Green function for the Dirichlet problem for the corresponding parabolic operator and makes use of the classical Hardy-Littlewood Tauberian theorem.Translated fromMatematicheskie Zametki, Vol. 64, No. 4, pp. 493–505, October, 1998.This research was supported by the Russian Foundation for Basic Research under grants No. 96-01-00443 and No. 96-01-00503.     

On superpositions of continuous functions defined on the baire space

For continuous functions specified on the Baire space, conditions for the representability of a function of several variables as a superposition of functions of a smaller number of variables are considered. With the use of linear functions of the form (1+α)t, a boundary value of the modulus of continuity separating the positive from the negative solution of the problem is found. For the case in which the problem has a negative solution, a constructive method for obtaining (n+1)-variable continuous functions with modulus of continuity ϕ(t) that are not representable as superposition ofn-variable continuous functions with the same modulus of continuity ϕ(t) is suggested. Translated fromMatermaticheskie Zametki, Vol. 66, No. 5, pp. 696–705, November, 1999.     

Analogues of Euler and Poisson summation formulae

Euler-Maclaurin and Poisson analogues of the summations ε _{a <n ≤b} χ(n)f(n), have been obtained in a unified manner, where (χ(n)) is a periodic complex sequence;d(n) is the divisor function andf(x) is a sufficiently smooth function on [a, b]. We also state a generalised Abel’s summation formula, generalised Euler’s summation formula and Euler’s summation formula in several variables.     

The index of singular integral operators in reflexive Orlicz spaces

We consider the Banach algebra of singular integral operators with matrix piecewise continuous coefficients in the reflexive Orlicz spaceL _M ⁿ (Γ). We assume that Γ belongs to a certain wide subclass of the class of Carleson curves; this subclass includes curves with cusps, as well as curves of the logarithmic spiral type. We obtain an index formula for an arbitrary operator from the algebra in terms of the symbol of this operator. Translated fromMatematicheskie Zametki, Vol. 64, No. 3, pp. 383–396, September, 1998.     

Darboux’s identity and its analogs

Let us choose a positive integern and denoteF(x, y)= , wheref(·) andg(·) are arbitrary sufficiently smooth functions. Three different proofs of the validity of the relation

Lefschetz Formulae for p -Adic Groups

Abstract In this paper, Lefschetz formulae for torus actions on p-adic groups are proven. These are similar to comparable formulae for real Lie groups. Applications lie in the realm of dynamical zeta functions.     

A new Leray formula for smooth functions on bounded domains in Cn

By means of a new technique of integral representations in C ⁿ given by the authors, we establish a new abstract formula with a vector function W for smooth functions on bounded domains in C ⁿ , which is different from the well-known Leray formula. This new formula eliminates the term that contains the parameter A from the classical Leray formula, and especially on some domains the uniform estimates for the are very simple. From the new Leray formula, we can obtain correspondingly many new formulas for smooth functions on many domains in C ⁿ , which are different from the classical ones, when we properly select the vector function W.     

Semigroups presented by one relation and satisfying the Church-Rosser property

This paper is devoted to the study of semigroups presented by a single defining relationA=B and satisfying the Church-Rosser property. Translated fromMatematicheskie Zametki, Vol. 61, No. 1, pp. 114–118, January, 1997. Translated by A. I. Shtern     

From Tanaka's formula to Ito's formula: The fundamental theorem of stochastic calculus

In this article we give a new proof of Ito's formula inR ⁿ starting from the one-dimensional Tanaka formula. The proof is algebraic and does not use any limiting procedure. It uses the integration by parts formula, Fubini's theorem for stochastic integrals and essential properties of local times.