Interpolation in superintuitionistic predicate logics with equality

It is shown that the Craig interpolation property and the Beth property are preserved under passage from a superintuitionistic predicate logic to its extension via standard axioms for equality, and under adding formulas of pure equality as new axioms. We find an infinite independent set of formulas which, though not equivalent to formulas of pure equality, may likewise be added as new axiom schemes without loss of the interpolation, or Beth, property. The formulas are used to construct a continuum of logics with equality, which are intermediate between the intuitionistic and classical ones, having the interpolation property. Moreover, an equality-free fragment of the logics constructed is an intuitionistic predicate logic, and formulas of pure equality satisfy all axioms of the classical predicate logic. Supported by RFFR grant No. 96-01-01552. Translated fromAlgebra i Logika, Vol. 36, No. 5, pp. 543–561, September–October, 1997.     

Linear differential operators with unbounded operator coefficients and semigroups of bounded operators

Associated with a family of evolution operators in a complex Banach space is a linear unbounded operator, which is studied with the aid of a semigroup of difference operators and a difference operator in a sequence space. Some formulas for the spectra of the linear operators in question (in particular, for abstract hyperbolic differential operators) and the spectrum mapping theorem for the semigroup of difference operators are obtained. Translated fromMatematicheskie Zametki, Vol. 59, No. 6, pp. 811–820, June, 1996. This research was supported by the Russian Foundation for Basic Research under grant No. 95-01-00032 and by the International Science Foundation under grant No. NZA000 and grant No. NZA300.     

Algebraic semantics for superintuitionistic predicate logics

We construct a unified algebraic semantics for superintuitionistic predicate logics. Assigned to each predicate logic is some deductive system of a propositional language which is kept fixed throughout all predicate superintuitionistic ones. Given that system, we build up a variety of algebras w.r.t. which a given logic is proved to be strongly complete. Supported by the Russian Arts Foundation (RAF), grant No. 97-03-04089a. Translated fromAlgebra i Logika, Vol. 38, No. 1, pp. 68–95, January–February, 1999.     

K-groups of quadratic extensions of rings

We construct the Levin braid connecting the Tate cohomology ofK-groups for quadratic extensions of rings with antistructures. Explicit formulas for isomorphisms of relative cohomology groups for the induced mapping and for the transfer mapping are obtained; these formulas are necessary in the construction of the Levin braid.Translated fromMatematicheskie Zametki, Vol. 58, No. 2, pp. 272–280, August, 1995.This research was partially supported by the Russian Foundation for Basic Research under grant No. 93-011-1402.     

The number of periodic solutions of polynomial differential equations

We estimate the number of periodic solutions for special classes ofnth-order ordinary differential equations with variable coefficients. Translated fromMatematicheskie Zametki, Vol. 64, No. 5, pp. 720–727, November, 1998. The author thanks Yu. S. Il'yashenko for setting the problems, permanent advice, and overall support. The author is also thankful to D. A. Panov for numerous discussions. This research was supported by the CRDF Foundation under grant MR1-220, by the INTAS Foundation under grant No. 93-05-07, and by the Russian Foundation for Basic Research under grant No. 95-01-01258.     

Use of the Lagrange method in nonlinear analysis

The problem of the spectrum of a nonlinear system and its relation to ellipsoid transform widths are studied.Translated fromMatematicheskie Zametki, Vol. 59, No. 3, pp. 334–342, March, 1996.This research was partially supported by the Russian Foundation for Basic Research under grant No. 93-01-00237 and by the International Science Foundation under grant No. MP1000.     

Infinite arithmetic formulas and the reflection principle

An extension of the language of arithmetic is constructed such that it allows us to work with recursive sequences of arithmetic formulas as if it were a single formula. It is proved that Feferman's reflection principles are inferable in the extension obtained. Supported by the Competitive Center for Basic Research (CCBR), grant No. 93-1-88-12. Translated fromAlgebra i Logika, Vol. 36, No. 3, pp. 245–258, May–June, 1997.     

Asymptotics ast → ∞ of the solutions of nonlinear equations with nonsmall initial perturbations

A new method is proposed for finding asymptotics ast → ∞ of the solutions of the Cauchy problem for nonlinear evolution equations with nonsmall initial data. Translated fromMatematicheskie Zametki, Vol. 59, No. 6, pp. 855–864, June, 1996. This research was supported by the Russian Foundation for Basic Research under grant No. 93-011-134 and by the International Science Foundation under grant No. BX000.     

Differentiability of convex measures

In this paper we consider convex measures on finite-dimensional spaces. We prove the differentiability of convex measures in the Skorokhod sense (and under some natural conditions, in the Fomin sense also). Simultaneously we give some additional results on differentiability of convex measures.Translated fromMatematicheskie Zametki, Vol. 58, No. 6, pp. 862–871, December, 1995.This research was partially supported by the Russian Foundation for Basic Research under grant No. 94-01-01556, by the Ministry of Science under grant No. M38300 and by the International Science Foundation under grant No. M38000.     

Integral characteristics in the inverse problem of magnetotelluric sounding

This article examines some aspects of the one-dimensional inverse problem of magnetotelluric sounding. A uniqueness theorem is proved in the presence ofS-surfaces. A numerical algorithm based on transformation formulas is proposed. This research was partially supported by Russian Foundation for Basic Research (grant No. 96-01-00410) and by the State Scientific-Technical Program “Future Information Technologies” (grant No. 0201.06.010). Translated from Chislennye Metody v Matematicheskoi Fizike, Moscow State University, pp. 53–66, 1998.     

Integrable boundary conditions for many-component burgers equations

Infinite series of boundary conditions that are consistent with even-order higher symmetries and ensure the integrability of a Burgers type equation are constructed. Translated fromMatematicheskie Zametki, Vol. 60, No. 6, pp. 888–901, December, 1996. This research was supported by the Russian Foundation for Basic Research under grant No. 93-011-16008 and by the International Science Foundation under grant Nos. MLY00 and RK2000.     

Volumes of hyperbolic Löbell 3-manifolds

In 1931 F. Löbell constructed the first example of a closed orientable three-dimensional hyperbolic manifold. In the present paper we study properties of closed hyperbolic 3-manifolds generalizing Löbell's classical example. Explicit formulas for the volumes of these manifolds in terms of the Lobachevski function are obtained.Translated fromMatematicheskie Zametki, Vol. 64, No. 1, pp. 17–23, July, 1998.This research was partially supported by GARC-KOSEF (Global Analysis Research Center of National Seoul University) and by the Russian Foundation for Basic Research under grant No. 95-01-01410.     

Solutions of the heat equation in domains with singularities

We study the asymptotics of solutions to the Dirichlet problem for the heat equation in time-dependent domains with singular points.Translated fromMatematicheskie Zametki, Vol. 64, No. 2, pp. 163–179, August, 1998.This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-00504 and by INTAS under grant No. 93-351.     

Modules over quantum polynomials

We describe the structure of finitely generated modules over general quantum Laurent polynomials and prove that Artin modules over general quantum Laurent polynomials are cyclic. Translated fromMatematicheskie Zametki, Vol. 59, No. 4, pp. 497–503, April, 1996. This research was partially supported by the Russian Foundation for the Basic Research under grant No. 93-011-1544 and by the INTAS program under grant No. 93-2618.     

Critical cases of stability. Converse implicit function theorem for dynamical systems with cosymmetry

Stability criteria of boundary equilibria for dynamical systems in the three critical cases, (n, k)=(3, 0), (2, 1), and (1, 1), are obtained.Translated fromMatematicheskie Zametki, Vol. 63, No. 4, pp. 572–578, April, 1998.The author wishes to thank V. I. Yudovich for useful discussions.This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-01791 and by the International Science Foundation under grant NRQ000.     

Regular sets in residual classes

Sets regular modulo a fixed odd prime power are explicitly constructed under the condition that their cardinalities do not exceed an arbitrarily small positive power of the modulus.Translated fromMatematicheskie Zametki, Vol. 64, No. 2, pp. 224–228, August, 1998.This research was supported by the Russian Foundation for Basic Research under grant No. 97-01-00721 and by the Professor B. Novak grant (Karlov University, Prague).     

Extremum problems for functions with small support

In this paper we consider an extremum problem for even periodic functions having small intervals as their supports and subject to restrictions on the sum of the absolute values of their Fourier coefficients. Translated fromMatematicheskie Zametki, Vol. 60, No. 3, pp. 323–332, September, 1996. We wish to express our gratitude to S. B. Stechkin for his interest in our work. This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-00378, and the work of the first and second authors also by the International Science Foundation under grant No. MC5300.     

Some conic bundless

A representation of the anticanonical K3 surface of a singular pencil of conics is described. This generalizes the well-known Shokurov theorem.Translated fromMatematicheskie Zametki, Vol. 63, No. 6, pp. 903–910, June, 1998.The author is greatly indebted to V. A. Iskovskikh, Yu. G. Prokhorov, and I. A. Chel'tsov for fruitful discussions.This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-00820 and by INTAS under grant No. 93-2805.00-00-00.     

Growth functions,rewriting systems,and the Euler characteristic

We present a formula that expresses the number of admissible words for an arbitrary system of prohibitions in terms of the generating series. We also indicate applications of this formula to the calculation of the topological entropy of symbolic systems and to the calculation of growth functions of groups and algebras. Translated fromMatematicheskie Zametki, Vol. 58, No. 5, pp. 653–668, November, 1995. This research was partially supported by the Russian Foundation for Basic Research under grant No. 94-01-00820 and by the International Science Foundation grant MV1000.     

Estimates of trigonometric sums by the third derivative

The well-known Van der Corput estimate of a trigonometric sum by the third derivative is sharpened. Translated fromMatematicheskie Zametki, Vol. 60, No. 3, pp. 383–389, September, 1996. This research was supported by the Russian Foundation for Basic Research under grant No. 94-01-00002.