Modal positive operators

For every modal positive operator, there exists a formula defining a least fixed point for that operator in partially ordered Kripke models with the property of being cofinal for every infinite ascending chain. A similar result obtains also for strict partially ordered Kripke models with the same property. Supported by the Russian Humanitarian Science Foundation (RHSF), grant No. 97-03-04089, and by the Siberian Branch of the Russian Academy of Science through PSO RAN No. 473, grant No. 3. Translated fromAlgebra i Logika, Vol. 38, No. 5, pp. 585–597, September–October, 1999.     

Endomorphisms of the group of all recursive permutations

We describe all endomorphisms of the group Aut_T ɛ of all recursive permutations. It is proved that the family of these endomorphisms is countable, and that they all are continuous and may be defined by some natural recursive operators. Orbits relative to the image of Aut_Tω prove to be recursive, and there exists a recursive model M such that this image is exactly its recursive automorphism group. There exists a universal endomorphism which contains, in a sense, all endomorphisms of that group. The universal endomorphism is unique with respect to some natural recursive equivalence. Supported by RFFR grant No. 093-01-01525. Translated fromAlgebra i Logika, Vol. 36, No. 1, pp. 54–76, January–February, 1997.     

Intermediate predicate logic without the beth property

It is shown that a logic J _fd ^* characterized by all Kripke frames the domains of all nonmaximal worlds of which are finite lacks the Beth property. The logic is the first example of an intermediate superintuitionistic logic without the Beth property. The interpolation and the Beth properties are also proved missing in all predicate superintuitionistic logics which contain J _fd ^* and are contained in a logic characterized by frames of the form〈N _n , ≤,{Dk}k∈N n〉. Supported by the Russian Foundation for Humanities, grant No. 97-03-04089. Translated fromAlgebra i Logika, Vol. 37, No. 1, pp. 107–117, January–February, 1998.     

On the uniform convergence of wavelet expansions of random processes from Orlicz spaces of random variables. I

We establish conditions under which there exists a function c(t) > 0 such that {fx1850-01}, where X(t) is a random process from an Orlicz space of random variables. We obtain estimates for the probabilities {fx1850-02}. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 12, pp. 1647–1660, December, 2007.     

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.     

Representation of green’s function for the heat equation on a compact lie group

We obtain an explicit formula which presents the solution of the heat equation on a compact Lie group as the limit of finite-to-one convolutions of Green’s function for the heat equation in Euclidean space. Translated fromMatematicheskie Zametki, Vol. 67, No. 3, pp. 397–413, March, 2000.     

Spectra of computable models for Ehrenfeucht theories

We construct an example of a theory with a finite (greater than one) number of isomorphism types of countable models such that its prime and saturated models have computable presentations and there exists a model which lacks in such. Supported by the Council for Grants (under RF President) and State Aid of Fundamental Science Schools via project NSh-4413.2006.1. __________ Translated from Algebra i Logika, Vol. 46, No. 3, pp. 275–289, May–June, 2007.     

The Cauchy Boundary Value Problems on Closed Piecewise Smooth Manifolds in <Emphasis Type="Italic">C</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Suppose that D is a bounded domain with a piecewise C^1 smooth boundary in C^n. Let ψ∈C^1 α(δD). By using the Hadamard principal value of the higher order singular integral and solid angle coefficient method of points on the boundary, we give the Plemelj formula of the higher order singular integral with the Boehner-Martinelli kernel, which has integral density ψ. Moreover, by means of the Plemelj formula and methods of complex partial differential equations, we discuss the corresponding Cauehy boundary value problem with the Boehner-Martinelli kernel on a closed piecewise smooth manifold and obtain its unique branch complex harmonic solution.     

On the inverse problem of finding the coefficient of heat transfer

We establish conditions for the existence and uniqueness of a solution to the problem of finding the coefficient of heat transfer in a boundary condition of third kind. We give an explicit formula for computing this coefficient. Translated fromMatermaticheskie Metody i Fiziko-Mekhanicheskie Polya, No. 37, 1994, pp. 45–50.     

Barwise's information frames and modal logics

 The paper studies Barwise's information frames and answers the John Barwise question: to find axiomatizations for the modal logics generated by information frames. We find axiomatic systems for (i) the modal logic of all complete information frames, (ii) the logic of all sound and complete information frames, (iii) the logic of all hereditary and complete information frames, (iv) the logic of all complete, sound and hereditary information frames, and (v) the logic of all consistent and complete information frames. The notion of weak modal logics is also proposed, and it is shown that the weak modal logics generated by all information frames and by all hereditary information frames are K and K4 respectively. To develop general theory, we prove that (i) any Kripke complete modal logic is the modal logic of a certain class of information frames and that (ii) the modal logic generated by any given class of complete, rarefied and fully classified information frames is Kripke complete. This paper is dedicated to the memory of talented mathematician John Barwise. Received: 7 May 2000 Published online: 10 October 2002 Key words or phrases: Knowledge presentation – Information – Information flow – Information frames – Modal logic-Kripke model     

Submodels of Kripke models

A Kripke model ? is a submodel of another Kripke model ℳ if ? is obtained by restricting the set of nodes of ℳ. In this paper we show that the class of formulas of Intuitionistic Predicate Logic that is preserved under taking submodels of Kripke models is precisely the class of semipositive formulas. This result is an analogue of the Łoś-Tarski theorem for the Classical Predicate Calculus. In Appendix A we prove that for theories with decidable identity we can take as the embeddings between domains in Kripke models of the theory, the identical embeddings. This is a well known fact, but we know of no correct proof in the literature. In Appendix B we answer, negatively, a question posed by Sam Buss: whether there is a classical theory T, such that ℋT is HA. Here ℋT is the theory of all Kripke models ℳ such that the structures assigned to the nodes of ℳ all satisfy T in the sense of classical model theory. Received: 4 February 1999 / Published online: 25 January 2001     

The mean square of the DirichletL-functions in the critical strip

We study an asymptotic formula of the DirichletL-functions in the critical strip. This is an analogy of the Atkinson-type formula for DirichletL-functions. Published in Lietuvos Matematikos Rinkinys, Vol. 40, No. 2, pp. 201–213, April–June, 2000.     

Recognition of simple groups U 3(Q) by element orders

An exhaustive solution is given to the recognition-by-spectrum problem for finite, simple, three-dimensional unitary groups. For every such group, the number of non-isomorphic, finite, isospectral groups is determined. In particular, a new counterexample to Problem 13.63 in the Kourovka Notebook is furnished. Supported by RFBR grant No. 05-01-00797, and by SB RAS Young Researchers Support grant No. 29 and Integration Project No. 2006.1.2. __________ Translated from Algebra i Logika, Vol. 45, No. 2, pp. 185–202, March–April, 2006.     

Problem without initial conditions for linear and almost linear degenerate operator differential equations

We study the problem without initial conditions for linear and almost linear degenerate operator differential equations in Banach spaces. The uniqueness of a solution of this problem is proved in the classes of bounded functions and functions with exponential behavior as t → –∞. We also establish sufficient conditions for initial data under which there exists a solution of the considered problem in the class of functions with exponential behavior at infinity.     

A singularly perturbed boundary value problem for a second-order equation in the case of variation of stability

A boundary value problem for a second-order nonlinear singularly perturbed differential equation is considered for the case in which there is variation of stability caused by the intersection of roots of the degenerate equation. By the method of differential inequalities, we prove the existence of a solution such that the limit solution is nonsmooth. Translated fromMatematicheskie Zametki, Vol. 63, No. 3, pp. 354–362, March, 1998. This research was partially supported by the Russian Foundation for Basic Research under grant No. 96-01-00694.     

An asymptotic formula of the Hellinger integral for renewal processes

Asymptotic formulas of the Hellinger integral are used in the investigation of properties of optimal estimates and statistical criteria. For a certain class of renewal processes, this formula was obtained by the author in [Lith. Math. J.,38(2), 131–143 (1998)]. In this paper, we obtain such a formula for all renewal processes whose intermediate renewal moments have absolutely continuous distributions. We use the traditional representation of the Hellinger integral and the theory of large deviations. Šiauliai University, Višinskio 25, 5400 Šiauliai, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 39, No. 4, pp. 493–497, October–December, 1999. Translated by V. Mackevičius     

Oseen and Stokes asymptotics for the problem on stationary motion of two immiscible liquids

The main result is an asymptotic formula for a solution to the conjugation problem for the Navier-Stokes equations describing the slow motion of two immiscible liquids such that one of them occupies a bounded domain Ω₁ ⊂ ℝ³, whereas the other occupies the exterior domain Ω₂=ℝ⁴∖Ω. Such a formula was obtained for a solution to the exterior problem with sticking conditions on the boundary in the works of Fischer, Hsiao, and Wendland. The result obtained is applied to the proof of the solvability of a free-boundary problem describing a uniform drop in an infinite liquid. Bibliography: 10 titles. Translated fromProblemy Matematicheskogo Analiza, No. 16. 1997, pp. 208–238.     

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.