Functional equations for Hecke-Maaass series

The Dirichlet (Hecke-Maass) series associated with the eigenfuctionsf andg of the invariant differential operator Δ_k=−y²(∂²/∂x²)+iky∂/∂x of weightk are investigated. It is proved that any relation of the form (f/kM)=g for thek-action of the groupSL ₂ SL ₂(ℝ) is equivalent to a pair of functional equations relating the Hecke-Maass series forf andg and involving only traditional gamma factors. This work was supported by the Russian Foundation for Basic Research (grant No. 96-01-10439). Institute of Applied Mathematics, Far East Division of Russian Academy of Sciences. Translated from Funktional'nyi Analiz i Ego Prilozheniya, Vol. 34, No. 2, pp. 23–32, April–June, 2000. Translated by V. M. Volosov     

Isomorphism types of Rogers semilattices for families from different levels of the arithmetical hierarchy

We investigate differences in isomorphism types for Rogers semilattices of computable numberings of families of sets lying in different levels of the arithmetical hierarchy. Supported by RFBR grant No. 05-01-00819 and by INTAS grant No. 00-499. Supported by NSFC grant No. 60310213. __________ Translated from Algebra i Logika, Vol. 45, No. 6, pp. 637–654, November–December, 2006.     

Δ-spaces

We introduce the notion of a Δ-space and argue that a complete subcategory of the categoryTOP ₀of all topological T₀-spaces, defined by Δ-spaces, is a subdirectly closed subcategory ofTOP ₀that contains many of the known denotational semantic categories of topological spaces as subdirectly closed subcategories. As a consequence, the affirmative answer is given to Scott’s question which inquires whether the category of bifinite domains is a complete subdirectly closed subcategory ofEQU. Supported jointly by RFFR grant No. 96-0-00976 and by DFG grant No. 436-11312670. Translated fromAlgebra i Logika, Vol. 38, No. 6, pp. 667–679, November–December, 1999.     

An autostable 1-decidable model without a computable Scott family of ∃-formulas

We disprove the assumption that the existence of a computable Scott family of ∃-formulas with a finite tuple of parameters from a 1-decidable model is a necessary condition of its being autostable. Given a family S of general recursive functions, we construct a unar m_s and find the family S for which m_s is the desired counterexample. Supported by RFFR grant No. 093-01-01506 and by ISF grant NQ6000. Translated fromAlgebra i Logika, Vol. 35, No. 4, pp. 458–467, July–August, 1996.     

On the separation of spectra in the analysis of Berezin kernels

The problem of restricting a highest weight representation of the group U(p, q) to the subgroup O(p, q) is considered. This restriction has an intricate spectrum that contains representations of different types. We construct a decomposition of this representation into reducible representations each of which has a single-type spectrum. Some integrals over classical groups are also calculated; these integrals generalize those of Hua. Partially supported by grant RFBR 98-01-00303 and the Russian program of support of leading scientific schools (grant RFBR 96-01-96249). Moscow Institute of Electronics and Mathematics. Translated from Funktsional'nyi analiz i Ego Prilozheniya, Vol. 34, No. 3, pp. 49–62, July–September, 2000. Translated by Yu. A. Neretin     

A Modal Logic That is Complete with Respect to Strictly Linearly Ordered <Emphasis Type="Italic">A</Emphasis>-Models

An axiomatization is furnished for a polymodal logic of strictly linearly ordered A-frames: for frames of this kind, we consider a language of polymodal logic with two modal operators, □_< and □_≺. In the language, along with the operators, we introduce a constant β, which describes a basis subset. In the language with the two modal operators and constant β, an Lα-calculus is constructed. It is proved that such is complete w.r.t. the class of all strictly linearly ordered A-frames. Moreover, it turns out that the calculus in question possesses the finite-model property and, consequently, is decidable. __________ Translated from Algebra i Logika, Vol. 44, No. 5, pp. 560–582, September–October, 2005. Supported by RFBR grant No. 03-06-80178, by the Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-2069.2003.1, and by INTAS grant No. 04-77-7080.     

Refining the polynomial hierarchy

By a result of Kadin, the difference hierarchy over NP does not collapse if the polynomial hierarchy does not. We extend this result to a natural refinement of the polynomial hierarchy. The refinement is obtained from levels of the polynomial hierarchy by validly applying addition modulo 2, and so here we call it the plus-hierarchy. Also, we consider two refinements of the plus-hierarchy, which may be relevant to a classification of some languages. Supported by the A. von Humboldt Foundation and by RFFR grant No. 96-01-00257. Translated fromAlgebra i Logika, Vol. 38, No. 4, pp. 456–475, July–August, 1999.     

Minimal permutation representations of finite simple exceptional groups of typesG 2 andF 4

A minimal permutation representation of a group is a faithful permutation representation of least degree. Well-studied to date are the minimal permutation representations of finite sporadic and classical groups for which degrees, point stabilizers, as well as ranks, subdegrees, and double stabilizers, have been found. Here we attempt to provide a similar account for finite simple ezceptional groups of types G₂ and F₄. Supported by RFFR grant No. 96-01-01893, the program “Universities of Russia,” and by International Science Foundation and Government of Russia grant No. RPC300. Translated fromAlgebra i Logika, Vol. 35, No. 6, pp. 663–684, November–December, 1996.     

Embeddings of the classesH ω

In this paper we study functions belonging to the classesV _ε and ΛBV, which are encountered in the theory of Fourier trigonometric series. Necessary and sufficient conditions for the embedding of the classesH _ω in the classesV _ϕ and ABV are obtained. Translated fromMatematicheskie Zametki, Vol. 64, No. 5, pp. 713–719, November, 1998. This research was supported by the program “Leading Scientific Schools” under grant No. 96/97-15-96073.     

Algebraic atomistic lattices of quasivarieties

The Gorbunov-Tumanov conjecture on the structure of lattices of quasivarieties is proved true for the case of algebraic lattices. Namely, for an algebraic atomistic lattice L, the following conditions are equivalent: (1) L is represented as L_q(K) for some algebraic quasivariety K; (2) L is represented as S_Λ (A) for some algebraic lattice A which satisfies the minimality condition and nearly satisfies the maximality conditions; (3) L is a coalgebraic lattice admitting an equaclosure operator. Supported by RFFR grants Nos. 96-01-01525 and 96-0-000976, and by DFG grant No. 436 (RUS) 113/2670. Translated from Algebra i Logika, Vol. 36, No. 4, pp. 363–386, July–August, 1997.     

Lower central series of sylow 2-subgroups of symplectic and orthogonal groups over ℤ2 m

Lower central series are constructed for 2-subgroups of orthogonal and symplectic groups over the ring of residue classes of integers modulo 2^m, treated as exceptional in the previous accounts. Nilpotency classes are found out. Supported by RFFR grant No. 96-01-00409, and by the Krasnoyarsk Foundation for Sciences, grant No. 5f0107. Translated fromAlgebra i Logika, Vol. 37, No. 4, pp. 413–431, July–August, 1998.     

Local description of homogeneous real hypersurfaces of the two-dimensional complex space in terms of their normal equations

In the paper, a classification of real hypersurfaces of the space ℂ² that admit transitive actions of local Lie groups of holomorphic transformations is constructed. Any nonspherical Levi nondegenerate homogeneous surface is determined by the triple of real coefficientsN ₅₂₀ ² ,N ₄₄₀, ImN ₄₂₁ of a Moser normal equation. All such surfaces are described by several quadratic curves in the space of above coefficcients. This work was partially supported by RFBR grant 96-01-01002. Voronezh State Academy of Architecture and Civil Engineering. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 34, No. 2, pp. 33–42, April–June, 2000. Translated by A. V. Loboda     

A superrank of varieties of Lie algebras

A variety of Lie algebras over a field of characteristic 0 has a finite superrank if it is generated by the Grassmann envelope of a finitely generated Lie superalgebra. We prove that every commutator variety not in N_cA has infinite superrank. Consequently, infinite are superranks of all polynilpotent varieties of Lie algebras except N_c and N_cA. Supported by RFFR grant No. 96-01-00146. Translated fromAlgebra i Logika, Vol. 37, No. 4, pp. 394–412, July–August. 1998.     

Extension by zero of functions of several variables

We obtain sufficient conditions on a domainG ⊂ ℝn for functions defined onG to be extendable by zero to the entire space ℝn with smoothness preserved in an integral norm. Translated fromMatematicheskie Zametki, Vol. 64, No. 3, pp. 351–365, September, 1998. This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-00243 and by program “Leading Science Schools” under grant No. 96-15-96102.     

Weight functions on groups and an amenability criterion for Beurling algebras

The paper is devoted to the study of weights on groups. A connection between weight functions and harmonic functions is established. A relationship between the weight theory on groups with the “Tychonoff property” and the theory of bounded cohomology is presented. It is proved that the Beurling algebraℓ1 (G, ω) constructed for the weightω is amenable if and only if the groupG is amenable and the weightω is equivalent to a multiplicative characterχ:G→ℝ₊. Translated fromMatematicheskie Zametki, Vol. 60, No. 3, pp. 448–460, September, 1996. This research was partially supported by the Russian Foundation for Basic Research under grant No. 96-01-00974 and by the INTAS Foundation under grant No. 94-3420.     

Δ 2 0 -copies of linear orderings

It is proved that, for any n ∈ ω, there exist countable linear orderings L_n whose Δ ₂ ⁰ -spectrum consists of exactly all non n-low Δ ₂ ⁰ -degrees. Properties of such orderings are examined, for n = 1 and n = 2. Supported by RFBR grant No. 02-01-00169 and by RF Ministry of Education grant No. E02-1.0-177. __________ Translated from Algebra i Logika, Vol. 45, No. 3, pp. 354–370, May–June, 2006.     

Solvable elimination of ramification in extensions of discretely valued fields

An analog of the Abhyankar lemma is proved for an arbitrary finite separable extensions of discretely valued fields with a functional residue field. Supported by RFFR grants Nos. 96-01-01675 and 96-01-01678, and by RF Goskomvuz grant No. 2. Translated fromAlgebra i Logika, Vol. 37, No. 1, pp. 63–87., January–February, 1998.     

Minimal permutation representations of finite simple exceptional groups of typesE 6,E 7, andE 8

A minimal permutation representation of a group is its faithful permutation representation of least degree. We will find degrees and point stabilizers, as well as ranks, subdegrees, and double stabilizers, for groups of types E₆, E₇, and E₈. This brings to a close the study of minimal permutation representations of finite simple Chevalley groups. Supported by RFFR grant No. 93-01-01501, through the program “Universities of Russia,” and by grant No. RPC300 of ISF and the Government of Russia. Translated fromAlgebra i Logika, Vol. 36, No. 5, pp. 518–530, 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.     

Asymptotics of the parabolic Green function for an elliptic operator on a manifold with conical points

We construct the asymptotics ast→0 of the trace of the operator exp(−tP) for an elliptic operatorP on a manifold with conical points. Translated fromMatematicheskie Zametki, Vol. 63, No. 1, pp. 28–36, January, 1998. This research was supported by the Belarus Foundation for Basic Research under grant No. 96-01 00790.