Solutions for the prescribing mean curvature equation

By variational methods, for a kind of Yamabe problem whose scalar curvature vanishes in the unit ball BN and on the boundary S^N-1 the mean curvature is prescribed, we construct multi-peak solutions whose maxima are located on the boundary as the parameter tends to 0^＋ under certain assumptions. We also obtain the asymptotic behaviors of the solutions.     

Asymptotic Smoothing Effect of Solutions to Davey-Stewartson Systems on the Whole Plane

This paper discusses the Cauchy problem of elliptic-elliptic-type Davey-Stewartson systems with zero-order dissipation on R^2. Making use of the Fourier spectral projector, together with a long time comparison between the solutions to the Davey-Stewartson systems and to an auxiliary problem, we prove that the global attractor in H^1 （R^2） for the addressed Davey-Stewartson systems is a compact subset of H^2（R^2）, and thus reveal the asymptotic smoothing effect of the solutions for the systems.     

Recognition of finite simple groupsL 3(2 m ) ANDU 3(2 m ) by their element orders

It is proved that a finite group isomorphic to a simple non-Abelian group L₃(2^m) or U₃(2^m) is, up to isomorphism, recognizable by a set of its element orders. On the other hand, for every simple group S=S₄(2^m), there exist infinitely many pairwise non-isomorphic groups G with w(G)=w(S). As a consequence, we present a list of all recognizable finite simple groups G, for which 4t ∉ ω(G) with t>1. Supported by RFFR grant No. 99-01-00550, by the National Natural Science Foundation of China (grant No. 19871066), and by the State Education Ministry of China (grant No. 98083). Translated fromAlgebra i Logika, Vol. 39, No. 5, pp. 567–585, September–October, 2000.     

Relations and deformations of odd Hamiltonian superalgebras

The following (co)homology groups of the Lie superalgebraH(0,n) are calculated: theH ² (H, H) cohomology with coefficients in the adjoint module and theH ₂ (LH) homology of the nitpotent subalgebra. It is shown that dimH ² (H,H) = 1. Translated fromMatematicheskie Zametki, Vol. 63, No. 3, pp. 391–401, March, 1998. This research was supported by the Russian Foundation for Basic Research under grant No. 95-01-01263 and by the International Science Foundation under grant No. RO 4300.     

A note on hypersurfaces in a Euclidean space

We obtain a characterization for a compact and connected hypersurface of R ⁿ⁺¹ to be a sphere in terms of the lower bound on the Ricci curvature.This work is supported by the research grant No. (Math/1409/05) of the Research Center, College of Science, King Saud University.     

Integral formulas for closed spacelike hypersurfaces in anti-de Sitter space <Emphasis Type="Italic">H</Emphasis><Stack><Subscript>1</Subscript><Superscript><Emphasis Type="Italic">n</Emphasis>+1</Superscript></Stack> (−1)

In this paper, we study closed k-maximal spacelike hypersurfaces M ⁿ in anti-de Sitter space H ₁ ⁿ⁺¹ (−1) with two distinct principal curvatures and give some integral formulas about these hypersurfaces. The first author was supported by Japan Society for Promotion of Science. The third author was supported by grant Proj. No. R17-2008-001-01000-0 from Korea Science & Engineering Foundation.     

Polynomial bounds for Large Bernoulli sections of ℓ 1 N

We present a quantitative form of the result of Bai and Yin from [2], and use to show that the section of ℓ ₁ ^(1+δ)n spanned byn random independent sign vectors is with high probability isomorphic to euclidean with isomorphism constant polynomial in δ⁻¹. Partially supported by BSF grant 2002-006. Supported by the National Science Foundation under agreement No. DMS-0111298. Supported in part by the Israel Science Academy.     

The finite quotients of the multiplicative group of a division algebra of degree 3 are solvable

LetD be a finite dimensional division algebra. It is known that in a variety of cases, questions about the normal subgroup structure ofD ^x (the multiplicative group ofD) can be reduced to questions about finite quotients ofD ^x. In this paper we prove that when deg(D)=3, finite quotients ofD ^x are solvable. the proof uses Wedderburn’s Factorization Theorem. Partially supported by grant no. 427-97-1 from the Israeli Science Foundation and by grant no. 6782-1-95 from the Israeli Ministry of Science and Art.     

Coefficient conditions for the asymptotic stability of solutions of systems of linear difference equations with continuous time and delay

We establish sufficient algebraic coefficient conditions for the asymptotic stability of solutions of systems of linear difference equations with continuous time and delay in the case of a rational correlation between delays. We use (n ² + m)-parameter Lyapunov functions (n is the dimension of the system of equations and m is the number of delays). Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 4, pp. 516–522, April, 1998. This work was partially supported by the Joint Foundation of the Ukrainian Government and the Soros International Science Foundation (grant No. K42199).     

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.     

Co-volume methods for degenerate parabolic problems

Summary A complementary volume (co-volume) technique is used to develop a physically appealing algorithm for the solution of degenerate parabolic problems, such as the Stefan problem. It is shown that, these algorithms give rise to a discrete semigroup theory that parallels the continuous problem. In particular, the discrete Stefan problem gives rise to nonlinear semigroups in both the discreteL ¹ andH ^–1 spaces.The first author was supported by a grant from the Hughes foundation, and the second author was supported by the National Science Foundation Grant No. DMS-9002768 while this work was undertaken. This work was supported by the Army Research Office and the National Science Foundation through the Center for Nonlinear Analysis.     

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.     

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.     

Efficient construction of a small hitting set for combinatorial rectangles in high dimension

We describe a deterministic algorithm which, on input integersd, m and real number (0,1), produces a subset S of [m] ^d ={1,2,3,...,m} ^d that hits every combinatorial rectangle in [m] ^d of volume at least , i.e., every subset of [m] ^d the formR ₁×R ₂×...×R _d of size at least m ^d . The cardinality of S is polynomial inm(logd)/, and the time to construct it is polynomial inmd/. The construction of such sets has applications in derandomization methods based on small sample spaces for general multivalued random variables.A preliminary version of this paper appeared in Proceedings of the 25th Annual ACM Symposium on Theory of Computing, 1993.Research partially done while visiting the International Computer Science Institute. Research supported in part by a grant from the Israel-USA Binational Science Foundation.A large portion of this research was done while still at the International Computer Science Institute in Berkeley, California. Research supported in part by National Science Foundation operating grants CCR-9304722 and NCR-9416101, and United States-Israel Binational Science Foundation grant No. 92-00226.Supported in part by NSF under grants CCR-8911388 and CCR-9215293 and by AFOSR grants AFOSR-89-0512 AFOSR-90-0008, and by DIMACS, which is supported by NSF grant STC-91-19999 and by the New Jersey Commission on Science and Technology. Research partially done while visiting the International Computer Science Institute.Partially supported by NSF NYI Grant No. CCR-9457799. Most of this research was done while the author was at MIT, partially supported by an NSF Postdoctoral Fellowship. Research partially done while visiting the International Computer Science Institute.     

Holomorphic functions and embedded real surfaces

The paper is devoted to the study of necessary and sufficient topological conditions for an embedded real surface to lie in a strictly pseudoconvex domain on a complex surface. These results are used to construct Stein domains on algebraic manifolds and to describe envelopes of holomorphy of real surfaces in P ² and in some other complex surfaces.Translated fromMatematicheskie Zametki, Vol. 63, No. 4, pp. 599–606, April, 1998.The author is grateful to A. Vitushkin for the statement of the problem and for constant encouragement.This research was partially supported by the Russian Foundation for Basic Research under grant No. 96-01-01002 and by the International Science Foundation under grant No. a97-1850.     

Solution to the lighting problem for zone bodies

In the paper, the class of zone bodies, which includes, in particular, zonoids (and zonohedrons) is introduced. The lighting problem for this class is solved, thus generalizing earlier results for zonohedrons (H.Martini) and zonoids (V. G. Boltyanskii and P. S. Soltan). Namely, it is proved that the boundary of anyn-dimensional zone body other than a parallelepiped can be lit by 3·2 ^n–2 pencils of parallel rays or less.Translated fromMatematicheskie Zametki, Vol. 58, No. 4, pp. 505–511, November, 1995.This work was partially supported by the International Science Foundation under grant No. MAT6565-0925.     

Lattices That are Embeddable in Suborder Lattices

Various types of lattices are embedded in suborder lattices of posets possessing certain properties. In particular, it is shown that the class of lattices isomorphic to sublattices of suborder lattices of posets of length at most n is a variety, for any n < ω.Supported by INTAS-YSF grant 2001/1-65, by RFBR-DFG grant No. 01-01-04003, by RF Ministry of Education grant No. E02-1.0-32, by the Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-2112.2003.1, and by a grant from the Russian Science Support Foundation.__________Translated from Algebra i Logika, Vol. 44, No. 4, pp. 483–511, July–August, 2005.     

Freedom from the interpolation property for tense calculi associated with Ershov spaces

We study into the question whether calculi associated with Ershov topological spaces possess Craig’s interpolation property. Supported by RFBR grant No. 06-01-00358, by INTAS grant No. 04-77-7080, and by the Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-4787.2006.1. __________ Translated from Algebra i Logika, Vol. 46, No. 6, pp. 745–762, November–December, 2007.     

Invariant symmetric restrictions of a self-adjoint operator. II

We give a criterion of invariance and symmetry of the restriction of an arbitrary unbounded self-adjoint operator in the space L ₂(ℝⁿ, dx) by using the introduced notion of support of an arbitrary vector and the notion of capacity of a subspace N ⊂ ℝⁿ. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 6, pp. 781–791, June, 1998. This work was partially supported by the Foundation for Fundamental Research of the Ministry of Science and Technology of the Ukraine (grant No. 1/238 “Operator”).     

Asymptotic coarsening of proper metric spaces and index problems

 We introduce an asymptotic coarse structure on proper metric spaces and study the associated C ^* -algebras and assembly maps. We establish an asymptotic Lipschitz homotopy invariance theorem for the K-theory of these C ^* -algebras and the K-homology of the metric space, and show that the assembly map is an isomorphism over an asymptotically scaleable space. Received: 12 July 2001 / Revised version: 29 October 2002 Published online: 3 March 2003 The first author is supported in part by the National Basic Research Project (973), NSF and the Educational Ministry of P. R. China. The second author is supported by the NSF grant 10201007, P. R. China, and a grant from Shanghai Science and Technology Commission, No. 01ZA14003. Mathematics Subject Classification (2000): Primary 46L80