Siegmund Duality for Continuous Time Markov Chains on ℤ<Stack><Subscript>+</Subscript><Superscript>d</Superscript></Stack>

For the continuous time Markov chain with transition function P(t) on Z₊^d, we give the necessary and sufficient conditions for the existence of its Siegmund dual with transition function P(t). If Q, the q-matrix of P(t), is uniformly bounded, we show that the Siegmund dual relation can be expressed directly in terms of q-matrices, and a sufficient condition under which the Q-function is the Siegmund dual of some Q-function is also given.     

Conformally flat Lorentzian hypersurfaces in ℝ<Stack><Subscript>1</Subscript><Superscript>4</Superscript></Stack> with three distinct principal curvatures

A three dimensional Lorentzian hypersurface x: M ₁ ³ → ? ₁ ⁴ is called conformally flat if its induced metric is conformal to the flat Lorentzian metric, and this property is preserved under the conformal transformation of ? ₁ ⁴ . Using the projective light-cone model, for those whose shape operators have three distinct real eigenvalues, we calculate the integrability conditions by constructing a scalar conformal invariant and a canonical moving frame in this paper. Similar to the Riemannian case, these hypersurfaces can be determined by the solutions to some system of partial differential equations.     

Quincunx multiresolution analysis for <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>(ℚ<Stack><Subscript>2</Subscript><Superscript>2</Superscript></Stack>)

With an eye on applications in quantum mechanics and other areas of science, much work has been done to generalize traditional analytic methods to p-adic systems. In 2002 the first paper on p-adic wavelets was published. Since then p-adic wavelet sets, multiresolution analyses, and wavelet frames have all been introduced. However, so far all constructions have involved dilations by p. This paper presents the first construction of a p-adic wavelet system with a more general matrix dilation, laying the foundation for further work in this direction.     

Complex lines in complex hyperbolic space <Emphasis Type="Italic">H</Emphasis><Stack><Subscript>ℂ</Subscript><Superscript>2</Superscript></Stack>

Let p ₁, p ₂, p ₃, p ₄ be four pairwise distinct points in the boundary of complex hyperbolic 2-space H _ℂ² and any three points do not lie in the same C-circle. We show that we are always able to group the four points into two classes such that each class contains two points, the two complex lines spanned by each class are ultra-parallel or intersect. As an application, we can simplify the discussion in the paper [7], in which Parker and Platis used the global geometry coordinates to describe the Falbel’s cross-ratio variety of the four pairwise distinct points on the ∂H _ℂ².     

Disjoint <Emphasis Type="Italic">K</Emphasis><Stack><Subscript>4</Subscript><Superscript>−</Superscript></Stack> in claw-free graphs with minimum degree at least five

A graph is said to be claw-free if it does not contain an induced subgraph isomorphic to K _1,3. Let K ₄ ^? be the graph obtained by removing exactly one edge from K ₄ and let k be an integer with k ? 2. We prove that if G is a claw-free graph of order at least 13k ? 12 and with minimum degree at least five, then G contains k vertex-disjoint copies of K ₄ ^? . The requirement of number five is necessary.     

Approximation of the classes <Emphasis Type="Italic">C</Emphasis><Stack><Subscript>β</Subscript><Superscript>ψ</Superscript></Stack><Emphasis Type="Italic">H</Emphasis><Subscript>ω</Subscript> by generalized Zygmund sums

We obtain asymptotic equalities for the least upper bounds of approximations by Zygmund sums in the uniform metric on the classes of continuous 2π-periodic functions whose (ψ, β)-derivatives belong to the set H _ω in the case where the sequences ψ that generate the classes tend to zero not faster than a power function.     

Discrete subgroups of <Emphasis Type="Italic">PU</Emphasis> (2, 1) acting on P<Stack><Subscript>ℂ</Subscript><Superscript>2</Superscript></Stack> and the Kobayashi metric

In this paper, we prove following: If G ⊂ PU (2, 1) is an infinite, discrete group, acting on P_ℂ² without complex invariant lines, then the component containing ℍP_ℂ² of the domain of discontinuity Ω(G) = PP_ℂ²∖ Λ (G), according to Kulkarni, is G-invariant complete Kobayashi hyperbolic. The authors were supported by the Universidad Autónoma de Yucatán and the Universidad Nacional Autónoma de México.     

On Dirichlet Type Spaces <Emphasis Type="Italic">A</Emphasis><Stack><Subscript><Emphasis Type="Italic">ω</Emphasis></Subscript><Superscript>2</Superscript></Stack> Over the Half–Plane

Some extensions of the results of the first author related with the Hilbert spaces A _ω,0 ² of functions holomorphic in the half–plane are proved. Some new Hilbert spaces A _ω ² of Dirichlet type are introduced, which are included in the Hardy space H2 over the half–plane. Several results on representations, boundary properties, isometry, interpolation, biorthogonal systems and bases are obtained for the spaces A _ω ² ? H₂.     

<Emphasis Type="Italic">OD</Emphasis>-Characterization of the automorphism groups of <Emphasis Type="Italic">O</Emphasis><Stack><Subscript>10</Subscript><Superscript>±</Superscript></Stack>(2)

The prime graph of a finite group was introduced by Gruenberg and Kegel. The degree pattern of a finite group G associated to its prime graph was introduced in [1] and denoted by D(G). The group G is called k-fold OD-characterizable if there exist exactly k non-isomorphic groups H satisfying conditions (1) |G| = |H| and (2) D(G) = D(H). Moreover, a 1-fold OD-characterizable group is simply called an OD-characterizable group. Till now a lot of finite simple groups were shown to be OD-characterizable, and also some finite groups especially the automorphism groups of some finite simple groups were shown not being OD-characterizable but k-fold OD-characterizable for some k > 1. In the present paper, the authors continue this topic and show that the automorphism groups of orthogonal groups O ₁₀⁺(2) and O ₁₀⁻(2) are OD-characterizable.     

Density of the Medvedev lattice of Π<Superscript>0</Superscript><Subscript>1</Subscript> classes

The partial ordering of Medvedev reducibility restricted to the family of ⁰₁ classes is shown to be dense. For two disjoint computably enumerable sets, the class of separating sets is an important example of a ⁰₁ class, which we call a ``c.e. separating class'. We show that there are no non-trivial meets for c.e. separating classes, but that the density theorem holds in the sublattice generated by the c.e. separating classes. Mathematics Subject Classification (2000): 03D30, 03D25     

The number of S-precomplete classes in the functional system <Emphasis Type="Italic">P</Emphasis><Stack><Subscript><Emphasis Type="Italic">κ</Emphasis></Subscript><Superscript><Emphasis Type="Italic">τ</Emphasis></Superscript></Stack>

The problem on the number of precomplete classes in the functional system P _κ ^τ is considered, elements of P _κ ^τ are deterministic S-functions defined on words of length τ composed from letters of an alphabet of cardinality κ. An asymptotics for the number of S-precomplete classes in P _κ ^τ is obtained for arbitrary fixed κ and τ tending to infinity.     

Local structure of Rogers semilattices of Σ<Subscript>n</Subscript><Superscript>0</Superscript>-computable numberings

We deal in specific features of the algebraic structure of Rogers semilattices of _n ⁰-computable numberings, for n 2. It is proved that any Lachlan semilattice is embeddable (as an ideal) in such every semilattice, and that over an arbitrary non 0-principal element of such a lattice, any Lachlan semilattice is embeddable (as an interval) in it.Supported by INTAS grant No. 00-499, by FP Universities of Russia grant UR.04.01.013, and by the Grant Center for Fundamental Research (GCFR), project PD02-1.1-475.__________Translated from Algebra i Logika, Vol. 44, No. 2, pp. 148–172, March–April, 2005.     

Embeddings into the Medvedev and Muchnik lattices of Π<Superscript>0</Superscript><Subscript>1</Subscript> classes

Let _w and _M be the countable distributive lattices of Muchnik and Medvedev degrees of non-empty ₁⁰ subsets of 2, under Muchnik and Medvedev reducibility, respectively. We show that all countable distributive lattices are lattice-embeddable below any non-zero element of _w. We show that many countable distributive lattices are lattice-embeddable below any non-zero element of _M.Simpsons research was partially supported by NSF Grant DMS-0070718. We thank the anonymous referee for a careful reading of this paper and helpful comments.     

A Ring Theoretic Construction of Hadamard Difference Sets in ℤ<Subscript>8</Subscript><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>×ℤ<Subscript>2</Subscript><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Let be the Galois ring of characteristic 2³ and rank n and let . We give an explicit construction of Hadamard difference sets in .}Research supported by NSA grant MDA 904-02-1-0080.     

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.     

On Some Properties of Functions in the Sobolev-Morrey-Type Spaces <Emphasis Type="Italic">W</Emphasis><Stack><Subscript>p,a,ϰ,τ</Subscript><Superscript>l</Superscript></Stack>(<Emphasis Type="Italic">G</Emphasis>)

In the framework of embedding theory, we study some properties of the functions of the Sobolev-Morrey-type spaces and local smoothness of solutions to one class of quasielliptic equations.Original Russian Text Copyright © 2005 Nadzhafov A. M.__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 3, pp. 634–648, May–June, 2005.     

Spin structures and spectra of ℤ<Subscript>2</Subscript><Superscript><Emphasis Type="Italic">k</Emphasis></Superscript>-manifolds

We give necessary and sufficient conditions for the existence of pin^± and spin structures on Riemannian manifolds with holonomy group ₂^k. For any n4 (resp. n6) we give examples of pairs of compact manifolds (resp. compact orientable manifolds) M₁, M₂, non homeomorphic to each other, that are Laplace isospectral on functions and on p-forms for any p and such that M₁ admits a pin^± (resp. spin) structure whereas M₂ does not.Mathematics Subject Classification (2000):58J53, 57R15, 20H15Partially supported by Conicet and grants from SecytUNC, Foncyt and AgCba.     

A norm of partial sums of Fourier-Jacobi series for functions from <Emphasis Type="Italic">L</Emphasis><Stack><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript><Superscript>(<Emphasis Type="Italic">α,β</Emphasis>)</Superscript></Stack>

Asymptotic estimates for a norm of partial sums of Fourier-Jacobi series of functions from L _p ^(α,β) are obtained.     

Semiclassical short strings in AdS<Subscript>5</Subscript> × <Emphasis Type="Italic">S</Emphasis><Superscript>5</Superscript>

In the short-string limit, we present results for the one-loop correction to the energy of string solutions in AdS ₅ × S ⁵ that belong to a certain class. The computations are based on the observation that the fluctuation operators, just as for rigid spinning-string elliptic solutions, can be written in the single-gap Lamé form. Based on these computations, we reveal a remarkable universality of the expression for the energy of short semiclassical strings, which may help in better understanding the structure of the strongcoupling expansion of the anomalous dimensions of dual gauge theory operators.