Verifying Huppert’s Conjecture for <Superscript>2</Superscript><Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript>(<Emphasis Type="Italic">q</Emphasis><Superscript>2</Superscript>)

Let G be a finite group and cd(G) be the set of irreducible character degrees of G. Bertram Huppert conjectured that if H is a finite nonabelian simple group such that cd(G) = cd(H), then G ≅ H×A, where A is an abelian group. In this paper, we verify the conjecture for the twisted Ree groups ² G ₂(q ²) for q ² = 3^2m + 1, m ≥ 1. The argument involves verifying five steps outlined by Huppert in his arguments establishing his conjecture for many of the nonabelian simple groups.     

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.     

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.     

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.     

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.     

The Algorithm Z and Ramanujan’s <Subscript>1</Subscript><Emphasis Type="Italic">ψ</Emphasis><Subscript>1</Subscript> summation

We use the Algorithm Z on partitions due to Zeilberger, in a variant form, to give a combinatorial proof of Ramanujan’s ₁ ψ ₁ summation formula.     

<Emphasis Type="Italic">q</Emphasis>-extensions of Dougall’s bilateral <Subscript>2</Subscript>H<Subscript>2</Subscript>-series

Transformations for a bilateral ₂ ψ ₂-series are investigated by means of Abel’s lemma on summation by parts. Two q-extensions of Dougall’s classical identity for bilateral ₂H₂-sum are established. As by-products, the Rogers–Fine identity is recovered and a new proof is presented for Bailey’s identity of bilateral well-poised ₆ ψ ₆-series.     

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.     

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.     

Pointwise multipliers and decomposition theorems in <Emphasis Type="Italic">F</Emphasis><Superscript><Emphasis Type="Italic">∞,q</Emphasis></Superscript><Subscript><Emphasis Type="Italic">s</Emphasis></Subscript>

In this paper we give a characterization of the pointwise multipliers of the holomorphic Triebel-Lizorkin spaces F^{, q}_s, and we apply these results to prove a corona decomposition theorem for these spaces.Mathematics Subject Classification (1991):32A37Both authors partially supported by DGICYT Grant BMF2002-04072-C02-01 and CIRIT Grant 2001-SGR-00172     

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.     

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.     

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     

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.     

Multi-self-similar Markov processes on ℝ<Subscript>+</Subscript><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> and their Lamperti representations

 A classical result, due to Lamperti, establishes a one-to-one correspondence between a class of strictly positive Markov processes that are self-similar, and the class of one-dimensional Lévy processes. This correspondence is obtained by suitably time-changing the exponential of the Lévy process. In this paper we generalise Lamperti's result to processes in n dimensions. For the representation we obtain, it is essential that the same time-change be applied to all coordinates of the processes involved. Also for the statement of the main result we need the proper concept of self-similarity in higher dimensions, referred to as multi-self-similarity in the paper. The special case where the Lévy process ξ is standard Brownian motion in n dimensions is studied in detail. There are also specific comments on the case where ξ is an n-dimensional compound Poisson process with drift. Finally, we present some results concerning moment sequences, obtained by studying the multi-self-similar processes that correspond to n-dimensional subordinators. Received: 22 August 2002 / Revised version: 10 February 2003 Published online: 15 April 2003 RID="*" ID="*" MaPhySto – Centre for Mathematical Physics and Stochastics, funded by a grant from the Danish National Research Foundation Mathematics Subject Classification (2000): 60G18, 60G51, 60J25, 60J60, 60J75 Key words or phrases: Lévy process – Self-similarity – Time-change – Exponential functional – Brownian motion – Bessel process – Piecewise deterministic Markov process – Moment sequence     

Jackson inequality in <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript>(ℝ<Superscript><Emphasis Type="Italic">N</Emphasis></Superscript>) with generalized modulus of continuity

The sharp Jackson inequality is proved for the space of entire functions of many variables with mean-square norm for an arbitrary modulus of continuity generated by a difference operator with constant coefficients.     

<Emphasis Type="Italic">π</Emphasis>-formulas implied by Dougall’s summation theorem for <Subscript>5</Subscript><Emphasis Type="Italic">F</Emphasis><Subscript>4</Subscript>-series

The general summation theorem for well-poised ₅ F ₄-series discovered by Dougall (Proc. Edinb. Math. Soc. 25:114–132, 1907) is shown to imply several infinite series of Ramanujan-type for 1/π and 1/π ², including those due to Bauer (J. Reine Angew. Math. 56:101–121, 1859) and Glaisher (Q. J. Math. 37:173–198, 1905) as well as some recent ones by Levrie (Ramanujan J. 22:221–230, 2010).     

The Δ<Superscript>0</Superscript><Subscript><Emphasis Type="Italic">α</Emphasis></Subscript>-Computable Enumerations of the Classes of Projective Planes

Studying computable representations of projective planes, for the classes K of pappian, desarguesian, and all projective planes, we prove that K ^c /? admits no hyperarithmetical Friedberg enumeration and admits a Friedberg Δ⁰_α+3-computable enumeration up to a Δ⁰ _α -computable isomorphism.     

A new characterization of Anderson’s inequality in <Emphasis Type="Italic">C</Emphasis><Subscript>1</Subscript>-classes

Let ℋ be a separable infinite dimensional complex Hilbert space, and let ℒ(H) denote the algebra of all bounded linear operators on ℋ into itself. Let A = (A ₁, A ₂,..., A _n), B = (B ₁, B ₂,..., B _n) be n-tuples of operators in ℒ(H); we define the elementary operators Δ _A,B : ℒ(H) ↦ ℒ(H) by

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.