Rate of convergence to the semi-circle law for the Deformed Gaussian Unitary Ensemble

It is shown that the Kolmogorov distance between the expected spectral distribution function of an n × n matrix from the Deformed Gaussian Ensemble and the distribution function of the semi-circle law is of order O(n ^−2/3+v ). Research supported by the DFG-Forschergruppe FOR 399/1. Partially supported by INTAS grant N 03-51-5018, by RFBF grant N 02-01-00233, by RFBR-DFG grant N 04-01-04000, by RF grant of the leading scientific schools NSh-4222.2006.1.     

Orthogonal surfaces with constant mean curvature in the Euclidean 4-space

We classify orthogonal surfaces ofE ⁴ whose mean curvature is constant by proving that they are either of 1-type or of 2-type.Partially supported by a ME/Fulbright grant and by a DGICYT grant PB91-0705-C02-01     

On the complexity of the approximate table representation of discrete analogs of functions of finite smoothness in the metric ofL P

For discrete analogs of classes of functions of finite smoothness, we study the quantity log Approx characterizing the minimal necessary length of tables that allow us to reconstruct functions from these classes with error not exceeding 1 in the metric of the spaceL ^p . Translated fromMatematicheskie Zametki, Vol. 64, No. 5, pp. 643–647, November, 1998. The author wishes to express his gratitude to his scientific adviser O. B. Lupanov for setting the problem and attention to the author's work. This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-01068.     

Values of <Emphasis Type="Italic">L</Emphasis>-functions at integers outside the critical strip

We incorporate the non-critical values of L-functions of cusp forms into a cohomological set-up analogous to the one of Eichler, Manin and Shimura. We use the 1-cocycles we associate in this way to non-critical values to prove an expression for such values which is similar in structure to Manin’s formula for the critical value of the L-function of a weight 2 cusp form. YoungJu Choie is partially supported by KOSEF R01-2003-00011596-0 and by ITRC Research Fund. N. Diamantis is partially supported by EPSRC grant EP/D032350/1.     

Groups with small Dehn functions and bipartite chord diagrams

We introduce a new invariant of bipartite chord diagrams and use it to construct the first examples of groups with Dehn function n²log n. Some of these groups have undecidable conjugacy problem. Our groups are multiple HNN extensions of free groups. We show that n²log n is the smallest Dehn function of a multiple HNN extension of a free group with undecidable conjugacy problem. Both authors were supported in part by the NSF grants DMS 0245600 and DMS 0455881. In addition, the research of the first author was supported in part by the Russian Fund for Basic Research 05-01-00895, the research of the second author was supported in part by the NSF grant DMS 9978802 and the US-Israeli BSF grant 1999298. Received: February 2005; Revision: September 2005; Accepted: September 2005     

Strongly-cyclic branched coverings of knots via (g, 1)-decompositions

Strongly-cyclic branched coverings of knots are studied by using their (g, 1)-decompositions. Necessary and sufficient conditions for the existence and uniqueness of such coverings are obtained. It is also shown that their fundamental groups admit geometric g-words cyclic presentations. Work performed under the auspices of G.N.S.A.G.A. of I.N.d.A.M. of Italy and supported by M.U.R.S.T., by the University of Bologna, funds for selected research topics. The third named author was also supported by the INTAS project “CalcoMet-GT” 03-51-3663, the grant of RFRB, and the grant of SB RAN.     

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.     

Boundary Harnack Principle for <Emphasis Type="Italic">p</Emphasis>-harmonic Functions in Smooth Euclidean Domains

We establish a scale-invariant version of the boundary Harnack principle for p-harmonic functions in Euclidean C ^1,1-domains and obtain estimates for the decay rates of positive p-harmonic functions vanishing on a segment of the boundary in terms of the distance to the boundary. We use these estimates to study the behavior of conformal Martin kernel functions and positive p-superharmonic functions near the boundary of the domain. H. A. was partially supported by Grant-in-Aid for (B) (2) (No. 15340046) Japan Society for the Promotion of Science. N. S. was partially supported by NSF grant DMS-0355027. X. Z. was partially supported by the Taft foundation.     

Crossed Product of a C *-Algebra by a Semigroup of Endomorphisms Generated by Partial Isometries

The paper presents a construction of the crossed product of a C^*- algebra by a semigroup of endomorphisms generated by partial isometries. This work was in part supported by Polish Ministry of Science and High Education grant number N N201 382634.     

Rate of convergence to the semi-circular law

A stochastic bound of order O _P (n ^–1/2 ) for the Kolmogorov distance between the spectral distribution function of an n×n matrix from Wigner ensemble and the distribution function of the semi-circular law is obtained. The result holds assuming that the twelfth moment of the entries of the matrix is uniformly bounded.Research supported by the DFG-Forschergruppe FOR 399/1-1 ``Spektrale Analyse, Asymptotische Verteilungen und Stochastische Dynamiken'.Research supported by the DFG-Forschergruppe FOR 399/1-1 ``Spektrale Analyse, Asymptotische Verteilungen und Stochastische Dynamiken'.Partially supported by Russian Foundation for Fundamental Research Grants NN02-01-00233, 00-15-96019. Partially supported by INTAS N99-01317, DFG-RFBR N99-01-04027. Mathematics Subject Classification (2000): 60F05     

A reciprocity theorem for certain q-series found in Ramanujan’s lost notebook

Three proofs are given for a reciprocity theorem for a certain q-series found in Ramanujan’s lost notebook. The first proof uses Ramanujan’s ₁ψ₁ summation theorem, the second employs an identity of N. J. Fine, and the third is combinatorial. Next, we show that the reciprocity theorem leads to a two variable generalization of the quintuple product identity. The paper concludes with an application to sums of three squares. Dedicated to Richard Askey on the occasion of his 70th birthday. 2000 Mathematics Subject Classification Primary—33D15 B. C. Berndt: Research partially supported by grant MDA904-00-1-0015 from the National Security Agency. A. J. Yee: Research partially supported by a grant from The Number Theory Foundation.     

Weak-type Endpoint Estimates for Multilinear Singular Integral Operators

In this paper, the authors discuss a class of multilinear singular integrals and obtain that the operators are bounded from H^1(R^n) to weak L^1(R^n). Using this result, we can directly prove a main theorem in [5].     

<Emphasis Type="Italic">Φ</Emphasis>-Variation of a bifractional Brownian motion

Let B _H,K = {B _H,K (t)} _t⩾0 be a bifractional Brownian motion with parameters H ∈ (0, 1) and K ∈ (0, 1]. For a function Φ: [0, ∞) → [0, ∞) and for a partition κ = {t _i }_n ⁱ⁼⁰ of an interval [0, T] with T > 0, let {ie418-01}. We prove that, for a suitable Φ depending on H and K, {ie418-02} almost surely. The research was partially supported by the Lithuanian State Science and Studies Foundation, grant No. T-16/08     

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.     

Infinitesimally homogeneous submanifolds of euclidean spaces

For a class of submanifolds of ^N, the infinitesimally homogeneous ones, the second fundamental form and itss-times iterated derivativessk+1 at any fixed point determine the immersion uniquely. The integerk>0 will be called the extrinsic Singer invariant. Any infinitesimally homogeneous submanifoldM (which is not necessarily complete) is an open part of a globally homogeneous (complete) submanifold. Indeed, the infinitesimal data at any pointp, determine, canonically, a Lie subgroupG of the isometry group of ^N , whose orbit atp is a complete submanifold that extendsM. Work partially supported by the GNSAGA of CNR and by the MURST of Italy     

Real hypersurfaces in complex two-plane Grassmannians with commuting normal Jacobi operator

We give a complete classification of -invariant real hypersurfaces in complex two-plane Grassmannians G ₂(C ^m+2) with commuting normal Jacobi operator . The first author was supported by MCYT-FEDER grant BFM 2001-2871-C04-01, the second author by grant Proj. No. KRF-2006-351-C00004 from Korea Research Foundation and the third author by grant Proj. No. R14-2002-003-01001-0 from Korea Research Foundation, Korea 2006 and Proj. No. R17-2007-006-01000-0 from KOSEF.     

Generalized Fredholm property and a priori estimates for linear operators on tensor products of Hilbert spaces

For a linear operator acting in a Hilbert space, the generalized Fredholm property (invertibility modulo a certain ideal) is proved to be equivalent to certaina priori estimates. This result is applied to establish a connection between properties of linear operators on tensor products of Hilbert spaces, such asn- andd-normality, the (generalized and ordinary) Fredholm property, and appropriatea priori estimates.Translated fromMatematicheskie Zametki, Vol. 64, No. 6, pp. 902–912, December, 1998.The author is grateful to V. M. Deundyak for useful discussion of this work.This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-01195.     

On the computation of the projective surgery obstruction groups

The computation of the projective surgery obstruction groupsL _n ^p (ZG), forG a hyperelementary finite group, is reduced to standard calculations in number theory, mostly involving class groups. Both the exponent of the torsion subgroup and the precise divisibility of the signatures are determined. ForG a 2-hyperelementary group, theL _n ^p (ZG) are detected by restriction to certain subquotients ofG, and a complete set of invariants is given for oriented surgery obstructions.Partially supported by NSERC grant A4000.Partially supported by NSF grant DMS-8610730(1) and the Danish Research Council.     

A note on approximation of semigroups of contractions on Hilbert spaces

We show that every contractive C ₀-semigroup on a separable, infinite-dimensional Hilbert space X can be approximated by unitary C ₀-groups in the weak operator topology uniformly on compact subsets of ℝ₊. As a consequence we get a new characterization of a bounded H ^∞-calculus for the negatives of generators of bounded holomorphic semigroups. Applications of our results to the study of a topological structure of the set of (almost) weakly stable contractive C ₀-semigroups on X are also discussed. The author was partially supported by the Marie Curie “Transfer of Knowledge” programme, project “TODEQ”, and by a MNiSzW grant Nr. N201384834.     

p-Adic Automorphic Functions for the Unitary Group in Three Variables

Let G be a special unitary group in three variables over a local field and, a co-compact subgroup of G. We construct, -automorphic functions on the rigid analytic space of points in the projective plane stable under the action of tori in G.This research was partially supported by an ARC grant.2000 Mathematics Subject Classification: 11F55, 11F85, 20G25, 22E35