On the supersingular reduction of elliptic curves

Let a∈Q and denote byE _a the curvey ² = (x ² + l)(x + a). We prove thatE _a(F_p) is cyclic for infinitely many primesp. This fact was known previously only under the assumption of the generalized Riemann hypothesis. Research partially supported by NSERC grant A9418.     

On the existence of periodic complementary binary sequences

Using the connection between periodic complementary binary sequences and difference families given by Bömer and Antweiler [2], we use number theoretic techniques to obtain necessary conditions for their existence when two or three sequences are involved. Our results provide theoretical proof for the nonexistence of PCS₃ ²⁰(a _i) and settle the open case PCS₂ ³⁶(a _i) of [2], negatively.Research partially supported by National Security Agency grant #MDA 904-90-H-4008.     

A geometrical property ofC(K) spaces

We introduce a geometrical property of norm one complemented subspaces ofC(K) spaces which is useful for computing lower bounds on the norms of projections onto subspaces ofC(K) spaces. Loosely speaking, in the dual of such a space ifx* is a w* limit of a net (x _a ^* ) andx*=x*₁+x*₂ with ‖x*‖=‖x*₁‖ + ‖x*₂‖, then we measure how efficiently thex _a ^* 's can be split into two nets converging tox*₁ andx*₂, respectively. As applications of this idea we prove that if for everyε>0,X is a norm (1+ε) complemented subspace of aC(K) space, then it is norm one complemented in someC(K) space, and we give a simpler proof that a slight modification of anl ₁-predual constructed by Benyamini and Lindenstrauss is not complemented in anyC(K) space. Research partially supported by a grant of the U.S.-Israel Binational Science Foundation. Research of the first-named author is supported in part by NSF grant DMS-8602395. Research of the second-named author was partially supported by the Fund for the Promotion of Research at the Technion, and by the Technion VPR-New York Metropolitan Research Fund.     

Hankel and Toeplitz Transforms on H 1: Continuity, Compactness and Fredholm Properties

We revisit the boundedness of Hankel and Toeplitz operators acting on the Hardy space H ¹ and give a new proof of the old result stating that the Hankel operator H _a is bounded if and only if a has bounded logarithmic mean oscillation. We also establish a sufficient and necessary condition for H _a to be compact on H ¹. The Fredholm properties of Toeplitz operators on H ¹ are studied for symbols in a Banach algebra similar to C + H ^∞ under mild additional conditions caused by the differences in the boundedness of Toeplitz operators acting on H ¹ and H ². The first author was partially supported by the European Commission IHP Network “Harmonic Analysis and Related Problems” (Contract Number: HPRN-CT-2001-00273-HARP) and by the Greek Research Program “Pythagoras 2” (75% European funds and 25 National funds). The second author was fully supported by the European Commission IHP Network “Harmonic Analysis and Related Problems” (Contract Number: HPRN-CT-2001-00273-HARP) while he visited the first author at the University of Crete and later by the Academy of Finland Project 207048.     

The packing measure of a class of generalized sierpinski carpet

For 1/4 ＜ a ＜√2/4, let S1(x) = ax, S2(x) = 1 - a ax, x ∈ [0,1]. Ca is the attractor of the iterated function system {S1, S2}, then the packing measure of Ca × Ca is Ps(a)(Ca × Ca) = 4.2s(a)(1 - a)s(a),where s(a) = -loga4.     

L 1-convergence of double cosine- and Walsh-Fourier series

We prove the convergence inL ¹([−gp, π)²)-norm of the double Fourier series of an integrable functionf(x, y) which is periodic and even with respect tox andy, with coefficientsa _jk satisfying certain conditions of Hardy-Karamata kind, and such thata _jk logj logk→0 asj, k→∞. These sufficient conditions become quite natural in particular cases. Then we extend these results to the convergence of double Walsh-Fourier series inL ¹ (0, 1)²)- norm. As a by-product, we obtain Tauberian conditions ensuring the convergence of a double numerical series provided it is Cesàro summable. This research was partially supported by the Hungarian National Foundation for Scientific Research under Grant # 234.     

Branching in the enumeration degrees of the Σ 2 0 sets

We show that every incomplete Σ ₂ ⁰ enumeration degree is meet-reducible in the structure of the enumeration degrees of the Σ ₂ ⁰ sets. The first author was partially supported by NSF-grant DMS-9500983. The second author was partially supported by the HC&M research networkComplexity, Logic and Recursion Theory (COLORET), contract no. ERBCHRXCT930415, and by MURST 60%.     

Chains of baire class 1 functions and various notions of special trees

Following Laczkovich we consider the partially ordered setB ₁(ℝ) of Baire class 1 functions endowed with the pointwise order, and investigate the order types of the linearly ordered subsets. Answering a question of Komjáth and Kunen we show (inZFC) that special Aronszajn lines are embeddable intoB ₁(ℝ). We also show that under Martin's Axiom a linearly ordered set ℒ with |ℒ| < 2^ω is embeddable intoB ₁(ℝ) iff ℒ does not contain a copy of ω₁ or ω _* ¹ . We present aZFC example of a linear order of size 2^ω showing that this characterisation is not valid for orders of size continuum. These results are obtained using the notion of a compact-special tree; that is, a tree that is embeddable into the class of compact subsets of the reals partially ordered under reverse inclusion. We investigate how this notion is related to the well-known notion of an ℝ-special tree and also to some other notions of specialness. Partially supported by Hungarian Scientific Foundation grant no. 37758, 49786 and F 43620. The second author's research for this paper was partially supported by NSERC of Canada.     

Pseudo Riemannian manifolds whose generalized Jacobi operator has constant characteristic polynomial

Letg be a non-degenerate innerproduct of signature (p,q) onR ^m . LetGr _r,s (g) be the Grassmanian of planes so the restriction of g to is non-degenerate and has signature (r, s). IfR is an algebraic curvature tensor onR ^m , we define a generalized Jacobi operator onGr _r,s (g) and study when the characteristic polynomial of this operator is constant.Dedicated to Professor Helmut Karzel on his 70th birthdayResearch partially supported by the NSF (USA).Research partially supported by the NFSI (Bulgaria).     

Unique ergodicity of flows on homogeneous spaces

LetG be a unimodular Lie group, Γ a co-compact discrete subgroup ofG and ‘a’ a semisimple element ofG. LetT _a be the mapgΓ →ag Γ:G/Γ →G/Γ. The following statements are pairwise equivalent: (1) (T _a, G/Γ,θ) is weak-mixing. (2) (T _a, G/Γ) is topologically weak-mixing. (3) (G ^u, G/Γ) is uniquely ergodic. (4) (G ^u, G/Γ,θ) is ergodic. (5) (G ^u, G/Γ) is point transitive. (6) (G ^u, G/Γ) is minimal. If in additionG is semisimple with finite center and no compact factors, then the statement “(T _a, G/Γ,θ) is ergodic” may be added to the above list. The authors were partially supported by NSF grant MCS 75-05250.     

Self-similar sets of zero Hausdorff measure and positive packing measure

We prove that there exist self-similar sets of zero Hausdorff measure, but positive and finite packing measure, in their dimension; for instance, for almost everyu ∈ [3, 6], the set of all sums ∑ ₀ ⁸ a _n 4⁻ⁿ a _n 4⁻ⁿ with digits witha _n ∈ {0, 1,u} has this property. Perhaps surprisingly, this behavior is typical in various families of self-similar sets, e.g., for projections of certain planar self-similar sets to lines. We establish the Hausdorff measure result using special properties of self-similar sets, but the result on packing measure is obtained from a general complement to Marstrand’s projection theorem, that relates the Hausdorff measure of an arbitrary Borel set to the packing measure of its projections. Research of Y. Peres was partially supported by NSF grant #DMS-9803597. Research of K. Simon was supported in part by the OTKA foundation grant F019099. Research of B. Solomyak was supported in part by NSF grant #DMS 9800786, the Fulbright Foundation, and the Institute of Mathematics at The Hebrew University of Jerusalem.     

Connected components in the space of composition operators on<Emphasis Type="Italic">H∞</Emphasis> functions of many variables

LetE be a complex Banach space with open unit ballB _e. The structure of the space of composition operators on the Banach algebra H∞, of bounded analytic functions onB _e with the uniform topology, is studied. We prove that the composition operators arising from mappings whose range lies strictly insideB _e form a path connected component. WhenE is a Hilbert space or aC _o(X)- space, the path connected components are shown to be the open balls of radius 2. The research of this author was supported by grant number SAB1999-0214 from the Ministerio de Educación, Cultura y Deporte during his stay at the Universidad de Valencia. The research of this author was partially supported DGES(Spain) pr. 96-0758. The research of this author was partially supported by Magnus Ehrnrooths stiftelse.     

Hyperbolic linear Weingarten surfaces in ℝ<Superscript>3</Superscript>

A hyperbolic linear Weingarten surface in ℝ³ is a surface M whose mean and Gaussian curvatures satisfy the relationship 2aH +bK = c for real numbers a, b, c such that a²+bc < 0. In this work we obtain a representation for such a surface in terms of its Gauss map when, more generally, a, b, c are functions on M. We also study the completeness of such surfaces and describe a procedure to construct complete examples from solutions of the sine-Gordon equation. The first author is partially supported by Junta de Comunidades de Castilla-La Mancha, Grant no. PAI-05-034. The first and second authors are partially supported by Ministerio de Education y Ciencia Grant No. MTM2004-02746.     

A coefficient inequality for Bloch functions with applications to uniformly locally univalent functions

We give a Fekete-Szeg? type inequality for an analytic function on the unit disk with Bloch seminorm ≤1. As an application of it, we derive a sharp inequality for the third coefficient of a uniformly locally univalent function f(z) = z + a ₂ z ² + a ₃ z ³ + ⋯ on the unit disk with pre-Schwarzian norm ≤λ for a given λ > 0. The first author was partially supported by the JSPS Grant-in-Aid for Scientific Research (B), 17340039. Authors’ addresses: T. Sugawa and T. Terada, Department of Mathematics, Graduate School of Science, Hiroshima University, Higashi-Hiroshima 739-8526, Japan Current address: T. Sugawa, Division of Mathematics, Graduate School of Information Sciences, Tohoku University, Sendai 980-8579, Japan     

Hankel-Toeplitz type operators onL a 1 (Ω)

This paper studies Hankel and Toeplitz operators on the Bergman spaceL _a ¹ () of bounded symmetric domains. These operators are defined in terms of a certain bounded projection onL ¹(,dV). The main results of the paper include several characterizations for the boundedness and (weak-star) compactness of these Hankel-Toeplitz type operators. When the symbol is conjugate holomorphic, our results here are similar to those obtained by Békollé, Berger, Coburn, and Zhu [2] in theL ²-Bergman space context.Research partially supported by the National Science Foundation     

Toeplitz-Hankel type operators on product spaces

Via a series of orthogonal two-dimensional wavelets, an orthogonal decomposition of the space of square integral functions on U×U (U is the upper half-plane) with the measurey ^a ₁ y ^a ₁ dx ₁ dx ₂ dx ₁ dx ₂ is given. Four kinds of Toeplitz-Hankel type operators between the decomposition components are defined and boundedness, S_p properties of them are established. Research was supported by the National Natural Science Foundation of China.     

Contractivity properties of Ornstein-Uhlenbeck semigroup for general commutation relations

We study a certain class of von Neumann algebras generated by selfadjoint elements ω_i=a_i+a_i⁺, where a_i, a_i⁺ satisfy the general commutation relations:We assume that operator T for which the constants are matrix coefficients satisfies the braid relation. Such algebras were investigated in [BSp] and [K] where the positivity of the Fock representation and factoriality were shown. In this paper we prove that T-Ornstein-Uhlenbeck semigroup U_t=Γ_T(e^−t), t>0 arising from the second quantization procedure is hyper- and ultracontractive. The optimal bounds for hypercontractivity are also discussed.This paper was partially supported by KBN grant no 2P03A00732 and also by RTN grant HPRN-CT-2002-00279.     

Almost poised basic hypergeometric series

Given a basic hypergeometric series with numerator parametersa ₁,a ₂, ...,a _r and denominator parametersb ₂, ...,b _r, we say it isalmost poised ifb _i, =a ₁ q ^δ,i a _i,δ_i = 0, 1 or 2, for 2 ≤i ≤r. Identities are given for almost poised series withr = 3 andr = 5 when a₁, =q ⁻²ⁿ. Partially supported by N.S.F. Grant No. DMS-8521580.