Computing the Ramanujan tau function

We show that the Ramanujan tau function τ(n) can be computed by a randomized algorithm that runs in time for every O( ) assuming the Generalized Riemann Hypothesis. The same method also yields a deterministic algorithm that runs in time O( ) (without any assumptions) for every ε > 0 to compute τ(n). Previous algorithms to compute τ(n) require Ω(n) time. Research supported in part by NSF grant CCR-9988202 2000 Mathematics Subject Classification Primary—11-04; Secondary—11Y55, 11F11     

Variations on a Theme of Hardy’s

Let θ > 1 and let ϕ : [0,1] → ℂ be such that the two-sided series converges for all x ∊ [0,1], (then necessarily φ(0) = φ(1) = 0).Suppose Define For different classes of functions φ we show that À notre ami, Jean-Louis Nicolas2000 Mathematics Subject Classification: Primary—11B83, 11B99     

Spherical Monogenics: An Algebraic Approach

The space of spherical monogenics in can be regarded as a model for the irreducible representation of Spin(m) with weight . In this paper we construct an orthonormal basis for . To describe the symmetry behind this procedure, we define certain Spin(m − 2)-invariant representations of the Lie algebra (2) on . Received: October, 2007. Accepted: February, 2008.     

Standing waves in the Maxwell-Schrödinger equation and an optimal configuration problem

We study the following system of Maxwell-Schrödinger equations $ \Delta u - u - \delta u \psi+ f(u)=0, \quad \Delta \psi + u^2 = 0 \mbox{in} {\mathbb R}^N , u, \;\psi > 0, \quad u, \;\psi \to 0 \ \mbox{as} \ |x| \to + \infty, $ where δ > 0, u, ψ : $\psi: {\mathbb R}^N \to {\mathbb R}We study the following system of Maxwell-Schr?dinger equations where δ > 0, u, ψ : , f : , N ≥ 3. We prove that the set of solutions has a rich structure: more precisely for any integer K there exists δ_K > 0 such that, for 0 < δ < δ_K, the system has a solution (u_δ, ψ_δ) with the property that u_δ has K spikes centered at the points . Furthermore, setting , then, as δ → 0, approaches an optimal configuration for the following maximization problem: Subject class: Primary 35B40, 35B45; Secondary 35J55, 92C15, 92C40     

On Some Sets with Even Valued Partition Function

Let IN be the set of positive integers, = {b₁ < ⋅s < b_h}⊂IN, N∊IN and N≥ b_h. = ₀( ,N) is the set (introduced by J.-L. Nicolas, I.Z. Ruzsa and A. Sárközy) such that {1,..., N} = and p( ,n)≡ 0 (mod 2) for n∊IN and n > N, where p( ,n) denotes the number of partitions of n with parts in . Let us denote by σ ( ,n) the sum of the divisors of n belonging to . In a paper jointly written with J.-L. Nicolas, we have recently proved that, for all k≥ 0, the sequence (σ( ,2^k n))_{n≥ 1}mod 2^k+1 is periodic with an odd period q_k. In this paper, we will characterize for any fixed odd positive integer q, the sets and the integers N such that q₀ = q, and those for which q_k = q for all k≥ 0. Moreover, a set = ₀( ,N) is constructed with the property that its period, i.e. the period of (σ( ,n))_{n≥ 1}mod 2, is 217, and for which the counting function is asymptotically equal to that of ₀({1,2,3,4,5},5) which is a set of period 31.Dedicated to Professor J.-L. Nicolas on the occasion of his 60th birthday2000 Mathematics Subject Classification: Primary—11P81, 11P83Research supported by MIRA 2002 program n^o 0203012701, Number Theory, Lyon-Monastir.     

Subharmonic Functions in the Unit Ball

For functions u subharmonic in the unit ball B_N of , this paper compares the growth of the repartition function of their Riesz measure μ with the growth of u near the boundary of B_N. Cases under study are: and , with A, B, γ positive constants and if N=2 or if N≥ 3. This paper contains several integral results, as for instance: when ∫_BN u⁺(x)[-ω^′(|x|²)]dx < +∞ for some positive decreasing C¹ function ω, it is proved that .     

A generalized Jensen’s mapping and linear mappings between Banach modules

Let X and Y be vector spaces. It is shown that a mapping f : X → Y satisfies the functional equation

Quantitative irrationality for sums of reciprocals of Fibonacci and Lucas numbers

Irrationality measures are given for the values of the series , where and W_n is a rational valued Fibonacci or Lucas form, satisfying a second order linear recurrence. In particular, we prove irrationality of all the numbers

A Vector-valued H p Corona Theorem on the Polydisk

For the corona problem on the bidisk, we find analytic solutions belonging to the Orlicz-type space In addition, for 1 ≤ p < ∞, an corona theorem is established. Similar techniques can be used for the polydisk.     

On the k-error linear complexity over \mathbb{F}_p of Legendre and Sidelnikov sequences

We determine exact values for the k-error linear complexity L _k over the finite field of the Legendre sequence of period p and the Sidelnikov sequence of period p ^m  − 1. The results are

Arithmetic mean of differences of Dedekind sums

Recently, Girstmair and Schoissengeier studied the asymptotic behavior of the arithmetic mean of Dedekind sums , as N → ∞. In this paper we consider the arithmetic mean of weighted differences of Dedekind sums in the form , where is a continuous function with , runs over , the set of Farey fractions of order Q in the unit interval [0,1] and are consecutive elements of . We show that the limit lim _Q→∞ A _h (Q) exists and is independent of h.     

On the Range of the Aluthge Transform

Let be the algebra of all bounded linear operators on a complex separable Hilbert space For an operator let be the Aluthge transform of T and we define for all where T = U|T| is a polar decomposition of T. In this short note, we consider an elementary property of the range of Δ. We prove that R(Δ) is neither closed nor dense in However R(Δ) is strongly dense if is infinite dimensional. An erratum to this article is available at .     

Gauss Sum of Index 4： （2） Non-cyclic Case

Assume that m ≥ 2, p is a prime number, （m,p（p - 1）） = 1,-1 not belong to 〈p〉 belong to （Z/mZ）^＊ and [（Z/mZ）^＊：〈p〉]=4.In this paper, we calculate the value of Gauss sum G（X）=∑x∈F^＊x（x）ζp^T（x） over Fq,where q=p^f,f=φ（m）/4,x is a multiplicative character of Fq and T is the trace map from Fq to Fp.Under our assumptions,G（x） belongs to the decomposition field K of p in Q（ζm） and K is an imaginary quartic abelian unmber field.When the Galois group Gal（K/Q） is cyclic,we have studied this cyclic case in anotyer paper：＂Gauss sums of index four：（1）cyclic case＂（accepted by Acta Mathematica Sinica,2003）.In this paper we deal with the non-cyclic case.     

Lower bounds of some singular integrals and their applications

Let and . We are interested in the lower bounds of the integral:

Explicit Bounded-Degree Unique-Neighbor Concentrators

Here we solve an open problem considered by various researchers by presenting the first explicit constructions of an infinite family of bounded-degree ‘unique-neighbor’ concentrators Γ; i.e., there are strictly positive constants α and ε, such that all Γ = (X,Y,E(Γ)) ∈ satisfy the following properties. The output-set Y has cardinality times that of the input-set X, and for each subset S of X with no more than α|X| vertices, there are at least ε|S| vertices in Y that are adjacent in Γ to exactly one vertex in S. Also, the construction of is simple to specify, and each has fewer than edges. We then modify to obtain explicit unique-neighbor concentrators of maximum degree 3. * Supported by NSF grant CCR98210-58 and ARO grant DAAH04-96-1-0013.     

On Confining Potentials and Essential Self-Adjointness for Schrödinger Operators on Bounded Domains in {\mathbb{R}}^n

Let Ω be a bounded domain in with C²-smooth boundary, , of co-dimension 1, and let be a Schr?dinger operator on Ω with potential . We seek the weakest conditions we can find on the rate of growth of the potential V close to the boundary which guarantee essential self-adjointness of H on . As a special case of an abstract condition, we add optimal logarithmic type corrections to the known condition where . More precisely, we show that if, as x approaches ,

Nonlinear Elliptic Problems Above Criticality

We consider the elliptic problem Δu + u^p = 0, u > 0 in an exterior domain, under zero Dirichlet and vanishing conditions, where is smooth and bounded, and p is supercritical, namely . We prove that this problem has infinitely many solutions with slow decay at infinity. In addition, a fast decay solution exists if p is close enough to the critical exponent. If p differs from certain sequence of resonant values which tends to infinity, then the Dirichlet problem is also solvabe in a bounded domain Ω with a sufficiently small spherical hole. Lecture held in the Seminario Matematico e Fisico on Januay 23, 2006 Received: July 2006     

Mittelwertoperatoren und Charakterisierung des Kotangens durch Funktionalgleichungen

Zusammenfassung  Die Funktion genügt für jedes der Funktionalgleichung

The probability of generating a permutation group

It is well known that a permutation group of degree can be generated by elements. In this paper we study the asymptotic behavior of the probability of generating a permutation group of degree n with elements. In particular we prove that if n is large enough and elements generate a permutation group G of degree n modulo G G ², then almost certainly these elements generate G itself. Received: 2 January 2002