Quaternary universal forms over \mathbf{Q}(\sqrt{13})

Let be a real quadratic field over Q with m a square-free positive rational integer and be the integer ring in F. A totally positive definite integral n-ary quadratic form f=f(x ₁,…,x _n )=∑_1≤i,j≤n α _ij x _i x _j ( ) is called universal if f represents all totally positive integers in . Chan, Kim and Raghavan proved that ternary universal forms over F exist if and only if m=2,3,5 and determined all such forms. There exists no ternary universal form over real quadratic fields whose discriminants are greater than 12. In this paper we prove that there are only two quaternary universal forms (up to equivalence) over . For the proof of universality we apply the theory of quadratic lattices.      

Sharp estimates for maximal operators associated to the wave equation

The wave equation, ∂ _tt u=Δu, in ℝ ⁿ⁺¹, considered with initial data u(x,0)=f∈H ^s (ℝ ⁿ ) and u’(x,0)=0, has a solution which we denote by . We give almost sharp conditions under which and are bounded from H ^s (ℝ ⁿ ) to L ^q (ℝ ⁿ ).     

Grandes valeurs du nombre de factorisations d’un entier en produit ordonné de facteurs premiers

Among various functions used to count the factorizations of an integer n, we consider here the number of ways of writing n as an ordered product of primes, which, if , is equal to the multinomial coefficient . The function P(s)=∑ _{p prime} p ^−s , sometimes called the prime zeta function, plays an important role in the study of the function h. We denote by λ=1.399433… the real number defined by P(λ)=1. The mean value of the function h satisfies . In this paper, we study how large h(n) can be. We prove that there exists a constant C ₁>0 such that, for all n≥3, holds. We also prove that there exists a constant C ₂ such that, for all n≥3, there exists m≤n satisfying . Let us call h-champion an integer N such that M<N implies h(M)<h(N). S. Ramanujan has called highly composite a τ-champion number, where τ(n)=∑ _d∣n 1 is the number of divisors of n. We give several results about the number of prime factors of an h-champion number N, about the exponents in the standard factorization into primes of such an N and about the number Q(X) of h-champion numbers N≤X. At the end of the paper, several open problems are listed. Recherche partiellement financée par le CNRS, Institut Camille Jordan, UMR 5208 et par l’action de coopération franco-algérienne 01 MDU 514, Arithmétique, Géométrie Algébrique et Applications.     

A Characterization of Quadric Constant Mean Curvature Hypersurfaces of Spheres

Let be an immersion of a complete n-dimensional oriented manifold. For any v∈ℝ ⁿ⁺², let us denote by ℓ _v :M→ℝ the function given by ℓ _v (x)=〈φ(x),v〉 and by f _v :M→ℝ, the function given by f _v (x)=〈ν(x),v〉, where is a Gauss map. We will prove that if M has constant mean curvature, and, for some v≠0 and some real number λ, we have that ℓ _v =λ f _v , then, φ(M) is either a totally umbilical sphere or a Clifford hypersurface. As an application, we will use this result to prove that the weak stability index of any compact constant mean curvature hypersurface M ⁿ in which is neither totally umbilical nor a Clifford hypersurface and has constant scalar curvature is greater than or equal to 2n+4. A. Brasil Jr. was partially supported by CNPq, Brazil, 306626/2007-1.     

On sum-product bases

We investigate additive-multiplicative bases in . Let , s>2, and . It is proved that , provided min {|B| ^s/2|A|^(s−2)/2,|A| ^s/2|B|^(s−2)/2}>p ^s/2. This note is supported by “Balaton Program Project” and OTKA grants K 61908, K 67676.     

Iterated Brownian Motion in Parabola-Shaped Domains

Iterated Brownian motion Z_t serves as a physical model for diffusions in a crack. If τ_D(Z) is the first exit time of this processes from a domain D⊂ℝⁿ, started at z∈D, then P_z[τ_D(Z)>t] is the distribution of the lifetime of the process in D. In this paper we determine the large time asymptotics of which gives exponential integrability of for parabola-shaped domains of the form P_α={(x,Y)∈ℝ×ℝⁿ⁻¹:x>0, |Y|<Ax^α}, for 0<α<1, A>0. We also obtain similar results for twisted domains in ℝ² as defined in DeBlassie and Smits: Brownian motion in twisted domains, Preprint, 2004. In particular, for a planar iterated Brownian motion in a parabola we find that for z∈℘

Diophantine quadruples in $\mathbb {Z}[\sqrt {4k+3}]$

It this paper, we study the existence of Diophantine quadruples with property D(z) in the ring , where d is such that the Pellian equation x ²−dy ²=±2 is solvable. This existence is characterized by the representability of z as a difference of two squares.      

Further properties of a function of Ogg and Ligozat

Certain identities of Ramanujan may be succinctly expressed in terms of the rational function on the modular curve X ₀(N), where and f _χ is a certain modular unit on the Nebentypus cover X _χ (N) introduced by Ogg and Ligozat for prime and w _N is the Fricke involution. These correspond to levels N=5,13, where the genus g _N of X ₀(N) is zero. In this paper we study slightly more general kind of relations for each such that X ₀(N) has genus g _N =1,2, and also for each such that the Atkin–Lehner quotient X ₀⁺(N) has genus g _N ⁺=1,2. Finally we study the normal closure of the field of definition of the zeros of the latter.      

Toeplitz operators related to strongly singular Calderón-Zygmund operators

In this paper, the boundedness of Toeplitz operator T _b(f) related to strongly singular Calderón-Zygmund operators and Lipschitz function b ε (ℝⁿ) is discussed from L ^p(ℝⁿ) to L ^q(ℝⁿ), , and from L ^p(ℝⁿ) to Triebel-Lizorkin space . We also obtain the boundedness of generalized Toeplitz operator Θ _α0 ^b from L ^p(ℝⁿ) to L ^q(ℝⁿ), . All the above results include the corresponding boundedness of commutators. Moreover, the boundedness of Toeplitz operator T _b(f) related to strongly singular Calderón-Zygmund operators and BMO function b is discussed on L ^p(ℝⁿ), 1 < p < ∞.     

Diophantine exponents for mildly restricted approximation

We are studying the Diophantine exponent μ _n,l defined for integers 1≤l<n and a vector α∈ℝ ⁿ by letting

Closure Properties of Solutions to Heat Inequalities

We prove that if u ₁,u ₂:(0,∞)×ℝ ^d →(0,∞) are sufficiently well-behaved solutions to certain heat inequalities on ℝ ^d then the function u:(0,∞)×ℝ ^d →(0,∞) given by also satisfies a heat inequality of a similar type provided . On iterating, this result leads to an analogous statement concerning n-fold convolutions. As a corollary, we give a direct heat-flow proof of the sharp n-fold Young convolution inequality and its reverse form. Both authors were supported by EPSRC grant EP/E022340/1.     

Mean-value theorems and extensions of the Elliott-Daboussi theorem on additive arithmetic semigroups

We present more general forms of the mean-value theorems established before for multiplicative functions on additive arithmetic semigroups and prove, on the basis of these new theorems, extensions of the Elliott-Daboussi theorem. Let be an additive arithmetic semigroup with a generating set ℘ of primes p. Assume that the number G(m) of elements a in with “degree” ∂(a)=m satisfies

A central limit theorem for convex sets

We show that there exists a sequence for which the following holds: Let K⊂ℝⁿ be a compact, convex set with a non-empty interior. Let X be a random vector that is distributed uniformly in K. Then there exist a unit vector θ in ℝⁿ, t₀∈ℝ and σ>0 such that

On Convergence and Convolutions of Random Signed Measures

Let μ _n be a sequence of random finite signed measures on the locally compact group G equal to either or ℝ ^d . We give weak conditions on the sequence μ _n and on functions K such that the convolution product μ _n *K, and its derivatives, converge in law, in probability, or almost surely in the Banach spaces or L ^p (G). Examples for sequences μ _n covered are the empirical process (possibly arising from dependent data) and also random signed measures where is some (nonparametric) estimator for the measure ℙ, including the usual kernel and wavelet based density estimators with MISE-optimal bandwidths. As a statistical application, we apply the results to study convolutions of density estimators.      

Riesz transformations of distributions and a generalized Hardy space

LeiE(ℝⁿ) be the space of all functions on ℝⁿ which can continue to the entire holomorphic functions on ℂⁿ. We define Riesz transformation R_j of distributions as a linear transformation of the quotient spaceD′(ℝⁿ)/E(ℝⁿ) to itself, j=1,2,..., n. These generalized Riesz transformations share the same properties with the classical ones, such as . As applications we generalize further a theorem of F. & M. Riesz generalized by Stein and Weiss, and then define a generalized Hardy space, of which some properties are studied.     

On Normal Approximation for Strongly Mixing Random Variables

In this paper, we estimate the difference , where Z _n is the sum of n centered and normalized random variables (without the stationarity assumption) satisfying the strong mixing condition, N is a standard normal random variable, and h:ℝ→ℝ is a Lipschitz function. In particular cases, the obtained upper bounds are of order O(n ^−1/2). The research was partially supported by the Lithuanian State Science and Studies Foundation, grant No. T-10/06.     

Complete asymptotic expansions associated with Epstein zeta-functions

Let Q(u,v)=|u+vz|² be a positive-definite quadratic form with a complex parameter z=x+iy in the upper-half plane. The Epstein zeta-function attached to Q is initially defined by for Re s>1, where the term with m=n=0 is to be omitted. We deduce complete asymptotic expansions of as y→+∞ (Theorem 1 in Sect. 2), and of its weighted mean value (with respect to y) in the form of a Laplace-Mellin transform of (Theorem 2 in Sect. 2). Prior to the proofs of these asymptotic expansions, the meromorphic continuation of over the whole s-plane is prepared by means of Mellin-Barnes integral transformations (Proposition 1 in Sect. 3). This procedure, differs slightly from other previously known methods of the analytic continuation, gives a new alternative proof of the Fourier expansion of (Proposition 2 in Sect. 3). The use of Mellin-Barnes type of integral formulae is crucial in all aspects of the proofs; several transformation properties of hypergeometric functions are especially applied with manipulation of these integrals. Research supported in part by Grant-in-Aid for Scientific Research (No. 13640041), the Ministry of Education, Culture, Sports, Science and Technology of Japan.