The general dévissage theorem for Witt groups of schemes

Let be a closed subscheme of the noetherian scheme X. We show that if X has a dualizing complex then there exists a dualizing complex of Z such that there is an isomorphism of coherent Witt groups for all . Received: 3 March 2006     

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.     

Pointwise estimates for relative fundamental solutions of heat equations in $${\mathbb{R} \times \mathbb{C}}$$

Let be a subharmonic, nonharmonic polynomial and a parameter. Define , a closed, densely defined operator on . If and , we solve the heat equations , u(0,z) = f(z) and , . We write the solutions via heat semigroups and show that the solutions can be written as integrals against distributional kernels. We prove that the kernels are C ^∞ off of the diagonal {(s, z, w) : s = 0 and z = w} and find pointwise bounds for the kernels and their derivatives.      

Asymptotic zero distribution of hypergeometric polynomials

We show that the zeros of the hypergeometric polynomials , , cluster on the loop of the lemniscate as . We also state the equations of the curves on which the zeros of , lie asymptotically as . Auxiliary results for the asymptotic zero distribution of other functions related to hypergeometric polynomials are proved, including Jacobi polynomials with varying parameters and associated Legendre functions. Graphical evidence is provided using Mathematica. This revised version was published online in June 2006 with corrections to the Cover Date.     

Quelques Résultats sur les Algèbres Flexibles Préhilbertiennes sans Diviseurs de Zéro Vérifiant ∥a 2∥ = ∥a∥2

Résumé. Soit A une algèbre réelle. On suppose que l’espace vectoriel A est muni d’une norme ∥.∥ préhilbertienne vérifiant ∥a ²∥ = ∥a∥² pour tout . Si A est flexible, sans diviseurs de zéro et de dimension ≤ 4, alors A est isomorphe à ou , ce qui généralise un théorème d’El-Mallah [1]. Si A est flexible, sans diviseurs de zéro, contenant un idempotent central et vérifiant la propriété d’Osborn, alors A est de dimension finie et isomorphe à , ou . Enfin nous montrons qu’une algèbre normée préhilbertienne unitaire d’unité e telle que ∥e∥ = 1 est flexible et vérifie ∥a ²∥ = ∥ a∥².

---

Let A be a real algebra. Assuming that a vector space A is endowed with a pre-Hilbert norm ∥.∥ satisfying ∥a ²∥ = ∥a∥² for all . If A is flexible, without divisor of zero and of a dimension ≤ 4, then A is isomorphic to or , which generalize El-Mallah’s theorem [1]. If A is flexible, without divisor of zero, containing a central idempotent and satisfying Osborn’s properties, then A is finite dimensional and isomorphic to , or . Finally we prove that a normed pre-Hilbert algebra with unit e such that ∥e∥ = 1 is flexible and satisfies ∥a ²∥ = ∥a∥².

     

On the fourier coefficients of functions concentrated on a subset of the circle

We consider ,mE > 0,G(E) is a certain subspace of L ¹ (E) consisting of functions concentrated on E and integrable, and {d_k}, (k ∈ ℤ) in a summable sequence of positive numbers. It is proved that if G(E)=L^p(E), p≥2, then there exists f∈G(E) such that |f(n)|≥d_n, (one of the questions involved in the majorization problem). Sufficient conditions are obtained for certain other function classes G(E). We study the question of partial majorization. Bibliography: 2 titles. Translated fromProblemy Matematicheskogo Analiza, No. 13, 1992, pp. 42–48.     

Trigonometric series of classesL^p (\mathbb{T}), p \in ]1;\infty [ and their conservative meansand their conservative means

Suppose that a lower triangular matrix μ:[μ _m ⁽ⁿ⁾ ] defines a conservative summation method for series, i.e.,

Bifurcations of Positive and Negative Continua in Quasilinear Elliptic Eigenvalue Problems

The main result of this work is a Dancer-type bifurcation result for the quasilinear elliptic problem

On the Existence of Trace for Elements of \mathbb{C}_p

Let T be a transcendental element of and the orbit of T. On we have a Haar measure . The goal of this paper is to characterize all the elements of for which the integral , called the trace of T, is well defined.Presented by A. Verschoren     

Remarks on ample vector bundles of curve genus two

Let be an ample vector bundle of rank n – 1 on a smooth complex projective variety X of dimension n≥ 3 such that X is a -bundle over and that for any fiber F of the bundle projection . The pairs with = 2 are classified, where is the curve genus of . This allows us to improve some previous results. Received: 13 June 2006     

Asymptotic behaviour for a nonlocal diffusion equation on a lattice

In this paper we study the asymptotic behaviour as t → ∞ of solutions to a nonlocal diffusion problem on a lattice, namely, with t ≥ 0 and . We assume that J is nonnegative and verifies . We find that solutions decay to zero as t → ∞ and prove an optimal decay rate using, as our main tool, the discrete Fourier transform.      

Gaussian radial‐basis functions: Cardinal interpolation of ℓp and power‐growth data

Suppose λ is a positive number. Basic theory of cardinal interpolation ensures the existence of the Gaussian cardinal function \(L_\lambda (x) = \sum\nolimits_{k \in \mathbb{Z}} {c_k \exp ( - \lambda (x - k)^2 ),x \in \mathbb{R}} ,\) satisfying the interpolatory conditions \(L_\lambda (j) = \delta _{0j} ,j \in \mathbb{Z}.\) The paper considers the Gaussian cardinal interpolation operator

$(\mathcal{L}_\lambda {\text{y}})(x): = \sum\limits_{k \in \mathbb{Z}} {y_k L_\lambda (x - k),{\text{ y}} = (y_k )_{k \in \mathbb{Z}} ,{\text{ }}x \in \mathbb{R}} ,$

as a linear mapping from ℓ^p(ℤ) into L ^p(ℝ), 1≤ p ∞, and in particular, its behaviour as λ→0⁺. It is shown that \(\left\| {\mathcal{L}_\lambda } \right\|_p \) is uniformly bounded (in λ) for 1 < p < ∞, and that \(\left\| {\mathcal{L}_\lambda } \right\|_1 \asymp \log (1/\lambda )\) as λ→0⁺. The limiting behaviour is seen to be that of the classical Whittaker operator

$\mathcal{W}:{\text{y}} \mapsto \sum\limits_{k \in \mathbb{Z}} {y_k \frac{{\sin \pi (x - k)}}{{\pi (x - k)}}} ,$

in that \(\lim _{\lambda \to 0^ + } \left\| {\mathcal{L}_\lambda {\text{y}} - \mathcal{W}{\text{y}}} \right\|_p = 0,\) for every \({\text{y}} \in \ell ^p (\mathbb{Z}){\text{ and }}1 < p < \infty .\) It is further shown that the Gaussian cardinal interpolants to a function f which is the Fourier transform of a tempered distribution supported in (-π,π) converge locally uniformly to f as λ→0⁺. Multidimensional extensions of these results are also discussed.

     

The integral monodromy of hyperelliptic and trielliptic curves

We compute the and monodromy of every irreducible component of the moduli spaces of hyperelliptic and trielliptic curves. In particular, we provide a proof that the monodromy of the moduli space of hyperelliptic curves of genus g is the symplectic group . We prove that the monodromy of the moduli space of trielliptic curves with signature (r,s) is the special unitary group . Rachel Pries was partially supported by NSF grant DMS-04-00461.     

On groups of type Φ

We define a group G to be of type Φ if it has the property that for every -module G, proj. G < ∞ iff proj. H G < ∞ for every finite subgroup H of G. We conjecture that the type Φ is an algebraic characterization of those groups G which admit a finite dimensional model for , the classifying space for the family of the finite subgroups of G. We also conjecture that the type Φ is equivalent to spli being finite, where spli is the supremum of the projective lengths of the injective -modules. Here we prove certain parts of these conjectures. The project is cofounded by the European Social Fund and National Resources–EPEAK II–Pythagoras. Received: 21 June 2006     

A class number formula for the p-cyclotomic field

Let p be an odd prime number and . Let be the classical Stickelberger ideal of the group ring . Iwasawa [6] proved that the index equals the relative class number of . In [2], [4] we defined for each subgroup H of G a Stickelberger ideal of , and studied some of its properties. In this note, we prove that when mod 4 and [G : H] = 2, the index equals the quotient . Received: 13 January 2006     

Existence and Weyl’s law for spherical cusp forms

Let G be a split adjoint semisimple group over and a maximal compact subgroup. We shall give a uniform, short and essentially elementary proof of the Weyl law for cusp forms on congruence quotients of . This proves a conjecture of Sarnak for -split groups, previously known only for the case G = PGL(n). The key idea amounts to a new type of simple trace formula. Received: April 2005 Revision: June 2006 Accepted: October 2006     

Lifshits Tails for Random Schrödinger Operators with Nonsign Definite Potentials

This paper is devoted to the study of Lifshits tails for random Schr?dinger operator acting on of the form , where H ₀ is a -periodic Schr?dinger operator, λ is a positive coupling constant, are i.i.d and bounded random variables and V is the single site potential with changing sign. We prove that, in the weak disorder regime, at an open band edge, a true Lifshits tail for the random Schr?dinger operator occurs under a certain set of conditions on H ₀ and on V. Submitted: April 17, 2007. Accepted: December 13, 2007.     

Wreath products with the integers,proper actions and Hilbert space compression

We prove that the properties of acting metrically properly on some space with walls or some CAT(0) cube complex are closed by taking the wreath product with . We also give a lower bound for the (equivariant) Hilbert space compression of in terms of the (equivariant) Hilbert space compression of H. A la mémoire de Michel Matthey