Pairs of Convex Bodies with Centrally Symmetric Intersections of Translates

For a pair of convex bodies $K$ and $K$ in $E^d$, the $d$-dimensional intersections $K \cap (x + K),$ $x \in E^d,$ are centrally symmetric if and only if $K$ and $K$ are represented as direct sums $K = R \oplus P$ and $K = R \oplus P$ such that: (i) $R$ is a compact convex set of some dimension $m$, $0 \le m \le d,$ and $R = z - R$ for a suitable vector $z \in E^d$, (ii) $P$ and $P$ are isothetic parallelotopes, both of dimension $d-m$.     

On the Rubin–Stark Conjecture for a special class of CM extensions of totally real number fields

We use Greithers recent results on Brumers Conjecture to prove Rubins integral version of Starks Conjecture, up to a power of 2, for an infinite class of CM extensions of totally real number fields, called nice extensions under the assumption that a certain Iwasawa –invariant vanishes. As a consequence and under the same assumption, we show that the Brumer–Stark Conjecture is true for nice extensions, up to a power of 2.Mathematical Subject Classification (1991): 11R42, 11R58, 11R27Research on this project was partially supported by NSF grants DMS–9801267 and DMS–0196340.in final form: 16 July 2003     

Imperfect and Nonideal Clutters: A Common Approach

We prove three theorems. First, Lovászs theorem about minimal imperfect clutters, including also Padbergs corollaries. Second, Lehmans result on minimal nonideal clutters. Third, a common generalization of these two. The endeavor of working out a common denominator for Lovászs and Lehmans theorems leads, besides the common generalization, to a better understanding and simple polyhedral proofs of both.* Visiting of the French Ministry of Research and Technology, laboratoire LEIBNIZ, Grenoble, November 1995—April 1996.     

New elliptic potentials

We construct all tangential covers as zeroes of a particular set of polynomials. The resulting pointed curves give rise, through the Krichever dictionary, to elliptic KP solitons. We then give a criterion to detect the so-called hyperelliptic-tangential covers, which give rise to elliptic KdV solitons. Finally we construct new examples of the latter covers and write down the corresponding (doubly periodic finite-gap) source potentials.Dedicated to the memory of J.-L. Verdier     

Bruck-Ryser abstract theorem and symmetric designs

In [3] U. Ott introduced a Bruck-Ryser abstract theorem concerning lattices of an R-module. As one of the applications of this theorem, he obtained a new proof of the Bruck-Ryser theorem for finite projective planes and studied p-curves in a projective plane. In this paper, we apply the Bruck-Ryser abstract theorem in order to give a new proof of the Bruck-Ryser-Chowla theorem for symmetric (v, k, )-design with (v, k, )=1, and to study p-curves in arbitrary symmetric designs.     

On Aperiodic Relational Morphisms

We characterize aperiodic relational morphisms as those that are injective on regular $\H$-classes. This result is applied to obtain simple proofs and generalizations of McAlisters results on joins of aperiodic semigroups and groups. Also, we show that if $\pv H$ is a proper, non-trivial pseudovariety of groups, then \[\pv A\ast \pv H\subsetneq (\pv A\ast \pv G)\cap \ov {\pv H}.\] We provide coordinate-free formulations and proofs of Rhodess Presentation Lemma and generalizations. As an application, we give simpler proofs of Tilsons theorem on the complexity of semigroups with at most $2$ non-zero $\J$-classes and Rhodess theorem that complexity is not local.     

A new example of ‘independence’ and ‘white noise’

Summary We examine the notion of free independence according to Voiculescu. This form of independence is used for defining free white noise or process with stationary and freely independent increments. We prove a general limit theorem giving the combinatorics of infinitely freely divisible states and thus of free white noises with the help of admissible partitions. We realize the free analogues of the Wiener process and of the Poisson process as processes on the full Fock space ofL ² (—).     

Modular operads

We develop a higher genus analogue of operads, which we call modular operads, in which graphs replace trees in the definition. We study a functor F on the category of modular operads, the Feynman transform, which generalizes Kontsevichs graph complexes and also the bar construction for operads. We calculate the Euler characteristic of the Feynman transform, using the theory of symmetric functions: our formula is modelled on Wicks theorem. We give applications to the theory of moduli spaces of pointed algebraic curves.     

A commutativity criterion for certain algebras of invariant differential operators on nilpotent homogeneous spaces

Let G be a connected, simply connected real nilpotent Lie group with Lie algebra , H a connected closed subgroup of G with Lie algebra and f a linear form on satisfying f([, ]) = {0} Let _f be the unitary character of H with differential at the origin. Let _f be the unitary representation of G induced from the character _f of H. We consider the algebra (, , f) of differential operators invariant under the action of G on the bundle with basis G/H associated to these data. We show that (, , f) is commutative if and only if _f is of finite multiplicities. This proves a conjecture of Corwin-Greenleaf and Duflo. Mathematics Subject Classification (1991):43A80, 43A85, 22E25, 22E27, 22E30UMR n 7539 du CNRS, Analyse, Géométrie et Applications.UMR n 7586 du CNRS, Théorie des Groupes, Représentations, Applications.     

Fonctions holomorphes à croissance modérée et vecteurs distributions

The space of distribution vectors of a holomorphic discrete series representation of scalar type can be identified to a space of holomorphic functions with moderate growth. This is the main result of the paper. We give an application of it to the Hankel transform and to multivariate Laguerre series expansions.

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Resume. Lespace des vecteurs distributions dune représentation de la série discrète holomorphe scalaire dun domaine borné symétrique sidentifie à lespace des fonctions holomorphes à croissance modérée. Cest le principal résultat de cet article. Nous en donnons une application à la transformation de Hankel et aux développements en série de fonctions de Laguerre de plusieurs variables.

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Mathematics Subject Classification (2000): 32M15, 22E46in final form: 19 January 2004     

Sur les puissances de l’idéal fondamental de l’anneau de Witt

We reformulate a recent result of Arason and Elman by giving a very simple presentation of the powers of the fundamental ideal of the Witt ring of a field of characteristic 2.

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Sur les puissances de lidéal fondamental de lanneau de Witt

Nous reformulons un résultat récent de Arason et Elman en donnant une présentation très simple des puissances de lidéal fondamental de lanneau de Witt dun corps de caractéristique 2.

     

The role of Fourier modes in extension theorems of Hartogs-Chirka type

We generalize Chirkas theorem on the extension of functions holomorphic in a neighbourhood of (F)(D×D) – where D is the open unit disc and (F) is the graph of a continuous D-valued function F – to the bidisc. We extend holomorphic functions by applying the Kontinuitätssatz to certain continuous families of analytic annuli, which is a procedure suited to configurations not covered by Chirkas theorem.Mathematics Subject classification (2000): 32D15Work supported by NSF Grant DMS-0072237.     

A Time Domain Characterization of 2-Microlocal Spaces

In Kolwankar and Lévy Véhel, new functional spaces, denoted $K^{s,s}_{x_0}$, were introduced. These spaces characterize the fine local regularity of functions, much in the spirit of 2-microlocal spaces $C^{s,s}_{x_0}$. In contrast with $\C^{s,s}_{x_0}$ spaces, however, $K^{s,s}_{x_0}$ spaces are defined through simple estimations on the pointwise values of the functions. In this work, we generalize the definition of $K^{s,s}_{x_0}$ spaces and prove the equality $C^{s,s}_{x_0}=K^{s,s}_{x_0}$ for $s+s>0$, $s>0$. Using this result, we propose an algorithm able to estimate a part of the 2-microlocal frontier. Experiments on sampled data show that reasonable accuracy is achieved even for difficult functions such as continuous but nowhere differentiable ones. As a by-product, robust estimators of both the pointwise and the local exponents are obtained.     

A Note on π-Engel Conditions

In this paper we show that the weakly -Engel conditions are closely related to the existance of normal -complements; while the -Engel conditions are closely related to the -nilpotent groups.AMS Subject Classification (2000): 20D20     

Local symmetries and conservation laws

Starting with Lie's classical theory, we carefully explain the basic notions of the higher symmetries theory for arbitrary systems of partial differential equations as well as the necessary calculation procedures. Roughly speaking, we explain what analogs of higher KdV equations are for an arbitrary system of partial differential equations and also how one can find and use them. The cohomological nature of conservation laws is shown and some basic results are exposed which allow one to calculate, in principle, all conservation laws for a given system of partial differential equations. In particular, it is shown that symmetry and conservation law are, in some sense, the dual conceptions which coincides in the self-dual case, namely, for Euler-Lagrange equations. Training examples are also given.Translated from the Russian by B. A. Kuperschmidt.     

Definability and automorphisms in abstract logics

In any model theoretic logic, Beths definability property together with Feferman-Vaughts uniform reduction property for pairs imply recursive compactness, and the existence of models with infinitely many automorphisms for sentences having infinite models. The stronger Craigs interpolation property plus the uniform reduction property for pairs yield a recursive version of Ehrenfeucht-Mostowskis theorem. Adding compactness, we obtain the full version of this theorem. Various combinations of definability and uniform reduction relative to other logics yield corresponding results on the existence of non-rigid models.     

Blow-up rates for parabolic systems

Let ⁿ be a bounded domain andB _R be a ball in ⁿ of radiusR. We consider two parabolic systems: ut=u +f(), i= +g(u) in × (0,T) withu=v=0 on × (0,T) andu _t =u, v _t =v inB _r × (0,T) withe/v=f (v), e/v=g(u) onB _R × (0,T). Whenf(v) andg(u) are power law or exponential functions, we establish estimates on the blow-up rates for nonnegative solutions of the systems.     

A certain periodic control problem for differential equations with impulses in the space of bounded numerical sequences

One considers the differential equation dx/dt=f(t, x) with the impulse action ¦_t=t_i=H_i(t_i,x) in the space of bounded numerical sequences, where f(t, x), H_i(t, x) are T-periodic, countable-dimensional vector-valued functions, is a positive parameter. One gives conditions for the existence of a control (₁,₂) such that the solution of the equation dx/dt=f(t, x)–₁ with impulse action x¦_t=t_i=H_i(t_i,x)–₂ assuming for t= the value x=x₀, be T-periodic.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 2, pp. 271–275, February, 1990.     

Détecteur de tridents dans les émulsions photographiques nucléaires

Summary The direct creation of an electron-positron pair by a high-energy charged particle manifests itself in a certain type of track configuration which is known as a trident. A device for automatic scanning and detection of tridents in nuclear emulsion is described. The device consists of two main parts: a TV camera which analyses the magnified image of the emulsion and a logical detection unit which counts all the observed tracks and also counts the track configurations which may correspond to tridents. The performance of the apparatus is discussed.     

Mutational equations in metric spaces

This paper summarizes an extension of differential calculus to a mutational calculus for maps from one metric space to another. The simple idea is to replace half-lines allowing to define difference quotients of maps and their various limits in the case of vector space by transitions with which we can also define differential quotients of a map. Their various limits are called mutations of a map. Many results of differential calculus and set-valued analysis, including the Inverse Function Theorem, do not really rely on the linear structure and can be adapted to the nonlinear case of metric spaces and exploited. Furthermore, the concept of differential equation can be extended tomutational equation governing the evolution in metric spaces. Basic Theorems as the Nagumo Theorem, the Cauchy-Lipschitz Theorem, the Center Manifold Theorem and the second Lyapunov Method hold true for mutational equations.This work was motivated by evolution equations of tubes in visual servoing on one hand, mathematical morphology on the other, when the metric spaces are power spaces. This paper begins by listing some consequences of general theorems concerning mutational equations for tubes.