Invariant polynomials under the maximal unipotent subgroup of a classical group

Let N be a maximal unipotent subgroup of a classical complex Lie group G, whose Lie algebra we denote by g. Inside the ring of N-invariant polynomials S′(g)^N we consider the subring generated by the polynomials of weight e^{m(Λ + Λ*)}, where Λ and Λ^* are the highest weights corresponding to the natural representation of g and to its dual respectively, and m ε Z.We prove that this subring is a polynomial ring and we explicitly give a set of generators.     

Some Consequences of Schanuel's Conjecture in Exponential Rings

In last years Schanuel's Conjecture has played a fundamental role in Transcendental Number Theory and in decidability issues.

In this article we investigate algebraic relations among the elements of the exponential field (?, e ^x ) modulo Schanuel's Conjecture. We prove that there are no further relations between π and i assuming Schanuel's Conjecture except the known ones, e ^πi  = ?1 and i ² = ?1. Moreover, modulo Schanuel's Conjecture we prove that the E-subring of ? generated by π is isomorphic to the free exponential ring on π.     

A generalization of Fermat's little theorem

Fermat's Little Theorem states that if p is a prime number and gcd (x,p) = 1, then x^p?1 ≡ 1 (modp) If the requirement that gcd (x,p) = 1 is dropped, we can say x^p ≡ x(modp)for any integer x. Euler generalized Fermat's Theorem in the following way: if gcd (x,n) = 1 then x^φ(n) ≡ 1(modn), where φ is the Euler phi-function. It is clear that Euler's result cannot be extended to all integers x in the same way Fermat's Theorem can; that is, the congruence x^φ(n)+1 ≡ x(modn)is not always valid. In this note we show exactly when the congruence x^φ(n)+1 ≡ x(modn) is valid.     

Equivalence of weak Dirichlet's principle,the method of weak solutions and Perron's method towards classical solutions of Dirichlet's problem for harmonic functions

For boundary data ? ∈ W ^1,2(G ) (where G ? ?^N is a bounded domain) it is an easy exercise to prove the existence of weak L ²‐solutions to the Dirichlet problem “Δu = 0 in G, u |_?G = ? |_?G ”. By means of Weyl's Lemma it is readily seen that there is ? ∈ C ^∞(G ), Δ? = 0 and ? = u a.e. in G . On the contrary it seems to be a complicated task even for this simple equation to prove continuity of ? up to the boundary in a suitable domain if ? ∈ W ^1,2(G ) ∩ C ⁰( ). The purpose of this paper is to present an elementary proof of that fact in (classical) Dirichlet domains. Here the method of weak solutions (resp. Dirichlet's principle) is equivalent to the classical approaches (Poincaré's “sweeping‐out method”, Perron's method). (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Immanants

Matrix-valued versions of the usual numerical ranges for operators, matrices, and elements of C ^*-algebras are discussed. Particular attention is paid to those aspects of the theory that parallel the classical case. Also included are some of the applications of matricial ranges to issues involving unitary equivalence and C ^*-convexity.

Résumé. Nous discutons les "ranges" matriciels d'operateurs, de matrices et d'éléments d'une C ^*-algèbre. Nous portons une attention particulière aux aspects de la théorie qui suivent le cas classique. Nous avons aussi considéré applications à des problèmes incluant l'equivalence unitaire aussi que la C ^*-convexité.     

Solution of a restricted Hadamard problem on Minkowski spaces

We give a complete solution of Hadamard's problem concerning Huygens' principle on even-dimensional Minkowski spaces 𝕄ⁿ⁺¹ for a restricted class of linear, second-order, normal hyperbolic operators ℒ︁ = □_n+1 + u(x¹, x²) with real (locally) analytic potentials u = u(x¹, x²) depending on two spatial variables and homogeneous of degree −2. © 1997 John Wiley & Sons, Inc.     

Un Algorithme pour le Calcul des Syzygies sur V[X] dans le cas où V est un Domaine de Valuation

Soit V un domaine de valuation. Nous donnons un algorithme pour calculer une base du V-saturé d'un sous-module de type fini d'un V-module libre (avec une base éventuellement infinie). Nous l'appliquons pour calculer le V-saturé d'un sous-V[X]-module de type fini de V[X] ⁿ (n ∈ ?*). Ceci permet enfin de calculer un système générateur fini pour les syzygies sur V[X] d'une famille finie de vecteurs de V[X] ^k .

We give an algorithm for computing the V-saturation of any finitely generated submodule of V[X] ⁿ (n ∈ ?*), where V is a valuation domain. This allows us to compute a finite system of generators for the syzygy module of any finitely generated submodule of V[X] ^k .     

与次不变凸函数有关的拟平衡问题研究

本文研究了拟平衡问题解的存在性问题.利用对Subinvex函数在Rl空间上的类变分问题的讨论及凸规划问题与平衡问题的同解性理论,把次不变凸(Subinvex)函数特征由原Rl空间推广到一般拓扑线性空间的平衡问题上,得到一类称之为拟平衡问题的解的存在性问题相关理论.     

The Pseudo-differential Operator hμ,a on Hankel Invariant Spaces

Using Hankel transform the symbol 'a' is defined and the pseudo-differential operator (p.d.o.) h_μ,a associated with the Bessel operator d ²/dx ² + (1 ? 4μ ²)/4x ² in terms of this symbol is defined. It is shown that the operator h_μ,a is a continuous linear map of a Hankel invariant space into itself. A special pseudo-differential operator called the Hankel potential is defined and some of its properties are investigated.     

SHAPE AND INDUCED REPRESENTATIONS

Abstract

Let K: P → T be a fixed functor. A criterion is given for a functor M': T → V to be a (right) Kan extension along K of some functor M: P → V. The functors M having a given M' as Kan extension are, in general, classified by continuous functors (V ^P)^o → V. We introduce a notion of system of imprimitivity, generalizing that of Mackey; when the shape category of K is codense in the systems of imprimitivity classify the functors H having M' as Kan extension. As a special case one obtains Mackey's Imprimitivity Theorem for finite groups.     

指数Diophantine方程4~x+b~y=(b+4)~2

本文研究了指数Diophantine方程4~x+b~y=(b+4)~2的解.设b1是给定的正奇数,运用有关指数Diophantine方程的已知结果以及有关Pell方程的Stormer定理的推广,证明了方程4~x+b~y=(b+4)~2仅有正整数解(x,y,z)=(1,1,1).     

Density of Gabor Frames

A Gabor system is a set of time-frequency shifts S(g, Λ) ={e^{2 π ibx}g(x − a)}_{(a, b) Λ} of a function g L²(R^d). We prove that if a finite union of Gabor systems _{k = 1}^rS(g_k, Λ_k) forms a frame for L²(R^d) then the lower and upper Beurling densities of Λ = _{k = 1}^r Λ_k satisfy D⁻(Λ) ≥ 1 and D ⁺ (Λ) < ∞. This extends recent work of Ramanathan and Steger. Additionally, we prove the conjecture that no collection _{k = 1}^r{g_k(x − a)}_{a Γk} of pure translates can form a frame for L²(R^d).     

On the powers of 3/2 and other rational numbers

Letp > q > 1 be two coprime integers. In this paper, we prove several results about subsets of the interval [0, 1) which does or does not contain all the fractional parts {ξ (p /q)ⁿ }, n = 0, 1, 2, …, for certain non‐zero real number ξ. We show, for instance, that there are no real ξ for which the union of two intervals [8/39, 18/39] ∪ [21/39, 31/39] contains the set {ξ (3/2)ⁿ }, n ∈ N . The most important aspect of this result is that the total length of both intervals 20/39 is greater than 1/2: the same result as above for [0, 1/2) would imply that there are no Mahler's Z ‐numbers which the best known unsolved problem in this area. On the other hand, it is shown that there are infinitely many ξ for which {ξ (3/2)ⁿ } ∈ (5/48, 43/48) for each integer n ≥ 0. We also give simpler proofs of few recent results in this area. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Le n-ondes de lax pour le probleme mixte (the n-waves of lax for initial-boundary value problem)

We Confirm two conjectures of Lax about Glimm's weak solutions for initial boundary value problem whose data have compact support; asmptotic decay as t^-1/2 of the total variation in the genuinely nonlinear case and L¹ -approximation by N-waves or travelling-waves in general case; it is boundary's version of Liu's results [Li2].     

Maximal commutative subalgebras of matrix algebras

Maximal commutative subalgebras of the algebra of n by n matrices over a field k very rarely have dimension smaller than n. There is a (B, N)-construction which yields subalgebras of this kind. The Courter's algebra that is of this kind was shown a (B, N)-construction where B is the Schur algebra of size 4 and N = k ⁴. That is, the Courter's algebra is isomorphic to B ? (k ⁴)², the idealization of (k ⁴)². It was questioned how many isomorphism classes can be produced by varying the finitely generated faithful B-module N. In this paper, we will show that the set of all algebras B ? N ² fall into a single isomorphism class, where B is the Schur algebra of size 4 and N a finitely generated faithful B-module.     

Mixed finite element methods for generalized Forchheimer flow in porous media

Mixed finite element methods are analyzed for the approximation of the solution of the system of equations that describes the flow of a single‐phase fluid in a porous medium in ?^d, d ≤ 3, subject to Forchhheimer's law—a nonlinear form of Darcy's law. Existence and uniqueness of the approximation are proved, and optimal order error estimates in L^∞(J; L²(Ω)) and in L^∞(J; H(div; Ω)) are demonstrated for the pressure and momentum, respectively. Error estimates are also derived in L^∞(J; L^∞(Ω)) for the pressure. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

一类带食饵趋向的Beddington-DeAngelis捕食者-食饵扩散模型整体解的有界性

本文研究一类带食饵趋向的Beddington-DeAngelis捕食者-食饵扩散模型,其中食饵趋向性描述的是捕食者对食饵数量变化而产生的一种正向迁移.利用Neumann热半群的L^p-L^q估计和带抛物型方程Moser迭代的L^p估计,获得了该模型经典解的整体有界性.     

De Morgan (2, n)-Semigroups of n-Place Functions

An abstract characterization of sets of some partial functions from Aⁿ to A closed with respect to the Menger's and Mann's superpositions and a quasicomplementation is presented.     

Cantor exchange systems and renormalization

We prove a one-to-one correspondence between (i) C¹⁺ conjugacy classes of C^1+H Cantor exchange systems that are C^1+H fixed points of renormalization and (ii) C¹⁺ conjugacy classes of C^1+H diffeomorphisms f with a codimension 1 hyperbolic attractor Λ that admit an invariant measure absolutely continuous with respect to the Hausdorff measure on Λ. However, we prove that there is no C^1+α Cantor exchange system, with bounded geometry, that is a C^1+α fixed point of renormalization with regularity α greater than the Hausdorff dimension of its invariant Cantor set.     

A Nonlinear Version of Noether's Type Theorem

Abstract

Let L be a line bundle on a smooth curve C, which defines a birational morphism onto Φ(C) ? P ^r . We prove that, under suitable assumptions on L, which are satisfied by Castelnuovo's curves, a generic section in H ⁰(C, L ²) can be written as α² + β² + γ², with α, β, γ ∈ H ⁰(C, L). If there are no quadrics of rank 3 containing Φ(C), this is true for any section. For canonical curves, this gives a non linear version of Noether's Theorem.