The Hopf algebra Rep U_{q} \widehat{\frak g \frak l}_\infty

We define the Hopf algebra structure on the Grothendieck group of finite-dimensional polynomial representations of in the limitN→∞. The resulting Hopf algebra Rep is a tensor product of its Hopf subalgebras Rep_a ,a ∈ ℂ^×/q^2ℤ. Whenq is generic (resp.,q ² is a primitive root of unity of orderl), we construct an isomorphism between the Hopf algebra Rep _a and the algebra of regular functions on the prounipotent proalgebraic group (resp., ). Whenq is a root of unity, this isomorphism identifies the Hopf subalgebra of Rep _a spanned by the modules obtained by pullback with respect to the Frobenius homomorphism with the algebra generated by the coefficients of the determinant of an element of considered as anl×l matrix over the Taylor series. This gives us an explicit formula for the Frobenius pullbacks of the fundamental representations. In addition, we construct a natural action of the Hall algebra associated to the infinite linear quiver (resp., the cyclic quiver withl vertices) on Rep _a and describe the span of tensor products of evaluation representations taken at fixed points as a module over this Hall algebra.     

On the finiteness of optimal partitions

We prove that, under appropriate assumptions on the domain Ω and on the datumg, any optimal partition of Ω (minimizing the sum of the total perimeter and the approximation term is finite. Finiteness result for the problem of image segmentation in Artificial Vision can be deduced.

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Sunto Dimostriamo che, in opportune ipotesi sul dominio Ω e sul datog, ogni partizione ottimale di Ω (minimizzante il perimetro totale in Ω più il termine di approssimazione è finita. Se ne deducono risultati di finitezza per il problema della segmentazione di immagini in Visione Artificiale.

     

On non-closure of range of values of elliptic operator for a plane angle

Summary Let a plane angle of opening α∈(π, 2π). LetP _D andP _N the Dirichlet and Neumann problems associated to the Poisson equation in . ForP _D andP _N it is proved non existence of solution in L _p ( ) whenp=2/(1±π/α). In other words, the ranges of elliptic operators naturally associated toP _D andP _N are not-closed in L _p ( ) forp=2/(1±π/α).

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Sunto Sia } un angolo piano di apertura α∈(π, 2π). SianoP _D eP _N i problemi di Dirichlet e di Neumann associati all'equazione di Poisson in . PerP _D eP _N si prova non esistenza di soluzioni in L _p ( ) quandop=2/(1±π/α). Vale a dire i ranges degli operatori ellittici naturalmente associati aP _D eP _N sono non-chiusi in π^--AgBrα _K L _p ( ) perp=2/(1±π/α).

     

Stability of the topological pressure for piecewise monotonic maps underC 1-perturbations

Assume thatX is a finite union of closed intervals and consider aC ¹-mapX→ℝ for which {c∈X: T′c=0} is finite. Set . Fix ann ∈ ℕ. For ε>0, theC ¹-map is called an ε-perturbation ofT if is a piecewise monotonic map with at mostn intervals of monotonicity and is ε-close toT in theC ¹-topology. The influence of small perturbations ofT on the dynamical system (R(T),T) is investigated. Under a certain condition on the continuous functionf:X → ℝ, the topological pressure is lower semi-continuous. Furthermore, the topological pressure is upper semi-continuous for every continuous functionf:X → ℝ. If (R(T),T) has positive topological entropy and a unique measure μ of maximal entropy, then every sufficiently small perturbation ofT has a unique measure of maximal entropy, and the map is continuous atT in the weak star-topology.     

Regolarità hölderiana delle derivate dei minimi di funzionali con parte principale quadratica

Riassunto In questo lavoro si prova la regolarità h?lderiana delle derivate, fino all'ordinek, dei minimi locali dei funzionali sotto opportune ipotesi suA _ij ^αβ e sug.

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Summary In this paper we prove h?lder-continuity of the derivates, up to orderk, of local minima of functionals under suitable hypotheses forA _ij ^αβ andg.

     

Generalized interval exchanges and the 2–3 conjecture

We introduce the notion of a generalized interval exchange induced by a measurable k-partition of [0,1). can be viewed as the corresponding restriction of a nondecreasing function on ℝ with . A is called λ-dense if λ(A _i ∩(a, b))>0 for each i and any 0≤ a< b≤1. We show that the 2–3 Furstenberg conjecture is invalid if and only if there are 2 and 3 λ-dense partitions A and B of [0,1), such that . We give necessary and sufficient conditions for this equality to hold. We show that for each integer m≥2, such that 3∤2m+1, there exist 2 and 3 non λ-dense partitions A and B of [0,1), corresponding to the interval exchanges on 2m intervals, for which and commute.     

A note on symbolic and ordinary powers of homogeneous ideals

In this note we are interested in the graded modulesM _k=I^(k)/I^k and , whereI is a saturated ideal in the homogeneous coordinate ringS=K[x₀,…,x_n] of ℙⁿ,I ^(k) is the symbolic power and is the saturation of the ordinary power. Very little is known about these modules, and we provide a bound on their diameters, we compute the Hilbert functions and we study some characteristic submodules in the special case ofn+1 general points in ℙⁿ.

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Sunto In questa nota siamo interessati ai moduli graduatiM _k=I(k)/I^k e , doveI è un ideale saturato nell'anello delle coordinate omogeneeS:=K[x₀,…,x_n] di ℙⁿ,I ^(k) è la potenza simbolica e è la saturazione della potenza ordinaria. Poco è noto su questi moduli e qui viene fornito un limite superiore ai loro diametri. Ne calcoliamo inoltre le funzioni di Hilbert e studiamo alcuni sottomoduli caratteristici nel caso speciale din+1 punti in posizione generale, in ℙⁿ.

     

The spectrum of compact hypersurface in sphere

Let M be a compact minimal hypersurface of sphere Sn 1(1). Let (M) be H (r)-torus of sphere Sn 1 (1).Assume they have the same constant mean curvature H, the result in [1] is that ifSpec0(M, g) =Spec0((M), g),then for 3≤ n ≤ 6, r2≤n-1/n or n ≥ 6, r2 ≥ n-1, then M is isometric to (M). We improved the result and prove that: if Spec0(M,g) =Spec0((M),g), then M is isometric to (M). Generally, if Specp(M,g) =Specp((M),g), here p is fixed and satisfies that n(n - 1) ≠ 6p(n - p), then M is isometric to (M).     

Topological and metric product\mathbb{Z}^d -actions and their applications-actions and their applications

We show that for a -action Ψ being the Kronecker sum of a symbolic strictly ergodic -actionT and a Chacon -actionS, the rank (covering number) of Ψ is the same as that forT. Using this result we construct, for a given natural numberr≥2 and a real numberb∈(0,1) withr\b≥1, a -action with rankr, covering numberb and a simple spectrum. On the other hand, for any positive integersr, m with 1≤m≤r≤∞ we construct a -action with rankr and spectral multiplicitym.     

Some existence and nonexistence theorems for compact graphs of constant mean curvature with boundary in parallel planes

Given H≥0 and bounded convex curves α₁, ...,⇌_n, α in the plane z=0 bounding domains D₁, …, D_n, D, respectively, with if i ∈ j and with D_i ⊂ D, we obtain several results proving the existence of a constanth depending only on H and on the geometry of the curves α_i, α such that the Dirichlet problem for the constant mean curvature H equation: where may accept or not a solution.     

Baer cotorsion Pairs

LetR be a unital associative ring and two classes of leftR-modules. In [St3] the notion of a ( ) pair was introduced. In analogy to classical cotorsion pairs, a pair (V,W) of subclasses is called a ( ) pair if it is maximal with respect to the classes and the condition Ext _R ¹ (V, W)=0 for all . In this paper we study pairs whereR = ℤ and is the class of all torsion-free abelian groups andT is the class of all torsion abelian groups. A complete characterization is obtained assumingV=L. For example, it is shown that every pair is singly cognerated underV=L. The author was supported by a DFG grant.     

The closure diagram for nilpotent orbits of the split real form of E8

Let and be adjoint nilpotent orbits in a real semisimple Lie algebra. Write ≥ if is contained in the closure of . This defines a partial order on the set of such orbits, known as the closure ordering. We determine this order for the split real form of the simple complex Lie algebra, E ₈. The proof is based on the fact that the Kostant-Sekiguchi correspondence preserves the closure ordering. We also present a comprehensive list of simple representatives of these orbits, and list the irreeducible components of the boundaries and of the intersections .     

Forbidden paths and cycles in ordered graphs and matrices

Assume % MathType!End!2!1! and let Ω⊂R ^N(N≥4) be a smooth bounded domain, 0∈Ω. We study the semilinear elliptic problem: % MathType!End!2!1!. By investigating the effect of the coefficientQ, we establish the existence of nontrivial solutions for any λ>0 and multiple positive solutions with λ,μ>0 small.     

A class of multipliers for $$\mathcal{W}^ \bot $$

Let denote the class of ergodic probability preserving transformations which are disjoint from every weakly mixing system. Let be the class of multipliers for , i.e. ergodic transformations whose all ergodic joinings with any element of are also in . Fix an ergodic rotationT, a mildly mixing actionS of a locally compact second countable groupG and an ergodic cocycle ϕ forT with values inG. The main result of the paper is a sufficient (and also necessary by [LeP] whenG is countable Abelian andS is Bernoullian) condition for the skew product build fromT, ϕ andS to be an element of . Moreover, the self-joinings of such extensions ofT are described with an application to study semisimple extensions of rotations. Dedicated to Hillel Furstenberg on the occasion of his retirement The first-named author was supported in part by CRDF, grant UM1-2546-KH-03. The second-named author was supported in part by KBN grant 1P03A 03826.     

Défaut d’estimation a priori pour un problème de dirichlet

Résumé  Dans un célèbre papier ([3]), B. GIDAS et J.SPRUCK ont utilisé-sous des hypothèses adéquates- la technique du “blow up” pour montrer que les solutionsu ∈C ⁰ ∩C ¹ (Ω) du problème admettent une estimation a priori dansC ⁰ . Dans ce travail, on montre que, si les solutionsu sont juste supposéesC ⁰ , une telle estimation a priori n’existe plus. In a famous paper ([3]), B. GIDAS and J. SPRUCK used a “blow-up” argument to show that, under appropriate assumptions, all the solutionsu ∈C ⁰ ∩C ¹ (Ω) of the problem admit an a priori estimate inC ⁰ . In this work, we show that, if one supposes the solutions are only inC ⁰ , such an a priori estimate does not hold.     

Sharp estimates for\bar \partial on convex domains of finite typeon convex domains of finite type

Let Ω be a bounded convex domain in C _n , with smooth boundary of finite typem. The equation is solved in Ω with sharp estimates: iff has bounded coefficients, the coefficients of our solutionu are in the Lipschitz space Λ. Optimal estimates are also given when data have coefficients belonging toL ^p(Ω),p≥1. We solve the -equation by means of integral operators whose kernels are not based on the choice of a “good” support function. Weighted kernels are used; in order to reflect the geometry ofbΩ, we introduce a weight expressed in terms of the Bergman kernel of Ω.     

On generalized Beurling algebras

We consider the space, where Ω is a family of weights. We give a sufficient condition, on Ω, for to be locally convex algebra with continuous multiplication. Examples of such weights are also provided.     

A local duality theorem for categories of modules

Let Λ be an associative ring with identity and let be the category of left unitary Λ-modules. A subcateqory of the category is said to be small if the pairwise nonisomorphic objects of form a set. The main result of this paper consists of the fact that for every small full subcategory , there exists a ring Γ such that is dual to a small full subcategory of the category . Some applications of this result are indicated. Bibliography: 3 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 227, 1995, pp. 66–73.