Norm-preserving extensions of functionals and denting points of convex sets

Let W and Z be Banach spaces, and let and be closed subspaces. Let be a subspace of , the Banach space of bounded linear operators from W* to Z**, containing . We describe, for and , all norm-preserving extensions of to the space in terms of convergence of convex combinations. We also characterize denting points of bounded convex subsets of Banach spaces in similar terms. Various applications are presented. Supported by Estonian Science Foundation Grant 5704.     

$${\mathcal{R}}$$ -boundedness,pseudodifferential operators,and maximal regularity for some classes of partial differential operators

It is shown that an elliptic scattering operator A on a compact manifold with boundary with operator valued coefficients in the morphisms of a bundle of Banach spaces of class () and Pisier’s property (α) has maximal regularity (up to a spectral shift), provided that the spectrum of the principal symbol of A on the scattering cotangent bundle avoids the right half-plane. This is accomplished by representing the resolvent in terms of pseudodifferential operators with -bounded symbols, yielding by an iteration argument the -boundedness of λ(A−λ)⁻¹ in for some . To this end, elements of a symbolic and operator calculus of pseudodifferential operators with -bounded symbols are introduced. The significance of this method for proving maximal regularity results for partial differential operators is underscored by considering also a more elementary situation of anisotropic elliptic operators on with operator valued coefficients.     

Nested set complexes of Dowling lattices and complexes of Dowling trees

Given a finite group G and a natural number n, we study the structure of the complex of nested sets of the associated Dowling lattice (Proc. Internat. Sympos., 1971, pp. 101–115) and of its subposet of the G-symmetric partitions which was recently introduced by Hultman (, 2006), together with the complex of G-symmetric phylogenetic trees . Hultman shows that the complexes and are homotopy equivalent and Cohen–Macaulay, and determines the rank of their top homology. An application of the theory of building sets and nested set complexes by Feichtner and Kozlov (Selecta Math. (N.S.) 10, 37–60, 2004) shows that in fact is subdivided by the order complex of . We introduce the complex of Dowling trees and prove that it is subdivided by the order complex of . Application of a theorem of Feichtner and Sturmfels (Port. Math. (N.S.) 62, 437–468, 2005) shows that, as a simplicial complex, is in fact isomorphic to the Bergman complex of the associated Dowling geometry. Topologically, we prove that is obtained from by successive coning over certain subcomplexes. It is well known that is shellable, and of the same dimension as . We explicitly and independently calculate how many homology spheres are added in passing from to . Comparison with work of Gottlieb and Wachs (Adv. Appl. Math. 24(4), 301–336, 2000) shows that is intimely related to the representation theory of the top homology of . Research partially supported by the Swiss National Science Foundation, project PP002-106403/1.     

On pointed Hopf algebras associated with the symmetric groups

Let G be the symmetric group . It is an important open problem whether the dimension of the Nichols algebra is finite when is the class of the transpositions and ρ is the sign representation, with m ≥ 6. In the present paper, we discard most of the other conjugacy classes showing that very few pairs might give rise to finite-dimensional Nichols algebras. This work was partially supported by CONICET, ANPCyT and Secyt (UNC).     

A note on simultaneous Diophantine approximation on planar curves

Let denote the set of simultaneously - approximable points in and denote the set of multiplicatively ψ-approximable points in . Let be a manifold in . The aim is to develop a metric theory for the sets and analogous to the classical theory in which is simply . In this note, we mainly restrict our attention to the case that is a planar curve . A complete Hausdorff dimension theory is established for the sets and . A divergent Khintchine type result is obtained for ; i.e. if a certain sum diverges then the one-dimensional Lebesgue measure on of is full. Furthermore, in the case that is a rational quadric the convergent Khintchine type result is obtained for both types of approximation. Our results for naturally generalize the dimension and Lebesgue measure statements of Beresnevich et al. (Mem AMS, 179 (846), 1–91 (2006)). Moreover, within the multiplicative framework, our results for constitute the first of their type. The research of Victor V. Beresnevich was supported by an EPSRC Grant R90727/01. Sanju L. Velani is a Royal Society University Research Fellow. For Iona and Ayesha on No. 3.     

Rigidity and holomorphic Segre transversality for holomorphic Segre maps

Let and denote the complexifications of Heisenberg hypersurfaces in and , respectively. We show that non-degenerate holomorphic Segre mappings from into with possess a partial rigidity property. As an application, we prove that the holomorphic Segre non-transversality for a holomorphic Segre map from into with propagates along Segre varieties. We also give an example showing that this propagation property of holomorphic Segre transversality fails when N > 2n − 2.     

Tits topology and fundamental groups of compact Riemannian manifolds

Let M be a compact Riemannian manifold without conjugate points. We generalize the Tits topology on the ideal boundary of the universal covering space of M. Then we show that if π₁(M) is amenable and is compact with respect to the Tits topology, then M is flat. This work was supported by Grant No.R01-2006-000-10047-0(2006) from the Basic Research Program of the Korea Science & Engineering Foundation.     

Reduced Minimum Modulus Preserving in Banach Space

Let be the algebra of all bounded linear operators on a complex Banach space X and γ(T) be the reduced minimum modulus of operator . In this work, we prove that if , is a surjective linear map such that is an invertible operator, then , for every , if and only if, either there exist two bijective isometries and such that for every , or there exist two bijective isometries and such that for every . This generalizes for a Banach space the Mbekhta’s theorem [12].      

Formal structure of direct image of holonomic $${\mathcal{D}}$$ -modules of exponential type

We compute formal invariants associated with the cohomology sheaves of the direct image of holonomic -modules of exponential type. We also prove that every formal -modules is isomorphic, after a ramification, to a germ of formalized direct image of analytic -module of exponential type.     

Formule de torsion pour le facteur epsilon d’un caractère sur une surface

Let be a smooth proper surface over a finite field of characteristic p > 2, and let be a rank one smooth l-adic sheaf (l ≠ p) on a dense open subset . In this paper, under some assumptions on the wild ramification of , we prove a torsion formula for the epsilon factor (that is the global constant) of the functional equation of the L-function . Our torsion formula is a generalization to higher dimension of the classical torsion formula for the local constants.

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Résumé  Soit une surface propre et lisse sur un corps fini de caractéristique p > 2, et un caractère l-adique (avec l ≠ p) lisse sur un ouvert dense . Sous certaines hypothèses sur la ramification sauvage de , on prouve une formule de torsion pour le facteur epsilon (i.e. la constante globale) de l’équation fonctionnelle de la fonction . Notre formule de torsion est une généralisation en dimension supérieure de la formule de torsion pour les constantes locales, qui est à la base de la théorie des constantes locales.

     

On injective modules over Azumaya algebras over locally noetherian schemes

Let be an Azumaya algebra over a locally noetherian scheme X. We describe in this work quasi-coherent -bimodules which are injective in the category of sheaves of left -modules     

Uniform central limit theorems for kernel density estimators

Let be the classical kernel density estimator based on a kernel K and n independent random vectors X _i each distributed according to an absolutely continuous law on . It is shown that the processes , , converge in law in the Banach space , for many interesting classes of functions or sets, some -Donsker, some just -pregaussian. The conditions allow for the classical bandwidths h _n that simultaneously ensure optimal rates of convergence of the kernel density estimator in mean integrated squared error, thus showing that, subject to some natural conditions, kernel density estimators are ‘plug-in’ estimators in the sense of Bickel and Ritov (Ann Statist 31:1033–1053, 2003). Some new results on the uniform central limit theorem for smoothed empirical processes, needed in the proofs, are also included.      

Another hybrid conjugate gradient algorithm for unconstrained optimization

Another hybrid conjugate gradient algorithm is subject to analysis. The parameter β _k is computed as a convex combination of (Hestenes-Stiefel) and (Dai-Yuan) algorithms, i.e. . The parameter θ _k in the convex combination is computed in such a way so that the direction corresponding to the conjugate gradient algorithm to be the Newton direction and the pair (s _k , y _k ) to satisfy the quasi-Newton equation , where and . The algorithm uses the standard Wolfe line search conditions. Numerical comparisons with conjugate gradient algorithms show that this hybrid computational scheme outperforms the Hestenes-Stiefel and the Dai-Yuan conjugate gradient algorithms as well as the hybrid conjugate gradient algorithms of Dai and Yuan. A set of 750 unconstrained optimization problems are used, some of them from the CUTE library.      

The Reduced Minimum Modulus in <Emphasis>C</Emphasis><Superscript>*</Superscript>-Algebras

We define the reduced minimum modulus of a nonzero element a in a unital C ^*-algebra by . We prove that . Applying this result to and its closed two side ideal , we get that dist , and for any if RR = 0, where and is the quotient homomorphism and . These results generalize corresponding results in Hilbert spaces.     

Algebraic-geometry codes, one-point codes, and evaluation codes

One-point codes are those algebraic-geometry codes for which the associated divisor is a non-negative multiple of a single point. Evaluation codes were defined in order to give an algebraic generalization of both one-point algebraic-geometry codes and Reed–Muller codes. Given an -algebra A, an order function on A and given a surjective -morphism of algebras , the ith evaluation code with respect to is defined as the code . In this work it is shown that under a certain hypothesis on the -algebra A, not only any evaluation code is a one-point code, but any sequence of evaluation codes is a sequence of one-point codes. This hypothesis on A is that its field of fractions is a function field over and that A is integrally closed. Moreover, we see that a sequence of algebraic-geometry codes G _i with associated divisors is the sequence of evaluation codes associated to some -algebra A, some order function and some surjective morphism with if and only if it is a sequence of one-point codes.      

On <Emphasis Type="Italic">t</Emphasis><Superscript>2</Superscript>-like solutions of certain second order differential equations

Via an integral transformation, we establish two embedding results between the Emden-Fowler type equation , t ≥ t ₀ > 0, with solutions x such that as , , and the equation , u > 0, with solutions y such that for given k > 0. The conclusions of our investigation are used to derive conditions for the existence of radial solutions to the elliptic equation , , that blow up as in the two dimensional case.      

Inflectional loci of scrolls

Let be a scroll over a smooth curve C and let denote the hyperplane bundle. The special geometry of X implies that certain sheaves related to the principal part bundles of are locally free. The inflectional loci of X can be expressed in terms of these sheaves, leading to explicit formulas for the cohomology classes of the loci. The formulas imply that the only uninflected scrolls are the balanced rational normal scrolls.      

Notes on estimating inverse-Gaussian and gamma subordinators under high-frequency sampling

We study joint efficient estimation of two parameters dominating either the inverse-Gaussian or gamma subordinator, based on discrete observations sampled at satisfying as . Under the condition that as we have two kinds of optimal rates, and . Moreover, as in estimation of diffusion coefficient of a Wiener process the -consistent component of the estimator is effectively workable even when T _n does not tend to infinity. Simulation experiments are given under several h _n ’s behaviors.     

Relations between different types of spectra and spectral characterizations

In the study of the asymptotic behaviour of solutions of differential-difference equations the -spectrum has been useful, where and implies Fourier transform , with given , φ∈L ^∞(ℝ,X), X a Banach space, (half)line. Here we study and related concepts, give relations between them, especially weak Laplace half-line spectrum of φ, and thus ⊂ classical Beurling spectrum = Carleman spectrum =  ; also  = Beurling spectrum of “φ modulo ” (Chill-Fasangova). If satisfies a Loomis type condition (L _U ), then countable and uniformly continuous ∈U are shown to imply ; here (L _U ) usually means , indefinite integral Pf of f in U imply Pf in (the Bohl-Bohr theorem for = almost periodic functions, U=bounded functions). This spectral characterization and other results are extended to unbounded functions via mean classes , ℳ ^m U ((2.1) below) and even to distributions, generalizing various recent results for uniformly continuous bounded φ. Furthermore for solutions of convolution systems S*φ=b with in some we show . With these above results, one gets generalizations of earlier results on the asymptotic behaviour of solutions of neutral integro-differential-difference systems. Also many examples and special cases are discussed.