2009

Alternative iterative methods for nonexpansive mappings, rates of convergence and applications

Colao, Vittorio — Thu, 01 Jan 2009 00:00:00 GMT

Alternative iterative methods for nonexpansive mappings, rates of convergence and applications Colao, Vittorio; Leuştean, Laurenţiu; López, Genaro; Martín Márquez, Victoria Alternative iterative methods for a nonexpansive mapping in a Banach space are proposed and proved to be convergent to a common solution to a fixed point problem and a variational inequality. We give rates of asymptotic regularity for such iterations using proof-theoretic techniques. Some applications of the convergence results are presented. OWLF 2008

Proof mining in metric fixed point theory and ergodic theory

Leuştean, Laurenţiu — Thu, 01 Jan 2009 00:00:00 GMT

Proof mining in metric fixed point theory and ergodic theory Leuştean, Laurenţiu In this survey we present some recent applications of proof mining to the fixed point theory of (asymptotically) nonexpansive mappings and to the metastability (in the sense of Terence Tao) of ergodic averages in uniformly convex Banach spaces. OWLF 2009

Minimal Riesz energy on the sphere for axis-supported external fields

Brauchart, Johann S. — Thu, 01 Jan 2009 00:00:00 GMT

Minimal Riesz energy on the sphere for axis-supported external fields Brauchart, Johann S.; Dragnev, Peter D.; Saff, Edward B. Abstract. We investigate the minimal Riesz $s$-energy problem for positive measures on the d-dimensional unit sphere $\mathbb{S}^d$ in the presence of an external field induced by a point charge, and more generally by a line charge. The model interaction is that of Riesz potentials $|x−y|^{−s}$ with $d−2 ≤ s < d$. For a given axis-supported external field, the support and the density of the corresponding extremal measure on $\mathbb{S}^d$ is determined. The special case $s = d − 2$ yields interesting phenomena, which we investigate in detail. A weak$^∗$ asymptotic analysis is provided as $s → (d − 2)^+$. OWLF 2008

Heisenberg-Weyl algebra revisited: combinatorics of words and paths

Blasiak, Pawel — Thu, 01 Jan 2009 00:00:00 GMT

Heisenberg-Weyl algebra revisited: combinatorics of words and paths Blasiak, Pawel; Duchamp, Gérard H. E.; Horzela, Andrzej; Penson, Karol A.; Solomon, Allan I. The Heisenberg–Weyl algebra, which underlies virtually all physical representations of Quantum Theory, is considered from the combinatorial point of view. We provide a concrete model of the algebra in terms of paths on a lattice with some decomposition rules. We also discuss the rook problem on the associated Ferrers board; this is related to the calculus in the normally ordered basis. From this starting point we explore a combinatorial underpinning of the Heisenberg–Weyl algebra, which offers novel perspectives, methods and applications. OWLF 2008

Geometric quantization of integrable systems with hyperbolic singularities

Hamilton, Mark D. — Thu, 01 Jan 2009 00:00:00 GMT

Geometric quantization of integrable systems with hyperbolic singularities Hamilton, Mark D.; Miranda, Eva We construct the geometric quantization of a compact surface using a singular real polarization coming from an integrable system. Such a polarization always has singularities, which we assume to be of nondegenerate type. In particular, we compute the effect of hyperbolic singularities, which make an infinite-dimensional contribution to the quantization, thus showing that this quantization depends strongly on polarization. OWLF 2008

Optimal bounds for the colored Tverberg Problem

Blagojevic, Pavle V. M. — Fri, 20 Mar 2009 00:00:00 GMT

Optimal bounds for the colored Tverberg Problem Blagojevic, Pavle V. M.; Matschke, Benjamin; Ziegler, Günter M. We prove a "Tverberg type" multiple intersection theorem. It strengthens the prime case of the original Tverberg theorem from 1966, as well as the topological Tverberg theorem of Bárány et al. (1980), by adding color constraints. It also provides an improved bound for the (topological) colored Tverberg problem of Bárány & Larman (1992) that is tight in the prime case and asymptotically optimal in the general case. The proof is based on relative equivariant obstruction theory Research in Pairs 2009

Weighted Fourier inequalities for radial functions

Gorbachev, D. — Thu, 19 Mar 2009 00:00:00 GMT

Weighted Fourier inequalities for radial functions Gorbachev, D.; Liflyand, E.; Tichonovič, S. V. Weighted $L^p(\mathbb{R}^n) \to L^q(\mathbb{R}^n)$ Fourier inequalities are studied. We prove Pitt-Boas type results on integrability with power weights of the Fourier transform of a radial function. Research in Pairs 2009

A new counting function for the zeros of holomorphic curves

Anderson, J. M. — Wed, 18 Mar 2009 00:00:00 GMT

A new counting function for the zeros of holomorphic curves Anderson, J. M.; Hinkkanen, Aimo Let $f_1,..., f_p$ be entire functions that do not all vanish at any point, so that $(f_1,..., f_p)$ is a holomorphic curve in $\mathbb{CP}^{p-1}$. We introduce a new and more careful notion of counting the order of the zero of a linear combination of the functions $f_1,..., f_p$ at any point where such a linear combination vanishes, and, if all the $f_1,..., f_p$ are polynomials, also at infinity. This enables us to formulate an inequality, which sometimes holds as an identity, that sharpens the classical results of Cartan and others. Research in Pairs 2009

Discrete non-commutative integrability: the proof of a conjecture by M. Kontsevich

Di Francesco, Philippe — Tue, 17 Mar 2009 00:00:00 GMT

Discrete non-commutative integrability: the proof of a conjecture by M. Kontsevich Di Francesco, Philippe; Kedem, Rinat We prove a conjecture of Kontsevich regarding the solutions of rank two recursion relations for non-commutative variables which, in the commutative case, reduce to rank two cluster algebras of affine type. The conjecture states that solutions are positive Laurent polynomials in the initial cluster variables. We prove this by use of a non-commutative version of the path models which we used for the commutative case. Research in Pairs 2009

On Siegel modular forms of level p and their properties mod p

Böcherer, Siegfried — Mon, 16 Mar 2009 00:00:00 GMT

On Siegel modular forms of level p and their properties mod p Böcherer, Siegfried; Nagaoka, Shoyu Using theta series we construct Siegel modular forms of level p which behave well modulo $p$ in all cusps. This construction allows us to show (under a mild condition) that all Siegel modular forms of level p and weight 2 are congruent mod $p$ to level one modular forms of weight $p+1$; in particular, this is true for Yoshidal lifts of level $p$. Research in Pairs 2008