On generalized gamma functions related to the Laurent coefficients of the Riemann zeta function

Summary We study a class of generalized gamma functions _k (z) which relate to the generalized Euler constants _k (basically the Laurent coefficients of(s)) as (z) does to the Euler constant. A new series expansion for _k is derived, and the constant term in the asymptotic expansion for log _k (z) is studied in detail. These and related constants are numerically computed for 1 k 15.     

Convergence des solutions formelles de certaines équations fonctionnelles

Résumé Soitq un nombre algébrique de module 1, qui ne soit pas une racine de l'unité, etP [X, Y ₀,Y ₁] un polynôme non nul. Dans cet article, nous montrons que toute solution de l'équation fonctionnelleP(z, (z), (qz))=0, qui est une série formelle (z) dansQ[[z]], a un rayon de convergence non nul.

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Summary Letq Q be an algebraic number of modulus one that is not a root of unity. LetP Q[X, Y ₀,Y ₁] be a non zero polynomial. In this paper, we show that every formal power series,(z) Q[[z]], solution of the functional equationP(z), (z), (qz)) = 0 has a non zero radius of convergence.

     

From Ginzburg-Landau to Gross-Pitaevskii

In this note we consider the Gross-Pitaevskii equation i _t ++(1–²)=0, where is a complex-valued function defined on ^N×, and study the following 2-parameters family of solitary waves: (x, t)=e ^it v(x ₁–ct, x), where and x denotes the vector of the last N–1 variables in ^N . We prove that every distribution solution , of the considered form, satisfies the following universal (and sharp) L -bound:

A general minimax theorem

This paper is concerned with minimax theorems for two-person zero-sum games (X, Y, f) with payofff and as main result the minimax equality inf supf (x, y)=sup inff (x, y) is obtained under a new condition onf. This condition is based on the concept of averaging functions, i.e. real-valued functions defined on some subset of the plane with min {x, y}< (x, y)x, y} forx y and (x, x)=x. After establishing some simple facts on averaging functions, we prove a minimax theorem for payoffsf with the following property: Forf there exist averaging functions and such that for any x₁, x₂ X, > 0 there exists x₀ X withf (x₀, y) > f (x₁,y),f (x₂,y))– for ally Y, and for any y₁, y₂ Y, > 0 there exists y₀ Y withf (x, y₀) (f (x, y₁),f (x, y₂))+. This result contains as a special case the Fan-König result for concave-convex-like payoffs in a general version, when we take linear averaging with (x, y)=x+(1–)y, (x, y)=x+(1–)y, 0 <, < 1.Then a class of hide-and-seek games is introduced, and we derive conditions for applying the minimax result of this paper.

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Zusammenfassung In dieser Arbeit werden Minimaxsätze für Zwei-Personen-Nullsummenspiele (X, Y,f) mit Auszahlungsfunktionf behandelt, und als Hauptresultat wird die Gültigkeit der Minimaxgleichung inf supf (x, y)=sup inff (x, y) unter einer neuen Bedingung an f nachgewiesen. Diese Bedingung basiert auf dem Konzept mittelnder Funktionen, d.h. reellwertiger Funktionen, welche auf einer Teilmenge der Ebene definiert sind und dort der Eigenschaft min {x, y} < < (x, y)x, y} fürx y, (x, x)=x, genügen. Nach der Herleitung einiger einfacher Aussagen über mittelnde Funktionen beweisen wir einen Minimaxsatz für Auszahlungsfunktionenf mit folgender Eigenschaft: Zuf existieren mittelnde Funktionen und, so daß zu beliebigen x₁, x₂ X, > 0 mindestens ein x₀ X existiert mitf (x₀,y) (f (x ₁,y),f (x₂,y)) – für alley Y und zu beliebigen y₁, y₂ Y, > 0 mindestens ein y₀ Y existiert mitf (x, y₀) (f (x, y₁),f (x, y ₂))+ für allex X. Dieses Resultat enthält als Spezialfall den Fan-König'schen Minimaxsatz für konkav-konvev-ähnliche Auszahlungsfunktionen in einer allgemeinen Version, wenn wir lineare Mittelung mit (x, y)=x+(1–)y, (x, y)= x+(1–)y, 0 <, < 1, betrachten.Es wird eine Klasse von Suchspielen eingeführt, welche mit dem vorstehenden Resultat behandelt werden können.

     

Points cônes du mouvement brownien plan,le cas critique

Summary LetB=(B _t,t0) be a planar Brownian motion and let >0. For anyt0, the pointz=B _t is called a one-sided cone point with angle if there exist >0 and a wedgeW(,z) with vertexz and angle such thatB _sW(,z) for everys[t, t+]. Burdzy and Shimura have shown independently that one-sided cone points with angle exist when >/2 but not when      

Asymptotic Formulae and Divisor Problems

We investigate the asymptotic behaviour of the summatory functions of _z(n, ), _k(n, ) z ⁽ⁿ⁾ and _k(n, ) z ⁽ⁿ⁾.     

Certain Convolution Operators for Meromorphic Functions

Let (p N) be the class of functions analytic in 0 < |z| < 1. A convolution operator L_p(a, c) on p is introduced. This paper gives some sharp inequalities for f(z) satisfying Re{(1 – )z^pL_p(a, c) f(z) + z^pL_p(a + 1, c) f(z)} > , where 0, < 1, a > 0 and c 0, –1, –2,....AMS Subject Classification (1991) 30C45 30A10     

The Whitehead Group of the Novikov Ring

The Bass–Heller–Swan–Farrell–Hsiang–Siebenmann decomposition of the Whitehead group K ₁(A[z,z^-1]) of a twisted Laurent polynomial extension A[z,z^-1] of a ring A is generalized to a decomposition of the Whitehead group K ₁(A((z))) of a twisted Novikov ring of power series A((z))=A[[z]][z^-1]. The decomposition involves a summand W₁(A, ) which is an Abelian quotient of the multiplicative group W(A,) of Witt vectors 1+a₁z+a₂z²+ ··· A[[z]]. An example is constructed to show that in general the natural surjection W(A, )^ab W₁(A, ) is not an isomorphism.     

Subsets with Restricted Movement

Let G be a finite permutation group on a set with no fixed points in and let m and k be integers with 0 < m < k. For a finite subset of the movement of is defined as move() = max_gG| ^g \ |. Suppose further that G is not a 2-group and that p is the least odd prime dividing |G| and move() m for all k-element subsets of . Then either || k + m or k (7m – 5) / 2, || (9m – 3)/2. Moreover when || > k + m, then move() m for every subset of .     

Intuitionistic weak arithmetic

We construct -framed Kripke models of i₁ and i₁ non of whose worlds satisfies xy(x=2yx=2y+1) and x,yzExp(x, y, z) respectively. This will enable us to show that i₁ does not prove ¬¬xy(x=2yx=2y+1) and i₁ does not prove ¬¬x, yzExp(x, y, z). Therefore, i₁¬¬lop and i₁¬¬i₁. We also prove that HAl₁ and present some remarks about i₂. Mathematics Subject Classification (2000):03F30, 03F55, 03H15.     

Convolution Operators with Fractional Measures Associated to Holomorphic Functions

Let be an open set in the complex plane and let be a holomorphic function on . Let K be a compact subset of with nonempty interior such that 0 K. Let be the Borel measure of R ⁴ C ² given by(E = _K E(z, (z))|z|^–2 d(z)where 0 < 2 and d(x ₁ + ix ₂) = dx ₁ dx ₂ denotes the Lebesgue measure on C. Let T be the convolution operator T f = * f. In this paper we characterize the type set E associated to T .     

A characterization of Thalesian orthogonality in Pappian planes of characteristic 2

In an affine plane over a field K, any Thalesian orthogonality relation is equivalent to _d for some dK{0}, where _d denotes the relation with constant of orthogonality d/it (i.e., after suitable coordinatization the slopes m, m*K{0} of orthogonal lines satisfy m·m*=d) (cf. [1], [5]). In the present paper we show that in a Pappian plane of characteristic two any orthogonality relation admitting the same group as ₁ is equivalent to ₁. This gives a characterization of Thalesian orthogonality over perfect fields of characteristic two.     

Ein Kontinuitätssatz für k-holomorphe Mengen

Let k denote a non-trivial non-archimedean complete valuated field and X an irreducible k-affinoid space. We discuss the Hartog's domain H^*:=(X×Eⁿ) (U×Eⁿ) where øUX is an affinoid subdomain, Eⁿ is the n-dimensional unit-polydisc over k and Eⁿ is the ringdomain of all z==(z₁,...,z_n)Eⁿ with some coordinate |z_i|=1. The main result is the non-archimedean version of Rothstein's extensiontheorem for analytic subvarieties: Every k-holomorphic subvariety AH^* whose every branch has dimension (dim X + 1) can be extended to a k-holomorphic subvariety X×Eⁿ such that every branch of has dimension (dim X + l).     

Minimization of the Conformal Radius under Circular Cutting of a Domain

Let D be a simply connected domain on the complex plane such that 0 D. For r > 0 , let D _r be the connected component of D {z : |z| < r} containing the origin. For fixed r, we solve the problem on minimization of the conformal radius R(D _r, 0) among all domains D with given conformal radius R(D, 0). This also leads to the solution of the problem on maximization of the logarithmic capacity of the local -extension E (a) of E among all continua E with given logarithmic capacity. Here, E (a) = E {z : |z – a| }, a E, > 0. Bibliography: 12 titles.     

Quadratically constrained quadratic programming: Some applications and a method for solution

The problem (QPQR) considered here is: minimizeQ ₁ (x) subject toQ _i (x) 0,i M ₁ {2,...,m},x P R ⁿ, whereQ _i (x), i M {1} M ₁ are quadratic forms with positive semi-definite matrices, andP a compact nonempty polyhedron of Rⁿ. Applications of (QPQR) and a new method to solve it are presented.Letu S={u R ^m;u 0, u _i= l}be fixed;then the problem:iM minimize u _iQ_i (x (u)) overP, always has an optimal solutionx (u), which is either feasible, iM i.e. u C₁ {u S;Q _i (x (u)) 0,i M ₁} or unfeasible, i.e. there exists ani M ₁ withu C {u S; Q_i(x(u)) 0}.Let us defineC _i C_i S _i withS _i {u S; u _i=0}, i M. A constructive method is used to prove that C _i is not empty and thatx (û) withiM û C _i characterizes an optimal solution to (QPQR). Quite attractive numerical results have been reached with this method.

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Zusammenfassung Die vorliegende Arbeit befaßt sich mit Anwendungen und einer neuen Lösungsmethode der folgenden Aufgabe (QPQR): man minimiere eine konvexe quadratische ZielfunktionQ _i (x) unter Berücksichtigung konvexer quadratischer RestriktionenQ _i (x) 0, iM ₁ {2,...,m}, und/oder linearer Restriktionen.·Für ein festesu S {u R ^m;u 0, u _i=1},M {1} M₁ besitzt das Problem:iM minimiere die konvexe quadratische Zielfunktion u _i Q_i (x (u)) über dem durch die lineareniM Restriktionen von (QPQR) erzeugten, kompakten und nicht leeren PolyederP R ⁿ, immer eine Optimallösungx (u), die entweder zulässig ist: u C₁ {u S;Q ₁ (x (u)) 0,i M ₁} oder unzulässig ist, d.h. es existiert eini M ₁ mitu C_i {u S;Q _i (x(u))0}.Es seien folgende MengenC _i C_i S _i definiert, mitS _i {u S;u _i=0}, i M. Es wird konstruktiv bewiesen, daß C _i 0 undx (û) mitû C _i eine Optimallösung voniM iM (QPQR) ist; damit ergibt sich eine Methode zur Lösung von (QPQR), die sich als sehr effizient erwiesen hat. Ein einfaches Beispiel ist angegeben, mit dem alle Schritte des Algorithmus und dessen Arbeitsweise graphisch dargestellt werden können.

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An earlier version of this paper was written during the author's stay at the Institute for Operations Research, Swiss Federal Institute of Technology, Zürich.     

Critical sets in parametric optimization

We deal with one-parameter families of optimization problems in finite dimensions. The constraints are both of equality and inequality type. The concept of a generalized critical point (g.c. point) is introduced. In particular, every local minimum, Kuhn-Tucker point, and point of Fritz John type is a g.c. point. Under fairly weak (even generic) conditions we study the set consisting of all g.c. points. Due to the parameter, the set is pieced together from one-dimensional manifolds. The points of can be divided into five (characteristic) types. The subset of nondegenerate critical points (first type) is open and dense in (nondegenerate means: strict complementarity, nondegeneracy of the corresponding quadratic form and linear independence of the gradients of binding constraints). A nondegenerate critical point is completely characterized by means of four indices. The change of these indices along is presented. Finally, the Kuhn-Tucker subset of is studied in more detail, in particular in connection with the (failure of the) Mangasarian-Fromowitz constraint qualification.     

Hyponormality of Toeplitz Operators with Polynomial and Symmetric-Type Symbols

Suppose that is a trigonometric polynomial of the form (z) = ^N_n=-N a_n zⁿ. It is well-known that T is normal if and only if | a_{– N}| =  | a_N| and the Fourier coefficients of satisfy the following symmetry condition:

On the order of approximation to functions ofH p (R) (0<p≦1) by certain means of Fourier integrals

H ^P (R ₊ ² ) — R ₊ ² ={zC: Imz>0} ^p (R) — H ^p (R ₊ ² ). P ^k (f,x) — ë- — ,W ^k (f,x) — — R ^k, (f,x) — f H (R) (. §1,1)–3)); _k (, f) _p — - . , fH ^p (R) 0<p1,kN; (1+)^–1<p1, 0<<,kN.     

Asymptotics of the Discrete Spectrum of the Three-Particle Schrödinger Difference Operator on a Lattice

We consider the Hamiltonian H (K) of a system consisting of three bosons that interact through attractive pair contact potentials on a three-dimensional integer lattice. We obtain an asymptotic value for the number N(K,z) of eigenvalues of the operator H₀(K) lying below z0 with respect to the total quasimomentum K0 and the spectral parameter z–0.