Realizations of regular polytopes

Let be a finite regular incidence-polytope. A realization of is given by an imageV of its vertices under a mapping into some euclidean space, which is such that every element of the automorphism group () of induces an isometry ofV. It is shown in this paper that the family of all possible realizations (up to congruence) of forms, in a natural way, a closed convex cone, which is also denoted by The dimensionr of is the number of equivalence classes under () of diagonals of , and is also the number of unions of double cosets ^{** *–1*} ( ^*), where ^* is the subgroup of () which fixes some given vertex of . The fine structure of corresponds to the irreducible orthogonal representations of (). IfG is such a representation, let its degree bed _G , and let the subgroup ofG corresponding to ^* have a fixed space of dimensionw _G . Then the relations

Comparing different classes of absolutely summing multilinear operators

Let C denote the composition operator defined on the standard Hardy spaces H^p as where is an analytic self-map of the unit disk in the complex plane. In this paper we discuss those invariant subspaces of C in H^p which are invariant under the shift operator, We restrict our attention to the case where is an inner function. Our main result characterises these invariant subspaces. We also consider C when restricted to such an invariant subspace and we describe the structure of the operator and find a formula for the essential spectral radius.Received: 27 January 2004     

Some remarks on Turán's inequality. III: The completion

. 0pq, 1–1/p+1/p0. f(x) — n, [–1,1],

On the Pólya conjecture concerning the maximum and minimum of the modulus of an entire function of finite order given by a lacunary power series

, (fz) , ,

Sharpening of Stečkin's theorem to strong approximation

. . , , : f — ,

On multivariate polynomial interpolation

We provide a map which associates each finite set in complexs-space with a polynomial space from which interpolation to arbitrary data given at the points in is possible and uniquely so. Among all polynomial spacesQ from which interpolation at is uniquely possible, our is of smallest degree. It is alsoD- and scale-invariant. Our map is monotone, thus providing a Newton form for the resulting interpolant. Our map is also continuous within reason, allowing us to interpret certain cases of coalescence as Hermite interpolation. In fact, our map can be extended to the case where, with eachgq, there is associated a polynomial space P, and, for given smoothf, a polynomialqQ is sought for which

Global properties of solutions to 1D-viscous compressible barotropic fluid equations with density dependent viscosity

The Navier-Stokes equations for compressible barotropic fluid in 1D with the mass force under zero velocity boundary conditions are studied. We prove the uniform upper and lower bounds for the density as well as the uniform in time L ²()-estimates for _x and u _x (u is the velocity). Moreover, a collection of the decay rate estimates for - (with being the stationary density) and u in ²()-norm and H ¹()-norm as time t are established. The results are given for general state function p() (but mainly monotone) and viscosity coefficient µ() of arbitrarily fast (or slow) growth as well as for the large data.     

A test of convergence of Fourier series with respect to multiplicative systems,analogous to the Jordan test

, {p _n} _n=0 (p₀=1, _n2 n2). : x f(t) V(G)

Large-time behavior of perturbed diffusion markov processes with applications to the second eigenvalue problem for fokker-planck operators and simulated annealing

We study the large-time behavior and rate of convergence to the invariant measures of the processes dX (t)=b(X) (t)) dt + (X (t)) dB(t). A crucial constant appears naturally in our study. Heuristically, when the time is of the order exp( – )/² , the transition density has a good lower bound and when the process has run for about exp( – )/², it is very close to the invariant measure. LetL =(²/2) – U · be a second-order differential operator on ^d. Under suitable conditions,L ^z has the discrete spectrum

Entropy control costs and entropic equilibria

Strategic errors may occur because of player trembles or because of noise in the communication channels. Suppose that the probability distribution of potential errors can be influenced at a cost proportional to the entropy reduction. This modified strategic-form game involves the choice of a probability distribution over strategies and a probability distribution over potential errors with weight on the latter. Each modified game has a Nash Equilibrium (NE), and any limit as 0 is called an -entropic equilibrium, -entropic equilibria always exist and constitute a subset of trembling-hand-perfect equilibria, but otherwise -EE are independent of other refinements such as Proper NE.The author is grateful to Larry Samuelson and an anonymous referee for helpful comments, but retains sole responsibility for any error.     

Hypercentral Series and Paired Intersections of Sylow Subgroups of Chevalley Groups

Let G(k) be the Chevalley group of normal type associated with a root system G = , or of twisted type G = ^m,m = 2,3, over a field K. Its root subgroups X_s, for all possible s G⁺, generate a maximal unipotent subgroup U = UG(k) if p = charK < 0, U is a Sylow p-subgroup of G(K). We examine G and K for which there exists a paired intersection U U⁹, g G(K), which is not conjugate in G(K) to a normal subgroup of U. If K is a finite field, this is equivalent to a condition that the normalizer of U U⁹ in G(K)has a p-multiple index. Put p() = max(r,r)/(s,s) | r,s . We prove a statement (Theorem 1) saying the following. Let G(K) be a Chevalley group of Lie rank greater than 1 over a finite field K of characteristic p and U be its Sylow p-subgroup equal to UG(K); also, either G = and p() is distinct from p and 1, or G(K) is a twisted group. Then G(K) contains a monomial element n such that the normalizer U of Uⁿ in G(K) has a p-multiple index. Let K be an associative commutative ring with unity and (K,J) be a congruence subgroup of the Chevalley group (K) modulo a nilpotent ideal J. We examine an hypercentral series 1 Z₁ Z₂ ... Z_c-1 of the group U(K) (K,J). Theorem 2 shows that under an extra restriction on the quotient (J_t : J) of ideals, central series are related via Z_i = T_c-iC, 1 i < c, where C is a subgroup of central diagonal elements. Such a connection exists, in particular, if K = Z_pm and J = (p^d), 1 d < m, d| m.     

On a fixed-point theorem on locally convex linear topological spaces

Iff is a self mapping on a closed convex subsetK of a separated quasicomplete locally convex linear topological spaceE such that (i)E is strictly convex, (ii)f (K) is contained in a compact subset ofK and (iii)f satisfies a contraction condition, then it is shown that for eachxK, the sequence of {U ⁿ (x)} _n =1 of iterates, whereU KK is defined byU (y)=f(y)+(1-) y, yK, converges to a fixed point off.     

Imbedding theorems for the invariant subspaces of the backward shift operator

For subspaces K ^p of the form of the Hardy space H^p and for measures with support in the closed unit circleclos , one finds conditions that ensure the imbedding KL^p(). One considers measures with support inclos , satisfying the following condition: for some number >0 and for all circles with center on the circumference, intersecting the set , we have the inequality ()C(). Here C does not depend on , while () is the radius of the circle . For such measures one has the imbedding K ^p L^p(). From here one derives a criterion for the imbedding K _A ² L²(), found by B. Cohn for inner functions , such that the set is connected for some positive . In the paper one also proves that a condition on , necessary and sufficient for the imbedding of K ^p into L^p(), must depend on p.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 149, pp. 38–51, 1986.     

Cooperation in an asymmetric Volunteer's dilemma game theory and experimental evidence

The symmetric Volunteer's dilemma game (VOD) models a situation in which each ofN actors faces the decision of either producing a step-level collective good (action C) or freeriding (D). One player's cooperative action suffices for producing the collective good. Unilateral cooperation yields a payoffU forD-players andU-K for the cooperative player(s). However, if all actors decide for freeriding, each player's payoff is zero (U>K>0). In this article, an essential modification is discussed. In an asymmetric VOD, the interest in the collective good and/or, the production costs (i.e. U or K) may vary between actors. The generalized asymmetric VOD is similar to market entry games. Alternative hypotheses about the behaviour of subjects are derived from a game-theoretical analysis. They are investigated in an experimental setting. The application of the mixed Nash-equilibrium concept yields a rather counter-intuitive prediction which apparently contradicts the empirical data. The predictions of the Harsanyi-Selten-theory and Schelling's focal point theory are in better accordance with the data. However, they do not account for the diffusion-of-responsibility-effect also observable in the context of an asymmetric VOD game.I am indebted to Wulf Albers, Norman Braun, Werner Güth, Norbert L. Kerr, Reinhard Selten, and the participants of the V^th International Social Dilemma Conference in Bielefeld for critical and helpful comments. I am grateful to Axel Franzen who organized the experiment at the University of Mannheim. This work was supported by a grant of the Deutsche Forschungsgemeinschaft (DFG).     

Subgroups of the groupZ n and a generalization of the Gołąb-Schinzel functional equation

Letn be a positive integer and letX be a linear space over a commutative fieldK. In the set = (K\{0}) × X we define a binary operation ·: × by