Intersection Cohomology on Nonrational Polytopes⋆

We consider a fan as a ringed space (with finitely many points). We develop the corresponding sheaf theory and functors, such as direct image R_* ( is a subdivision of a fan), Verdier duality, etc. The distinguished sheaf , called the minimal sheaf plays the role of an equivariant intersection cohomology complex on the corresponding toric variety (which exists if is rational). Using we define the intersection cohomology space IH(). It is conjectured that a strictly convex piecewise linear function on acts as a Lefschetz operator on IH(). We show that this conjecture implies Stanley's conjecture on the unimodality of the generalized h-vector of a convex polytope.     

Comparison of moments for tangent sequences of random variables

Summary It is shown that for all tangent sequences (d _n) and (e _n) of nonnegative or conditionally symmetric random variables and for every function satisfying the growth condition (2x)(x) the following inequality holds: . This generalizes results of J. Zinn and proves a conjecture of S. Kwapie and W.A. Woyczyski.     

On a zero-crossing probability

Let {X(t), 0E{exp (–sX(t))}=exp (–t(s)), where (s)=(1–(s)), is the intensity of the Poisson process, and (s) is the Laplace transform of the distribution of nonnegative jumps. Consider the zero-crossing probability =P{X(t)–t=0 for some t,0<t<}. We show that =() where is the largest nonnegative root of the equation (s)=s. It is conjectured that this result holds more generally for any stochastic process with stationary independent increments and with sample paths that are nondecreasing step functions vanishing at 0.     

On a Family of Hyperplane Arrangements Related to the Affine Weyl Groups

Let be an irreducible crystallographic rootsystem in a Euclidean space V, with ⁺ theset of positive roots. For , , let be the hyperplane . We define a set of hyperplanes . This hyperplane arrangement is significant inthe study of the affine Weyl groups. In this paper it is shown that thePoincaré polynomial of is , where n is the rank of and h is the Coxeter number of the finiteCoxeter group corresponding to .     

Circularity of Finite Groups without Fixed Points

Let be a fixed point free group given by the presentation where and are relative prime numbers, t = /s and s = gcd( – 1,), and is the order of modulo . We prove that if (1) = 2, and (2) is embeddable into the multiplicative group of some skew field, then is circular. This means that there is some additive group N on which acts fixed point freely, and |((a)+b)((c)+d)| 2 whenever a,b,c,d N, a0c, are such that (a)+b(c)+d.     

On an isomorphism problem on the closed-set lattice of a graph

Let (G) and V(G) be, respectively, the closed-set lattice and the vertex set of a graph G. Any lattice isomorphism : V(G)(G) induces a bijection : V(G)V(G) such that for each x in V(G), (x)=x' in V(G') iff ({x})={x'}. A graph G is strongly sensitive if for any graph G' and any lattice isomorphism : (G)(G), the bijection induced by is a graph isomorphism of G onto G'. In this paper we present some sufficient conditions for graphs to be strongly sensitive and prove in particular that all C ₄-free graphs and all covering graphs of finite lattices are strongly sensitive.     

Über die Minimalflächen des isotropen Raumes,welche zugleich Affinminimalflächen sind

One determines all the minimal surfaces of the isotropic space, which simultaneously are affinminimal surfaces. A characteristic property of those surfaces is that the isotropic spherical imagines of the asymptotic lines of form two orthogonal pencils of circles. There are three types of such surfaces : first the well known right helicoid _I , second an interesting transcendental surface _II , and third the isotropic analogy _III of the minimal surface ofEnneper. The surfaces permit cinematic generations. Especially _II and _III can be generated byClifford screws in a certain indefinite quasielliptic space.In the isotropic space conjugate to the surfaces are isotropic minimal surfaces ^* with plane lines of curvature. There are also three types of such surfaces: _I ^* is a logarithmic surface of revolution, _II ^* is an interesting transcendental surface, and _III ^* is again the isotropic minimal surface ofEnnerper.     

Steepest Descent with Curvature Dynamical System

Let H be a real Hilbert space and let <..,.> denote the corresponding scalar product. Given a function that is bounded from below, we consider the following dynamical system:

Galois Graphs: Walks, Trees and Automorphisms

Given a symmetric polynomial (x, y) over a perfect field k of characteristic zero, the Galois graph G() is defined by taking the algebraic closure as the vertex set and adjacencies corresponding to the zeroes of (x, y). Some graph properties of G(), such as lengths of walks, distances and cycles are described in terms of . Symmetry is also considered, relating the Galois group Gal( /k) to the automorphism group of certain classes of Galois graphs. Finally, an application concerning modular curves classifying pairs of isogeny elliptic curves is revisited.     

On the Binomial Asymptotics of an Entire Dirichlet Series

Let M() be the maximum modulus and let () be the maximum term of an entire Dirichlet series with nonnegative exponents _n increasing to . We establish a condition for _n under which the relations

Generalized ornstein-uhlenbeck process having a characteristic operator with polynomial coefficients

Summary Let be a weighted Schwartz's space of rapidly decreasing functions, the dual space and (t) a perturbed diffusion operator with polynomial coefficients from into itself. It is proven that (t) generates the Kolmogorov evolution operator from into itself via stochastic method. As applications, we construct a unique solution of a Langevin's equation on : whereW(t) is a Brownian motion and ^*(t) is the adjoint of (t) and show a central limit theorem for interacting multiplicative diffusions.     

An Inequality for Tail Probabilities of Martingales with Bounded Differences

Let M _n =X₁+...+X_n be a martingale with bounded differences X_m=M_m-M_m-1 such that {|X_m| _m}=1 with some nonnegative _m. Write ²= ₁ ² + ... + _n ² . We prove the inequalities {M _nx}c(1-(x/)), {M _n x} 1- c(1- (-x/)) with a constant . The result yields sharp inequalities in some models related to the measure concentration phenomena.     

A strong renewal theorem for generalized renewal functions in the infinite mean case

Summary LetF(x) be a nonarithmetic c.d.f. on (0, ) such that 1 —F(x)=x ^– L(x), whereL(x) is slowly varying and 01. Leta(x) be regularly varying with exponent 1. A strong renewal theorem (of Blackwell type) for generalized renewal functions of the form is proved here, thus extending the recent work of Embrechts, Maejima and Omey [1] and that of Erickson [4].Kevin K. Anderson is now at Lawrence Livermore National Laboratory, P.O. Box 808, Livermore, CA 914550 USA. His research was performed in part under the auspices of the U.S. Department of Energy at LLNL under Contract W-7405-Eng-48.The research of Krishna B. Athreya was supported in part by NSF Grants DMS-8502311 and DMS-8706319.     

Formal Flows on the Non-Archimedean Open Unit Disk

This paper deals with group actions of one-dimensional formal groups defined over the ring of integers in a finite extension of the p-adic field, where the space acted upon is the maximal ideal in the ring of integers of an algebraic closure of the p-adic field. Given a formal group F as above, a formal flow is a series (t,x) satisfying the conditions (0,x)=x and (F(s,t),x)=(s,(t,x)). With this definition, any formal group will act on the disk by left translation, but this paper constructs flows with any specified divisor of fixed points, where a point of the open unit disk is a fixed point of order n if (x–) ⁿ |((t,x)–x). Furthermore, if is an analytic automorphism of the open unit disk with only finitely many periodic points, then there is a flow , an element of the maximal ideal of the ring of constants, and an integer m such that the m-fold iteration of (x) is equal to (,x). All the formal flows constructed here are actions of the additive formal group on the unit disk. Indeed, if the divisor of fixed points of a formal flow is of degree at least two, then the formal group involved must become isomorphic to the additive group when the base is extended to the residue field of the constant ring.     

Finitely generated relatively universal varieties of Heyting algebras

Given distinct varieties and of the same type, we say that is relatively -universal if there exists an embedding :K from a universal categoryK such that for every pairA, B ofK-objects, a homomorphismf:A B has the formf=_g for someK-morphismg:A B if and only if Im(f) . Finitely generated relatively -universal varieties of Heyting algebras are described for the variety of Boolean algebras, the variety generated by a three element chain, and for the variety generated by the four element Boolean algebra with an added greatest element.Dedicated to the memory of Alan DayPresented by J. Sichler.The support of the NSERC is gratefully acknowledged.     

Generalized Variation of Mappings with Applications to Composition Operators and Multifunctions

We study (set-valued) mappings of bounded -variation defined on the compact interval I and taking values in metric or normed linear spaces X. We prove a new structural theorem for these mappings and extend Medvedev's criterion from real valued functions onto mappings with values in a reflexive Banach space, which permits us to establish an explicit integral formula for the -variation of a metric space valued mapping. We show that the linear span GV (I;X) of the set of all mappings of bounded -variation is automatically a Banach algebra provided X is a Banach algebra. If h:I× X Y is a given mapping and the composition operator is defined by (f)(t)=h(t,f(t)), where tI and f:I X, we show that :GV (I;X) GV (I;Y) is Lipschitzian if and only if h(t,x)=h₀(t)+h₁(t)x, tI, xX. This result is further extended to multivalued composition operators with values compact convex sets. We prove that any (not necessarily convex valued) multifunction of bounded -variation with respect to the Hausdorff metric, whose graph is compact, admits regular selections of bounded -variation.     

An existence theorem for generalized quasi-variational inequalities

In this paper, we deal with the following generalized quasi-variational inequality problem: given a closed convex subsetX ⁿ , a multifunction :X 2 ⁿ and a multifunction :X 2 ^X , find a point ( ) X × ⁿ such that We prove an existence theorem in which, in particular, the multifunction is not supposed to be upper semicontinuous.     

Extensions and selections in subspaces ofC(K)

LetK be a compact Hausdorff space and letFK be a peak interpolation set for a function algebraAC(K). Let be a map fromK to the family of all convex subsets of such that the set {(z, x)zK, x(z)} is open inK×C and such thatg(z)(z) (zK) for somegA. We prove that everyfC(F) satisfyingf(s)(s) (sF) (f(s)closure (s) (sF)) admits an extensionfAA} satisfyingf(z)(z) (zK) (f(z))}closure (z) (zK), respectively). We prove a more general theorem of this kind and present various applications which generalize known dominated interpolation theorems for subspaces ofC(K).     

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.

     

On local oscillatory integrals with variable Calderón-Zygmund kernels, II

Let , be a real analytic function or a real-C function on ⁿ andk be a variable Calderón-Zygmund kernel. Define the oscillatory singular integral operatorT by