Ultrafilter translations

We develop a method for extending results about ultrafilters into a more general setting. In this paper we shall be mainly concerned with applications to cardinality logics. For example, assumingV=L, Gödel's Axiom of Constructibility, we prove that if > then the logic with the quantifier there exist many is (,)-compact if and only if either is weakly compact or is singular of cofinality<. As a corollary, for every infinite cardinals and , there exists a (,)-compact non-(,)-compact logic if and only if either < orcf<cf or < is weakly compact.Counterexamples are given showing that the above statements may fail, ifV=L is not assumed.However, without special assumptions, analogous results are obtained for the stronger notion of [,]-compactness.     

Finitely subadditive and countably subadditive outer measures

Results are given comparing countably subadditive (csa) outer measures and finitely subadditive (fsa) outer measures, especially relating to regularity and measurability conditions such as (*) condition:A setE (of an arbitrary setX), is measurable ( an outer measure),ES (the collection of measurable sets) iff (X)=(E)+(E). Specific examples are given contrasting csa and fsa outer measures. In particular fsa and csa outer measures derived from finitely additive measures defined on an algebra of sets generated by a lattice of sets, are investigated in some detail.     

Effilement minimal en une singularité isolée de l'équation de Schrödinger et application au principe de Picard

In this paper, the generalized Schrödinger equation (–)u=0 on the punctured unit disk of ² is investigated. If is rotation free and satisfies the Picard principle at the origin, it is shown that if a setE is minimal thin relatively to an extremal harmonic functionh with zero boundary values at {|x|=1}, there exists a sequence (r _n ) converging to zero such that B(O,r _n ) C E. Lete be the -unit. It is proved that if a measure satisfies _\E e h d<, for a minimal thin, relatively toh , setE then the Picard principle is valid for the measure + .

     

Quasianapole Interaction of Charged Fermions

We obtain the analytic expression for the total cross section of the reaction e ^– e ⁺l ^– l ⁺ (l=,) taking possible quasianapole interaction effects into account. We find numerical restrictions on the interaction parameter value from data for the reaction e ^– e ^+–+ in the energy domain below the Z ⁰ peak.     

Measures not charging semipolars and equations of schrödinger type

SupposeX is a Borel right process andm is a -finite excessive measure forX. Given a positive measure not chargingm-semipolars we associate an exact multiplicative functionalM(). No finiteness assumptions are made on . Given two such measures and ,M()=M() if and only if and agree on all finely open measurable sets. The equation (q–L)u+u=f whereL is the generator of (a subprocess of)X may be solved for appropriatef by means of the Feynman-Kac formula based onM(). Both uniqueness and existence are considered.Supported in part by NSF Grant DMS 92-24990.     

Vector-valued stochastic processes. I. Vector measures and vector-valued stochastic processes with finite variation

In this paper we study the relationship _V (M)=E(1 _M dV _S ) between operatorvalued processesV with finite variation V and operator-valued stochastic measures _V with finite variation | _V |. The variations satisfy the inequality | _V | _|V|, which, under certain conditions, is an equality (for example, ifV is measurable).     

Fubini theorem for anticipating stochastic integrals in Hilbert space

Let (X,l,) be a measure space, letW be a cylindrical Hilbert-Wiener process, and let be an anticipating integrable process-valued function onX. We prove, under natural assumptions on, that there exists a measurable version Y_x,x X, of the anticipating integral of(x) such that the integral _x Y_x(dx) is a version of the anticipating integral of _X (x)(dx). We apply this anticipating Fubini theorem to study solutions of a class of stochastic evolution equations in Hilbert space.     

The martin boundary for harmonic functions on groups of automorphisms of a homogeneous tree

Consider a closed subgroup of the automorphism group of a homogeneous treeT, and assume that acts transitively on the vertex set. Suppose that is a probability measure on which has continuous density with respect to Haar measure and whose support is compact open and generates as a closed semigroup. It is shown that the Martin boundary of with respect to the random walk with law coincides with the space of ends ofT. This extends known results for free groups and applies, for example, to the affine group over a non archimedean local field.     

The Colin de Verdière number and sphere representations of a graph

Colin de Vedière introduced an interesting linear algebraic invariant (G) of graphs. He proved that (G)2 if and only ifG is outerplanar, and (G)3 if and only ifG is planar. We prove that if the complement of a graphG onn nodes is outerplanar, then (G)n–4, and if it is planar, then (G)n–5. We give a full characterization of maximal planar graphs whose complementsG have (G)=n–5. In the opposite direction we show that ifG does not have twin nodes, then (G)n–3 implies that the complement ofG is outerplanar, and (G)n–4 implies that the complement ofG is planar.Our main tools are a geometric formulation of the invariant, and constructing representations of graphs by spheres, related to the classical result of Koebe about representing planar graphs by touching disks. In particular we show that such sphere representations characterize outerplanar and planar graphs.     

Generalised Holley-Preston inequalities on measure spaces and their products

Summary It is shown that if (X, ) is a product of totally ordered measure spaces andf _j (j=1,2,3,4) are measurable non-negative functions onX satisfyingf ₁(x)f₂(y)f₃(xy)f₄(xy), where (, ) are the lattice operations onX, then (f ₁ d)(f ₂ d)(f ₃ d)(f ₄ d). This generalises results of Ahlswede and Daykin (for counting measure on finite sets) and Preston (for special choices off _j).     

On the existence of wave operators for the Klein-Gordon equation

The problem of existence of wave operators for the Klein-Gordon equation ( _t ² –+²+iV_1t+V₂)u(x,t)=0 (x R ⁿ,t R, n3, >0) is studied where V₁ and V₂ are symmetric operators in L₂(R ⁿ) and it is shown that conditions similar to those of Veseli-Weidmann (Journal Functional Analysis 17, 61–77 (1974)) for a different class of operators are also sufficient for the Klein-Gordon equation.     

μ-Reducible Languages

A word w is said to be a primitive word if it cannot be expressed as a power of any other word. A language L consisting of non-empty words is called -reducible if there exists a non-empty word w such that Lw contains only finitely many powers of each primitive word. We show that every regular component, context-free component, local language and every regular language containing no primitive words are -reducible. Languages which are not -reducible are investigated and characterized. We show that every code is -reducible. But there are 2-codes which are not -reducible. The -annihilator of a language L is the set of all non-empty words w such that Lw contains only finitely many powers of each primitive word. This paper also concerns the properties of the -annihilators of languages. The -annihilators of 2-codes and some other languages are investigated and characterized in this paper. The results provide an outline of the relationship between the catenation of languages and the powers of primitive words.     

Future independent times and Markov chains

Summary A random timeT is a future independent time for a Markov chain (X _n ) ₀ ifT is independent of (X _T+n ) _n ^/ =0 and if (X _T+n ) _n ^/ =0 is a Markov chain with initial distribution and the same transition probabilities as (X _n ) ₀ . This concept is used (with the conditional stationary measure) to give a new and short proof of the basic limit theorem of Markov chains, improving somewhat the result in the null-recurrent case.This work was supported by the Swedish Natural Science Research Council and done while the author was visiting the Department of Statistics, Stanford University     

Limiting behavior of weighted central paths in linear programming

We study the limiting behavior of the weighted central paths{(x(), s())} > 0 in linear programming at both = 0 and = . We establish the existence of a partition (B ,N ) of the index set { 1, ,n } such thatx _i() ands _j () as fori B , andj N , andx _N (),s _B () converge to weighted analytic centers of certain polytopes. For allk 1, we show that thekth order derivativesx ^(k) () ands ^(k) () converge when 0 and . Consequently, the derivatives of each order are bounded in the interval (0, ). We calculate the limiting derivatives explicitly, and establish the surprising result that all higher order derivatives (k 2) converge to zero when .     

Some extensions of the Hewitt-Savage zero-one law

Summary Let denote the class of infinite product probability measures = ₁× ₂× defined on an infinite product of replications of a given measurable space (X, A), and let denote the subset of for which (A) =0 or 1 for each permutation invariant event A. Previous works by Hewitt and Savage, Horn and Schach, Blum and Pathak, and Sendler (referenced in the paper) discuss very restrictive sufficient conditions under which a given member , of belongs to . In the present paper, the class is shown to possess several closure properties. E.g., if and _{0 n} for some n 1, then ₀× ₁× ₂×.... While the current results do not permit a complete characterization of they demonstrate conclusively that is a much larger subset of than previous results indicated. The interesting special case X={0,1} is discussed in detail.Research supported by the National Science Foundation under grant No. MCS75-07556     

Necessary and sufficient conditions for the convergence of convolution products of non-identical distributions on finite abelian semigroups

Given a sequence of probability measures ( _n ) on a finite abelian semigroup, we present necessary and sufficient conditions which guarantee the weak convergence of the convolution products _{k,n k+1}*···* _n (k<n), asn for allk0. These conditions are verifiable in the sense that they are based entirely on the individual measures in the sequence ( _n ).     

Mean quadratic variations and Fourier asymptotics of self-similar measures

We show that the mean quadratic variation of a self-similar measure under certain open set condition exhibits asymptotic periodicity. Through a generalized Wiener's Tauberian Theorem, we obtain some new identities and equivalences of the mean quadratic variation of a bounded measurev and its Fourier average . They are used to sharpen some recent results of Strichartz concerning the asymptotic behavior ofH _a(T); ) asT, where is the self-similar measure as above. In the development some results concerning the open set condition are also obtained.     

Decay of Concentration Functions for Adapted Probabilities on Discrete Groups

Given a probability measure, , on a locally compact group, necessary and sufficient support conditions which ensure that the concentration functions associated with converge to zero have previously been determined. In this note the rate of this convergence when is adapted on a discrete group G is shown to depend on the volume growth rate of N , the smallest normal subgroup a coset of which contains the support of .     

Asymptotics of the partial sums of a set of integral transforms

In this paper, we explore the asymptotic distribution of the zeros of the partial sums of the family of entire functions of order 1 and type 1, defined by G(,,z)=₀ ¹(t)t ^–1×(1–t)^–1e ^zt dt, where Re,Re>0, is Riemann-integrable on [0,1], continuous at t=0, 1 and satisfies (0)(1)0.     

Generalizations of Tarski's fixed point theorem for order varieties of complete meet semilattices

A poset P is -conditionally complete ( a cardinal) if every set X P all of whose subsets of cardinality < have an upper bound has a least upper bound. For we characterize the subposets of a -complete poset which can occur as the set of fixed points of some montonic function on P. This yields a generalization of Tarski's fixed point theorem. We also show that for every the class of -conditionally complete posets forms an order variety and we exhibit a simple generating poset for each such class.Research supported in part by NSERC while the author was visiting Professor Ivo Rosenberg at the Université de Montreal.Research supported in part by NSF-grant DMS-8703743.