Lower functions for asymmetric Lévy processes

Summary Let {X _t } be a ¹ process with stationary independent increments and its Lévy measurev be given byv{yy>x}=x ^–L ₁ (x), v{yy<–x}=x ^–L ₂ (x) whereL ₁,L ₂ are slowly varying at 0 and and 0<1. We construct two types of a nondecreasing functionh(t) depending on 0<<1 or =1 such that lim inf a.s. ast 0 andt for some positive finite constantC.This research is partialy supported by a grant from Korea University     

Entropy of some automorphisms of theII 1-factor of the free group in infinite number of generators

Summary LetL(F ) be theII ₁-factor defined by the free groupF in infinite number of generators. It is shown that for a class of automorphisms ofL(F ) arising from bijections of the set of generators ofF on itself, and including the free shift, the entropy is zero.Oblatum 15-III-1992     

Some Conditions for a C0-Semigroup to Be Asymptotically Finite-Dimensional

We study the class of bounded C ₀-semigroups T=(T _t ) _t0 on a Banach space X satisfying the asymptotic finite dimensionality condition: codim X ₀(T)<, where X ₀(T):={x X:lim_t T _t x=0}. We prove a theorem which provides some necessary and sufficient conditions for asymptotic finite dimensionality.     

The Ramsey property for collections of sequences not containing all arithmetic progressions

A family of sequences has the Ramsey property if for every positive integerk, there exists a least positive integerf (k) such that for every 2-coloring of {1,2, ...,f (k)} there is a monochromatick-term member of . For fixed integersm > 1 and 0 q < m, let _q(m) be the collection of those increasing sequences of positive integers {x ₁,..., x_k} such thatx _i+1 – x_i q(modm) for 1 i k – 1. Fort a fixed positive integer, denote byA _t the collection of those arithmetic progressions having constant differencet. Landman and Long showed that for allm 2 and 1 q < m, _q(m) does not have the Ramsey property, while _q(m) A _m does. We extend these results to various finite unions of _q(m) 's andA _t 's. We show that for allm 2, _q=1 ^m–1 _q(m) does not have the Ramsey property. We give necessary and sufficient conditions for collections of the form _q(m) ( _{t T} A _t) to have the Ramsey property. We determine when collections of the form _a(m₁) _b(m₂) have the Ramsey property. We extend this to the study of arbitrary finite unions of _q(m)'s. In all cases considered for which has the Ramsey property, upper bounds are given forf .     

Fractional dimension of partial orders

Given a partially ordered setP=(X, ), a collection of linear extensions {L ₁,L ₂,...,L _r } is arealizer if, for every incomparable pair of elementsx andy, we havex<y in someL _i (andy<x in someL _j ). For a positive integerk, we call a multiset {L ₁,L ₂,...,L _t } ak-fold realizer if for every incomparable pairx andy we havex<y in at leastk of theL _i 's. Lett(k) be the size of a smallestk-fold realizer ofP; we define thefractional dimension ofP, denoted fdim(P), to be the limit oft(k)/k ask. We prove various results about the fractional dimension of a poset.Research supported in part by the Office of Naval Research.     

Embeddings of Banach Spaces Into Banach Lattices and the Gordon–Lewis Property

In this paper we first show that if X is a Banach space and is a left invariant crossnorm on lX, then there is a Banach lattice L and an isometric embedding J of X into L, so that I J becomes an isometry of lX onto l_m J(X). Here I denotes the identity operator on l and l_m J(X) the canonical lattice tensor product. This result is originally due to G. Pisier (unpublished), but our proof is different. We then use this to prove the main results which characterize the Gordon–Lewis property GL and related structures in terms of embeddings into Banach lattices.     

Large Deviations of Local Times of Lévy Processes

For X(t) a real-valued symmetric Lévy process, its characteristic function is E(e ^iX(t))=exp(–t()). Assume that is regularly varying at infinity with index 1<2. Let L ^x _t denote the local time of X(t) and L* _t =sup _xR L ^x _t . Estimates are obtained for P(L ⁰ _t y) and P(L* _t y) as y and t fixed.     

On the equation of similar profiles

Summary Considerf+ ff+ (1–f²)+ f=0 together with the boundary conditionsf(0)=f(0)=0,f ()=1. If=–1,>0, arbitrary there is at least one solution which satisfies 0<f<1 on (0, ). By the additional conditionf>0 on (0, ) or, alternately 0<1, the uniqueness of the solution is demonstrated.If=1,<0, arbitrary the existence of solutions for which –1<f<0 in some initial interval (0,t) and satisfying generallyf>1 is established. In both problems, bounds forf (0) and qualitative behavior of the solutions are shown.

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Sommario Si consideri il problema definito dall'equazionef+ f f+ (1–f²)+ f=0 e dalle condizioni al contornof(0)=f (0)=0,f()=1. Assumendo=–1,>0, arbitrario si dimostra che esiste almeno una soluzione che soddisfa 0<f<1 nell'intervallo (0, ). Se in aggiunta si ipotizzaf>0 in (0, ), oppure 0<=1, l'unicità délia soluzione è assicurata.Successivamente si considéra il problema di valori al contorno con=1,<0, arbitrario. In questo caso esiste un'intera classe di soluzioni che soddisfano –1<f<0 in un intorno dell'origine e tali chef>1, in generale.Di detti problemi viene studiato il comportamento délle soluzioni e vengono determinate dalle maggiorazioni e minorazioni del valoref(0).

     

BC-type interpolation Macdonald polynomials and binomial formula for Koornwinder polynomials

We consider 3-parametric polynomialsP ^* (x; q, t, s) which replace theA _n-series interpolation Macdonald polynomialsP ^* (x; q, t) for theBC _n-type root system. For these polynomials we prove an integral representation, a combinatorial formula, Pieri rules, Cauchy identity, and we also show that they do not satisfy any rationalq-difference equation. Ass the polynomialsP ^* (x; q, t, s) becomeP ^* (x; q, t). We also prove a binomial formula for 6-parametric Koornwinder polynomials.     

On the postulation of a general projection of a curve inP 3

Summary We say that a curve C in P ³ has maximal rank if for every integer k the restriction map r_c(k):H ⁰(P ³, O_P3(k)) H⁰ (C, O_C(k))has maximal rank. Here we prove the following results. Theorem 1Fix integers g, d with 0g3,dg+3.Fix a curve X of genus g and L Pic^d (X).If g=3and X is hyperelliptic, assume d8. Let _L(X)be the image of X by the complete linear system H ⁰(X, L). Then a general projection of _L(X)into P ³ has maximal rank. Theorem 2For every integer g0,there exists an integer d(g, 3)such that for every dd(g, 3),for every smooth curve X of genus g and every LPic^d (X) the general projection of _L(X)into P ³ has maximal rank.     

Estimates of Entire Functions of Exponential Type Less than π in Terms of Logarithmic Sums over Real Duffin and Schaeffer Sequences

We give uniform estimates of entire functions of exponential type less than having sufficiently small logarithmic sums over real sequences { _n } satisfying | _n –n|L and _n+1– _n for fixed positive constants L and . We thereby generalize results about logarithmic sums over the set of integers and so-called relatively h-dense sequences.     

An infinite dimensional analogue of the Albeverio-Röckner's theorem on Fukushima's conjecture

LetX be the solution of the SDE:dX _t = (X _t)dB _t +b(X _t)dt, with andb C _b (R) such that >0 for some constant , andB a real Brownian motion. Let be the law ofX onE=C([0, 1],R) andk E* – {0}, whereE* is the topological dual space ofE. Consider the classical form: _k (u, v)=u / kv / kd, whereu andv are smooth functions onE. We prove that, if _k is closable for anyk in a dense subset ofE* and if the smooth functions are contained in the domain of the generator of the closure of _k , must be a constant function.     

Central limit theorem for stochastically continuous processes. Convergence to stable limit

LetX={X(t), t[0, 1]} be a stochastically continuous cadlag process. Assume that thek dimensional finite joint distributions ofX are in the domain of normal attraction of strictlyp-stable, 0<p<2, measure onR ^k for all 1k<. For functionsf, g such that _p (|X(x–X(u)|) >g(u–s) and _p (|X(s–X(t|)|X(t)–X(u|)>f(u–s), 0 s t u 1, conditions are found which imply that the distributions –(n ^–1/p (X ₁+···+X _n )),n1, converge weakly inD[0, 1] to the distribution of ap-stable process. HereX ₁,X ₂, ... are independent copies ofX and _p (Z)=sup _t<0 t ^pP{|Z|<t} denotes the weakpth moment of a random variable Z.     

A lower closure theorem for abstract control problems withL p-bounded controls

A lower closure theorem for an abstract control problem is proved. The functional isJ(,u)= _G f ⁰(t, (M)(t),u(t))dt and the state equations areN(t)=f(t, (M)(t),u(t)). It is shown that, if {( _k ,u _k)} is a sequence of admissible controlsu _k and corre-sponding trajectories _k such that lim infJ( _k ,u _k)<+ and such that _k weakly,M _k M strongly,N _k N weakly, and {u _k} is bounded in someL _p norm, then there is a controlu such that (,u) is admissible and lim infJ( _k ,u _k)J(,u).Dedicated to Professor M. R. HestenesThis research was supported by the National Science Foundation, Grant No. GP-33551X.     

Multivalued Stochastic Differential Equations: Convergence of a Numerical Scheme

In this paper we show the strong mean square convergence of a numerical scheme for a R ^d -multivalued stochastic differential equation: dX _t +A(X _t )dtb(t,X _t )dt+(t,X _t )dW _t and obtain the rate of convergence O(( log(1/)^1/2) when the diffusion coefficient is bounded. By introducing a discrete Skorokhod problem, we establish L ^p -estimates (p2) for the solutions and prove the convergence by using a deterministic result. Numerical experiments for the rate of convergence are presented.     

Behavior on the real line of entire functions represented by Dirichlet series

Conditions are found under which for an entire function f represented by a Dirichlet series with finite Ritt order on some sequence (x_k), 0 < x_k , as k one has ¦f(x_k)¦=M_t((1 + 0(1) x_k), M_f(x)=sup {¦ f (z) ¦:Re z x}.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 2, pp. 265–269, February, 1991.     

Randomly Weighted Sums of Subexponential Random Variables with Application to Ruin Theory

Let {X _k , 1 k n} be n independent and real-valued random variables with common subexponential distribution function, and let {k, 1 k n} be other n random variables independent of {X _k , 1 k n} and satisfying a _k b for some 0 < a b < for all 1 k n. This paper proves that the asymptotic relations P (max_{1 m n k=1} ^m _k X _k > x) P (sum _k=1 ⁿ _k X _k > x) sum _k=1 ⁿ P ( _k X _k > x) hold as x . In doing so, no any assumption is made on the dependence structure of the sequence { _k , 1 k n}. An application to ruin theory is proposed.     

A preconditioner for constrained and weighted least squares problems with Toeplitz structure

We study methods for solving the constrained and weighted least squares problem min _x by the preconditioned conjugate gradient (PCG) method. HereW = diag (₁, , _m ) with _{1 m} 0, andA ^T = [T ₁ ^T , ,T _k ^T ] with Toeplitz blocksT _l R ^{n × n} ,l = 1, ,k. It is well-known that this problem can be solved by solving anaugmented linear 2 × 2 block linear systemM +Ax =b, A ^T = 0, whereM =W ^–1. We will use the PCG method with circulant-like preconditioner for solving the system. We show that the spectrum of the preconditioned matrix is clustered around one. When the PCG method is applied to solve the system, we can expect a fast convergence rate.Research supported by HKRGC grants no. CUHK 178/93E and CUHK 316/94E.     

Spectrum of multidimensional periodic operators

Let be a lattice in the n-dimensional Euclidean space Rⁿ and let F be the fundamental domain of the lattice . We denote by H the Schrödinger operator generated in L₂(Rⁿ) by the expression –u + q(x)u(1), and by H^t the operator generated in L₂(F) by the expression (1) and by quasiperiodic boundary conditions, where q(x) is a periodic (with respect to the lattice ) function. Asymptotic formulas for the eigenvalues of the operator H_t are obtained and with the aid of these formulas it is proved that there exists a number (q) such that the interval [(q), ] belongs to the spectrum of the operator H [for n3 in the case of sufficiently smooth potentials q(x), while for n=2 for any potential q(x) from L₂(F)], i.e., the Bethe-Sommerfeld conjecture is proved for arbitrary lattices.Translated from Teoriya Funktsii, Funktsionali'nyi Analiz i Ikh Prilozheniya, No. 49, pp. 17–34, 1988.     

On a family of quasi-arithmetic means

Summary The main goal of this paper is to solve the idempotency equationF(x, x) = x, x [0, 1] for a class of functions of the type convex linear combination of at-norm and at-conorm. In the non-strict Archimedean case and for eachk (0, 1), we obtain a unique solutionF _k for this equation. While these functionsF _k are not associative, we also prove that they satisfy the bisymmetry condition.