Optimal Stopping in the L log L-Inequality of Hardy and Littlewood

Let B = (B_t)_t0 be standard Brownian motion started at zero.We prove for all c > 1and all stopping times for B satisfying E(^r) < for somer > 1/2. This inequality is sharp, and equality is attainedat the stopping time whereu_* = 1 + 1/e^c(c – 1) and = (c – 1)/c for c >1, with X_t = |B_t| and S_t = max_0rt|B_r|. Likewise, we prove for all c > 1 and all stopping times for B satisfying E(^r < for some r > 1/2. This inequalityis sharp, and equality is attained at the stopping time where v_* = c/e(c – 1) and =(c – 1)/c for c > 1. These results contain and refinethe results on the L log L-inequality of Gilat [6] which areobtained by analytic methods. The method of proof used hereis probabilistic and is based upon solving the optimal stoppingproblem with the payoff whereF(x) equals either xlog⁺ x or x log x. This optimal stoppingproblem has some new interesting features, but in essence issolved by applying the principle of smooth fit and the maximalityprinciple. The results extend to the case when B starts at anygiven point (as well as to all non-negative submartingales).1991 Mathematics Subject Classification 60G40, 60J65, 60E15.     

Restricted Set Addition in Groups I: The Classical Setting

The existing results are surveyed and several new results areproved for the cardinality of the restricted doubling 2^A ={a' + a':a',a' A,a' a'}, where A G is a subset of theset of elements of an (additively written) group G. In particular,known estimates for G = Z and G = Z/pZ are improved and a first-of-a-kindgeneral estimate valid for arbitrary G is given.     

Prime Divisors of Shifted Factorials

For any positive integer n we let P(n) be the largest primefactor of n. We improve and generalize several results of P.Erds and C. Stewart on P(n!+1). In particular, we show thatlim sup_ninfin; P(n!+1)/n2.5, which improves their lower boundof lim sup_ninfin; P(n!+1)/n2. 2000 Mathematics Subject Classification11A05, 11A07, 11J86.     

Multiple Positive Solutions of Semi-Positone Sturm-Liouville Boundary Value Problems

This paper treats the existence of multiple positive solutionsof the semi-positone Sturm–Liouville boundary value problem. almost everywhere on [R₀, R₁], where and f is allowed to take negative values (that is, f is semi-positone).When = 1, new results on the existence of one or two nonzeropositive solutions are obtained. These results generalize previousresults for positone cases (that is, f 0) to the semi-positonecases. We illustrate our results with an explicit example whichhas two nonzero positive solutions. These results are used todeduce results on intervals of eigenvalues for which there existone or two nonzero positive eigenfunctions. Applications ofthese eigenvalue results are provided. 2000 Mathematics SubjectClassification 34B18 (primary), 34B15, 34B16, 47H10, 47H30 (secondary).     

ZERO-SUM PROBLEMS IN FINITE ABELIAN GROUPS AND AFFINE CAPS

For a finite abelian group G, let (G) denote the smallest integer l such that every sequence Sover G of length | S| l has a zero-sum subsequence of lengthexp (G). We derive new upper and lower bounds for (G), and all our bounds are sharp for special typesof groups. The results are not restricted to groups G of theform , but they respect the structure of the group. In particular, we show for all odd n, which is sharp if n is a power of3. Moreover, we investigate the relationship between extremalsequences and maximal caps in finite geometry.     

Differential Simplicity in Polynomial Rings and Algebraic Independence of Power Series

Let k be a field of characteristic zero, f(X,Y), g(X,Y)k[X,Y],g(X,Y) (X,Y) and d:=g(X,Y)/X + f(X,Y)/Y. A connection is establishedbetween the d-simplicity of the local ring k[X,Y]_(X,Y) and thetranscendency of the solution in tk[[t]] of the algebraic differentialequation g(t,y(t))·(/t)y(t)+f(t,y(t)). This connectionis used to obtain some interesting results in the theory ofthe formal power series and to construct new examples of differentiallysimple rings.     

On the rationality of moduli spaces of pointed curves

Motivated by several recent results on the geometry of the modulispaces of stable curves of genus g with n marked points, we determine the birational structureof these spaces for small values of g and n by exploiting suitableplane models of a general curve. More precisely, _g,n is shownto be rational for g = 2 and 1 n 12, g = 3 and 1 n 14, g= 4 and 1 n 15, and g = 5 and 1 n 12     

On The Curvature of the Real Milnor Fiber

Let C be a germ at O R² of a real analytic plane curve, andC^C its complexification; let C_t B be a fiber of a real smoothdeformation of C in the ball B = B(O,). The following inequalityis proved between the integrals of real curvature k of C_t andthose of Gaussian curvature K of : The sharpness of this inequality is proved in the case whereC is a real irreducible germ. Similar results are proved foran affine algebraic curve C R² of degree d. 2000 MathematicsSubject Classification 14H20, 14H50, 53A04.     

Boundedness and Asymptotic Behaviour of Solutions of a Second-Order Nonlinear System

Consider the second-order nonlinear system where a is a positive and continuous function onR:(– , ), b, c and h are continuous functions on R, ande is a continuous function on I:[0, ). We obtain a sufficientcondition for all solutions of (*) to be bounded, and obtaina sufficient condition for all solutions of (*) to tend to zero.Our results can be applied to the well-known equation which substantially improves and extendsseveral known results in the literature.     

Regularity Conditions and Bernoulli Properties of Equilibrium States and g-Measures

When T : X X is a one-sided topologically mixing subshift offinite type and : X R is a continuous function, one can definethe Ruelle operator L : C(X) C(X) on the space C(X) of real-valuedcontinuous functions on X. The dual operator always has a probability measure as an eigenvectorcorresponding to a positive eigenvalue ( = with > 0). Necessary and sufficient conditionson such an eigenmeasure are obtained for to belong to twoimportant spaces of functions, W(X, T) and Bow (X, T). For example, Bow(X, T) if and only if is a measure with a certain approximateproduct structure. This is used to apply results of Bradleyto show that the natural extension of the unique equilibriumstate µ of Bow(X, T) has the weak Bernoulli propertyand hence is measure-theoretically isomorphic to a Bernoullishift. It is also shown that the unique equilibrium state ofa two-sided Bowen function has the weak Bernoulli property.The characterizations mentioned above are used in the case ofg-measures to obtain results on the ‘reverse’ ofa g-measure.