On the Waiting Time for the First Success Run

Let k and m are positive integers with k ≥ m. The probability generating function of the waiting time for the first occurrence of consecutive k successes in a sequence of m-th order Markov dependent trials is given as a function of the conditional probability generating functions of the waiting time for the first occurrence of consecutive m successes. This provides an efficient algorithm for obtaining the probability generating function when k is large. In particular, in the case of independent trials a simple relationship between the geometric distribution of order k and the geometric distribution of order k−1 is obtained. This research was partially supported by the ISM Cooperative Research Program(2004-ISM-CRP-2006) and by a Grant-in-Aid for Scientific Research (C) of the JSPI (Grant Number 16500183)     

Continuity in weak topology:higher order linear systems of ODE

We will introduce a type of Fredholm operators which are shown to have a certain con- tinuity in weak topologies.From this,we will prove that the fundamental matrix solutions of k-th, k≥2,order linear systems of ordinary differential equations are continuous in coefficient matrixes with weak topologies.Consequently,Floquet multipliers and Lyapunov exponents for periodic systems are continuous in weak topologies.Moreover,for the scalar Hill's equations,Sturm-Liouville eigenvalues, periodic and anti-periodic eigenvalues,and rotation numbers are all continuous in potentials with weak topologies.These results will lead to many interesting variational problems.     

Semi-local units modulo Gauss sums

For thep-th cyclotomic fieldk, Iwasawa proved thatp does not divide the class number of its maximal real subfield if and only if the odd part of the group of local units coincides with its subgroup generated by Jacobi sums related tok. We refine and give a quantitative version of this result for more general imaginary abelian fields. Our result is an analogy of the famous result on “semi-local units modulo cyclotomic units”. Partially supported by Grant-in-Aid for Scientific Research (C), Grant 09640054.     

Generalized chen submanifolds

As a generalization of Chen submanifolds,k-th Chen submanifolds are defined. A characterization for them is proved. Spherical 2nd Chen submanifolds are discussed. For a compact submanifoldM with parallel second fundamental form it is proved thatM is ak-th Chen submanifold if and only ifM is ofk-type.Dedicated to Prof. A. Barlotti for his 70- th birthdayThe first author was partially supported by the Provincial Scientific Research Fund from Guangdong Province, China.     

On thek-th largest eigenvalue of the Laplacian matrix of a graph

In this paper, we give the upper bound and lower bound ofk-th largest eigenvalue λk of the Laplacian matrix of a graphG in terms of the edge number ofG and the number of spanning trees ofG. This research is supported by the National Natural Science Foundation of China (Grant No.19971086) and the Doctoral Program Foundation of State Education Department of China.     

A priori estimates for truncation error of continued fractionsK(1/b n )

A priori estimates are obtained for the truncation error of continued fractions of the formK(1/b _n ), with complex elementsb _n . The method employed is based on the calculation of bounds for successive diameters of a sequence of nested disks, where then-th approximant of the continued fraction is contained in then-th disk. Numerical examples are given to illustrate useful procedures and typical error estimates for continued fraction expansions of the complex logarithm and the ratio of consecutive Bessel functions.This research was supported by the National Science Foundation under Grant No. GP-9009 and by the United States Air Force through the Air Force Office of Scientific Research under Grant No. AFOSR-70-1888.     

Factoring polynomials modulo special primes

We consider the problem of factoring polynomials overGF(p) for those prime numbersp for which all prime factors ofp– 1 are small. We show that if we have a primitivet-th root of unity for every primet dividingp– 1 then factoring polynomials overGF(p) can be done in deterministic polynomial time.Research partially supported by Hungarian National Foundation for Scientific Research, Grant 1812.     

What portion of the sample makes a partial sum asymptotically stable or normal?

Summary Let a sequence of independent and identically distributed random variables with the common distribution function in the domain of attraction of a stable law of index 0<2 be given. We show that if at each stage n a number k _n depending on n of the lower and upper order statistics are removed from the n-th partial sum of the given random variables then under appropriate conditions on k _n the remaining sum can be normalized to converge in distribution to a standard normal random variable. A further analysis is given to show which ranges of the order statistics contribute to asymptotic stable law behaviour and which to normal behaviour. Our main tool is a new Brownian bridge approximation to the uniform empirical process in weighted supremum norms.Work done while visiting the Bolyai Institute, Szeged University, partially supported by a University of Delaware Research Foundation Grant     

On the possible number of elements of given order in a finite group

The main motivation of this paper is to introduce a problem of some combinatorial flavor about finite groups which seems to be new in the literature. Letk>1 be a fixed positive integer and denote byf(k, G) the number of elements of orderk in the groupG. We examine the setF(k)={f(k, G)| G a finite group}/{0}. We give a complete characterization ofF(k) if 4|k ork=6 and show some modest partial results for certain other values ofk. It seems to us that the question is surprisingly difficult even in such simple cases ask=3, which we investigate in detail. Research (partially) supported by Hungarian National Foundation for Scientific Research (OTKA), Grant No. 1901. Research (partially) supported by Hungarian National Foundation for Scientific Research (OTKA), Grant No. 1903.     

Higher order valued reduction theorems for classical connections

We generalize reduction theorems for classical connections to operators with values in k-th order natural bundles. Using the 2nd order valued reduction theorems we classify all (0,2)-tensor fields on the cotangent bundle of a manifold with a linear (non-symmetric) connection. This paper has been supported by the Grant Agency of the Czech Republic under the Project number GA 201/02/0225.     

Sooner and later waiting time problems for success and failure runs in higher order Markov dependent trials

The probability generating functions of the waiting times for the first success run of length k and for the sooner run and the later run between a success run of length k and a failure run of length r in the second order Markov dependent trials are derived using the probability generating function method and the combinatorial method. Further, the systems of equations of 2^.m conditional probability generating functions of the waiting times in the m-th order Markov dependent trials are given. Since the systems of equations are linear with respect to the conditional probability generating functions, they can be solved exactly, and hence the probability generating functions of the waiting time distributions are obtained. If m is large, some computer algebra systems are available to solve the linear systems of equations.This research was partially supported by the Natural Sciences and Engineering Research Council of Canada.     

On n-Contractive and n-Hypercontractive Operators, II

This paper studies the n-contractive and n-hypercontractive Hilbert space operators (n = 1, 2, . . .), classes weaker than, but related to, the class of subnormal operators. The k-hyponormal operators are the more thoroughly explored examples of classes weaker than subnormal; we show that k-hyponormality implies 2k-contractivity. Turning to weighted shifts, it is shown that if a weighted shift is extremal in the sense that the general nonnegativity test for n-contractivity is satisfied with equality to zero, then the shift is necessarily the unweighted unilateral shift. Also considered are the n-contractivity of back step extensions and perturbations of subnormal weighted shifts and some connections with the Berger measure of a subnormal shift. The second author was supported by the Korean Research Foundation Grant funded by the Korean Government (MOEHRD) (KRF-2006-312-C00027). The third author was supported by the Korean Research Foundation Grant funded by the Korean Government (MOEHRD) (KRF- 2007-359-C00005.     

Hua’s theorem with <Emphasis Type="Italic">s</Emphasis> almost equal prime variables

It is proved unconditionally that every sufficiently large positive integer satisfying some necessary congruence conditions can be represented as the sum of s almost equal k-th powers of prime numbers for 2 ≤ k ≤ 10 and s =2k ＋ 1, which gives a short interval version of Hun＇s theorem.     

Minimum cutsets for an element of a boolean lattice

An informative new proof is given for the theorem of Nowakowski that determines for all n and k the minimum size of a cutset for an element A with |A|=k of the Boolean algebra B _n of all subsets of {1,...,n}, ordered by inclusion. An inequality is obtained for cutsets for A that is reminiscent of Lubell's inequality for antichains in B _n. A new result that is provided by this approach is a list of all minimum cutsets for A.Research supported in part by NSF Grant DMS 87-01475.Research supported in part by NSF Grant DMS 86-06225 and Air Force OSR-86-0076.     

Laws of iterated logarithm for state variables with applications

Letx _k be the state variable (solution) of a stochastic difference equation. This paper gives the laws of iterated logarithm for {x _k} and {x _kx _k ^τ }, which being strongly correlated, are neither stationary nor ergodic. The results obtained are then applied to Kalman filter and LQG control problem. Work supported by National Natural Science Foundation of China and the TWAS Research Grant No. 87-43.     

On approximation of analytic sets

We give conditions sufficient for the existence of a sequence of Nash sets convergent to a given analytic set. Moreover, for limit sets which are k-sheeted analytic covers we additionally require that sets in approximating sequence are also k-sheeted analytic covers. Then we present examples of applications of the developed theory.Research partially supported by KBN Grant no. 2P03A 015 22Mathematics Subject Classification (2000): 32E30, 32C07, 32C25     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> Bounds for the Commutator of Parabolic Singular Integral with Rough Kernels

In this paper, the authors give the L ^p (1 < p < ∞ ) boundedness of the k-th order commutator of parabolic singular integral with the kernel function Ω ∈ L(log⁺ L) ^k + 1(S ^n − 1). The result in this paper is an extension of some known results. The research was supported by NSF of China (Grant: 10571015) and SRFDP of China (Grant: 20050027025).     

Complete rotation hypersurfaces with Hk constant in space forms

In this paper we classify all complete rotation hypersurfaces withH _k constant in ⁿ⁺¹ andH ⁿ⁺¹, is the normalizedk-th symmetric function of the principal curvatures. Partial results are also given forH ⁿ⁺¹.Partially supported by DGAPA-UNAM, México, CONACYT, México, under Project 1068P, and CNPp, Brazil.     

Independent sets in regular graphs and sum-free subsets of finite groups

It is shown that there exists a function∈(k) which tends to 0 ask tends to infinity, such that anyk-regular graph onn vertices contains at most 2^{(1/2+∈(k))n} independent sets. This settles a conjecture of A. Granville and has several applications in Combinatorial Group Theory. Research supported in part by the United States-Israel Binational Science Foundation and by a Bergmann Memorial Grant.     

An upper bound on the number of planarK-sets

Given a setS ofn points, a subsetX of sizek is called ak-set if there is a hyperplane that separatesX fromS–X. We prove thatO(nk/log*k) is an upper bound for the number ofk-sets in the plane, thus improving the previous bound of Erdös, Lovász, Simmons, and Strauss by a factor of log*k.The research of J. Pach was supported in part by NSF Grant CCR-8901484 and by Grant OTKA-1418 from the Hungarian Foundation for Scientific Research. The research of W. Steiger and E. Szemerédi was supported in part by NSF Grant CCR-8902522. All authors express gratitude to the NSF DIMACS Center at Rutgers.