Determinants of Matrices Associated with Incidence Functions on Posets

Let S = x ₁,...,x _n} be a finite subset of a partially ordered set P. Let f be an incidence function of P. Let denote the n × n matrix having f evaluated at the meet of x _i and x _j as its i, j-entry and denote the n × n matrix having f evaluated at the join of x _i and x _j as its i, j-entry. The set S is said to be meet-closed if for all 1 i, j n. In this paper we get explicit combinatorial formulas for the determinants of matrices and on any meet-closed set S. We also obtain necessary and sufficient conditions for the matrices and on any meet-closed set S to be nonsingular. Finally, we give some number-theoretic applications.     

Singular Dirichlet boundary value problems II: Resonance case

Existence results are established for the resonant problem y + _m a y = f(t, y) a.e. on [0, 1] with y satisfying Dirichlet boundary conditions. The problem is singular since f is a Carathéodory function, with a > 0 a.e. on [0, 1] and      

Notes on estimating inverse-Gaussian and gamma subordinators under high-frequency sampling

We study joint efficient estimation of two parameters dominating either the inverse-Gaussian or gamma subordinator, based on discrete observations sampled at satisfying as . Under the condition that as we have two kinds of optimal rates, and . Moreover, as in estimation of diffusion coefficient of a Wiener process the -consistent component of the estimator is effectively workable even when T _n does not tend to infinity. Simulation experiments are given under several h _n ’s behaviors.     

On Asymptotic Properties of Solutions of Diffusion Equations

In this work the authors study the conditions for the existence of diffusion equations

Invariance principles for random walks on hypergroups on ℝ2 and ℕ

Let (X _n:n) be i.i.d. with finite variance and values in a hypergroupK:=₊ or and _j=1 ⁿ X _j be the randomized sum of these random variables. It is shown that the processes converge in distribution to a Gaussian process in the caseK=₊, that the processes converge towards a Bessel process on ₊ in the case of polynomial growth of the hypergroupK=₊ or , and that in the case of exponential growth converges towards a Brownian motion asn.     

On the Existence of Solutions for Some Nondegenerate Nonlinear Wave Equations of Kirchhoff Type

Let be a bounded domain in #x211D;ⁿ with a smooth boundary . In this work we study the existence of solutions for the following boundary value problem:

On the Structure of a Certain Class of Mixed Tsirelson Spaces

We study Banach spaces of the form We call such a space a p-space, p[1,), if for every k the space is isomorphic to _pk and the sequence (p_k) strictly decreases to p. We examine the finite block representability of the spaces _r in a p-space proving that it depends not only on p but also on the sequences (p_k) and (n_k). Assuming that _i n_i ^1/q decreases to 0, where q is the conjugate exponent of p, we prove the existence of an asymptotic biorthogonal system in X and also that c ₀ is finitely representable in X. Moreover we investigate the modified versions of p-spaces proving that, if n_km1/pkm-1/pk_m-1 increases to infinity for a subsequence (n_km) , then ₁ embeds into X. We also investigate complemented minimality for the class of spaces where is either a subsequence of the sequence of Schreier classes ( _n)_{n N} or a subsequence of ( _n)_{n N}.     

Limit Theorems for Sums of Heavy-tailed Variables with Random Dependent Weights

Let be independent and identically distributed random variables with heavy-tailed distributions. Consider a sequence of random weights , independent of and focus on the weighted sums , where μ involves a suitable centering. We establish sufficient conditions for these weighted sums to converge to non-trivial limit processes, as n→∞, when appropriately normalized. The convergence holds, for example, if is strictly stationary, dependent, and W ₁ has lighter tails than U ₁. In particular, the weights W _j s can be strongly dependent. The limit processes are scale mixtures of stable Lévy motions. We establish weak convergence in the Skorohod J ₁-topology. We also consider multivariate weights and show that they converge weakly in the strong Skorohod M ₁-topology. The M ₁-topology, while weaker than the J ₁-topology, is strong enough for the supremum and infimum functionals to be continuous. This research was partially supported by a fellowship of the Horace H. Rackham School of Graduate Studies at the University of Michigan and the NSF Grants BCS-0318209 and DMS-0505747 at Boston University.     

Scalar Operators Representable as a Sum of Projectors

We study sets there exist n projectors P₁,...,P_n such that . We prove that if n 6, then .     

Weighted inequalities for integral operators with some homogeneous kernels

In this paper we study integral operators of the form

On the Normal Order of ϕk+1(n)/ϕk(n), where ϕk is the k-Fold Iterate of Euler's Function

Let be the j-fold iterated function of . Let and > 0 be fixed, Q be a prime, and let N _k(Q|x) denote the number of those nx for which Q . We give the asymptotics of N _k(Q|x) in the range .     

An Extension of a Class of Iterative Procedures for Nonlinear Quasivariational Inequalities

Consider the convergence of the projection methods based on an extension of a special class of algorithms for the approximation--solvability of the following class of nonlinear quasivariational inequality (NQVI) problems: find an element such that and

Convergence of series of translations

For a mean zero norm one sequence (f _n )L ₂[0, 1], the sequence (f _n {nx+y}) is an orthonormal sequence inL ₂([0, 1]²); so if , then converges for a.e. (x, y)[0, 1]² and has a maximal function inL ₂([0, 1]²). But for a mean zerofL ₂[0, 1], it is harder to give necessary and sufficient conditions for theL ₂-norm convergence or a.e. convergence of . Ifc _n 0 and , then this series will not converge inL ₂-norm on a denseG subset of the mean zero functions inL ₂[0, 1]. Also, there are mean zerofL[0, 1] such that never converges and there is a mean zero continuous functionf with a.e. However, iff is mean zero and of bounded variation or in some Lip() with 1/2<1, and if |c _n | = 0(n ^–) for >1/2, then converges a.e. and unconditionally inL ₂[0, 1]. In addition, for any mean zerof of bounded variation, the series has its maximal function in allL _p[0, 1] with 1p<. Finally, if (f _n )L [0, 1] is a uniformly bounded mean zero sequence, then is a necessary and sufficient condition for to converge for a.e.y and a.e. (x _n )[0, 1]. Moreover, iffL [0, 1] is mean zero and , then for a.e. (x _n )[0, 1], converges for a.e.y and in allL _p [0, 1] with 1p<. Some of these theorems can be generalized simply to other compact groups besides [0, 1] under addition modulo one.     

Coconvex Approximation of Functions with More than One Inflection Point

Assume that f C[–1, 1] belongs to C[–1, 1] and changes its convexity at s > 1 different points y _i, , from (–1, 1). For n N, n 2, we construct an algebraic polynomial P _n of order n that changes its convexity at the same points y _i as f and is such that

On <Emphasis Type="Italic">t</Emphasis><Superscript>2</Superscript>-like solutions of certain second order differential equations

Via an integral transformation, we establish two embedding results between the Emden-Fowler type equation , t ≥ t ₀ > 0, with solutions x such that as , , and the equation , u > 0, with solutions y such that for given k > 0. The conclusions of our investigation are used to derive conditions for the existence of radial solutions to the elliptic equation , , that blow up as in the two dimensional case.      

Two consistency results concerning thin-tall Boolean algebras

In this paper we show that the following is relatively consistent withZFC +CH: There is no superatomic Boolean algebra of height ₂+1 and width, and there is no superatomic Boolean algebraA with for 0<<₁ and Presented by J. Mycielski.     

On Interpolation Sequences of One Class of Functions Analytic in the Unit Disk

We establish a criterion for the existence of a solution of the interpolation problem f( _n ) = b _n in the class of functions f analytic in the unit disk and satisfying the relation

Rate of Convergence of Increments for Random Fields

We obtain exact convergence rate in limit theorems for the maximal increments of random fields

On the Existence of Positive Eigenvalues for Semilinear Elliptic Equation on all of R d

In this paper, we consider a problem of the form

Convex Embeddings and Bisections of 3-Connected Graphs1

Given two disjoint subsets T ₁ and T ₂ of nodes in an undirected 3-connected graph G = (V, E) with node set V and arc set E, where and are even numbers, we show that V can be partitioned into two sets V ₁ and V ₂ such that the graphs induced by V ₁ and V ₂ are both connected and holds for each j = 1,2. Such a partition can be found in time. Our proof relies on geometric arguments. We define a new type of convex embedding of k-connected graphs into real space R ^k-1 and prove that for k = 3 such an embedding always exists. ¹ A preliminary version of this paper with title Bisecting Two Subsets in 3-Connected Graphs appeared in the Proceedings of the 10th Annual International Symposium on Algorithms and Computation, ISAAC 99, (A. Aggarwal, C. P. Rangan, eds.), Springer LNCS 1741, 425&ndash;434, 1999.