Bestness of the parabola theorem for continued fractions

We show that the simple conditional convergence set

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fixed is maximal in the sense that if b is an arbitrary point in , then P{b} is not conditional convergence set.     

Atoms of set systems with a fixed number of pairwise unions

For a finite set system with ground set X, we let

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. An atom of H is a nonempty maximal subset C of X such that for all A H, either C A or C ∩ A = 0. We obtain a best possible upper bound for the number of atoms determined by a set system H with H = k and H H = u for all integers k and u. This answers a problem posed by Sós.     

Directed triangles in directed graphs

We show that each directed graph on n vertices, each with indegree and outdegree at least n/t, where

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, contains a directed circuit of length at most 3.     

Longest cycles in 3-connected graphs

In this paper, we get the following result: Let G be a 3-connected graph with n vertices. Then , where

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. This is a new lower bound for the circumference c(G) of a 3-connected graph G.     

First inverse moment of a generalized quadratic form

In this paper an approximate expression for the first inverse moment of

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where _k is a Gaussian stationary vector process is derived. This generalized quadratic form is the estimate of the information matrix when using the Recursive Least Squares (RLS) algorithm with forgetting factor. This estimator is commonly used when estimating parameters in time-varying linear stochastic systems.     

Comparison of two norms of matrices

Any complex n × n matrix A satisfies the inequality

A ₁ ≤ n 1/2 A _d

where .₁ is the trace norm and ._d is the norm defined by

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,

where B is the set of orthonormal bases in the space of n × 1 matrices. The present work is devoted to the study of matrices A satisfying the identity:

A₁ = n1/2 A _d

This paper is a first step towards a characterization of matrices satisfying this identity. Actually, a workable characterization of matrices subject to this condition is obtained only for n = 2. For n = 3, a partial result on nilpotent matrices is presented. Like our previous study (J. Dazord, Linear Algebra Appl. 254 (1997) 67), this study is a continuation of the work of M. Marcus and M. Sandy (M. Marcus and M. Sandy, Linear and Multilinear Algebra 29 (1991) 283). Also this study is related to the work of R. Gabriel on classification of matrices with respect to unitary similarity (see R. Gabriel, J. Riene Angew, Math. 307/308 (1979) 31; R. Gabriel, Math. Z. 200 (1989) 591).     

On approximation of solutions of multidimensional SDE's with reflecting boundary conditions

Let D be either a convex domain in ^d or a domain satisfying the conditions (A) and (B) considered by Lions and Sznitman [7] and Saisho [11]. We estimate the rate of L^p convergence for Euler and Euler–Peano schemes for stochastic differential equations in D with normal reflection at the boundary of the form

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, where W is a d-dimensional Wiener process. As a consequence we give the rate of almost sure convergence for these schemes.     

Large 2P3-free graphs with bounded degree

Let ex^* (D;H) be the maximum number of edges in a connected graph with maximum degree D and no induced subgraph H; this is finite if and only if H is a disjoint union of paths. If the largest component of such an H has order m, then ex^*(D; H) = O(D²ex^*(D; P_m)). Constructively, ex^*(D;qP_m) = Θ(gD²ex^*(D;P_m)) if q>1 and m> 2(Θ(gD²) if m = 2). For H = 2P₃ (and D 8), the maximum number of edges is if D is even and

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if D is odd, achieved by a unique extremal graph.     

A note on unsolvable systems of max–min (fuzzy) equations

For unsolvable systems of linear equations of the form Ax=b over the max–min (fuzzy) algebra

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we propose an efficient method for finding a Chebychev-best approximation of the matrix in the set

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.     

Strong continuity of the relative rearrangement maps and application to a Galerkin approach for nonlocal problems

We study the strong continuity of the map u   (b_*u, b_*u(| > u(·)|)). Here,

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for σ]0 means Ω[, u_* (respectively, (b|_{u=u*(σ)})*) denotes the decreasing rearrangement of u (respectively b restricted to the set {u = u_*(σ)}) and |E| denotes the Lebesgue measure of a set E included in a domain Ω. The results are useful for solving plasmas physics equations or any nonlocal problems involving the monotone rearrangement, its inverse or its derivatives.     

Universal H-colorable graphs without a given configuration

For every pair of finite connected graphs F and H, and every positive integer k, we construct a universal graph U with the following properties:

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Particularly, this solves a problem presented in [1] and [2] regarding the chromatic number of a universal graph.     

Extremal regular graphs for the achromatic number

The problem of constructing (m, n) cages suggests the following class of problems. For a graph parameter θ, determine the minimum or maximum value of p for which there exists a k-regular graph on p points having a given value of θ. The minimization problem is solved here when θ is the achromatic number, denoted by ψ. This result follows from the following main theorem. Let M(p, k) be the maximum value of ψ(G) over all k-regular graphs G with p points, let {x} be the least integer of size at least x, and let be given by ω(k) = {i(ik+1)+1:1i<∞}. Define the function ƒ(p, k) by

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. Then for fixed k2 we have M(p, K=ƒ(p, k) if pω(k) and M(p, k)=ƒ(p,k-1 if pε ω(k) for all p sufficiently large with respect to k.     

Another view on martingale central limit theorems

Based on the martingale version of the Skorokhod embedding Heyde and Brown (1970) established a bound on the rate of convergence in the central limit theorem (CLT) for discrete time martingales having finite moments of order 2+2δ with 0<δ1. An extension for all δ>0 was proved in Haeusler (1988). This paper presents a rather quick access based solely on truncation, optional stopping, and prolongation techniques for martingale difference arrays to obtain other upper bounds for sup

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(φbeing the standard normal d.f.) yielding weak sufficient conditions for the asymptotic normality of . It is shown that our approach also yields two types of martingale central limit theorems with random norming.     

Tree groups and the 4-string pure braid group

Given a graph Г, undirected, with no loops or multiple edges, we define the graph group on Г, F_Г, as the group generated by the vertices of Г, with one relation xy = xy for each pair x and y of adjacent vertices of Г.

In this paper we will show that the unpermuted braid group on four strings is an HNN-extension of the graph group F_s, where

S =

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The form of the extension will resolve a conjecture of Tits for the 4-string braid group. We will conclude, by analyzing the subgroup structure of graph groups in the case of trees, that for any tree T on a countable vertex set, F_t is a subgroup of the 4-string braid group.

We will also show that this uncountable collection of subgroups of the 4-string braid group is linear, that is, each subgroup embeds in GL(3, ), as well as embedding in Aut(F), where F is the free group of rank 2.     

Bounds on the number of isolates in sum graph labeling

A simple undirected graph H is called a sum graph if there is a labeling L of the vertices of H into distinct positive integers such that any two vertices u and v of H are adjacent if and only if there is a vertex w with label L(w)=L(u)+L(v). The sum number σ(G) of a graph G=(V,E) is the least integer r such that the graph H consisting of G and r isolated vertices is a sum graph. It is clear that σ(G)|E|. In this paper, we discuss general upper and lower bounds on the sum number. In particular, we prove that, over all graphs G=(V,E) with fixed |V|3 and |E|, the average of σ(G) is at least

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. In other words, for most graphs, σ(G)Ω(|E|).     

On AR(1) models with periodic and almost periodic coefficients

This paper is concerned with the existence of bounded solutions to the system of equations X_n=a_nX_n−1+ξ_n, nZ, where ξ_n are uncorrelated constant variance zero mean random variables. We give necessary and sufficient conditions for boundedness in the general case and then specifically for periodic and almost periodic (a_n). This provides the first step in extending the periodic autoregressive models, for which boundedness is equivalent to the stationarity of the blocked vector sequence

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to the almost periodic case.     

Iterative methods for the extremal positive definite solution of the matrix equation

In this paper, the inversion free variant of the basic fixed point iteration methods for obtaining the maximal positive definite solution of the nonlinear matrix equation X+A^*X^-A=Q with the case 0<1 and the minimal positive definite solution of the same matrix equation with the case 1 are proposed. Some necessary conditions and sufficient conditions for the existence of positive definite solutions for the matrix equation are derived. Numerical examples to illustrate the behavior of the considered algorithms are also given.     

Uniform asymptotic expansions of the Pollaczek polynomials

Two uniform asymptotic expansions are obtained for the Pollaczek polynomials P_n(cosθ;a,b). One is for , , in terms of elementary functions and in descending powers of . The other is for , in terms of a special function closely related to the modified parabolic cylinder functions, in descending powers of n. This interval contains a turning point and all possible zeros of P_n(cosθ) in θ(0,π/2].     

On a singular perturbation problem with two second-order turning points

In this paper, we study the singular perturbation problem

where 0<ε1 is a small positive parameter, p(x) and q(x) are sufficiently smooth and strictly positive functions. The main feature of this equation is that there are two second-order turning points in the interval (0,1). Based on the rigorous results on singular perturbation problems with one second-order turning point in our previous work, we obtain a uniform asymptotic approximation for the general solution of the above equation by means of a matching technique.     

Some results on a heat conduction problem by Myshkis

An infinite homogeneous d-dimensional medium initially is at zero temperature. A heat impulse is applied at the origin, raising the temperature there to a value greater than a constant value u₀>0. The temperature at the origin then decays, and when it reaches u₀, another equal-sized heat impulse is applied at a normalized time τ₁=1. Subsequent equal-sized heat impulses are applied at the origin at the normalized times τ_n, n=2,3,…, when the temperature there has decayed to u₀. This sequence of normalized waiting times τ_n can be defined recursively by a difference equation and its asymptotic behavior was known recently. This heat conduction problem was first studied in [J. Difference Equations Appl. 3 (1997) 89–91].

A natural subsequent question is what happens if the problem is set in a finite region, like in a laboratory, with the temperature at the boundary being kept zero forever. In this paper we obtain the asymptotic behavior of the heating times for the one-dimensional case.