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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.     

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.     

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.     

On the stability and boundedness of solutions of a kind of third order delay differential equations

In this paper, we study Eq. (1.1) for asymptotic stability of the zero solution when and uniformly bounded and uniformly ultimate bounded of all solutions when      

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.     

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.     

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.     

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.     

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.     

Invariant variational principles and conservation laws for some nonlinear partial differential equations with constant coefficients – I

The invariance under a one-parameter infinitesimal transformation groups [1] has been proven for a number of nonlinear partial differential equations (NLPDEs) with constant coefficients, which appear in a wide variety of modelling physical phenomena/applications. The invariance identities of Rund [2] involving the Lagrangian and the generators of the infinitesimal Lie groups are utilized, for writing down the conservation laws via Noether's theorem. In order that the study becomes more exhaustive, we have applied the above technique to the cases arising from the generalized Klein–Gordon equation by transforming it to ordinary differential equation (ODE), to get on exact solution for it.     

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.     

The number of spanning forests of a graph

In this paper we study the number of spanning forests of a graph. Let G be a connected simple graph. (1) We give a lower bound for the number of spanning forests of G in terms of the edge connectivity of G. (2) We give an upper bound for the number of rooted spanning forests of G. (3) We describe the elementary symmetric functions of inverse positive Laplacian eigenvalues of a tree. (4) We determine all Laplacian integral graphs with prime number of spanning trees. (5) We give a simple proof of a theorem of K. Hashimoto on Ihara zeta function.     

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.     

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.     

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).     

Domination graphs: examples and counterexamples

We give a counterexample to a conjecture of Dahlhaus et al. [3] claiming that the Special Quadratic Consensus Method yields a polynomial-time recognition for domination graphs, and discuss several new properties of domination graphs.     

Diameter preserving surjections in the geometry of matrices

We consider a class of graphs subject to certain restrictions, including the finiteness of diameters. Any surjective mapping φ:Γ→Γ^′ between graphs from this class is shown to be an isomorphism provided that the following holds: Any two points of Γ are at a distance equal to the diameter of Γ if, and only if, their images are at a distance equal to the diameter of Γ^′. This result is then applied to the graphs arising from the adjacency relations of spaces of rectangular matrices, spaces of Hermitian matrices, and Grassmann spaces (projective spaces of rectangular matrices).     

Almost periodic solutions of differential equations with piecewise constant argument of generalized type

We consider the existence and stability of an almost periodic solution of the following hybrid system:

(1)

where if θ_i≤t<θ_i+1,i=…−2,−1,0,1,2,…, is an identification function, θ_i is a strictly ordered sequence of real numbers, unbounded on the left and on the right, p_j,j=1,2,…,m, are fixed integers, and the linear homogeneous system associated with (1) satisfies exponential dichotomy. The deviations of the argument are not restricted by any sign assumption when existence is considered. A new technique of investigation of equations with piecewise argument, based on integral representation, is developed.     

The Aharonov-Bohm Solenoids in a Constant Magnetic Field

We study the spectral properties of a two-dimensional magnetic Schrödinger operator The magnetic field is given by where B > 0 is a constant, and the points are uniformly separated. We give an upper bound for the number of eigenvalues of H_N between two Landau levels or below the lowest Landau level, when N is finite. We prove the spectral localization of H_N near the spectrum of the single solenoid operator, when are far from each other, all the values are the same, and the boundary conditions at z_j are uniform. We determine the deficiency indices of the minimal operator and give a characterization of self-adjoint extensions of the minimal operator.submitted 28/05/04, accepted 23/07/04     

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.