Some lower bounds for the spectral radius of matrices using traces

Let A be an n×n matrix with eigenvalues λ₁,λ₂,…,λ_n, and let m be an integer satisfying rank(A)?m?n. If A is real, the best possible lower bound for its spectral radius in terms of m, trA and trA² is obtained. If A is any complex matrix, two lower bounds for are compared, and furthermore a new lower bound for the spectral radius is given only in terms of trA,trA²,‖A‖,‖A^∗A-AA^∗‖,n and m.     

Some explicit solutions of the additive Deligne-Simpson problem and their applications

In this paper we construct three infinite series and two extra triples (E₈ and ) of complex matrices B, C, and A=B+C of special spectral types associated to Simpson's classification in Amer. Math. Soc. Proc. 1 (1992) 157 and Magyar et al. classification in Adv. Math. 141 (1999) 97. This enables us to construct Fuchsian systems of differential equations which generalize the hypergeometric equation of Gauss-Riemann. In a sense, they are the closest relatives of the famous equation, because their triples of spectral flags have finitely many orbits for the diagonal action of the general linear group in the space of solutions. In all the cases except for E₈, we also explicitly construct scalar products such that A, B, and C are self-adjoint with respect to them. In the context of Fuchsian systems, these scalar products become monodromy invariant complex symmetric bilinear forms in the spaces of solutions.When the eigenvalues of A, B, and C are real, the matrices and the scalar products become real as well. We find inequalities on the eigenvalues of A, B, and C which make the scalar products positive-definite.As proved by Klyachko, spectra of three hermitian (or real symmetric) matrices B, C, and A=B+C form a polyhedral convex cone in the space of triple spectra. He also gave a recursive algorithm to generate inequalities describing the cone. The inequalities we obtain describe non-recursively some faces of the Klyachko cone.     

Polynomials satisfied by two linked matrices

Polynomials in two variables, evaluated at A and with A being a square complex matrix and being its transform belonging to the set {A⁼, A^†, A^∗}, in which A⁼, A^†, and A^∗ denote, respectively, any reflexive generalized inverse, the Moore-Penrose inverse, and the conjugate transpose of A, are considered. An essential role, in characterizing when such polynomials are satisfied by two matrices linked as above, is played by the condition that the column space of A is the column space of . The results given unify a number of prior, isolated results.     

The distance trisector curve

Given points p and q in the plane, we are interested in separating them by two curves C₁ and C₂ such that every point of C₁ has equal distance to p and to C₂, and every point of C₂ has equal distance to C₁ and to q. We show by elementary geometric means that such C₁ and C₂ exist and are unique. Moreover, for p=(0,1) and q=(0,−1), C₁ is the graph of a function , C₂ is the graph of −f, and f is convex and analytic (i.e., given by a convergent power series at a neighborhood of every point). We conjecture that f is not expressible by elementary functions and, in particular, not algebraic. We provide an algorithm that, given x∈R and ε>0, computes an approximation to f(x) with error at most ε in time polynomial in . The separation of two points by two “trisector” curves considered here is a special (two-point) case of a new kind of Voronoi diagram, which we call the zone diagram and which we investigate in a companion paper.     

A generalization of Kneser's Addition Theorem

Let A=(A₁,…,Am) be a sequence of finite subsets from an additive abelian group G. Let Σ?(A) denote the set of all group elements representable as a sum of ? elements from distinct terms of A, and set . Our main theorem is the following lower bound:

     

On the uniqueness of overcomplete dictionaries, and a practical way to retrieve them

A full-rank under-determined linear system of equations Ax = b has in general infinitely many possible solutions. In recent years there is a growing interest in the sparsest solution of this equation—the one with the fewest non-zero entries, measured by ∥x∥₀. Such solutions find applications in signal and image processing, where the topic is typically referred to as “sparse representation”. Considering the columns of A as atoms of a dictionary, it is assumed that a given signal b is a linear composition of few such atoms. Recent work established that if the desired solution x is sparse enough, uniqueness of such a result is guaranteed. Also, pursuit algorithms, approximation solvers for the above problem, are guaranteed to succeed in finding this solution.Armed with these recent results, the problem can be reversed, and formed as an implied matrix factorization problem: Given a set of vectors {b_i}, known to emerge from such sparse constructions, Ax_i = b_i, with sufficiently sparse representations x_i, we seek the matrix A. In this paper we present both theoretical and algorithmic studies of this problem. We establish the uniqueness of the dictionary A, depending on the quantity and nature of the set {b_i}, and the sparsity of {x_i}. We also describe a recently developed algorithm, the K-SVD, that practically find the matrix A, in a manner similar to the K-Means algorithm. Finally, we demonstrate this algorithm on several stylized applications in image processing.     

Hypergeometric Solutions of Trigonometric KZ Equations Satisfy Dynamical Difference Equations

The trigonometric KZ equations associated to a Lie algebra g depend on a parameter λ∈h where h⊂g is a Cartan subalgebra. A system of dynamical difference equations with respect to λ compatible with the KZ equations is introduced by V. Tarasov and the second author (2000, Internat. Math. Res. Notices15, 801-829). We prove that the standard hypergeometric solutions of the trigonometric KZ equations associated to sl_N also satisfy the dynamical difference equations.     

A polynomial bound on the number of comaximal localizations needed in order to make free a projective module

Let A be a commutative ring and M be a projective module of rank k with n generators. Let h=n-k. Standard computations show that M becomes free after localizations in comaximal elements (see Theorem 5). When the base ring A contains a field with at least hk+1 non-zero distinct elements we construct a comaximal family G with at most (hk+1)(nk+1) elements such that for each g∈G, the module M_g is free over A[1/g].     

Gaussian bounds for degenerate parabolic equations

Let A be a real symmetric, degenerate elliptic matrix whose degeneracy is controlled by a weight w in the A₂ or QC class. We show that there is a heat kernel Wt(x,y) associated to the parabolic equation wut=divA∇u, and Wt satisfies classic Gaussian bounds:

     

Boundedness of solutions for asymmetric Duffing equations

In this paper, we are concerned with the boundedness of all the solutions and the existence of quasiperiodic solutions for some Duffing equations , where e(t) is of period 1, and g : R → R possesses the characters: g(x) is superlinear when x ? d₀, d₀ is a positive constant and g(x) is semilinear when x ? 0.     

Bounds for the error of linear systems of equations using the theory of moments

Consider the system, of linear equations Ax = b where A is an n × n real symmetric, positive definite matrix and b is a known vector. Suppose we are given an approximation to x, ξ, and we wish to determine upper and lower bounds for ∥ x − ξ ∥ where ∥ ··· ∥ indicates the euclidean norm. Given the sequence of vectors {r_i}_{i^k = 0}, where r_i = Ar_{i − 1} and r₀ = b − Aξ, it is shown how to construct a sequence of upper and lower bounds for ∥ x − ξ ∥ using the theory of moments.     

On approximation of functions from Sobolev spaces on metric graphs

Some results on the approximation of functions from the Sobolev spaces on metric graphs by step functions are obtained. In particular, we show that the approximation numbers a_n of the embedding operator of the Sobolev space on a graph G of finite length |G| into the space , where μ is an arbitrary finite Borel measure on G, satisfy the inequality

     

Existence, regularity and local uniqueness of the solutions to the Maxwell-Landau-Lifshitz system in three dimensions

We study a model of ferromagnetism governed by Maxwell's equations coupled with the non-linear Landau-Lifshitz equation of micromagnetism. We are interested in the cases of space-periodic solutions for 3D domains. Obtaining the regularity of the solution m in space and of the solutions E, H in space we state the existence theorem. Finally, we prove the local uniqueness of the solutions.     

On random perturbation of some intersective sets

Given a subset S of Z and a sequence I = (I_n)_n=1^∞ of intervals of increasing length contained in Z, let

     

Absolute continuity of convolutions of orbital measures on Riemannian symmetric spaces

We study the absolute continuity of the measures and of on the Riemannian symmetric spaces X of noncompact type for nonzero elements Xj, X∈a. For m,l?r+1, where r is the rank of X, the considered convolutions have a density. We conjecture that the condition m,l?r+1 is necessary. The conjecture is proved for the symmetric spaces of type An−1. Moreover, the minimal value of l is determined, in function of the irregularity of X.     

A remark on Ricceri's conjecture for a class of nonlinear eigenvalue problems

Consider the eigenvalue problem : −Δu=λf(x,u) in Ω, u=0 on ∂Ω, where Ω is a bounded smooth domain in RN. Denote by the set of all Carathéodory functions f:Ω×R→R such that for a.e. x∈Ω, f(x,⋅) is Lipschitzian with Lipschitz constant L, f(x,0)=0 and , and denote by (resp. ) the set of λ>0 such that has at least one nonzero classical (resp. weak) solution. Let λ₁ be the first eigenvalue for the Laplacian-Dirichlet problem. We prove that and . Our result is a positive answer to Ricceri's conjecture if use f(x,u) instead of f(u) in the conjecture.     

Reduced Weyl asymptotics for pseudodifferential operators on bounded domains II. The compact group case

Let G⊂O(n) be a compact group of isometries acting on n-dimensional Euclidean space Rn, and X a bounded domain in Rn which is transformed into itself under the action of G. Consider a symmetric, classical pseudodifferential operator A₀ in L²(Rn) that commutes with the regular representation of G, and assume that it is elliptic on X. We show that the spectrum of the Friedrichs extension A of the operator is discrete, and using the method of the stationary phase, we derive asymptotics for the number Nχ(λ) of eigenvalues of A equal or less than λ and with eigenfunctions in the χ-isotypic component of L²(X) as λ→∞, giving also an estimate for the remainder term for singular group actions. Since the considered critical set is a singular variety, we recur to partial desingularization in order to apply the stationary phase theorem.