A full row-rank system matrix generated by the strip-based projection model in discrete tomography

Let Cu = k be an underdetermined linear system generated by the strip-based projection model in discrete tomography, where C is row-rank deficient. In the case of one scanning direction the linear dependency of the rows of C is studied in this paper. An index set H is specified such that if all rows of C with row indices in H are deleted then the rows of resultant matrix F are maximum linearly independent rows of C. Therefore, the corresponding system is equivalent to Cu = k and consequently, the cost of an image reconstruction from is reduced.     

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.     

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

     

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

     

Convex solutions of a functional equation arising in information theory

Given a convex function f defined for positive real variables, the so-called Csiszár f-divergence is a function If defined for two n-dimensional probability vectors p=(p₁,…,pn) and q=(q₁,…,qn) as . For this generalized measure of entropy to have distance-like properties, especially symmetry, it is necessary for f to satisfy the following functional equation: for all x>0. In the present paper we determine all the convex solutions of this functional equation by proposing a way of generating all of them. In doing so, existing usual f-divergences are recovered and new ones are proposed.     

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.     

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:

     

Mean value property and a Berezin-type transform on the half-space

Let B be the open unit ball in Rn. Liu (2007) [6] has shown that if then the iterates of a Berezin-type transform of f converge to the Poisson extension of the boundary values of f, as k→∞. In this paper, we extend this to the half-space setting. First, we obtain the mean value property for harmonic functions on the half-space H. Based on this property, we define a Berezin-type transform BH and investigate the limit of the iterates of BH.     

Transportation cost inequalities on path and loop groups

Let G be a connected Lie group with the Lie algebra . The action of Cameron-Martin space on the path space P_e(G) introduced by L. Gross (Illinois J. Math. 36 (1992) 447) is free. Using this fact, we define the H-distance on P_e(G), which enables us to establish a transportation cost inequality on P_e(G). This method will be generalized to the path space over the loop group , so that we obtain a transportation cost inequality for heat measures on .     

On the Spaltenstein correspondence

In the article [Spa1], N. Spaltenstein has established a bijection between the irreducible components of _χ, the space of full flags fixed by a nilpotent element χ ? M(n, k), where k is an algebraically closed field, and the standard tableaux associated to the Young diagram of χ. In this present work we determine, when χ is of hook type, for each irreducible component X of _χ, the unique Schubert cell _X of the full flag manifold = (V) (where V is vector space of dimension n over k), such that X ∩ _X is a dense subspace in X. This result will allow us to optimize the computation of _χ and when k = is the complex field, to see that the graph resolution of the partition (2, 1, …, 1) of n is related to the Dynkin diagram of sl(n, ).     

Piecewise linear maps, Liapunov exponents and entropy

Let be a class of piecewise linear maps associated with a transition matrix A. In this paper, we prove that if fA,x∈LA, then the Liapunov exponent λ(x) of fA,x is equal to a measure theoretic entropy hmA,x of fA,x, where mA,x is a Markov measure associated with A and x. The Liapunov exponent and the entropy are computable by solving an eigenvalue problem and can be explicitly calculated when the transition matrix A is symmetric. Moreover, we also show that maxxλ(x)=maxxhmA,x=log(λ₁), where λ₁ is the maximal eigenvalue of A.     

Dynamics of composite functions meromorphic outside a small set

Let M denote the class of functions f meromorphic outside some compact totally disconnected set E=E(f) and the cluster set of f at any a∈E with respect to is equal to . It is known that class M is closed under composition. Let f and g be two functions in class M, we study relationship between dynamics of f○g and g○f. Denote by F(f) and J(f) the Fatou and Julia sets of f. Let U be a component of F(f○g) and V be a component of F(g○f) which contains g(U). We show that under certain conditions U is a wandering domain if and only if V is a wandering domain; if U is periodic, then so is V and moreover, V is of the same type according to the classification of periodic components as U unless U is a Siegel disk or Herman ring.     

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.     

Fredholm differential operators with unbounded coefficients

We prove that a first-order linear differential operator G with unbounded operator coefficients is Fredholm on spaces of functions on with values in a reflexive Banach space if and only if the corresponding strongly continuous evolution family has exponential dichotomies on both and and a pair of the ranges of the dichotomy projections is Fredholm, and that the Fredholm index of G is equal to the Fredholm index of the pair. The operator G is the generator of the evolution semigroup associated with the evolution family. In the case when the evolution family is the propagator of a well-posed differential equation u′(t)=A(t)u(t) with, generally, unbounded operators , the operator G is a closure of the operator . Thus, this paper provides a complete infinite-dimensional generalization of well-known finite-dimensional results by Palmer, and by Ben-Artzi and Gohberg.