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.     

Continuous Dependence Estimates for Viscosity Solutions of Fully Nonlinear Degenerate Parabolic Equations

Using the maximum principle for semicontinuous functions (Differential Integral Equations3 (1990), 1001-1014; Bull. Amer. Math. Soc. (N.S)27 (1992), 1-67), we establish a general “continuous dependence on the non- linearities” estimate for viscosity solutions of fully nonlinear degenerate parabolic equations with time- and space-dependent nonlinearities. Our result generalizes a result by Souganidis (J. Differential Equations56 (1985), 345-390) for first- order Hamilton-Jacobi equations and a recent result by Cockburn et al. (J. Differential Equations170 (2001), 180-187) for a class of degenerate parabolic second-order equations. We apply this result to a rather general class of equations and obtain: (i) Explicit continuous dependence estimates. (ii) L^∞ and Hölder regularity estimates. (iii) A rate of convergence for the vanishing viscosity method. Finally, we illustrate results (i)-(iii) on the Hamilton-Jacobi- Bellman partial differential equation associated with optimal control of a degenerate diffusion process over a finite horizon. For this equation such results are usually derived via probabilistic arguments, which we avoid entirely here.     

Orbifolds and groupoids

We define a 2-category structure (Pre-Orb) on the category of reduced complex orbifold atlases. We construct a 2-functor F from (Pre-Orb) to the 2-category (Grp) of proper étale effective groupoid objects over the complex manifolds. Both on (Pre-Orb) and (Grp) there are natural equivalence relations on objects: (a natural extension of) equivalence of orbifold atlases on (Pre-Orb) and Morita equivalence in (Grp). We prove that F induces a bijection between the equivalence classes of its source and target.     

On Taylor and other joint spectra for commuting n-tuples of operators

Let R and S be commuting n-tuples of operators. We will give some spectral relations between RS and SR that extend the case of single operators. We connect the Taylor spectrum, the Fredholm spectrum and some other joint spectra of RS and SR. Applications to Aluthge transforms of commuting n-tuples are also provided.     

Generalizations of a property of orthogonal projectors

Generalizing the result in Lemma of Baksalary and Baksalary [J.K. Baksalary, O.M. Baksalary, Commutativity of projectors, Linear Algebra Appl. 341 (2002) 129-142], Baksalary et al. [J.K. Baksalary, O.M. Baksalary, T. Szulc, Linear Algebra Appl. 354 (2002) 35-39] have shown that if P₁ and P₂ are orthogonal projectors, then, in all nontrivial situations, a product of any length having P₁ and P₂ as its factors occurring alternately is equal to another such product if and only if P₁ and P₂ commute, in which case all products involving P₁ and P₂ reduce to the orthogonal projector P₁P₂ (= P₂P₁). In the present paper, further generalizations of this property are established. They consist in replacing a product of the type specified above, appearing on the left-hand side (say) of the equality under considerations, by an affine combination of two or three such products. Comments on the problem when the number of components in a combination exceeds three are also given.     

Hopf bifurcation control in the XCP for the Internet congestion control system

In this paper, we investigated the Hopf bifurcation of the eXplicit Control Protocol (XCP) for the Internet congestion control system. These bifurcation behaviors may cause heavy oscillation of average queue length and induce network instability. A time-delayed feedback control method was proposed for controlling Hopf bifurcation in the XCP system. Numerical simulation results are presented to show that the time-delayed feedback controller is efficient in controlling Hopf bifurcation.     

Asymptotics of Green functions and Martin boundaries for elliptic operators with periodic coefficients

We give the asymptotics at infinity of a Green function for an elliptic equation with periodic coefficients on R^d. Basic ingredients in establishing the asymptotics are an integral representation of the Green function and the saddle point method. We also completely determine the Martin compactification of R^d with respect to an elliptic equation with periodic coefficients by using the exact asymptotics at infinity of the Green function.     

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.     

On commutativity of projectors

The purpose of this paper is to revisit two problems discussed previously in the literature, both related to the commutativity property P₁P₂ = P₂P₁, where P₁ and P₂ denote projectors (i.e., idempotent matrices). The first problem was considered by Baksalary et al. [J.K. Baksalary, O.M. Baksalary, T. Szulc, A property of orthogonal projectors, Linear Algebra Appl. 354 (2002) 35-39], who have shown that if P₁ and P₂ are orthogonal projectors (i.e., Hermitian idempotent matrices), then in all nontrivial cases a product of any length having P₁ and P₂ as its factors occurring alternately is equal to another such product if and only if P₁ and P₂ commute. In the present paper a generalization of this result is proposed and validity of the equivalence between commutativity property and any equality involving two linear combinations of two any length products having orthogonal projectors P₁ and P₂ as their factors occurring alternately is investigated. The second problem discussed in this paper concerns specific generalized inverses of the sum P₁ + P₂ and the difference P₁ − P₂ of (not necessary orthogonal) commuting projectors P₁ and P₂. The results obtained supplement those provided in Section 4 of Baksalary and Baksalary [J.K. Baksalary, O.M. Baksalary, Commutativity of projectors, Linear Algebra Appl. 341 (2002) 129-142].     

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.     

Function spaces and one point extensions for the construct of metered spaces

The construct M of metered spaces and contractions is known to be a superconstruct in which all metrically generated constructs can be fully embedded. We show that M has one point extensions and that quotients in M are productive. We construct a Cartesian closed topological extension of M and characterize the canonical function spaces with underlying sets Hom(X,Y) for metered spaces X and Y. Finally we obtain an internal characterization of the objects in the Cartesian closed topological hull of M.     

Induced nilpotent orbits and birational geometry

In general, a nilpotent orbit closure in a complex simple Lie algebra g, does not have a crepant resolution. But, it always has a Q-factorial terminalization by the minimal model program. According to B. Fu, a nilpotent orbit closure has a crepant resolution only when it is a Richardson orbit, and the resolution is obtained as a Springer map for it. In this paper, we shall generalize this result to Q-factorial terminalizations when g is classical. Here, the induced orbits play an important role instead of Richardson orbits.     

The main invariants of a complex finsler space

In this paper we extend the results obtained in [3], where are investigated the general settings of the two-dimensional complex Finsler manifolds, with respect to a local complex Berwald frame. The geometry of such manifolds is controlled by three real invariants which live on T'M: two horizontal curvature invariants K and W and one vertical curvature invariant I. By means of these invariants are defined both the horizontal and the vertical holomorphic sectional curvatures. The complex Landsberg and Berwald spaces are of particular interest. Complex Berwald spaces coincide with Kähler spaces, in the two – dimensional case. We establish the necessary and sufficient condition under which K is a constant and we obtain a characterization for the Kähler purely Hermitian spaces by the fact K = W = constant and I = 0. For the class of complex Berwald spaces we have K = W = 0. Finally, a classification of two-dimensional complex Finsler spaces for which the horizontal curvature satisfies a special property is obtained.     

Theta Series of Quaternion Algebras over Function Fields

Let K be the function field over a finite field of odd order, and let H be a definite quaternion algebra over K. If Λ is an order of level M in H, we define theta series for each ideal I of Λ using the reduced norm on H. Using harmonic analysis on the completed algebra H_∞ and the arithmetic of quaternion algebras, we establish a transformation law for these theta series. We also define analogs of the classical Hecke operators and show that in general, the Hecke operators map the theta series to a linear combination of theta series attached to different ideals, a generalization of the classical Eichler Commutation Relation.     

A geometric realization of quantum groups of type D

We geometrize quantum groups of type D in the spirit of Beilinson et al. (1990) [1].     

Gradient based and least squares based iterative algorithms for matrix equations AXB + CXD = F

This paper develops a gradient based and a least squares based iterative algorithms for solving matrix equation AXB + CX^TD = F. The basic idea is to decompose the matrix equation (system) under consideration into two subsystems by applying the hierarchical identification principle and to derive the iterative algorithms by extending the iterative methods for solving Ax = b and AXB = F. The analysis shows that when the matrix equation has a unique solution (under the sense of least squares), the iterative solution converges to the exact solution for any initial values. A numerical example verifies the proposed theorems.     

On the projectors FF and FF

A particular version of the singular value decomposition is exploited for an extensive analysis of two orthogonal projectors, namely FF^† and F^†F, determined by a complex square matrix F and its Moore-Penrose inverse F^†. Various functions of the projectors are considered from the point of view of their nonsingularity, idempotency, nilpotency, or their relation to the known classes of matrices, such as EP, bi-EP, GP, DR, or SR. This part of the paper was inspired by Benítez and Rako?evi? [J. Benítez, V. Rako?evi?, Matrices A such that AA^† − A^†A are nonsingular, Appl. Math. Comput. 217 (2010) 3493-3503]. Further characteristics of FF^† and F^†F, with a particular attention paid on the results dealing with column and null spaces of the functions and their eigenvalues, are derived as well. Besides establishing selected exemplary results dealing with FF^† and F^†F, the paper develops a general approach whose applicability extends far beyond the characteristics provided therein.     

Projective sup-algebras: a general view

We begin with the notion of K-flat projectivity. For each sup-algebra L we then introduce a binary relation L_? on it. The K-flat projective sup-algebras are exactly such sup-algebras with each element a approximated by the element x, xL_?a and the relation L_? being stable with respect to the operations on L. Further on, we introduce the notion of a K-comonad and characterize K-flat projective sup-algebras as such sup-algebras having a coalgebra structure for the K-comonad.