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.     

LSQR iterative common symmetric solutions to matrix equations AXB = E and CXD = F

A new matrix based iterative method is presented to compute common symmetric solution or common symmetric least-squares solution of the pair of matrix equations AXB = E and CXD = F. By this iterative method, for any initial matrix X₀, a solution X^∗ can be obtained within finite iteration steps if exact arithmetic was used, and the solution X^∗ with the minimum Frobenius norm can be obtained by choosing a special kind of initial matrix. In addition, the unique nearest common symmetric solution or common symmetric least-squares solution to given matrix in Frobenius norm can be obtained by first finding the minimum Frobenius norm common symmetric solution or common symmetric least-squares solution of the new pair of matrix equations. The given numerical examples show that the matrix based iterative method proposed in this paper has faster convergence than the iterative methods proposed in [1] and [2] to solve the same problems.     

Least squares solutions with special structure to the linear matrix equation AXB = C

In this article we give some formulas for the maximal and minimal ranks of the submatrices in a least squares solution X to AXB = C. From these formulas, we derive necessary and sufficient conditions for the submatrices to be zero and other special forms, respectively. Finally, some Hermitian properties for least squares solution to matrix equation AXB = C are derived.     

Exponentiation in V-categories

For a Heyting algebra V which, as a category, is monoidal closed, we obtain characterizations of exponentiable objects and morphisms in the category of V-categories and apply them to some well-known examples. In the case these characterizations of exponentiable morphisms and objects in the categories (P)Met of (pre)metric spaces and non-expansive maps show in particular that exponentiable metric spaces are exactly the almost convex metric spaces, while exponentiable complete metric spaces are the complete totally convex ones.     

Squarefree P-modules and the cd-index

In this paper, we introduce a new algebraic concept, which we call squarefree P-modules. This concept is inspired from Karu's proof of the non-negativity of the cd-indices of Gorenstein* posets, and supplies a way to study cd-indices from the viewpoint of commutative algebra. Indeed, by using the theory of squarefree P-modules, we give several new algebraic and combinatorial results on CW-posets. First, we define an analogue of the cd-index for any CW-poset and prove its non-negativity when a CW-poset is Cohen–Macaulay. This result proves that the h-vector of the barycentric subdivision of a Cohen–Macaulay regular CW-complex is unimodal. Second, we prove that the Stanley–Reisner ring of the barycentric subdivision of an odd dimensional Cohen–Macaulay polyhedral complex has the weak Lefschetz property. Third, we obtain sharp upper bounds of the cd-indices of Gorenstein* posets for a fixed rank generating 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.     

Power series solutions to Volterra integral equations

Power series type solutions are given for a wide class of linear and q-dimensional nonlinear Volterra equations on R^p. The basic assumption on the kernel K(x, y) is that K(x, xt) has a power series in x. For example, this holds for any analytic kernel.The kernel may be strongly singular, provided certain constants are finite. One and only one such power series solution exists. Its coefficients are given by a simple iterative formula. In many cases this may be solved explicitly. In particular an explicit formula for the resolvent is given.     

The characterization of a class of multivariate MRA and semi-orthogonal parseval frame wavelets

Let A be a d × d expansive matrix with ∣detA∣ = 2. This paper addresses Parseval frame wavelets (PFWs) in the setting of reducing subspaces of L²(R^d). We prove that all semi-orthogonal PFWs (semi-orthogonal MRA PFWs) are precisely the ones with their dimension functions being non-negative integer-valued (0 or 1). We also characterize all MRA PFWs. Some examples are provided.     

Topological sums and products in ZF-set theory

In ZF, i.e., Zermelo-Fraenkel set theory without the axiom of choice, the category Top of topological spaces and continuous maps is well-behaved. In particular, Top has sums (=coproducts) and products. However, it may happen that for families (Xi)_i∈I and (Yi)_i∈I with the property that each Xi is homeomorphic to the corresponding Yi neither their sums _⊕i∈IXi and _⊕i∈IYi nor their products _∏i∈IXi and _∏i∈IYi are homeomorphic. It will be shown that the axiom of choice is not only sufficient but also necessary to rectify this defect.     

Large time behavior to the system of incompressible non-Newtonian fluids in R

In this paper, the authors study the large time behavior for the weak solutions to a class system of the incompressible non-Newtonian fluids in R². It is proved that the weak solutions decay in L² norm at (1+t)^−1/2 and the estimate for the decay rate is sharp in the sense that it coincides with the decay rate of a solution to the heat equation.     

Positive definite solutions of the nonlinear matrix equation X + AXA = Q (q > 0)

In this paper, the nonlinear matrix equation X + A^∗X^qA = Q (q > 0) is investigated. Some necessary and sufficient conditions for existence of Hermitian positive definite solutions of the nonlinear matrix equations are derived. An effective iterative method to obtain the positive definite solution is presented. Some numerical results are given to illustrate the effectiveness of the iterative methods.     

Two local operators and the BLUE

For a given matrix A, a matrix P such that PA = A is said to be a local identity, and such that P²A = PA is said to be a local idempotent. In the paper, some simple properties of such operators are presented. Their relation to the best linear unbiased estimation in the general Gauss-Markov model is also demonstrated.     

Solutions of symmetry‐constrained least‐squares problems

In this paper, two new matrix‐form iterative methods are presented to solve the least‐squares problem: and matrix nearness problem: where matrices and are given; ??₁ and ??₂ are the set of constraint matrices, such as symmetric, skew symmetric, bisymmetric and centrosymmetric matrices sets and S_XY is the solution pair set of the minimum residual problem. These new matrix‐form iterative methods have also faster convergence rate and higher accuracy than the matrix‐form iterative methods proposed by Peng and Peng (Numer. Linear Algebra Appl. 2006; 13 : 473–485) for solving the linear matrix equation AXB+CYD=E. Paige's algorithms, which are based on the bidiagonalization procedure of Golub and Kahan, are used as the framework for deriving these new matrix‐form iterative methods. Some numerical examples illustrate the efficiency of the new matrix‐form iterative methods. Copyright © 2008 John Wiley & Sons, Ltd.     

Extending compact topologies to compact Hausdorff topologies in ZF

Within the framework of Zermelo-Fraenkel set theory ZF, we investigate the set-theoretical strength of the following statements:

(1)

For every family(Ai)_i∈Iof sets there exists a family(Ti)_i∈Isuch that for everyi∈I(Ai,Ti)is a compactT₂space.

(2)

For every family(Ai)_i∈Iof sets there exists a family(Ti)_i∈Isuch that for everyi∈I(Ai,Ti)is a compact, scattered, T₂space.

(3)

For every set X, every compactR₁topology (itsT₀-reflection isT₂) on X can be enlarged to a compactT₂topology.

We show:

(a)

(1) implies every infinite set can be split into two infinite sets.

(b)

(2) iff AC.

(c)

(3) and “there exists a free ultrafilter” iff AC.

We also show that if the topology of certain compact T₁ spaces can be enlarged to a compact T₂ topology then (1) holds true. But in general, compact T₁ topologies do not extend to compact T₂ ones.     

Two, more readily computable equivariant Nielsen numbers I. Nielsen theory for M-ads

In this, the first of two papers outlining a Nielsen theory for “two, more readily computable equivariant numbers”, we define and study two Nielsen type numbers N(f,k;X−{Xν}_ν∈M) and N(f,k;X,{Xν}_ν∈M), where f and k are M-ad maps. While a Nielsen theory of M-ads is of interest in its own right, our main motivation lies in the fact that maps of M-ads accurately mirror one of two fundamental structures of equivariant maps. Being simpler however, M-ad Nielsen numbers are easier to study and to compute than equivariant Nielsen numbers. In the sequel, we show our M-ad numbers can be used to form both upper and lower bounds on their equivariant counterparts.The numbers N(f,k;X−{Xν}_ν∈M) and N(f,k;X,{Xν}_ν∈M), generalize the generalizations to coincidences, of Zhao's Nielsen number on the complement N(f;X−A), respectively Schirmer's relative Nielsen number N(f;X,A). Our generalizations are from the category of pairs, to the category of M-ads. The new numbers are lower bounds for the number of coincidence points of all maps f^′ and k^′ which are homotopic as maps ofM-ads to f, respectively k firstly on the complement of the union of the subspaces Xν in the domain M-ad X, and secondly on all of X. The second number is shown to be greater than or equal to a sum of the first of our numbers. Conditions are given which allow for both equality, and Möbius inversion. Finally we show that the fixed point case of our second number generalizes Schirmer's triad Nielsen number N(f;X₁∪X₂).Our work is very different from what at first sight appears to be similar partial results due to P. Wong. The differences, while in some sense subtle in terms of definition, are profound in terms of commutability. In order to work in a variety of both fixed point and coincidence points contexts, we introduce in this first paper and extend in the second, the concept of an essentiality on a topological category. This allows us to give computational theorems within this diversity. Finally we include an introduction to both papers here.     

Strong linear preservers of rank reverse permutability on triangular matrices

Let F be a field with ∣F∣ > 2 and T_n(F) be the set of all n × n upper triangular matrices, where n ? 2. Let k ? 2 be a given integer. A k-tuple of matrices A₁, …, A_k ∈ T_n(F) is called rank reverse permutable if rank(A₁ A₂ ? A_k) = rank(A_k A_k−1 ? A₁). We characterize the linear maps on T_n(F) that strongly preserve the set of rank reverse permutable matrix k-tuples.     

On the Hermitian positive definite solutions of nonlinear matrix equation X + A∗XA = Q

Nonlinear matrix equation X^s + A∗X^−tA = Q, where A, Q are n × n complex matrices with Q Hermitian positive definite, has widely applied background. In this paper, we consider the Hermitian positive definite solutions of this matrix equation with two cases: s ? 1, 0 < t ? 1 and 0 < s ? 1, t ? 1. We derive necessary conditions and sufficient conditions for the existence of Hermitian positive definite solutions for the matrix equation and obtain some properties of the solutions. We also propose iterative methods for obtaining the extremal Hermitian positive definite solution of the matrix equation. Finally, we give some numerical examples to show the efficiency of the proposed iterative methods.     

On matrix equations and

With the help of the concept of Kronecker map, an explicit solution for the matrix equation X−AXF=C is established. This solution is neatly expressed by a symmetric operator matrix, a controllability matrix and an observability matrix. In addition, the matrix equation is also studied. An explicit solution for this matrix equation is also proposed by means of the real representation of a complex matrix. This solution is neatly expressed by a symmetric operator matrix, two controllability matrices and two observability matrices.