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.     

De Vries Algebras and Compact Regular Frames

By the de Vries theorem, the category DeV of de Vries algebras is dually equivalent to the category KHaus of compact Hausdorff spaces. By the Isbell theorem, the category KRFrm of compact regular frames is dually equivalent to KHaus. The proofs of both theorems employ the axiom of choice. It is a consequence of the de Vries and Isbell theorems that DeV is equivalent to KRFrm. We give a direct proof of this result, which is choice-free. In the absence of the axiom of countable dependent choice (CDC), the category KCRFrm of compact completely regular frames is a proper subcategory of KRFrm. We introduce the category cDeV of completely regular de Vries algebras, which in the absence of (CDC) is a proper subcategory of DeV, and show that cDeV is equivalent to KCRFrm. Finally, we show how the restriction of the equivalence of DeV and KRFrm works in the zero-dimensional and extremally disconnected cases.     

Applications of a theorem by Ky Fan in the theory of graph energy

The energy of a graph G is equal to the sum of the absolute values of the eigenvalues of G, which in turn is equal to the sum of the singular values of the adjacency matrix of G. Let X, Y, and Z be matrices, such that X+Y=Z. The Ky Fan theorem establishes an inequality between the sum of the singular values of Z and the sum of the sum of the singular values of X and Y. This theorem is applied in the theory of graph energy, resulting in several new inequalities, as well as new proofs of some earlier known inequalities.     

Some new extensions of Edelstein-Suzuki-type fixed point theorem to G-metric and G-cone metric spaces

In this paper, we prove some fixed point theorems for generalized contractions in the setting of G-metric spaces. Our results extend a result of Edelstein [M. Edelstein, On fixed and periodic points under contractive mappings, J. London Math. Soc., 37 (1962), 74–79] and a result of Suzuki [T. Suzuki, A new type of fixed point theorem in metric spaces, Nonlinear Anal., 71 (2009), 5313–5317]. We prove, also, a fixed point theorem in the setting of G-cone metric spaces.     

Variational Inequalities: an Elementary Approach

We will deal with the following problem: Let M be an n×n matrix with real entries. Under which conditions the family of inequalities: x∈? ⁿ ;x?0;M·x?0has non–trivial solutions? We will prove that a sufficient condition is given by m_i,j+m_j,i?0 (1?i,j?n); from this result we will derive an elementary proof of the existence theorem for Variational Inequalities in the framework of Monotone Operators.     

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.     

Some axioms for constructive analysis

This note explores the common core of constructive, intuitionistic, recursive and classical analysis from an axiomatic standpoint. In addition to clarifying the relation between Kleene’s and Troelstra’s minimal formal theories of numbers and number-theoretic sequences, we propose some modified choice principles and other function existence axioms which may be of use in reverse constructive analysis. Specifically, we consider the function comprehension principles assumed by the two minimal theories EL and M, introduce an axiom schema CF_d asserting that every decidable property of numbers has a characteristic function, and use it to describe a precise relationship between the minimal theories. We show that the axiom schema AC₀₀ of countable choice can be decomposed into a monotone choice schema AC ₀₀ ^m (which guarantees that every Cauchy sequence has a modulus) and a bounded choice schema BC₀₀. We relate various (classically correct) axiom schemas of continuous choice to versions of the bar and fan theorems, suggest a constructive choice schema AC_1/2,0 (which incidentally guarantees that every continuous function has a modulus of continuity), and observe a constructive equivalence between restricted versions of the fan theorem and correspondingly restricted bounding axioms ${AB_{1/2,0}^{2^{\mathbb{N}}}}$ . We also introduce a version WKL!! of Weak K?nig’s Lemma with uniqueness which is intermediate in strength between WKL and the decidable fan theorem FT_d.     

Between perception and intuition: Learning about infinity

Based on an empirical study, we explore children’s primary and secondary perceptions on infinity. When discussing infinity, children seem to highlight three categories of primary perceptions: processional, topological, and spiritual. Based on their processional perception, children see the set of natural numbers as being infinite and endow Q with a discrete structure by making transfers from N to Q. In a continuous context, children are more likely to mobilize a topological perception. Evidence for a secondary perception of N arises from students’ propensities to develop infinite sequences of natural numbers, and from their ability to prove that N is infinite. Children’s perceptions on infinity change along the school years. In general, the perceptual dominance moves from sequential (processional) to topological across development. However, we found that around 11-13 years old, processional and topological perceptions interfere with each other, while before and after this age they seem to coexist and collaborate, one or the other being specifically activated by the nature of different tasks.     

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].     

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.     

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 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.     

More on the combinatorial invariance of Kazhdan-Lusztig polynomials

We prove that the Kazhdan-Lusztig polynomials are combinatorial invariants for intervals up to length 8 in Coxeter groups of type A and up to length 6 in Coxeter groups of type B and D. As a consequence of our methods, we also obtain a complete classification, up to isomorphism, of Bruhat intervals of length 7 in type A and of length 5 in types B and D, which are not lattices.     

Particular formulae for the Moore-Penrose inverse of a columnwise partitioned matrix

An essential part of Cegielski’s [Obtuse cones and Gram matrices with non-negative inverse, Linear Algebra Appl. 335 (2001) 167-181] considerations of some properties of Gram matrices with nonnegative inverses, which are pointed out to be crucial in constructing obtuse cones, consists in developing some particular formulae for the Moore-Penrose inverse of a columnwise partitioned matrix A = (A₁ : A₂) under the assumption that it is of full column rank. In the present paper, these results are generalized and extended. The generalization consists in weakening the assumption mentioned above to the requirement that the ranges of A₁ and A₂ are disjoint, while the extension consists in introducing the conditions referring to the class of all generalized inverses of A.     

A remark on conservative cocompletions of categories

For a set F of small categories, F-conservative cocompletions of a category are cocompletions preserving all existing colimits of type F. We prove that every category has a free F-conservative cocompletion. However, unless F is trivial, this cocompletion fails in general to be locally small.     

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.     

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.