Optimal estimates on rotation number of almost periodic systems

In this paper, we will give some optimal estimates on the rotation number of the linear equation and that of the asymmetric equation: where p(t) and q(t) are almost periodic functions and These estimates are obtained by introducing some kind of new norms in the spaces of almost periodic functions. Supported by the National Natural Science Foundation of China (no. 10325102), TRAPOYT-M.O.E. of China (2001), and the National 973 Project of China (no. G1999075108). Received: April 6, 2004; revised: July 7, 2004     

Invertibility symbol for a Banach algebra generated by two idempotents and a shift

A Banach algebra generated by two idempotentsp, r, identitye and a shiftv which satisfy the conditionspv=vp andrv=v(e–r) is investigated. It is proved that all irreducible representations of algebraA are two- and four-dimensional. The explicit form of these representations is obtained. An Invertibility Symbol is constructed. Some examples are considered.     

Completely positive mappings and mean matrices

Some functions f:R⁺→R⁺ induce mean of positive numbers and the matrix monotonicity gives a possibility for means of positive definite matrices. Moreover, such a function f can define a linear mapping on matrices (which is basic in the constructions of monotone metrics). The present subject is to check the complete positivity of in the case of a few concrete functions f. This problem has been motivated by applications in quantum information.     

On Some Integral Equations in Hilbert Space with an Application to the Theory of Elasticity

A Volterra type integral equation in a Hilbert space with an additional linear operator L and a spectral parameter depending on time is considered. If the parameter does not belong to the spectrum of L unconditional solvability of the considered problem is proved. In the case where the initial value of the parameter coincides with some isolated point of the spectrum of the operator L sufficient conditions for solvability are established. The obtained results are applied to the partial integral equations associated with a contact problem of the theory of elasticity.     

A characterization of Leonard pairs using the notion of a tail

Let V denote a vector space with finite positive dimension. We consider an ordered pair of linear transformations A:V→V and A^∗:V→V that satisfy (i) and (ii) below:

(i)

There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A^∗ is diagonal.

(ii)

There exists a basis for V with respect to which the matrix representing A^∗ is irreducible tridiagonal and the matrix representing A is diagonal.

We call such a pair a Leonard pair on V. In this paper, we characterize the Leonard pairs using the notion of a tail. This notion is borrowed from algebraic graph theory.     

Boundary Behavior of Functions in the de Branges–Rovnyak Spaces

This paper deals with the boundary behavior of functions in the de Branges–Rovnyak spaces. First, we give a criterion for the existence of radial limits for the derivatives of functions in the de Branges–Rovnyak spaces. This criterion generalizes a result of Ahern–Clark. Then we prove that the continuity of all functions in a de Branges–Rovnyak space on an open arc I of the boundary is enough to ensure the analyticity of these functions on I. We use this property in a question related to Bernstein’s inequality. Received: May 10, 2007. Revised: August 8, 2007. Accepted: August 8, 2007.     

The automorphism group of a split metacyclic p-group

This paper finds the order, structure and presentation for the automorphism group of a split metacyclic p-group, where p is odd. Received: 8 February 2006     

Point Evaluations and Polynomial Approximation in the Mean

We construct an example of a Borel subset E of the unit disk such that area measure restricted to E, which we denote A _E , has the property that the set of bounded point evaluations for the polynomials with respect to the L ^t (A _E ) norm varies with t. We further show that E can be chosen to be a simply connected region. In the context of smooth measures, like area measure, examples of this type were unexpected.      

Matrix versions of the Cauchy and Kantorovich inequalities

Summary A version of Cauchy's inequality is obtained which relates two matrices by an inequality in the sense of the Loewner ordering. In that ordering a symmetric idempotent matrix is dominated by the identity matrix and this fact yields a simple proof.A consequence of this matrix Cauchy inequality leads to a matrix version of the Kantorovich inequality, again in the sense of Loewner.     

Collectionwise Hausdorff versus collectionwise normal with respect to compact sets

Collectionwise normal (CWN) and collectionwise Hausdorff (CWH) spaces have played an increasingly important role in topology since the introduction of these concepts by R.H. Bing in 1951 [3]. It has remained an open and frequently raised question as to whether CWH T₃-spaces are CWN with respect to compact sets. Recently, a counterexample requiring the existence of measurable cardinals and having little additional topological structure was constructed by W.G. Fleissner and the author. In this paper, the author gives a simple example in ZFC of a CWH, first countable, perfect T₃-space that is not CWN with respect to compact, metrizable sets, and, under Martin's Axiom, such an example that is also a Moore space. In addition, the author considers the analogous question for strongly collectionwise Hausdorff (SCWH) T₃-spaces and characterizes the existence of SCWH T₃-spaces that are not CWN with respect to compact sets in set-theoretic and box product formulations. The constructions utilized throughout the paper are of a general nature and several apparently new set-theoretic techniques for interchanging ‘points’ and ‘sets’ are introduced.     

Multiplicity Results for Asymptotically Linear Equations, Using the Rotation Number Approach

By using a topological approach and the relation between rotation numbers and weighted eigenvalues, we give some multiplicity results for the boundary value problem u′′ + f(t, u) = 0, u(0) = u(T) = 0, under suitable assumptions on f(t, x)/x at zero and infinity. Solutions are characterized by their nodal properties. Supported by MIUR, GNAMPA and FCT.     

Mock tridiagonal systems

We introduce the notion of a mock tridiagonal system. This is a generalization of a tridiagonal system in which the irreducibility assumption is replaced by a certain nonvanishing condition. We show how mock tridiagonal systems can be used to construct tridiagonal systems that meet certain specifications. This paper is part of our ongoing project to classify the tridiagonal systems up to isomorphism.     

A classification of sharp tridiagonal pairs

Let F denote a field and let V denote a vector space over F with finite positive dimension. We consider a pair of linear transformations A:V→V and A^∗:V→V that satisfy the following conditions: (i) each of A,A^∗ is diagonalizable; (ii) there exists an ordering of the eigenspaces of A such that A^∗V_i⊆V_i-1+V_i+V_i+1 for 0?i?d, where V_-1=0 and V_d+1=0; (iii) there exists an ordering of the eigenspaces of A^∗ such that for 0?i?δ, where and ; (iv) there is no subspace W of V such that AW⊆W, A^∗W⊆W, W≠0, W≠V. We call such a pair a tridiagonal pair on V. It is known that d=δ and for 0?i?d the dimensions of coincide. The pair A,A^∗ is called sharp whenever . It is known that if F is algebraically closed then A,A^∗ is sharp. In this paper we classify up to isomorphism the sharp tridiagonal pairs. As a corollary, we classify up to isomorphism the tridiagonal pairs over an algebraically closed field. We obtain these classifications by proving the μ-conjecture.     

Topo-canonical completions of closure algebras and Heyting algebras

We introduce and investigate topo-canonical completions of closure algebras and Heyting algebras. We develop a duality theory that is an alternative to Esakia’s duality, describe duals of topo-canonical completions in terms of the Salbany and Banaschewski compactifications, and characterize topo-canonical varieties of closure algebras and Heyting algebras. Consequently, we show that ideal completions preserve no identities of Heyting algebras. We also characterize definable classes of topological spaces. Received January 20, 2006; accepted in final form September 12, 2006.     

On the tychonoff functor and w-compactness

The main purpose of this paper is to settle the following problem concerning a product formula for the Tychonoff functor τ, by introducing the notion of w-compact spaces: Characterize a topological space X such that τ(X×Y)=τ(X)×τ(Y) for any topological space Y. We also study the properties of w-compact spaces, and it is proved that, for any family {X_α} of w-compact spaces, the product ΠX_α is also w-compact and τ(ΠX_α)=Πτ(X_α).     

A class of loops with right alternative loop rings

The purpose of this paper is to exhibit a class of loops which have strongly right alternative loop rings that are not alternative..     

Intégrales-fibres pour une formule de Chowla - Selberg

The Chowla-Selberg formula links the class number of negative fundamental quadratic discriminant to values of the Dedekind eta function evaluated over representatives of the class group. It enables the explicit calculation of a large class of elliptic integrals, especially the complete elliptic integrals of the first kind. In this work, we use the integral-fibre theory to offer a new proof of a significant case of this fundamental formula. The method we present here seems susceptible of getting a direct analytic access to a generalization of the explicit instances of the Chowla-Selberg formula in higher dimensions.