Mathematics Commons^™

Notes On Interpolation In The Generalized Schur Class. Ii. Nudelman's Problem, Daniel Alpay, T. Constantinescu, A. Dijksma, J. Rovnyak, A. Dijksma

Mathematics, Physics, and Computer Science Faculty Articles and Research

An indefinite generalization of Nudel′man’s problem is used in a systematic approach to interpolation theorems for generalized Schur and Nevanlinna functions with interior and boundary data. Besides results on existence criteria for Pick-Nevanlinna and Carath´eodory-Fej´er interpolation, the method yields new results on generalized interpolation in the sense of Sarason and boundary interpolation, including properties of the finite Hilbert transform relative to weights. The main theorem appeals to the Ball and Helton almost-commutant lifting theorem to provide criteria for the existence of a solution to Nudel′man’s problem.

Mathematisation Of The Science Of Motion And The Birth Of Analytical Mechanics : A Historiographical Note, Marco Panza

MPP Published Research

Usually, one speaks of mathematization of a natural or social science to mean that mathematics comes to be a tool of such a science: the language of mathematics is used to formulate its results, and/or some mathematical techniques is employed to obtain these results.

Preface, Alexander Kurz

Engineering Faculty Articles and Research

No abstract provided.

Modal Predicates And Coequations, Alexander Kurz, Jiří Rosický

Engineering Faculty Articles and Research

We show how coalgebras can be presented by operations and equations. This is a special case of Linton’s approach to algebras over a general base category X, namely where X is taken as the dual of sets. Since the resulting equations generalise coalgebraic coequations to situations without cofree coalgebras, we call them coequations. We prove a general co-Birkhoff theorem describing covarieties of coalgebras by means of coequations. We argue that the resulting coequational logic generalises modal logic.

Definability, Canonical Models, And Compactness For Finitary Coalgebraic Modal Logic, Alexander Kurz, Dirk Pattinson

Engineering Faculty Articles and Research

This paper studies coalgebras from the perspective of the finitary observations that can be made of their behaviours. Based on the terminal sequence, notions of finitary behaviours and finitary predicates are introduced. A category Behω(T) of coalgebras with morphisms preserving finitary behaviours is defined. We then investigate definability and compactness for finitary coalgebraic modal logic, show that the final object in Behω(T) generalises the notion of a canonical model in modal logic, and study the topology induced on a coalgebra by the finitary part of the terminal sequence.

A Note On Interpolation In The Generalized Schur Class. I. Applications Of Realization Theory, Daniel Alpay, T. Constantinescu, A. Dijksma, J. Rovnyak, A. Dijksma

Mathematics, Physics, and Computer Science Faculty Articles and Research

Realization theory for operator colligations on Pontryagin spaces is used to study interpolation and factorization in generalized Schur classes. Several criteria are derived which imply that a given function is almost the restriction of a generalized Schur function. The role of realization theory in coefficient problems is also discussed; a solution of an indefinite Carathéodory-Fejér problem is obtained, as well as a result that relates the number of negative (positive) squares of the reproducing kernels associated with the canonical coisometric, isometric, and unitary realizations of a generalized Schur function to the number of negative (positive) eigenvalues of matrices derived from …

Some Extensions Of Loewner's Theory Of Monotone Operator Functions, Daniel Alpay, Vladimir Bolotnikov, A. Dijksma, J. Rovnyak, A. Dijksma

Mathematics, Physics, and Computer Science Faculty Articles and Research

Several extensions of Loewner’s theory of monotone operator functions are given. These include a theorem on boundary interpolation for matrix-valued functions in the generalized Nevanlinna class. The theory of monotone operator functions is generalized from scalar- to matrix-valued functions of an operator argument. A notion of -monotonicity is introduced and characterized in terms of classical Nevanlinna functions with removable singularities on a real interval. Corresponding results for Stieltjes functions are presented.

Strictly Positive Definite Functions On A Compact Group, Mohamed Allali, Tomasz Przebinda

Mathematics, Physics, and Computer Science Faculty Articles and Research

We recognize a result of Schreiner, concerning strictly positive definite functions on a sphere in an Euclidean space, as a generalization of Bochner's theorem for compact groups.

Modal Rules Are Co-Implications, Alexander Kurz

Engineering Faculty Articles and Research

In [13], it was shown that modal logic for coalgebras dualises—concerning definability— equational logic for algebras. This paper establishes that, similarly, modal rules dualise implications:It is shown that a class of coalgebras is definable by modal rules iff it is closed under H (images) and Σ (disjoint unions). As a corollary the expressive power of rules of infinitary modal logic on Kripke frames is characterised.

La Révolution Scientifique, Les Révolutions, Et L'Histoire Des Sciences. Comment Ernest Coumet Nous A Libérés De L'Héritage D'Alexandre Koyré, Marco Panza

MPP Published Research

Dans son intervention au colloque Koyré (Paris, 1986), Ernest Coumet a suggéré que le terme «révolution scientifique» ne désigne pas chez Koyré un événement historique, mais un idéaltype, au sens de Max Weber. L'auteur discute d'abord cette thèse de Coumet et expose les arguments que ce dermier apporte pour la soutenir. Dans la deuxième partie de l'article, il critique l'usage de la notion de révolution en histoire des sciences, en s'opposant en particulier à la possibilité de distinguer dans les productions des savants une «pensée scientifique» qui serait influencée par la «pensée philosophique» et dont les bouleversements marqueraient l'avènement d'une …

Notes On Coalgebras, Cofibrations And Concurrency, Alexander Kurz, Dirk Pattinson

Engineering Faculty Articles and Research

We consider categories of coalgebras as (co)-fibred over a base category of parameters and analyse categorical constructions in the total category of deterministic and non-deterministic coalgebras.

(Ω, Ξ)-Logic: On The Algebraic Extension Of Coalgebraic Specifications, Rolf Hennicker, Alexander Kurz

Engineering Faculty Articles and Research

We present an extension of standard coalgebraic specification techniques for statebased systems which allows us to integrate constants and n-ary operations in a smooth way and, moreover, leads to a simplification of the coalgebraic structure of the models of a specification. The framework of (Ω,Ξ)-logic can be considered as the result of a translation of concepts of observational logic (cf. [9]) into the coalgebraic world. As a particular outcome we obtain the notion of an (Ω, Ξ)- structure and a sound and complete proof system for (first-order) observational properties of specifications.

Hartogs' Phenomenon For Polyregular Functions And Projective Dimension Of Related Modules Over A Polynomial Ring, W. W. Adams, P. Loustaunau, V. P. Palamadov, Daniele C. Struppa

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper we prove that the projective dimension of Mn = R^4/(An) is 2n -1, where R is the ring of polynomials in 4n variables with complex coefficients and (An) is the module generated by the columns of a 4x4n matrix which arises as the Fourier transform of the matrix of differential operators associated with the regularity condition for a function of n quaternionic variables. As a corollary we show that the sheaf R of regular functions has flabby dimension 2n -1, and we prove a cohomology vanishing theorem for open sets in the space Hn of quaternions. We …

A Theorem On Reproducing Kernel Hilbert Spaces Of Pairs, Daniel Alpay

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper we study reproducing kernel Hilbert and Banach spaces of pairs. These are a generalization of reproducing kernel Krein spaces and, roughly speaking, consist of pairs of Hilbert (or Banach) spaces of functions in duality with respect to a sesquilinear form and admitting a left and right reproducing kernel. We first investigate some properties of these spaces of pairs. It is then proved that to every function K(z, ω) analytic in z and ω* there is a neighborhood of the origin that can be associated with a reproducing kernel Hilbert space of pairs with left reproducing kernel K(z, …

Some Remarks On Reproducing Kernel Krein Spaces, Daniel Alpay

Mathematics, Physics, and Computer Science Faculty Articles and Research

The one-to-one correspondence between positive functions and reproducing kernel Hilbert spaces was extended by L. Schwartz to a (onto, but not one-to-one) correspondence between difference of positive functions and reproducing kernel Krein spaces. After discussing this result, we prove that matrix value function K(z,ω) symmetric and jointly analytic in z and ω in a neighborhood of the origin is the reproducing kernel of a reproducing kernel Krein space. We conclude with an example showing that such a function can be the reproducing kernel of two different Krein spaces.

Dilatations Des Commutants D'Opérateurs Pour Des Espaces De Krein De Fonctions Analytiques, Daniel Alpay

Mathematics, Physics, and Computer Science Faculty Articles and Research

Let K1 and K2 be two Krein spaces of functions analytic in the unit disk and invariant for the left shift operator R0(R0f(z)=(f(z)−f(0))/z), and let A be a linear continuous operator from K1 into K2 whose adjoint commutes with R0. We study dilations of A which preserve this commuting property and such that the Hermitian forms defined by I−AA∗ and I−BB∗ have the same number of negative squares. We thus obtain a version of the commutant lifting theorem in the framework of Krein spaces of analytic functions. To prove this result we suppose that the graph of the operator A∗, …

On A Generalization Of The Corona Problem, Graziano Gentili, Daniele C. Struppa

Mathematics, Physics, and Computer Science Faculty Articles and Research

Let g, f_l,...., f_m EH ^(A). We provide conditions on f_l,...,f_m in order that Ig(z) lIf_i(z)l+...+If_m (z)I, for all z in 4, imply that g, or g², belong to the ideal generated by f_l,....,f_m in H.

Reproducing Kernel Krein Spaces Of Analytic Functions And Inverse Scattering, Daniel Alpay

Mathematics, Physics, and Computer Science Faculty Articles and Research

The purpose of this thesis is to study certain reproducing kernel Krein spaces of analytic functions, the relationships between these spaces and an inverse scattering problem associated with matrix valued functions of bounded type, and an operator model.

Roughly speaking, these results correspond to a generalization of earlier investigations on the applications of de Branges' theory of reproducing kernel Hilbert spaces of analytic functions to the inverse scattering problem for a matrix valued function of the Schur class.

The present work considers first a generalization of a portion of de Branges' theory to Krein spaces. We then formulate a general …

On Some Dependencies Between Functional Forms In Functional Programming Systems, Atanas Radenski

Mathematics, Physics, and Computer Science Faculty Articles and Research

In the present paper, some dependencies between functional forms in functional programming systems are demonstrated. It is shown that any functional form can be expressed by using only the forms composition, condition and construction.