Algebra Commons^™

Bitopological Duality For Distributive Lattices And Heyting Algebras, Guram Bezhanishvili, Nick Bezhanishvili, David Gabelaia, Alexander Kurz

Engineering Faculty Articles and Research

We introduce pairwise Stone spaces as a natural bitopological generalization of Stone spaces—the duals of Boolean algebras—and show that they are exactly the bitopological duals of bounded distributive lattices. The category PStone of pairwise Stone spaces is isomorphic to the category Spec of spectral spaces and to the category Pries of Priestley spaces. In fact, the isomorphism of Spec and Pries is most naturally seen through PStone by first establishing that Pries is isomorphic to PStone, and then showing that PStone is isomorphic to Spec. We provide the bitopological and spectral descriptions of many algebraic concepts important for the study …

Equational Coalgebraic Logic, Alexander Kurz, Raul Leal

Engineering Faculty Articles and Research

Coalgebra develops a general theory of transition systems, parametric in a functor T; the functor T specifies the possible one-step behaviours of the system. A fundamental question in this area is how to obtain, for an arbitrary functor T, a logic for T-coalgebras. We compare two existing proposals, Moss’s coalgebraic logic and the logic of all predicate liftings, by providing one-step translations between them, extending the results in [21] by making systematic use of Stone duality. Our main contribution then is a novel coalgebraic logic, which can be seen as an equational axiomatization of Moss’s logic. The three logics are …

Transformée En Échelle De Signaux Stationnaires, Daniel Alpay, Mamadou Mboup

Mathematics, Physics, and Computer Science Faculty Articles and Research

Using the scale transform of a discrete time signal we define a new family of linear systems. We focus on a particular case related to function theory in the bidisk.

Generalized Q-Functions And Dirichlet-To-Neumann Maps For Elliptic Differential Operators, Daniel Alpay, Jussi Behrndt

Mathematics, Physics, and Computer Science Faculty Articles and Research

The classical concept of Q-functions associated to symmetric and selfadjoint operators due to M.G. Krein and H. Langer is extended in such a way that the Dirichlet-to-Neumann map in the theory of elliptic differential equations can be interpreted as a generalized Q-function. For couplings of uniformly elliptic second order differential expression on bounded and unbounded domains explicit Krein type formulas for the difference of the resolvents and trace formulas in an H2-framework are obtained.

The Schur Transformation For Nevanlinna Functions: Operator Representations, Resolvent Matrices, And Orthogonal Polynomials, Daniel Alpay, A. Dijksma, H. Langer

Mathematics, Physics, and Computer Science Faculty Articles and Research

A Nevanlinna function is a function which is analytic in the open upper half plane and has a non-negative imaginary part there. In this paper we study a fractional linear transformation for a Nevanlinna function n with a suitable asymptotic expansion at ∞, that is an analogue of the Schur transformation for contractive analytic functions in the unit disc. Applying the transformation p times we find a Nevanlinna function np which is a fractional linear transformation of the given function n. The main results concern the effect of this transformation to the realizations of n and np, by which we …

Krein Systems, Daniel Alpay, I. Gohberg, M. A. Kaashoek, L. Lerer, A. Sakhnovich

Mathematics, Physics, and Computer Science Faculty Articles and Research

In the present paper we extend results of M.G. Krein associated to the spectral problem for Krein systems to systems with matrix valued accelerants with a possible jump discontinuity at the origin. Explicit formulas for the accelerant are given in terms of the matrizant of the system in question. Recent developments in the theory of continuous analogs of the resultant operator play an essential role.

Functorial Coalgebraic Logic: The Case Of Many-Sorted Varieties, Alexander Kurz, Daniela Petrişan

Engineering Faculty Articles and Research

Following earlier work, a modal logic for T-coalgebras is a functor L on a suitable variety. Syntax and proof system of the logic are given by presentations of the functor. This paper makes two contributions. First, a previous result characterizing those functors that have presentations is generalized from endofunctors on one-sorted varieties to functors between many-sorted varieties. This yields an equational logic for the presheaf semantics of higher-order abstract syntax. As another application, we show how the move to functors between many-sorted varieties allows to modularly combine syntax and proof systems of different logics. Second, we show how to associate …

A Characterization Of Schur Multipliers Between Character-Automorphic Hardy Spaces, Daniel Alpay, M. Mboup

Mathematics, Physics, and Computer Science Faculty Articles and Research

We give a new characterization of character-automorphic Hardy spaces of order 2 and of their contractive multipliers in terms of de Branges Rovnyak spaces. Keys tools in our arguments are analytic extension and a factorization result for matrix-valued analytic functions due to Leech.

On The Reproducing Kernel Hilbert Spaces Associated With The Fractional And Bi-Fractional Brownian Motions, Daniel Alpay, David Levanony

Mathematics, Physics, and Computer Science Faculty Articles and Research

We present decompositions of various positive kernels as integrals or sums of positive kernels. Within this framework we study the reproducing kernel Hilbert spaces associated with the fractional and bi-fractional Brownian motions. As a tool, we define a new function of two complex variables, which is a natural generalization of the classical Gamma function for the setting we consider.

Rational Functions Associated To The White Noise Space And Related Topics, Daniel Alpay, David Levanony

Mathematics, Physics, and Computer Science Faculty Articles and Research

Motivated by the hyper-holomorphic case we introduce and study rational functions in the setting of Hida’s white noise space. The Fueter polynomials are replaced by a basis computed in terms of the Hermite functions, and the Cauchy-Kovalevskaya product is replaced by the Wick product.

Carathéodory Functions In The Banach Space Setting, Daniel Alpay, Olga Timoshenko, Dan Volok

Mathematics, Physics, and Computer Science Faculty Articles and Research

We prove representation theorems for Carathéodory functions in the setting of Banach spaces.

Pi-Calculus In Logical Form, Marcello M. Bonsangue, Alexander Kurz

Engineering Faculty Articles and Research

Abramsky’s logical formulation of domain theory is extended to encompass the domain theoretic model for picalculus processes of Stark and of Fiore, Moggi and Sangiorgi. This is done by defining a logical counterpart of categorical constructions including dynamic name allocation and name exponentiation, and showing that they are dual to standard constructs in functor categories. We show that initial algebras of functors defined in terms of these constructs give rise to a logic that is sound, complete, and characterises bisimilarity. The approach is modular, and we apply it to derive a logical formulation of pi-calculus. The resulting logic is a …

François Viète, Between Analysis And Cryptanalysis, Marco Panza

MPP Published Research

François Viète is considered the father both of modern algebra and of modern cryptanalysis. The paper outlines Viète's major contributions in these two mathematical fields and argues that, despite an obvious parallel between them, there is an essential difference. Viète's 'new algebra' relies on his reform of the classical method of analysis and synthesis, in particular on a new conception of analysis and the introduction of a new formalism. The procedures he suggests to decrypt coded messages are particular forms of analysis based on the use of formal methods. However, Viète's algebraic analysis is not an analysis in the same …

Matrix-J-Unitary Non-Commutative Rational Formal Power Series, Daniel Alpay, D. S. Kalyuzhnyi-Verbovetzkii

Mathematics, Physics, and Computer Science Faculty Articles and Research

Formal power series in N non-commuting indeterminates can be considered as a counterpart of functions of one variable holomorphic at 0, and some of their properties are described in terms of coefficients. However, really fruitful analysis begins when one considers for them evaluations on N-tuples of n × n matrices (with n = 1, 2, . . .) or operators on an infinite-dimensional separable Hilbert space. Moreover, such evaluations appear in control, optimization and stabilization problems of modern system engineering.

In this paper, a theory of realization and minimal factorization of rational matrix-valued functions which are J-unitary on the imaginary …

Coalgebras And Their Logics, Alexander Kurz

Engineering Faculty Articles and Research

"Transition systems pervade much of computer science. This article outlines the beginnings of a general theory of specification languages for transition systems. More specifically, transition systems are generalised to coalgebras. Specification languages together with their proof systems, in the following called (logical or modal) calculi, are presented by the associated classes of algebras (e.g., classical propositional logic by Boolean algebras). Stone duality will be used to relate the logics and their coalgebraic semantics."

Rational Hyperholomorphic Functions In R4, Daniel Alpay, Michael Shapiro, Dan Volok

Mathematics, Physics, and Computer Science Faculty Articles and Research

We introduce the notion of rationality for hyperholomorphic functions (functions in the kernel of the Cauchy-Fueter operator). Following the case of one complex variable, we give three equivalent definitions: the first in terms of Cauchy-Kovalevskaya quotients of polynomials, the second in terms of realizations and the third in terms of backward-shift invariance. Also introduced and studied are the counterparts of the Arveson space and Blaschke factors.

Point Evaluation And Hardy Space On A Homogeneous Tree, Daniel Alpay, Dan Volok

Mathematics, Physics, and Computer Science Faculty Articles and Research

We consider stationary multiscale systems as defined by Basseville, Benveniste, Nikoukhah and Willsky. We show that there are deep analogies with the discrete time non stationary setting as developed by the first author, Dewilde and Dym. Following these analogies we define a point evaluation with values in a C*–algebra and the corresponding “Hardy space” in which Cauchy’s formula holds. This point evaluation is used to define in this context the counterpart of classical notions such as Blaschke factors.

Weak Factorizations, Fractions And Homotopies, Alexander Kurz, Jiří Rosický

Engineering Faculty Articles and Research

We show that the homotopy category can be assigned to any category equipped with a weak factorization system. A classical example of this construction is the stable category of modules. We discuss a connection with the open map approach to bisimulations proposed by Joyal, Nielsen and Winskel.

Coalgebras And Modal Expansions Of Logics, Alexander Kurz, Alessandra Palmigiano

Engineering Faculty Articles and Research

In this paper we construct a setting in which the question of when a logic supports a classical modal expansion can be made precise. Given a fully selfextensional logic S, we find sufficient conditions under which the Vietoris endofunctor V on S-referential algebras can be defined and we propose to define the modal expansions of S as the logic that arises from the V-coalgebras. As an example, we also show how the Vietoris endofunctor on referential algebras extends the Vietoris endofunctor on Stone spaces. From another point of view, we examine when a category of ‘spaces’ (X,A), ie sets X …

Algebraic Semantics For Coalgebraic Logics, Clemens Kupke, Alexander Kurz, Dirk Pattinson

Engineering Faculty Articles and Research

With coalgebras usually being defined in terms of an endofunctor T on sets, this paper shows that modal logics for T-coalgebras can be naturally described as functors L on boolean algebras. Building on this idea, we study soundness, completeness and expressiveness of coalgebraic logics from the perspective of duality theory. That is, given a logic L for coalgebras of an endofunctor T, we construct an endofunctor L such that L-algebras provide a sound and complete (algebraic) semantics of the logic. We show that if L is dual to T, then soundness and completeness of the algebraic semantics immediately yield the …

Preface, Thomas Hildebrandt, Alexander Kurz

Engineering Faculty Articles and Research

No abstract provided.

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.

Stone Coalgebras, Clemens Kupke, Alexander Kurz, Yde Venema

Engineering Faculty Articles and Research

In this paper we argue that the category of Stone spaces forms an interesting base category for coalgebras, in particular, if one considers the Vietoris functor as an analogue to the power set functor. We prove that the so-called descriptive general frames, which play a fundamental role in the semantics of modal logics, can be seen as Stone coalgebras in a natural way. This yields a duality between the category of modal algebras and that of coalgebras over the Vietoris functor. Building on this idea, we introduce the notion of a Vietoris polynomial functor over the category of Stone spaces. …

Preface, Alexander Kurz

Engineering Faculty Articles and Research

No abstract provided.

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.

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.

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.

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.