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Articles 1 - 9 of 9
Full-Text Articles in Algebra
A Generalized White Noise Space Approach To Stochastic Integration For A Class Of Gaussian Stationary Increment Processes, Daniel Alpay, Alon Kipnis
A Generalized White Noise Space Approach To Stochastic Integration For A Class Of Gaussian Stationary Increment Processes, Daniel Alpay, Alon Kipnis
Mathematics, Physics, and Computer Science Faculty Articles and Research
Given a Gaussian stationary increment processes, we show that a Skorokhod-Hitsuda stochastic integral with respect to this process, which obeys the Wick-Itô calculus rules, can be naturally defined using ideas taken from Hida’s white noise space theory. We use the Bochner-Minlos theorem to associate a probability space to the process, and define the counterpart of the S-transform in this space. We then use this transform to define the stochastic integral and prove an associated Itô formula.
On Discrete Analytic Functions: Products, Rational Functions, And Reproducing Kernels, Daniel Alpay, Palle Jorgensen, Ron Seager, Dan Volok
On Discrete Analytic Functions: Products, Rational Functions, And Reproducing Kernels, Daniel Alpay, Palle Jorgensen, Ron Seager, Dan Volok
Mathematics, Physics, and Computer Science Faculty Articles and Research
We introduce a family of discrete analytic functions, called expandable discrete analytic functions, which includes discrete analytic polynomials, and define two products in this family. The first one is defined in a way similar to the Cauchy-Kovalevskaya product of hyperholomorphic functions, and allows us to define rational discrete analytic functions. To define the second product we need a new space of entire functions which is contractively included in the Fock space. We study in this space some counterparts of Schur analysis.
Non-Commutative Stochastic Distributions And Applications To Linear Systems Theory, Daniel Alpay, Guy Salomon
Non-Commutative Stochastic Distributions And Applications To Linear Systems Theory, Daniel Alpay, Guy Salomon
Mathematics, Physics, and Computer Science Faculty Articles and Research
In this paper, we introduce a non-commutative space of stochastic distributions, which contains the non-commutative white noise space, and forms, together with a natural multiplication, a topological algebra. Special inequalities which hold in this space allow to characterize its invertible elements and to develop an appropriate framework of non-commutative stochastic linear systems.
Residuated Frames With Applications To Decidability, Nikolaos Galatos, Peter Jipsen
Residuated Frames With Applications To Decidability, Nikolaos Galatos, Peter Jipsen
Mathematics, Physics, and Computer Science Faculty Articles and Research
Residuated frames provide relational semantics for substructural logics and are a natural generalization of Kripke frames in intuitionistic and modal logic, and of phase spaces in linear logic. We explore the connection between Gentzen systems and residuated frames and illustrate how frames provide a uniform treatment for semantic proofs of cut-elimination, the finite model property and the finite embeddability property, which imply the decidability of the equational/universal theories of the associated residuated lattice-ordered groupoids. In particular these techniques allow us to prove that the variety of involutive FL-algebras and several related varieties have the finite model property.
Relation Algebras As Expanded Fl-Algebras, Nikolaos Galatos, Peter Jipsen
Relation Algebras As Expanded Fl-Algebras, Nikolaos Galatos, Peter Jipsen
Mathematics, Physics, and Computer Science Faculty Articles and Research
This paper studies generalizations of relation algebras to residuated lattices with a unary De Morgan operation. Several new examples of such algebras are presented, and it is shown that many basic results on relation algebras hold in this wider setting. The variety qRA of quasi relation algebras is defined and shown to be a conservative expansion of involutive FL-algebras. Our main result is that equations in qRA and several of its subvarieties can be decided by a Gentzen system, and that these varieties are generated by their finite members.
Convex Cones Of Generalized Positive Rational Functions And Nevanlinna-Pick Interpolation, Daniel Alpay, Izchak Lewkowicz
Convex Cones Of Generalized Positive Rational Functions And Nevanlinna-Pick Interpolation, Daniel Alpay, Izchak Lewkowicz
Mathematics, Physics, and Computer Science Faculty Articles and Research
Scalar rational functions with a non-negative real part on the right half plane, called positive, are classical in the study of electrical networks, dissipative systems, Nevanlinna-Pick interpolation and other areas. We here study generalized positive functions, i.e with a non-negative real part on the imaginary axis. These functions form a Convex Invertible Cone, cic in short, and we explore two partitionings of this set: (i) into (infinitely many non-invertible) convex cones of functions with prescribed poles and zeroes in the right half plane and (ii) each generalized positive function can be written as a sum of even and odd parts. …
Topological Convolution Algebras, Daniel Alpay, Guy Salomon
Topological Convolution Algebras, Daniel Alpay, Guy Salomon
Mathematics, Physics, and Computer Science Faculty Articles and Research
In this paper we introduce a new family of topological convolution algebras of the form ⋃p∈NL2(S,μp), where S is a Borel semi-group in a locally compact group G, which carries an inequality of the type ∥f∗g∥p≤Ap,q∥f∥q∥g∥p for p>q+d where d pre-assigned, and Ap,q is a constant. We give a sufficient condition on the measures μp for such an inequality to hold. We study the functional calculus and the spectrum of the elements of these algebras, and present two examples, one in the setting of non commutative stochastic distributions, and the other related to Dirichlet series.
Representation Formulas For Hardy Space Functions Through The Cuntz Relations And New Interpolation Problems, Daniel Alpay, Palle Jorgensen, Izchak Lewkowicz, Itzik Marziano
Representation Formulas For Hardy Space Functions Through The Cuntz Relations And New Interpolation Problems, Daniel Alpay, Palle Jorgensen, Izchak Lewkowicz, Itzik Marziano
Mathematics, Physics, and Computer Science Faculty Articles and Research
We introduce connections between the Cuntz relations and the Hardy space H2 of the open unit disk D. We then use them to solve a new kind of multipoint interpolation problem in H2, where for instance, only a linear combination of the values of a function at given points is preassigned, rather than the values at the points themselves.
Pontryagin De Branges-Rovnyak Spaces Of Slice Hyperholomorphic Functions, Daniel Alpay, Fabrizio Colombo, Irene Sabadini
Pontryagin De Branges-Rovnyak Spaces Of Slice Hyperholomorphic Functions, Daniel Alpay, Fabrizio Colombo, Irene Sabadini
Mathematics, Physics, and Computer Science Faculty Articles and Research
We study reproducing kernel Hilbert and Pontryagin spaces of slice hyperholomorphic functions which are analogs of the Hilbert spaces of analytic functions introduced by de Branges and Rovnyak. In the first part of the paper we focus on the case of Hilbert spaces, and introduce in particular a version of the Hardy space. Then we define Blaschke factors and Blaschke products and we consider an interpolation problem. In the second part of the paper we turn to the case of Pontryagin spaces. We first prove some results from the theory of Pontryagin spaces in the quaternionic setting and, in particular, …