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Articles 1 - 30 of 120
Full-Text Articles in Other Mathematics
Generalized Q-Fock Spaces And Structural Identities, Daniel Alpay, Paula Cerejeiras, Uwe Kaehler, Baruch Schneider
Generalized Q-Fock Spaces And Structural Identities, Daniel Alpay, Paula Cerejeiras, Uwe Kaehler, Baruch Schneider
Mathematics, Physics, and Computer Science Faculty Articles and Research
Using đ-calculus we study a family of reproducing kernel Hilbert spaces which interpolate between the Hardy space and the Fock space. We give characterizations of these spaces in terms of classical operators such as integration and backward-shift operators, and their đ-calculus counterparts. Furthermore, these new spaces allow us to study intertwining operators between classic backward-shift operators and the q-Jackson derivative.
On Axially Rational Regular Functions And Schur Analysis In The Clifford-Appell Setting, Daniel Alpay, Fabrizio Colombo, Antonino De Martino, Kamal Diki, Irene Sabadini
On Axially Rational Regular Functions And Schur Analysis In The Clifford-Appell Setting, Daniel Alpay, Fabrizio Colombo, Antonino De Martino, Kamal Diki, Irene Sabadini
Mathematics, Physics, and Computer Science Faculty Articles and Research
In this paper we start the study of Schur analysis for CauchyâFueter regular quaternionic-valued functions, i.e. null solutions of the CauchyâFueter operator in . The novelty of the approach developed in this paper is that we consider axially regular functions, i.e. functions spanned by the so-called Clifford-Appell polynomials. This type of functions arises naturally from two well-known extension results in hypercomplex analysis: the Fueter mapping theorem and the generalized CauchyâKovalevskaya (GCK) extension. These results allow one to obtain axially regular functions starting from analytic functions of one real or complex variable. Precisely, in the Fueter theorem two operators play a âŠ
Pseudo-Differential Operators On The Circle, Bernoulli Polynomials, Roger Gay, Ahmed Sebbar
Pseudo-Differential Operators On The Circle, Bernoulli Polynomials, Roger Gay, Ahmed Sebbar
Mathematics, Physics, and Computer Science Faculty Articles and Research
We show how the classical polylogarithm function Lis (z) and its relatives, the Hurwitz zeta function and the Lerch function are all of a spectral nature, and can explain many properties of the complex powers of the Laplacian on the circle and of the distribution (x +i0)s .We also make a relation with a result of Keiper [Fractional Calculus and its relationship to Riemannâs zeta function, Master of Science, Ohio State University, Mathematics (1975)].
Spacetime Geometry Of Acoustics And Electromagnetism, Lucas Burns, Tatsuya Daniel, Stephon Alexander, Justin Dressel
Spacetime Geometry Of Acoustics And Electromagnetism, Lucas Burns, Tatsuya Daniel, Stephon Alexander, Justin Dressel
Mathematics, Physics, and Computer Science Faculty Articles and Research
Both acoustics and electromagnetism represent measurable fields in terms of dynamical potential fields. Electromagnetic force-fields form a spacetime bivector that is represented by a dynamical energyâmomentum 4-vector potential field. Acoustic pressure and velocity fields form an energyâmomentum density 4-vector field that is represented by a dynamical action scalar potential field. Surprisingly, standard field theory analyses of spin angular momentum based on these traditional potential representations contradict recent experiments, which motivates a careful reassessment of both theories. We analyze extensions of both theories that use the full geometric structure of spacetime to respect essential symmetries enforced by vacuum wave propagation. The âŠ
Superoscillations And Fock Spaces, Daniel Alpay, Fabrizio Colombo, Kamal Diki, Irene Sabadini, Daniele C. Struppa
Superoscillations And Fock Spaces, Daniel Alpay, Fabrizio Colombo, Kamal Diki, Irene Sabadini, Daniele C. Struppa
Mathematics, Physics, and Computer Science Faculty Articles and Research
In this paper we use techniques in Fock spaces theory and compute how the Segal-Bargmann transform acts on special wave functions obtained by multiplying superoscillating sequences with normalized Hermite functions. It turns out that these special wave functions can be constructed also by computing the approximating sequence of the normalized Hermite functions. First, we start by treating the case when a superoscillating sequence is multiplied by the Gaussian function. Then, we extend these calculations to the case of normalized Hermite functions leading to interesting relations with Weyl operators. In particular, we show that the Segal-Bargmann transform maps superoscillating sequences onto âŠ
An Extension Of The ComplexâReal (CâR) Calculus To The Bicomplex Setting, With Applications, Daniel Alpay, Kamal Diki, Mihaela Vajiac
An Extension Of The ComplexâReal (CâR) Calculus To The Bicomplex Setting, With Applications, Daniel Alpay, Kamal Diki, Mihaela Vajiac
Mathematics, Physics, and Computer Science Faculty Articles and Research
In this paper, we extend notions of complex âââ-calculus to the bicomplex setting and compare the bicomplex polyanalytic function theory to the classical complex case. Applications of this theory include two bicomplex least mean square algorithms, which extend classical real and complex least mean square algorithms.
Operators Induced By Certain Hypercomplex Systems, Daniel Alpay, Ilwoo Choo
Operators Induced By Certain Hypercomplex Systems, Daniel Alpay, Ilwoo Choo
Mathematics, Physics, and Computer Science Faculty Articles and Research
In this paper, we consider a family {Ht}tâR of rings of hypercomplex numbers, indexed by the real numbers, which contain both the quaternions and the split-quaternions. We consider natural Hilbert-space representations {(C2, Ït)}tâR of the hypercomplex system {Ht}tâR, and study the realizations Ït(h) of hypercomplex numbers h â Ht, as (2 Ă 2)-matrices acting on C2, for an arbitrarily fixed scale t â R. Algebraic, operator-theoretic, spectral-analytic, and free-probabilistic properties of them are considered.
A HörmanderâFock Space, Daniel Alpay, Fabrizio Colombo, Kamal Diki, Irene Sabadini, Daniele C. Struppa
A HörmanderâFock Space, Daniel Alpay, Fabrizio Colombo, Kamal Diki, Irene Sabadini, Daniele C. Struppa
Mathematics, Physics, and Computer Science Faculty Articles and Research
In a recent paper we used a basic decomposition property of polyanalytic functions of order 2 in one complex variable to characterize solutions of the classical â-problem for given analytic and polyanalytic data. Our approach suggested the study of a special reproducing kernel Hilbert space that we call the Hörmander-Fock space that will be further investigated in this paper. The main properties of this space are encoded in a specific moment sequence denoted by η= (ηn)nâ„0 leading to a special entire function E(z) that is used to express the kernel function of the Hörmander-Fock space. We âŠ
HörmanderâS L2 -Method, â-Problem And Polyanalytic Function Theory In One Complex Variable, Daniel Alpay, Fabrizio Colombo, Kamal Diki, Irene Sabadini, Daniele C. Struppa
HörmanderâS L2 -Method, â-Problem And Polyanalytic Function Theory In One Complex Variable, Daniel Alpay, Fabrizio Colombo, Kamal Diki, Irene Sabadini, Daniele C. Struppa
Mathematics, Physics, and Computer Science Faculty Articles and Research
In this paper we consider the classical â-problem in the case of one complex variable both for analytic and polyanalytic data. We apply the decomposition property of polyanalytic functions in order to construct particular solutions of this problem and obtain new Hörmander type estimates using suitable powers of the Cauchy-Riemann operator. We also compute particular solutions of the â-problem for specific polyanalytic data such as the ItĂŽ complex Hermite polynomials and polyanalytic Fock kernels.
The Structure Of Locally Integral Involutive Po-Monoids And Semirings, José Gil-Férez, Peter Jipsen, Siddhartha Lodhia
The Structure Of Locally Integral Involutive Po-Monoids And Semirings, José Gil-Férez, Peter Jipsen, Siddhartha Lodhia
Mathematics, Physics, and Computer Science Faculty Articles and Research
We show that every locally integral involutive partially ordered monoid (ipo-monoid) A = (A,â©œ, ·, 1,âŒ,â), and in particular every locally integral involutive semiring, decomposes in a unique way into a family {Ap : p â A+} of integral ipo-monoids, which we call its integral components. In the semiring case, the integral components are semirings. Moreover, we show that there is a family of monoid homomorphisms Ί = {Ïpq : Ap â Aq : p â©œ q}, indexed on the positive cone (A+,â©œ), so that the structure of A can be recovered as a glueing R ΊAp of its integral âŠ
Music: Numbers In Motion, Graziano Gentili, Luisa Simonutti, Daniele C. Struppa
Music: Numbers In Motion, Graziano Gentili, Luisa Simonutti, Daniele C. Struppa
Mathematics, Physics, and Computer Science Faculty Articles and Research
Music develops and appears as we allow numbers to acquire a dynamical aspect and create, through their growth, the various keys that permit the richness of the musical texture. This idea was simply adumbrated in Platoâs work, but its importance to his philosophical worldview cannot be underestimated. In this paper we begin by discussing what is probably the first written record of an attempt to create a good temperament and then follow the Pythagoreans approach, whose problems forced musicians, over the next several centuries up to the Renaissance and early modern times, to come up with many different variations.
The Merchant And The Mathematician: Commerce And Accounting, Graziano Gentili, Luisa Simonutti, Daniele C. Struppa
The Merchant And The Mathematician: Commerce And Accounting, Graziano Gentili, Luisa Simonutti, Daniele C. Struppa
Mathematics, Physics, and Computer Science Faculty Articles and Research
In this article we describe the invention of double-entry bookkeeping (or partita doppiaas it was called in Italian), as a fertile intersection between mathematics and early commerce. We focus our attention on this seemingly simple technique that requires only minimal mathematical expertise, but whose discovery is clearly the result of a mathematical way of thinking, in order to make a conceptual point about the role of mathematics as the humus from which disciplines as different as operations research, computer science, and data science have evolved.
On Superoscillations And Supershifts In Several Variables, Yakir Aharonov, Fabrizio Colombo, Andrew N. Jordan, Irene Sabadini, Tomer Shushi, Daniele C. Struppa, Jeff Tollaksen
On Superoscillations And Supershifts In Several Variables, Yakir Aharonov, Fabrizio Colombo, Andrew N. Jordan, Irene Sabadini, Tomer Shushi, Daniele C. Struppa, Jeff Tollaksen
Mathematics, Physics, and Computer Science Faculty Articles and Research
The aim of this paper is to study a class of superoscillatory functions in several variables, removing some restrictions on the functions that we introduced in a previous paper. Since the tools that we used with our approach are not common knowledge we will give detailed proof for the case of two variables. The results proved for superoscillatory functions in several variables can be further extended to supershifts in several variables.
Fock And Hardy Spaces: Clifford Appell Case, Daniel Alpay, Kamal Diki, Irene Sabadini
Fock And Hardy Spaces: Clifford Appell Case, Daniel Alpay, Kamal Diki, Irene Sabadini
Mathematics, Physics, and Computer Science Faculty Articles and Research
In this paper, we study a specific system of CliffordâAppell polynomials and, in particular, their product. Moreover, we introduce a new family of quaternionic reproducing kernel Hilbert spaces in the framework of Fueter regular functions. The construction is based on a general idea which allows us to obtain various function spaces by specifying a suitable sequence of real numbers. We focus on the Fock and Hardy cases in this setting, and we study the action of the Fueter mapping and its range.
Superoscillating Sequences And Supershifts For Families Of Generalized Functions, F. Colombo, I. Sabadini, Daniele Carlo Struppa, A. Yger
Superoscillating Sequences And Supershifts For Families Of Generalized Functions, F. Colombo, I. Sabadini, Daniele Carlo Struppa, A. Yger
Mathematics, Physics, and Computer Science Faculty Articles and Research
We construct a large class of superoscillating sequences, more generally of F-supershifts, where F is a family of smooth functions in (t, x) (resp. distributions in (t, x), or hyperfunctions in x depending on the parameter t) indexed by λ â R. The frame in which we introduce such families is that of the evolution through Schrödinger equation (iâ/âtâH (x))(Ï) = 0 (H (x) = â(â2/âx2)/2+V (x)), V being a suitable potential). If F = {(t, x) â Ïλ(t, x) ; λ â R}, where Ïλ is evolved from the initial datum x â eiλx , F-supershifts will be of âŠ
Algorithmic Correspondence For Relevance Logics, Bunched Implication Logics, And Relation Algebras Via An Implementation Of The Algorithm Pearl, Willem Conradie, Valentin Goranko, Peter Jipsen
Algorithmic Correspondence For Relevance Logics, Bunched Implication Logics, And Relation Algebras Via An Implementation Of The Algorithm Pearl, Willem Conradie, Valentin Goranko, Peter Jipsen
Mathematics, Physics, and Computer Science Faculty Articles and Research
The non-deterministic algorithmic procedure PEARL (acronym for âPropositional variables Elimination Algorithm for Relevance Logicâ) has been recently developed for computing first-order equivalents of formulas of the language of relevance logics LR in terms of the standard Routley-Meyer relational semantics. It succeeds on a large class of axioms of relevance logics, including all so called inductive formulas. In the present work we re-interpret PEARL from an algebraic perspective, with its rewrite rules seen as manipulating quasi-inequalities interpreted over Urquhartâs relevant algebras, and report on its recent Python implementation. We also show that all formulae on which PEARL succeeds are canonical, i.e., âŠ
Unary-Determined Distributive â -Magmas And Bunched Implication Algebras, Natanael Alpay, Peter Jipsen, Melissa Sugimoto
Unary-Determined Distributive â -Magmas And Bunched Implication Algebras, Natanael Alpay, Peter Jipsen, Melissa Sugimoto
Mathematics, Physics, and Computer Science Faculty Articles and Research
A distributive lattice-ordered magma (dâ-magma) (A,â§,âš,â ) is a distributive lattice with a binary operation â that preserves joins in both arguments, and when â is associative then (A,âš,â ) is an idempotent semiring. A dâ-magma with a top †is unary-determined if xâ y=(xâ â€â§y)âš(xâ§â€â y). These algebras are term-equivalent to a subvariety of distributive lattices with †and two join-preserving unary operations p, q. We obtain simple conditions on p, q such that xâ y=(pxâ§y)âš(xâ§qy) is associative, commutative, idempotent and/or has an identity element. This generalizes previous results on the structure of doubly idempotent semirings and, in the case when the âŠ
Generalized Grassmann Algebras And Applications To Stochastic Processes, Daniel Alpay, Paula Cerejeiras, Uwe Kaehler
Generalized Grassmann Algebras And Applications To Stochastic Processes, Daniel Alpay, Paula Cerejeiras, Uwe Kaehler
Mathematics, Physics, and Computer Science Faculty Articles and Research
In this paper, we present the groundwork for an ItĂŽ/Malliavin stochastic calculus and Hida's white noise analysis in the context of a supersymmetry with Z3-graded algebras. To this end, we establish a ternary Fock space and the corresponding strong algebra of stochastic distributions and present its application in the study of stochastic processes in this context.
On The Application Of Principal Component Analysis To Classification Problems, Jianwei Zheng, Cyril Rakovski
On The Application Of Principal Component Analysis To Classification Problems, Jianwei Zheng, Cyril Rakovski
Mathematics, Physics, and Computer Science Faculty Articles and Research
Principal Component Analysis (PCA) is a commonly used technique that uses the correlation structure of the original variables to reduce the dimensionality of the data. This reduction is achieved by considering only the first few principal components for a subsequent analysis. The usual inclusion criterion is defined by the proportion of the total variance of the principal components exceeding a predetermined threshold. We show that in certain classification problems, even extremely high inclusion threshold can negatively impact the classification accuracy. The omission of small variance principal components can severely diminish the performance of the models. We noticed this phenomenon in âŠ
Trilinear Smoothing Inequalities And A Variant Of The Triangular Hilbert Transform, Michael Christ, Polona Durcik, Joris Roos
Trilinear Smoothing Inequalities And A Variant Of The Triangular Hilbert Transform, Michael Christ, Polona Durcik, Joris Roos
Mathematics, Physics, and Computer Science Faculty Articles and Research
Lebesgue space inequalities are proved for a variant of the triangular Hilbert transform involving curvature. The analysis relies on a crucial trilinear smoothing inequality developed herein, and on bounds for an anisotropic variant of the twisted paraproduct.
The trilinear smoothing inequality also leads to Lebesgue space bounds for a corresponding maximal function and a quantitative nonlinear Roth-type theorem concerning patterns in the Euclidean plane.
On A Polyanalytic Approach To Noncommutative De BrangesâRovnyak Spaces And Schur Analysis, Daniel Alpay, Fabrizio Colombo, Kamal Diki, Irene Sabadini
On A Polyanalytic Approach To Noncommutative De BrangesâRovnyak Spaces And Schur Analysis, Daniel Alpay, Fabrizio Colombo, Kamal Diki, Irene Sabadini
Mathematics, Physics, and Computer Science Faculty Articles and Research
In this paper we begin the study of Schur analysis and of de BrangesâRovnyak spaces in the framework of Fueter hyperholomorphic functions. The difference with other approaches is that we consider the class of functions spanned by Appell-like polynomials. This approach is very efficient from various points of view, for example in operator theory, and allows us to make connections with the recently developed theory of slice polyanalytic functions. We tackle a number of problems: we describe a Hardy space, Schur multipliers and related results. We also discuss Blaschke functions, Herglotz multipliers and their associated kernels and Hilbert spaces. Finally, âŠ
A New Method To Generate Superoscillating Functions And Supershifts, Yakir Aharonov, Fabrizio Colombo, Irene Sabadini, Tomer Shushi, Daniele C. Struppa, Jeff Tollaksen
A New Method To Generate Superoscillating Functions And Supershifts, Yakir Aharonov, Fabrizio Colombo, Irene Sabadini, Tomer Shushi, Daniele C. Struppa, Jeff Tollaksen
Mathematics, Physics, and Computer Science Faculty Articles and Research
Superoscillations are band-limited functions that can oscillate faster than their fastest Fourier component. These functions (or sequences) appear in weak values in quantum mechanics and in many fields of science and technology such as optics, signal processing and antenna theory. In this paper, we introduce a new method to generate superoscillatory functions that allows us to construct explicitly a very large class of superoscillatory functions.
Krein Reproducing Kernel Modules In Clifford Analysis, Daniel Alpay, Paula Cerejeiras, Uwe KĂ€hler
Krein Reproducing Kernel Modules In Clifford Analysis, Daniel Alpay, Paula Cerejeiras, Uwe KĂ€hler
Mathematics, Physics, and Computer Science Faculty Articles and Research
Classic hypercomplex analysis is intimately linked with elliptic operators, such as the Laplacian or the Dirac operator, and positive quadratic forms. But there are many applications like the crystallographic X-ray transform or the ultrahyperbolic Dirac operator which are closely connected with indefinite quadratic forms. Although appearing in many papers in such cases Hilbert modules are not the right choice as function spaces since they do not reflect the induced geometry. In this paper we are going to show that Clifford-Krein modules are naturally appearing in this context. Even taking into account the difficulties, e.g., the existence of different inner products âŠ
Superoscillations And Analytic Extension In Schur Analysis, Daniel Alpay, Fabrizio Colombo, Irene Sabadini
Superoscillations And Analytic Extension In Schur Analysis, Daniel Alpay, Fabrizio Colombo, Irene Sabadini
Mathematics, Physics, and Computer Science Faculty Articles and Research
We give applications of the theory of superoscillations to various questions, namely extension of positive definite functions, interpolation of polynomials and also of Rfunctions; we also discuss possible applications to signal theory and prediction theory of stationary stochastic processes. In all cases, we give a constructive procedure, by way of a limiting process, to get the required results.
Total Differentiability And Monogenicity For Functions In Algebras Of Order 4, I. Sabadini, Daniele C. Struppa
Total Differentiability And Monogenicity For Functions In Algebras Of Order 4, I. Sabadini, Daniele C. Struppa
Mathematics, Physics, and Computer Science Faculty Articles and Research
In this paper we discuss some notions of analyticity in associative algebras with unit. We also recall some basic tool in algebraic analysis and we use them to study the properties of analytic functions in two algebras of dimension four that played a relevant role in some work of the Italian school, but that have never been fully investigated.
Boxes, Extended Boxes And Sets Of Positive Upper Density In The Euclidean Space, Polona Durcik, Vjekoslav KovaÄ
Boxes, Extended Boxes And Sets Of Positive Upper Density In The Euclidean Space, Polona Durcik, Vjekoslav KovaÄ
Mathematics, Physics, and Computer Science Faculty Articles and Research
We prove that sets with positive upper Banach density in sufficiently large dimensions contain congruent copies of all sufficiently large dilates of three specific higher-dimensional patterns. These patterns are: 2n vertices of a fixed n-dimensional rectangular box, the same vertices extended with n points completing three-term arithmetic progressions, and the same vertices extended with n points completing three-point corners. Our results provide common generalizations of several Euclidean density theorems from the literature.
Green's Function For The Schrodinger Equation With A Generalized Point Interaction And Stability Of Superoscillations, Yakir Aharonov, Jussi Behrndt, Fabrizio Colombo, Peter Schlosser
Green's Function For The Schrodinger Equation With A Generalized Point Interaction And Stability Of Superoscillations, Yakir Aharonov, Jussi Behrndt, Fabrizio Colombo, Peter Schlosser
Mathematics, Physics, and Computer Science Faculty Articles and Research
In this paper we study the time dependent Schrödinger equation with all possible self-adjoint singular interactions located at the origin, which include the ÎŽ and ÎŽ'-potentials as well as boundary conditions of Dirichlet, Neumann, and Robin type as particular cases. We derive an explicit representation of the time dependent Green's function and give a mathematical rigorous meaning to the corresponding integral for holomorphic initial conditions, using Fresnel integrals. Superoscillatory functions appear in the context of weak measurements in quantum mechanics and are naturally treated as holomorphic entire functions. As an application of the Green's function we study the stability âŠ
Quantitatively Hyper-Positive Real Functions, Daniel Alpay, Izchak Lewkowicz
Quantitatively Hyper-Positive Real Functions, Daniel Alpay, Izchak Lewkowicz
Mathematics, Physics, and Computer Science Faculty Articles and Research
Hyper-positive real, matrix-valued, rational functions are associated with absolute stability (the Lurie problem). Here, quantitative subsets of Hyper-positive functions, related through nested inclusions, are introduced. Structurally, this family of functions turns out to be matrix-convex and closed under inversion.
A state-space characterization of these functions through a corresponding Kalman-Yakubovich-Popov Lemma, is given. Technically, the classical Linear Matrix Inclusions, associated with passive systems, are here substituted by Quadratic Matrix Inclusions.
De Branges Spaces On Compact Riemann Surfaces And A Beurling-Lax Type Theorem, Daniel Alpay, Ariel Pinhas, Victor Vinnikov
De Branges Spaces On Compact Riemann Surfaces And A Beurling-Lax Type Theorem, Daniel Alpay, Ariel Pinhas, Victor Vinnikov
Mathematics, Physics, and Computer Science Faculty Articles and Research
Using the notion of commutative operator vessels, this work investigates de Branges-Rovnyak spaces whose elements are sections of a line bundle of multiplicative half-order differentials on a compact real Riemann surface. As a special case, we obtain a Beurling-Lax type theorem in the setting of the corresponding Hardy space on a finite bordered Riemann surface.
Exact And Strongly Exact Filters, M. A. Moshier, A. Pultr, A. L. Suarez
Exact And Strongly Exact Filters, M. A. Moshier, A. Pultr, A. L. Suarez
Mathematics, Physics, and Computer Science Faculty Articles and Research
A meet in a frame is exact if it join-distributes with every element, it is strongly exact if it is preserved by every frame homomorphism. Hence, finite meets are (strongly) exact which leads to the concept of an exact resp. strongly exact filter, a filter closed under exact resp. strongly exact meets. It is known that the exact filters constitute a frame FiltE(L) somewhat surprisingly isomorphic to the frame of joins of closed sublocales. In this paper we present a characteristic of the coframe of meets of open sublocales as the dual to the frame of strongly exact filters FiltsE(L).