Mathematics Commons^™

The Structure Of Generalized Bi-Algebras And Weakening Relation Algebras, Nikolaos Galatos, Peter Jipsen

Mathematics, Physics, and Computer Science Faculty Articles and Research

Generalized bunched implication algebras (GBI-algebras) are defined as residuated lattices with a Heyting implication, and are positioned between Boolean algebras with operators and lattices with operators. We characterize congruences on GBI-algebras by filters that are closed under Gumm–Ursini terms, and for involutive GBI-algebras these terms simplify to a dual version of the congruence term for relation algebras together with two more terms. We prove that representable weakening relation algebras form a variety of cyclic involutive GBI-algebras, denoted by RWkRA, containing the variety of representable relation algebras. We describe a double-division conucleus construction on residuated lattices and on (cyclic involutive) GBI-algebras …

Realizations Of Holomorphic And Slice Hyperholomorphic Functions: The Krein Space Case, Daniel Alpay, Fabrizio Colombo, Irene Sabadini

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper we treat realization results for operator-valued functions which are analytic in the complex sense or slice hyperholomorphic over the quaternions. In the complex setting, we prove a realization theorem for an operator-valued function analytic in a neighborhood of the origin with a coisometric state space operator thus generalizing an analogous result in the unitary case. A main difference with previous works is the use of reproducing kernel Krein spaces. We then prove the counterpart of this result in the quaternionic setting. The present work is the first paper which presents a realization theorem with a state space …

Acoustic Versus Electromagnetic Field Theory: Scalar, Vector, Spinor Representations And The Emergence Of Acoustic Spin, Lucas Burns, Konstantin Y. Bliokh, Franco Nori, Justin Dressel

Mathematics, Physics, and Computer Science Faculty Articles and Research

We construct a novel Lagrangian representation of acoustic field theory that describes the local vector properties of longitudinal (curl-free) acoustic fields. In particular, this approach accounts for the recently-discovered nonzero spin angular momentum density in inhomogeneous sound fields in fluids or gases. The traditional acoustic Lagrangian representation with a scalar potential is unable to describe such vector properties of acoustic fields adequately, which are however observable via local radiation forces and torques on small probe particles. By introducing a displacement vector potential analogous to the electromagnetic vector potential, we derive the appropriate canonical momentum and spin densities as conserved Noether …

Structure Theorems For Idempotent Residuated Lattices, José Gil-Férez, Peter Jipsen, George Metcalfe

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper we study structural properties of residuated lattices that are idempotent as monoids. We provide descriptions of the totally ordered members of this class and obtain counting theorems for the number of finite algebras in various subclasses. We also establish the finite embeddability property for certain varieties generated by classes of residuated lattices that are conservative in the sense that monoid multiplication always yields one of its arguments. We then make use of a more symmetric version of Raftery’s characterization theorem for totally ordered commutative idempotent residuated lattices to prove that the variety generated by this class has …

Stochastic Wiener Filter In The White Noise Space, Daniel Alpay, Ariel Pinhas

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper we introduce a new approach to the study of filtering theory by allowing the system's parameters to have a random character. We use Hida's white noise space theory to give an alternative characterization and a proper generalization to the Wiener filter over a suitable space of stochastic distributions introduced by Kondratiev. The main idea throughout this paper is to use the nuclearity of this space in order to view the random variables as bounded multiplication operators (with respect to the Wick product) between Hilbert spaces of stochastic distributions. This allows us to use operator theory tools and …

Semi De Morgan Logic Properly Displayed, Giuseppe Greco, Fei Qin, M. Andrew Moshier, Alessandra Palmigiano

Mathematics, Physics, and Computer Science Faculty Articles and Research

In the present paper, we endow semi De Morgan logic and a family of its axiomatic extensions with proper multi-type display calculi which are sound, complete, conservative, and enjoy cut elimination and subformula property. Our proposal builds on an algebraic analysis of the variety of semi De Morgan algebras, and applies the guidelines of the multi-type methodology in the design of display calculi.

A General Setting For Functions Of Fueter Variables: Differentiability, Rational Functions, Fock Module And Related Topics, Daniel Alpay, Ismael L. Paiva, Daniele C. Struppa

Mathematics, Physics, and Computer Science Faculty Articles and Research

We develop some aspects of the theory of hyperholomorphic functions whose values are taken in a Banach algebra over a field—assumed to be the real or the complex numbers—and which contains the field. Notably, we consider Fueter expansions, Gleason’s problem, the theory of hyperholomorphic rational functions, modules of Fueter series, and related problems. Such a framework includes many familiar algebras as particular cases. The quaternions, the split quaternions, the Clifford algebras, the ternary algebra, and the Grassmann algebra are a few examples of them.

Pseudo-Contractions, Rigidity, Fixed Points And Related Questions, Daniel Alpay, Vladimir Bolotnikov, David Shoikhet

Mathematics, Physics, and Computer Science Faculty Articles and Research

The class of holomorphic self-mappings of the open unit disk (which are contractions with respect to the Poincaré metric) admits a natural extension to the class of holomorphic pseudo-contractions. In this paper, we study various inequalities involving the values of derivatives of holomorphic pseudo-contractions at fixed points (particularly, at the Denjoy–Wolff fixed point).

The Infinite Is The Chasm In Which Our Thoughts Are Lost: Reflections On Sophie Germain's Essays, Adam Glesser, Bogdan D. Suceavă, Mihaela Vajiac

Mathematics, Physics, and Computer Science Faculty Articles and Research

"Sophie Germain (1776–1831) is quite well-known to the mathematical community for her contributions to number theory [17] and elasticity theory (e.g., see [2, 5]). On the other hand, there have been few attempts to understand Sophie Germain as an intellectual of her time, as an independent thinker outside of academia, and as a female mathematician in France, facing the prejudice of the time of the First Empire and of the Bourbon Restoration, while pursuing her thoughts and interests and writing on them. Sophie Germain had to face a double challenge: the mathematical difficulty of the problems she approached and the …

Distributive Laws In Residuated Binars, Wesley Fussner, Peter Jipsen

Mathematics, Physics, and Computer Science Faculty Articles and Research

In residuated binars there are six non-obvious distributivity identities of ⋅,/,∖ over ∧,∨. We show that in residuated binars with distributive lattice reducts there are some dependencies among these identities; specifically, there are six pairs of identities that imply another one of these identities, and we provide counterexamples to show that no other dependencies exist among these.

The Fock Space As A De Branges–Rovnyak Space, Daniel Alpay, Fabrizio Colombo, Irene Sabadini

Mathematics, Physics, and Computer Science Faculty Articles and Research

We show that de Branges–Rovnyak spaces include as special cases a number of spaces, such as the Hardy space, the Fock space, the Hardy–Sobolev space and the Dirichlet space. We present a general framework in which all these spaces can be obtained by specializing a sequence that appears in the construction. We show how to exploit this approach to solve interpolation problems in the Fock space.

A Panorama On Superoscillations, Fabrizio Colombo, Irene Sabadini, Daniele Carlo Struppa

Mathematics, Physics, and Computer Science Faculty Articles and Research

Purpose of this note is to give an overview on superoscillating sequences and some of their properties. We discuss their persistence in time under Schrödinger equation, we propose various classes of superoscillating functions and we also briefly mention how they can be used to approximate some generalized functions.

Realization Of Tensor Product And Of Tensor Factorization Of Rational Functions, Daniel Alpay, Izchak Lewkowicz

Mathematics, Physics, and Computer Science Faculty Articles and Research

We study the state space realization of a tensor product of a pair of rational functions. At the expense of “inflating” the dimensions, we recover the classical expressions for realization of a regular product of rational functions. Under an additional assumption that the limit at infinity of a given rational function exists and is equal to identity, we introduce an explicit formula for a tensor factorization of this function.

A More Powerful Unconditional Exact Test Of Homogeneity For 2 × C Contingency Table Analysis, Louis Ehwerhemuepha, Heng Sok, Cyril Rakovski

Mathematics, Physics, and Computer Science Faculty Articles and Research

The classical unconditional exact p-value test can be used to compare two multinomial distributions with small samples. This general hypothesis requires parameter estimation under the null which makes the test severely conservative. Similar property has been observed for Fisher's exact test with Barnard and Boschloo providing distinct adjustments that produce more powerful testing approaches. In this study, we develop a novel adjustment for the conservativeness of the unconditional multinomial exact p-value test that produces nominal type I error rate and increased power in comparison to all alternative approaches. We used a large simulation study to empirically estimate the …

Positivity, Rational Schur Functions, Blaschke Factors, And Other Related Results In The Grassmann Algebra, Daniel Alpay, Ismael L. Paiva, Daniele C. Struppa

Mathematics, Physics, and Computer Science Faculty Articles and Research

We begin a study of Schur analysis in the setting of the Grassmann algebra when the latter is completed with respect to the 1-norm. We focus on the rational case. We start with a theorem on invertibility in the completed algebra, and define a notion of positivity in this setting. We present a series of applications pertaining to Schur analysis, including a counterpart of the Schur algorithm, extension of matrices and rational functions. Other topics considered include Wiener algebra, reproducing kernels Banach modules, and Blaschke factors.

Positive And Generalized Positive Real Lemma For Slice Hyperholomorphic Functions, Daniel Alpay, Fabrizio Colombo, Izchak Lewkowicz, Irene Sabadini

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper we prove a quaternionic positive real lemma as well as its generalized version, in case the associated kernel has negative squares for slice hyperholomorphic functions. We consider the case of functions with positive real part in the half space of quaternions with positive real part, as well as the case of (generalized) Schur functions in the open unit ball.

Extending Set Functors To Generalised Metric Spaces, Adriana Balan, Alexander Kurz, Jiří Velebil

Mathematics, Physics, and Computer Science Faculty Articles and Research

For a commutative quantale V, the category V-cat can be perceived as a category of generalised metric spaces and non-expanding maps. We show that any type constructor T (formalised as an endofunctor on sets) can be extended in a canonical way to a type constructor T_V on V-cat. The proof yields methods of explicitly calculating the extension in concrete examples, which cover well-known notions such as the Pompeiu-Hausdorff metric as well as new ones.

Conceptually, this allows us to to solve the same recursive domain equation X ≅ TX in different categories (such as sets and metric spaces) and …

Distribution Spaces And A New Construction Of Stochastic Processes Associated With The Grassmann Algebra, Daniel Alpay, Ismael L. Paiva, Daniele C. Struppa

Mathematics, Physics, and Computer Science Faculty Articles and Research

We associate with the Grassmann algebra a topological algebra of distributions, which allows the study of processes analogous to the corresponding free stochastic processes with stationary increments, as well as their derivatives.

Positive Subreducts In Finitely Generated Varieties Of Mv-Algebras, Leonardo M. Cabrer, Peter Jipsen, Tomáš Kroupa

Mathematics, Physics, and Computer Science Faculty Articles and Research

Positive MV-algebras are negation-free and implication-free subreducts of MV-algebras. In this contribution we show that a finite axiomatic basis exists for the quasivariety of positive MV-algebras coming from any finitely generated variety of MV-algebras.

About A Non-Standard Interpolation Problem, Daniel Alpay, Alain Yger

Mathematics, Physics, and Computer Science Faculty Articles and Research

Using algebraic methods, and motivated by the one variable case, we study a multipoint interpolation problem in the setting of several complex variables. The duality realized by the residue generator associated with an underlying Gorenstein algebra, using the Lagrange interpolation polynomial, plays a key role in the arguments.

The Chapman Bone Algorithm: A Diagnostic Alternative For The Evaluation Of Osteoporosis, Elise Levesque, Anton Ketterer, Wajiha Memon, Cameron James, Noah Barrett, Cyril Rakovski, Frank Frisch

Mathematics, Physics, and Computer Science Faculty Articles and Research

Osteoporosis is the most common metabolic bone disease and goes largely undiagnosed throughout the world, due to the inaccessibility of DXA machines. Multivariate analyses of serum bone turnover markers were evaluated in 226 Orange County, California, residents with the intent to determine if serum osteocalcin and serum pyridinoline cross-links could be used to detect the onset of osteoporosis as effectively as a DXA scan. Descriptive analyses of the demographic and lab characteristics of the participants were performed through frequency, means and standard deviation estimations. We implemented logistic regression modeling to find the best classification algorithm for osteoporosis. All calculations and …

Partially-Ordered Multi-Type Algebras, Display Calculi And The Category Of Weakening Relations, Peter Jipsen, Fei Liang, M. Andrew Moshier, Apostolos Tzimoulis

Mathematics, Physics, and Computer Science Faculty Articles and Research

"We define partially-ordered multi-type algebras and use them as algebraic semantics for multi-type display calculi that have recently been developed for several logics, including dynamic epistemic logic [7], linear logic[10], lattice logic [11], bilattice logic [9] and semi-De Morgan logic [8]."

Fast Adjustable Npn Classification Using Generalized Symmetries, Xuegong Zhou, Lingli Wang, Peiyi Zhao, Alan Mishchenko

Mathematics, Physics, and Computer Science Faculty Articles and Research

NPN classification of Boolean functions is a powerful technique used in many logic synthesis and technology mapping tools in FPGA design flows. Computing the canonical form of a function is the most common approach of Boolean function classification. In this paper, a novel algorithm for computing NPN canonical form is proposed. By exploiting symmetries under different phase assignments and higher-order symmetries of Boolean functions, the search space of NPN canonical form computation is pruned and the runtime is dramatically reduced. The algorithm can be adjusted to be a slow exact algorithm or a fast heuristic algorithm with lower quality. For …

Generalized Fock Spaces And The Stirling Numbers, Daniel Alpay, Motke Porat

Mathematics, Physics, and Computer Science Faculty Articles and Research

The Bargmann-Fock-Segal space plays an important role in mathematical physics and has been extended into a number of directions. In the present paper, we imbed this space into a Gelfand triple. The spaces forming the Fréchet part (i.e., the space of test functions) of the triple are characterized both in a geometric way and in terms of the adjoint of multiplication by the complex variable, using the Stirling numbers of the second kind. The dual of the space of test functions has a topological algebra structure, of the kind introduced and studied by the first named author and Salomon.

Herglotz Functions Of Several Quaternionic Variables, Khaled Abu-Ghanem, Daniel Alpay, Fabrizio Colombo, Izchak Lewkowicz, Irene Sabadini

Mathematics, Physics, and Computer Science Faculty Articles and Research

We first review realizations of Herglotz functions in the unit ball of C^N and provide new insights. Then, we define the corresponding class and prove the extend the results in the case of several quaternionic variables.

Nonassociative Right Hoops, Peter Jipsen, Michael Kinyon

Mathematics, Physics, and Computer Science Faculty Articles and Research

The class of nonassociative right hoops, or narhoops for short, is defined as a subclass of right-residuated magmas, and is shown to be a variety. These algebras generalize both right quasigroups and right hoops, and we characterize the subvarieties in which the operation x ^^ y = (x/y)y is associative and/or commutative. Narhoops with a left unit are proved to be integral if and only if ^ is commutative, and their congruences are determined by the equivalence class of the left unit. We also prove that the four identities defining narhoops are independent.

On Lattices Of Z-Ideals Of Function Rings, Themba Dube, Oghenetega Ighedo

Mathematics, Physics, and Computer Science Faculty Articles and Research

An ideal I of a ring A is a z-ideal if whenever a, b ∈ A belong to the same maximal ideals of A and a ∈ I, then b ∈ I as well. On the other hand, an ideal J of A is a d-ideal if Ann²(a) ⊆ J for every a ∈ J. It is known that the lattices Z(L) and D(L) of the ring 𝓡L of continuous real-valued functions on a frame L, consisting of z-ideals and d-ideals of 𝓡 …

Distributive Residuated Frames And Generalized Bunched Implication Algebras, Nikolaos Galatos, Peter Jipsen

Mathematics, Physics, and Computer Science Faculty Articles and Research

We show that all extensions of the (non-associative) Gentzen system for distributive full Lambek calculus by simple structural rules have the cut elimination property. Also, extensions by such rules that do not increase complexity have the finite model property, hence many subvarieties of the variety of distributive residuated lattices have decidable equational theories. For some other extensions, we prove the finite embeddability property, which implies the decidability of the universal theory, and we show that our results also apply to generalized bunched implication algebras. Our analysis is conducted in the general setting of residuated frames.

“Wrong” Side Interpolation By Positive Real Rational Functions, Daniel Alpay, Izchak Lewkowicz

Mathematics, Physics, and Computer Science Faculty Articles and Research

Using polynomial interpolation, along with structural properties of the family of rational positive real functions, we here show that a set of m nodes in the open left half of the complex plane, can always be mapped to anywhere in the complex plane by rational positive real functions whose degree is at most m. Moreover we introduce an easy-to-find parametrization in R2m+3 of a large subset of these interpolating functions.

Evolution Of Superoscillations For Schrödinger Equation In A Uniform Magnetic Field, Fabrizio Colombo, Jonathan Gantner, Daniele C. Struppa

Mathematics, Physics, and Computer Science Faculty Articles and Research

Aharonov-Berry superoscillations are band-limited functions that oscillate faster than their fastest Fourier component. Superoscillations appear in several fields of science and technology, such as Aharonov’s weak measurement in quantum mechanics, in optics, and in signal processing. An important issue is the study of the evolution of superoscillations using the Schrödinger equation when the initial datum is a weak value. Some superoscillatory functions are not square integrable, but they are real analytic functions that can be extended to entire holomorphic functions. This fact leads to the study of the continuity of a class of convolution operators acting on suitable spaces of …