Other Mathematics Commons^™

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 …

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 …

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.

Asymptotic Quasi-Completeness And Zfc, Mirna Džamonja, Marco Panza

MPP Published Research

The axioms ZFC of first order set theory are one of the best and most widely accepted, if not perfect, foundations used in mathematics. Just as the axioms of first order Peano Arithmetic, ZFC axioms form a recursively enumerable list of axioms, and are, then, subject to Gödel’s Incompleteness Theorems. Hence, if they are assumed to be consistent, they are necessarily incomplete. This can be witnessed by various concrete statements, including the celebrated Continuum Hypothesis CH. The independence results about the infinite cardinals are so abundant that it often appears that ZFC can basically prove very little about such cardinals. …

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.

Was Frege A Logicist For Arithmetic?, Marco Panza

MPP Published Research

The paper argues that Frege’s primary foundational purpose concerning arithmetic was neither that of making natural numbers logical objects, nor that of making arithmetic a part of logic, but rather that of assigning to it an appropriate place in the architectonics of mathematics and knowledge, by immersing it in a theory of numbers of concepts and making truths about natural numbers, and/or knowledge of them transparent to reason without the medium of senses and intuition.

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 …

Enthymemathical Proofs And Canonical Proofs In Euclid’S Plane Geometry, Abel Lassalle, Marco Panza

MPP Published Research

Since the application of Postulate I.2 in Euclid’s Elements is not uniform, one could wonder in what way should it be applied in Euclid’s plane geometry. Besides legitimizing questions like this from the perspective of a philosophy of mathematical practice, we sketch a general perspective of conceptual analysis of mathematical texts, which involves an extended notion of mathematical theory as system of authorizations, and an audience-dependent notion of proof.

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.

Review Of G. Israel, Meccanicismo. Trionfi E Miserie Della Visione Meccanica Del Mondo, Marco Panza

MPP Published Research

"This is Giorgio's Israel last book, which appeared only a few weeks after his untimely death, in September 2015. For many reasons, it can be considered as his intellectual legacy, since it comes back, in a new and organic way, to many of the research topics to which he devoted his life and his many publications, which include several papers in Historia Mathematica. One of these papers, co-authored with M. Menghini, appeared in vol. 25/4, 1998 and was devoted to Poincaré's and Enriques's opposite views on qualitative analysis, which is a theme also dealt with in this book (pp. 117–122)."

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 …

On A Class Of Quaternionic Positive Definite Functions And Their Derivatives, Daniel Alpay, Fabrizio Colombo, Irene Sabadini

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper, we start the study of stochastic processes over the skew field of quaternions. We discuss the relation between positive definite functions and the covariance of centered Gaussian processes and the construction of stochastic processes and their derivatives. The use of perfect spaces and strong algebras and the notion of Fock space are crucial in this framework.

On Benacerraf’S Dilemma, Again, Marco Panza

MPP Published Research

In spite of its enormous influence, Benacerraf’s dilemma admits no standard unanimously accepted formulation. This mainly depends on Benacerraf’s having originally presented it in a quite colloquial way, by avoiding any compact, somehow codified, but purportedly comprehensive formulation (Benacerraf 1973 cf. p. 29).

The Mathematics Of Superoscillations, Yakir Aharonov, Fabrizio Colombo, Irene Sabadini, Daniele C. Struppa, Jeff Tollaksen

Mathematics, Physics, and Computer Science Faculty Articles and Research

In the past 50 years, quantum physicists have discovered, and experimentally demonstrated, a phenomenon which they termed superoscillations. Aharonov and his collaborators showed that superoscillations naturally arise when dealing with weak values, a notion that provides a fundamentally different way to regard measurements in quantum physics. From a mathematical point of view, superoscillating functions are a superposition of small Fourier components with a bounded Fourier spectrum, which result, when appropriately summed, in a shift that can be arbitrarily large, and well outside the spectrum. Purpose of this work is twofold: on one hand we provide a self-contained survey of the …

Features Of Agent-Based Models, Reiko Heckel, Alexander Kurz, Edmund Chattoe-Brown

Engineering Faculty Articles and Research

The design of agent-based models (ABMs) is often ad-hoc when it comes to defining their scope. In order for the inclusion of features such as network structure, location, or dynamic change to be justified, their role in a model should be systematically analysed. We propose a mechanism to compare and assess the impact of such features. In particular we are using techniques from software engineering and semantics to support the development and assessment of ABMs, such as graph transformations as semantic representations for agent-based models, feature diagrams to identify ingredients under consideration, and extension relations between graph transformation systems to …

On A Class Of Quaternionic Positive Definite Functions And Their Derivatives, Daniel Alpay, Fabrizio Colombo, Irene Sabadini

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper, we start the study of stochastic processes over the skew field of quaternions. We discuss the relation between positive definite functions and the covariance of centered Gaussian processes and the construction of stochastic processes and their derivatives. The use of perfect spaces and strong algebras and the notion of Fock space are crucial in this framework.

Characterizations Of Families Of Rectangular, Finite Impulse Response, Para-Unitary Systems, Daniel Alpay, Palle Jorgensen, Izchak Lewkowicz

Mathematics, Physics, and Computer Science Faculty Articles and Research

We here study Finite Impulse Response (FIR) rectangular, not necessarily causal, systems which are (para)-unitary on the unit circle (=the class U). First, we offer three characterizations of these systems. Then, introduce a description of all FIRs in U, as copies of a real polytope, parametrized by the dimensions and the McMillan degree of the FIRs.

Finally, we present six simple ways (along with their combinations) to construct, from any FIR, a large family of FIRs, of various dimensions and McMillan degrees, so that whenever the original system is in U, so is the whole family.

A key role is …

Platonismes, Marco Panza

MPP Published Research

Selon la vulgata philosophique, le platonisme concernant un certain domaine de recherche est la thèse affirmant que ce domaine concerne des objets qui lui sont propres, dont l’existence est indépendante de l’activité cognitive humaine. Souvent, dans la même vulgata on parle aussi de platonisme pour se référer à une thèse un peu différente, d’après laquelle ce qu’on dit concernant ce domaine est vrai ou faux indépendamment de toute justification ou réfutation que l’on puisse apporter. Naturellement, si parmi les énoncées ayant trait à ce demain, il y en a qu’on peut prendre comme particulièrement surs du fait d’en avoir une …

Adaptive Orthonormal Systems For Matrix-Valued Functions, Daniel Alpay, Fabrizio Colombo, Tao Qian, Irene Sabadini, Tao Qian

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper we consider functions in the Hardy space Hp×q2 defined in the unit disc of matrix-valued. We show that it is possible, as in the scalar case, to decompose those functions as linear combinations of suitably modified matrix-valued Blaschke product, in an adaptive way. The procedure is based on a generalization to the matrix-valued case of the maximum selection principle which involves not only selections of suitable points in the unit disc but also suitable orthogonal projections. We show that the maximum selection principle gives rise to a convergent algorithm. Finally, we discuss the case of real-valued signals.

Functions Of The Infinitesimal Generator Of A Strongly Continuous Quaternionic Group, Daniel Alpay, Fabrizio Colombo, Jonathan Gantner, David P. Kimsey

Mathematics, Physics, and Computer Science Faculty Articles and Research

The analogue of the Riesz-Dunford functional calculus has been introduced and studied recently as well as the theory of semigroups and groups of linear quaternionic operators. In this paper we suppose that T is the infinitesimal generator of a strongly continuous group of operators (ZT (t))t2R and we show how we can define bounded operators f(T ), where f belongs to a class of functions which is larger than the class of slice regular functions, using the quaternionic Laplace-Stieltjes transform. This class will include functions that are slice regular on the S-spectrum of T but not necessarily at infinity. Moreover, …