Mathematics Commons^™

Conditionals And Modalities In Constructive Quantum Logics, Juan P. Aguilera, Guillaume Massas

Mathematics, Physics, and Computer Science Faculty Articles and Research

We investigate logics that generalize both intuitionistic logic and quantum logic. In earlier work, we introduced Ex-logic, an extension of Holliday's fundamental logic that coincides with the intersection of orthologic and the implication-free fragment of intuitionistic logic. In this paper, we add an implication connective to Ex-logic and axiomatize iEx-logic, the intersection of full intuitionistic logic and orthomodular logic with the implication connective interpreted as the Sasaki hook. As a consequence, we obtain a characterization of the lattice of logics extending iEx-logic as the product of the lattice of intermediate logics and the lattice of orthomodular logics. We also explore …

Quantitatively Hyper-Positive Real Rational Functions Iii, 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.

On Links Between A Theorem Of Schoenberg, Rohlin Decompositions Of Measures, The Bochner-Minlos Theorem And The Fock Space, Daniel Alpay, Paula Cerejeiras, Palle Jorgensen, Uwe Kaehler

Mathematics, Physics, and Computer Science Faculty Articles and Research

The main goal of this paper is to gain new results in stochastics by drawing on, and combining, different areas that are normally not considered to be related. Thus, in this paper we extend the previous class of Gaussian-like functions ML which will allow for future generalized stochastic processes in infinite dimensional analysis. We show that an approach similar to the one by the classical Bochner-Minlos theorem for the white-noise case can be achieved by using Gaussian-like functions belonging to a large family -the ML_r classes (0 < r ≤∞). We show how Schoenberg’s theorem for positive definite functions on a Hilbert space allows to go beyond the classical setting of Bochner-Milnos theorem. Furthermore, we show that the application of the Rohlin’s disintegration theorem allows for a decomposition of the associated probability measures , see Theorems 3.2 and 4.3. We end this paper with several important examples of functions in these classes ML_r and provide some interesting counterexamples, e.g. Theorem 7.4, to get a …

Norm-Variation Of Triple Ergodic Averages For Commuting Transformations, Polona Durcik, Lenka Slavíková, Christoph Thiele

Mathematics, Physics, and Computer Science Faculty Articles and Research

We prove an r-variation estimate, r>4, in the norm for ergodic averages with respect to three commuting transformations. It is not known whether such estimates hold for all r≥2 as in the analogous cases for one or two commuting transformations, or whether such estimates hold for any r< ∞ for more than three commuting transformations.

Mckinsey-Tarski Algebras And Raney Extensions, G. Bezhanishvili, R. Raviprakash, A. L. Suarez, Joanne Walters-Wayland

Mathematics, Physics, and Computer Science Faculty Articles and Research

We introduce the notion of Raney morphism between MT-algebras and show that the resulting category is equivalent to the category of Raney extensions. This is done by generalizing the construction of the Funayama envelope of a frame. The resulting notion of the T₀-hull of a Raney extension generalizes that of the T_D-hull of a frame.

Unitary Rational Functions: The Scaled Quaternion Case, Daniel Alpay, Ilwoo Choo, Mihaela Vajiac

Mathematics, Physics, and Computer Science Faculty Articles and Research

We develop the theory of minimal realizations and factorizations of rational functions where the coefficient space is a ring of the type introduced in our previous work, the scaled quaternions, which includes as special cases the quaternions and the split quaternions. The methods involved are not a direct generalization of the complex or quaternionic settings, and in particular, the adjoint is not the classical adjoint and we use properties of real Hilbert spaces. This adjoint allows to define the counterpart of unitarity for matrix-rational functions, and we develop the corresponding theories of realizations and unitary factorizations. We also begin a …

Locally Integral Involutive Po-Semigroups, José Gil-Férez, Peter Jipsen, Melissa Sugimoto

Mathematics, Physics, and Computer Science Faculty Articles and Research

We show that every locally integral involutive partially ordered semigroup (ipo-semigroup) A=(A,≤,⋅,∼,−), and in particular every locally integral involutive semiring, decomposes in a unique way into a family {Ap:p∈A+} of integral ipo-monoids, which we call its integral components. In the semiring case, the integral components are unital semirings. Moreover, we show that there is a family of monoid homomorphisms Φ={φpq:Ap→Aq:p≤q}, indexed over the positive cone (A+,≤), so that the structure of A can be recovered as a glueing ∫ΦAp of its integral components along Φ. Reciprocally, we give necessary and sufficient conditions so that the Płonka sum of any family …

There Is No Obstruction To A Euclidean Proof For The Fourth Postulate, José Gil-Férez, Piotr Błaszczyk, M. Andrew Moshier, Alberto Naibo, Jean-Michel Salanskis

Mathematics, Physics, and Computer Science Faculty Articles and Research

In Gil-Férez et al. (2025), we presented a proof of Postulate 4 using purely Euclidean techniques, against which Blåsjö (2025) raised several objections. In this note, we offer linguistic, textual, historical, and mathematical evidence that demonstrate that all these objections are baseless.

Free Probability Theory Over The Scaled Hyperbolic Numbers, Daniel Alpay, Ilwoo Choo

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper, we introduce a notion of free probability over the scaled hyperbolic numbers. Scaled hypercomplex numbers {D_t} _t∈R are constructed as sub-structures of scaled hypercomplex numbers {H_t} _t∈R under the scales (or, the moments) of the set R of real numbers.We show that if t < 0, then the classical free probability theory covers our free probability on {D_t} _{t< 0}; if t > 0, then our free probability on {D_t} _t>0 is represented by the free probability over the classical hyperbolic numbers D = D₁; and if t = 0, then the free probability on D₀ is actually over the …

The Kaczmarz Algorithm In Hilbert C*-Modules, Daniel Alpay, Chad Berner, Eric S. Weber

Mathematics, Physics, and Computer Science Faculty Articles and Research

The Kaczmarz algorithm in Hilbert spaces is a classical iterative method for stably recovering vectors from inner product data. In this paper, we extend the algorithm to the setting of Hilbert C*-modules and establish analogues of its effectiveness in both finite-dimensional and stationary cases. Consequently, we demonstrate that continuous families of elements in a Hilbert space can be uniformly recovered using the Kaczmarz algorithm. Additionally, we develop a normalized Cauchy transform for continuous families of measures and use it to provide sufficient conditions under which standard frames in Hilbert C(X)-modules can be generated by the Kaczmarz …

Matrices Induced By Scaled Hypercomplex Numbers Over The Real Field R, Daniel Alpay, Ilwoo Choo

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper, we construct, and study a certain type of definite, or indefinite inner product spaces over the real field R, induced by the scaled hypercomplex numbers H_t for a fixed scale t ∈ R, and some bounded operators acting on such vector spaces. In particular, we are interested in the vector spaces H^N_t consisting of all N-tuples of scaled hypercomplex numbers of H_t, and the (N x N)-matrices acting on H^N_t whose entries are from H_t, i.e., H_t-matrices, for all N ∈ N. For an arbitrarily fixed …

Differentiation And Certain Operators On Scaled Sectional Hypercomplex Numbers, Daniel Alpay, Ilwoo Choo

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper, we study certain sectional structures of the t-scaled hypercomplex numbers H_t for a scale t ∈ R, including the quaternions H_-1, and the split quaternions H₁. For a fixed scale t ∈ R, by defining the collection S_t of certain pureimaginary t-scaled hypercomplex number in H_t , we sectionize H_t from the imaginaries of S_t. We concentrate on a section SH_It for an arbitrarily fixed imaginary I_t ∈ S_t , called the t-scaled section for I_t. Differentiation theory on the …

Probabilities With Values In Scaled Hyperbolic Numbers, Daniel Alpay, Ilwoo Choo

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper, we introduce a notion of a probabilistic measure which takes values in t-scaled hyperbolic numbers for t ∈ R, with a system of axioms generalizing directly Kolmogorov’s axioms. i.e., we establish a suitable measure theory in the set Dt of all t-scaled hyperbolic numbers for arbitrarily fixed t ∈ R.

Similarity Metrics, Metrics, And Conditionally Negative Definite Functions, Daniel Alpay, Liora Mayats-Alpay

Mathematics, Physics, and Computer Science Faculty Articles and Research

We give an example of a similarity metric which is not positive definite, and present a general theorem which provides a large family of similarity metrics which are positive definite.

Analyticity, Superoscillations And Supershifts In Several Variables, Fabrizio Colombo, Irene Sabadini, Daniele C. Struppa, Alain Yger

Mathematics, Physics, and Computer Science Faculty Articles and Research

Superoscillations have roots in various scientific disciplines, including optics, signal processing, radar theory, and quantum mechanics. This intriguing mathematical phenomenon permits specific functions to oscillate at a rate surpassing their highest Fourier component. A different way of thinking about superoscillations consists in realizing that it is possible to reproduce the exponential function far away from the origin by only knowing its value in a countable set of points near the origin. By using this perspective, one can extend the idea of superoscillations to functions that are not a sum of exponential functions, namely to the notion of supershift. The study …

Supershift Properties For Nonanalytic Signals, Fabrizio Colombo, Irene Sabadini, Daniele Carlo Struppa, Alain Yger

Mathematics, Physics, and Computer Science Faculty Articles and Research

The phenomenon of superoscillations is of great interest in microscopy, antenna design, and material sciences. This phenomenon has been generalized and has given rise to the concept of supershift, which is a far reaching extension that applies to functions that may present discontinuous derivatives. From this perspective, this is a notion that might have significant applications. This paper will provide an up to date report on the complex connections between the concept of supershift and that of analyticity.

Analyticity And Supershift With Regular Sampling, Fabrizio Colombo, Irene Sabadini, Daniele C. Struppa, Alain Yger

Mathematics, Physics, and Computer Science Faculty Articles and Research

The notion of supershift (in itself a generalization of the notion of superoscillation arising in quantum mechanics) expresses the fact that the sampling of a function in an interval allows to compute the values of the function far from the interval. In this paper, we study the relation between supershift and real analyticity. We use a classical result due to Serge Bernstein to show that real analyticity for a complex-valued function implies a strong form of supershift. On the other hand, we use a parametric version of a result by Leonid Kantorovitch to show that the converse is not true. …

Dedekind-Macneille And Related Completions: Subfitness, Regularity, And Booleanness, G. Bezhanishvili, F. Dashiell Jr., M. A. Moshier, Joanne Walters-Wayland

Mathematics, Physics, and Computer Science Faculty Articles and Research

Completions play an important rôle for studying structure by supplying elements that in some sense “ought to be.” Among these, the Dedekind-MacNeille completion is of particular importance. In 1968 Janowitz provided necessary and sufficient conditions for it to be subfit or Boolean. Another natural separation axiom connected to these is regularity. We explore similar characterizations of when closely related completions are subfit, regular, or Boolean. We are mainly interested in the Bruns-Lakser, ideal, and canonical completions, which (unlike the Dedekind-MacNeille completion) satisfy stronger forms of distributivity. The first two are widely used in pointfree topology, while the latter is of …

Stochastic Bundles, New Classes Of Gaussian Processes And White Noise-Space Analysis Indexed By Measures, Daniel Alpay, Palle Jorgensen

Mathematics, Physics, and Computer Science Faculty Articles and Research

Starting from a fixed measure space (X,F,μ), with μ a positive sigma-finite measure defined on the sigma-algebra F, we continue here our study of a generalization W(μ) of Brownian motion, and introduce a corresponding white-noise process. In detail, the generalized Brownian motion is a centered Gaussian process W^(μ), indexed by the elements A in F of finite μ measure, and with covariance function μ(A ∩ B). The purpose of our present paper is to make precise and study the corresponding whitenoise process, i.e., a point-wise process which is indexed by X, and which arises …

Subfitness In Distributive (Semi)Lattices, G. Bezhanishvili, J. Madden, M. A. Moshier, M. Tressl, Joanne Walters-Wayland

Mathematics, Physics, and Computer Science Faculty Articles and Research

We investigate whether the set of subfit elements of a distributive semilattice is an ideal. This question was raised by the second author at the BLAST conference in 2022. We show that in general it has a negative solution, however if the semilattice is a lattice, then the solution is positive. This is somewhat unexpected since, as we show, a semilattice is subfit if and only if so is its distributive lattice envelope.

From Λ-Hollow Frames To Λ-Repletions In W: Ii. Λ-Repletions In W, Richard N. Ball, Anthony W. Hager, Joanne Walters-Wayland

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this article we analyze the fine structure of the essential extensions of an object of W, the category of divisible archimedean lattice ordered groups with designated weak units. In particular, we show that an object Ghas an ordinally indexed sequence {τ^α_G}_δG of essential extensions with the following features. τ⁰_G is (isomorphic to) the identity function on G.

• For every α>0, τ^α_G is an essential extension of G into a W-object which is of the form RLfor some frame L, and which is λ-replete for some λ.

• Every such …

Vieta’S Formula, The Fabius Function And The Partition Function, Ahmed Sebbar

Mathematics, Physics, and Computer Science Faculty Articles and Research

We use an idea of Pólya and Szegö to give a common basis to Vieta’s formula, Fabius function and the partition function. Moreover our construction leads also to a function considered by Hallström, Bowen and Macintyre, which has, as a particular value, the Kepler-Bouwkamp constant, and to a function considered by Zondadari, that vanishes only at prime numbers.

New Fueter-Type Variables Associated To The Global Operator In The Quaternionic Case, Daniel Alpay, Kamal Diki, Mihaela Vajiac

Mathematics, Physics, and Computer Science Faculty Articles and Research

The purpose of this paper is to develop a new theory of three non-commuting quaternionic variables and its related Schur analysis theory for a modified version of the quaternionic global operator.

Scaled Global Operators And Fueter Variables On Non-Zero Scaled Hypercomplex Numbers, Daniel Alpay, Ilwoo Choo, Mihaela Vajiac

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper we describe the rise of global operators in the scaled quaternionic case, an important extension from the quaternionic case to the family of scaled hypercomplex numbers H_t, t ∈ R^∗ , of which the H₋₁ = H is the space of quaternions and H₁ is the space of split quaternions.We also describe the scaled Fueter-type variables associated to these operators, developing a coherent theory in this field. We use these types of variables to build different types of function spaces on H_t. Counterparts of the Hardy space and of the …

Q-Rational Functions And Interpolation With Complete Nevanlinna–Pick Kernels, Daniel Alpay, Paula Cerejeiras, Uwe Kaehler, Baruch Schneider

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper we introduce the concept of matrix-valued q-rational functions. In comparison to the classical case, we give different characterizations with principal emphasis on realizations and discuss algebraic manipulations. We also study the concept of Schur multipliers and complete Nevanlinna–Pick kernels in the context of q-deformed reproducing kernel Hilbert spaces and provide first applications in terms of an interpolation problem using Schur multipliers and complete Nevanlinna–Pick kernels.

The Bicomplex Tensor Product And A Bicomplex Choi Theorem, Daniel Alpay, Antonino De Martino, Kamal Diki, Mihaela Vajiac

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper we extend the concept of tensor product to the bicomplex case and use it to prove the bicomplex counterpart of the classical Choi theorem in the theory of complex matrices and operators. The concept of hyperbolic tensor product is also discussed, and we link these results to the theory of quantum channels in the bicomplex and hyperbolic case.

Frames And Spaces For Distributive Quasi Relation Algebras And Distributive Involutive Fl-Algebras, Andrew Craig, Peter Jipsen, Claudette Robinson

Mathematics, Physics, and Computer Science Faculty Articles and Research

Analogous to atom structures for relation algebras, we define partially ordered frames and prove they are duals for complete perfect distributive quasi relation algebras and distributive involutive FL-algebras. We then extend this dual representation to all algebras and their corresponding frames with a Priestley topology.

For relation algebras up to size 16 it has been determined which algebras are representable by binary relations. We compute all finite distributive quasi relation algebras up to 8 elements and provide representations for some of them.

On The Structure Of Balanced Residuated Partially Ordered Monoids, Stefano Bonzio, José Gil-Férez, Peter Jipsen, Adam Přenosil, Melissa Sugimoto

Mathematics, Physics, and Computer Science Faculty Articles and Research

A residuated poset is a structure ⟨A,⩽, ·, \, /, 1⟩ where ⟨A,⩽⟩ is a poset and ⟨A, ·, 1⟩ is a monoid such that the residuation law x · y ⩽ z ⇐⇒ x ⩽ z/y ⇐⇒ y ⩽ x\z holds. A residuated poset is balanced if it satisfies the identity x\x ≈ x/x. By generalizing the well-known construction of Płonka sums, we show that a specific class of balanced residuated posets can be decomposed into such a sum indexed by the set of positive idempotent elements. Conversely, given a semilattice directed system of residuated posets equipped with two …

Short-Time Fourier Transform And Superoscillations, Daniel Alpay, Antonino De Martino, Kamal Diki, Daniele C. Struppa

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper we investigate new results on the theory of superoscillations using time-frequency analysis tools and techniques such as the short-time Fourier transform (STFT) and the Zak transform. We start by studying how the short-time Fourier transform acts on superoscillation sequences. We then apply the supershift property to prove that the short-time Fourier transform preserves the superoscillatory behavior by taking the limit. It turns out that these computations lead to interesting connections with various features of time-frequency analysis such as Gabor spaces, Gabor kernels, Gabor frames, 2D-complex Hermite polynomials, and polyanalytic functions. We treat different cases depending on the …

S-Preclones And The Galois Connection SPol–SInv, Part I, Peter Jipsen, Erkko Lehtonen, Reinhard Pöschel

Mathematics, Physics, and Computer Science Faculty Articles and Research

We consider S-operations f : Aⁿ → A in which each argument is assigned a signum s ∈ S representing a “property” such as being order- preserving or order-reversing with respect to a fixed partial order on A. The set S of such properties is assumed to have a monoid structure reflecting the behaviour of these properties under the composition of S-operations (e.g., order-reversing composed with order-reversing is order- preserving). The collection of all S-operations with prescribed properties for their signed arguments is not a clone (since it is not closed under arbitrary identification of arguments), …