Geometry and Topology Commons^™

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.

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.

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 …

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 …

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 …

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 energy–momentum 4-vector potential field. Acoustic pressure and velocity fields form an energy–momentum 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 …

A Strong-Type Furstenberg–Sárközy Theorem For Sets Of Positive Measure, Polona Durcik, Vjekoslav Kovač, Mario Stipčić

Mathematics, Physics, and Computer Science Faculty Articles and Research

For every β ∈ (0,∞), β ≠ 1, we prove that a positive measure subset A of the unit square contains a point (x₀, y₀) such that A nontrivially intersects curves y − y0 = a(x −x0)^β for a whole interval I ⊆ (0,∞) of parameters a ∈ I . A classical Nikodym set counterexample prevents one to take β = 1, which is the case of straight lines. Moreover, for a planar set A of positive density, we show that the interval I can be arbitrarily large on the logarithmic scale. These results can …

New Characterizations Of Reproducing Kernel Hilbert Spaces And Applications To Metric Geometry, Daniel Alpay, Palle E. T. Jorgensen

Mathematics, Physics, and Computer Science Faculty Articles and Research

We give two new global and algorithmic constructions of the reproducing kernel Hilbert space associated to a positive definite kernel. We further present a general positive definite kernel setting using bilinear forms, and we provide new examples. Our results cover the case of measurable positive definite kernels, and we give applications to both stochastic analysis and metric geometry and provide a number of examples.

On The Menger And Almost Menger Properties In Locales, Tilahun Bayih, Themba Dube, Oghenetega Ighedo

Mathematics, Physics, and Computer Science Faculty Articles and Research

The Menger and the almost Menger properties are extended to locales. Regarding the former, the extension is conservative (meaning that a space is Menger if and only if it is Menger as a locale), and the latter is conservative for sober T_D-spaces. Non-spatial Menger (and hence almost Menger) locales do exist, so that the extensions genuinely transcend the topological notions. We also consider projectively Menger locales, and show that, as in spaces, a locale is Menger precisely when it is Lindelöf and projectively Menger. Transference of these properties along localic maps (via direct image or pullback) is considered.

Boxes, Extended Boxes And Sets Of Positive Upper Density In The Euclidean Space, Polona Durcik, Vjekoslav Kovač

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.

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 …

Topological Duality And Lattice Expansions, Ii: Lattice Expansions With Quasioperators, M. Andrew Moshier, Peter Jipsen

Mathematics, Physics, and Computer Science Faculty Articles and Research

The main objective of this paper (the second of two parts) is to show that quasioperators can be dealt with smoothly in the topological duality established in Part I. A quasioperator is an operation on a lattice that either is join preserving and meet reversing in each argument or is meet preserving and join reversing in each argument. The paper discusses several common examples, including orthocomplementation on the closed subspaces of a fixed Hilbert space (sending meets to joins), modal operators auS and a- on a bounded modal lattice (preserving joins, resp. meets), residuation on a bounded residuated lattice (sending …

On Clifford Analysis For Holomorphic Mappings, M. E. Luna-Elizarrarás, M. Shapiro, Daniele C. Struppa

Mathematics, Physics, and Computer Science Faculty Articles and Research

In the classical theory of several complex variables, holomorphic mappings are just n-tuples of holomorphic functions in m variables, with arbitrary n and m, and no relations between these functions are assumed. Some 30 years ago John Ryan introduced complex, or complexified, Clifford analysis which is, in a sense, the study of certain classes of holomorphic mappings where the components are not independent, and instead obey the relations generated by the Cauchy– Riemann and Dirac-type operators. In this paper, we take a closer look at this theory emphasizing some additional properties that holomorphic mappings satisfy in this context. Our attention …

Topological Duality And Lattice Expansions, I: A Topological Construction Of Canonical Extensions, M. Andrew Moshier, Peter Jipsen

Mathematics, Physics, and Computer Science Faculty Articles and Research

The two main objectives of this paper are (a) to prove purely topological duality theorems for semilattices and bounded lattices, and (b) to show that the topological duality from (a) provides a construction of canonical extensions of bounded lattices. In previously known dualities for semilattices and bounded lattices, the dual spaces are compact 0-dimensional spaces with additional algebraic structure. For example, semilattices are dual to 0-dimensional compact semilattices. Here we establish dual categories in which the spaces are characterized purely in topological terms, with no additional algebraic structure. Thus the results can be seen as generalizing Stone's duality for distributive …

Integral Conditions For The Vanishing Of The Cohomology Of Open Sets In Cn, Fabrizio Colombo, M. E. Luna-Elizarrarás, Irene Sabadini, M. Shapiro, Daniele C. Struppa

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper we develop and extend some techniques introduced in [1] to find integral conditions for the vanishing of the cohomology of open bounded sets in Cn with values in the sheaf of holomorphic functions.

Strictly Positive Definite Functions On A Compact Group, Mohamed Allali, Tomasz Przebinda

Mathematics, Physics, and Computer Science Faculty Articles and Research

We recognize a result of Schreiner, concerning strictly positive definite functions on a sphere in an Euclidean space, as a generalization of Bochner's theorem for compact groups.

Hartogs' Phenomenon For Polyregular Functions And Projective Dimension Of Related Modules Over A Polynomial Ring, W. W. Adams, P. Loustaunau, V. P. Palamadov, Daniele C. Struppa

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper we prove that the projective dimension of Mn = R^4/(An) is 2n -1, where R is the ring of polynomials in 4n variables with complex coefficients and (An) is the module generated by the columns of a 4x4n matrix which arises as the Fourier transform of the matrix of differential operators associated with the regularity condition for a function of n quaternionic variables. As a corollary we show that the sheaf R of regular functions has flabby dimension 2n -1, and we prove a cohomology vanishing theorem for open sets in the space Hn of quaternions. We …