Mathematics Commons^™

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.

Euler Archive Spotlight: Ed Sandifer's Influence, Erik Tou

Euleriana

The Euler Archive owes its existence to the work and enthusiasm of C. Edward Sandifer. In this issue, we give a history of Ed's work and influence on the creation and growth of the Euler Archive.

Translating Scientific Latin Texts With Artificial Intelligence: The Works Of Euler And Contemporaries, Sylvio R. Bistafa

Euleriana

The major hindrance in the study of earlier scientific literature is the availability of Latin translations into modern languages. This is particularly true for the works of Euler who authored about 850 manuscripts and wrote a thousand letters and received back almost two thousand more. The translation of many of these manuscripts, books and letters have been published in various sources over the last two centuries, but many more have not yet appeared. Fortunately, nowadays, artificial intelligence (AI) translation can be used to circumvent the challenges of translating such substantial number of texts. To validate this tool, benchmark tests have …

On The Vibration Of Strings: An English Translation Of Leonhard Euler’S `Sur La Vibration Des Cordes' (E140), Reilly R. Fortune

Euleriana

We present an English translation of E140 - 'Sur la Vibration des cordes.'

About The Cases In Which The Formula X^4+Mxxyy+Y^4 Can Be Reduced To A Square, Georg Ehlers

Euleriana

Euler continues a previous study on the title Quartic (E696) with a new approach. His starting point here is the observation that, when the title Quartic is solved for m, the resulting fraction becomes an integer when z=ax^2y^2-(x^2±y^2). He provides many quadratic forms for m that allow special solutions, and tables for |m|≤200. Even though the tables are known today to be incomplete, they allow an insight into the enormous amount of work that was needed for their compilation.

The interest in the title Quartic predates Euler, and the fact that he resumed the study indicates a …

Ed Sandifer: A Running Mathematician And Mathematical Runner, Rick Cleary

Euleriana

Historian of mathematics C. Edward Sandifer was an outstanding marathon runner as well as a first rate mathematician. This note is a review of Prof. Sandifer's athletic successes, and a look at the attributes that he brought to both his professional and running careers.

How Ed Did It - A Memorial Conference To Honor Ed Sandifer, Lawrence D'Antonio

Euleriana

I look back at the virtual conference from February 2023 that was organized to honor the memory of the historian of mathematics Ed Sandifer, who had died in August 2022. The program of the conference is given at the end of the article.

Looking Back And Looking Forward, Christopher Goff, Erik Tou

Euleriana

An introduction to the contents in Issue 2, Volume 4 of Euleriana.

Some Studies On Mathematical Morphology In Remotely Sensed Data Analysis, Geetika Barman

Doctoral Theses

The application of Mathematical Morphology (MM) techniques has proven to be beneficial in the extraction of shapebased and texture-based features during remote sensing image analysis. The characteristics of these techniques, such as nonlinear adaptability and comprehensive lattice structure, make them useful for contextual spatial feature analysis. Despite the advancements, there are still persistent challenges, including the curse of dimensionality, maintaining spatial correlation, and the adaptability of morphological operators in higher dimensions. The focus of this thesis is to explore the potential of MM-based methods to analyse spatial features in addressing these challenges, specifically in the context of spatialcontextual feature analysis …

A Correspondence Between Deep Boltzmann Machines And P-Adic Statistical Field Theories, Wilson A. Zuniga-Galindo

School of Mathematical & Statistical Sciences Faculty Publications

There is a strong interest in studying the correspondence between Euclidean quantum fields and neural networks. This correspondence takes different forms depending on the type of networks considered. In this work, we study this correspondence in the case of deep Boltzmann machines (DBMs) having a tree-like topology. We use p-adic numbers to encode this type of topology. A p-adic continuous DBM is a statistical field theory (SFT) defined by an energy functional on the space of square-integrable functions defined on a p-adic N-dimensional ball. The energy functionals are non-local, meaning they depend on the interaction of all the neurons forming …

Ideals In Semigroups Of Partial Transformations With Invariant Set, Jitsupa Srisawat, Yanisa Chaiya

Turkish Journal of Mathematics

This paper explores the ideals and their structural properties in two generalizations of the partial transformationsemigroup. Furthermore, principal, maximal, and minimal ideals within these semigroups are elucidated.

A Non-Newtonian Conics In Multiplicative Analytic Geometry, Aykut Has, Beyhan Yilmaz

Turkish Journal of Mathematics

In this study, conics (circle, ellipse, hyperbola) are characterized by taking into account basic multiplicationoperations in multiplicative space. For this purpose, firstly multiplicative axes and regions are introduced. Additionally,the multiplicative cone definition is given and visualized on the figure. General definitions and theorems of non-Newtonianconics are given. Additionally, examples were given and drawings were made to make the resulting characterizations andtheorems more memorable.

The Heat Kernel On A Finite Graph In Different Time-Scales, Yang Chen, Jay Jorgenson, Luis Lopez, Lejla Smajlovic

Turkish Journal of Mathematics

Let G be a finite, weighted graph, and let [[EQUATION]] be a Time-scale with a fixed point [[EQUATION]] such that sup[[EQUATION]]. In this paper we construct the heat kernel on G in Time-scale [[EQUATION]] in terms of a certain convolution series involving he heat operator acting on a parametrix, which is a fairly general function depending on the vertex set of G and the time variable [[EQUATION]]. We develop some applications by choosing different parametrices and various Time-scales. The results we obtain here do extend, in part, aspects of the recent articles in that the Time-scale considered in this paper …

Transmission Eigenvalues Problem Of A Schrödinger Equation, Emel Yildirim, Elgiz Bairamov

Turkish Journal of Mathematics

In this paper, transmission eigenvalues of a Schrödinger equation have been studied by constructing a new inner product and using Weyl theory. Necessary conditions for these eigenvalues to be negative, real and finite have been examined. This method has provided a new framework related to transmission eigenvalue problems and investigation of their properties. The conclusions has been verified for special case of the problem.

Individual Stability Of Representations Of Abelian Semigroups, Heybetkulu Mustafayev

Turkish Journal of Mathematics

Let S be a suitable subsemigroup of a locally compact abelian group and let T={T(s)}s(-S be a bounded and strongly continuous trepresentation of S on a Banach space X. In this note, we study the spectral conditions on T and the ergodic conditios on x in X which will imply that T(s)x-->0 strongly as s--> infinity through S.

Rings And Finite Fields Whose Elements Are Sums Or Differences Of Tripotents And Potents, Adel Abyzov, Stephen Cohen, Peter Danchev, Daniel Tapkin

Turkish Journal of Mathematics

We significantly strengthen results on the structure of matrix rings over finite fields and applythem to describe the structure of the so-called weakly n-torsion clean rings. Specifically, we establish that, forany field F with either exactly seven or strictly more than nine elements, each matrix over F is presentableas a sum of of a tripotent matrix and a q-potent matrix if and only if each element in F is presentable as asum of a tripotent and a q-potent, whenever q > 1 is an odd integer. In addition, if Q is a power of an oddprime and F is a field …

Spherical Product Hypersurfaces In Euclidean Spaces, Sezgi̇n Büyükkütük, Günay Öztürk

Turkish Journal of Mathematics

Spherical product surfaces are obtained with the help of a special product by considering two curves inn−dimensional space. One of their special cases is rotational surface. The reason why the present study is significantthat the spherical product is used to construct hypersurfaces. (n−1)−curves are needed during this construction. Firstly,the spherical product hypersurfaces are defined in E4 , Gaussian and mean curvature are yielded and then conditionsbeing flat or minimal are examined. Moreover, superquadrics, which are associated with spherical product, are handledfor the first time in hypersurface form and give some examples. Finally, spherical product hypersurfaces are generalizedto n−dimensional Euclidean space …

A Sufficient Condition For The Wildness Of An Automorphism Of A Free Leibnizalgebra, Zeynep Özkurt

Turkish Journal of Mathematics

In this paper, we apply the criterion of Mikhalev and Umirbaev for the invertibility of an endomorphismof a finitely generated free Leibniz algebra via its Jacobian matrix to determine whether a given endomorphism is anautomorphism. Moreover, it is shown that the invertibility of the determinant of the Jacobian matrix of an automorphismimplies its wildness.

A Short Note On Generalized Robertson Walker Spacetimes, Uday Chand De, Aydin Gezer

Turkish Journal of Mathematics

In this article, generalized Robertson Walker spacetimes are investigated in light of perfect fluid spacetimes. First, we establish that a perfect fluid spacetime with non-vanishing vorticity whose associated scalars are constant along the velocity vector field becomes a generalized Robertson Walker spacetime. Among others, it is also shown that a Ricci parallel perfect fluid spacetime is either a generalized Robertson Walker spacetime or a static spacetime. Finally, we acquire that in a conformally semi-symmetric generalized Robertson Walker spacetime of dimension $4$, the scalar curvature vanishes and the spacetime is locally isometric to the Minkowski spacetime, provided the electric part of …

A Note On The Hull And Linear Complementary Pair Of Cyclic Codes, Zohreh Aliabadi, Tekgül Kalayci

Turkish Journal of Mathematics

The Euclidean hull of a linear code C is defined as C ∩ C⊥ , where C⊥ denotes the dual of C underthe Euclidean inner product. A linear code with the trivial hull is called a linear complementary dual (LCD) code. Apair (C,D) of linear codes of length n over the finite field Fq is called a linear complementary pair (LCP) of codes ifC ⊕ D = Fnq. More generally, a pair (C,D) of linear codes of the same length over Fq is called a linear ℓ -intersectionpair of codes if C ∩D has dimension ℓ as a vector space …