Rings And Finite Fields Whose Elements Are Sums Or Differences Of Tripotents And Potents, 2024 Plovdiv University, Plovdiv 4000, Bulgaria

#### 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, 2024 TÜBİTAK

#### 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, 2024 TÜBİTAK

#### 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 Note On The Hull And Linear Complementary Pair Of Cyclic Codes, 2024 TÜBİTAK

#### 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 …

Errata: The Product Of Distributions And Stochastic Differential Equations Arising From Powers Of Infinite Dimensional Brownian Motions, 2024 Institute for Industrial and Applied Mathematics, Chungbuk National University, Cheongju 28644, Korea

#### Errata: The Product Of Distributions And Stochastic Differential Equations Arising From Powers Of Infinite Dimensional Brownian Motions, Un Cig Ji, Hui-Hsiung Kuo, Hara-Yuko Mimachi, Kimiaki Saito

*Journal of Stochastic Analysis*

No abstract provided.

Coarse-Gridded Simulation Of The Nonlinear Schrödinger Equation With Machine Learning, 2024 Air Force Institute of Technology

#### Coarse-Gridded Simulation Of The Nonlinear Schrödinger Equation With Machine Learning, Benjamin F. Akers, Kristina O. F. Williams

*Faculty Publications*

A numerical method for evolving the nonlinear Schrödinger equation on a coarse spatial grid is developed. This trains a neural network to generate the optimal stencil weights to discretize the second derivative of solutions to the nonlinear Schrödinger equation. The neural network is embedded in a symmetric matrix to control the scheme’s eigenvalues, ensuring stability. The machine-learned method can outperform both its parent finite difference method and a Fourier spectral method. The trained scheme has the same asymptotic operation cost as its parent finite difference method after training. Unlike traditional methods, the performance depends on how close the initial data …

Minimal Separating Sets In Surfaces, 2024 Portland State University

#### Minimal Separating Sets In Surfaces, Christopher Nelson Aagaard

*Dissertations and Theses*

Given a connected topolgical space *X*, we say that *L ⊆ X* is a *minimal separating set* if removing *L* from *X* gives a disconnected surface, butremoving any proper subset of *L* leaves the surface connected. We classify which embeddings of topological graphs are minimal separating in an orientable surface *X* with genus *g*, and construct a computer program to compute the number of such embeddings, and the number of topological graphs which admit such an embedding for *g* ≤ 5.

Meta-Analysis Of Set-Based Multiple Phenotype Association Test Based On Gwas Summary Statistics From Different Cohorts, 2024 Michigan Technological University

#### Meta-Analysis Of Set-Based Multiple Phenotype Association Test Based On Gwas Summary Statistics From Different Cohorts, Lirong Zhu, Shuanglin Zhang, Qiuying Sha

*Michigan Tech Publications, Part 2*

Genome-wide association studies (GWAS) have emerged as popular tools for identifying genetic variants that are associated with complex diseases. Standard analysis of a GWAS involves assessing the association between each variant and a disease. However, this approach suffers from limited reproducibility and difficulties in detecting multi-variant and pleiotropic effects. Although joint analysis of multiple phenotypes for GWAS can identify and interpret pleiotropic loci which are essential to understand pleiotropy in diseases and complex traits, most of the multiple phenotype association tests are designed for a single variant, resulting in much lower power, especially when their effect sizes are small and …

Twisted Alexander Polynomials And Ptolemy Varieties Of Knots And Surface Bundles, 2024 The Graduate Center, City University of New York

#### Twisted Alexander Polynomials And Ptolemy Varieties Of Knots And Surface Bundles, Michael R. Marinelli

*Dissertations, Theses, and Capstone Projects*

The first focus of this dissertation is to compute Ptolemy varieties for triangulations of two infinite families of manifolds. Given an ideal triangulation of a cusped manifold, one can compute the Ptolemy variety and using it, obtain parabolic representations of the fundamental group. We compute certain obstruction classes for these manifolds, which are necessary to obtain the discrete faithful representation. This leads to our second focus of the dissertation, the twisted Alexander polynomial. The twisted Alexander polynomial (TAP) is a variation of the classical Alexander polynomial twisted by a representation of the fundamental group into a linear group. It was …

Limit Theorems For L-Functions In Analytic Number Theory, 2024 The Graduate Center, City University of New York

#### Limit Theorems For L-Functions In Analytic Number Theory, Asher Roberts

*Dissertations, Theses, and Capstone Projects*

We use the method of Radziwill and Soundararajan to prove Selberg’s central limit theorem for the real part of the logarithm of the Riemann zeta function on the critical line in the multivariate case. This gives an alternate proof of a result of Bourgade. An upshot of the method is to determine a rate of convergence in the sense of the Dudley distance. This is the same rate Selberg claims using the Kolmogorov distance. We also achieve the same rate of convergence in the case of Dirichlet L-functions. Assuming the Riemann hypothesis, we improve the rate of convergence by using …

Categorical Chain Conditions For Étale Groupoid Algebras, 2024 The Graduate Center, City University of New York

#### Categorical Chain Conditions For Étale Groupoid Algebras, Sunil Philip

*Dissertations, Theses, and Capstone Projects*

Let R be a unital commutative ring and G an ample groupoid. Using the topology of the groupoid G, Steinberg defined an étale groupoid algebra RG. These étale groupoid algebras generalize various algebras, including group algebras, commutative algebras over a field generated by idempotents, traditional groupoid algebras, Leavitt path algebras, higher-rank graph algebras, and inverse semigroup algebras. Steinberg later characterized the classical chain conditions for étale groupoid algebras. In this work, we characterize categorically noetherian and artinian, locally noetherian and artinian, and semisimple étale groupoid algebras, thereby generalizing existing results for Leavitt path algebras and introducing new results for inverse …

Generalized Periodicity And Applications To Logistic Growth, 2024 Missouri University of Science and Technology

#### Generalized Periodicity And Applications To Logistic Growth, Martin Bohner, Jaqueline Mesquita, Sabrina Streipert

*Mathematics and Statistics Faculty Research & Creative Works*

Classically, a continuous function f:R→R is periodic if there exists an ω>0 such that f(t+ω)=f(t) for all t∈R. The extension of this precise definition to functions f:Z→R is straightforward. However, in the so-called quantum case, where f:qN0→R (q>1), or more general isolated time scales, a different definition of periodicity is needed. A recently introduced definition of periodicity for such general isolated time scales, including the quantum calculus, not only addressed this gap but also inspired this work. We now return to the continuous case and present the concept of ν-periodicity that connects these different formulations of periodicity for …

The Cubic-Quintic Nonlinear Schrödinger Equation With Inverse-Square Potential, 2024 Missouri University of Science and Technology

#### The Cubic-Quintic Nonlinear Schrödinger Equation With Inverse-Square Potential, Alex H. Ardila, Jason Murphy

*Mathematics and Statistics Faculty Research & Creative Works*

We consider the nonlinear Schrödinger equation in three space dimensions with a focusing cubic nonlinearity and defocusing quintic nonlinearity and in the presence of an external inverse-square potential. We establish scattering in the region of the mass-energy plane where the virial functional is guaranteed to be positive. Our result parallels the scattering result of [11] in the setting of the standard cubic-quintic NLS.

A New Fractional Derivative Extending Classical Concepts: Theory And Applications, 2024 Zayed University

#### A New Fractional Derivative Extending Classical Concepts: Theory And Applications, Mutaz Mohammad, Mohamed Saadaoui

*All Works*

In this paper, a novel general definition for the fractional derivative and fractional integral based on an undefined kernel function is introduced. For 0<α≤1, this definition aligns with classical interpretations and is applicable for calculating the derivative in an open negative interval I⊆[a,+∞),a∈R. Additionally, when α=1, the definition coincides with the classical derivative. Fundamental properties of the fractional integral and derivative, including the product rule, quotient rule, chain rule, Rolle's theorem, and the mean value theorem, are derived. These properties are illustrated through various applications to demonstrate their applicability. Furthermore, some applications of solving fractional nonlinear systems of integro-differential equations using framelets are presented.

Why Green Wavelength Is Closer To Blue Than To Red And How It Is Related To Computations: Information-Based Explanation, 2024 Admiral Makarov National Shipbuilding University

#### Why Green Wavelength Is Closer To Blue Than To Red And How It Is Related To Computations: Information-Based Explanation, Victor L. Timchenko, Yury P. Kondratenko, Olga Kosheleva, Vladik Kreinovich, Nguyen Hoang Phuong

*Departmental Technical Reports (CS)*

In our previous papers, we analyzed the idea of using light signals of three basic color -- red, green, and blue -- to speed up computations, in particular fuzzy-related computations. A natural question is: why red, green, and blue? Why not select some other colors: e.g., from the wavelength viewpoint, green is much closer to blue than to green, so why not select colors whose distribution is more even? In this paper, we show that if we consider this problem from the information viewpoint, then the corresponding equal-information criterion indeed implies that the intermediate wavelength should be closer to the …

Why Decisions Based On The Results Of Worst-Case, Most Realistic, And Best-Case Scenarios Work Well?, 2024 Czech Technical University in Prague

#### Why Decisions Based On The Results Of Worst-Case, Most Realistic, And Best-Case Scenarios Work Well?, Miroslav Svitek, Olga Kosheleva, Vladik Kreinovich, Chon Van Le

*Departmental Technical Reports (CS)*

Often, to make an appropriate decision, people try three scenarios: the worst case, the most realistic case, and the best case. This three-scenarios approach often leads to reasonable decisions. A natural question is: why worst case and best case? These extreme cases mean that all numerous independent random factors work in the same direction: either are all stacked for or are all stacked against. Such stacking of random factors is highly improbable. So, at first glance, it would be more beneficial to use more realistic scenarios than the worst case and the best case. However, empirically, decisions based on the …

To Which Interdisciplinary Research Collaborations Should We Pay More Attention?, 2024 Czech Technical University in Prague

#### To Which Interdisciplinary Research Collaborations Should We Pay More Attention?, Miroslav Svitek, Olga Kosheleva, Vladik Kreinovich, Nguyen Hoang Phuong

*Departmental Technical Reports (CS)*

Interdisciplinary research is very important in modern science. However, such a research is not easy, it often needs support and help. Resources that can be used for such a support are limited, so we need to decide which of many possible collaborations we should support. In this paper, we provide a natural simple model of collaboration effectiveness. Based on this model, we conclude that we should support collaborations for which the vector product of the participants' knowledge vectors attains the largest values.

A Full Description Of All Commutative Associative Polynomial Operations On Probabilities, 2024 Czech Technical University in Prague

#### A Full Description Of All Commutative Associative Polynomial Operations On Probabilities, Miroslav Svitek, Olga Kosheleva, Vladik Kreinovich, Nguyen Hoang Phuong

*Departmental Technical Reports (CS)*

When two events are independent, the probability that both events occur is equal to the product p1 * p2 of the probabilities of each of these events. The probability that at least one of these events will occur is equal to p1 + p2 − p1 * p2. In both cases, we have a commutative associative polynomial operation. A natural question is: how can we describe all possible operations of this type? These operations are described in this paper.

Why Kolmolgorov-Arnold Networks (Kan) Work So Well: A Qualitative Explanation, 2024 New Mexico State University

#### Why Kolmolgorov-Arnold Networks (Kan) Work So Well: A Qualitative Explanation, Hung T. Nguyen, Vladik Kreinovich, Olga Kosheleva

*Departmental Technical Reports (CS)*

In the usual deep neural network, weights are adjusted during training, but the activation function remains the same. Lately, it was experimentally shown that if, instead of using the same activation function always, we train the activation functions as well, we get a much better results -- i.e., for the networks with the same number of parameters, we get a much better accuracy. Such networks are called Kolmogorov-Arnold networks. In this paper, we provide a general explanation of why these new networks work so well.

How To Check Continuity Based On Approximate Measurement Results, 2024 University of Latvia

#### How To Check Continuity Based On Approximate Measurement Results, Inese Bula, Vladik Kreinovich

*Departmental Technical Reports (CS)*

In many practical situations, a reasonable conjecture is that, e.g., the dependence of some quantity on the spatial location is continuous, with an appropriate bounds on the difference between the values at nearby points. If we knew the exact values of the corresponding quantity, checking this conjecture would be very straightforward. In reality, however, measurement results are only approximations to the actual values. In this paper, we show how to check continuity based on the approximate measurement results.