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Rigidity Of Ext And Tor Via Flat–Cotorsion Theory, Lars Winther Christensen, Luigi Ferraro, Peder Thompson Nov 2023

Rigidity Of Ext And Tor Via Flat–Cotorsion Theory, Lars Winther Christensen, Luigi Ferraro, Peder Thompson

School of Mathematical & Statistical Sciences Faculty Publications

Let p be a prime ideal in a commutative noetherian ring R and denote by k(p) the residue field of the local ring R_p. We prove that if an R-module M satisfies Ext_R^n(k(p),M) = 0 for some n >= dim R, then Ext_R^i(k(p),M) = 0 holds for all i >= n. This improves a result of Christensen, Iyengar, and Marley by lowering the bound on n. We also improve existing results on Tor-rigidity. This progress is driven by the existence of minimal semi-flat-cotorsion replacements in the derived category as recently proved by Nakamura and Thompson.

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Eigenvalue And Singular Value Inequalities Via Extreme Principles, Fuzhen Zhang Nov 2023

Eigenvalue And Singular Value Inequalities Via Extreme Principles, Fuzhen Zhang

Mathematics Colloquium Series

Given two square matrices of the same order, we consider the eigenvalues and singular values of the sum and product of the matrices. For example, what can be said about the sum of the largest and smallest eigenvalues of the product of two positive semidefinite matrices? This talk reviews some eigenvalue and singular value inequalities recently obtained via minimax principles. In particular, we present singular value inequalities of log-majorization type.

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From Type-2 Fuzzy To Type-2 Intervals And Type-2 Probabilities, Vladik Kreinovich, Olga Kosheleva, Luc Longpré Nov 2023

From Type-2 Fuzzy To Type-2 Intervals And Type-2 Probabilities, Vladik Kreinovich, Olga Kosheleva, Luc Longpré

Departmental Technical Reports (CS)

Our knowledge comes from observations, measurements, and expert opinions. Measurements and observations are never 100% accurate, there is always a difference between the measurement result and the actual value of the corresponding quantity. We gauge the resulting uncertainty either by an interval of possible values, or by a probability distribution on the set of possible values, or by a membership function that describes to what extent different values are possible. The information about uncertainty also comes either from measurements or from expert estimates and is, therefore, also uncertain. It is important to take such "type-2" uncertainty into account. This is …

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Giant Footprints Of Buddha And Generalized Limits, Julio C. Urenda, Vladik Kreinovich Nov 2023

Giant Footprints Of Buddha And Generalized Limits, Julio C. Urenda, Vladik Kreinovich

Departmental Technical Reports (CS)

In many places in Asia, there are footprints claimed to be left by Buddha. Many of them are much larger than the usual size of human feet, up to 150 cm and more in length. In this paper, we provide a possible mathematical explanation for such unusual sizes.

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How To Efficiently Propagate P-Box Uncertainty, Olga Kosheleva, Vladik Kreinovich Nov 2023

How To Efficiently Propagate P-Box Uncertainty, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, to get the desired estimate or prediction, we need to process existing data. This data usually comes from measurements, and measurements are never 100% accurate. Because we only know the input values with uncertainty, the results of processing this data also comes with uncertainty. To make an appropriate decision, we need to know how accurate is the resulting estimate, i.e., how the input uncertainty "propagates" through the data processing algorithm. In the ideal case, when we know the probability distribution of each measurement error, we can, in principle, use Monte-Carlo simulations to describe the uncertainty of …

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Uncertainty Quantification For Results Of Ai-Based Data Processing: Towards More Feasible Algorithms, Christoph Q. Lauter, Martine Ceberio, Vladik Kreinovich, Olga Kosheleva Nov 2023

Uncertainty Quantification For Results Of Ai-Based Data Processing: Towards More Feasible Algorithms, Christoph Q. Lauter, Martine Ceberio, Vladik Kreinovich, Olga Kosheleva

Departmental Technical Reports (CS)

AI techniques have been actively and successfully used in data processing. This tendency started with fuzzy techniques, now neural network techniques are actively used. With each new technique comes the need for the corresponding uncertainty quantification (UQ). In principle, for both fuzzy and neural techniques, we can use the usual UQ methods -- however, these techniques often require an unrealistic amount of computation time. In this paper, we show that in both cases, we can use specific features of the corresponding techniques to drastically speed up the corresponding computations.

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Usually, Either Left And Right Brains Are Equally Active Or Only One Of Them Is Active: First-Principles Explanation, Julio C. Urenda, Vladik Kreinovich Nov 2023

Usually, Either Left And Right Brains Are Equally Active Or Only One Of Them Is Active: First-Principles Explanation, Julio C. Urenda, Vladik Kreinovich

Departmental Technical Reports (CS)

It is known that in most practical situations, either both left and right brains are equally active, or only one of them is active. A recent paper showed that this empirical phenomenon can be explained by a realistic model of the brain effectiveness. In this paper, we show that this conclusion can be made without any specific assumptions about the brain, based on first principles.

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Which Random-Set Representation Of A Fuzzy Set Is The Simplest?, Vladik Kreinovich, Olga Kosheleva, Hung T. Nguyen Nov 2023

Which Random-Set Representation Of A Fuzzy Set Is The Simplest?, Vladik Kreinovich, Olga Kosheleva, Hung T. Nguyen

Departmental Technical Reports (CS)

One of the ways to elicit membership degrees is by polling. For example, we ask a group of people how many believe that 30 C is hot. If 8 out of ten say that it is hot, we assign the degree 8/10 to the statement "30 C is hot". In precise mathematical terms, polling can be described via so-called random sets. It is known that every fuzzy set can be obtained this way, i.e., that every fuzzy set can be represented by an appropriate random set. Moreover, it is known that for many fuzzy sets, there are several different random-set …

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Unexpectedness Stratified By Codimension, Frank Zimmitti Nov 2023

Unexpectedness Stratified By Codimension, Frank Zimmitti

Dissertations and Doctoral Documents, University of Nebraska-Lincoln, 2023–

A recent series of papers, starting with the paper of Cook, Harbourne, Migliore and Nagel on the projective plane in 2018, studies a notion of unexpectedness for finite sets Z of points in N-dimensional projective space. Say the complete linear system L of forms of degree d vanishing on Z has dimension t yet for any general point P the linear system of forms vanishing on Z with multiplicity m at P is nonempty. If the dimension of L is more than the expected dimension of t−r, where r is N+m−1 choose N …

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Game-Theoretic Approaches To Optimal Resource Allocation And Defense Strategies In Herbaceous Plants, Molly R. Creagar Nov 2023

Game-Theoretic Approaches To Optimal Resource Allocation And Defense Strategies In Herbaceous Plants, Molly R. Creagar

Dissertations and Doctoral Documents, University of Nebraska-Lincoln, 2023–

Empirical evidence suggests that the attractiveness of a plant to herbivores can be affected by the investment in defense by neighboring plants, as well as investment in defense by the focal plant. Thus, allocation to defense may not only be influenced by the frequency and intensity of herbivory but also by defense strategies employed by other plants in the environment. We incorporate a neighborhood defense effect by applying spatial evolutionary game theory to optimal resource allocation in plants where cooperators are plants investing in defense and defectors are plants that do not. We use a stochastic dynamic programming model, along …

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A Natural Pseudometric On Homotopy Groups Of Metric Spaces, Jeremy Brazas, Paul Fabel Nov 2023

A Natural Pseudometric On Homotopy Groups Of Metric Spaces, Jeremy Brazas, Paul Fabel

Mathematics Faculty Publications

For a path-connected metric space (X, d), the n-th homotopy group π n ( X) inherits a natural pseudometric from the n-th iterated loop space with the uniform metric. This pseudometric gives π n ( X) the structure of a topological group and when X is compact, the induced pseudometric topology is independent of the metric d. In this paper, we study the properties of this pseudometric and how it relates to previously studied structures on π n ( X). Our main result is that the pseudometric topology agrees with the shape topology on π n ( X) if X …

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Some Properties Of Conjunctivity (Subfitness) In Generalized Settings, M. Andrew Moshier, Jorge Picado, Aleš Pultr Nov 2023

Some Properties Of Conjunctivity (Subfitness) In Generalized Settings, M. Andrew Moshier, Jorge Picado, Aleš Pultr

Mathematics, Physics, and Computer Science Faculty Articles and Research

The property of subfitness used in point-free topology (roughly speaking) to replace the slightly stronger T1-separation, appeared (as disjunctivity) already in the pioneering Wallman’s [16], then practically disappeared to reappear again (conjunctivity, subfitness), until it was in the recent decades recognized as an utmost important condition playing a very special role. Recently, it was also observed that this property (or its dual) appeared independently in general poset setting (e.g. as separativity in connection with forcing). In a recent paper [2], Delzell, Ighedo and Madden discussed it in the context of semilattices. In this article we discuss …

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An Analysis Of Plant Response To Herbivory, Lawrence Gustavo Seminario-Romero Nov 2023

An Analysis Of Plant Response To Herbivory, Lawrence Gustavo Seminario-Romero

Dissertations and Doctoral Documents, University of Nebraska-Lincoln, 2023–

Herbivory is often believed to always negatively affect the growth of plants, but there are plant species that can actually benefit from being consumed by herbivores. These plants have a phenotypic trait known as “compensatory growth.” A plant that exhibits compensatory growth is able to increase its intrinsic growth rate as a response to taking damage caused by herbivory, and as a result, produce more biomass. Very few mathematical models used to describe plant-herbivore interactions have accounted for plant compensatory growth. Moreover, these models tend to assume that herbivores follow a Holling type II functional response, which may not necessarily …

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Conceptualizing Ethics, Authenticity, And Efficacy Of Simulations In Teacher Education, Carrie Wilkerson Lee, Liza Bondurant, Bima Sapkota, Heather Howell, Yvonne Lai Nov 2023

Conceptualizing Ethics, Authenticity, And Efficacy Of Simulations In Teacher Education, Carrie Wilkerson Lee, Liza Bondurant, Bima Sapkota, Heather Howell, Yvonne Lai

School of Mathematical & Statistical Sciences Faculty Publications

This working group was a continuation of working groups in 2019 and 2021 that initially aimed to focus on equity in simulations of practice in mathematics teacher education. We began by discussing our conceptualizations of simulations and equity. Next, we reflected on the lack of work that currently exists at the intersection of simulations and equity as well as our limited collective expertise in this space. We proposed the following areas of potential research: Access,Design, Affective Domains, Teaching Practices, Assessment, Critical Conversations. Attendees self-selected into focus groups and met to discuss their current work and how future work could focus …

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Geometric Aspects Of Quantization And Relationship To Integrability, Paul Bracken Nov 2023

Geometric Aspects Of Quantization And Relationship To Integrability, Paul Bracken

School of Mathematical & Statistical Sciences Faculty Publications

It is the case that quantum mechanics has a deep geometric structure and can be presented accordingly. Quantum mechanics is to a certain degree foreshadowed by the geometry inherent in the geometric structure of classical mechanics. The purpose here is to present some new results and proofs which impact the mathematical structure of quantum mechanics. The relationship between integrability and quantum physics is investigated in terms of geometric ideas and structures. Two physical examples are drawn from these mathematical ideas which directly relate to physics. An introduction as to how these ideas can be extended to infinite degrees of freedom …

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Interactions Of Traveling Waves In Low Order Approximations For The Two-Dimensional Euler Equations, Julio C. Paez, Zhijun Qiao, Vesselin Vatchev Nov 2023

Interactions Of Traveling Waves In Low Order Approximations For The Two-Dimensional Euler Equations, Julio C. Paez, Zhijun Qiao, Vesselin Vatchev

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, we study traveling waves with elastic and inelastic interactions in a low order of approximation. The waves are derived in quadratic or linear order of approximation from the two-dimensional Euler equations. A new type of inelastic interaction that is closely related to numerical study observations of the Euler equations is obtained in a linear order.

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A Review Of Cyber Attacks On Sensors And Perception Systems In Autonomous Vehicle, Taminul Islam, Md. Alif Sheakh, Anjuman Naher Jui, Omar Sharif, Md Zobaer Hasan Nov 2023

A Review Of Cyber Attacks On Sensors And Perception Systems In Autonomous Vehicle, Taminul Islam, Md. Alif Sheakh, Anjuman Naher Jui, Omar Sharif, Md Zobaer Hasan

School of Mathematical & Statistical Sciences Faculty Publications

Vehicle automation has been in the works for a long time now. Automatic brakes, cruise control, GPS satellite navigation, etc. are all common features seen in today's automobiles. Automation and artificial intelligence breakthroughs are likely to lead to an increase in the usage of automation technologies in cars. Because of this, mankind will be more reliant on computer-controlled equipment and car systems in our daily lives. All major corporations have begun investing in the development of self-driving cars because of the rapid advancement of advanced driver support technologies. However, the level of safety and trustworthiness is still questionable. Imagine what …

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The Algorithm For The Design Of Fine Granular Substances’ Smart-Type Heat And Moisture Converters Based On Their Accuracy And Speed Criteria, Erkin Uljaev, Ali Abduakhatovich Abduraxmanov Oct 2023

The Algorithm For The Design Of Fine Granular Substances’ Smart-Type Heat And Moisture Converters Based On Their Accuracy And Speed Criteria, Erkin Uljaev, Ali Abduakhatovich Abduraxmanov

Chemical Technology, Control and Management

The paper describes a technique and algorithm allowing to perform parametric design of smart-type heat and moisture converters (hereinafter SHMC) of fine-grained dispersive materials based on their criteria of accuracy and speed. The proposed algorithm optimizes the process of design of smart-type switches and ensure optimal performance of the switches. The method of calculation and selection of optimal parameters of smart-type heat and moisture converters intended to be used in the measurement of parameters such as heat and humidity of fine dispersive substances are aimed at boosting two parameters, i.e., the accuracy and speed. Also, the design stages have been …

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Aspects Of Non-Associative Gauge Theory, Sergey Grigorian Oct 2023

Aspects Of Non-Associative Gauge Theory, Sergey Grigorian

School of Mathematical & Statistical Sciences Faculty Publications

A smooth loop is the direct non-associative generalization of Lie group. In this paper, we review the theory of smooth loops and smooth loop bundles. This is then used to define a non-associative analog of the Chern-Simons functional.

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Further Generalizations Of Happy Numbers, E. Simonton Williams Oct 2023

Further Generalizations Of Happy Numbers, E. Simonton Williams

Rose-Hulman Undergraduate Mathematics Journal

A positive integer n is defined to be happy if iteration of the function taking the sum of the squares of the digits of n eventually reaches 1. In this paper we generalize the concept of happy numbers in several ways. First we confirm known results of Grundman and Teeple and establish further results extending the known structure of happy numbers to higher powers. Then we construct a similar function expanding the definition of happy numbers to negative integers. Working with this function, we prove a range of results paralleling those already proven for traditional and generalized happy numbers. Finally, …

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Fibonacci Differential Equation And Associated Spiral Curves, Mehmet Pakdemirli Oct 2023

Fibonacci Differential Equation And Associated Spiral Curves, Mehmet Pakdemirli

CODEE Journal

The Fibonacci differential equation is defined with analogy from the Fibonacci difference equation. The linear second order differential equation is solved for suitable initial conditions. The solutions constitute spirals in the polar coordinates. The properties of the spirals with respect to the Fibonacci numbers and the differences between the new spirals and classical spirals are discussed.

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Divisibility Probabilities For Products Of Randomly Chosen Integers, Noah Y. Fine Oct 2023

Divisibility Probabilities For Products Of Randomly Chosen Integers, Noah Y. Fine

Rose-Hulman Undergraduate Mathematics Journal

We find a formula for the probability that the product of n positive integers, chosen at random, is divisible by some integer d. We do this via an inductive application of the Chinese Remainder Theorem, generating functions, and several other combinatorial arguments. Additionally, we apply this formula to find a unique, but slow, probabilistic primality test.

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Elliptic Triangles Which Are Congruent To Their Polar Triangles, Jarrad S. Epkey, Morgan Nissen, Noelle K. Kaminski, Kelsey R. Hall, Nicholas Grabill Oct 2023

Elliptic Triangles Which Are Congruent To Their Polar Triangles, Jarrad S. Epkey, Morgan Nissen, Noelle K. Kaminski, Kelsey R. Hall, Nicholas Grabill

Rose-Hulman Undergraduate Mathematics Journal

We prove that an elliptic triangle is congruent to its polar triangle if and only if six specific Wallace-Simson lines of the triangle are concurrent. (If a point projected onto a triangle has the three feet of its projections collinear, that line is called a Wallace-Simson line.) These six lines would be concurrent at the orthocenter. The six lines come from projecting a vertex of either triangle onto the given triangle. We describe how to construct such triangles and a dozen Wallace-Simson lines.

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Supply Chain Simulation Of Manufacturing Shirts Using System Dynamics For Sustainability, Gurinder Kaur, Ron Kander Oct 2023

Supply Chain Simulation Of Manufacturing Shirts Using System Dynamics For Sustainability, Gurinder Kaur, Ron Kander

School of Design and Engineering Papers

In supply chain management (SCM), goods and services flow from the raw materials stage to the end user with complexities and uncertainty at each stage. Computer modeling and simulation is a particularly useful method to examine supply chain operational issues because it can solve operational complexities that are challenging and time consuming to analyze. Manufacturing companies fear losing valuable time and assets during the manufacturing process; the inaccurate estimation of raw materials, human capital, or physical infrastructure not only leads to monetary loss for the manufacturing unit, but also has a detrimental effect on the environment. The purpose of this …

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Digitisation Of Classical Exercise Practices With Stack: Management Of Large Mathematics Classes In Higher Education Institutions, Idrissa S. Amour, Fatuma Simba, Septimi Kitta, Abdi T. Abdalla Oct 2023

Digitisation Of Classical Exercise Practices With Stack: Management Of Large Mathematics Classes In Higher Education Institutions, Idrissa S. Amour, Fatuma Simba, Septimi Kitta, Abdi T. Abdalla

Tanzania Journal of Science

Management of large classes’ tutorials is a known problem in Mathematics. Engineering mathematics classes enrol around 1000 students at the University of Dar es Salaam. For effective learning, each tutorial session should register not more than 40 students. This requires at least 25 sessions of tutorials per week, which may not be feasible due to both scarce human resources and venues. In this work, we developed online Mathematics exercises based on the System for Teaching and Assessment using Computer Algebra Kernel (STACK). About 300 STACK questions in linear algebra, calculus, complex numbers, and numerical analysis were constructed for weekly tutorials …

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In Euler’S Footsteps: The Enduring Appeal Of Special Functions And Special Problems, Lubomir Markov Oct 2023

In Euler’S Footsteps: The Enduring Appeal Of Special Functions And Special Problems, Lubomir Markov

Mathematics Colloquium Series

We denote the Euler-Riemann zeta function by ζ(x) and the dilogarithm by (x). The question of determining the exact value of ζ(2) (known as the Basel Problem), the one of obtaining as much information as possible about ζ(3), and a host of other related problems have been of unwavering interest for over 300 years. Several other special functions arise from the consideration of series similar to (x). Two of them are Ramanujan's inverse tangent integral and Legendre's chi-function . In our talk we shall derive the power series expansion for the function and use it to obtain several rapidly convergent …

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Structure Of A Total Independent Set, Lewis Stanton Oct 2023

Structure Of A Total Independent Set, Lewis Stanton

Rose-Hulman Undergraduate Mathematics Journal

Let $G$ be a simple, connected and finite graph with order $n$. Denote the independence number, edge independence number and total independence number by $\alpha(G), \alpha'(G)$ and $\alpha''(G)$ respectively. This paper establishes an upper bound for $\alpha''(G)$ in terms of $\alpha(G)$, $\alpha'(G)$ and $n$. We also describe the possible structures for a total independent set containing a given number of elements.

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Nuclear Dimension Of Graph C∗-Algebras With Condition (K), Gregory Faurot, Christopher Schafhauser Oct 2023

Nuclear Dimension Of Graph C∗-Algebras With Condition (K), Gregory Faurot, Christopher Schafhauser

Department of Mathematics: Faculty Publications

We prove that for any countable directed graph E with Condition (K), the associated graph C*-algebra C*(E) has nuclear dimension at most 2. Furthermore, we provide a sufficient condition producing an upper bound of 1.

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On A Stationary Random Knot, Andrey A. Dorogovtsev Oct 2023

On A Stationary Random Knot, Andrey A. Dorogovtsev

Journal of Stochastic Analysis

No abstract provided.

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Certain Invertible Operator-Block Matrices Induced By C*-Algebras And Scaled Hypercomplex Numbers, Daniel Alpay, Ilwoo Choo Oct 2023

Certain Invertible Operator-Block Matrices Induced By C*-Algebras And Scaled Hypercomplex Numbers, Daniel Alpay, Ilwoo Choo

Mathematics, Physics, and Computer Science Faculty Articles and Research

The main purposes of this paper are (i) to enlarge scaled hypercomplex structures to operator-valued cases, where the operators are taken from a C*-subalgebra of an operator algebra on a separable Hilbert space, (ii) to characterize the invertibility conditions on the operator-valued scaled-hypercomplex structures of (i), (iii) to study relations between the invertibility of scaled hypercomplex numbers, and that of operator-valued cases of (ii), and (iv) to confirm our invertibility of (ii) and (iii) are equivalent to the general invertibility of (2×2)-block operator matrices.

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