Digital Commons Network^™

Norm And Numerical Radius Of Single Operators Through Tools And Techniques From Multivariable Operator Theory, Raul E. Curto, Sang Hoon Lee, Jasang Yoon

School of Mathematical & Statistical Sciences Faculty Publications

We employ tools and techniques from multivariable operator theory to obtain new proofs and extensions of well known inequalities regarding the norm and the numerical radius of elementary operators defined on the C⁎" role="presentation" style="box-sizing: border-box; margin: 0px; padding: 0px; position: relative;">C⁎C⁎–algebra of all bounded operators on Hilbert space, or on the ⁎–ideal of Hilbert-Schmidt operators. In the process, we provide new insights on the study of Heinz-type inequalities related to the arithmetic-geometric mean inequality, and generalize results of several authors, including R. Bhatia, G. Corach, C. Davis, F. Kittaneh, and M.S. Moslehian. To estimate the norm, our approach …

Eigen’S Paradox And The Quasispecies Model In A Non-Archimedean Framework, Wilson A. Zuniga-Galindo

School of Mathematical & Statistical Sciences Faculty Publications

In this article we present a new p -adic generalization of the Eigen–Schuster model where the genomes (sequences) are represented by words written in the alphabet { 0 , 1 , … , p − 1 } , where p is a prime number, with a time variable length. The time evolution of the concentration of a given sequence is controlled by a p -adic evolution equation. The long term behavior of the concentration of a sequence depends on a fitness function f , a mutation measure Q , and an initial concentration distribution. The new model provides essentially two …

Leveraging The "Large" In Large Lecture Statistics Classes, Kady Schneiter, Kimberleigh Felix Hadfield, Jenny Lee Clements

Mathematics and Statistics Faculty Publications

Being a teacher or a student in a class with a large enrollment can be intimidating. Often, teachers view comforts that are common to small classes as unattainable in a larger class, including knowing students’ names, using active learning, employing group work, and creating group discussion. Students in large classes may find that the class size leads to isolation. At Utah State University, we offer introductory statistics classes for various audiences using a large lecture format. The authors have collectively led these large lectures dozens of times and found that, despite its shortcomings, the large lecture format can be an …

On Cantor Sets Defined By Generalized Continued Fractions, Danielle Hedvig, Masha Gorodetski

Rose-Hulman Undergraduate Mathematics Journal

We study a special class of generalized continuous fractions, both in real and complex settings, and show that in many cases, the set of numbers that can be represented by a continued fraction for that class form a Cantor set. Specifically, we study generalized continued fractions with a fixed absolute value and a variable coefficient sign. We ask the same question in the complex setting, allowing the coefficient's argument to be a multiple of \pi/2. The numerical experiments we conducted showed that in these settings the set of numbers formed by such continued fractions is a Cantor set for large …

The New Era In American Mathematics, 1920-1950 (Book Review), Calvin Jongsma

Faculty Work Comprehensive List

Reviewed Title: The New Era in American Mathematics, 1920-1950 by Karen Hunger Parshall. Princeton University Press, 2022. 640 pages. ISBN: 9780691197555.

Sd Prism: Visualizing The Standard Deviation As The Size Of A Prism Using R/Rstudio, Hieu Nguyen, Tom Nguyen, Mamunur Rashid, Jyotirmoy Sarkar

Student Research

SD Prism is a graphical R package used for visualizing the standard deviation of a data set. Given a raw data set, the standard deviation (SD) is defined as the square-root of twice the mean square of all pairwise half deviations between any two sample observations. This interpretation leads to a geometric visualization of the sample SD, and a more elementary explanation as to why the denominator in the sample variance is one less than the sample size. In this article we will explain step by step how to understand it mathematically and how the package implements the methodology to …

Numerical Analysis Of A Second Order Ensemble Method For Evolutionary Magnetohydrodynamics Equations At Small Magnetic Reynolds Number, John Carter, Nan Jiang

Mathematics and Statistics Faculty Research & Creative Works

We study a second order ensemble method for fast computation of an ensemble of magnetohydrodynamics flows at small magnetic Reynolds number. Computing an ensemble of flow equations with different input parameters is a common procedure for uncertainty quantification in many engineering applications, for which the computational cost can be prohibitively expensive for nonlinear complex systems. We propose an ensemble algorithm that requires only solving one linear system with multiple right-hands instead of solving multiple different linear systems, which significantly reduces the computational cost and simulation time. Comprehensive stability and error analyses are presented proving conditional stability and second order in …

Why Seneca Effect?, Sean R. Aguilar, Vladik Kreinovich

Departmental Technical Reports (CS)

Already ancients noticed that decrease is usually faster than growth -- whether we talk about companies or empires. A modern researcher Ugo Bardi confirmed that this phenomenon is still valid today. He called it Seneca effect, after the ancient philosopher Seneca -- one of those who observed this phenomenon. In this paper, we provide a natural explanation for the Seneca effect.

Why Smaller-Size Objects Affect The Flow Much More Than Larger Ones: A Geometric Explanation With Applications Ranging From Volcanoes And Tornadoes To Blood, Fish, And Building Preservation, Laxman Bokati, Vladik Kreinovich

Departmental Technical Reports (CS)

At first glance, the larger the object, the larger should be its effect on the surroundings -- in particular, the larger should be its effect on the surrounding flow. However, in many practical situations, we observe the opposite effect: smaller-size particles affect the flow much more than larger-size particles. This seemingly counterintuitive phenomena has been observed in many situations: lava flow in the volcanoes, air circulation in tornadoes, blood flow in a body, the effect of fish on water circulation in the ocean, and the effect of added particles on seeping water that damages historic buildings. In this paper, we …

What Is The Most Adequate Fuzzy Methodology?, Noah Velasco, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In practice, there is often a need to describe the relation y = f(x) between two quantities in algorithmic form: e.g., we want to describe the control value y corresponding to the given input x, or we want to predict the future value y based on the current value x. In many such cases, we have expert knowledge about the desired dependence, but experts can only describe their knowledge by using imprecise ("fuzzy") words from a natural language. Methodologies for transforming such knowledge into an algorithm y = f(x) are known as fuzzy methodologies. There exist several fuzzy methodologies, a …

A General Commonsense Explanation Of Several Medical Results, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In this paper, we show that many recent experimental medical results about the effect of different factors on our health can be explained by common sense ideas.

On Some Mixing Properties Of Copula-Based Markov Chains, Martial Longla, Hamadou Mous-Abou, Isidore Seraphin Ngongo

Faculty and Student Publications

This paper brings some insights of ψ′-mixing, ψ∗-mixing and ψ-mixing for copula-based Markov chains and the perturbations of their copulas. We provide new tools to check Markov chains for ψ-mixing or ψ′-mixing. We show that perturbations of ψ′-mixing copula-based Markov chains are ψ′-mixing while perturbations of ψ-mixing Markov chains are not necessarily ψ-mixing Markov chains, even when the perturbed copula generates ψ-mixing. The Farlie–Gumbel–Morgenstern, gaussian and Ali-Mikhail-Haq copula families are considered among other examples. A statistical study is provided to emphasize the impact of perturbations on copula-based Markov chains in a simulation study. Moreover, we provide a correction to a …

Why Best-Worst Method Works Well, Sean Aguilar, Vladik Kreinovich

Departmental Technical Reports (CS)

In many cases, experts are much more accurate when they estimate the ratio of two quantities than when they estimate the actual values. For example, if it difficult to accurately estimate the height of a person on a photo, but if we have two people standing side by side, we can easily estimate to what extent one of them is taller than the other one. To get accurate estimates, it is therefore desirable to use such ratio estimates. Empirical analysis shows that to obtain the most accurate results, we need to compare all the objects with either the "best" object …

How Hot Is Too Hot, Sofia Holguin, Vladik Kreinovich

Departmental Technical Reports (CS)

A recent study has shown that the temperature threshold -- after which even young healthy individuals start feeling the effect of heat on their productivity -- is 30.5 ± 1 C. In this paper, we use decision theory ideas to provide a theoretical explanation for this empirical finding.

Invariance Explains Empirical Success Of Many Intelligent Techniques, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many applications of intelligent computing, we need to choose an appropriate function -- e.g., an appropriate re-scaling function, or an appropriate aggregation function. In applications of intelligent techniques, the problem of selecting an optimal function is usually too complex or too imprecise to be solved analytically, so the best functions are found empirically, by trying a large number of alternatives. In this paper, we show that in many such cases, the resulting empirical choice can be explained by natural invariance ideas. Example range from applications to building blocks of intelligent techniques -- such as aggregation (including hierarchical aggregation) and …

Why Exponential Almon Lag Works Well In Econometrics: An Invariance-Based Explanation, Laxman Bokati, Vladik Kreinovich

Departmental Technical Reports (CS)

In many econometric situations, we can predict future values of relevant quantities by using an empirical formula known as exponential Almon lag. While this formula is empirically successful, there have been no convincing theoretical explanation for this success. In this paper, we provide such a theoretical explanation based on general invariance ideas.

Seemingly Counter-Intuitive Features Of Good-To-Great Companies Actually Make Perfect Sense: Possible Algorithmics-Based Explanations, Francisco Zapata, Eric Smith, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In the late 1990s, researchers analyzed what distinguishes great companies from simply good ones. They found several features that are typical for great companies. Interestingly, most of these features seem counter-intuitive. In this paper, we show that from the algorithmic viewpoint, many of these features make perfect sense. Some of the resulting explanations are simple and straightforward, other explanations rely on complex not-well-publicized results from theoretical computer science.

Teaching Euclidean Geometry Using Proof And Dynamic Geometry Software, Cheryl Stoddard

Mathematics Graduate Theses

This paper examines incorporating proofs and dynamic geometry software when teaching Euclidean geometry to produce deeper understanding of geometric relationships for middle and secondary level students. The utilization of technology through classroom investigations and activities has shown to have an impact on the achievement scores on posttests and in the development of mathematical proof.

Introduction To Mathematical Analysis I - 3rd Edition, Beatriz Lafferriere, Gerardo Lafferriere, Mau Nam Nguyen

PDXOpen: Open Educational Resources

Video lectures explaining problem solving strategies are available.

This set of lecture notes is meant to provide students with a strong foundation in mathematical analysis. Such a foundation is crucial for future study of deeper topics of analysis. Students should be familiar with most of the concepts presented here after completing the calculus sequence. However, these concepts will be reinforced through rigorous proofs.

Introduction to Mathematical Analysis II contains lecture notes that complement this book, with sections on integration, numerical series, series of functions, and selected advanced topics. Together, both volumes support a one semester or two-quarter course on introductory …

Generalization Of Bi-Canonical Degrees, Joseph Brennan, Laura Ghezzi, Jooyoun Hong, Wolmer Vasconcelos

Publications and Research

We discuss invariants of Cohen-Macaulay local rings that admit a canonical module ω. Attached to each such ring R, when ω is an ideal, there are integers–the type of R, the reduction number of ω–that provide valuable metrics to express the deviation of R from being a Gorenstein ring. In (Ghezzi et al. in JMS 589:506–528, 2017) and (Ghezzi et al. in JMS 571:55–74, 2021) we enlarged this list with the canonical degree and the bi-canonical degree. In this work we extend the bi-canonical degree to rings where ω is not necessarily an ideal. We also discuss generalizations to rings …

Darboux Transformation And Solitonic Solution To The Coupled Complex Short Pulse Equation, Bao-Feng Feng, Liming Ling

School of Mathematical & Statistical Sciences Faculty Publications

The Darboux transformation (DT) for the coupled complex short pulse (CCSP) equation is constructed through the loop group method. The DT is then utilized to construct various exact solutions including bright soliton, dark-soliton, breather and rogue wave solutions to the CCSP equation. In case of vanishing boundary condition (VBC), we perform the inverse scattering analysis to understand the soliton solution better. Breather and rogue wave solutions are constructed in case of non-vanishing boundary condition (NVBC). Moreover, we conduct a modulational instability (MI) analysis based on the method of squared eigenfunctions, whose result confirms the condition for the existence of rogue …

Definite Condition Of The Evolutionary (P)Over-Right-Arrow(X)-Laplacian Equation, Huashui Zhan, Zhaosheng Feng

School of Mathematical & Statistical Sciences Faculty Publications

For the nonlinear degenerate parabolic equations, how to find an appropriate boundary value condition to ensure the well-posedness of weak solution has been an interesting and challenging problem. In this paper, we develop the general characteristic function method to study the stability of weak solutions based on a partial boundary value condition.

Fabrications And Applications Of Micro/Nanofluidics In Oil And Gas Recovery: A Comprehensive Review, Junchen Liu, Yandong Zhang, Mingzhen Wei, Xiaoming He, Baojun Bai

Geosciences and Geological and Petroleum Engineering Faculty Research & Creative Works

Understanding fluid flow characteristics in porous medium, which determines the development of oil and gas oilfields, has been a significant research subject for decades. Although using core samples is still essential, micro/nanofluidics have been attracting increasing attention in oil recovery fields since it offers direct visualization and quantification of fluid flow at the pore level. This work provides the latest techniques and development history of micro/nanofluidics in oil and gas recovery by summarizing and discussing the fabrication methods, materials and corresponding applications. Compared with other reviews of micro/nanofluidics, this comprehensive review is in the perspective of solving specific issues in …

The Local Lifting Problem For Curves With Quaternion Actions, George Mitchell

Dissertations, Theses, and Capstone Projects

The lifting problem asks whether one can lift Galois covers of curves defined over positive characteristic to Galois covers of curves over characteristic zero. The lifting problem has an equivalent local variant, which asks if a Galois extension of complete discrete valuation rings over positive characteristic, with algebraically closed residue field, can be lifted to characteristic zero. In this dissertation, we content ourselves with the study of the local lifting problem when the prime is 2, and the Galois group of the extension is the group of quaternions. In this case, it is known that certain quaternion extensions cannot be …

Asymptotic Classes, Pseudofinite Cardinality And Dimension, Alexander Van Abel

Dissertations, Theses, and Capstone Projects

We explore the consequences of various model-theoretic tameness conditions upon the behavior of pseudofinite cardinality and dimension. We show that for pseudofinite theories which are either Morley Rank 1 or uncountably categorical, pseudofinite cardinality in ultraproducts satisfying such theories is highly well-behaved. On the other hand, it has been shown that pseudofinite dimension is not necessarily well-behaved in all ultraproducts of theories which are simple or supersimple; we extend such an observation by constructing simple and supersimple theories in which pseudofinite dimension is necessarily ill-behaved in all such ultraproducts. Additionally, we have novel results connecting various forms of asymptotic classes …

Computation Of Risk Measures In Finance And Parallel Real-Time Scheduling, Yajuan Li

Dissertations

Many application areas employ various risk measures, such as a quantile, to assess risks. For example, in finance, risk managers employ a quantile to help determine appropriate levels of capital needed to be able to absorb (with high probability) large unexpected losses in credit portfolios comprising loans, bonds, and other financial instruments subject to default. This dissertation discusses the computation of risk measures in finance and parallel real-time scheduling.

Firstly, two estimation approaches are compared for one risk measure, a quantile, via randomized quasi-Monte Carlo (RQMC) in an asymptotic setting where the number of randomizations for RQMC grows large, but …

Low-Reynolds-Number Locomotion Via Reinforcement Learning, Yuexin Liu

Dissertations

This dissertation summarizes computational results from applying reinforcement learning and deep neural network to the designs of artificial microswimmers in the inertialess regime, where the viscous dissipation in the surrounding fluid environment dominates and the swimmer’s inertia is completely negligible. In particular, works in this dissertation consist of four interrelated studies of the design of microswimmers for different tasks: (1) a one-dimensional microswimmer in free-space that moves towards the target via translation, (2) a one-dimensional microswimmer in a periodic domain that rotates to reach the target, (3) a two-dimensional microswimmer that switches gaits to navigate to the designated targets in …

Parameter Selection In Fully Homomorphic Encryption Schemes And Fhe Applications, Cavidan Yakupoglu

Dissertations

Quantum computing has been gaining momentum as a result of recent technological advances. Existing cryptographic systems rely on the difficult problems that can be solved by sufficiently powerful quantum computers. As the quantum age approaches, the desire to discover new difficult problems that cannot be solved by quantum systems has increased. Lattice-based cryptography is a prominent tool for the post-quantum era that facilitates the implementation of encryption systems for practical applications.

The Learning with Error (LWE) and Ring-LWE problems introduce new lattice hardness assumptions that have been incorporated into public-key cryptosystems to facilitate the implementation of numerous privacy-enhancing applications. Fully …

Computing Rational Powers Of Monomial Ideals, Pratik Dongre, Benjamin Drabkin, Josiah Lim, Ethan Partida, Ethan Roy, Dylan Ruff, Alexandra Seceleanu, Tingting Tang

Department of Mathematics: Faculty Publications

This paper concerns fractional powers of monomial ideals. Rational powers of a monomial ideal generalize the integral closure operation as well as recover the family of symbolic powers. They also highlight many interesting connections to the theory of convex polytopes. We provide multiple algorithms for computing the rational powers of a monomial ideal. We also introduce a mild generalization allowing real powers of monomial ideals. An important result is that given any monomial ideal I, the function taking a real number to the corresponding real power of I is a step function which is left continuous and has rational …

Frobenius Objects In The Category Of Spans, Ivan Contreras, Molly Keller, Rajan Amit Mehta

Mathematics Sciences: Faculty Publications

We consider Frobenius objects in the category Span, where the objects are sets and the morphisms are isomorphism classes of spans of sets. We show that such structures are in correspondence with data that can be characterized in terms of simplicial sets. An interesting class of examples comes from groupoids.

Our primary motivation is that Span can be viewed as a set-theoretic model for the symplectic category, and thus Frobenius objects in Span provide set-theoretic models for classical topological field theories. The paper includes an explanation of this relationship.

Given a finite commutative Frobenius object in Span, one can obtain …