Mathematics Commons^™

Why Bernstein Polynomials: Yet Another Explanation, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many computational situations -- in particular, in computations under interval or fuzzy uncertainty -- it is convenient to approximate a function by a polynomial. Usually, a polynomial is represented by coefficients at its monomials. However, in many cases, it turns out more efficient to represent a general polynomial by using a different basis -- of so-called Bernstein polynomials. In this paper, we provide a new explanation for the computational efficiency of this basis.

How To Make A Decision Under Interval Uncertainty If We Do Not Know The Utility Function, Jeffrey Escamilla, Vladik Kreinovich

Departmental Technical Reports (CS)

Decision theory describes how to make decisions, in particular, how to make decisions under interval uncertainty. However, this theory's recommendations assume that we know the utility function -- a function that describes the decision maker's preferences. Sometimes, we can make a recommendation even when we do not know the utility function. In this paper, we provide a complete description of all such cases.

Shall We Place More Advanced Students In A Separate Class?, Shahnaz Shahbazova, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In every class, we have students who are more advanced and students who are more behind. From this viewpoint, it seems reasonable to place more advanced students in a separate class. This should help advanced students progress faster, and it should help other students as well, since the teachers in the remaining class can better attend to their needs. However, empirically, this does not work: when we form a separate class, the overall amount of gained knowledge decreases. In this paper, we provide a possible explanation for this seemingly counterintuitive phenomenon.

Towards An Optimal Design: What Can We Recommend To Elon Musk?, Martine Ceberio, Olga Kosheleva, Vladik Kreinovich, Hung T. Nguyen

Departmental Technical Reports (CS)

Elon Musk's successful "move fast and break things" strategy is based on the fact that in many cases, we do not need to satisfy all usual constraints to be successful. By sequentially trying smaller number of constraints, he finds the smallest number of constraints that are still needed to succeed -- and using this smaller number of constrains leads to a much cheaper (and thus, more practical) design. In this strategy, Musk relies on his intuition -- which, as all intuitions, sometimes works and sometimes doesn't. To replace this intuition, we propose an algorithm that minimizes the worst-case cost of …

Faithful Representation Of Free Groups And Congruent Subgroups Of Sl3(Z), Julius Kurian

Theses

This thesis is concerned with the matrix representation of a free nonabelian group by matrices of size ≥ 3. We proceed from defining an equivalence class and then transitioning to free groups.We discuss in details the group G_n(k) which is the group generated by the matrices filled with first, (second, etc.) column, except for the intersection with the diagonal, and we have ones on the diagonal and zeros at the other places. The filled places are occupied by the same parameter k. An alternative proof for the known fact that G_n(3) is not …

Pricing Asian Options During Crises With The Impact Of Exogenous Event, Moath Ali Bakour

Theses

Asian options are financial derivatives products whose value depends on the value of underlying asset prices. These options are called path-dependent since their payoff is built on the prices of the underlying asset over some time. As in the case of the European options, the Asian option pricing problem is primarily subject to the prediction model for asset prices. The pioneer Black-Scholes model in the paper [2] suggests a GBM-Geometric Brownian motion. The Black-Scholes formula has several shortcomings. For instance, the Geometric Brownian motion does not take into consideration crises. Another problem in the Black- Scholes model is that it …

Syllabus For Oer Version Of Math 20100, Thea Pignataro

Open Educational Resources

Syllabus for OER version of first semester calculus with analytic geometry for physical sciences and engineering.

Mediating Effect Of Bmi On The Association Of Economic Status And Coexistence Of Hypertension And Diabetes In Bangladesh: A Counterfactual Framework-Based Weighting Approach, Foyez , Md. Jamal Hossain Ahmmed, Md. Jamal Hossain, Md Tareq Ferdous Khan, Muhammad Mahabub Rahaman Manik, Saimon Shahriar, Dulal Chandra Nandi, Md Parvej Hussain

Mathematics and Statistics Faculty Publications

Background and Aims

Non-communicable diseases such as hypertension and diabetes are matters of huge concern worldwide, with an increasing trend in prevalence over the previous decade. First of all, this study aimed to evaluate the association between economic status (ES) and body mass index (BMI), ES and comorbidity of hypertension and diabetes, and BMI and comorbidity independently. Second, it explored the mediating role of BMI in the association between ES and comorbidity of hypertension and diabetes. Finally, it investigated whether the mediating effect differs with the place of residence, gender, and education levels.

Methods

A total of 11,291 complete cases …

Circle Packings From Tilings Of The Plane, Philip Rehwinkel , '22, Ian Whitehead, David Yang , '24, Mengyuan Yang , '25

Mathematics & Statistics Faculty Works

We introduce a new class of fractal circle packings in the plane, generalizing the polyhedral packings defined by Kontorovich and Nakamura. The existence and uniqueness of these packings are guaranteed by infinite versions of the Koebe–Andreev–Thurston theorem. We prove structure theorems giving a complete description of the symmetry groups for these packings. And we give several examples to illustrate their number-theoretic and group-theoretic significance.

Discrete Macaulay-Steiner Geometry, Nikola Kuzmanovski

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

This thesis is concerned with discrete isoperimetric inequalities and Hilbert functions. Two generalizations of the Ahlswede-Cai local global principle are presented. These results give positive answers to two questions posed by Harper. One of these results is achieved by proving uniqueness of the lexicographic and colexicographic orders in two dimensions. The other result generalizes the technique which is commonly known as compression and includes almost all previously published results in this direction. The Ahlswede-Cai local global principle is a direct corollary of this result. Optimal downsets are studied in rectangles and triangles. All optimal downsets are found. The main result …

Visualizing The Standard Deviation Via Revolution Using R/Rstudio, Hieu Nguyen '25, Trung Pham '25, Mamunur Rashid, Jyotirmoy Sarkar

Student Research

The standard deviation is a commonly used statistical measure to quantify the level of variation present in a set of numbers or in a random variable. Sarkar and Rashid (2016) introduced an interpretation of the population standard deviation as the radius of a cylinder with a volume equivalent to that of the solid of revolution when the 2-D graph of the empirical cumulative distribution function is revolved about the vertical line through the mean. This article demonstrates step-by-step how to use the RevSD package in R/RStudio to visualize the standard deviation of data using this innovative technique. The RevSD package …

The Inextricability Of Students’ Mathematical And Physical Reasoning In Quantum Mechanics Problems, Kaitlyn Stephens Serbin, Megan Wawro

School of Mathematical & Statistical Sciences Faculty Publications

Reasoning with mathematics plays an important role in university students’ learning throughout their courses in the scientific disciplines, such as physics. In addition to understanding mathematical concepts and procedures, physics students often must mathematize physical constructs in terms of their associated mathematical structures and interpret mathematical entities in terms of the physical context. In this study, we investigate physics students’ reasoning about mathematics in relation to physics content addressed in two quantum mechanics problems. Through qualitative analysis of interview data from twelve students, results show that 1) students use intricate, nonuniform problem-solving methods with reasoning that moves fluidly between structural …

Normalized Ground State Of A Mixed Dispersion Nonlinear Schrodinger Equation With Combined Power-Type Nonlinearities, Zhouji Ma, Xiaojun Chang, Zhaosheng Feng

School of Mathematical & Statistical Sciences Faculty Publications

We study the existence of normalized ground state solutions to a mixed dispersion fourth-order nonlinear Schrodinger equation with combined power-type nonlinearities. By analyzing the subadditivity of the ground state energy with respect to the prescribed mass, we employ a constrained minimization method to establish the existence of ground state that corresponds to a local minimum of the associated functional. Under certain conditions, by studying the monotonicity of ground state energy as the mass varies, we apply the constrained minimization arguments on the Nehari-Pohozaev manifold to prove the existence of normalized ground state solutions.

The Eliahou-Kervaire Resolution Over A Skew Polynomial Ring, Luigi Ferraro, Alexis Hardesty

School of Mathematical & Statistical Sciences Faculty Publications

In a 1987 paper, Eliahou and Kervaire constructed a minimal resolution of a class of monomial ideals in a polynomial ring, called stable ideals. As a consequence of their construction they deduced several homological properties of stable ideals. Furthermore they showed that this resolution admits an associative, graded commutative product that satisfies the Leibniz rule. In this paper we show that their construction can be extended to stable ideals in skew polynomial rings. As a consequence we show that the homological properties of stable ideals proved by Eliahou and Kervaire hold also for stable ideals in skew polynomial rings.

On The Projections And Unitary Groups Of Unital C*-Algebras, Fouzia Shaheen

Dissertations

H. Dye proved that the unitary group in a factor determines the algebraic type of that factor. Al-Rawashdeh, Booth and Giordano established that, for a large class of simple unital C*-algebras, an isomorphism between the unitary groups induces an isomorphism between their K₀-ordered group and 1-groups. Then using the results of Dadarlat-Elliot-Gong and Kirchberg-Phillips, the C*-algebras are isomorphic. Dye introduced special projections P_i,j (a) of the matrix algebra M_n(A), and he used it as a main tool to establish his results in the case of von Neumann factors. Precisely, in case of von Neumann algebra, …

On Generating Bijections For Permutations And Inversion Sequences, Melanie J. Ferreri

Dartmouth College Ph.D Dissertations

Given an algebraic proof of a combinatorial identity, we use recursive methods to construct a bijection demonstrating the identity.

Our first application centers around derangements and nonderangements. A derangement is a permutation with no fixed point, and a nonderangement is a permutation with at least one fixed point. There is a one-term recurrence for the number of derangements of n elements, and we describe a bijective proof of this recurrence which can be found using a recursive map. We then show the combinatorial interpretation of this bijection and how it compares with other known bijections, and show how this extends …

Exploring Quaternion Neural Network Loss Surfaces, Jeremiah Bill, Bruce A. Cox

Faculty Publications

This paper explores the superior performance of quaternion multi-layer perceptron (QMLP) neural networks over real-valued multi-layer perceptron (MLP) neural networks, a phenomenon that has been empirically observed but not thoroughly investigated. The study utilizes loss surface visualization and projection techniques to examine quaternion-based optimization loss surfaces for the first time. The primary contribution of this research is the statistical evidence that QMLP models yield smoother loss surfaces than real-valued neural networks, which are measured and compared using a robust quantitative measure of loss surface “goodness” based on estimates of surface curvature. Extensive computational testing validates the effectiveness of these surface …

Doubly Almost Bipartite Leonard Pairs, Shuichi Masuda

Dissertations and Theses

In Linear algebra, the concept of Leonard pair (LP) was motivated by the theory of Q-polynomial distance-regular graphs. In this dissertation, we will first give a brief introduction to LPs and to two closely-related classes of objects: (i) bipartite Leonard pairs (BLPs) and (ii) almost bipartite Leonard pairs (ABLPs). Taking these as departure points, we will introduce a new class of object - doubly almost bipartite Leonard pairs (DABLPs). The primary aim of our work is to fully classify (up to isomorphism) this new family. In addition, since there is known to be a natural correspondence between Leonard pairs …

Analyzing The Influence Of Design And Operating Conditions On Combustion And Emissions In Premixed Turbulent Flames: A Comprehensive Review, Medhat Elkelawy Prof. Dr. Eng., E. A. El Shenawy Prof. Dr., Hagar Alm-Eldin Bastawissi, Ibrahim Ali Mousa Eng., Mohamed M. Abdel-Raouf Ibrahim Dr. Eng.

Journal of Engineering Research

Recently, premixed combustion has dominated the field of combustion research worldwide. The current work is a review that addresses the effects of design and operating regimes on the combustion and emission characteristics of premixed turbulent flames. The study accounts for recent developments aimed at overcoming combustor operability issues that influence emissions and flame stability. Various experimental setups have been utilized in investigations, with results pertaining to performance and emissions concerning premixed turbulent flames. Thus, the objective of this paper is to provide a comprehensive review of the effects of swirl vane angles and equivalence fuel-air ratios for tests conducted both …

An Exploration Of Secondary School Science Teacher’S Epistemological Belief In Malang, Anindyta Nur Rizkyana Safitri, Endang Purwaningsih, Nuril Munfaridah, Alif Darmawan

Jurnal Pendidikan Sains

The objective of this study is to delve into the epistemological convictions of secondary school science teachers in Malang across various tenures, namely senior teachers, junior teachers, and pre-service teachers. Employing an explanatory sequential design model, the research employs a mixed-method approach. Quantitative data, gathered through the administration of the Scientific Epistemology Beliefs Questionnaire (n = 30), are complemented by qualitative insights drawn from interviews with teachers (n = 6) concerning their epistemological orientations. Descriptive statistics and qualitative analyses, utilizing an a priori coding technique aligned with the study’s conceptual framework, were conducted on the interview data. Quantitative analysis reveals …