Mathematics Commons^™

Euler’S Original Derivation Of Elastica Equation, Shigeki Matsutani

Euleriana

Euler derived the differential equations of elastica by the variational method in 1744, but his original derivation has never been properly interpreted or explained in terms of modern mathematics. We elaborate Euler's original derivation of elastica and show that Euler used Noether's theorem concerning the translational symmetry of elastica, although Noether published her theorem in 1918. It is also shown that his equation is essentially the static modified KdV equation which is obtained by the isometric and isoenergy conditions, known as the Goldstein-Petrich scheme.

Early Theories On Fluid Resistance And Translation Of Euler’S “Dilucidationes De Resistentia Fluidorum”, Sylvio R. Bistafa

Euleriana

In 1763, Euler published Dilucidationes de resistentia fluidorum (Explanations on the resistance of fluids), a memoir that challenges the fluid resistance theories proposed by Isaac Newton and d’Alembert. Euler's work explores the resistance experienced by solid bodies moving through fluids, critiquing both Newton's "common rule" and d’Alembert's paradox, which predicted zero resistance for non-viscous fluids. Euler's treatise is divided into two parts: the first focuses on the mathematical modeling of fluid flow patterns, while the second addresses the calculation of fluid resistance on surfaces. Despite significant advancements, Euler's work remains constrained by the limitations of non-viscous fluid assumptions, ultimately grappling …

Assistance For The Calculation Of Sines (Translation Of E246), Julian Schennach

Euleriana

Paralleling his famous relation e^iφ = cos(φ) + i sin(φ), Euler establishes the equality (cos φ + i sin φ)ⁿ = (cos nφ + i sin nφ). He uses it to comprehensively derive trigonometric identities that convert arbitrary powers of sines and cosines of an angle (and products thereof) into sums of sines and cosines of multiples of that angle. Some negative and fractional powers are shown to yield infinite series. Euler further describes a general method to evaluate various infinite series involving weighted trigonometric functions. These results foreshadow Fourier series. As Euler points out, the scope of …

On Amicable Numbers, Jonathan David Evans

Euleriana

This is an English translation of Euler's 1750 paper "De numeris amicabilibus" (E152), the most substantial of his three works with this name. In it, he expounds at great length the ad hoc methods he has developed to search for pairs of amicable numbers, concluding with a list of around 60 new pairs.

A Reader's Guide To Daniel Bernoulli's "Recurrent Series", Stacy Langton

Euleriana

This is a commentary and reader’s guide to Daniel Bernoulli’s article “Observations concerning recurrent series”. It is intended to help the modern reader understand what Bernoulli is doing in that article.

Daniel Bernoulli's "Observations Concerning Recurrent Series", Stacy Langton

Euleriana

Daniel Bernoulli’s article "Observations concerning Recurrrent Series" is one of the gems of the mathematical literature. It describes a numerical method for solving polynomial equations. This method, now known as “Bernoulli’s method”, is still in use today. The basis of the method is the study of certain sequences of numbers that satisfy what we now call linear recurrence relations. Bernoulli’s article shows how to find solutions of these recurrence relations. A nice application is a now well-known formula for the Fibonacci numbers. Bernoulli also uses his theory of recurrence relations to give a proof of an early form of De …

Comprehensive Conversations, Christopher Goff, Erik Tou

Euleriana

An introduction to the contents in Volume 5 (Issue 1) of Euleriana.

Generalized Transversality Conditions For Fuzzy Quantum-Symmetric Variational Problems Via Granular Approach, Martin Bohner, Ewa Girejko, Agnieszka B. Malinowska, Linh Nguyen, Baruch Schneider, Tri Truong

Mathematics and Statistics Faculty Research & Creative Works

This paper investigates fuzzy q-symmetric variational problems with natural boundary conditions. Based on the relative distance measure fuzzy arithmetic and horizontal membership functions (HMFs), we propose novel concepts of differentiability and integrability for fuzzy functions on quantum geometric subsets of real numbers. Then, fundamental foundations of q-symmetric calculus of variations based on HMFs are provided. With the help of HMFs and granular q-symmetric differentiability, we derive necessary optimality conditions for fuzzy q-symmetric variational problems that depend on free endpoints. Moreover, sufficient conditions for minimizers of q-symmetric variational problems are obtained. Some numerical examples illustrating the proposed approach are presented.

Novel Statistical And Topological Data Analyses Of 2d Electronic Images, Robert L. Paige, Vic Patrangenaru

Mathematics and Statistics Faculty Research & Creative Works

In this paper, novel statistical and topological data analyses of 2D images are developed. One considers methodologies based on the Region Covariance Descriptor (RCD) and Topological Data Analysis (TDA) rooted in the simplicial as well as cubical persistent homologies. These methods provide statistical methods for data from populations of complex data objects that are elements of non-Euclidean spaces. The 2D image data considered consist of pictures of two leaves—A and B—from the same tree, twenty of each leaf, from different perspectives. The novel statistical procedures developed are used for correctly determining that leaf A images and leaf B images are …

Strong External Difference Families And Classification Of Α-Valuations, Donald L. Kreher, Maura B. Paterson, Douglas R. Stinson

Michigan Tech Publications

One method of constructing (Formula presented.) -SEDFs (i.e., strong external difference families) in (Formula presented.) makes use of (Formula presented.) -valuations of complete bipartite graphs (Formula presented.). We explore this approach and we provide a classification theorem which shows that all such (Formula presented.) -valuations can be constructed recursively via a sequence of “blow-up” operations. We also enumerate all (Formula presented.) -SEDFs in (Formula presented.) for (Formula presented.) and we show that all these SEDFs are equivalent to (Formula presented.) -valuations via affine transformations. Whether this holds for all (Formula presented.) as well is an interesting open problem. We also …

The Cannon-Thurston Map For Relatively Hyperbolic Free-By-Cyclic Groups, Kailey Beth Perry

Graduate Theses and Dissertations

Let F = H1 ∗ · · · ∗ Hk ∗ Fr be a splitting of a finitely generated free group, H = {H1 , . . . , Hk } a set of subgroups of F , and Φ ∈ Aut(F ) an automorphism such that for any i, Φ([Hi ]) = [Hj ] for some j. Consider the free-by-cyclic group Gϕ = ⟨F, t | t−1 xt = Φ(x)⟩. Suppose Gϕ is strongly hyperbolic relative to H̃ = {H̃1 , . . . , H̃k }, where H̃i = ⟨Hi , t | t−1 xt = Φ(x)⟩. In …

An Interpolating Regime For The Extreme Values Of A Random Zeta Model, Christine K. Chang

Dissertations, Theses, and Capstone Projects

We study the large values of a random model of the Riemann zeta function over short intervals. The extreme value statistics depend on the interval size: the log-correlated regime governs intervals of order one while the i.i.d. regime emerges over longer intervals. The main focus is to describe the transition between these two well-understood regimes as the interval varies in length. This thesis shows that there is an intermediate regime where the behavior of the zeta model’s maxima cannot be entirely captured by either extreme— i.i.d. or fully log-correlated. This suggests that the Riemann zeta function exhibits correlations around its …

The Global Phase Space Of The Three-Vortex Interaction System And Its Application To Vortex-Dipole Scattering, Atul Anurag

Dissertations

This dissertation presents a global reduction of the classical three-vortex problem that is free from coordinate singularities, enabling a comprehensive analysis of the system's dynamics across all circulation regimes.

To achieve this, a two-step symplectic reduction procedure is developed. The first step introduces Jacobi coordinates adapted to the symplectic structure of the vortex system, and the second applies a Lie-Poisson reduction to the resulting system. This formulation eliminates the non-physical singularities associated with collinear vortex configurations and facilitates a global phase space analysis, including a detailed and novel investigation of bifurcations.

Within this reduced framework, all relative fixed points are …

Reduced Order Models Of Hydrodynamically Interacting Flapping Wings, Jose Pabon

Dissertations

Fish schools exhibit a collective behavior and self-organization that is mediated by hydrodynamic interactions between individual fish. However, the long-time evolution of hydrodynamically interacting collectives is challenging to investigate due to the persistent influence of long-lived vortical structures, and the high-resolution requirements of direct numerical simulation at large Reynolds numbers. Reduced-order models have therefore played an important role in theoretical investigations of collectives of swimming bodies. The main results detailed herein are several new reduced-order models of swimmers that self-propel by flapping, i.e., by executing a prescribed periodic rigid body motion. The models are extensions of a discrete-time dynamical system …

Sugar: A Sequence Unfolding Based Transformer Model For Group Activity Recognition, Yash U. Gondkar

Graduate Masters Theses

Large Language Models have improved significantly in the past couple of years due to the adoption of transformers. However, transformers still find it challenging to process videos due to limited context size caused by their quadratic computing cost. Therefore, we studied a booming field in machine learning which powers applications like social scene analysis and video surveillance systems called Group Activity Recognition (GAR). We found that recent models were able to achieve more than 90% accuracy on popular datasets like the Volleyball dataset, however, it turned out that even they relied on transformers.

Therefore, in this work, we developed a …

A Partial Resolution Of Hedden’S Conjecture On Satellite Homomorphisms, Randall Johanningsmeier , '24, Hillary Kim , '25, Allison N. Miller

Mathematics & Statistics Faculty Works

A pattern knot in a solid torus defines a self-map of the smooth knot concordance group. We prove that if the winding number of a pattern is even but not divisible by 8, then the corresponding map is not a homomorphism, thus partially establishing a conjecture of Hedden.

Spectrum Analysis Of Thermally Driven Curvature Inversion In Strained Graphene Ripples For Energy Conversion Applications Via Molecular Dynamics, James M. Mangum, Md R. Kabir, Tamzeed B. Amin, Syed M. Rahman, Ashaduzzaman, Paul M. Thibado

Physics Faculty Publications and Presentations

The extraordinary mechanical flexibility, high electrical conductivity, and nanoscale instability of freestanding graphene make it an excellent candidate for vibration energy harvesting. When freestanding graphene is stretched taut and subject to external forces, it will vibrate like a drum head. Its vibrations occur at a fundamental frequency along with higher-order harmonics. Alternatively, when freestanding graphene is compressed, it will arch slightly out of the plane or buckle under the load. Remaining flat under compression would be energetically too costly compared to simple bond rotations. Buckling up or down, also known as ripple formation, naturally creates a bistable situation. When the …

Corrigendum To "Face-Magic Labelings Of Polygonal Graphs", Wai Chee Shiu, Richard M. Low, Andy K. Liu

Theory & Applications of Graphs

In this note, we correct a misstatement of a theorem.

Trigonometry: A Brief Conversation, Carolyn D. King Phd, Evelyn Tam, Fei Ye Phd, Beata Carvajal

Open Educational Resources

These five units are specifically tailored to foster the mastery of a few selected trigonometry topics that comprise the one credit MA-121 Elementary Trigonometry course.  Each unit introduces the topic, provides space for practice, but more importantly, provides opportunities for students to reflect on the work in order to deepen their conceptual understanding.

These units have also been assigned to students of other courses such as pre-calculus and calculus as a review of trigonometric basics essential to those courses.  This is the third edition of the materials.  Based on suggestions from instructors Units 2 and 3 have been edited to …

Murmurations Of Modular Forms And P-Power Coefficients, Debanjana Kundu, Katharina Müller

School of Mathematical & Statistical Sciences Faculty Publications

We extend the work of N. Zubrilina on murmuration of modular forms to the case when prime-indexed coefficients are replaced by squares of primes. Our key observation is that the shape of the murmuration density is the same.

Structural And Computational Results On Equitably Dissectible Graphs, Ann Wells Clifton

Doctoral Dissertations

This dissertation introduces the class of equitably dissectible graphs — graphs whose vertex set can be recursively partitioned into two parts whose sizes differ by at most one where the two induced subgraphs are connected. In Chapter 2, we show that a graph is equitably dissectible if and only if it has a spanning tree that is equitably dissectible, and we use this result to establish connections between the class of equitably dissectible graphs and Hamiltonian graphs. We also fully characterize the equitably dissectible spiders of the form S3(a) and S3(a, b), identifying a family of unruly spiders that are …

A Backtracking Algorithm For Determining The Existence Of Regular Graphs Of Specified Girth And Excess, Stetson Ray Bosecker

Doctoral Dissertations

The study of cages focuses on finding (k, g)-graphs of minimal order. This dissertation generalizes the problem of finding cages to the determination of graphs with specified excess, thereby broadening the significance of the results. The (k, g, ε)-graph problem seeks to determine the existence or nonexistence of k-regular graphs with girth g and excess ε = n(G)−M(k, g) (where M(k, g) represents the Moore bound for cage graphs). Motivated by heuristic methods used to determine properties within the study of cages, we present a backtracking algorithm capable of constructing (k, g, ε)-graphs or determining their nonexistence. Chapter 2 provides …

Privacy-Preserving Structure Learning For Geospatial Data Using Information-Theoretic Dependency Measures, Ahmed Mudhish

Doctoral Dissertations

This dissertation proposes a privacy-preserving framework for structure learning in Bayesian networks (BNs) that addresses the challenges of distributed geospatial data face. Geospatial datasets often exhibit region-specific patterns such as sparsity and nonlinear dependencies. These patterns undermine the effectiveness of traditional machine learning models. Additionally, learned BN structures may reveal sensitive relationships in the generated graph by BNs. These relationships pose a significant privacy risk if reverse-engineered. To address these issues, three novel algorithms are introduced. First, the Selective Naïve Bayes with HSIC (SNB-HSIC) algorithm applies a kernel-based dependency measure to filter redundant and irrelevant features in sparse datasets, improving …

Betti Numbers For Modules Over Artinian Local Rings, Kaiyue He

Dissertations - ALL

We study the behavior of Betti growth of modules over Artinian rings under some $\Tor$ vanishing conditions. Let $(R,\fm)$ be a local Artinian ring with a fixed ideal $I$, and $M$ a finite module. We extend the framework developed in \cite{huneke2014vanishing} by introducing and analyzing a new invariant, $\gamma_I(M)$ (see Definition \ref{definition}), associated with the ideal $I$ of the ring $R$ and a finitely generated $R$-module $M$. It turns out that this invariant $\gamma_I(M)$ provides an upper bound for the ratio of certain consecutive Betti numbers of the finitely generated modules $N$ such that $\Tor_i(M,N) = 0$ for some $i …

Universal Centralizers, Morita Abelianization, And Wonderful Models In Lie Theory, Peter Crooks

Funded Research Records

No abstract provided.

Project-Based Learning With Odes: Modeling Straw Rocket Motion With Air Resistance, Viktoria Savatorova, Ethan Dyer, Aleksei Talonov

CODEE Journal

This paper presents a hands-on project that guides students through building and validating a mathematical model of projectile motion. The project starts with the idealized case of motion under gravity without air resistance and then introduces air drag : first as a linear force, and then as a nonlinear quadratic force, with the Reynolds number providing the justification for the quadratic model. Students perform experiments with vertical and angled launches, capturing and analyzing motion data using video analysis software. Vertical launch data allows parameter estimation via least squares fitting of the nonlinear drag model, yielding values for initial velocity and …

Degeneracies Of Triangulated Graphs, Allan Bickle

Theory & Applications of Graphs

A graph $G$ is $k$-degenerate if each subgraph has minimum degree

at most $k$. The degeneracy\textbf{ }$D\left(G\right)$ is the smallest

$k$ such that $G$ is $k$-degenerate. We determine the truth values

of four statements (using different quantifiers) about when a planar

graph $G$ with degeneracy $k$ has a triangulation with degeneracy

$l$. We characterize which 3-connected planar graphs can only be

triangulated to degeneracy 3. Then we consider analogous questions

for maximal planar bipartite graphs. We prove some structural results

on these graphs, including results on decomposition of planar graphs

into various types of bipartite graphs.

Numerical Results Of New And Modified Heuristics For The Vertex Coloring Problem, Elliot Hanson

Summer Research Showcase

The chromatic number, χ(G) of an undirected graph G=(V,E) is the minimum number of colors required to color its vertices so that no two adjacent vertices have the same color. Given a graph, G, finding its chromatic number is useful for solving scheduling problems and other combinatorial optimization problems. However, determining the chromatic number of a connected graph is NP-Hard, meaning there is no known polynomial time algorithm which solves it. Thus, we are interested in heuristic solutions which give approximations for the chromatic number in polynomial time. There are well-known heuristics for finding χ(G) for any graph G. …

The Fermat Quartic X4 + Y4 = 2m In Quadratic Number Fields, Jayanth Athipatla

Mathematics Faculty Publications

In this paper, we generalise to the family of Fermat quartics X4+Y4=2m,m∈Z, a result of Aigner [‘Über die Möglichkeit von x4+y4=z4 in quadratischen Körpern’, Jahresber. Deutsch. Math.-Ver. 43 (1934), 226–228], which proves that there is only one quadratic field, namely Q(−7−−−√), that contains solutions to the Fermat quartic X4+Y4=1. The m≡0(mod4) case is due to Aigner. The m≡2(mod4) case follows from a result of Emory [‘The Diophantine equation X4+Y4=D2Z4 in quadratic fields’, Integers 12 (2012), Article no. A65, 8 pages]. This paper focuses on the two cases m≡1,3(mod4), classifying for m≡1(mod4) the infinitely many quadratic number fields that contain solutions, …

Career: Algebra And Representation Theory For Non-Semisimple Topological Field Theory, Matthew Young

Funded Research Records

No abstract provided.