Physical Sciences and Mathematics Commons^™

Bernstein-Sato Theory For Singular Rings In Positive Characteristic, Jack Jack, Luis Núñez-Betancourt, Eamon Quinlan-Gallego

Department of Mathematics: Faculty Publications

The Bernstein-Sato polynomial is an important invariant of an element or an ideal in a polynomial ring or power series ring of characteristic zero, with interesting connections to various algebraic and topological aspects of the singularities of the vanishing locus. Work of Mustaţă, later extended by Bitoun and the third author, provides an analogous Bernstein-Sato theory for regular rings of positive characteristic.

In this paper, we extend this theory to singular ambient rings in positive characteristic. We establish finiteness and rationality results for Bernstein-Sato roots for large classes of singular rings, and relate these roots to other classes of numerical …

A Mathematical Investigation Of Landauer’S Principle, Paul Bracken

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

A minimal mathematical approach is used to state Landauer’s principle in a precise, general way. The results are obtained by means of a rigorous development which is based on the use of quantum statistical mechanics. A mathematical form of the principle results as an equality rather than an inequality. The equality version does imply the original statement of the principle as introduced by Landauer.

Rediscovering The Artistic Side Of Mathematics, Bogdan G. Nita, Ashwin Vaidya

LASER Journal

Welcome to the inaugural issue of the LASER, a journal devoted to the problems at the interface of math and art. The terms ’math’ and ’art’ are to be broadly construed to encompass all quantitative sciences and forms of art. The journal’s name, acronym for Linking Art and Science through Education and Research, suggests our interest in the theory, practice and pedagogy of this interdisciplinary subject.

Tri-Plane Diagrams For Simple Surfaces In S4, Manuel Aragón, Zack Dooley, Alexander Goldman, Yucong Lei, Isaiah Martinez, Nicholas Meyer, Devon Peters, Scott Warrander, Ana Wright, Alex Zupan

Department of Mathematics: Faculty Publications

Meier and Zupan proved that an orientable surface K in S⁴ admits a tri-plane diagram with zero crossings if and only if K is unknotted, so that the crossing number of K is zero. We determine the minimal crossing numbers of nonorientable unknotted surfaces in S⁴, proving that c(P^n,m) = max{1, |n−m|}, where P^n,m denotes the connected sum of n unknotted projective planes with normal Euler number +2 and m unknotted projective planes with normal Euler number −2. In addition, we convert Yoshikawa’s table of knotted surface ch-diagrams to tri-plane …

Outer Independent Double Italian Domination Of Some Graph Products, Rouhollah Jalaei, Doost Ali Mojdeh

Theory and Applications of Graphs

An outer independent double Italian dominating function on a graph G is a function f:V(G) →{0,1,2,3} for which each vertex x ∈ V(G) with {f(x)∈ {0,1} then Σ_{y ∈ N[x]}f(y) ⩾ 3 and vertices assigned 0 under f are independent. The outer independent double Italian domination number γ_oidI(G) is the minimum weight of an outer independent double Italian dominating function of graph G. In this work, we present some contributions to the study of outer independent double Italian domination of three graph products. We characterize the Cartesian product, lexicographic product and direct product of custom …

P-Adic Statistical Field Theory And Deep Belief Networks, Wilson A. Zuniga-Galindo

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

In this work we initiate the study of the correspondence between p-adic statistical field theories (SFTs) and neural networks (NNs). In general quantum field theories over a p-adic spacetime can be formulated in a rigorous way. Nowadays these theories are considered just mathematical toy models for understanding the problems of the true theories. In this work we show these theories are deeply connected with the deep belief networks (DBNs). Hinton et al. constructed DBNs by stacking several restricted Boltzmann machines (RBMs). The purpose of this construction is to obtain a network with a hierarchical structure (a deep learning architecture). An …

Stable Trace Ideals And Applications, Haydee Lindo, Hai Long Dao

All HMC Faculty Publications and Research

We study stable trace ideals in one dimensional local Cohen–Macaulay rings and give numerous applications.

Robust Explicit Estimation Of The Log-Logistic Distribution With Applications, Zhuanzhuan Ma, Min Wang, Chanseok Park

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

The parameters of the log-logistic distribution are generally estimated based on classical methods such as maximum likelihood estimation, whereas these methods usually result in severe biased estimates when the data contain outliers. In this paper, we consider several alternative estimators, which not only have closed-form expressions, but also are quite robust to a certain level of data contamination. We investigate the robustness property of each estimator in terms of the breakdown point. The finite sample performance and effectiveness of these estimators are evaluated through Monte Carlo simulations and a real-data application. Numerical results demonstrate that the proposed estimators perform favorably …

Decomposition Rate As An Emergent Property Of Optimal Microbial Foraging, Stefano Manzoni, Arjun Chakrawal, Glenn Ledder

Department of Mathematics: Faculty Publications

Decomposition kinetics are fundamental for quantifying carbon and nutrient cycling in terrestrial and aquatic ecosystems. Several theories have been proposed to construct process-based kinetics laws, but most of these theories do not consider that microbial decomposers can adapt to environmental conditions, thereby modulating decomposition. Starting from the assumption that a homogeneous microbial community maximizes its growth rate over the period of decomposition, we formalize decomposition as an optimal control problem where the decomposition rate is a control variable. When maintenance respiration is negligible, we find that the optimal decomposition kinetics scale as the square root of the substrate concentration, resulting …

Current Scenario Of Solar Energy Applications In Bangladesh: Techno-Economic Perspective, Policy Implementation, And Possibility Of The Integration Of Artificial Intelligence, Monirul Islam Miskat, Protap Sarker, Hemal Chowdhury, Tamal Chowdhury, Md Salman Rahman, Nazia Hossain, Piyal Chowdhury, Sadiq M. Sait

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

Bangladesh is blessed with abundant solar resources. Solar power is considered the most desirable energy source to mitigate the high energy demand of this densely populated country. Although various articles deal with solar energy applications in Bangladesh, no detailed review can be found in the literature. Therefore, in this study, we report on the current scenario of renewable energy in Bangladesh and the most significant potential of solar energy’s contribution among multiple renewable energy resources in mitigating energy demand. One main objective of this analysis was to outline the overall view of solar energy applications in Bangladesh to date, as …

Log-Optimal (D + 2)-Configurations In D–Dimensions, Peter D. Dragnev, Oleg R. Musin

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

We enumerate and classify all stationary logarithmic configurations of d + 2 points on the unit sphere in d–dimensions. In particular, we show that the logarithmic energy attains its local minima at configurations that consist of two orthogonal to each other regular simplexes of cardinality m and n. The global minimum occurs when m = n if d is even and m = n + 1 otherwise. This characterizes a new class of configurations that minimize the logarithmic energy on Sd−1 for all d. The other two classes known in the literature, the regular simplex (d + 1 points on …

When Plaquing Is Not Possible: Computational Methods For Detecting Induced Phages, Taylor Miller-Ensminger, Genevieve Johnson, Swarnali Banerjee, Catherine Putonti

Mathematics and Statistics: Faculty Publications and Other Works

High-throughput sequencing of microbial communities has uncovered a large, diverse population of phages. Frequently, phages found are integrated into their bacterial host genome. Distinguishing between phages in their integrated (lysogenic) and unintegrated (lytic) stage can provide insight into how phages shape bacterial communities. Here we present the Prophage Induction Estimator (PIE) to identify induced phages in genomic and metagenomic sequences. PIE takes raw sequencing reads and phage sequence predictions, performs read quality control, read assembly, and calculation of phage and non-phage sequence abundance and completeness. The distribution of abundances for non-phage sequences is used to predict induced phages with statistical …

Between Heaven And Earth! A Poem-Collage Pair About Hypatia Of Alexandria, Sarah Glaz, Mark Sanders

Journal of Humanistic Mathematics

The poem-collage pair presented here is a work of collaboration between the mathematician and poet, Sarah Glaz, and the collage and ceramic artist, Mark Sanders. The piece is part of their larger joint poem-collage project involving the history of mathematics. Included as background is a brief discussion on the history and mathematics involved, and a reflection on several landmark locations and some of the relevant imagery appearing in the poem and the collage.

The Babelogic Of Mathematics, Vijay Fafat

Journal of Humanistic Mathematics

How would the Bible written about a Mathematical God start, describing the Creation of Mathematics and Logic? How would Rigveda's "Nasadiya sukta" read if it were describing the Void before mathematics was "born"? Here is an attempt at a partial answer, one which takes the original Genesis chapter and the Nasadiya sukta and makes suitable changes to create a fairly consistent, if somewhat anachronistic narrative (with the slight mixing up of Bertrand Russell and Lobachevsky / Bolyai attributable to "Babelogic"), along with a new ending to the Beginning...

Astor Place Barber, Audrey Nasar

Journal of Humanistic Mathematics

"Astor Place Barber" is a short story about a math professor and a barber. It plays with the logical concept of a paradox via the Barber's Paradox, which, made famous by Bertrand Russell, tells the story of a barber who both shaves himself and does not shave himself.

Locked In Functions: A Short Poem For Robert Langlands, Virgilio A. Rivas

Journal of Humanistic Mathematics

This short poem is inspired by Robert Langlands, recipient of the 2018 Abel Prize. The poem tries to sum up in poetic language, as brief but substantial as it can be, the philosophical and rhetorical connotation of his contributions to mathematics, from automorphic forms to number theory, and the famous Langlands programme, among others. Also partly inspired by Edward Frenkel's tribute to Langlands, the book Love and Mathematics, the poem seeks to capture the philosophical beauty of mathematics that privileges the importance of 'functions' over 'passions', consistent with Langlands' purely mathematical side.

Unsolved Haiku, Scott W. Williams

Journal of Humanistic Mathematics

This poem describes the still unsolved 1937 conjecture of Lloyd Collatz: Do repeated applications of the algorithm described yield the number 1?

Mathematics, Kim Regnier Jongerius

Journal of Humanistic Mathematics

Inspired by the song "Memories" from the musical Cats, this work describes some of the frustrations and joys inherent in mathematical work.

Self-Reference And Diagonalisation, Joël A. Doat

Journal of Humanistic Mathematics

This poem is an exercise on self-reference and diagonalisation in mathematics featuring Turing’s proof of the undecidability of the halting problem, Cantor’s cardinality argument, the Burali-Forti paradox, and Epimenides' liar paradox.

Wartime Logic, Tony Bedenikovic

Journal of Humanistic Mathematics

No abstract provided.

Book Review: Algebra The Beautiful: An Ode To Math’S Least-Loved Subject By G. Arnell Williams, Judith V. Grabiner

Journal of Humanistic Mathematics

In his book Algebra the Beautiful, G. Darnell Williams has undertaken a challenging job – to show the importance, deep structure, intellectual connections, and sheer beauty of classroom algebra. This review describes some of the questions the book raises, the historical and cultural context it provides, and the intellectual apparatus it deploys.

Poetry Folder: Mathematical Constants Beyond The Half-Circle

Journal of Humanistic Mathematics

In our July 2022 issue, we announced an open call for poetry about mathematical constants other than pi. And you delivered. This folder contains five eclectic poems celebrating those constants beyond the half-circle, written by Robin Chapman, John Donoghue, Kevin Farey, Lawrence M. Lesser, and E. R. Lutken.

Enjoy!

Anneli Lax: They Think, Therefore We Are, Elena Anne Corie Marchisotto

Journal of Humanistic Mathematics

No abstract provided.

Walking Alone: My Career In Mathematics, Maohua Le, Yongzhong Hu

Journal of Humanistic Mathematics

In this article, dictated by Maohua Le and arranged by Yongzhong Hu, Professor Le briefly recounts his legendary experience of self-study mathematics, which reflects the life experiences of his generation of Chinese people.

Blending Mathematics Teaching With Kindness, Kien H. Lim, Anthony Matsuura

Journal of Humanistic Mathematics

Mathematics can be intellectually demanding, engaging, and fulfilling. Learning mathematical concepts adequately warrants an environment where students can err without penalty, shame, or hurtful consequences. Teaching mathematics efficaciously depends on the trusting relationship between the teacher and the students. We advocate blending mathematics teaching with kindness because it benefits the teacher, the students, and society. Kindness, niceness, caring, and benevolence are interrelated but not synonymous. We outline four progressive levels of kindness: conditional, superficial, optimal, and genuine. Blending mathematics teaching and kindness effectively requires the teacher to decenter from their own perspectives and adopt the student’s perspective as the student …

Mathaphor As A Literary Tool, Sarah Voss

Journal of Humanistic Mathematics

Drawing from one of my recent sermons, I chart the path of mathaphor (metaphor drawn from mathematics) as a literary tool. Following a short history of the concept, I detail contemporary author Colum McCann’s significant use of a geometric term to unveil and encourage compassion.

Figure-Ground Perception: A Poem Proof, Richard Delaware

Journal of Humanistic Mathematics

This is a proof, in poetic form, of a bit of real analysis, more specifically involving the topology of accumulation points, that exploits the human optical phenomenon of figure-ground perception. Sometimes it is not a change in content, but a snap shift in point of view that yields a proof.

Using Bloom's Taxonomy For Math Outreach Within And Outside The Classroom, Manmohan Kaur

Journal of Humanistic Mathematics

Not everyone is a great artist, but we don’t often hear, “I dislike art.” Most people are able to appreciate visual arts, music and sports, without necessarily excelling in it themselves. On the other hand, the phrase “I dislike math” is widely prevalent. This is especially ironic in our current society, where mathematics affects our day-to-day activities in essential ways such as e-commerce and e-mail. This paper describes the opportunity to popularize mathematics by focusing on its fun and creative aspects, and illustrates this opportunity through a brief discussion of interdisciplinary topics that expose the beauty, elegance and value of …

The Mathematics Of The Harp: Modeling The Classical Instrument And Designing Futuristic Ones, Cristina Carr, Daniel Chioffi, Maya Glenn, Stefan O. Nita, Vlad N. Nita, Bogdan G. Nita

Journal of Humanistic Mathematics

We analyze and model the neck of the classical harp based on the length of the strings, their tension and density. We then use the results to design new and innovative harp shapes by adjusting the parameters of the model.

Lessons From Human Experience: Teaching A Humanities Course Made Me A Better Math Teacher, Erin Griesenauer

Journal of Humanistic Mathematics

As a professor at a Liberal Arts college, I recently taught a General Education course called Human Experience. Far from my normal experiences in the mathematics classroom, in Human Experience I was tasked with teaching topics from the humanities, including art, philosophy, history, and political science. Teaching this course was challenging, but it was also transformative. Teaching a course so far from my background gave me the opportunity to experiment with different pedagogical techniques and to reflect on how I set up my math classes. I learned many lessons that I have brought back to my math classes—lessons that have …