Mathematics Commons^™

Positive Solutions For Nonlinear Fractional Differential Equations With A Disturbance Parameter On An Infinite Interval, Zhaocai Hao, Lu Song, Martin Bohner, Qianqian Gu, Panzhen Liu

Mathematics and Statistics Faculty Research & Creative Works

This paper studies a class of integral boundary value problems of fractional differential equations with a disturbance parameter on an infinite interval. By applying the upper-lower solution method and fixed point index theory, the influence of the disturbance parameter on the existence of positive solutions is rigorously analyzed. We have obtained that the boundaryvalue problem admits at least twopositive solutions, one positivesolution or no solution. An example is provided to validate the results. The results of this study provide a significant complement and refinement to some existing findings.

Limit Theorems For Andrews’ Restricted Overpartitions, Tapas Bhowmik, Alex Cao, Jack Frew, John Lehman, Wei-Lun Tsai

Faculty Publications

The study of overpartitions in recent years has been used to great effect in various fields, including hypergeometric series, q-series identities, and mathematical physics. We investigate the limiting distributions of the number of parts in a family of overpartitions of n,  introduced by Andrews, where parts are counted with two different weights. Using Andrews’ identities and the saddle-point method, we establish two central limit theorems (CLTs) for the number of parts as n → ∞, corresponding to these weightings. We also derive explicit formulas for the mean and variance in each case.

A Virtual Community Math Circle, Skona Brittain, Sayonita Ghosh Hajra, Steve Heller, Daniel Hodgins, Peter Petto, Gabriella Pinter, Lauren Rose, A. Gwinn Royal, Asmita Sodhi

Journal of Math Circles

The Julia Robinson Mathematics Festival (JRMF) Community Math Circle is free, volunteer-run, and online. We collaborate with JRMF and use their activities in our events. Started during the pandemic, the Community Math Circle continues to thrive. We have about 40 participants attending every month, typically kids aged 6 to 13, teachers, facilitators, and other adults. This article describes how our event is organized, planned, and executed, and how we train facilitators. We will also offer reflections on our successes and challenges.

On Cartan’S Examples Of Isoparametric Hypersurfaces And Their Focal Submanifolds, Thomas E. Cecil, Patrick J. Ryan

Mathematics and Computer Science Department Faculty Scholarship

This paper is a survey of Cartan’s examples of isoparametric hypersurfaces in spheres and their focal submanifolds that were described in his fundamental work on the subject, which appeared in four papers [2]–[5] published during the period 1938–1940.

The Butterfly Effect In Economics: Exploring Chaos With A Simple Financial Model, Sandy Hs Herho

CODEE Journal

Wild swings in financial markets need not result from external shocks like earthquakes or wars—they can emerge from deterministic chaos. This article introduces kalimusada, an open-source Python library that lets students and instructors explore this phenomenon through a simple three- equation model of financial dynamics. The model couples interest rates, investment, and prices through nonlinear feedback, generating bounded but unpredictable oscillations characteristic of chaos. Tiny differences in starting conditions—smaller than any measurement could detect—grow exponentially until two initially identical economies follow completely different paths. The library provides ready-to-use tools for visualizing this “butterfly effect” in economics, computing divergence metrics, and …

Effective Wordle Heuristics, Ronald I. Greenberg

Computer Science: Faculty Publications and Other Works

While previous researchers have performed an exhaustive search to determine an optimal Wordle strategy, that computation is very time consuming and produced a strategy using words that are unfamiliar to most people. With Wordle solutions being gradually eliminated (with a new puzzle each day and no reuse), an improved strategy could be generated each day, but the computation time makes a daily exhaustive search impractical. This paper shows that simple heuristics allow for fast generation of effective strategies and that little is lost by guessing only words that are possible solution words rather than more obscure words.

Systematic Synthesis Of Crispr/Cas Applications For Enhancing Salt Tolerance In Crops: A Decade Of Progress And Challenges, Xindi Sun, Fusheng Wu, Zhuanzhuan Ma, Guohao Liang, Shumei Chen, Xutong Hu, Shugao Fan, Ying Zhao

School of Mathematical & Statistical Sciences Faculty Publications

Soil salinity is a major constraint on global crop productivity, driving the need for salt-tolerant varieties. While CRISPR-Cas genome editing offers targeted solutions for trait improvement, significant biological and technical bottlenecks limit its application in conferring salt stress resilience. This systematic summarizes findings from 83 peer-reviewed studies (2015–2024) employing CRISPR/Cas technologies to improve salt tolerance in five major crops (rice, wheat, maize, sorghum, barley). Our systematic review reveals that early single-gene edits achieved modest gains (30–50% Na⁺ exclusion) but often showed limited yield gains in field settings, potentially due to compensatory regulation and environmental variation. The literature suggests that multiplex …

Unitary Rational Functions: The Scaled Quaternion Case, Daniel Alpay, Ilwoo Choo, Mihaela Vajiac

Mathematics, Physics, and Computer Science Faculty Articles and Research

We develop the theory of minimal realizations and factorizations of rational functions where the coefficient space is a ring of the type introduced in our previous work, the scaled quaternions, which includes as special cases the quaternions and the split quaternions. The methods involved are not a direct generalization of the complex or quaternionic settings, and in particular, the adjoint is not the classical adjoint and we use properties of real Hilbert spaces. This adjoint allows to define the counterpart of unitarity for matrix-rational functions, and we develop the corresponding theories of realizations and unitary factorizations. We also begin a …

Corrigendum To “Interim Rationalizable Implementation Of Functions” (Kunimoto T, Saran R, Serrano R (2024) Mathematics Of Operations Research 49(3):1791–1824), Takashi Kunimoto, Rene Saran, Roberto Serrano

Research Collection School Of Economics

This is the brief corrigendum to “Interim rationalizable implementation of functions” [Kunimoto T, Saran R, Serrano R (2024) Interim rationalizable implementation of functions. Math. Oper Res. 49(3):1791–1824].

Positive Solutions Of Semipositone Singular Three-Points Boundary Value Problems For Nonlinear Fractional Differential Equations, Xueyan Zhang, Zhaocai Hao, Martin Bohner

Mathematics and Statistics Faculty Research & Creative Works

This study introduces the existence of one positive solution for a specific category of semipositive singular three-point boundary value problems associated with Caputo fractional differential equations. The proof relies on the application of the Guo–Krasnosel'skii fixed point theorem. In the end, we provide an illustrative example.

Flips In Two-Dimensional Hypertriangulations, Herbert Edelsbrunner, Alexey Garber, Mohadese Ghafari, Teresa Heiss, Morteza Saghafian

School of Mathematical & Statistical Sciences Faculty Publications

We study flips in hypertriangulations of planar points sets. Here a level-k hypertriangulation of n points in the plane is a subdivision induced by the projection of a k-hypersimplex, which is the convex hull of the barycenters of the (k−1)-dimensional faces of the standard (n−1)-simplex. In particular, we introduce four types of flips and prove that the level-2 hypertriangulations are connected by these flips.

Constrained Quantization For Probability Distributions, Megha Pandey, Mrinal Kanti Roychowdhury

School of Mathematical & Statistical Sciences Faculty Publications

In this work, we extend the classical framework of quantization for Borel probability measures defined on normed spaces ℝ𝑘 by introducing and analyzing the notions of the nth constrained quantization error, constrained quantization dimension, and constrained quantization coefficient. These concepts generalize the well-established nth quantization error, quantization dimension, and quantization coefficient, which are traditionally considered in the unconstrained setting and thereby broaden the scope of quantization theory. A key distinction between the unconstrained and constrained frameworks lies in the structural properties of optimal quantizers. In the unconstrained setting, if the support of P contains at least n elements, then the …

Demographic And Other Correlates Of Non-Prescription Drug Use Among College Students During The Covid-19 Pandemic, Subi Gandhi, Sidketa Fofana, Md Rafiul Islam, Tamer Oraby

School of Mathematical & Statistical Sciences Faculty Publications

Background and objectives: Substance use among college students in the U.S. remains a pressing concern and may have intensified during the COVID-19 pandemic due to increased stress, uncertainty, and academic disruptions. This study investigates the relationship between non-prescription drug use and various demographic, mental health, and behavioral factors among college students during the pandemic's early stages.

Methods: Data were collected through online and in-person surveys in the summer semester of 2021. Behavioral health was assessed using validated instruments: the Patient Health Questionnaire-9 (PHQ-9) for depression and the Drug Abuse Screening Test-20 (DAST-20) for substance use. Demographic and behavioral variables were …

One-Step Purification Of A Bioactive Pak1-Derived Peptide, Djamali Muhoza, Emily P. Esquivel, Stacy R. Hunter, Pateince S. Okoto, Thallapuranam K.S. Kumar, Paul D. Adams

Chemistry & Biochemistry Faculty Publications and Presentations

The serine/threonine kinase PAK1 serves as a mediator of cytoskeletal reorganization and cancer-related signaling downstream of the small GTPases. Due to the challenges in purifying PAK1 complexes, a 46-residue peptide from PAK1, is widely used to study PAK1-Cdc42 signaling. Traditionally, this purification involved multi-step chromatography of recombinant GST-PBD46 complexes, yielding approximately 1 mg per 1.5 L culture. In this study, a 30 min heat treatment step after thrombin cleavage was used to precipitate GST while leaving pure PBD46 in solution. This step eliminated the need for further affinity and size-exclusion chromatography steps. This improved protocol produces proteins with a 6.5-fold …

Anxiety Gets All The Likes, Saniye Nur Ergan

Journal of Humanistic Mathematics

This internet meme (created by me:)) emerges from a simple yet persistent question that underlies much of my work: Why do we continue to treat certain emotions in mathematics as visible, nameable, and research-worthy, while others linger in the background as faint, unarticulated traces?

Being Different Doesn’T Make You Odd, Raffaella Mulas

Journal of Humanistic Mathematics

The number 2 is beautiful and it has many unique properties. It is the only even prime number, and being different from the other prime numbers doesn't make it odd. This is true also for people: being different doesn't make you odd. It makes you beautiful and unique.

Logical Conclusion, Robert Dawson

Journal of Humanistic Mathematics

There is a well-known anecdote about Kurt Gödel, just before his citizenship hearing, telling Einstein that the Constitution was logically flawed. This alternate-history story explored the possible ramifications.

Nightmare In The Library, Charles A. Coppin

Journal of Humanistic Mathematics

Students of real analysis and calculus find that the completeness property of the real numbers is difficult to understand, especially, its importance. The word numbers denote real numbers throughout this piece. After all, the real numbers are not any less imaginary than the so-called imaginary numbers. Although, sometimes, it does gain some mention in calculus courses, its presence as a topic is a mere will-o’-the-wisp of bygone days when teachers would often reach deep into the big ideas of calculus as a mainstay of their courses. For the sake of cultural literacy and the development of mathematical maturity, we believe …

To Those Who Ask "What's The Name Of Your Book, Again?", Marion D. Cohen

Journal of Humanistic Mathematics

No abstract provided.

Yet No Ramanujan, Db Jonas

Journal of Humanistic Mathematics

The poem Yet No Ramanujan conveys the musings of a layman in his garden, obsessed by the way that mundane things, the small experiences of everyday life, suggest the ways in which, as Baudelaire says, "everything is number." This meditation ends with the recognition that the putative stability of a self, a thinking subject, is after all, however complex and unique, like all matter, a kind of oscillation, a complex product of number.

Missing, James S. Huntley

Journal of Humanistic Mathematics

Fifty-five word stories have achieved a certain currency as a reflective format, perhaps especially in medicine. Many of them turn out to be poems. Here is one such story inviting the reader to consider the nature of the absent, of the 'missing', of zero, and the philosophical underpinnings of mathematics.

To The Mysterious Gamma: Euler- Mascheroni Constant, Md Sadikur Rahman

Journal of Humanistic Mathematics

No abstract provided.

Poems For A Calculus Class (After Reading Midlife Calculus And At The Dimensional Border), Kevin Farey

Journal of Humanistic Mathematics

This poetry folder is made up of two short trios. The first begins with “Methods,” a thirteen-line poem by Britt Kaufmann, and continues with two thirteen-line responses to it. The folder’s second half consists of three (non-thirteen-line) responses to a collection of thirteen-line-almost-sonnets by Philip Fried, who had in turn been inspired, during the pandemic, by Edwin Abbot’s Flatland.

Both Kaufmann and Fried are authors of previous JHM poetry folders. Poems from Kaufmann’s book Midlife Calculus appeared in the January 2025 issue and poems from Fried’s series At the Dimensional Border appeared in the July 2024 issue. In addition, …

Plato's God, Vijay Fafat

Journal of Humanistic Mathematics

A poem expressing a thought about recursion in the Platonic Realm, of mathematical archetypes and the dreaming conception of "God"...

The Tool/Weapon Duality Of Mathematics, Alexandre Borovik

Journal of Humanistic Mathematics

This paper outlines the principal moral and ethical issues of mathematics and existential choices a mathematician may face in his professional life. My text is a manifesto; it is a list of declarations rather than a systematic development of the theme. Indeed each section could be expanded into a paper on its own. I hope to write fairly soon a more detailed exposition.

Sumthin: A Context-Dependent African American Vernacular English Word For Variables, Mark Apedo

Journal of Humanistic Mathematics

If mathematics is a language, then African American Vernacular English (AAVE) could hold the key to a fascinating understanding of the evolution of algebra from human speech in North America. The use of the pronoun something (sumthin) to depict an unknown amount of money that varies implies that algebra developed as a logical necessity in human communication. Exploring this phenomenon and leveraging it to demonstrate the concept of algebra could enhance the comprehension of the subject among speakers of AAVE.

The Math Teaching Atlas: Trails, Anchors, And Compass In Action, Ivan Z. Feng

Journal of Humanistic Mathematics

This paper advocates a shift grounded in mathematical reasoning: teach and write math through explicit routeways rather than narratives, anchor new ideas to familiar concepts on a roadmap, and use motivation and concrete examples as a compass and simulation for discovery.

Reminiscences Of My Past Life As A High School Math Teacher, Wayne Nirode

Journal of Humanistic Mathematics

In this article, I reflect on my twenty years as a high school math teacher. I describe four distinct phases of my career in five-year increments. I aim to make public both what makes teaching high school students rewarding and what makes it difficult. Also, my writing illuminates how indelible the first five years were to my development as a teacher. I hope that this brief memoir helps early-career teachers see past the tumultuous beginnings while also inspiring veteran teachers to reflect on their own careers.

Creating Art With The Generalized Pythagorean Theorem, Ángel Colín

Journal of Humanistic Mathematics

The Pythagorean Theorem and its generalizations have been widely studied. In this article we focus on one particular generalization that asserts that the area of a polygon with one side the hypotenuse of a right triangle is the sum of the areas of similar polygons whose corresponding sides are the other two sides [4]. Using this theorem together with interactive software or simply pencil and paper, we create some visually enticing artwork. This activity, we believe, can contribute positively to geometry teaching at all academic levels.

Let's Play Uno! Training Mathematical Thinking With An Inclusive Card Game, Thierry Meyrath, Clara-Ioana Mincu, Antonella Perucca

Journal of Humanistic Mathematics

We explore probability exercises based on the game UNO. Excluding the Wild cards, the deck contains precisely 100 cards, which makes it convenient to express probabilities as percentages. We discuss the mathematical relation between cards that can be played after other cards. We also propose an algebraic exercise that relates several quantities, such as the number of cards left in the losing player’s hand. Finally, we argue that UNO is a very inclusive game: it is logistically easy to play, allows for players of different skill levels, and can be adapted to meet special needs. Beyond being playful exercises, our …