Mathematics Commons^™

A Note On Modeling Prejudice, Alan Lockard, Edwin Harcourt

Journal of Humanistic Mathematics

We model prejudice in the context of an iterated prisoner’s dilemma tournament. Prejudice here is defined as the inability to distinguish amongst members of an identifiable group. We run a computer simulation where agents of one of two groups with defined strategies are randomly matched against each other. The agents either cooperate with or defect against the agent they are matched with. Our focus is on agents who play an unprejudiced version of the Tit-for-Tat strategy (cooperate with any player in the first encounter, and then apply the same strategy [defect or cooperate] that the opposing player played against them …

How Did Cauchy Prove His Integral Theorem? A Historical Reconstruction For Education Research, José Gerardo Piña-Aguirre, Rosa María Farfán Márquez

Journal of Humanistic Mathematics

In his article “Mémoire sur les Intégrales Définies, prises entre des limites imaginaires” of 1825, Cauchy proves that the integral of a complex function f defined on complex numbers is independent of the path of integration if the function f remains finite and continuous along the chosen path. In contemporary literature that analyzes the mathematics of Cauchy, the specific mathematical process that allowed Cauchy to prove this result is not explicitly presented. The purpose of this manuscript is to communicate a symbolic-algebraic derivation that justifies the narrative argument used by Cauchy to support his proof, thus hopefully filling this gap. …

Gold And Silver In Balance: Khayyam’S Mathematical Analysis Of Binary Alloys, Yousef Yassi, Reza Yassi

Journal of Humanistic Mathematics

The history of titration dates back to Archimedes who established that an object submerged in a liquid displaces an amount of liquid whose weight is equal to the buoyant force acting on the object. Since then, many scientists and engineers have tried to optimize his approach or devise new instruments for titration purposes. Omar Khayyam (1048–1123), in addition to designing a hydro-static balance, developed mathematical approaches for titration of binary alloys, that is, alloys made up of only two elements. Here we explore the different versions of Khayyam’s work on titration given by various sources. We also compare the accuracy …

A Just Balance And The Teaching And Learning Of Mathematics, Marshall Gordon

Journal of Humanistic Mathematics

Society’s response to the question of what is worth sharing and how it best can be provided as conceived by the educational establishment shapes the education of every individual in school. As the classroom experience is a commitment to the future of society, it is good and just for society to support the educational development of each person in order that all can participate productively. In this way a “just balance” is established. In the particular case of educating students in mathematics, however, there is apparently an unjust balance, for despite the considerable time and effort students and teachers in …

Polygonal Number Similarity, Gunhan Caglayan

Journal of Humanistic Mathematics

This note takes an exploratory approach to define and then visualize the notion of polygonal number similarity between pairs of k-gonal numbers P_k(αn) and P_k(n) , where α is an integer scale factor of 2 or greater.

Translating Trattato Dell'alcibra Amuchabile, Olympia Nicodemi

Journal of Humanistic Mathematics

We present a detailed look at one small section of a premodern algebra manuscript from circa 1365 C.E. The manuscript, Tratatto dell’Alcibra Amuchabile (Anonimo), is written in the Tuscan vernacular, with no symbols, no punctuation, and wildly inconsistent (pre-printing press) spellings. We present some of the hurdles to translating this “rhetorical algebra” followed by an annotated translation of one of its algebra rules, Rule 5, through which we learn the methods and mindset of algebra that preceded and seeded the breakthroughs in algebra of the 16^th century. We also point to some of the significant differences between algebra then …

Mathematics And Ethics: Narratives And Conversations, David Pritchard

Journal of Humanistic Mathematics

We provide an eclectic overview of various narratives that mathematicians use to describe mathematics. We discuss some of the ethical issues and approaches that each narrative tends to emphasise, and how these can lead to productive conversation in the absence of a universally agreed philosophical or ethical framework. We also discuss the dangers of a deficit model of ethical engagement, and other threats to conversation.

A Schur-Fire Way To Make Matrix Products Coincide, Keith Copenhaver, Josh Hiller, Christopher Sciortino

Journal of Humanistic Mathematics

When are the Schur product and the matrix product of two matrices the same? With the help of Morpheus (from The Matrix) and more, we explore this question and find both mathematical answers and math-memes.

Exploring The Dynamics Of Romantic Relationships Through The Lens Of Prem Rog, Umang Jain, Dheeraj Sharma, Pranay Goswami, Kuldeep Malik

Journal of Humanistic Mathematics

Romantic relationships are dynamic events that begin, grow, and frequently remain for a long time in a stagnant or fluctuating state until possibly dissipating. Although they are unquestionably the most significant dynamic events in our lives, dynamic systems theory has only recently included them in its formal framework. Without a mathematical model, it would be impossible to analyze and comprehend the dynamics because, in general, love stories are too brief to allow things to stabilize and are affected by the ups and downs of the surrounding community. In this paper, we set up models made up of four ordinary differential …

The More Things Change..., Mark Huber, Gizem Karaali

Journal of Humanistic Mathematics

No abstract provided.

Front Matter

Journal of Humanistic Mathematics

No abstract provided.

Patient-Specific Lumped-Parameter Model For Quantifying Vessel-Specific Remodeling And Predicting Right Ventricular Function In Pulmonary Hypertension, Christopher G. Lechuga, Amirreza Kachabi, Mitchel J. Colebank, Claudia E. Korcarz, Farhan Raza

Faculty Publications

Purpose: Pulmonary hypertension (PH) is a heterogeneous disease with patient-specific variability and vessel-specific remodeling, which eventually lead to right ventricular (RV) failure. The gold standard for RV assessment—pressure–volume (PV) loop acquisition—is invasive and limited to specialized settings. This study aims to develop a patient-specific lumped-parameter model that quantifies vessel-specific remodeling and simulates RV PV loops across PH phenotypes using routine clinical data.

Methods: A lumped-parameter model was calibrated using right heart catheterization and echocardiography data. Model agreement was assessed by R2 values for pressure and flow goodness-of-fit, and model-derived hemodynamic metrics were comparedwith clinical values. A dimensionality reduction approach was …

Recurrence Of Locally Perturbed Random Walks With Long Jumps, Tamer Oraby, András Telcs

School of Mathematical & Statistical Sciences Faculty Publications

We study the effect of local perturbations on the recurrence of random walks with long jumps. Such walks serve as discrete models for infinite-horizon Lorentz processes, in which a particle can take arbitrarily long steps in specific directions. Motivated by a question of Sinai in the finite-horizon case and its extension by Szász to the infinite-horizon setting, we give recurrence and transience criteria for long-jump walks on Z2 and certain classes of graphs, and we prove that local perturbations in a bounded region do not change the recurrence property. Our proofs combine the Markov chain approach with the electrical network …

Compilation Of A Nationwide River Image Dataset For Identifying River Channels And River Rapids Via Deep Learning, Nicholas Brimhall, Kelvyn K. Bladen, Thomas Kerby, Carl J. Legleiter, Cameron Swapp, Hannah Fluckiger, Julie Bahr, Makenna Roberts, Kaden Hart, Christina L. Stegman, Brennan L. Bean, Kevin R. Moon

Mathematics and Statistics Student Research and Class Projects

Remote sensing enables large-scale, image-based assessments of river dynamics, offering new opportunities for hydrological monitoring. We present a publicly available dataset consisting of 281,024 satellite and aerial images of U.S. rivers, constructed using an Application Programming Interface (API) and the U.S. Geological Survey’s National Hydrography Dataset. The dataset includes images, primary keys, and ancillary geospatial information. We use a manually labeled subset of the images to train models for detecting rapids, defined as areas where high velocity and turbulence lead to a wavy, rough, or even broken water surface visible in the imagery. To demonstrate the utility of this dataset, …

Projective Corepresentations And Cohomology Of Compact Quantum Groups, Kiran Maity

Doctoral Theses

In this thesis, we briefly review various types of projective corepresentations of compact quantum groups and prove the existence of suitable envelopes for them. We also study the associated invariant (dual) 2-cohomology and calculate it in a few concrete examples

High Frobenius Pushforwards Generate The Bounded Derived Category, Matthew R. Ballard, Srikanth B. Iyengar, Pat Lank, Alapan Mukhopadhyay, Josh Pollitz

Faculty Publications

This work concerns generators for the bounded derived category of coherent sheaves over a noetherian scheme X of prime characteristic. The main result is that when the Frobenius map on X is finite, for any compact generator G of D(X) the Frobenius pushforward F^e_*G generates the bounded derived category whenever p^e is larger than the codepth of X, an invariant that is a measure of the singularity of X. The conclusion holds for all positive integers e when X is locally complete intersection. The question of when one can take G = …

Does Sequence Matter? Impact Of Redesigning Sequential Calculus Course On Students’ Learning Outcomes, Chantal Levesque-Bristol Dr., Wonki Lee Dr., Emily M. Bonem, Benjamin C. Wiles, Jennifer D. Moss, Wilella D. Burgess, Weiling Li

International Journal of Teaching and Learning in Higher Education

As colleges and universities increasingly transform their STEM courses through the adoption of more active, student-centered pedagogies, there is a growing need to understand the impact of these educational innovations on student outcomes. This study is conducted in the context of a university-wide faculty development and course redesign project, IMPACT (Instruction Matters, Purdue Academic Course Transformation). IMPACT supports faculty in implementing student-centered pedagogical practices and creating equitable and inclusive learning environments. This study focuses on a sequence of two introductory calculus courses (Calculus 1 and Calculus 2, hereafter CALC 1 and CALC 2) that were transformed as part of the …

Arbitrage-Free Pricing With Diffusion-Dependent Jumps, Hamza A. Virk, Yihren Wu, Majnu John

Journal of Stochastic Analysis

Standard jump-diffusion models assume independence between jumps and diffusion components. We develop a multi-type jump-diffusion model where jump occurrence and magnitude depend on contemporaneous diffusion movements. Unlike previous one-sided models that create arbitrage opportunities, our framework includes upward and downward jumps triggered by both large upward and large downward diffusion increments. We derive the explicit no-arbitrage condition linking the physical drift to model pa- rameters and market risk premia by constructing an Equivalent Martingale Measure using Girsanov’s theorem and a normalized Esscher transform. This condition provides a rigorous foundation for arbitrage-free pricing in models with diffusion-dependent jumps.

Rogue Peakon, Posedness And Blow-Up Phenomenon For An Integrable Camassa–Holm Type Equation, Mingxuan Zhu, Zhenteng Zeng, Zaihong Jiang, Baoqiang Xia, Zhijun Qiao

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, we study an integrable Camassa–Holm (CH) type equation with quadratic nonlinearity. The CH type equation is shown integrable through a Lax pair, and particularly the equation is found to possess a new kind of peaked soliton (peakon) solution – called rogue peakon, that is given in a rational form with some logarithmic function, but not a regular traveling wave. We also provide multi-rogue peakon solutions. Furthermore, we discuss the local well-posedness of the solution in the Besov space 𝐵𝑠𝑝,𝑟 with 1 ≤ p, r ≤ ∞, 𝑠 >max⁡{1+1/𝑝,3/2} or 𝐵3/22,1 , and then prove the ill-posedness …

Nonresonance For Problems Involving (P, Q)-Laplacian Equations With Nonlinear Perturbations, E. Lopera, Nsoki Mavinga, D. Sánchez

Mathematics & Statistics Faculty Works

We are concerned with the existence of solutions of (p, q)-Laplacian problems that include nonlinear perturbation terms in both the differential equations and the boundary. Using variational methods and critical point theory, we prove the existence of weak solutions for the nonlinear problem when the nonlinearities involved remain asymptotically below the infimum of the set of eigenvalues of the (p, q)-Laplacian problem with weights and a spectral parameter present in both the differential equation and the boundary. Additionally, we establish an existence result for the nonlinear problem when the nonlinearities involved remain asymptotically below …