Mathematics Commons^™

Time-Dependent Shear Flows And Their Applications In Parabolic–Parabolic Patlak–Keller–Segel Systems*, Siming He

Faculty Publications

In this study, we investigate the behavior of three-dimensional parabolic–parabolic Patlak–Keller–Segel systems in the presence of ambient shear flows. Our findings demonstrate that when the total mass of the cell density is below a specific threshold, the solution remains globally regular as long as the flow is sufficiently strong. The primary difficulty in our analysis stems from the fast creation of chemical gradients due to strong shear advection.

A Student Guide To Using Ai To Enhance Algebraic Understanding, Karan Puri

Open Educational Resources

This is a step-by-step guide that students can follow to use AI tools to check their understanding of concepts that have been tested in the introductory/college algebra classroom.

Counting Rotational Sets For Laminations Of The Unit Disk From First Principles, Michael J. Moorman, Gabriel B. Quijano, Matthew C. Williams Jr.

Rose-Hulman Undergraduate Mathematics Journal

By studying laminations of the unit disk, we can gain insight into the structure of Julia sets of polynomials and their dynamics in the complex plane. The polynomials of a given degree, d, have a parameter space. The hyperbolic components of such parameter spaces are in correspondence to rotational polygons, or classes of "rotational sets'', which we study in this paper. By studying the count of such rotational sets, and therefore the underlying structure of these rotational sets and polygons, we can gain insight into the interrelationship among hyperbolic components of the parameter space of these polynomials.

These rotational sets …

Finite Posets As Prime Spectra Of Commutative Noetherian Rings, David T. Alkass

Rose-Hulman Undergraduate Mathematics Journal

We study finite partially ordered sets of prime ideals as found in commutative Noetherian rings. In doing so, we establish that these posets have a bipartite structure and devise a construction for finding ring spectra that are order-isomorphic to many such posets. Specifically, we prove that any finite complete bipartite graph is order-isomorphic to the spectrum of a ring of essentially finite type over the field of rational numbers. Furthermore, we prove that prime spectra of such rings can also depict any finite path or even cycle.

Proportion Of P-Adic Polynomials Which Are Irreducible, Isaac Rajagopal

Rose-Hulman Undergraduate Mathematics Journal

We attempt to quantify the exact proportion of p-adic polynomials of degree n which are irreducible. We find an exact answer to this when n is prime and p != n, and also when n = 4 and p != 2. Our answers are rational functions in p. This relates to previous work done to find exact proportions of p-adic polynomials of degree n which have k roots.

Heart Disease Prediction Using Ensemble Tree Algorithms: A Supervised Learning Perspective, Enoch Sakyi-Yeboah, Edmund F. Agyemang, Vincent Agbenyeavu, Akua Osei- Nkwantabisa, Priscilla Kissi-Appiah, Lateef Moshood, Lawrence Agbota, Ezekiel N.N. Nortey

School of Mathematical & Statistical Sciences Faculty Publications

Heart disease stands as a leading cause of morbidity and mortality globally, presenting a significant public health challenge. Therefore, early prediction and detection are critical, leading to timely and appropriate interventions at early stages. Four ensemble tree-based algorithms were used in this study: adaptive boosting, extreme gradient boosting, random forest, and extremely randomized trees, investigating their ability to predict heart disease. Data related to heart disease clinical features was obtained from the open Kaggle Machine Learning Dataset repository. Adaptive Boosting stands out as the highest performer, achieving an average testing accuracy of 93.70%, precision of 93.71%, recall of 93.70%, and …

Extended High-Frequency Hearing And Suprathreshold Neural Synchrony In The Auditory Brainstem, Jithin Raj Balan, Sri Mishra, Hansapani Rodrigo

School of Mathematical & Statistical Sciences Faculty Publications

Elevated hearing thresholds in the extended high frequencies (EHFs) (>8 kHz) are often associated with poorer speech-in-noise recognition despite a clinically normal audiogram. However, whether EHF hearing loss is associated with disruptions in neural processing within the auditory brainstem remains uncertain. The objective of the present study was to investigate whether elevated EHF thresholds influence neural processing at lower frequencies in individuals with normal audiograms. Auditory brainstem responses (ABRs) were recorded at a suprathreshold level (80 dB normal hearing level) from 45 participants with clinically normal hearing. The recording protocol was optimized to obtain robust wave I of the …

Analysis Of Systematic Trade-Offs Between Military And Healthcare Expenditure Alongside Gdp Growth Of Select Asian And Western Exporting Economies In The 21st Century, Rahul Balamurugan, Carlos Gershenson, Preethi Nanjundan, Hiroki Sayama

Northeast Journal of Complex Systems (NEJCS)

This study explores the complexity in the trade-offs between military expenditure, healthcare expenditure, and GDP growth across select Asian nations and major weapon-exporting countries, examining how nations allocate finite resources between national security and human well-being over the past two decades. Using a systems science approach, the research integrates Granger causality testing to analyze temporal and directional relationships among GDP growth, military expenditure, and healthcare expenditure, uncovering their dynamic interdependencies. The methodology includes trend and slope analysis, Granger causality testing, outlier detection, and clustering to identify heterogeneity in resource allocation strategies. Developed, weapon-exporting nations exhibit complementary trends, with strong causality …

From Machine Learning To Human Learning: What Can Pedagogy Learn From Ai Successes, Victor L. Timchenko, Yury P. Kondratenko, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Many machine learning techniques -- including many techniques behind the current AI-based boom in machine learning -- come from the analysis of successful human learning strategies (and researchers expect that other human learning experiences can lead to even more effective AI-based systems). At this moment, so much experience have been accumulated in AI-based machine learning that it is time to start the analysis in the opposite direction -- to see what can human-based pedagogy learn from AI successes. In this chapter, we provide the first results of such an analysis -- some of which go somewhat against the current pedagogical …

Gurevich's Quizani Dialogs As An Example Of Explainable Mathematics, And How This Is Related To Quantum Space-Time Ideas That Can Speed Up Computations, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Everyone talks about the need for Explainable AI -- when, to supplement a long difficult-to-understand sequence of computational steps leading to AI's decision, we are looking for a shorter and understandable more-informal explanation for this decision. In this paper, we argue that this need is a particular case of what we call Explainable Mathematics -- when we want to supplement a long sequence of arguments and/or computations with a shorter and understandable more-informal explanation. Important instances of Explainable Mathematics are Yuri Gurevich's Quizani dialogs that help explain complex results from theoretical computer science and physicists' more-informal explanations of complex physical …

Unfortunately, The Universal Predictor Cannot Be Made Constructive, Julio C. Urenda, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

A recent article in the Notices of the American Mathematical Society reminded the mathematics community that, under the Axiom of Choice, it is possible to have a universal predictor: if we input, into this predictor, the values of a function for all moments t < to for some to, then, for almost all to, this predictor correctly predicts the next values of this function on some interval [to, to + ε). This predictor cannot be used for actual predictions: it is based on the Axiom of Choice and is, therefore, not constructive. A natural question is: maybe it is possible to have another universal predictor, which is constructive? In this paper we show that, unfortunately, it is not possible to have a constructive universal predictor. In other words, the above universal predictor result cannot be used for actual predictions.

Discrete Math For Computer Science - Chapter 8: Union And Intersection And Complement: Set Identities, Houman Kamran Habibkhani

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Discrete Math For Computer Science - Chapter 15: Mathematical Induction, Houman Kamran Habibkhani

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Discrete Math For Computer Science - Chapter 13: Sequences: Recurrence Relations, Houman Kamran Habibkhani

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Discrete Math For Computer Science - Chapter 26: Introduction To Graphs: Graph Representations, Houman Kamran Habibkhani

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Discrete Math For Computer Science - Chapter 2: Logical Equivalence: Laws Of Propositional Logic, Houman Kamran Habibkhani

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Discrete Math For Computer Science - Chapter 11: Inverse Of A Function: Composition Of Functions, Houman Kamran Habibkhani

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Discrete Math For Computer Science - Chapter 16: Recursive Definitions: Recursive Algorithms, Houman Kamran Habibkhani

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Discrete Math For Computer Science, Houman Kamran Habibkhani

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Discrete Mathematics and its Applications is a focused introduction to the primary themes in a discrete mathematics course, as introduced through extensive applications, expansive discussion, and detailed exercise sets. These themes include mathematical reasoning, combinatorial analysis, discrete structures, algorithmic thinking, and enhanced problem-solving skills through modeling. Its intent is to demonstrate the relevance and practicality of discrete mathematics to all students.

Discrete Math For Computer Science - Chapter 4: Logical Reasoning: Rules Of Inference, Houman Kamran Habibkhani

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