Digital Commons Network^™

A Comprehensive Study Of Clique Graphs And Clique Regular Graphs, Connor Phillips

Senior Honors Projects, 2020-current

If Γ is a graph for which every edge is in exactly one clique of order ω, then one can form a new graph with vertex set equal to these cliques.  This is a generalization of the line graph of Γ.  We discover many general results and classifications related to these clique graphs that will be useful to researchers studying graphs with this property. In particular, we find bounds on the spectrum of Γ (with exact results when Γ is k-regular) and some complete classifications when Γ is strongly regular. We apply our results to derive novel information about the …

Constructing Genotype And Phenotype Network Helps Reveal Disease Heritability And Phenome-Wide Association Studies, Xuewei Cao, Lirong Zhu, Xiaoyu Liang, Shuanglin Zhang, Qiuying Sha

Michigan Tech Publications

Background

Analyses of a bipartite Genotype and Phenotype Network (GPN), linking the genetic variants and phenotypes based on statistical associations, provide an integrative approach to elucidate the complexities of genetic relationships across diseases and identify pleiotropic loci. In this study, we assess contributions to constructing a well-defined GPN with a clear representation of genetic associations by comparing the network properties with a random network, including connectivity, centrality, and community structure. Then, we extend our discussion to include two applications of bipartite GPN in disease heritability enrichment analysis and phenome-wide association studies (PheWAS).

Results

We construct network topology annotations of genetic …

Free Probability Theory Over The Scaled Hyperbolic Numbers, Daniel Alpay, Ilwoo Choo

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper, we introduce a notion of free probability over the scaled hyperbolic numbers. Scaled hypercomplex numbers {D_t} _t∈R are constructed as sub-structures of scaled hypercomplex numbers {H_t} _t∈R under the scales (or, the moments) of the set R of real numbers.We show that if t < 0, then the classical free probability theory covers our free probability on {D_t} _{t< 0}; if t > 0, then our free probability on {D_t} _t>0 is represented by the free probability over the classical hyperbolic numbers D = D₁; and if t = 0, then the free probability on D₀ is actually over the …

From Textbook To Classroom: Gaps In Mathematical Representation And Translation In Nepali Education, Deepak Basyal

Mathematics and Statistics

Translating real-world problem contexts into mathematical symbols is a key skill in curricula worldwide. This study examines Nepali students’ abilities to translate words into symbols and how Nepali textbooks from earlier grades support the development of these skills. To investigate this phenomenon, a sequential explanatory study was conducted with a test for 10th graders and an analysis of Grades 6—9 textbooks. Results show that students struggle to convert real-world contexts into mathematical symbols, while textbooks provide limited opportunities for practice. In particular, students have difficulty translating symbols back into real-world situations, suggesting a link between textbook opportunities and achievement. These …

Orbital Stability Of Solitary Wave Solutions Of The Generalized Benjamin Equation, Leijin Cao, Binhua Feng, Zhaosheng Feng, Yichun Mo

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, we study the existence and stability of solitary wave solutions for the generalized Benjamin equation in both the L2-critical and L2-supercritical cases by applying the variational methods and the non-homogeneous Gagliardo–Nirenberg inequality. Our main results generalize and complement the existing results in the literature.

Degeneracies In A Weighted Sum Of Two Squares, Ishan V. Ramesh

Rose-Hulman Undergraduate Mathematics Journal

This work is an attempt to classify and quantify instances when a weighted sum of two squares of positive integers, 3n2 1 +n2 2, can be realized in more than one way. Our project was inspired by a particular study of two-dimensional quantum billiards [S. G. Jackson, H. Perrin, G. E. Astrakharchik, and M. Olshanii, SciPost Phys. Core 7, 062 (2024)] where the weighted sums of interest represents an energy level with the two integers being the billiard’s quantum numbers; there, the 3-fold degeneracies seem to dominate the energy spectrum. Interestingly, contrary to the conventional paradigm, these degeneracies are not …

Math Meets Climate: The Energy Balance Model, Maria I. Sanchez Muniz

Open Educational Resources

This assignment introduces students to the mathematics of Earth’s climate through the classical energy balance model. Students analyze how incoming solar radiation, outgoing thermal radiation, and temperature-dependent albedo interact to determine Earth’s equilibrium temperature. Using analytical calculations and computational tools, students identify equilibrium states, assess their stability, and interpret the results through the lens of dynamical systems and bifurcation theory. The activity builds conceptual understanding of climate feedbacks, greenhouse effects, and tipping behavior using a transparent, one-variable model. Designed for applied mathematics and interdisciplinary STEM courses, this assignment emphasizes computation, physical interpretation, and real-world relevance. It is released as a …

Collaborative Research: Cif: Medium: Foundations Of Robust Deep Learning Via Data Geometry And Dyadic Structure, Kevin Moon

Funded Research Records

No abstract provided.

Representation Theory In Unoriented And Non-Semisimple Physics, Matthew Bruce Young

Funded Research Records

No abstract provided.

Understanding Enso Through Mathematical Models, Maria I. Sanchez Muniz

Open Educational Resources

This assignment introduces students to conceptual models of the El Niño–Southern Oscillation (ENSO) and guides them through a structured investigation of their physical and mathematical foundations. Students analyze the recharge–oscillator and delayed–oscillator frameworks, explore how differential equations capture ocean–atmosphere interactions, and evaluate parameter-driven changes in oscillatory behavior. A key component of the work is the guided use of generative AI as a research tool: students employ AI models to locate peer-reviewed literature, interrogate model extensions, and refine their understanding of complex mechanisms, while synthesizing all final explanations in their own words. By blending classical climate modeling with modern AI-supported inquiry, …

Systematics And Systems Theory: Reconstructability Analysis Of The Tetrad, Martin Zwick

Complex Systems Faculty Publications and Presentations

This talk discusses the relationship between systems theory, specifically Reconstructability Analysis, and Systematics, a systems theory-like framework of number symbolism developed by John G. Bennett, which he presented in his four-volume magnum opus, The Dramatic Universe. The talk, given to a community of people interested in Bennett's ideas, focuses on Martin Zwick's paper "Ideas and Graphs: the Tetrad of Activity" archived at https://archives.pdx.edu/ds/psu/36249.

A (Mini) Mathematics Research Experience In A Math Teachers’ Circle Session, Michelle Manes, Linda Venenciano, Seanyelle Yagi

Journal of Math Circles

We describe a single Math Teachers’ Circle session during which a group of teachers collectively engaged in a “mini mathematics research experience.” This provides a model for providing research experiences for mathematics teachers through content-based professional development programs. We conjecture that these experiences will have many of the same benefits as research experiences for teachers in lab sciences.

Timss 2023 – Infographic: Future Pd Needs Of Maths Teachers, Jo Earp

Teacher infographics

The IEA’s Trends in International Mathematics and Science Study (TIMSS) is a global assessment measuring student achievement in years 4 and 8. Teachers from participating schools also complete a questionnaire. Today’s infographic looks at the topics that mathematics teachers reported needing professional development on in the future.

(R2127) On 2-Absorbing Hesitant Primary Fuzzy Ideals Of Rings, B. Anitha, M. Vidhya

Applications and Applied Mathematics: An International Journal (AAM)

By presenting 2-absorbing hesitant primary fuzzy ideals, we begin the investigation of a generalisation of hesitant primary fuzzy ideals (HPRFI) in rings in this research. The concepts of a weakly completely 2-absorbing hesitant primary fuzzy ideal (WC2-AHPRFI) and a Weakly completely 2-absorbing hesitant fuzzy ideal (WC2-AHFI) are developed, and their structural features and attributes are examined. We introduce the idea of a 2-absorbing hesitant K-fuzzy ideal (2-AHK-FI), 2-absorbing hesitant K-primary fuzzy ideal (2-AHK-PRFI) and examine a few of its characteristics.

(R2122) Analysis Of Halo Orbits In The Elliptical R3bp With Mass Variation, Majhar Ali, Abdullah .

Applications and Applied Mathematics: An International Journal (AAM)

The elliptic restricted three-body problem investigates the motion behaviour of the variable mass infinitesimal body under the gravitational forces of the radiated oblate primary and dipole secondary. The equations of motion of the infinitesimal body are determined using Jeans law and Meshcherskii space time transformations. Using the Lindstedt-Poincaré method, we perform the solutions of the equations of motion. With the use of these solutions and the equations of motion, we numerically illustrate the time series, phase spaces, projections and the Halo orbits.

(R2094) On Mds Bisymmetric Rhotrices Using Self-Dual Bases And Conjugate Elements Of Finite Fields, Shalini Gupta, Ruchi Narang, Manpreet Singh

Applications and Applied Mathematics: An International Journal (AAM)

Bisymmetric matrices have wide range of applications in statistics, engineering problems, information theory and computer science including coding theory and cryptography. In cryptography, a rhotrix being a couple matrix doubles the security of the cryptosystem. Here, we construct maximum distance separable (MDS) bisymmetric rhotrices using self-dual bases and conjugate elements of finite fields. MDS rhotrices are very crucial for the designing of block ciphers and hash functions in cryptography.

Data-Assimilation-Enabled Fast Convergence Of The Uzawa Scheme For Navier-Stokes Equations With Large Reynolds Numbers, Jessica C. Franklin

All Theses

This work studies efficient techniques utilized to solve the Navier-Stokes equations that model incompressible Newtonian flow in the setting where partial solution data is available. First, we analyze the Picard iteration and Picard with continuous data assimilation applied and test convergence. Then, we analyze the Arrow-Hurwicz (AH) and Uzawa iterative procedures and test convergence. Finally, we apply continuous data assimilation to Uzawa, and analytical and numerical results show that CDA improves Uzawa convergence for the NSE, similar to CDA-Picard but it is much more efficient.

A Persistent Homology Framework For Scrna-Seq: Assessing Clustering Robustness And Quantifying Preprocessing And Integration Effects On Topological Features., Jonah Daneshmand

Electronic Theses and Dissertations

As single-cell RNA sequencing (scRNA-seq) data expands, robust methods for integrating diverse datasets are critical. This dissertation applies Persistent Homology (PH), a technique from Topological Data Analysis (TDA), to a collection of scRNA-seq datasets spanning eight tissue types to quantify how data integration affects topological features and biological interpretability. We assessed global topological structure using Betti curves, Euler characteristics, and persistence landscapes across raw, normalized, and integrated data representations. Our analysis revealed a performance inversion: while conventional methods excelled on unintegrated data, high-granularity topological methods, particularly those sensitive to global data structure, became superior after integration. This suggests a synergy …

Preparing Confident Elementary Mathematics Teachers: Addressing Anxiety, Building Fluency, And Strengthening Practice., Amber Diamond Pendleton

Electronic Theses and Dissertations

This dissertation, presented in three articles, examines how elementary preservice teachers (PSTs) experience and develop as mathematics learners and future educators when methods coursework deliberately integrates mathematical reasoning with affective supports. Collectively, the studies investigate how research-informed instructional design may help address mathematics anxiety (MA), mathematics teaching anxiety (MTA), procedural fluency, and instructional practice. The first article, written for practitioners and elementary mathematics teachers, illustrates how well-established instructional routines, such as number talks and student work analysis, can be facilitated to support procedural fluency in multiplication. Using classroom vignettes, this article highlights instructional moves that make structure visible, promote strategy …

Wacsaw: An Adaptive, Statistical Method To Classify Movement Into Sleep And Wakefulness States, Austin Vandegriffe, V. A. Samaranayake, Matthew S. Thimgan

Mathematics and Statistics Faculty Research & Creative Works

Wearable actimeters can improve our understanding of sleep in the natural environments. Current algorithms may produce inaccuracies in specific individuals and circumstances, such as quiet wakefulness. New hardware allows data collection at higher frequencies enabling sophisticated analytical methods. We have developed a novel statistical algorithm, the Wasserstein Algorithm for Classifying Sleep and Wakefulness (WACSAW), to identify behavioral states from recordings of everyday movement. WACSAW employs optimal transport techniques to identify segments with differing activity variability. Functions characterizing the segments' movement distributions were clustered into two groups using a k-nearest neighbors and labeled as sleep or wake based on their proximity …

Coherent Span-Valued 2d Tqfts, Sophia Marx, Rajan Amit Mehta

Mathematics Sciences: Faculty Publications

We consider commutative Frobenius pseudomonoids in the bicategory of spans, and we show that they are in correspondence with 2-Segal cosymmetric sets. Such a structure can be interpreted as a coherent 2-dimensional topological quantum field theory taking values in the bicategory of spans. We also describe a construction that produces a 2-Segal cosymmetric set from any partial monoid equipped with a distinguished element.

Fast Computation Of The Friction Stir Welding Process With Model Order Reduction, Joshua Kay

All Graduate Theses and Dissertations, Fall 2023 to Present

Friction stir welding (FSW) is a manufacturing process used to join materials through intense heat and pressure. To better understand and optimize this FSW process, mathematical models incorporating non-Newtonian Navier-stokes equations for large plastic deformation and heat equation for the heat transfer are used to simulate the FSW process. However, solving these coupled and nonlinear equations with high accuracy requires significant computational power and time. This thesis has applied model order reduction, more precisely the Proper Orthogonal Decomposition (POD) and Discrete Empirical Interpolation Method (DEIM), to efficiently solve the FSW system in low-dimensional space. Slight modifications are made regarding the …

The Impact Of Transmission Thresholds Across Multiple Scales On The Spread Of Chronic Wasting Disease In Wisconsin, Jen Mcclure

All Graduate Theses and Dissertations, Fall 2023 to Present

Wildlife diseases can be difficult to control once they are established. This is especially true when they spread through contact with infectious material left in the environment. One such disease is chronic wasting disease (CWD), a fatal illness affecting deer and related species in North America and other regions. CWD is caused by prions, misfolded proteins that can remain infectious for years after shedding by infected hosts.

Recent research shows that CWD infection does not always follow from the gradual accumulation of prions through small contact events. Rather, an individual may need to encounter a certain prion dose all at …

Implications Of The Attenuated Allee Effect On Population Dynamics, Dana Strong

All Graduate Theses and Dissertations, Fall 2023 to Present

The Allee effect is an ecological phenomenon characterized by a per capita growth rate that increases as the population size increases in the context of low population density. The Allee effect can lead to rapid extinction events. Consequently, ecologists can attempt to control the strength of the Allee effect to help increase the population (in the case of endangered species) or decrease the population (in the case of parasites or pests).

This research aimed to study not the strength of the Allee effect, but the intensity of the Allee effect, or how rapidly cooperation between individuals causes the per capita …

By The Numbers: Numeracy, Religion, And The Quantitative Transformation Of Early Modern England (Book Review), Calvin Jongsma

Faculty Work Comprehensive List

Reviewed Title: By the Numbers: Numeracy, Religion, and the Quantitative Transformation of Early Modern England by Jessica Marie Otis. New York, NY: Oxford University Press, 2024. 264 pp. ISBN: 9780197608777.

Content And Consequences: Impact Of Representation In Stem Higher Education Instructional Content On Marginalized Students, Nichole Ventura

Doctorate in Education

This qualitative study examined representation of historically marginalized students in STEM instructional content at the higher education level and its impact on their learning experiences. Despite growing diversity initiatives in STEM enrollment, curricular materials often fail to reflect the identities of underrepresented students. Using critical theory and interpretivist approaches, this research investigated how representation—or its absence—shapes students' sense of belonging, academic identity formation, and persistence. Through semi-structured interviews with undergraduate students from historically marginalized backgrounds, and purposeful sampling, this study captured the lived experiences of students engaging with STEM instructional materials. Interview protocols explored how students perceive their representation in …

(R2137) Smarandache Curved Ruled Surfaces And Their Characterizations According To Modified Orthogonal Frame In E^3, Gülnur Şaffak Atalay

Applications and Applied Mathematics: An International Journal (AAM)

In this study, the ruled surfaces generated by Smarandache curves are expressed according to the modified orthogonal frame defined according to both curvature and torsion of the given unit speed curve in 3-dimensional Euclidean space; some special characterizations of these surfaces such as developability, striction curves, distribution parameters are given. In addition, some examples of these surfaces are given in graphical form. We use Maple to draw the graphs of the examples.

Scaling Wind Farm Layout Optimisation To Meet Uk Energy Demands, Grace Richardson

The Plymouth Student Scientist

This research aims to explore the Wind Farm Layout Optimisation Problem (WFLOP) through both single- and multi-objective optimisation, to show that it can be used to create farm layouts that provide enough power for all homes in the UK. We highlight the intricacies of the problem, including the application of different algorithms and models, and the challenges of scaling the models to achieve optimal results. We employ both a Hill Climbing Algorithm (HCA) and an evolutionary algorithm (NSGA-II) to obtain wind farm layouts, and compare these to the well-known results proposed by Mosetti et al. Moreover, we present a novel …

Connections On Manifolds: A Modern Approach To Classical Differential Geometry, Jed Allen Wilshire

Theses and Dissertations

The purpose of this thesis is to connect modern differential geometry concepts to the classical differential geometry study of surfaces, specifically spheres. This thesis will cover the topics of Riemannian manifolds, tangent spaces, vectors, vector fields, the connection of Levi-Civita, the Christoffel symbols, and Riemannian curvature. Each concept will be demonstrated on the well known unit sphere manifold.

A Computational Approach To Ramanujan-Type Congruences For Products Of Level 13, Aiden Michael Lalonde

Theses and Dissertations

In this thesis, we find products that are modular forms of level 13 whose Fourier coefficients satisfy certain congruences. These products are constructed in terms of integer powers of Klein forms and the Dedekind eta function. The products that are modular forms for Γ1 (13) have vectors of exponents that are elements of a bounded polytope. A parameterization for this polytope allows us to find a complete set of products that are modular forms for Γ1 (13). The corresponding graded ring of modular forms containing these products is constructed from a finite set of generators. These generators satisfy a finite …