Mathematics Commons^™

Cardiovascular Disease Subtypes And Alzheimer's Disease: Phenotypic And Genetic Associations In The Uk Biobank And All Of Us Research Program, Aili Toyli, Chen Zhao, Kuan Jui Su, Hui Shen, Hong Wen Deng, Qing Hui Chen, Qiuying Sha, Weihua Zhou

Michigan Tech Publications

BACKGROUND: Cardiovascular disease (CVD) and Alzheimer's disease (AD) are major public health concerns that share overlapping risk factors and potential mechanistic pathways. Although vascular contributions to cognitive decline are well documented, the specific relationships between AD and different CVD subtypes remain poorly understood. METHODS: In this cross-sectional study, we examined associations between AD and 11 CVD subtypes using logistic regression models in 2 large biobanks: the UK Biobank (n=502 133) and the All of Us Research Program (n=287 011). Models were adjusted for demographic, lifestyle, and clinical covariates. We also explored genetic overlap between AD and CVD traits through proximity-based …

Λ-Fold Near-Factorizations Of Groups, Donald L. Kreher, Shuxing Li, Douglas R. Stinson

Michigan Tech Publications

We initiate the study of λ-fold near-factorizations of groups with λ>1. While λ-fold near-factorizations of groups with λ=1 have been studied in numerous papers, this is the first detailed treatment for λ>1. We establish fundamental properties of λ-fold near-factorizations and introduce the notion of equivalence. We prove various necessary conditions of λ-fold near-factorizations, including upper bounds on λ. We present three constructions of infinite families of λ-fold near-factorizations, highlighting the characterization of two subfamilies of λ-fold near-factorizations. We discuss a computational approach to λ-fold near-factorizations and tabulate computational results for abelian groups of small order.

A Two-Phase Perspective On A Related-Rates Paradox, Charlie Vazquez Acosta

Honors Capstones

Related-rates problems are a standard topic in first-year calculus and have appeared in textbooks for over 150 years. These problems are used to teach implicit differentiation and the relationship between changing quantities. Common examples include the falling ladder, the fishing bobber, the melting snowball, and the leaking conical tank. In each of these problems, a quantity is changing at a constant rate, and students are asked to find the rate of change of another related quantity. While the computations themselves are usually straightforward, the standard models lead to unrealistic results near the end of the motion. For example, the falling …

New Identities In The Character Table Of Symmetric Groups Involving Riordan Numbers, David J. Hemmer, Armin Straub, Karlee J. Westrem

Michigan Tech Publications

Amdeberhan recently proposed certain equalities between sums in the character table of symmetric groups. These equalities are between signed column sums in the character table, summing over the rows labeled by partitions in Ev(λ), where λ is a partition of n with r nonzero parts and Ev(λ) is a multiset containing 2r partitions of 2n. While we observe that these equalities are not true in general, we prove that they do hold in interesting special cases. These lead to new equalities between sums of degrees of irreducible characters for the symmetric group and a new combinatorial interpretation for the Riordan …

Twas-Ctl: A Robust And Efficient Method For Multi-Tissue Transcriptome-Wide Association Studies Using Cross-Tissue Learners, Md Mutasim Billah, Hairong Wei, Fengzhu Sun, Kui Zhang

Michigan Tech Publications

The advent of transcriptome-wide association studies (TWAS) has expanded the classical genome-wide association study (GWAS) framework by integrating gene expression with genetic variation to identify trait-associated variants. While multi-tissue TWAS approaches improve statistical power over single-tissue models, existing methods often lose information during result aggregation and require intensive computation. Here, we present TWAS-CTL (Cross-Tissue Learner), a novel framework that leverages heterogeneous gene expression across tissues by adaptively reweighting and optimizing multiple single-tissue learners. Simulations demonstrate that TWAS-CTL achieves higher statistical power than the leading method, UTMOST, while maintaining proper type I error control and reducing computational time by over half. …

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 …

Gboost-Ctl: A Novel Method In Multi-Tissue Transcriptome-Wide Associations Studies In Cross-Tissue Learner Incorporating Gwas Information, Md Mutasim Billah, Hairong Wei, Fengzhu Sun, Kui Zhang

Michigan Tech Publications

Genome-wide association studies (GWAS) have uncovered numerous genetic variants linked to complex human diseases, yet linking these variants to transcripts and tissues that drive pathology remains difficult. Multi-tissue transcriptome-wide association studies (TWAS) offer a powerful bridge, but existing analytical methods have some limitations, either by discarding important signals by separately analyzing and then aggregating results across tissues, implying imputation models in individual tissues, or fusing them with weights that ignore how much GWAS signal each tissue actually carries. Therefore, most of the existing methods do not work uniformly across different GWAS cohorts. Here, we propose GBoost-CTL - a GWAS-boosted cross-tissue …

Strong External Difference Families And Classification Of Α-Valuations, Donald L. Kreher, Maura B. Paterson, Douglas R. Stinson

Michigan Tech Publications

One method of constructing (Formula presented.) -SEDFs (i.e., strong external difference families) in (Formula presented.) makes use of (Formula presented.) -valuations of complete bipartite graphs (Formula presented.). We explore this approach and we provide a classification theorem which shows that all such (Formula presented.) -valuations can be constructed recursively via a sequence of “blow-up” operations. We also enumerate all (Formula presented.) -SEDFs in (Formula presented.) for (Formula presented.) and we show that all these SEDFs are equivalent to (Formula presented.) -valuations via affine transformations. Whether this holds for all (Formula presented.) as well is an interesting open problem. We also …

Preservation Of The Bernstein Property For Sums Of Independent Random Variables, Iosif Pinelis

Michigan Tech Publications

It is shown that Bernstein-type conditions on independent random variables are preserved by their sum. Some optimality properties of such preservation are proved.

On The Hamilton-Waterloo Problem With A Single Factor Of 6-Cycles, Zazil Santizo Huerta, Melissa S. Keranen

Michigan Tech Publications

The uniform Hamilton-Waterloo Problem (HWP) asks for a resolvable (CM, CN)decomposition of Kv into α CM-factors and β CN-factors. We denote a solution to the uniform Hamilton-Waterloo problem by HWP(v, M, N, α, β). Our research concentrates on addressing some of the remaining unresolved cases, which pose a significant challenge to generalize. We place a particular emphasis on instances where the gcd(M, N) = {2, 3}, with a specific focus on the parameter M = 6. We introduce modifications to some known structures, and develop new approaches to resolving these outstanding challenges in the construction of uniform 2-factorizations. This innovative …

Stability Analysis Of The Eulerian-Lagrangian Finite Volume Methods For Nonlinear Hyperbolic Equations In One Space Dimension, Yang Yang, Jiajie Chen, Jing Mei Qiu

Michigan Tech Publications

In this paper, we construct a novel Eulerian-Lagrangian finite volume (ELFV) method for nonlinear scalar hyperbolic equations in one space dimension. It is well known that the exact solutions to such problems may contain shocks though the initial conditions are smooth, and direct numerical methods may suffer from restricted time step sizes. To relieve the restriction, we propose an ELFV method, where the space-time domain was separated by the partition lines originated from the cell interfaces whose slopes are obtained following the Rakine-Hugoniot junmp condition. Unfortunately, to avoid the intersection of the partition lines, the time step sizes are still …

Fixed Hooks In Arbitrary Columns Of Partitions, Philip Cuthbertson

Michigan Tech Publications

In a paper by the author, Hemmer, Hopkins, and Keith, the concept of a fixed point in a sequence was applied to the sequence of first column hook lengths of a partition. In this paper, we generalize this notion to fixed hook lengths in an arbitrary column of a partition. We establish combinatorial connections between these fixed hooks and colored partitions that have interesting gap and mex-like conditions. Additionally, we obtain several generating functions for hook lengths of a given fixedness by hook length or part size in unrestricted partitions, as well as some classical restrictions such as odd and …

Latent Characterization Of The Complete Batse Gamma-Ray Bursts Catalogue Using Gaussian Mixture Of Factor Analysers And Model-Estimated Overlap-Based Syncytial Clustering, Fan Dai, Ranjan Maitra

Michigan Tech Publications

Characterizing and distinguishing gamma-ray bursts (GRBs) has interested astronomers for many decades. While some authors have found two or three groups of GRBs by analysing only a few parameters, recent work identified five ellipsoidally shaped groups upon considering nine parameters T50, T90, F1, F2, F3, F4, P64, P256, P1024. Yet others suggest subclasses within the two or three groups found earlier. Using a mixture model of Gaussian factor analysers, we analysed 1150 GRBs, that had nine parameters observed, from the current Burst and Transient Source Experiment (BATSE) catalogue, and again established five ellipsoidal-shaped groups to describe the GRBs. These five …

Nestings Of Bibds With Block Size Four, Marco Buratti, Donald L. Kreher, Douglas R. Stinson

Michigan Tech Publications

In a nesting of a balanced incomplete block design (or BIBD), we wish to add a point (the nested point) to every block of a (Formula presented.) -BIBD in such a way that we end up with a partial (Formula presented.) -BIBD. In the case where the partial (Formula presented.) -BIBD is in fact a (Formula presented.) -BIBD, we have a perfect nesting. We show that a nesting is perfect if and only if (Formula presented.). Perfect nestings were previously known to exist in the case of Steiner triple systems (i.e., (Formula presented.) -BIBDs) when (Formula presented.), as well as …

A Novel Phenotype Imputation Method With Copula Model, Jianjun Zhang, Jane Zizhen Zhao, Samantha Gonzales, Xuexia Wang, Qiuying Sha

Michigan Tech Publications

BACKGROUND: Jointly analyzing multiple phenotype/traits may increase power in genetic association studies by aggregating weak genetic effects. The chance that at least one phenotype is missing increases exponentially as the number of phenotype increases especially for a real dataset. It is a common practice to discard individuals with missing phenotype or phenotype with a large proportion of missing values. Such a discarding method may lead to a loss of power or even an insufficient sample size for analysis. To our knowledge, many existing phenotype imputing methods are built on multivariate normal assumptions for analysis. Violation of these assumptions may lead …

Identifiability And Convergence Behavior For Markov Chain Monte Carlo Using Multivariate Probit Models, Xiao Zhang

Michigan Tech Publications

Multivariate probit models have been popularly utilized to analysis multivariate ordinal data. However, the identifiable multivariate probit models entail the covariance matrix for the underlying multivariate normal variables to be a correlation matrix, which brings a rigorous task to conduct efficient statistical analysis. Parameter expansion to make the identifiable model to be non-identifiable has been inevitably explored. However, the effect of the expanded parameters on the convergence of Markov chain Monte Carlo (MCMC) is seldomly investigated; in addition, the comparison of MCMC developed based on the identifiable model and that based on the non-identifiable model is hardly ever explored, especially …

A High-Order Eulerian–Lagrangian Runge–Kutta Finite Volume (El–Rk–Fv) Method For Scalar Nonlinear Conservation Laws, Jiajie Chen, Joseph Nakao, Jing Mei Qiu, Yang Yang

Michigan Tech Publications

We present a class of high-order Eulerian–Lagrangian Runge–Kutta finite volume methods that can numerically solve Burgers’ equation with shock formations, which could be extended to general scalar conservation laws. Eulerian–Lagrangian (EL) and semi-Lagrangian (SL) methods have recently seen increased development and have become a staple for allowing large time-stepping sizes. Yet, maintaining relatively large time-stepping sizes post shock formation remains quite challenging. Our proposed scheme integrates the partial differential equation on a space-time region partitioned by linear approximations to the characteristics determined by the Rankine–Hugoniot jump condition. We trace the characteristics forward in time and present a merging procedure for …

Meta-Analysis Of Set-Based Multiple Phenotype Association Test Based On Gwas Summary Statistics From Different Cohorts, Lirong Zhu, Shuanglin Zhang, Qiuying Sha

Michigan Tech Publications

Genome-wide association studies (GWAS) have emerged as popular tools for identifying genetic variants that are associated with complex diseases. Standard analysis of a GWAS involves assessing the association between each variant and a disease. However, this approach suffers from limited reproducibility and difficulties in detecting multi-variant and pleiotropic effects. Although joint analysis of multiple phenotypes for GWAS can identify and interpret pleiotropic loci which are essential to understand pleiotropy in diseases and complex traits, most of the multiple phenotype association tests are designed for a single variant, resulting in much lower power, especially when their effect sizes are small and …

A Staged Approach Using Machine Learning And Uncertainty Quantification To Predict The Risk Of Hip Fracture, Anjum Shaik, Kristoffer A. Larsen, Nancy E. Lane, Chen Zhao, Kuan Jui Su, Joyce H. Keyak, Qing Tian, Qiuying Sha, Hui Shen, Hong Wen Deng, Weihua Zhou

Michigan Tech Publications

Hip fractures present a significant healthcare challenge, especially within aging populations, where they are often caused by falls. These fractures lead to substantial morbidity and mortality, emphasizing the need for timely surgical intervention. Despite advancements in medical care, hip fractures impose a significant burden on individuals and healthcare systems. This paper focuses on the prediction of hip fracture risk in older and middle-aged adults, where falls and compromised bone quality are predominant factors. The study cohort included 547 patients, with 94 experiencing hip fracture. To assess the risk of hip fracture, clinical variables and clinical variables combined with hip DXA …

New Examples Of Self-Dual Near-Extremal Ternary Codes Of Length 48 Derived From 2-(47,23,11) Designs, Sanja Rukavina, Vladimir Tonchev

Michigan Tech Publications

In a recent paper (Araya and Harada, 2023), Araya and Harada gave examples of self-dual near-extremal ternary codes of length 48 for 145 distinct values of the number A12 of codewords of minimum weight 12, and raised the question about the existence of codes for other values of A12. In this note, we use symmetric 2-(47,23,11) designs with an automorphism group of order 6 to construct self-dual near-extremal ternary codes of length 48 for 150 new values of A12.

A Combinatorial Proof Of A Partition Perimeter Inequality, Hunter Waldron

Michigan Tech Publications

The partition perimeter is a statistic defined to be one less than the sum of the number of parts and the largest part. Recently, Amdeberhan, Andrews, and Ballantine proved the following analog of Glaisher’s theorem: for all m ≥ 2 and n ≥ 1, there are at least as many partitions with perimeter n and parts repeating fewer than m times as there are partitions with perimeter n with parts not divisible by m. In this work, we provide a combinatorial proof of their theorem by relating the combinatorics of the partition perimeter to that of compositions. Using this technique, …

Composition-Theoretic Series And False Theta Functions, William Keith, Robert Schneider, Andrew V. Sills

Michigan Tech Publications

Many natural partition-theoretic series can be equally readily interpreted as compo-sition-theoretic series, but this viewpoint seems to have not been much employed in either theory. We consider some of the consequences of this viewpoint. As examples, we give results concerning the reciprocals of Ramanujan’s theta functions and of the false theta functions of L. J. Rogers, and raise an array of questions related to these. Part of this study may be considered a natural dual of the truncated pentagonal number theorem of Andrews and Merca.

A Novel Method For Multiple Phenotype Association Studies Based On Genotype And Phenotype Network, Xuewei Cao, Shuanglin Zhang, Qiuying Sha

Michigan Tech Publications

Joint analysis of multiple correlated phenotypes for genome-wide association studies (GWAS) can identify and interpret pleiotropic loci which are essential to understand pleiotropy in diseases and complex traits. Meanwhile, constructing a network based on associations between phenotypes and genotypes provides a new insight to analyze multiple phenotypes, which can explore whether phenotypes and genotypes might be related to each other at a higher level of cellular and organismal organization. In this paper, we first develop a bipartite signed network by linking phenotypes and genotypes into a Genotype and Phenotype Network (GPN). The GPN can be constructed by a mixture of …

Error Estimates For A Mixed Finite Element Method For The Maxwell's Transmission Eigenvalue Problem, Chao Wang, Jintao Cui, Jiguang Sun

Michigan Tech Publications

In this paper, we analyze a numerical method combining the Ciarlet-Raviart mixed finite element formulation and an iterative algorithm for the Maxwell's transmission eigenvalue problem. The eigenvalue problem is first written as a nonlinear quad-curl eigenvalue problem. Then the real transmission eigenvalues are proved to be the roots of a non-linear function. They are the generalized eigenvalues of a related linear self-adjoint quad-curl eigenvalue problem. These generalized eigenvalues are computed by a mixed finite element method. We derive the error estimates using the spectral approximation of compact operators, the theory of mixed finite element method for quad-curl problems, and the …

Asymptotics Of The Rate Function In The Large Deviation Principle For Sums Of Independent Identically Distributed Random Variables, Iosif Pinelis

Michigan Tech Publications

Let Λ∗be the rate function in the large deviation principle for the sums X1 + · · · + Xn of independent identically distributed random variables X1, X2, …. It is shown that Λ∗(x) ∼ − ln P(X1 ≥ x) (as x → ∞) if and only if ln P(X1 ≥ x) ∼ L0(x) for some concave function L0. The main ingredient of the proof is the general, explicit expression of a suitable quasi-minimizer in t ≥ 0 of the Bernstein–Chernoff upper bound e−txEetX1 on P(X1 ≥ x), which is amenable to analysis and, at the same time, is close …

Hadamard Matrices Of Orders 60 And 64 With Automorphisms Of Orders 29 And 31, Makoto Araya, Masaaki Harada, Vladimir Tonchev

Michigan Tech Publications

A classification of Hadamard matrices of order 2p + 2 with an automorphism of order p is given for p = 29 and 31. The ternary self-dual codes spanned by the newly found Hadamard matrices of order 60 with an automorphism of order 29 are computed, as well as the binary doubly even self-dual codes of length 120 with generator matrices defined by related Hadamard designs. Several new ternary near-extremal self-dual codes, as well as binary near-extremal doubly even self-dual codes with previously unknown weight enumerators are found.

Deep Neural Network-Oriented Indicator Method For Inverse Scattering Problems Using Partial Data, Yule Lin, Xiaoyi Yan, Jiguang Sun, Juan Liu

Michigan Tech Publications

We consider the inverse scattering problem to reconstruct an obstacle using partial far-field data due to one incident wave. A simple indicator function, which is negative inside the obstacle and positive outside of it, is constructed and then learned using a deep neural network (DNN). The method is easy to implement and effective as demonstrated by numerical examples. Rather than developing sophisticated network structures for the classical inverse operators, we reformulate the inverse problem as a suitable operator such that standard DNNs can learn it well. The idea of the DNN-oriented indicator method can be generalized to treat other partial …

Low Shear In Short-Term Impacts Endothelial Cell Traction And Alignment In Long-Term, Mohanish K. Chandurkar, Nikhil Mittal, Shaina P. Royer-Weeden, Steven D. Lehmann, Yeonwoo Rho, Sangyoon J. Han

Michigan Tech Publications

Within the vascular system, endothelial cells (ECs) are exposed to fluid shear stress (FSS), a mechanical force exerted by blood flow that is critical for regulating cellular tension and maintaining vascular homeostasis. The way ECs react to FSS varies significantly; while high, laminar FSS supports vasodilation and suppresses inflammation, low or disturbed FSS can lead to endothelial dysfunction and increase the risk of cardiovascular diseases. Yet, the adaptation of ECs to dynamically varying FSS remains poorly understood. This study focuses on the dynamic responses of ECs to brief periods of low FSS, examining its impact on endothelial traction—a measure of …

Low Shear In Short-Term Impacts Endothelial Cell Traction And Alignment In Long-Term, Mohanish Chandurkar, Nikhil Mittal, Shaina P. Royer-Weeden, Steven D. Lehmann, Yeonwoo Rho, Sangyoon J. Han

Michigan Tech Publications

Within the vascular system, endothelial cells (ECs) are exposed to fluid shear stress (FSS), a mechanical force exerted by blood flow that is critical for regulating cellular tension and maintaining vascular homeostasis. The way ECs react to FSS varies significantly; while high, laminar FSS supports vasodilation and suppresses inflammation, low or disturbed FSS can lead to endothelial dysfunction and increase the risk of cardiovascular diseases. Yet, the adaptation of ECs to dynamically varying FSS remains poorly understood. This study focuses on the dynamic responses of ECs to brief periods of low FSS, examining its impact on endothelial traction-a measure of …

Integrating External Controls By Regression Calibration For Genome-Wide Association Study, Lirong Zhu, Shijia Yan, Xuewei Cao, Shuanglin Zhang, Qiuying Sha

Michigan Tech Publications

Genome-wide association studies (GWAS) have successfully revealed many disease-associated genetic variants. For a case-control study, the adequate power of an association test can be achieved with a large sample size, although genotyping large samples is expensive. A cost-effective strategy to boost power is to integrate external control samples with publicly available genotyped data. However, the naive integration of external controls may inflate the type I error rates if ignoring the systematic differences (batch effect) between studies, such as the differences in sequencing platforms, genotype-calling procedures, population stratification, and so forth. To account for the batch effect, we propose an approach …