College of Science and Engineering
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Browsing College of Science and Engineering by Department "Mathematics"
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Item Algebraic Algorithms for Even Circuits in Graphs(Multidisciplinary Digital Publishing Institute, 2019-09) Ha, Huy Tai; Morey, SusanWe present an algebraic algorithm to detect the existence of and to list all indecomposable even circuits in a given graph. We also discuss an application of our work to the study of directed cycles in digraphs.Item Cox Survival Analysis of Microarray Gene Expression Data Using Correlation Principal Component Regression(The Berkeley Electronic Press, 2007-05-29) Zhao, Qiang; Sun, JianguoStatistical analysis of microarray gene expression data has recently attracted a great deal of attention. One problem of interest is to relate genes to survival outcomes of patients with the purpose of building regression models for the prediction of future patients' survival based on their gene expression data. For this, several authors have discussed the use of the proportional hazards or Cox model after reducing the dimension of the gene expression data. This paper presents a new approach to conduct the Cox survival analysis of microarray gene expression data with the focus on models' predictive ability. The method modifies the correlation principal component regression (Sun, 1995) to handle the censoring problem of survival data. The results based on simulated data and a set of publicly available data on diffuse large B-cell lymphoma show that the proposed method works well in terms of models' robustness and predictive ability in comparison with some existing partial least squares approaches. Also, the new approach is simpler and easy to implement.Item DNA Methylation Heterogeneity Patterns in Breast Cancer Cell Lines(Libertas Academica, 2016-08) Tian, Sunny; Bertelsmann, Karina; Yu, Linda; Sun, ShuyingHeterogeneous DNA methylation patterns are linked to tumor growth. In order to study DNA methylation heterogeneity patterns for breast cancer cell lines, we comparatively study four metrics: variance, I (2) statistic, entropy, and methylation state. Using the categorical metric methylation state, we select the two most heterogeneous states to identify genes that directly affect tumor suppressor genes and high- or moderate-risk breast cancer genes. Utilizing the Gene Set Enrichment Analysis software and the ConsensusPath Database visualization tool, we generate integrated gene networks to study biological relations of heterogeneous genes. This analysis has allowed us to contribute 19 potential breast cancer biomarker genes to cancer databases by locating "hub genes" - heterogeneous genes of significant biological interactions, selected from numerous cancer modules. We have discovered a considerable relationship between these hub genes and heterogeneously methylated oncogenes. Our results have many implications for further heterogeneity analyses of methylation patterns and early detection of breast cancer susceptibility.Item Existence of Three Nonnegative Periodic Solutions for Functional Differential Equations and Applications to Hematopoiesis(The University of Central Florida, 2009-01) Padhi, Seshadev; Srivastava, Shilpee; Dix, Julio G.Using the Leggett-Wiliams fixed point theorem, we show the existence of at least three solutions to a system of first-order nonlinear functional differential equations. These solutions have non-negative components which makes them suitable for hematopoiesis models.Item Findings of the Dynamic Geometry Project at Texas State University(Texas State University, 2016-10) Dickey, Edwin; Jiang, Zhonghong; White, Alexander; Webre, BrittanyNo abstract prepared.Item Hemimethylation Patterns in Breast Cancer Cell Lines(Sage, 2019-01) Sun, Shuying; Lee, Yu Ri; Enfield, BrittanyDNA methylation is an epigenetic event that involves adding a methyl group to the cytosine (C) site, especially the one that pairs with a guanine (G) site (ie, CG or CpG site), in a human genome. This event plays an important role in both cancerous and normal cell development. Previous studies often assume symmetric methylation on both DNA strands. However, asymmetric methylation, or hemimethylation (methylation that occurs only on 1 DNA strand), does exist and has been reported in several studies. Due to the limitation of previous DNA methylation sequencing technologies, researchers could only study hemimethylation on specific genes, but the overall genomic hemimethylation landscape remains relatively unexplored. With the development of advanced next-generation sequencing techniques, it is now possible to measure methylation levels on both forward and reverse strands at all CpG sites in an entire genome. Analyzing hemimethylation patterns may potentially reveal regions related to undergoing tumor growth. For our research, we first identify hemimethylated CpG sites in breast cancer cell lines using Wilcoxon signed rank tests. We then identify hemimethylation patterns by grouping consecutive hemimethylated CpG sites based on their methylation states, methylation "M" or unmethylation "U." These patterns include regular (or consecutive) hemimethylation clusters (eg, "MMM" on one strand and "UUU" on another strand) and polarity (or reverse) clusters (eg, "MU" on one strand and "UM" on another strand). Our results reveal that most hemimethylation clusters are the polarity type, and hemimethylation does occur across the entire genome with notably higher numbers in the breast cancer cell lines. The lengths or sizes of most hemimethylation clusters are very short, often less than 50 base pairs. After mapping hemimethylation clusters and sites to corresponding genes, we study the functions of these genes and find that several of the highly hemimethylated genes may influence tumor growth or suppression. These genes may also indicate a progressing transition to a new tumor stage.Item Integrative Analysis of Gene Expression and Methylation Data for Breast Cancer Cell Lines(BioMed Central Ltd., 2018-06) Li, Catherine; Lee, Juyon; Ding, Jessica; Sun, ShuyingBackground: The deadly costs of cancer and necessity for an accurate method of early cancer detection have demanded the identification of genetic and epigenetic factors associated with cancer. DNA methylation, an epigenetic event, plays an important role in cancer susceptibility. In this paper, we use DNA methylation and gene expression data integration and pathway analysis to further explore and understand the complex relationship between methylation and gene expression. Results: Through linear modeling and analysis of variance, we obtain genes that show a significant correlation between methylation and gene expression. We then examine the functions and relationships of these genes using bioinformatic tools and databases. In particular, using ConsensusPathDB, we analyze the networks of statistically significant genes to identify hub genes, genes with a large number of links to other genes. We identify eight major hub genes, all in strong association with cancer susceptibility. Through further analysis of the function, gene expression level, and methylation level of these hub genes, we conclude that they are novel potential biomarkers for breast cancer. Conclusions: Our findings have various implications for cancer screening, early detection methods, and potential novel treatments for cancer. Researchers can also use our results to develop more effective methods for cancer study.Item Journaling to Support Student Learning: The Case of an Elementary Number Theory Course(University of Montana, 2019-01) Starkey, Christina M.; Warshauer, Hiroko; Warshauer, MaxThe use of journals in supporting student learning in elementary number theory is explored. Implications are made for the use of reflective writing for the teaching and learning of proofs and for undergraduate mathematics education.Item Mathematical Modeling Cycles as a Task Design Heuristic(University of Montana, 2017-01) Czocher, Jennifer A.There are many approaches to task design (Watson & Ohtani, 2015) from a large number of local and global design heuristics. The purpose of this paper is to present how mathematical modeling cycles, a popular way of describing mathematical modeling processes, were used as a task design heuristic.Item On Floating Equilibria in a Laterally Finite Container(Society for Industrial and Applied Mathematics, 2016-08) Treinen, Ray; McCuan, JohnThe main contribution of this paper is the precise numerical identification of a model set of parameters for a floating object/container system which admits three distinct equilibrium configurations, two of which are local energy minimizers among pseudo-equilibrium configurations. This numerical result strongly suggests the existence of a physical system in which an object can be observed to float in a centrally symmetric position in two geometrically distinct configurations, i.e., at two different heights. The numerical calculation relies on a fairly involved theoretical framework which can also be used to show uniqueness of equilibrium configurations for other parameters. Thus, the general dependence of observable stable equilibria on the physical parameters of the problem is both shown to be much more complicated than originally anticipated and likely to depend on additional information, e.g., the initial positioning of the floating object. In addition to providing a basis for numerical results, the theoretical framework developed here leads to two rigorous general results. The first is the existence of at least one equilibrium configuration when the density of the floating object is less than that of the liquid bath. The second is that all such equilibrium interfaces must project simply onto the base of the container.Item On mathematicians’ disagreements on what constitutes a proof(Taylor & Francis, 2019-04) Weber, Keith; Czocher, Jennifer A.We report the results of a study in which we asked 94 mathematicians to evaluate whether five arguments qualified as proofs. We found that mathematicians disagreed as to whether a visual argument and a computer-assisted argument qualified as proofs, but they viewed these proofs as atypical. The mathematicians were also aware that many other mathematicians might not share their judgment and viewed their own judgment as contextual. For typical proofs using standard inferential methods, there was a strong consensus amongst the mathematicians that these proofs were valid. An instructional consequence is that for the standard inferential methods covered in introductory proof courses, we should have the instructional goal that students appreciate why these inferential methods are valid. However, for controversial inferential methods such as visual inferences, students should understand why mathematicians have not reached a consensus on their validity.Item On the classification and asymptotic behavior of the symmetric capillary surfaces(Taylor & Francis, 2016-08-17) Treinen, Ray; Bagley, ZacharyWe consider the symmetric solutions to the Young-Laplace equation, and its extensions past vertical points. We provide a classification of all symmetric solutions using certain families of parameters. This classification produces a unified approach to fluid interfaces in capillary tubes, sessile and pendent drops, liquid bridges, as well as exterior and annular capillary surfaces. The generating curves for symmetric solutions have asymptotes for large arclengths, and the behavior of these asymptotes is analyzed.Item Oscillation of Functional Differential Equations of n-th Order with Distributed Deviating Arguments(2006-05) Padhi, Seshadev; Dix, Julio G.We establish conditions for the oscillation and asymptotic behavior of non-oscillatory solutions of the following functional differential equation with distributed deviating arguments. y(n)(t) + p(t)y(n-1)(t) + ∫bα q(t, ξ) ƒ(y(t), y(t - τ(t, ξ))) dσ(ξ) d = 0, We find explicit sufficient conditions for the oscillation as lower bounds for moments of the integral kernel q.Item Pairwise comparative analysis of six haplotype assembly methods based on users’ experience(BioMed Central Ltd., 2023-06-29) Sun, Shuying; Cheng, Flora; Han, Daphne; Wei, Sarah; Zhong, Alice; Massoudian, Sherwin; Johnson, Alison B.Background: A haplotype is a set of DNA variants inherited together from one parent or chromosome. Haplotype information is useful for studying genetic variation and disease association. Haplotype assembly (HA) is a process of obtaining haplotypes using DNA sequencing data. Currently, there are many HA methods with their own strengths and weaknesses. This study focused on comparing six HA methods or algorithms: HapCUT2, MixSIH, PEATH, WhatsHap, SDhaP, and MAtCHap using two NA12878 datasets named hg19 and hg38. The 6 HA algorithms were run on chromosome 10 of these two datasets, each with 3 filtering levels based on sequencing depth (DP1, DP15, and DP30). Their outputs were then compared. Result: Run time (CPU time) was compared to assess the efficiency of 6 HA methods. HapCUT2 was the fastest HA for 6 datasets, with run time consistently under 2 min. In addition, WhatsHap was relatively fast, and its run time was 21 min or less for all 6 datasets. The other 4 HA algorithms’ run time varied across different datasets and coverage levels. To assess their accuracy, pairwise comparisons were conducted for each pair of the six packages by generating their disagreement rates for both haplotype blocks and Single Nucleotide Variants (SNVs). The authors also compared them using switch distance (error), i.e., the number of positions where two chromosomes of a certain phase must be switched to match with the known haplotype. HapCUT2, PEATH, MixSIH, and MAtCHap generated output files with similar numbers of blocks and SNVs, and they had relatively similar performance. WhatsHap generated a much larger number of SNVs in the hg19 DP1 output, which caused it to have high disagreement percentages with other methods. However, for the hg38 data, WhatsHap had similar performance as the other 4 algorithms, except SDhaP. The comparison analysis showed that SDhaP had a much larger disagreement rate when it was compared with the other algorithms in all 6 datasets. Conclusion: The comparative analysis is important because each algorithm is different. The findings of this study provide a deeper understanding of the performance of currently available HA algorithms and useful input for other users.Item Parking Functions on Directed Graphs and Some Directed Trees(2020-06) King, Westin; Yan, Catherine H.Classical parking functions can be defined in terms of drivers with preferred parking spaces searching a linear parking lot for an open parking spot. We may consider this linear parking lot as a collection of n vertices (parking spots) arranged in a directed path. We generalize this notion to allow for more complicated “parking lots” and define parking functions on arbitrary directed graphs. We then consider a relationship proved by Lackner and Panholzer between parking functions on trees and “mapping digraphs” and we show that a similar relationship holds when edge orientations are reversed.Item Pattern Recognition in Epileptic EEG Signals via Dynamic Mode Decomposition(Multidisciplinary Digital Publishing Institute, 2020-04-01) Seo, Jong-Hyeon; Tsuda, Ichiro; Lee, Young Ju; Ikeda, Akio; Matsuhashi, Masao; Matsumoto, Riki; Kikuchi, Takayuki; Kang, HunseokIn this paper, we propose a new method based on the dynamic mode decomposition (DMD) to find a distinctive contrast between the ictal and interictal patterns in epileptic electroencephalography (EEG) data. The features extracted from the method of DMD clearly capture the phase transition of a specific frequency among the channels corresponding to the ictal state and the channel corresponding to the interictal state, such as direct current shift (DC-shift or ictal slow shifts) and high-frequency oscillation (HFO). By performing classification tests with Electrocorticography (ECoG) recordings of one patient measured at different timings, it is shown that the captured phenomenon is the unique pattern that occurs in the ictal onset zone of the patient. We eventually explain how advantageously the DMD captures some specific characteristics to distinguish the ictal state and the interictal state. The method presented in this study allows simultaneous interpretation of changes in the channel correlation and particular information for activity related to an epileptic seizure so that it can be applied to identification and prediction of the ictal state and analysis of the mechanism on its dynamics.Item Preliminary Analysis of Within-Sample Co-methylation Patterns in Normal and Cancerous Breast Samples(Sage, 2019-01) Sun, Lillian; Namboodiri, Surya; Chen, Emily; Sun, ShuyingDNA methylation plays a significant role in regulating the expression of certain genes in both cancerous and normal breast tissues. It is therefore important to study within-sample co-methylation, ie, methylation patterns between consecutive sites in a chromosome. In this article, we develop 2 new methods to compare co-methylation patterns between normal and cancerous breast samples. In particular, we investigate the co-methylation patterns of 4 different methylation states/levels separately. Using these 2 methods, we focus on addressing the following questions: How often does 1 methylation state change to other methylation states and how is this change dependent on chromosome distance? What co-methylation patterns do normal and cancerous breast samples have? Do genomic sites with different methylation states/levels have different co-methylation patterns? Our results show that cancerous and normal co-methylation patterns are significantly different. We find that this difference exists even when the physical distance of 2 sites are less than 50 bases. Breast cancer cell lines tend to remain in the same methylation state more often than normal samples, especially for the no/low or high/full methylation states. We also find that the co-methylation region lengths for various methylation states (no/low, partial, and high/full methylation states) are very different. For example, the co-methylation region lengths for partial methylation regions are shorter than the unmethylated or fully methylated regions. Our research may provide a deep understanding of co-methylation patterns. These co-methylation patterns will aid in discovering and understanding new methylation events that may be related to novel biomarkers.Item Regular Numbers and Mathematical Worlds(Flm Publishing Association, 2016-01) Whitacre, Ian; Bouhjar, Khalid; Bishop, Jessica Pierson; Philipp, Randolph A.; Schappelle, Bonnie P.; Lamb, Lisa L.Rather than describing the challenges of integer learning in terms of a transition from positive to negative numbers, we have arrived at a different perspective: We view students as inhabiting distinct mathematical worlds consisting of particular types of numbers (as construed by the students). These worlds distinguish and illuminate students' varied responses. Proficient students and adults may also inhabit multiple mathematical worlds from moment to moment. In our framework of mathematical worlds, we focus on the contrast between regular numbers and signed numbers in analyzing cases of student thinking. Implications of these ideas for educators and researchers are presented.Item Short and Long-Term Forecasting Using Artificial Neural Networks for Stock Prices in Palestine: A Comparative Study(Universita del Salento, 2017-04) Safi, Samir; White, AlexanderTo compare the forecast accuracy, Artificial Neural Networks, Autoregressive Integrated Moving Average and regression models were fit with training data sets and then used to forecast prices in a test set. Three different measures of accuracy were computed: Root Mean Square Error, Mean Absolute Error and Mean Absolute Percentage Error. To determine how the accuracy depends on sample size, models were compared between daily, monthly and quarterly time series of stock closing prices from Palestine.Item Strategies for Assessing Mathematical Knowledge for Teaching in Mathematics Content Courses(The University of Montana Department of Mathematical Sciences, 2020-06) Patterson, Cody L.; Parrott, Amy; Belnap, JasonIn their practice, teachers must not only know how to solve mathematics problems; they must also be able to make sense of students’ mathematical thinking, understand the organization and intent of curricular materials, and select contexts to motivate and highlight mathematical ideas. Similarly, mathematics content courses for prospective teachers (PTs) should not only seek to convey mathematical content; they should prepare PTs to use mathematical knowledge in ways that enhance school teaching and learning of the subject. Accordingly, mathematics teacher educators (MTEs) should assess not only the mathematics that PTs know but also whether this mathematical knowledge is organized in ways that are likely to support their teaching. In this article, we present some of the existing research on the assessment of mathematical knowledge for teaching and discuss ways in which MTEs can draw upon the work of elementary school teaching to help assess PTs’ content knowledge and habits of mind. These include assessments that focus on using representations that occur in elementary textbooks, building mathematical arguments, selecting problems to bring out important ideas, and making sense of students’ thinking.