Mathematics Faculty Publications and PresentationsCopyright (c) 2016 Boise State University All rights reserved.
http://scholarworks.boisestate.edu/math_facpubs
Recent documents in Mathematics Faculty Publications and Presentationsen-usFri, 23 Sep 2016 19:06:43 PDT3600Computing with Functions in Spherical and Polar Geometries I. The Sphere
http://scholarworks.boisestate.edu/math_facpubs/182
http://scholarworks.boisestate.edu/math_facpubs/182Wed, 24 Aug 2016 15:45:07 PDT
A collection of algorithms is described for numerically computing with smooth functions defined on the unit sphere. Functions are approximated to essentially machine precision by using a structure-preserving iterative variant of Gaussian elimination together with the double Fourier sphere method. We show that this procedure allows for stable differentiation, reduces the oversampling of functions near the poles, and converges for certain analytic functions. Operations such as function evaluation, differentiation, and integration are particularly efficient and can be computed by essentially one-dimensional algorithms. A highlight is an optimal complexity direct solver for Poisson's equation on the sphere using a spectral method. Without parallelization, we solve Poisson's equation with $100$ million degrees of freedom in 1 minute on a standard laptop. Numerical results are presented throughout. In a companion paper (part II) we extend the ideas presented here to computing with functions on the disk.
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Alex Townsend et al.CGISSOn Phase II Monitoring of the Probability Distributions of Univariate Continuous Processes
http://scholarworks.boisestate.edu/math_facpubs/181
http://scholarworks.boisestate.edu/math_facpubs/181Thu, 16 Jun 2016 12:17:03 PDT
Statistical process control (SPC) charts are widely used in industry for monitoring the stability of certain sequential processes like manufacturing, health care systems etc. Most SPC charts assume that the parametric form of the “in-control” process distribution F_{1} is available. However, it has been demonstrated in the literature that their performances are unreliable when the pre-specified process distribution is incorrect. Moreover, most SPC charts are designed to detect any shift in mean and/or variance. In real world problems, shifts in higher moments can happen without much change in mean or variance. If we fail to detect those and let the process run, it can eventually become worse and a shift in mean or variance can creep in. Moreover, the special cause that initiated the shift can inflict further damage to the system, and it may become a financial challenge to fix it. This paper provides an efficient and easy-to-use control chart for Phase II monitoring of univariate continuous processes when the parametric form of the “in-control” process distribution is unknown, but Phase I observations that are believed to be i.i.d. realizations from unknown F_{1} are available. Data-driven practical guidelines are also provided to choose the tuning parameter and the corresponding control limit of the proposed SPC chart. Numerical simulations and a real-life data analysis show that it can be used in many practical applications.
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Partha Sarathi MukherjeeTeaching Math to Deaf/Hard-of-Hearing (DHH) Children Using Mobile Games: Outcomes with Student and Teacher Perspectives
http://scholarworks.boisestate.edu/math_facpubs/180
http://scholarworks.boisestate.edu/math_facpubs/180Fri, 27 May 2016 14:36:39 PDT
Leveraging the use of mobile devices for education, such as instructional games, is an area of increasing interest for targeted subpopulations of students including those who are deaf/hard-of-hearing (DHH). This paper outlines the perspectives of Deaf Education teachers and DHH children who participated in the GeePerS*Math project. Interviews and surveys provide data from the primary implementation of the technology in an ecologically valid setting. Findings included similar results from both teachers and students with regard to attitudes and transfer of skills within the game to those in traditional curriculum. Unintended outcomes, such as gaining orienteering skills and peer-tutoring, were also noted. The results helped to inform the designers of educational technology with ways to relate with classroom instructors and children when creating advanced mobile applications.
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Brett E. Shelton et al.Using Ciliate Operations to Construct Chromosome Phylogenies
http://scholarworks.boisestate.edu/math_facpubs/179
http://scholarworks.boisestate.edu/math_facpubs/179Thu, 19 May 2016 13:43:37 PDT
Whole genome sequencing has revealed several examples where genomes of different species are related by permutation. The number of certain types of rearrangements needed to transform one permuted list into another can measure the distance between such lists. Using an algorithm based on three basic DNA editing operations suggested by a model for ciliate micronuclear decryption, this study defines the distance between two permutations to be the number of ciliate operations the algorithm performs during such a transformation. Combining well-known clustering methods with this distance function enables one to construct corresponding phylogenies. These ideas are illustrated by exploring the phylogenetic relationships among the chromosomes of eight fruit fly (Drosophila) species, using the well-known UPGMA algorithm on the distance function provided by the ciliate operations.
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Jacob L. Herlin et al.BRCBaire Spaces and Infinite Games
http://scholarworks.boisestate.edu/math_facpubs/178
http://scholarworks.boisestate.edu/math_facpubs/178Wed, 04 May 2016 10:47:40 PDT
It is well known that if the nonempty player of the Banach–Mazur game has a winning strategy on a space, then that space is Baire in all powers even in the box product topology. The converse of this implication may also be true: We know of no consistency result to the contrary. In this paper we establish the consistency of the converse relative to the consistency of the existence of a proper class of measurable cardinals.
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Fred Galvin et al.BRCDecompositions of Ideals of Minors Meeting a Submatrix
http://scholarworks.boisestate.edu/math_facpubs/177
http://scholarworks.boisestate.edu/math_facpubs/177Fri, 29 Apr 2016 10:02:53 PDT
We compute the primary decomposition of certain ideals generated by subsets of minors in a generic matrix or in a generic symmetric matrix, or subsets of Pfaffians in a generic skew-symmetric matrix. Specifically, the ideals we consider are generated by minors that have at least some given number of rows and columns in certain submatrices.
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Kent M. Neuerburg et al.A High-Order Radial Basis Function (RBF) Leray Projection Method for the Solution of the Incompressible Unsteady Stokes Equations
http://scholarworks.boisestate.edu/math_facpubs/176
http://scholarworks.boisestate.edu/math_facpubs/176Fri, 22 Apr 2016 11:22:49 PDT
A new projection method based on radial basis functions (RBFs) is presented for discretizing the incompressible unsteady Stokes equations in irregular geometries. The novelty of the method comes from the application of a new technique for computing the Leray-Helmholtz projection of a vector field using generalized interpolation with divergence-free and curl-free RBFs. Unlike traditional projection methods, this new method enables matching both tangential and normal components of divergence-free vector fields on the domain boundary. This allows incompressibility of the velocity field to be enforced without any time-splitting or pressure boundary conditions. Spatial derivatives are approximated using collocation with global RBFs so that the method only requires samples of the field at (possibly scattered) nodes over the domain. Numerical results are presented demonstrating high-order convergence in both space (between 5th and 6th order) and time (up to 4th order) for some model problems in two dimensional irregular geometries.
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Edward J. Fuselier et al.CGISSAn Efficient Generalized Least Squares Algorithm for Periodic Trended Regression with Autoregressive Errors
http://scholarworks.boisestate.edu/math_facpubs/175
http://scholarworks.boisestate.edu/math_facpubs/175Mon, 28 Mar 2016 11:10:18 PDT
Time series data with periodic trends like daily temperatures or sales of seasonal products can be seen in periods fluctuating between highs and lows throughout the year. Generalized least squares estimators are often computed for such time series data as these estimators have minimum variance among all linear unbiased estimators. However, the generalized least squares solution can require extremely demanding computation when the data is large. This paper studies an efficient algorithm for generalized least squares estimation in periodic trended regression with autoregressive errors. We develop an algorithm that can substantially simplify generalized least squares computation by manipulating large sets of data into smaller sets. This is accomplished by coining a structured matrix for dimension reduction. Simulations show that the new computation methods using our algorithm can drastically reduce computing time. Our algorithm can be easily adapted to big data that show periodic trends often pertinent to economics, environmental studies, and engineering practices.
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Jaechoul Lee et al.Civil EngineeringSelective Games on Binary Relations
http://scholarworks.boisestate.edu/math_facpubs/174
http://scholarworks.boisestate.edu/math_facpubs/174Wed, 23 Mar 2016 08:05:35 PDT
We present a unified approach, based on dominating families in binary relations, for the study of topological properties defined in terms of selection principles and the games associated to them. Among other applications, this general framework allows us to obtain a number of results on the preservation of such properties in products provided that the second player has a winning strategy in the associated game played on one of the coordinate spaces.
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Rodrigo R. Dias et al.BRCPrefrontal White Matter Pathology in Air Pollution Exposed Mexico City Young Urbanites and Their Potential Impact on Neurovascular Unit Dysfunction and the Development of Alzheimer’s Disease
http://scholarworks.boisestate.edu/math_facpubs/173
http://scholarworks.boisestate.edu/math_facpubs/173Fri, 18 Mar 2016 11:58:21 PDT
Millions of urban children are chronically exposed to high concentrations of air pollutants, i.e., fine particulate matter (PM_{2.5}) and ozone, associated with increased risk for Alzheimer’s disease. Compared with children living with clear air those in Mexico City (MC) exhibit systemic, brain and intrathecal inflammation, low CSF Aβ _{42}, breakdown of the BBB, attention and short-term memory deficits, prefrontal white matter hyperintensities, damage to epithelial and endothelial barriers, tight junction and neural autoantibodies, and Alzheimer and Parkinson's hallmarks. The prefrontal white matter is a target of air pollution. We examined by light and electron microscopy the prefrontal white matter of MC dogs (n: 15, age 3.17±0.74 years), children and teens (n: 34, age: 12.64±4.2 years) versus controls. Major findings in MC residents included leaking capillaries and small arterioles with extravascular lipids and erythrocytes, lipofuscin in pericytes, smooth muscle and endothelial cells (EC), thickening of cerebrovascular basement membranes with small deposits of amyloid, patchy absence of the perivascular glial sheet, enlarged Virchow-Robin spaces and nanosize particles (20-48 nm) in EC, basement membranes, axons and dendrites. Tight junctions, a key component of the neurovascular unit (NVU) were abnormal in MC versus control dogs (χ^{2} <0.0001), and white matter perivascular damage was significantly worse in MC dogs (p= 0.002). The integrity of the NVU, an interactive network of vascular, glial and neuronal cells is compromised in MC young residents. Characterizing the early NVU damage and identifying biomarkers of neurovascular dysfunction may provide a fresh insight into Alzheimer pathogenesis and open opportunities for pediatric neuroprotection.
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Partha S. MukherjeeLower Bound for Ranks of Invariant Forms
http://scholarworks.boisestate.edu/math_facpubs/172
http://scholarworks.boisestate.edu/math_facpubs/172Fri, 18 Mar 2016 11:30:46 PDT
We give a lower bound for the Waring rank and cactus rank of forms that are invariant under an action of a connected algebraic group. We use this to improve the Ranestad-Schreyer-Shafiei lower bounds for the Waring ranks and cactus ranks of determinants of generic matrices, Pfaffians of generic skew-symmetric matrices, and determinants of generic symmetric matrices.
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Harm Derksen et al.Evaluation of NASA's MERRA Precipitation Product in Reproducing the Observed Trend and Distribution of Extreme Precipitation Events in the United States
http://scholarworks.boisestate.edu/math_facpubs/171
http://scholarworks.boisestate.edu/math_facpubs/171Fri, 18 Mar 2016 11:19:55 PDT
This study evaluates the performance of NASA’s Modern-Era Retrospective Analysis for Research and Applications (MERRA) precipitation product in reproducing the trend and distribution of extreme precipitation events. Utilizing the extreme value theory, time-invariant and time-variant extreme value distributions are developed to model the trends and changes in the patterns of extreme precipitation events over the contiguous United States during 1979–2010. The Climate Prediction Center (CPC) U.S. Unified gridded observation data are used as the observational dataset. The CPC analysis shows that the eastern and western parts of the United States are experiencing positive and negative trends in annual maxima, respectively. The continental-scale patterns of change found in MERRA seem to reasonably mirror the observed patterns of change found in CPC. This is not previously expected, given the difficulty in constraining precipitation in reanalysis products. MERRA tends to overestimate the frequency at which the 99th percentile of precipitation is exceeded because this threshold tends to be lower in MERRA, making it easier to be exceeded. This feature is dominant during the summer months. MERRA tends to reproduce spatial patterns of the scale and location parameters of the generalized extreme value and generalized Pareto distributions. However, MERRA underestimates these parameters, particularly over the Gulf Coast states, leading to lower magnitudes in extreme precipitation events. Two issues in MERRA are identified: 1) MERRA shows a spurious negative trend in Nebraska and Kansas, which is most likely related to the changes in the satellite observing system over time that has apparently affected the water cycle in the central United States, and 2) the patterns of positive trend over the Gulf Coast states and along the East Coast seem to be correlated with the tropical cyclones in these regions. The analysis of the trends in the seasonal precipitation extremes indicates that the hurricane and winter seasons are contributing the most to these trend patterns in the southeastern United States. In addition, the increasing annual trend simulated by MERRA in the Gulf Coast region is due to an incorrect trend in winter precipitation extremes.
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Hamed Ashouri et al.A Critical Proton MR Spectroscopy Marker of Alzheimer Early Neurodegenerative Change: Low Hippocampal NAA/Cr Ratio Impacts APOE 4 Mexico City Children and Their Parents.
http://scholarworks.boisestate.edu/math_facpubs/170
http://scholarworks.boisestate.edu/math_facpubs/170Fri, 11 Mar 2016 08:23:37 PST
Severe air pollution exposures produce systemic, respiratory, myocardial, and brain inflammation and Alzheimer’s disease (AD) hallmarks in clinically healthy children. We tested whether hippocampal metabolite ratios are associated with contrasting levels of air pollution, APOE and BMI in paired healthy children and one parent sharing the same APOE alleles. We used (1) H-MRS to interrogate bilateral hippocampal single-voxel in 57 children (12.45± 3.4 years) and their 48 parents (37.5± 6.78 years) low pollution city v Mexico City (MC). NAA/Cr, Cho/Cr, and mI/Cr metabolite ratios were analysed. The right hippocampus N-acetylaspartate/creatine (NAA/Cr) was significantly different between cohorts (p=0.007). The NAA/Cr ratio in right hippocampus: controls v APOE 4 MC children and left hippocampus: MC APOE 4 parents v their children was significantly different after adjusting for age, gender and BMI (p=0.027 and 0.01, respectively). The NAA/Cr ratio is considered reflective of neuronal density/functional integrity /loss of synapses/higher pTau burden, thus a significant decrease in hippocampal NAA/Cr ratios may constitute a spectral marker of early neurodegeneration in young urbanites. Decreases in NAA/Cr correlate well with cognitive function, behavioural symptoms and dementia severity, thus since the progression of AD starts decades before clinical diagnosis, our findings support the hypothesis that under chronic exposures to fine particulate matter and ozone above the standards, neurodegenerative processes start in childhood and APOE 4 carriers are at higher risk. Gene /environmental factors are critical in the development of Alzheimer's disease and the identification and neuroprotection of young urbanites at high risk must become a public health priority.
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Partha S. MukherjeeArithmagons and Geometrically Invariant Multiplicative Integer Partitions
http://scholarworks.boisestate.edu/math_facpubs/169
http://scholarworks.boisestate.edu/math_facpubs/169Thu, 10 Mar 2016 13:52:47 PST
In this article, we introduce a formal definition for integral arithmagons. Informally, an arithmagon is a polygonal figure with integer labeled vertices and edges in which, under a binary operation, adjacent vertices equal the included edge. By considering the group of automorphisms for the associated graph, we count the number of integral arithmagons whose exterior sum or product equals a fixed number.
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J. A. Franco et al.Understanding Prospective Teachers’ Levels of Geometric Thoughts: Insights from a Discursive Analysis
http://scholarworks.boisestate.edu/math_facpubs/168
http://scholarworks.boisestate.edu/math_facpubs/168Tue, 01 Mar 2016 14:12:53 PST
The study investigates the characteristics of prospective teachers’ geometric discourses at the van Hiele model of thinking (1959/1985), using Sfard’s (2008) discursive framework. In this report, I align two prospective teachers’ pre- and post- van Hiele geometry test (Usiskin, 1982) results with the analyses of their geometric discourses from clinical interviews, to illustrate changes in geometric discourse when a student’s test results showed no change in van Hiele levels, and changes in geometric discourse when a student developed her thinking to the next van Hiele level. Revisiting the van Hiele model of thinking, complemented with a discursive lens, helped to understand learning as change in discourses, as prospective teacher develop thinking toward a higher van Hiele level.
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Sasha WangIdentifying Similar Polygons: Comparing Prospective Teachers’ Routines with a Mathematician’s
http://scholarworks.boisestate.edu/math_facpubs/167
http://scholarworks.boisestate.edu/math_facpubs/167Tue, 01 Mar 2016 13:25:17 PST
This paper reports two prospective teachers' and a mathematician's ways of identifying similar triangle and hexagons through the analysis of routines, a characteristic of geometric discourse. The findings show that visual recognition was a common approach for the mathematician as wells as the two prospective teachers. However, when asked for justification, their routines of identifying similar polygons diverged. The paper also discusses the implication of classroom discourse practices to enhance prospective teachers' communication and reasoning skills while learning geometric concepts such as similarity.
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Sasha WangStretching Probability Explorations with Geoboards
http://scholarworks.boisestate.edu/math_facpubs/166
http://scholarworks.boisestate.edu/math_facpubs/166Wed, 24 Feb 2016 11:22:16 PST
Peg students' knowledge about probability by using a common manipulative to connect number sense, geometry, and algebraic thinking.
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Ann Wheeler et al.Mexico City Normal Weight Children Exposed to High Concentrations of Ambient PM<sub>2.5</sub> Show High Blood Leptin and Endothelin-1, Vitamin D Deficiency, and Food Reward Hormone Dysregulation versus Low Pollution Controls. Relevance for Obesity and Alzheimer Disease
http://scholarworks.boisestate.edu/math_facpubs/165
http://scholarworks.boisestate.edu/math_facpubs/165Wed, 17 Feb 2016 13:57:11 PST
Millions of Mexico, US and across the world children are overweight and obese. Exposure to fossil-fuel combustion sources increases the risk for obesity and diabetes, while long-term exposure to fine particulate matter (PM _{2.5}) and ozone (O_{3}) above US EPA standards is associated with increased risk of Alzheimer’s disease (AD). Mexico City Metropolitan Area children are chronically exposed to PM_{2.5} and O_{3} concentrations above the standards and exhibit systemic, brain and intrathecal inflammation, cognitive deficits, and Alzheimer disease neuropathology. We investigated adipokines, food reward hormones, endothelial dysfunction, vitamin D and apolipoprotein E (APOE) relationships in 80 healthy, normal weight 11.1±3.2 years olds matched by age, gender, BMI and SES, low (n: 26) versus high (n:54) PM _{2.5} exposures. Mexico City children had higher leptin and endothelin-1 (p 2.5 exposed children. Mexico City APOE 4 v 3 children had higher glucose (p=0.009). Serum 25-hydroxyvitamin D < 30 ng/mL was documented in 87% of Mexico City children. Leptin is strongly positively associated to PM _{2.5} cumulative exposures. Residing in a high PM_{2.5} and O_{3} environment is associated with 12 h fasting hyperleptinemia, altered appetite-regulating peptides, vitamin D deficiency, and increases in ET-1 in clinically healthy children. These changes signal the future trajectory in urban children towards the development of insulin resistance, obesity, type II diabetes, premature cardiovascular disease, addiction-like behavior, cognitive impairment and Alzheimer’s disease. Increased efforts should be made to decrease pediatric PM _{2.5} exposures, to deliver health interventions prior to the development of obesity and to identify and mitigate environmental factors influencing obesity and Alzheimer disease.
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Partha S. MukherjeeUntyped λ-Calculus with Relative Typing
http://scholarworks.boisestate.edu/math_facpubs/164
http://scholarworks.boisestate.edu/math_facpubs/164Fri, 29 Jan 2016 09:10:54 PST
A system of untyped λ-calculus with a restriction on function abstraction using relative typing analogous to the restriction on set comprehension found in Quine's set theory “New Foundations” is discussed. The author has shown elsewhere that this system is equiconsistent with Jensen's NFU (“New Foundations” with urelements) + Infinity, which is in turn equiconsistent with the simple theory of types with infinity. The definition of the system is given and the construction of a model is described. A semantic motivation for the stratification criterion for function abstraction is given, based on an abstract model of computation. The same line of semantic argument is used to motivate an analogy between the notion of “strongly Cantorian set” found in “New Foundations” and the notion of “data type”; an implementation of absolute types as domains of retractions with strongly Cantorian ranges is described. The implementation of these concepts in a theorem prover developed by the author is sketched.
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M. Randall HolmesAugmenting the Immersed Boundary Method with Radial Basis Functions (RBFs) for the Modeling of Platelets in Hemodynamic Flows
http://scholarworks.boisestate.edu/math_facpubs/163
http://scholarworks.boisestate.edu/math_facpubs/163Mon, 11 Jan 2016 10:32:19 PST
We present a new computational method by extending the Immersed Boundary (IB) method with a geometric model based on parametric Radial Basis Function (RBF) interpolation of the Lagrangian structures. Our specific motivation is the modeling of platelets in hemodynamic flows, though we anticipate that our method will be useful in other applications involving surface elasticity. The efficacy of our new RBF-IB method is shown through a series of numerical experiments. Specifically, we test the convergence of our method and compare our method with the traditional IB method in terms of computational cost, maximum stable time-step size and volume loss. We conclude that the RBF-IB method has advantages over the traditional Immersed Boundary method, and is well-suited for modeling of platelets in hemodynamic flows.
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Varun Shankar et al.CGISS