Mathematics Faculty Publications and PresentationsCopyright (c) 2014 Boise State University All rights reserved.
http://scholarworks.boisestate.edu/math_facpubs
Recent documents in Mathematics Faculty Publications and Presentationsen-usWed, 17 Dec 2014 15:53:21 PST3600Clinical Study and Numerical Simulation of Brain Cancer Dynamics Under Radiotherapy
http://scholarworks.boisestate.edu/math_facpubs/148
http://scholarworks.boisestate.edu/math_facpubs/148Wed, 10 Dec 2014 14:40:26 PST
We perform a clinical and numerical study of the progression of brain cancer tumor growth dynamics coupled with the effects of radiotherapy. We obtained clinical data from a sample of brain cancer patients undergoing radiotherapy and compare it to our numerical simulations to a mathematical model of brain tumor cell population growth influenced by radiation treatment. We model how the body biologically receives a physically delivered dose of radiation to the affected tumorous area in the form of a generalized LQ model, modified to account for the conversion process of sublethal lesions into lethal lesions at high radiation doses. We obtain good agreement between our clinical data and our numerical simulations of brain cancer progression given by the mathematical model, which couples tumor growth dynamics and the effect of irradiation. The correlation, spanning a wide dataset, demonstrates the potential of the mathematical model to describe the dynamics of brain tumor growth influenced by radiotherapy.
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S. Nawrocki et al.The Axiom Scheme of Acyclic Comprehension
http://scholarworks.boisestate.edu/math_facpubs/147
http://scholarworks.boisestate.edu/math_facpubs/147Wed, 05 Nov 2014 10:25:16 PST
A “new” criterion for set existence is presented, namely, that a set {x|φ} should exist if the multigraph whose nodes are variables in φ and whose edges are occurrences of atomic formulas in φ is acyclic. Formulas with acyclic graphs are stratified in the sense of New Foundations, so consistency of the set theory with weak extensionality and acyclic comprehension follows from the consistency of Jensen’s system NFU. It is much less obvious, but turns out to be the case, that this theory is equivalent to NFU: it appears at first blush that it ought to be weaker. This paper verifies that acyclic comprehension and stratified comprehension are equivalent by verifying that each axiom in a finite axiomatization of stratified comprehension follows from acyclic comprehension.
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Zuhair Al-Johar et al.A Fast Parallel Algorithm for Delay Partial Differential Equations Modeling the Cell Cycle in Cell Lines Derived from Human Tumors
http://scholarworks.boisestate.edu/math_facpubs/146
http://scholarworks.boisestate.edu/math_facpubs/146Wed, 22 Oct 2014 09:27:10 PDT
We present a fast numerical algorithm for solving delay partial differential equations that model the growth of human tumor cells. The undetermined model parameters need to be estimated according to experimental data and it is desired to shorten the computational time needed in estimating them. To speed up the computations, we present an algorithm invoking parallelization designed for arbitrary numbers of available processors. The presented numerical results demonstrate the efficiency of the algorithm.
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Barbara Zubik-KowalOn the Sensitivity of 3-D Thermal Convection Codes to Numerical Discretization: A Model Intercomparison
http://scholarworks.boisestate.edu/math_facpubs/145
http://scholarworks.boisestate.edu/math_facpubs/145Fri, 12 Sep 2014 12:31:50 PDT
Fully 3-D numerical simulations of thermal convection in a spherical shell have become a standard for studying the dynamics of pattern formation and its stability under perturbations to various parameter values. The question arises as to how does the discretization of the governing equations affect the outcome and thus any physical interpretation. This work demonstrates the impact of numerical discretization on the observed patterns, the value at which symmetry is broken, and how stability and stationary behavior is dependent upon it. Motivated by numerical simulations of convection in the Earth's mantle, we consider isoviscous Rayleigh-Bénard convection at infinite Prandtl number, where the aspect ratio between the inner and outer shell is 0.55. We show that the subtleties involved in development mantle convection models are considerably more delicate than has been previously appreciated, due to the rich dynamical behavior of the system. Two codes with different numerical discretization schemes: an established, community-developed, and benchmarked finite element code (CitcomS) and a novel spectral method that combines Chebyshev polynomials with radial basis functions (RBF) are compared. A full numerical study is investigated for the following three cases. The first case is based on the cubic (or octahedral) initial condition (spherical harmonics of degree ℓ =4). How variations in the behavior of the cubic pattern to perturbations in the initial condition and Rayleigh number between the two numerical discrezations is studied. The second case investigates the stability of the dodecahedral (or icosahedral) initial condition (spherical harmonics of degree ℓ = 6). Although both methods converge first to the same pattern, this structure is ultimately unstable and systematically degenerates to cubic or tetrahedral symmetries, depending on the code used. Lastly, a new steady state pattern is presented as a combination of order 3 and 4 spherical harmonics leading to a five cell or a hexahedral pattern and stable up to 70 times the critical Rayleigh number. This pattern can provide the basis for a new accuracy benchmark for 3-D spherical mantle convection codes.
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P.-A. Arrial et al.Castelnuovo–Mumford Regularity and Arithmetic Cohen–Macaulayness of Complete Bipartite Subspace Arrangements
http://scholarworks.boisestate.edu/math_facpubs/144
http://scholarworks.boisestate.edu/math_facpubs/144Mon, 25 Aug 2014 16:17:33 PDT
We give the Castelnuovo–Mumford regularity of arrangements of (n−2)-planes in P^{n} whose incidence graph is a sufficiently large complete bipartite graph, and determine when such arrangements are arithmetically Cohen–Macaulay.
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Zach Teitler et al.Kernel Based Quadrature on Spheres and Other Homogeneous Spaces
http://scholarworks.boisestate.edu/math_facpubs/143
http://scholarworks.boisestate.edu/math_facpubs/143Mon, 18 Aug 2014 15:21:37 PDT
Quadrature formulas for spheres, the rotation group, and other compact, homogeneous manifolds are important in a number of applications and have been the subject of recent research. The main purpose of this paper is to study coordinate independent quadrature (or cubature) formulas associated with certain classes of positive definite and conditionally positive definite kernels that are invariant under the group action of the homogeneous manifold. In particular, we show that these formulas are accurate—optimally so in many cases—and stable under an increasing number of nodes and in the presence of noise, provided the set X of quadrature nodes is quasi-uniform. The stability results are new in all cases. In addition, we may use these quadrature formulas to obtain similar formulas for manifolds diffeomorphic to S^{n}, oblate spheroids for instance. The weights are obtained by solving a single linear system. For S^{2}, and the restricted thin plate spline kernel r^{2}log r, these weights can be computed for two-thirds of a million nodes, using a preconditioned iterative technique introduced by us.
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E. Fuselier et al.How Do They Know It Is a Parallelogram? Analysing Geometric Discourse at Van Hiele Level 3
http://scholarworks.boisestate.edu/math_facpubs/142
http://scholarworks.boisestate.edu/math_facpubs/142Fri, 08 Aug 2014 11:56:26 PDT
In this article, we introduce Sfard's discursive framework and use it to investigate prospective teachers' geometric discourse in the context of quadrilaterals. In particular, we focus on describing and analysing two participants' use of mathematical words and substantiation routines related to parallelograms and their properties at van Hiele level 3 thinking. Our findings suggest that a single van Hiele level of thinking encompasses a range of complexity of reasoning and differences in discourse and thus a deeper investigation of students' mathematical thinking within assigned van Hiele levels is warranted.
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Sasha Wang et al.Prospective Teachers’ Learning in Geometry: Changes in Discourse and Thinking
http://scholarworks.boisestate.edu/math_facpubs/141
http://scholarworks.boisestate.edu/math_facpubs/141Wed, 09 Jul 2014 08:46:59 PDT
This study investigates changes in prospective teachers’ levels of geometric thinking and the development of their geometric discourses in the classification of quadrilaterals. To examine prospective teachers’ thinking about geometry, this study connects Sfard’s discursive framework to another, namely the van Hiele theory. Findings of the study reveal discursive similarities and differences in participants’ geometric discourses within the same van Hiele level, as well as changes in geometric discourse as a result of changes in levels of geometric thinking. The study also investigates the usefulness of a discursive framework in providing rich descriptions of prospective teachers’ thinking processes.
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Sasha WangFlavonol-Rich Dark Cocoa Significantly Decreases Plasma Endothelin-1 and Improves Cognition in Urban Children
http://scholarworks.boisestate.edu/math_facpubs/140
http://scholarworks.boisestate.edu/math_facpubs/140Tue, 01 Jul 2014 12:42:37 PDT
Air pollution exposures are linked to systemic inflammation, cardiovascular and respiratory morbidity and mortality, neuroinflammation and neuropathology in young urbanites. In particular, most Mexico City Metropolitan Area (MCMA) children exhibit subtle cognitive deficits, and neuropathology studies show 40% of them exhibiting frontal tau hyperphosphorylation and 51% amyloid-β diffuse plaques (compared to 0% in low pollution control children). We assessed whether a short cocoa intervention can be effective in decreasing plasma endothelin 1 (ET-1) and/or inflammatory mediators in MCMA children. Thirty gram of dark cocoa with 680 mg of total flavonols were given daily for 10.11 ± 3.4 days (range 9–24 days) to 18 children (10.55 years, SD = 1.45; 11F/7M). Key metabolite ratios in frontal white matter and in hippocampus pre and during cocoa intervention were quantified by magnetic resonance spectroscopy. ET-1 significantly decreased after cocoa treatment (p = 0.0002). Fifteen children (83%) showed a marginally significant individual improvement in one or both of the applied simple short memory tasks. Endothelial dysfunction is a key feature of exposure to particulate matter (PM) and decreased endothelin-1 bioavailability is likely useful for brain function in the context of air pollution. Our findings suggest that cocoa interventions may be critical for early implementation of neuroprotection of highly exposed urban children. Multi-domain nutraceutical interventions could limit the risk for endothelial dysfunction, cerebral hypoperfusion, neuroinflammation, cognitive deficits, structural volumetric detrimental brain effects, and the early development of the neuropathological hallmarks of Alzheimer's and Parkinson's diseases.
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Partha S. MukherjeeTrends in Extreme U.S. Temperatures
http://scholarworks.boisestate.edu/math_facpubs/139
http://scholarworks.boisestate.edu/math_facpubs/139Tue, 24 Jun 2014 15:18:26 PDT
This paper develops trend estimation techniques for monthly maximum and minimum temperature time series observed in the 48 conterminous United States over the last century. While most scientists concur that this region has warmed on aggregate, there is no a priori reason to believe that temporal trends in extremes and averages will exhibit the same patterns. Indeed, under minor regularity conditions, the sample partial sum and maximum of stationary time series are asymptotically independent (statistically). Previous authors have suggested that minimum temperatures are warming faster than maximum temperatures in the United States; such an aspect can be investigated via the methods discussed in this study. Here, statistical models with extreme value and changepoint features are used to estimate trends and their standard errors. A spatial smoothing is then done to extract general structure. The results show that monthly maximum temperatures are not often greatly changing—perhaps surprisingly, there are many stations that show some cooling. In contrast, the minimum temperatures show significant warming. Overall, the southeastern United States shows the least warming (even some cooling), and the western United States, northern Midwest, and New England have experienced the most warming.
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Jaechoul Lee et al.The Prevalence of Antibodies Against Wheat and Milk Proteins in Blood Donors and Their Contribution to Neuroimmune Reactivities
http://scholarworks.boisestate.edu/math_facpubs/138
http://scholarworks.boisestate.edu/math_facpubs/138Thu, 12 Jun 2014 10:38:42 PDT
The aim of this study was to look for the presence of IgG, IgM, and IgA antibodies against two widely consumed foods, wheat and milk, in a relatively large number of specimens. As wheat, milk, and their antigens have been found to be involved in neuroimmune disorders, we measured the co-occurrence of their antibodies against various neural antigens. We assessed the reactivity of sera from 400 donors to wheat and milk proteins, GAD-65, cerebellar, MBP, and MOG. Statistical analysis showed significant clustering when certain wheat and milk protein antibodies were cross-referenced with neural antibodies. Approximately half of the sera with antibody elevation against gliadin reacted significantly with GAD-65 and cerebellar peptides; about half of the sera with elevated antibodies against α + β-casein and milk butyrophilin also showed antibody elevation against MBP and MOG. Inhibition studies showed that only two out of four of the samples with elevated cerebellar or MOG antibodies could be inhibited by gliadin or α + β-casein, confirming individual variation in epitope recognition. We conclude that a subgroup of blood donors, due to a breakdown in immunological tolerance, may react and produce significant levels of antibodies (p-values less than 0.05) against wheat and milk antigens that cross-react with different neural antigens, which may have broader implications in the induction of neuroimmune reactivities.
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Aristo Vojdani et al.On Hereditarily Small Sets in ZF
http://scholarworks.boisestate.edu/math_facpubs/137
http://scholarworks.boisestate.edu/math_facpubs/137Fri, 16 May 2014 10:02:21 PDT
We show in (the usual set theory without Choice) that for any set X, the collection of sets Y such that each element of the transitive closure of is strictly smaller in size than X (the collection of sets hereditarily smaller than X) is a set. This result has been shown by Jech in the case (where the collection under consideration is the set of hereditarily countable sets).
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M. Randall HolmesAutomorphisms of Cornoa Algebras, and Group Cohomology
http://scholarworks.boisestate.edu/math_facpubs/136
http://scholarworks.boisestate.edu/math_facpubs/136Tue, 06 May 2014 10:02:41 PDT
In 2007 Phillips and Weaver showed that, assuming the Continuum Hypothesis, there exists an outer automorphism of the Calkin algebra. (The Calkin algebra is the algebra of bounded operators on a separable complex Hilbert space, modulo the compact operators.) In this paper we establish that the analogous conclusion holds for a broad family of quotient algebras. Specifically, we will show that assuming the Continuum Hypothesis, if A is a separable algebra which is either simple or stable, then the corona of A has nontrivial automorphisms. We also discuss a connection with cohomology theory, namely, that our proof can be viewed as a computation of the cardinality of a particular derived inverse limit.
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Samuel Coskey et al.Ioana's Superrigidity Theorem and Orbit Equivalence Relations
http://scholarworks.boisestate.edu/math_facpubs/135
http://scholarworks.boisestate.edu/math_facpubs/135Fri, 14 Mar 2014 15:05:40 PDT
We give a survey of Adrian Ioana’s cocycle superrigidity theoremfor profinite actions of Property (T) groups and its applications to ergodic theory and set theory in this expository paper. In addition to a statement and proof of Ioana’s theorem, this paper features the following: (i) an introduction to rigidity, including a crash course in Borel cocycles and a summary of some of the best-known superrigidity theorems; (ii) some easy applications of superrigidity, both to ergodic theory (orbit equivalence) and set theory (Borel reducibility); and (iii) a streamlined proof of Simon Thomas’s theorem that the classification of torsion-free abelian groups of finite rank is intractable.
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Samuel CoskeyInfinite Time Turing Machines and an Application to the Hierarchy of Equivalence Relations on the Reals
http://scholarworks.boisestate.edu/math_facpubs/134
http://scholarworks.boisestate.edu/math_facpubs/134Fri, 28 Feb 2014 09:14:59 PSTSamuel Coskey et al.Borel Tukey Morphisms and Combinatorial Cardinal Invariants of the Continuum
http://scholarworks.boisestate.edu/math_facpubs/133
http://scholarworks.boisestate.edu/math_facpubs/133Thu, 27 Feb 2014 09:09:21 PST
We discuss the Borel Tukey ordering on cardinal invariants of the continuum. We observe that this ordering makes sense for a larger class of cardinals than has previously been considered. We then provide a Borel version of a large portion of van Douwen's diagram. For instance, although the usual proof of the inequality p≤b does not provide a Borel Tukey map, we show that in fact there is one. Afterwards, we revisit a result of Mildenberger concerning a generalization of the unsplitting and splitting numbers. Lastly, we use our results to give an embedding from the inclusion ordering on P(ω) into the Borel Tukey ordering on cardinal invariants.
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Samuel Coskey et al.Numerical Simulation of Diffusion MRI Signals Using an Adaptive Time-Stepping Method
http://scholarworks.boisestate.edu/math_facpubs/132
http://scholarworks.boisestate.edu/math_facpubs/132Wed, 26 Feb 2014 11:19:35 PST
The effect on the MRI signal of water diffusion in biological tissues in the presence of applied magnetic field gradient pulses can be modelled by a multiple compartment Bloch–Torrey partial differential equation. We present a method for the numerical solution of this equation by coupling a standard Cartesian spatial discretization with an adaptive time discretization. The time discretization is done using the explicit Runge–Kutta–Chebyshev method, which is more efficient than the forward Euler time discretization for diffusive- type problems. We use this approach to simulate the diffusion MRI signal from the extra-cylindrical compartment in a tissue model of the brain gray matter consisting of cylindrical and spherical cells and illustrate the effect of cell membrane permeability.
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Jing-Rebecca Li et al.Numerical Algorithm for the Growth of Human Tumor Cells and Their Responses to Therapy
http://scholarworks.boisestate.edu/math_facpubs/131
http://scholarworks.boisestate.edu/math_facpubs/131Wed, 26 Feb 2014 10:55:49 PST
We investigate a system of delay partial differential equations that models the growth of human tumor cells and their responses to therapy. The model includes unknown parameters that need to be estimated according to experimental data. We introduce a numerical algorithm, which shortens the computational time for solving the model equations and estimating their parameters. Numerical results demonstrate the efficiency of our algorithm and show correspondence between predicted and experimental data.
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Barbara Zubik-KowalFrom Whole Numbers to Invert and Multiply
http://scholarworks.boisestate.edu/math_facpubs/130
http://scholarworks.boisestate.edu/math_facpubs/130Fri, 21 Feb 2014 11:59:23 PST
An instructional sequence used in a course for prospective teachers directly relates to Common Core State Standards for grades 3–6.
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Laurie O. Cavey et al.Revisit Pattern Blocks to Develop Rational Number Sense
http://scholarworks.boisestate.edu/math_facpubs/129
http://scholarworks.boisestate.edu/math_facpubs/129Fri, 14 Feb 2014 13:57:34 PST
A classic manipulative, used since the 1960s, continues to offer opportunities for intriguing problem solving involving proportions.
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Joe Champion et al.