Mathematics Faculty Publications and Presentations Copyright (c) 2013 Boise State University All rights reserved. http://scholarworks.boisestate.edu/math_facpubs Recent documents in Mathematics Faculty Publications and Presentations en-us Thu, 16 May 2013 01:35:11 PDT 3600 Borel's Conjecture in Topological Groups http://scholarworks.boisestate.edu/math_facpubs/112 http://scholarworks.boisestate.edu/math_facpubs/112 Tue, 14 May 2013 10:30:03 PDT We introduce a natural generalization of Borel's Conjecture. For each infinite cardinal number κ, let BCκ denote this generalization. Then BCℵ0 is equivalent to the classical Borel conjecture. Assuming the classical Borel conjecture, ¬BCℵ1 is equivalent to the existence of a Kurepa tree of height ℵ1. Using the connection of BCκ with a generalization of Kurepa's Hypothesis, we obtain the following consistency results:

1. If it is consistent that there is a 1-inaccessible cardinal then it is consistent that BCℵ1.

2. If it is consistent that BCℵ1, then it is consistent that there is an inaccessible cardinal.

3. If it is consistent that there is a 1-inaccessible cardinal with ω inaccessible cardinals above it, then ¬BCℵω+(∀n<ω)BCℵn is consistent.

4. If it is consistent that there is a 2-huge cardinal, then it is consistent that BCℵω.

5. If it is consistent that there is a 3-huge cardinal, then it is consistent thatBCκ for a proper class of cardinals κ of countable cofinality.

]]> Fred Galvin et al. A High-Order Kernel Method for Diffusion and Reaction-Diffusion Equations on Surfaces http://scholarworks.boisestate.edu/math_facpubs/111 http://scholarworks.boisestate.edu/math_facpubs/111 Fri, 29 Mar 2013 14:01:23 PDT In this paper we present a high-order kernel method for numerically solving diffusion and reaction-diffusion partial differential equations (PDEs) on smooth, closed surfaces embedded in Rd . For two-dimensional surfaces embedded in R3 , these types of problems have received growing interest in biology, chemistry, and computer graphics to model such things as diffusion of chemicals on biological cells or membranes, pattern formations in biology, nonlinear chemical oscillators in excitable media, and texture mappings. Our kernel method is based on radial basis functions and uses a semi-discrete approach (or the method-of-lines) in which the surface derivative operators that appear in the PDEs are approximated using collocation. The method only requires nodes at “scattered” locations on the surface and the corresponding normal vectors to the surface. Additionally, it does not rely on any surfacebased metrics and avoids any intrinsic coordinate systems, and thus does not suffer from any coordinate distortions or singularities. We provide error estimates for the kernel-based approximate surface derivative operators and numerically study the accuracy and stability of the method. Applications to different non-linear systems of PDEs that arise in biology and chemistry are also presented.

]]> Edward J. Fuselier et al. The Usual Model Construction for NFU Preserves Information http://scholarworks.boisestate.edu/math_facpubs/110 http://scholarworks.boisestate.edu/math_facpubs/110 Tue, 26 Feb 2013 11:10:23 PST The usual construction of models of NFU (New Foundations with urelements, introduced by Jensen) is due to Maurice Boffa. A Boffa model is obtained from a model of (a fragment of) Zermelo-Fraenkel with Choice (ZFC) with an automorphism which moves a rank: the domain of the Boffa model is a rank that is moved. “Most” elements of the domain of the Boffa model are urelements in terms of the interpreted NFU. The main result of this paper is that the restriction of the membership relation of the original model of set theory with automorphism to the domain of the Boffa model is first-order definable in the language of NFU. In particular, all information about the extensions in the original model of the urelements of the model of NFU is definable in terms of NFU. A corollary (answering a question of Thomas Forster) is that the urelements in a Boffa model are not homogeneous.

]]> M. Randall Holmes Numerical Experiments with Model Equations of Cancer Invasion of Tissue http://scholarworks.boisestate.edu/math_facpubs/109 http://scholarworks.boisestate.edu/math_facpubs/109 Tue, 05 Feb 2013 10:52:50 PST M. Kolev et al. Waring Decompositions of Monomials http://scholarworks.boisestate.edu/math_facpubs/108 http://scholarworks.boisestate.edu/math_facpubs/108 Thu, 31 Jan 2013 09:59:09 PST A Waring decomposition of a polynomial is an expression of the polynomial as a sum of powers of linear forms, where the number of summands is minimal possible. We prove that any Waring decomposition of a monomial is obtained from a complete intersection ideal, determine the dimension of the set of Waring decompositions, and give the conditions under which the Waring decomposition is unique up to scaling the variables.

]]> Weronika Buczyńska et al. Discontinuous Parameter Estimates with Least Squares Estimators http://scholarworks.boisestate.edu/math_facpubs/107 http://scholarworks.boisestate.edu/math_facpubs/107 Wed, 23 Jan 2013 13:09:30 PST We discuss weighted least squares estimates of ill-conditioned linear inverse problems where weights are chosen to be inverse error covariance matrices. Least squares estimators are the maximum likelihood estimate for normally distributed data and parameters, but here we do not assume particular probability distributions. Weights for the estimator are found by ensuring its minimum follows a χ2 distribution. Previous work with this approach has shown that it is competitive with regularization methods such as the L-curve and Generalized Cross Validation (GCV) [20]. In this work we extend the method to find diagonal weighting matrices, rather than a scalar regularization parameter. Diagonal weighting matrices are advantageous because they give piecewise smooth least squares estimates and hence are a mechanism through which least squares can be used to estimate discontinuous parameters. This is explained by viewing least squares estimation as a constrained optimization problem. Results with diagonal weighting matrices are given for a benchmark discontinuous inverse problem from [13]. In addition, the method is used to estimate soil moisture from data collected in the Dry Creek Watershed near Boise, Idaho. Parameter estimates are found that combine two different types of measurements, and weighting matrices are found that incorporate uncertainty due to spatial variation so that the parameters can be used over larger scales than those that were measured.

]]> J. L. Mead Numerical Simulations for Tumor and Cellular Immune System Interactions in Lung Cancer Treatment http://scholarworks.boisestate.edu/math_facpubs/106 http://scholarworks.boisestate.edu/math_facpubs/106 Mon, 14 Jan 2013 13:24:24 PST We investigate a new mathematical model that describes lung cancer regression in patients treated by chemotherapy and radiotherapy. The model is composed of nonlinear integro-differential equations derived from the so-called kinetic theory for active particles and a new sink function is investigated according to clinical data from carcinoma planoepitheliale. The model equations are solved numerically and the data are utilized in order to find their unknown parameters. The results of the numerical experiments show a good correlation between the predicted and clinical data and illustrate that the mathematical model has potential to describe lung cancer regression.

]]> M. Kolev et al. A Study of Different Modeling Choices for Simulating Platelets Within the Immersed Boundary Method http://scholarworks.boisestate.edu/math_facpubs/105 http://scholarworks.boisestate.edu/math_facpubs/105 Mon, 03 Dec 2012 11:37:41 PST The Immersed Boundary (IB) method is a widely-used numerical methodology for the simulation of fluid–structure interaction problems. The IB method utilizes an Eulerian discretization for the fluid equations of motion while maintaining a Lagrangian representation of structural objects. Operators are defined for transmitting information (forces and velocities) between these two representations. Most IB simulations represent their structures with piecewise linear approximations and utilize Hookean spring models to approximate structural forces. Our specific motivation is the modeling of platelets in hemodynamic flows. In this paper, we study two alternative representations – radial basis functions (RBFs) and Fourier-based (trigonometric polynomials and spherical harmonics) representations – for the modeling of platelets in two and three dimensions within the IB framework, and compare our results with the traditional piecewise linear approximation methodology. For different representative shapes, we examine the geometric modeling errors (position and normal vectors), force computation errors, and computational cost and provide an engineering trade-off strategy for when and why one might select to employ these different representations.

]]> Varun Shankar et al. The Secant Conjecture in the Real Schubert Calculus http://scholarworks.boisestate.edu/math_facpubs/104 http://scholarworks.boisestate.edu/math_facpubs/104 Fri, 02 Nov 2012 11:36:54 PDT We formulate the Secant Conjecture, which is a generalization of the Shapiro Conjecture for Grassmannians. It asserts that an intersection of Schubert varieties in a Grassmannian is transverse with all points real if the flags defining the Schubert varieties are secant along disjoint intervals of a rational normal curve. We present theoretical evidence for this conjecture as well as computational evidence obtained in over one terahertz-year of computing, and we discuss some of the phenomena we observed in our data.

]]> Luis D. García-Puente et al. Weak Covering Properties and Infinite Games http://scholarworks.boisestate.edu/math_facpubs/103 http://scholarworks.boisestate.edu/math_facpubs/103 Thu, 25 Oct 2012 12:12:35 PDT We investigate game-theoretic properties of selection principles related to weaker forms of the Menger and Rothberger properties. For appropriate spaces some of these selection principles are characterized in terms of a corresponding game. We use generic extensions by Cohen reals to illustrate the necessity of some of the hypotheses in our theorems.

]]> L. Babinkostova et al. Scattered Data Interpolation on Embedded Submanifolds with Restricted Positive Definite Kernels: Sobolev Error Estimates http://scholarworks.boisestate.edu/math_facpubs/102 http://scholarworks.boisestate.edu/math_facpubs/102 Fri, 21 Sep 2012 14:51:27 PDT In this paper we present error estimates for kernel interpolation at scattered sites on manifolds. The kernels we consider will be obtained by the restriction of positive definite kernels on Rd, such as radial basis functions, to a smooth, compact embedded submanifold M ⊂ Rd with no boundary. For restricted kernels having finite smoothness, we provide a complete characterization of the native space on M. After this and some preliminary setup, we present Sobolev-type error estimates for the interpolation problem for smooth and non-smooth kernels. In the case of non-smooth kernels, we provide error estimates for target functions too rough to be within the native space of the kernel. Numerical results verifying the theory are also presented for a one-dimensional curve embedded in R3 and a two-dimensional torus.

]]> Edward J. Fuselier et al. Using Online Assessment and Practice to Achieve Better Retention and Placement in Precalculus and Calculus http://scholarworks.boisestate.edu/math_facpubs/101 http://scholarworks.boisestate.edu/math_facpubs/101 Tue, 04 Sep 2012 15:15:40 PDT In the fall of 2008 Boise State University began using an online assessment tool, ALEKS1, as an initial assignment in Precalculus and Calculus courses. This paper reports on the effectiveness of the ALEKS assessment as a self-placement tool, used in conjunction with standard placement tests and prerequisite courses. The benchmark levels of 40% and 70% of knowledge space in the ALEKS course: Preparation for Calculus for Precalculus and Calculus courses were used. The paper looks at the effectiveness of the assessment with these benchmark levels as a first student assignment, both as a tool for student success, and as an instrument for making efficient use of the university's resources. Although there are no hard answers, and although much information is anecdotal, we introduce a statistic that is pertinent to these questions and show that it indicates partial effectiveness of the ALEKS assessment.

]]> Alex Feldman et al. Analytical Upstream Collocation Solution of a Quadratically Forced Steady-State Convection-Diffusion Equation http://scholarworks.boisestate.edu/math_facpubs/100 http://scholarworks.boisestate.edu/math_facpubs/100 Thu, 05 Jul 2012 12:01:30 PDT Purpose – The purpose of this paper is to present the analytical solution to the Hermite collocation discretization of a quadratically forced steady-state convection-diffusion equation in one spatial dimension with constant coefficients, defined on a uniform mesh, with Dirichlet boundary conditions. To improve the accuracy of the method “upstream weighting” of the convective term is used in an optimal way. The authors also provide a method to determine where the forcing function should be optimally sampled. Computational examples are given, which support and illustrate the theory of the optimal sampling of the convective and forcing term.

Design/methodology/approach – The authors: extend previously published results (which dealt only with the case of linear forcing) to the case of quadratic forcing; prove the theorem that governs the quadratic case; and then illustrate the results of the theorem using computational examples.

Findings – The algorithm developed for the quadratic case dramatically decreases the error (i.e. the difference between the continuous and numerical solutions).

Research limitations/implications – Because the methodology successfully extends the linear case to the quadratic case, it is hoped that the method can, indeed, be extended further to more general cases. It is true, however, that the level of complexity rose significantly from the linear case to the quadratic case.

Practical implications – Hermite collocation can be used in an optimal way to solve differential equations, especially convection-diffusion equations.

Originality/value – Since convection-dominated convection-diffusion equations are difficult to solve numerically, the results in this paper make a valuable contribution to research in this field.

]]> Stephen H. Brill et al. On Primeness of Labeled Oriented Trees http://scholarworks.boisestate.edu/math_facpubs/99 http://scholarworks.boisestate.edu/math_facpubs/99 Fri, 22 Jun 2012 12:56:44 PDT Knot complements are aspherical. Whether this extends to ribbon disc complements, or, equivalently, to standard 2-complexes of labeled oriented trees, remains unresolved. It is known that prime injective labeled oriented trees are diagrammatically reducible, that is, aspherical in a strong combinatorial sense. We show that arbitrary prime labeled oriented trees need not be DR. We conjecture that all injective labeled oriented trees are aspherical and prove the conjecture under natural conditions.

]]> Jens Harlander et al. Hyperbolic Alternating Virtual Link Groups http://scholarworks.boisestate.edu/math_facpubs/98 http://scholarworks.boisestate.edu/math_facpubs/98 Fri, 11 May 2012 12:24:42 PDT We study the topology and geometry of virtual link complements and groups. We show that the groups defined by the Wirtinger presentation of certain prime dense alternating virtual links are CAT(0) and hyperbolic.

]]> Jens Harlander Quantum Deformations of Fundamental Groups of Oriented 3-Manifolds http://scholarworks.boisestate.edu/math_facpubs/97 http://scholarworks.boisestate.edu/math_facpubs/97 Wed, 25 Apr 2012 10:06:56 PDT We compute two-term skein modules of framed oriented links in oriented 3-manifolds. They contain the self-writhe and total linking number invariants of framed oriented links in a universal way. The relations in a natural presentation of the skein module are interpreted as monodromies in the space of immersions of circles into the 3-manifold.

]]> Uwe Kaiser Nonparametric Copula Density Estimation in Sensor Networks http://scholarworks.boisestate.edu/math_facpubs/96 http://scholarworks.boisestate.edu/math_facpubs/96 Tue, 27 Mar 2012 11:34:04 PDT Statistical and machine learning is a fundamental task in sensor networks. Real world data almost always exhibit dependence among different features. Copulas are full measures of statistical dependence among random variables. Estimating the underlying copula density function from distributed data is an important aspect of statistical learning in sensor networks. With limited communication capacities or privacy concerns, centralization of the data is often impossible. By only collecting the ranks of the data observed by different sensors, we estimate and evaluate the copula density on an equally spaced grid after binning the standardized ranks at the fusion center. Without assuming any parametric forms of copula densities, we estimate them nonparametrically by maximum penalized likelihood estimation (MPLE) method with a Total Variation (TV) penalty. Linear equality and positivity constraints arise naturally as a consequence of marginal uniform densities of any copulas. Through local quadratic approximation to the likelihood function, the constrained TV-MPLE problem is cast as a sequence of corresponding quadratic optimization problems. A fast gradient based algorithm solves the constrained TV penalized quadratic optimization problem. Numerical experiments show that our algorithm can estimate the underlying copula density accurately.

]]> Leming Qu et al. A Guide to RBF-Generated Finite Differences for Nonlinear Transport: Shallow Water Simulations on a Sphere http://scholarworks.boisestate.edu/math_facpubs/95 http://scholarworks.boisestate.edu/math_facpubs/95 Fri, 16 Mar 2012 13:58:56 PDT The current paper establishes the computational efficiency and accuracy of the RBF-FD method for large-scale geoscience modeling with comparisons to state-of-the-art methods as high-order discontinuous Galerkin and spherical harmonics, the latter using expansions with close to 300,000 bases. The test cases are demanding fluid flow problems on the sphere that exhibit numerical challenges, such as Gibbs phenomena, sharp gradients, and complex vortical dynamics with rapid energy transfer from large to small scales over short time periods. The computations were possible as well as very competitive due to the implementation of hyperviscosity on large RBF stencil sizes (corresponding roughly to 6th to 9th order methods) with up to O(105) nodes on the sphere. The RBF-FD method scaled as O(N) per time step, where N is the total number of nodes on the sphere. In Appendix A, guidelines are given on how to chose parameters when using RBF-FD to solve hyperbolic PDEs.

]]> Natasha Flyer et al. A Refined Efficiency Rate for Ordinary Least Squares and Generalized Least Squares Estimators for a Linear Trend with Autoregressive Errors http://scholarworks.boisestate.edu/math_facpubs/94 http://scholarworks.boisestate.edu/math_facpubs/94 Tue, 13 Mar 2012 12:29:12 PDT When a straight line is fitted to time series data, generalized least squares (GLS) estimators of the trend slope and intercept are attractive as they are unbiased and of minimum variance. However, computing GLS estimators is laborious as their form depends on the autocovariances of the regression errors. On the other hand, ordinary least squares (OLS) estimators are easy to compute and do not involve the error autocovariance structure. It has been known for 50 years that OLS and GLS estimators have the same asymptotic variance when the errors are second-order stationary. Hence, little precision is gained by using GLS estimators in stationary error settings. This article revisits this classical issue, deriving explicit expressions for the GLS estimators and their variances when the regression errors are drawn from an autoregressive process. These expressions are used to show that OLS methods are even more efficient than previously thought. Specifically, we show that the convergence rate of variance differences is one polynomial degree higher than that of least squares estimator variances. We also refine Grenander's (1954) variance ratio. An example is presented where our new rates cannot be improved upon. Simulations show that the results change little when the autoregressive parameters are estimated.

]]> Jaechoul Lee et al. BPFA and Projective Well-Orderings of the Reals http://scholarworks.boisestate.edu/math_facpubs/93 http://scholarworks.boisestate.edu/math_facpubs/93 Mon, 13 Feb 2012 12:06:33 PST If the bounded proper forcing axiom BPFA holds and ω1=ω1L, then there is a lightface Σ13 well-ordering of the reals. The argument combines a well-ordering due to Caicedo-Veličković with an absoluteness result for models of MA in the spirit of "David's trick." We also present a general coding scheme that allows us to show that BPFA is equiconsistent with R being lightface Σ14, for many "consistently locally certified" relations R on ℝ. This is accomplished through a use of David's trick and a coding through the Σ2 stable ordinals of L.

]]> Andrés Eduardo Caicedo et al.