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Articles 61 - 90 of 171
Full-Text Articles in Mathematics
Monomials And Basin Cylinders For Network Dynamics, Daniel Austin, Ian H. Dinwoodie
Monomials And Basin Cylinders For Network Dynamics, Daniel Austin, Ian H. Dinwoodie
Mathematics and Statistics Faculty Publications and Presentations
We describe methods to identify cylinder sets inside a basin of attraction for Boolean dynamics of biological networks. Such sets are used for designing regulatory interventions that make the system evolve towards a chosen attractor, for example initiating apoptosis in a cancer cell. We describe two algebraic methods for identifying cylinders inside a basin of attraction, one based on the Groebner fan that finds monomials that define cylinders and the other on primary decomposition. Both methods are applied to current examples of gene networks.
Robust Estimates For Hp-Adaptive Approximations Of Non-Self-Adjoint Eigenvalue Problems, Stefano Giani, Luka Grubišić, Agnieszka Międlar, Jeffrey S. Ovall
Robust Estimates For Hp-Adaptive Approximations Of Non-Self-Adjoint Eigenvalue Problems, Stefano Giani, Luka Grubišić, Agnieszka Międlar, Jeffrey S. Ovall
Mathematics and Statistics Faculty Publications and Presentations
We present new residual estimates based on Kato’s square root theorem for spectral approximations of non-self-adjoint differential operators of convection–diffusion–reaction type. These estimates are incorporated as part of an hp-adaptive finite element algorithm for practical spectral computations, where it is shown that the resulting a posteriori error estimates are reliable. Provided experiments demonstrate the efficiency and reliability of our approach.
Mathematical Model For Bone Mineralization, Svetlana V. Komarova, Lee Safranek, Jay Gopalakrishnan, Miao-Jung Yvonne Ou, Marc D. Mckee, Monzur Murshed, Frank Rauch, Erica Zuhr
Mathematical Model For Bone Mineralization, Svetlana V. Komarova, Lee Safranek, Jay Gopalakrishnan, Miao-Jung Yvonne Ou, Marc D. Mckee, Monzur Murshed, Frank Rauch, Erica Zuhr
Mathematics and Statistics Faculty Publications and Presentations
Defective bone mineralization has serious clinical manifestations, including deformities and fractures, but the regulation of this extracellular process is not fully understood. We have developed a mathematical model consisting of ordinary differential equations that describe collagen maturation, production and degradation of inhibitors, and mineral nucleation and growth. We examined the roles of individual processes in generating normal and abnormal mineralization patterns characterized using two outcome measures: mineralization lag time and degree of mineralization. Model parameters describing the formation of hydroxyapatite mineral on the nucleating centers most potently affected the degree of mineralization, while the parameters describing inhibitor homeostasis most effectively …
Nonsmooth Algorithms And Nesterov's Smoothing Technique For Generalized Fermat-Torricelli Problems, Nguyen Mau Nam, Nguyen Thai An, R. Blake Rector, Jie Sun
Nonsmooth Algorithms And Nesterov's Smoothing Technique For Generalized Fermat-Torricelli Problems, Nguyen Mau Nam, Nguyen Thai An, R. Blake Rector, Jie Sun
Mathematics and Statistics Faculty Publications and Presentations
We present algorithms for solving a number of new models of facility location which generalize the classical Fermat--Torricelli problem. Our first approach involves using Nesterov's smoothing technique and the minimization majorization principle to build smooth approximations that are convenient for applying smooth optimization schemes. Another approach uses subgradient-type algorithms to cope directly with the nondifferentiability of the cost functions. Convergence results of the algorithms are proved and numerical tests are presented to show the effectiveness of the proposed algorithms.
Transients In The Synchronization Of Oscillator Arrays, Carlos E. Cantos, J. J. P. Veerman
Transients In The Synchronization Of Oscillator Arrays, Carlos E. Cantos, J. J. P. Veerman
Mathematics and Statistics Faculty Publications and Presentations
The purpose of this note is threefold. First we state a few conjectures that allow us to rigorously derive a theory which is asymptotic in N (the number of agents) that describes transients in large arrays of (identical) linear damped harmonic oscillators in R with completely decentralized nearest neighbor interaction. We then use the theory to establish that in a certain range of the parameters transients grow linearly in the number of agents (and faster outside that range). Finally, in the regime where this linear growth occurs we give the constant of proportionality as a function of the signal velocities …
Exact Tests For Singular Network Data, Ian H. Dinwoodie, Kruti Pandya
Exact Tests For Singular Network Data, Ian H. Dinwoodie, Kruti Pandya
Mathematics and Statistics Faculty Publications and Presentations
We propose methodology for exact statistical tests of hypotheses for models of network dynamics. The methodology formulates Markovian exponential families, then uses sequential importance sampling to compute expectations within basins of attraction and within level sets of a sufficient statistic for an over-dispersion model. Comparisons of hypotheses can be done conditional on basins of attraction. Examples are presented.
Vanishing Configurations In Network Dynamics With Asynchronous Updates, Ian H. Dinwoodie
Vanishing Configurations In Network Dynamics With Asynchronous Updates, Ian H. Dinwoodie
Mathematics and Statistics Faculty Publications and Presentations
We consider Boolean dynamics for biological networks where stochasticity is introduced through asynchronous updates. An exact method is given for finding states which can reach a steady state with positive probability, and a method is given for finding states which cannot reach other steady states. These methods are based on computational commutative algebra. The algorithms are applied to dynamics of a cell survival network to determine node assignments that exclude termination in a cancerous state
Stochastic Order Relations Among Parallel Systems From Weibull Distributions, Nuria Torrado, Subhash C. Kochar
Stochastic Order Relations Among Parallel Systems From Weibull Distributions, Nuria Torrado, Subhash C. Kochar
Mathematics and Statistics Faculty Publications and Presentations
In this article, we focus on stochastic orders to compare the magnitudes of two parallel systems from Weibull distributions when one set of scale parameters majorizes the other. The new results obtained here extend some of those proved by Dykstra et al. (1997) and Joo and Mi (2010) from exponential to Weibull distributions. Also, we present some results for parallel systems from multiple-outlier Weibull models.
An Analysis Of The Practical Dpg Method, Jay Gopalakrishnan, Weifeng Qiu
An Analysis Of The Practical Dpg Method, Jay Gopalakrishnan, Weifeng Qiu
Mathematics and Statistics Faculty Publications and Presentations
We give a complete error analysis of the Discontinuous Petrov Galerkin (DPG) method, accounting for all the approximations made in its practical implementation. Specifically, we consider the DPG method that uses a trial space consisting of polynomials of degree p on each mesh element. Earlier works showed that there is a "trial-to-test" operator T, which when applied to the trial space, defines a test space that guarantees stability. In DPG formulations, this operator T is local: it can be applied element-by-element. However, an infinite dimensional problem on each mesh element needed to be solved to apply T. In practical computations, …
Conditional Tests On Basins Of Attraction With Finite Fields, Ian H. Dinwoodie
Conditional Tests On Basins Of Attraction With Finite Fields, Ian H. Dinwoodie
Mathematics and Statistics Faculty Publications and Presentations
An iterative method is given for computing the polynomials that vanish on the basin of attraction of a steady state in discrete polynomial dynamics with finite field coefficients. The algorithm is applied to dynamics of a T cell survival network where it is used to compare transition maps conditional on a basin of attraction.
Dispersive And Dissipative Errors In The Dpg Method With Scaled Norms For Helmholtz Equation, Jay Gopalakrishnan, Ignacio Muga, Nicole Olivares
Dispersive And Dissipative Errors In The Dpg Method With Scaled Norms For Helmholtz Equation, Jay Gopalakrishnan, Ignacio Muga, Nicole Olivares
Mathematics and Statistics Faculty Publications and Presentations
This paper studies the discontinuous Petrov--Galerkin (DPG) method, where the test space is normed by a modified graph norm. The modification scales one of the terms in the graph norm by an arbitrary positive scaling parameter. The main finding is that as the parameter approaches zero, better results are obtained, under some circumstances, when the method is applied to the Helmholtz equation. The main tool used is a dispersion analysis on the multiple interacting stencils that form the DPG method. The analysis shows that the discrete wavenumbers of the method are complex, explaining the numerically observed artificial dissipation in the …
Convergence Rates Of The Dpg Method With Reduced Test Space Degree, Timaeus Bouma, Jay Gopalakrishnan, Ammar Harb
Convergence Rates Of The Dpg Method With Reduced Test Space Degree, Timaeus Bouma, Jay Gopalakrishnan, Ammar Harb
Mathematics and Statistics Faculty Publications and Presentations
This paper presents a duality theorem of the Aubin-Nitsche type for discontinuous Petrov Galerkin (DPG) methods. This explains the numerically observed higher convergence rates in weaker norms. Considering the specific example of the mild-weak (or primal) DPG method for the Laplace equation, two further results are obtained. First, the DPG method continues to be solvable even when the test space degree is reduced, provided it is odd. Second, a non-conforming method of analysis is developed to explain the numerically observed convergence rates for a test space of reduced degree
Regularity Of Mediatrices In Surfaces, Pilar Herreros, Mario Ponce, J. J. P. Veerman
Regularity Of Mediatrices In Surfaces, Pilar Herreros, Mario Ponce, J. J. P. Veerman
Mathematics and Statistics Faculty Publications and Presentations
For distinct points p and q in a two-dimensional Riemannian manifold, one defines their mediatrix Lpq as the set of equidistant points to p and q. It is known that mediatrices have a cell decomposition consisting of a finite number of branch points connected by Lipschitz curves. This paper establishes additional geometric regularity properties of mediatrices. We show that mediatrices have the radial linearizability property, which implies that at each point they have a geometrically defined derivative in the branching directions. Also, we study the particular case of mediatrices on spheres, by showing that they are Lipschitz simple closed curves …
Supplementary Balance Laws For Cattaneo Heat Propagation, Serge Preston
Supplementary Balance Laws For Cattaneo Heat Propagation, Serge Preston
Mathematics and Statistics Faculty Publications and Presentations
In this work we determine for the Cattaneo heat propagation system all the supplementary balance laws (conservation laws ) of the same order (zero) as the system itself and extract the constitutive relations (expression for the internal energy) dictated by the Entropy Principle. The space of all supplementary balance laws (having the functional dimension 8) contains four original balance laws and their deformations depending on 4 functions of temperature (λ0(ϑ),KA (ϑ), A = 1, 2, 3). The requirements of the II law of thermodynamics leads to the exclusion of three functional degrees (KA= 0, A …
A Primal Dpg Method Without A First-Order Reformulation, L. Demkowicz, Jay Gopalakrishnan
A Primal Dpg Method Without A First-Order Reformulation, L. Demkowicz, Jay Gopalakrishnan
Mathematics and Statistics Faculty Publications and Presentations
We show that it is possible to apply the DPG methodology without reformulating a second-order boundary value problem into a first-order system, by considering the simple example of the Poisson equation. The result is a new weak formulation and a new DPG method for the Poisson equation, which has no numerical trace variable, but has a numerical flux approximation on the element interfaces, in addition to the primal interior variable.
Multigrid For An Hdg Method, Bernardo Cockburn, O. Bubois, Jay Gopalakrishnan
Multigrid For An Hdg Method, Bernardo Cockburn, O. Bubois, Jay Gopalakrishnan
Mathematics and Statistics Faculty Publications and Presentations
We analyze the convergence of a multigrid algorithm for the Hybridizable Discontinuous Galerkin (HDG) method for diffusion problems. We prove that a non-nested multigrid V-cycle, with a single smoothing step per level, converges at a mesh independent rate. Along the way, we study conditioning of the HDG method, prove new error estimates for it, and identify an abstract class of problems for which a nonnested two-level multigrid cycle with one smoothing step converges even when the prolongation norm is greater than one. Numerical experiments verifying our theoretical results are presented.
Building A Knowledge Base: Understanding Prospective Elementary Teachers’ Mathematical Content Knowledge, Eva Thanheiser, Christine Browning, Alden Edson, Signe Kastberg, Jane-Jane Lo
Building A Knowledge Base: Understanding Prospective Elementary Teachers’ Mathematical Content Knowledge, Eva Thanheiser, Christine Browning, Alden Edson, Signe Kastberg, Jane-Jane Lo
Mathematics and Statistics Faculty Publications and Presentations
This survey of the literature summarizes and reflects on research findings regarding elementary preservice teachers’ (PSTs’) mathematics conceptions and the development thereof. Despite the current focus on teacher education, peer-reviewed journals offer a surprisingly sparse insight in these areas. The limited research that exists chiefly presents views of PSTs’ reasoning at singular points during a term, thus focusing on conceptions almost to the exclusion of the their development. We summarize the current findings, which are a beginning of a collective understanding of PSTs’ mathematical content knowledge. We believe much more work is needed to understand how PSTs can best develop …
Some New Applications Of P-P Plots, Isha Dewan, Subhash C. Kochar
Some New Applications Of P-P Plots, Isha Dewan, Subhash C. Kochar
Mathematics and Statistics Faculty Publications and Presentations
The P-P plot is a powerful graphical tool to compare stochastically the magnitudes of two random variables. In this note, we introduce a new partial order, called P?P order based on P-P plots. For a pair of random variables (X 1, Y1) and (X 2, Y 2) one can see the relative precedence of Y 2 over X 2 versus that of Y 1 over X 1 using P-P order. We show that several seemingly very technical and difficult concepts like convex transform order and super-additive ordering can be easily explained with the …
On Continuously Defective Elastic Crystals, Marek Elźanowski, Serge Preston
On Continuously Defective Elastic Crystals, Marek Elźanowski, Serge Preston
Mathematics and Statistics Faculty Publications and Presentations
We analyze mathematical underpinnings of Davini's theory of defective crystals when the defectiveness of a kinematic state may be material point dependent. We show how the underlying space can be identified with a suitably chosen homogeneous space and how the uniformly defective structure is just a special case.
Nonnegativity Of Exact And Numerical Solutions Of Some Chemotactic Models, Patrick De Leenheer, Jay Gopalakrishnan, Erica Zuhr
Nonnegativity Of Exact And Numerical Solutions Of Some Chemotactic Models, Patrick De Leenheer, Jay Gopalakrishnan, Erica Zuhr
Mathematics and Statistics Faculty Publications and Presentations
We investigate nonnegativity of exact and numerical solutions to a generalized Keller–Segel model. This model includes the so-called “minimal” Keller–Segel model, but can cover more general chemistry. We use maximum principles and invariant sets to prove that all components of the solution of the generalized model are nonnegative. We then derive numerical methods, using finite element techniques, for the generalized Keller–Segel model. Adapting the ideas in our proof of nonnegativity of exact solutions to the discrete setting, we are able to show nonnegativity of discrete solutions from the numerical methods under certain standard assumptions. One of the numerical methods is …
Self-Assembled Nanoparticle Antiglare Coatings, Khalid Askar, Blayne M. Phillips, Xuan Dou, Juan Lopez, Carl Smith, Bin Jiang, Peng Jiang
Self-Assembled Nanoparticle Antiglare Coatings, Khalid Askar, Blayne M. Phillips, Xuan Dou, Juan Lopez, Carl Smith, Bin Jiang, Peng Jiang
Mathematics and Statistics Faculty Publications and Presentations
Here we report a simple and scalable bottom-up technology for assembling close-packed nanoparticle monolayers on both sides of a glass substrate as high-quality antiglare coatings. Optical measurements show that monolayer coatings consisting of 110 nm silica nanoparticles can significantly reduce optical reflectance and enhance specular transmittance of the glass substrate for a broad range of visible wavelengths. Both experiments and numerical simulations reveal that the antiglare properties of the self-assembled colloidal monolayers are significantly affected by the size of the colloidal particles.
Well Conditioned Boundary Integral Equations For Two-Dimensional Sound-Hard Scattering Problems In Domains With Corners, Akash Anand, Jeffrey S. Ovall, Catalin Turc
Well Conditioned Boundary Integral Equations For Two-Dimensional Sound-Hard Scattering Problems In Domains With Corners, Akash Anand, Jeffrey S. Ovall, Catalin Turc
Mathematics and Statistics Faculty Publications and Presentations
We present several well-posed, well-conditioned direct and indirect integral equation formulations for the solution of two-dimensional acoustic scattering problems with Neumann boundary conditions in domains with corners. We focus mainly on Direct Regularized Combined Field Integral Equation (DCFIE-R) formulations whose name reflects that (1) they consist of combinations of direct boundary integral equations of the second-kind and first-kind integral equations which are preconditioned on the left by coercive boundary single-layer operators, and (2) their unknowns are physical quantities, i.e., the total field on the boundary of the scatterer. The DCFIE-R equations are shown to be uniquely solvable in appropriate function …
Some Unified Results On Comparing Linear Combinations Of Independent Gamma Random Variables, Subhash C. Kochar, Maochao Xu
Some Unified Results On Comparing Linear Combinations Of Independent Gamma Random Variables, Subhash C. Kochar, Maochao Xu
Mathematics and Statistics Faculty Publications and Presentations
In this paper, a new sufficient condition for comparing linear combinations of independent gamma random variables according to star ordering is given. This unifies some of the newly proved results on this problem. Equivalent characterizations between various stochastic orders are established by utilizing the new condition. The main results in this paper generalize and unify several results in the literature including those of Amiri, Khaledi, and Samaniego [2], Zhao [18], and Kochar and Xu [9].
Convergence Analysis Of A Multigrid Algorithm For The Acoustic Single Layer Equation, Simon Gemmrich, Jay Gopalakrishnan, Nilima Nigam
Convergence Analysis Of A Multigrid Algorithm For The Acoustic Single Layer Equation, Simon Gemmrich, Jay Gopalakrishnan, Nilima Nigam
Mathematics and Statistics Faculty Publications and Presentations
We present and analyze a multigrid algorithm for the acoustic single layer equation in two dimensions. The boundary element formulation of the equation is based on piecewise constant test functions and we make use of a weak inner product in the multigrid scheme as proposed in Bramble et al. (1994) . A full error analysis of the algorithm is presented. We also conduct a numerical study of the effect of the weak inner product on the oscillatory behavior of the eigenfunctions for the Laplace single layer operator.
Instability In A Generalized Keller–Segel Model, Patrick De Leenheer, Jay Gopalakrishnan, Erica Zuhr
Instability In A Generalized Keller–Segel Model, Patrick De Leenheer, Jay Gopalakrishnan, Erica Zuhr
Mathematics and Statistics Faculty Publications and Presentations
We present a generalized Keller–Segel model where an arbitrary number of chemical compounds react, some of which are produced by a species, and one of which is a chemoattractant for the species. To investigate the stability of homogeneous stationary states of this generalized model, we consider the eigenvalues of a linearized system. We are able to reduce this infinite dimensional eigenproblem to a parametrized finite dimensional eigenproblem. By matrix theoretic tools, we then provide easily verifiable sufficient conditions for destabilizing the homogeneous stationary states. In particular, one of the sufficient conditions is that the chemotactic feedback is sufficiently strong. Although …
Partial Expansion Of A Lipschitz Domain And Some Applications, Weifeng Qiu, Jay Gopalakrishnan
Partial Expansion Of A Lipschitz Domain And Some Applications, Weifeng Qiu, Jay Gopalakrishnan
Mathematics and Statistics Faculty Publications and Presentations
We show that a Lipschitz domain can be expanded solely near a part of its boundary, assuming that the part is enclosed by a piecewise C1 curve. The expanded domain as well as the extended part are both Lipschitz. We apply this result to prove a regular decomposition of standard vector Sobolev spaces with vanishing traces only on part of the boundary. Another application in the construction of low-regularity projectors into finite element spaces with partial boundary conditions is also indicated
Supporting Implementation Of The Common Core State Standards For Mathematics: Recommendations For Professional Development, Paola Sztajn, Karen A. Marrongelle, Peg Smith, Bonnie L. Melton
Supporting Implementation Of The Common Core State Standards For Mathematics: Recommendations For Professional Development, Paola Sztajn, Karen A. Marrongelle, Peg Smith, Bonnie L. Melton
Mathematics and Statistics Faculty Publications and Presentations
In 2010, the National Governor’s Association and the Council of Chief State School Officers published the Common Core State Standards for Mathematics (CCSSM) and to date, 44 states, the District of Columbia, and the U.S. Virgin Islands have adopted the document. These content and practice standards, which specify what students are expected to understand and be able to do in K-12 mathematics, represent a significant departure from what mathematics is currently taught in most classrooms and how it is taught. Developing teachers’ capacity to enact these new standards in ways that support the intended student learning outcomes will require considerable …
Stochastic Comparisons Of Order Statistics And Spacings: A Review, Subhash C. Kochar
Stochastic Comparisons Of Order Statistics And Spacings: A Review, Subhash C. Kochar
Mathematics and Statistics Faculty Publications and Presentations
We review some of the recent developments in the area of stochastic comparisons of order statistics and sample spacings. We consider the cases when the parent observations are identically as well as nonidentically distributed. But most of the time we will be assuming that the observations are independent. The case of independent exponentials with unequal scale parameters as well as the proportional hazard rate model is discussed in detail.
Wavenumber Explicit Analysis Of A Dpg Method For The Multidimensional Helmholtz Equation, Leszek Demkowicz, Jay Gopalakrishnan, Ignacio Muga, Jeffrey Zitelli
Wavenumber Explicit Analysis Of A Dpg Method For The Multidimensional Helmholtz Equation, Leszek Demkowicz, Jay Gopalakrishnan, Ignacio Muga, Jeffrey Zitelli
Mathematics and Statistics Faculty Publications and Presentations
We study the properties of a novel discontinuous Petrov Galerkin (DPG) method for acoustic wave propagation. The method yields Hermitian positive definite matrices and has good pre-asymptotic stability properties. Numerically, we find that the method exhibits negligible phase errors (otherwise known as pollution errors) even in the lowest order case. Theoretically, we are able to prove error estimates that explicitly show the dependencies with respect to the wavenumber ω, the mesh size h, and the polynomial degree p. But the current state of the theory does not fully explain the remarkably good numerical phase errors. Theoretically, comparisons are made with …
Benchmark Results For Testing Adaptive Finite Element Eigenvalue Procedures Ii (Cluster Robust Eigenvector And Eigenvalue Estimates), Stefano Giani, Luka Grubisic, Jeffrey S. Ovall
Benchmark Results For Testing Adaptive Finite Element Eigenvalue Procedures Ii (Cluster Robust Eigenvector And Eigenvalue Estimates), Stefano Giani, Luka Grubisic, Jeffrey S. Ovall
Mathematics and Statistics Faculty Publications and Presentations
As a model benchmark problem for this study we consider a highly singular transmission type eigenvalue problem which we study in detail both analytically as well as numerically. In order to justify our claim of cluster robust and highly accurate approximation of a selected groups of eigenvalues and associated eigenfunctions, we give a new analysis of a class of direct residual eigenspace/vector approximation estimates. Unlike in the first part of the paper, we now use conforming higher order finite elements, since the canonical choice of an appropriate norm to measure eigenvector approximation by discontinuous Galerkin methods is an open problem.