Boundary Data Maps For Schrödinger Operators On A Compact Interval, 2010 Missouri University of Science and Technology

#### Boundary Data Maps For Schrödinger Operators On A Compact Interval, Stephen L. Clark, Fritz Gesztesy, M. Mitrea

*Mathematics and Statistics Faculty Research & Creative Works*

We provide a systematic study of boundary data maps, that is, 2 x 2 matrix-valued Dirichlet-to-Neumann and more generally, Robin-to-Robin maps, associated with one-dimensional Schrödinger operators on a compact interval [0, R] with separated boundary conditions at 0 and R. Most of our results are formulated in the non-self-adjoint context. Our principal results include explicit representations of these boundary data maps in terms of the resolvent of the underlying Schrödinger operator and the associated boundary trace maps, Krein-type resolvent formulas relating Schrödinger operators corresponding to different (separated) boundary conditions, and a derivation of the Herglotz property of boundary data maps …

Computational Issues In Sensitivity Analysis For 1d Interface Problems, 2010 Missouri University of Science and Technology

#### Computational Issues In Sensitivity Analysis For 1d Interface Problems, L. G. Davis, John R. Singler

*Mathematics and Statistics Faculty Research & Creative Works*

This paper is concerned with the construction of accurate and e cient computational algorithms for the numerical approximation of sensitivities with respect to a parameter dependent interface location. Motivated by sensitivity analysis with respect to piezoelectric actuator placement on an

The Hodrick-Prescott Filter: A Special Case Of Penalized Spline Smoothing, 2010 Missouri University of Science and Technology

#### The Hodrick-Prescott Filter: A Special Case Of Penalized Spline Smoothing, Robert Paige L., A. A. Trindade

*Mathematics and Statistics Faculty Research & Creative Works*

We prove that the Hodrick-Prescott Filter (HPF), a commonly used method for smoothing econometric time series, is a special case of a linear penalized spline model with knots placed at all observed time points (except the first and last) and uncorrelated residuals. This equivalence then furnishes a rich variety of existing data-driven parameter estimation methods, particularly restricted maximum likelihood (REML) and generalized cross-validation (GCV). This has profound implications for users of HPF who have hitherto typically relied on subjective choice, rather than estimation, for the smoothing parameter. By viewing estimates as roots of an appropriate quadratic estimating equation, we also …

High Accuracy Combination Method For Solving The Systems Of Nonlinear Volterra Integral And Integro-Differential Equations With Weakly Singular Kernels Of The Second Kind, 2010 Missouri University of Science and Technology

#### High Accuracy Combination Method For Solving The Systems Of Nonlinear Volterra Integral And Integro-Differential Equations With Weakly Singular Kernels Of The Second Kind, Xiaoming He, Lu Pan, Tao Lü

*Mathematics and Statistics Faculty Research & Creative Works*

This paper presents a high accuracy combination algorithm for solving the systems of nonlinear Volterra integral and integro-differential equations with weakly singular kernels of the second kind. Two quadrature algorithms for solving the systems are discussed, which possess high accuracy order and the asymptotic expansion of the errors. By means of combination algorithm, we may obtain a numerical solution with higher accuracy order than the original two quadrature algorithms. Moreover an a posteriori error estimation for the algorithm is derived. Both of the theory and the numerical examples show that the algorithm is effective and saves storage capacity and computational …

Weyl-Titchmarsh Theory For Hamiltonian Dynamic Systems, 2010 Missouri University of Science and Technology

#### Weyl-Titchmarsh Theory For Hamiltonian Dynamic Systems, Shurong Sun, Shaozhu Chen, Martin Bohner

*Mathematics and Statistics Faculty Research & Creative Works*

We establish the Weyl-Titchmarsh theory for singular linear Hamiltonian dynamic systems on a time scale T , which allows one to treat both continuous and discrete linear Hamiltonian systems as special cases for T= ℝ and T= ℤ within one theory and to explain the discrepancies between these two theories. This paper extends the Weyl-Titchmarsh theory and provides a foundation for studying spectral theory of Hamiltonian dynamic systems. These investigations are part of a larger program which includes the following: (i) M(λ) theory for singular Hamiltonian systems, (ii) on the spectrum of Hamiltonian systems, (iii) on boundary value problems for …

Nonlinear Model Reduction Using Group Proper Orthogonal Decomposition, 2010 Missouri University of Science and Technology

#### Nonlinear Model Reduction Using Group Proper Orthogonal Decomposition, Benjamin T. Dickinson, John R. Singler

*Mathematics and Statistics Faculty Research & Creative Works*

We propose a new method to reduce the cost of computing nonlinear terms in projec- tion based reduced order models with global basis functions. We develop this method by extending ideas from the group nite element (GFE) method to proper orthogonal decomposition (POD) and call it the group POD method. Here, a scalar two-dimensional Burgers' equation is used as a model problem for the group POD method. Numerical results show that group POD models of Burgers' equation are as accurate and are computationally more e cient than standard POD models of Burgers' equation.

Gronwall-Ouiang-Type Integral Inequalities On Time Scales, 2010 Missouri University of Science and Technology

#### Gronwall-Ouiang-Type Integral Inequalities On Time Scales, Ailian Liu, Martin Bohner

*Mathematics and Statistics Faculty Research & Creative Works*

We present several Gronwall-OuIang-type integral inequalities on time scales. Firstly, an OuIang inequality on time scales is discussed. Then we extend the Gronwall-type inequalities to multiple integrals. Some special cases of our results contain continuous Gronwall-type inequalities and their discrete analogues. Several examples are included to illustrate our results at the end.

Optimality Of Balanced Proper Orthogonal Decomposition For Data Reconstruction, 2010 Missouri University of Science and Technology

#### Optimality Of Balanced Proper Orthogonal Decomposition For Data Reconstruction, John R. Singler

*Mathematics and Statistics Faculty Research & Creative Works*

Proper orthogonal decomposition (POD) finds an orthonormal basis yielding an optimal reconstruction of a given dataset. We consider an optimal data reconstruction problem for two general datasets related to balanced POD, which is an algorithm for balanced truncation model reduction for linear systems. We consider balanced POD outside of the linear systems framework, and prove that it solves the optimal data reconstruction problem. the theoretical result is illustrated with an example.

On Hyperplanes And Semispaces In Max-Min Convex Geometry, 2010 West Chester University of Pennsylvania

#### On Hyperplanes And Semispaces In Max-Min Convex Geometry, Viorel Nitica, Sergeĭ Sergeev

*Mathematics Faculty Publications*

No abstract provided.

The Topology Of Incidence Pseudographs, 2010 Otterbein University

#### The Topology Of Incidence Pseudographs, Thomas R. James, Reinhard Klette

*Mathematics Faculty Scholarship*

Incidence pseudographs model a (re°exive and symmetric) inci- dence relation between sets of various dimensions, contained in a count- able family. Work by Klaus Voss in 1993 suggested that this general discrete model allows to introduce a topology, and some authors have done some studies into this direction in the past (also using alternative discrete models such as, for example, abstract complexes or orders on sets of cells). This paper provides a comprehensive overview about the topology of incidence pseudographs. This topology has various appli- cations, such as in modeling basic data in 2D or 3D digital picture analysis, or …

A Generalization Of Bernoulli's Inequality, 2010 Department of Mathematics and Statistics, Florida International University

#### A Generalization Of Bernoulli's Inequality, Laura De Carli, Steve M. Hudson

*Department of Mathematics and Statistics*

No abstract provided.

Curve Interpolation And Coding Theory, 2010 Gettysburg College

#### Curve Interpolation And Coding Theory, Darren B. Glass

*Math Faculty Publications*

Whether it is downloading files from the Internet, having conversations between cell phones, or sending information from a laptop to a printer, we often want to transmit data in situations where we need to worry about interference from other signals that may cause errors in the transmission. The branch of mathematics known as coding theory is dedicated to finding ways to tell when these are errors in transmission and, when possible, how to correct those errors. The goal of coding theory is to build as much redundancy as possible into a message without greatly increasing its length. [excerpt]

U-Singularity And T-Topos Theoretic Entropy, 2010 California Polytechnic State University - San Luis Obispo

#### U-Singularity And T-Topos Theoretic Entropy, Goro C. Kato

*Mathematics*

We will give descriptions of *u*-singularities as we introduce the notion of *t*-topos theoretic entropies. The unifying methodology for a *u*-singularity is the universal mapping property of an inverse or direct limit. The qualitative, conceptual, and structural analyses of u-singularities are given in terms of inverse and direct limits of micro decompositions of a presheaf and coverings of an object in t-site in the theory of temporal topos.

Urcohomologies And Cohomologies Of N -Complexes, 2010 Okayama University

#### Urcohomologies And Cohomologies Of N -Complexes, Naoya Hiramatsu, Goro Kato

*Mathematics*

For a general sequence of objects and morphisms, we construct two N-complexes. Then we can define cohomologies (i, k)-type of the N-complexes not only on a diagonal region but also in the triangular region. We obtain an invariant defined on a general sequence of objects and morphisms. For a short exact sequence of N-complexes, we get the associated long exact sequence generalizing the classical long exact sequence. Lastly, several properties of the vanishing cohomologies of N-complexes are given.

Using Correlation Coefficients To Estimate Slopes In Multiple Linear Regression, 2010 University of Montana, Missoula

#### Using Correlation Coefficients To Estimate Slopes In Multiple Linear Regression, Rudy Gideon

*Mathematical Sciences Faculty Publications*

This short note takes correlation coefficients as the starting point to obtain inferential results in linear regression. Under certain conditions, the population correlation coefficient and the sampling correlation coefficient can be related via a Taylor series expansion to allow inference on the coefficients in simple and multiple regression. This general method includes nonparametric correlation coefficients and so gives a universal way to develop regression methods. This work is part of a correlation estimation system that uses correlation coefficients to perform estimation in many settings, for example, time series, nonlinear and generalized linear models, and individual distributions.

Chvátal-Erdös Type Theorems, 2010 University of Memphis

#### Chvátal-Erdös Type Theorems, Ralph J. Faudree, Jill R. Faudree, Ronald J. Gould, Michael S. Jacobson, Colton Magnant

*Colton Magnant*

The Chvátal-Erdös theorems imply that if G is a graph of order n ≥ 3 with κ(G) ≥ α(G), then G is hamiltonian, and if κ(G) > α(G), then G is hamiltonian-connected. We generalize these results by replacing the connectivity and independence number conditions with a weaker minimum degree and independence number condition in the presence of sufficient connectivity. More specifically, it is noted that if G is a graph of order n and k ≥ 2 is a positive integer such that κ(G) ≥ k, δ(G) > (n+k2-k)/(k+1), and δ(G) ≥ α(G)+k-2, then G is hamiltonian. It is shown that if …

Incorporating Genome Annotation In The Statistical Analysis Of Genomic And Epigenomic Tiling Array Data, 2010 Missouri University of Science and Technology

#### Incorporating Genome Annotation In The Statistical Analysis Of Genomic And Epigenomic Tiling Array Data, Gayla R. Olbricht

*Mathematics and Statistics Faculty Research & Creative Works*

"A wealth of information and technologies are currently available for the genomewide investigation of many types of biological phenomena. Genomic annotation databases provide information about the DNA sequence of a particular organism and give locations of different types of genomic elements, such as the exons and introns of genes. Microarrays are a powerful type of technology that make use of DNA sequence information to investigate different types of biological phenomena on a genome-wide level. Tiling arrays are a unique type of microarray that provide unbiased, highdensity coverage of a genomic region, making them well suited for many applications, such as …

Numerical Investigation Of Aeroelastic Mode Distribution For Aircraft Wing Model In Subsonic Air Flow, 2010 University of New Hampshire

#### Numerical Investigation Of Aeroelastic Mode Distribution For Aircraft Wing Model In Subsonic Air Flow, Marianna A. Shubov, Stephen B. Wineberg, Robert Holt

*Mathematics & Statistics Scholarship*

In this paper, the numerical results on two problems originated in aircraft wing modeling have been presented. The first problem is concerned with the approximation to the set of the aeroelastic modes, which are the eigenvalues of a certain boundary-value problem. The affirmative answer is given to the following question: can the leading asymptotical terms in the analytical formulas be used as reasonably accurate description of the aeroelastic modes? The positive answer means that these leading terms can be used by engineers for practical calculations. The second problem is concerned with the flutter phenomena in aircraft wings in a subsonic, …

On K4 Of The Gaussian And Eisenstein Integers, 2010 University of Massachusetts - Amherst

#### On K4 Of The Gaussian And Eisenstein Integers, Mathieu Dutour Sikiric, Herbert Gangl, Paul Gunnells, Jonathan Hanke, Achill Schürmann, Dan Yasaki

*Paul Gunnells*

Abstract. In this paper we investigate the structure of the algebraic K-groups K4(Z[i]) and K4(Z[ρ]), where i := √ −1 and ρ := (1 + √ −3)/2. We exploit the close connection between homology groups of GLn(R) for n 6 5 and those of related classifying spaces, then compute the former using Voronoi’s reduction theory of positive definite quadratic and Hermitian forms to produce a very large finite cell complex on which GLn(R) acts. Our main results are (i) K4(Z[i]) is a finite abelian 3-group, and (ii) K4(Z[ρ]) is trivial.

Asymptotic Behavior Of The Finite-Size Magnetization As A Function Of The Speed Of Approach To Criticality, 2010 University of Massachusetts - Amherst

#### Asymptotic Behavior Of The Finite-Size Magnetization As A Function Of The Speed Of Approach To Criticality, Richard S. Ellis, Jonathan Machta, Peter Tak-Hun Otto

*Richard S. Ellis*

The main focus of this paper is to determine whether the thermodynamic magnetization is a physically relevant estimator of the finite-size magnetization. This is done by comparing the asymptotic behaviors of these two quantities along parameter sequences converging to either a second-order point or the tricritical point in the mean-field Blume–Capel model. We show that the thermodynamic magnetization and the finite-size magnetization are asymptotic when the parameter α governing the speed at which the sequence approaches criticality is below a certain threshold α0. However, when α exceeds α0, the thermodynamic magnetization converges to 0 much faster than the finite-size magnetization. …