Physical Sciences and Mathematics Commons^™

Multiboard Determinacy, Andrés E. Caicedo

Andrés E. Caicedo

No abstract provided.

Counting Pattern-Avoiding Permutations, Lara Pudwell

Lara K. Pudwell

No abstract provided.

An Introduction To Enumeration Schemes, Lara Pudwell

Lara K. Pudwell

No abstract provided.

Homological Dimensions And Regular Rings, Alina Iacob, Srikanth B. Iyengar

Alina Iacob

A question of Avramov and Foxby concerning injective dimension of complexes is settled in the affirmative for the class of noetherian rings. A key step in the proof is to recast the problem on hand into one about the homotopy category of complexes of injective modules. Analogous results for flat dimension and projective dimension are also established.

Simplify, Cancel, And Other Math Lingo With Multiple Meanings, Lisa Yocco

Lisa S. Yocco

No abstract provided.

Logically Rectangular Finite Volume Methods With Adaptive Refinement On The Sphere, Marsha Berger, Donna Calhoun, Christiane Helzel, Randall Leveque

Donna Calhoun

The logically rectangular finite volume grids for two-dimensional partial differential equations on a sphere and for three-dimensional problems in a spherical shell introduced recently have nearly uniform cell size, avoiding severe Courant number restrictions. We present recent results with adaptive mesh refinement using the GEOCLAW software and demonstrate well-balanced methods that exactly maintain equilibrium solutions, such as shallow water equations for an ocean at rest over arbitrary bathymetry.

Counting Pattern-Avoiding Permutations, Lara Pudwell

Lara K. Pudwell

No abstract provided.

Non-Smooth Solutions To Least Squares Problems, Jodi Mead

Jodi Mead

In an attempt to overcome the ill-posedness or illconditioning of inverse problems, regularization methods are implemented by introducing assumptions on the solution. Common regularization methods include total variation, L-curve, Generalized Cross Validation (GCV), and the discrepancy principle. It is generally accepted that all of these approaches except total variation unnecessarily smooth solutions, mainly because the regularization operator is in an L2 norm. Alternatively, statistical approaches to ill-posed problems typically involve specifying a priori information about the parameters in the form of Bayesian inference. These approaches can be more accurate than typical regularization methods because the regularization term is weighted with …

Copula Density Estimation By Total Variation Penalized Likelihood, Leming Qu, Yi Qian, Hui Xie

Leming Qu

Copulas are full measures of dependence among random variables. They are increasingly popular among academics and practitioners in financial econometrics for modeling comovements between markets, risk factors, and other relevant variables. A copula's hidden dependence structure that couples a joint distribution with its marginals makes a parametric copula non-trivial. An approach to bivariate copula density estimation is introduced that is based on a penalized likelihood with a total variation penalty term. Adaptive choice of the amount of regularization is based on approximate Bayesian Information Criterion (BIC) type scores. Performance are evaluated through the Monte Carlo simulation.

Problem #3468, Stan Wagon, Peter Saltzman, Joseph Devincentis

Stan Wagon, Retired

No abstract provided.

A Pair Of Operator Summation Formulas And Their Applications, Tian-Xiao He, Leetsch C. Hsu, Dongsheng Yin

Tian-Xiao He

Two types of symbolic summation formulas are reformulated using an extension of Mullin–Rota’s substitution rule in [R. Mullin, G.-C. Rota, On the foundations of combinatorial theory: III. Theory of binomial enumeration, in: B. Harris (Ed.), Graph Theory and its Applications, Academic Press, New York, London, 1970, pp. 167–213], and several applications involving various special formulas and identities are presented as illustrative examples.

Confidence Intervals For Long Memory Regressions, Kyungduk Ko, Jaechoul Lee, Robert Lund

Kyungduk Ko

This paper proposes an accurate confidence interval for the trend parameter in a linear regression model with long memory errors. The interval is based upon an equivalent sum of squares method and is shown to perform comparably to a weighted least squares interval. The advantages of the proposed interval lies in its relative ease of computation and should be attractive to practitioners.

First-Order Bias Correction For Fractionally Integrated Time Series, Jaechoul Lee, Kyungduk Ko

Kyungduk Ko

Most of the long memory estimators for stationary fractionally integrated time series models are known to experience non-negligible bias in small and finite samples. Simple moment estimators are also vulnerable to such bias, but can easily be corrected. In this paper, we propose bias reduction methods for a lag-one sample autocorrelation-based moment estimator. In order to reduce the bias of the moment estimator, we explicitly obtain the exact bias of lag-one sample autocorrelation up to the order n−1. An example where the exact first-order bias can be noticeably more accurate than its asymptotic counterpart, even for large samples, is presented. …

On The Issue Of Decoupled Decoding Of Codes Derived From Quaternion Orthogonal Designs, Tadeusz Wysocki, Beata Wysocki, Sarah Spence Adams

Sarah Spence Adams

Quaternion orthogonal designs (QODs) have been previously introduced as a basis for orthogonal space-time polarization block codes (OSTPBCs). This note will serve to correct statements concerning the optimality of a decoupled maximum-likelihood (ML) decoding algorithm. It will be shown that when compared to coupled decoding, the decoupled decoding is only optimal in certain cases. This raises several open problems concerning the decoding of OSTPBCs.

On Sequences Of Numbers And Polynomials Defined By Linear Recurrence Relations Of Order 2, Tian-Xiao He, Peter J.-S. Shiue

Tian-Xiao He

Here we present a new method to construct the explicit formula of a sequence of numbers and polynomials generated by a linear recurrence relation of order 2. The applications of the method to the Fibonacci and Lucas numbers, Chebyshev polynomials, the generalized Gegenbauer-Humbert polynomials are also discussed. The derived idea provides a generalmethod to construct identities of number or polynomial sequences defined by linear recurrence relations. The applications using the method to solve some algebraic and ordinary differential equations are presented.

A Finite Volume Method For Solving Parabolic Equations On Curved Surfaces, Donna Calhoun

Donna Calhoun

No abstract provided.

Computing Sequences And Series By Recurrence, Stephen J. Sugden

Stephen Sugden

Extract: Many commonly-used mathematical functions may be computed via carefully-constructed recurrence formulas. Sequences are typically defined by giving a formula for the general term. Series is the mathematical name given to partial sums of sequences. In either case we may often take advantage of the great expressive power of recurrence relations to create code which is both lucid and compact. Further, this does not necessarily mean that we must use recursive code. In many instances, iterative code is adequate, and often more efficient.

Correction To “The Theory Of Quaternion Orthogonal Designs”, Tadeusz Wysocki, Beata Wysocki, Sarah Spence Adams

Sarah Spence Adams

Seberry et al. claimed that even though the dual-polarized transmission channel cannot be considered as described by means of a single quaternionic gain, the maximum-likelihood (ML) decoding rule can be decoupled for orthogonal space–time-polarization block codes (OSTPBCs) derived from quaternion orthogonal designs (QODs) [1, Sec. IV]. Regretfully, a correction is necessary, and we will show that decoupled decoding using the method presented therein is only optimal for codes derived from certain QODs, not from arbitrary QODs as previously suggested.

Comic Books That Teach Mathematics, Bruce Kessler

Bruce Kessler

During the 2008--2009 academic year, the author embarked on an extremely non-standard curriculum path: developing comic books with embedded mathematics appropriate for 3rd through 6th grade students. With the help of an education professor to measure impact, an elementary-school principal, and talented undergraduate illustrators, this project came to fruition and the comics were implemented in elementary classrooms at Cumberland Trace Elementary in the Warren County School System in Bowling Green, Kentucky. This talk gives the motivation for the idea, introduces the characters, and how the comics integrated the math content into the stories.

Comic Books That Teach Mathematics, Bruce Kessler

Bruce Kessler

During the 2008--2009 academic year, the author embarked on an extremely non-standard curriculum path: developing comic books with embedded mathematics appropriate for 3rd through 6th grade students. With the help of an education professor to measure impact, an elementary-school principal, and talented undergraduate illustrators, this project came to fruition and the comics were implemented in elementary classrooms at Cumberland Trace Elementary in the Warren County School System in Bowling Green, Kentucky. This manuscript gives the history of this idea, the difficulties of developing the content of the comics and getting them illustrated, and the implementation plan in the school.

A …

Using Works Of Visual Art To Teach Matrix Transformations, James Luke Akridge, Rachel Bowman, Peter Hamburger, Bruce Kessler

Bruce Kessler

The authors present a modern technique for teaching matrix transformations on $\R^2$ that incorporates works of visual art and computer programming. Two of the authors were undergraduate students in Dr. Hamburger's linear algebra class, where this technique was implemented as a special project for the students. The two students generated the images seen in this paper, and the movies that can be found on the accompanying webpage www.wku.edu/\~bruce.kessler/.

Wavelet Deconvolution In A Periodic Setting Using Cross-Validation, Leming Qu, Partha Routh, Kyungduk Ko

Kyungduk Ko

The wavelet deconvolution method WaveD using band-limited wavelets offers both theoretical and computational advantages over traditional compactly supported wavelets. The translation-invariant WaveD with a fast algorithm improves further. The twofold cross-validation method for choosing the threshold parameter and the finest resolution level in WaveD is introduced. The algorithm’s performance is compared with the fixed constant tuning and the default tuning in WaveD.

Bayesian Wavelet-Based Methods For The Detection Of Multiple Changes Of The Long Memory Parameter, Kyungduk Ko

Kyungduk Ko

Long memory processes are widely used in many scientific fields, such as economics, physics, and engineering. Change point detection problems have received considerable attention in the literature because of their wide range of possible applications. Here we describe a wavelet-based Bayesian procedure for the estimation and location of multiple change points in the long memory parameter of Gaussian autoregressive fractionally integrated moving average models (ARFIMA(p, d, q)), with unknown autoregressive and moving average parameters. Our methodology allows the number of change points to be unknown. The reversible jump Markov chain Monte Carlo algorithm is used for posterior inference. The method …

Harmonic Functions On R-Covered Foliations And Group Actions On The Circle, Sergio Fenley, Renato Feres, Kamlesh Parwani

Kamlesh Parwani

Let (M,F) be a compact codimension-one foliated manifold whose leaves are equipped with Riemannian metrics, and consider continuous functions on M that are harmonic along the leaves of F. If every such function is constant on leaves we say that (M,F) has the Liouville property. Our main result is that codimension-one foliated bundles over compact negatively curved manifolds satisfy the Liouville property. Related results for R-covered foliations, as well as for discrete group actions and discrete harmonic functions, are also established.

Research On Fractal Mathematics And Some Application In Mechanics, Yang Xiaojun

Xiao-Jun Yang

Since Mandelbrot proposed the concept of fractal in 1970s’, fractal has been applied in various areas such as science, economics, cultures and arts because of the universality of fractal phenomena. It provides a new analytical tool to reveal the complexity of the real world. Nowadays the calculus in a fractal space becomes a hot topic in the world. Based on the established definitions of fractal derivative and fractal integral, the fundamental theorems of fractal derivatives and fractal integrals are investigated in detail. The fractal double integral and fractal triple integral are discussed and the variational principle in fractal space has …

Sequence Characterization Of Riordan Arrays, Tian-Xiao He, Renzo Sprugnoli

Tian-Xiao He

In the realm of the Riordan group, we consider the characterization of Riordan arrays by means of the A- and Z-sequences. It corresponds to a horizontal construction of a Riordan array, whereas the traditional approach is through column generating functions. We show how the A- and Z-sequences of the product of two Riordan arrays are derived from those of the two factors; similar results are obtained for the inverse. We also show how the sequence characterization is applied to construct easily a Riordan array. Finally, we give the characterizations relative to some subgroups of the Riordan group, in particular, of …

Elementary-Level Mathematics Content In Comic Book Format, Bruce Kessler, Janet Tassell, Mary Evans, Cathy Willoughby, Melissa Zimmer

Bruce Kessler

No abstract provided.

Numerical Studies Of A Nonlinear Heat Equation With Square Root Reaction Term, Ron Buckmire, Karl Mcmurtry, Ronald Mickens

Ron Buckmire

Interest in calculating numerical solutions of a highly nonlinear parabolic partial differential equation with fractional power diffusion and dissipative terms motivated our investigation of a heat equation having a square root nonlinear reaction term. The original equation occurs in the study of plasma behavior in fusion physics. We begin by examining the numerical behavior of the ordinary differential equation obtained by dropping the diffusion term. The results from this simpler case are then used to construct nonstandard finite difference schemes for the partial differential equation. A variety of numerical results are obtained and analyzed, along with a comparison to the …

Paradigms For Non-Classical Substitutions, Lawrence Stout, P. Eklund, M. Galan, J. Kortelainen

Lawrence N. Stout

We will present three paradigms for non-classical substitution. Firstly, we have the classical substitution of variables with terms. This is written in a strict categorical form supporting presentation of the other two paradigms. The second paradigm is substitutions of variables with many-valued sets of terms. These two paradigms are based on functors and monads over the category of sets. The third paradigm is the substitution of many-valued sets of variables with terms over many-valued sets of variables. The latter is based on functors and monads over the category of many-valued sets. This provides a transparency of the underlying categories and …

Probabilistic Analysis Of A Differential Equation For Linear Programming, Asa Ben-Hur, Joshua Feinberg, Shmuel Fishman, Hava Siegelmann

Hava Siegelmann

In this paper we address the complexity of solving linear programming problems with a set of differential equations that converge to a fixed point that represents the optimal solution. Assuming a probabilistic model, where the inputs are i.i.d. Gaussian variables, we compute the distribution of the convergence rate to the attracting fixed point. Using the framework of Random Matrix Theory, we derive a simple expression for this distribution in the asymptotic limit of large problem size. In this limit, we find the surprising result that the distribution of the convergence rate is a scaling function of a single variable. This …