Mathematics Commons^™

Commentary On Ballou's Paper: Galois - The Myths And The Man, Catherine Degrandpre

The Mathematics Enthusiast

mathematics curriculum. As seen in her paper, the solution of the cubic essentially involves the use of a clever substitution in the general cubic in order to reduce it into a cubic without the square term, which in turn is factorable as a quadratic provided one makes another substitution. A natural question to ask ourselves, which mathematicians in the colorful theory of equations undoubtedly did as well, is how far can we push this technique of cle ver substitutions to solve higher degree equations. These substitutions are called Tschirnhaus transformations and have a pattern of the form x = y …

The Slope Mean And Its Invariance Properties, Jun Ji, Charles Kicey

Faculty Articles

Discusses the slope mean and its invariance properties. Notion of invariance; Comparison of linear regression methods; Comparison with classic means; Characterization by invariance; Focus on quasi-arithmetic means; Theorems used.

Statistical Analysis Of Longitudinal And Multivariate Discrete Data, Deepak Mav

Mathematics & Statistics Theses & Dissertations

Correlated multivariate Poisson and binary variables occur naturally in medical, biological and epidemiological longitudinal studies. Modeling and simulating such variables is difficult because the correlations are restricted by the marginal means via Fréchet bounds in a complicated way. In this dissertation we will first discuss partially specified models and methods for estimating the regression and correlation parameters. We derive the asymptotic distributions of these parameter estimates. Using simulations based on extensions of the algorithm due to Sim (1993, Journal of Statistical Computation and Simulation, 47, pp. 1–10), we study the performance of these estimates using infeasibility, coverage probabilities of the …

The Straggling Green's Function Method For Ion Transport, Steven Andrew Walker

Mathematics & Statistics Theses & Dissertations

For many years work has been conducted on developing a concise theory and method for HZE ion transport capable of being validated in the laboratory. Previous attempts have ignored dispersion and energy downshift associated with nuclear fragmentation and energy and range straggling. Here we present a Green's function approach to ion transport that incorporates these missing elements. This work forms the basis for a new version of GRNTRN, a Green's function transport code. Comparisons of GRNTRN predictions and laboratory results for an ⁵⁶Fe ion beam with average energy at the target of one GeV/amu or more are presented for …

A Topological Representation Theorem For Oriented Matroids, Jürgen Bokowski, Simon King, Sussane Mock, Ileana Streinu

Computer Science: Faculty Publications

We present a new direct proof of a topological representation theorem for oriented matroids in the general rank case. Our proof is based on an earlier rank 3 version. It uses hyperline sequences and the generalized Schönflies theorem. As an application, we show that one can read off oriented matroids from arrangements of embedded spheres of codimension one, even if wild spheres are involved.

Some Sober Conceptions Of Mathematical Truth, Marco Panza

MPP Published Research

It is not sufficient to supply an instance of Tarski’s schema, ⌈“p” is true if and only if p⌉ for a certain statement in order to get a definition of truth for this statement and thus fix a truth-condition for it. A definition of the truth of a statement x of a language L is a bi-conditional whose two members are two statements of a meta-language L’. Tarski’s schema simply suggests that a definition of truth for a certain segment x of a language L consists in a statement of the form: ⌈v(x) is true if and only if τ(x)⌉, …

The Norm Of A Composition Operator With Linear Symbol Acting On The Dirichlet Space, Christopher Hammond

Mathematics Faculty Publications

We obtain a representation for the norm of a composition operator on the Dirichlet space induced by a map of the form φ(z)=az+b. We compare this result to an upper bound for ‖Cφ‖ that is valid whenever φ is univalent. Our work relies heavily on an adjoint formula recently discovered by Gallardo-Gutiérrez and Montes-Rodríguez.

Relative Equilibria For The Generalized Rigid Body, Antonio Hernandez-Garduno, Jeffrey K. Lawson, Jerrold E. Marsden

Mathematics Faculty Research

This paper gives necessary and sufficient conditions for the (n-dimensional) generalized free rigid body to be in a state of relative equilibrium. The conditions generalize those for the case of the three-dimensional free rigid body, namely that the body is in relative equilibrium if and only if its angular velocity and angular momentum align, that is, if the body rotates about one of its principal axes. For the n-dimensional rigid body in the Manakov formulation, these conditions have a similar interpretation. We use this result to state and prove a generalized Saari’s Conjecture (usually stated for the …

Combinatorial Proofs Of Fermat's, Lucas's, And Wilson's Theorems, Peter G. Anderson, Arthur T. Benjamin, Jeremy A. Rouse

All HMC Faculty Publications and Research

No abstract provided in this article.

Error Analysis Of Variable Degree Mixed Methods For Elliptic Problems Via Hybridization, Bernardo Cockburn, Jay Gopalakrishnan

Mathematics and Statistics Faculty Publications and Presentations

A new approach to error analysis of hybridized mixed methods is proposed and applied to study a new hybridized variable degree Raviart-Thomas method for second order elliptic problems. The approach gives error estimates for the Lagrange multipliers without using error estimates for the other variables. Error estimates for the primal and flux variables then follow from those for the Lagrange multipliers. In contrast, traditional error analyses obtain error estimates for the flux and primal variables first and then use it to get error estimates for the Lagrange multipliers. The new approach not only gives new error estimates for the new …

The Terwilliger Algebra Of An Almost-Bipartite P- And Q-Polynomial Association Scheme, John S. Caughman Iv, Mark S. Maclean, Paul M. Terwilliger

Mathematics and Statistics Faculty Publications and Presentations

Let Y denote a D-class symmetric association scheme with D≥3, and suppose Y is almost-bipartite P- and Q-polynomial. Let x denote a vertex of Y and let T=T(x) denote the corresponding Terwilliger algebra. We prove that any irreducible T-module W is both thin and dual thin in the sense of Terwilliger. We produce two bases for W and describe the action of T on these bases. We prove that the isomorphism class of W as a T-module is determined by two parameters, the dual endpoint and diameter of W. We find a recurrence which gives the multiplicities with which the …

Schwinger Pair Creation Of Kaluza-Klein Particles: Pair Creation Without Tunneling, Tamar Friedmann, Herman Verlinde

Mathematics Sciences: Faculty Publications

We study Schwinger pair creation of charged Kaluza-Klein (KK) particles from a static KK electric field. We find that the gravitational backreaction of the electric field on the geometry—which is incorporated via the electric KK-Melvin solution—prevents the electrostatic potential from overcoming the rest mass of the KK particles, thus impeding the tunneling mechanism which is often thought of as responsible for the pair creation. However, we find that pair creation still occurs with a finite rate formally similar to the classic Schwinger result, but via an apparently different mechanism, involving a combination of the Unruh effect and vacuum polarization due …

Some Contributions To Semidefine Linear Complementary Problem., D. Sampangi Raman Dr.

Doctoral Theses

This thesis is composed of chapters 1 to 5. In Chapter 1, we formally define SDLCP and give some examples of SDLCP and show that SDLCP is a special case of variational inequality problem. Also we show that LCP is indeed a special case of SDLCP. Later section of this chapter deals with definitions and notations.In Chapter 2, we are concerned with the results which have been obtained in an effort to generalize the P-matrix condition of LCP to SDLCP. It is known that a matrix M is a P-matrix if and only if M does not re verse the …

Cryptographic And Combinatorial Properties Of Boolean Functions And S-Boxes., Kishan Chand Gupta Dr.

Doctoral Theses

In this thesis we study combinatorial aspects of Boolean functions and S-boxes with impor- tant cryptographic properties and construct new functions possesing such properties. These have possible applications in the design of private key (symmetric key) cryptosystems.Symmetric key cryptosystems are broadly divided into two classes.1. Stream Ciphers,2. Block Ciphers.Some recent proposals of stream ciphers are SNOW [37], SCREAM [52], TURING (98], MUGI (117), HBB (102], RABBIT (9), HELIX (38] and some proposals of block ciphers are DES, AES, RC6 [97), MARS (12], SERPENT (6], TWOFISH (104].In stream cipher cryptography a pseudorandom sequence of bits of length cqual to the message …

Morphological Tower: A Tool For Multi-Scale Image Processing., Susanta Mukhopadhyay Dr.

Doctoral Theses

An image is a recorded replication of natural scene or objects using suitable sensor and recording media. The visual quality of the recorded image may be enhanced using various types of low-level processing namely noise smoothing, contrast enhancement. The images of the same scene recorded by several sensors reveal more inforination but in their own respective ways. This advantage of multi-sensor imaging system is real- ized through the fusion of the multimodal images. One more important higher-level processing is segmentation where the image is decomposed into a set of meaningful regions ( e.g. objects and background). An image, in general, …

Counting Lattice Points In Admissible Adelic Sets, Lenny Fukshansky

CMC Faculty Publications and Research

Lecture given at the Midwest Number Theory Conference for Graduate Students and Recent PhDs II, February 2005.

Graphs Whose Minimal Rank Is Two: The Finite Fields Case, Wayne Barrett, Hein Van Der Holst, Raphael Loewy

Faculty Publications

Let F be a finite field, G = (V,E) be an undirected graph on n vertices, and let S(F,G) be the set of all symmetric n × n matrices over F whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. Let mr(F,G) be the minimum rank of all matrices in S(F,G). If F is a finite field with p^t elements, p does not = 2, it is shown that mr(F,G) ≤ 2 if and only if the complement of G is the join of a complete graph with either the union of at most …

A Modular Integer Gcd Algorithm, Kenneth Weber, Vilmar Trevisan, Luiz Felipe Martins

Mathematics and Statistics Faculty Publications

This paper describes the first algorithm to compute the greatest common divisor (GCD) of two n-bit integers using a modular representation for intermediate values U, V and also for the result. It is based on a reduction step, similar to one used in the accelerated algorithm [T. Jebelean, A generalization of the binary GCD algorithm, in: ISSAC '93: International Symposium on Symbolic and Algebraic Computation, Kiev, Ukraine, 1993, pp. 111–116; K. Weber, The accelerated integer GCD algorithm, ACM Trans. Math. Softw. 21 (1995) 111–122] when U and V are close to the same size, that replaces U by (U-bV)/p, where …

Jump Systems And Laminated Manhattan Sets, Jessica Cuomo, Nkiruka Nwasokwa, Vadim Ponomarenko

Mathematics Faculty Research

A jump system is a set of lattice points satisfying a certain "two-step" axiom. A Manhattan set is the convex hull of a two-dimensional jump system. Taking multiple Manhattan sets, in layers, forms a three-dimensional object. We determine under what conditions this object is, in turn, a jump system.

The Aronsson-Euler Equation For Absolutely Minimizing Lipschitz Extensions With Respect To Carnot-Carathéodory Metrics, Thomas Bieske, Luca Capogna

Mathematics Sciences: Faculty Publications

We derive the Euler-Lagrange equation (also known in this setting as the Aronsson-Euler equation) for absolute minimizers of the L ∞ variational problem inf∥∇ 0u∥L∞(Ω), u = g ε Lip(∂Ω) on ∂Ω, where Ω ⊂ G is an open subset of a Carnot group, ∇ 0u denotes the horizontal gradient of u: Ω ℝ R, and the Lipschitz class is defined in relation to the Carnot-Carathéodory metric. In particular, we show that absolute minimizers are infinite harmonic in the viscosity sense. As a corollary we obtain the uniqueness of absolute minimizers in a large class of groups. This result extends …

Scattering Matrix Analysis Of Photonic Crystals, Valeriy Lukyanov

Dissertations

Using a scattering matrix approach we analyze and study the scattering and transmission of waves through a two-dimensional photonic crystal which consists of a periodic array of parallel rods with circular cross sections. Without making any assumptions about normal incidence, single mode propagation, and sufficient inter-scatter separation in the direction of propagation, we show how to compute the transmission and reflection coefficients of these periodic structures. The method is based on the computation of a generalized scattering matrix for one column of the periodic structure.

We also develop an analytical method to analyze and to study the scattering and transmission …

Balanced Scaling Vectors Using Linear Combinations Of Existing Scaling Vectors, Bruce Kessler

Mathematics Faculty Publications

The majority of the research done into creating balanced multiwavelets has involved establishing a series of conditions on the mask of the new scaling vector by solving a large nonlinear system. The result is a completely different new function vector solution to the dilation equation with the new matrix coefficients. The research presented here will show a way to use previously-constructed orthonormal scaling vectors to generate equivalent orthonormal scaling vectors that are balanced up to the approximation order of the previous scaling vector. The technique uses linear combinations of the integer translates of the previous-constructed scaling vector.

Essays On Dynamic Disequilibrium Models., Arpita Dhar Dr.

Doctoral Theses

The present dissertation is devoted to the analysis of some problems of economic growth in the framework of what has been called Disequilibrium model.Studies on economic growth have been mostly confined, with few exceptions, to the analysis of equilibrium models. But in reality there are cases where the concept of equilibrium may not be relevant and hence a disequilibrium approach is needed. The present thesis is concerned with analysing some theoretical and empirical aspects of the problem of economic growth in a disequilibrium framework. At the theoretical level, the objective has been to re-examine the conclusions of some existing equilibrium …

On Some Generalized Transforms For Signal Decomposition And Reconstruction., Yumnam Singh Dr.

Doctoral Theses

In this thesis, we propose two new subband transforms entitled ISITRA and YKSK transforms and their possible applications in image compression and encryption. Both these transforms are developed based on a common model of multiplication known as Bino’s model of multiplication. ISITRA is a convolution based transforms i.e., that both forward and inverse transform of ISITRA is based on convolution as in DWT or 2-channel filter bank. However, it is much more general than the existing DWT or 2-channel filter bank scheme in the sense that it we can get different kinds of filters in addition to the filters specified …

The Scintillation Index In Moderate To Strong Turbulence For The Gaussian Beam Wave Along A Slant Path, Fredrick Eugene Thomas

Electronic Theses and Dissertations

Scintillation is one of the most common statistics in the literature of mathematical modeling of laser propagation through random media. One approach to estimating scintillation is through the Rytov approximation, which is limited to weak atmospheric turbulence. Recently, an improvement of the Rytov approximation was developed employing a linear filter function technique. This modifies the Rytov approximation and extends the validity into the moderate to strong regime. In this work, an expression governing scintillation of a Gaussian beam along an uplink slant path valid in all regimes of turbulence is presented, as well as results for the limiting cases of …

On Properties Of Group Closures Of One-To-One Transformations., Inessa Levi

Faculty Bibliography

For a permutation group H on an infinite set X and a transformation / of X, let (/:// ) = ({hfh'1 : h € //)) be a group closure of / . We find necessary and sufficient conditions for distinct normal subgroups of the symmetric group on X and a one-to-one transformation / of X to generate distinct group closures of / . Amongst these group closures we characterize those that are left simple, left cancellative, idempotent-free semigroups, whose congruence lattice forms a chain and whose congruences are preserved under automorphisms.

A Platform-Independent Software Suite For Statistical Analysis Of High Dimensional Biology Data, David B. Allison, Jacob P. L. Brand, Jode W. Edwards, Gary L. Gadbury, Kyoungmi Kim, Tapan Mehta, Grier P. Page, Amit Patki, Vinodh Srinivasasainagendra, Prinal Trivedi, Jelai Wang, Stanislav O. Zakharkin

Mathematics and Statistics Faculty Research & Creative Works

Many efforts in microarray data analysis are focused on providing tools and methods for the qualitative analysis of microarray data. HDBStat! (High-Dimensional Biology-Statistics) is a software package designed for analysis of high dimensional biology data such as microarray data. It was initially developed for the analysis of microarray gene expression data, but it can also be used for some applications in proteomics and other aspects of genomics. HDBStat! provides statisticians and biologists a flexible and easy-to-use interface to analyze complex microarray data using a variety of methods for data preprocessing, quality control analysis and hypothesis testing.

Maximal Regular Boundary Value Problems In Banach-Valued Weighted Space, Ravi P. Agarwal, Veli B. Shakhmurov, Martin Bohner

Mathematics and Statistics Faculty Research & Creative Works

This study focuses on nonlocal boundary value problems for elliptic ordinary and partial differential-operator equations of arbitrary order, defined in Banach-valued function spaces. The region considered here has a varying bound and depends on a certain parameter. Several conditions are obtained that guarantee the maximal regularity and Fredholmness, estimates for the resolvent, and the completeness of the root elements of differential operators generated by the corresponding boundary value problems in Banach-valued weighted Lp spaces. These results are applied to nonlocal boundary value problems for regular elliptic partial differential equations and systems of anisotropic partial differential equations on cylindrical domain to …

Second Order Dynamic Inclusions, Christopher C. Tisdell, Martin Bohner

Mathematics and Statistics Faculty Research & Creative Works

The theory of dynamic inclusions on a time scale is introduced, hence accommodating the special cases of differential inclusions and difference inclusions. Fixed point theory for set-valued upper semicontinuous maps, Green's functions, and upper and lower solutions are used to establish existence results for solutions of second order dynamic inclusions.

Existence And Comparison Principles For General Quasilinear Variational-Hemivariational Inequalities, Siegfried Carl, Vy Khoi Le, Dumitru Motreanu

Mathematics and Statistics Faculty Research & Creative Works

We consider quasilinear elliptic variational-hemivariational inequalities involving convex, lower semicontinuous and locally Lipschitz functionals. We provide a generalization of the fundamental notion of sub- and supersolutions on the basis of which we then develop the sub-supersolution method for variational-hemivariational inequalities, including existence, comparison, compactness and extremality results.