Mathematics Commons^™

Evaluating Maximum Likelihood Estimation Methods To Determine The Hurst Coefficient, Christina Marie Kendziorski, J. B. Bassingthwaighte, Peter J. Tonellato

Mathematics, Statistics and Computer Science Faculty Research and Publications

A maximum likelihood estimation method implemented in S-PLUS (S-MLE) to estimate the Hurst coefficient (H) is evaluated. The Hurst coefficient, with 0.5<HS-MLE was developed to estimate H for fractionally differenced (fd) processes. However, in practice it is difficult to distinguish between fd processes and fractional Gaussian noise (fGn) processes. Thus, the method is evaluated for estimating H for both fd and fGn processes. S-MLE gave biased results of H for fGn processes of any length and for fd processes of lengths less than 2¹⁰. A modified method is proposed to correct for …

Generalized Averages For Solutions Of Two-Point Dirichlet Problems, Philip Korman, Yi Li

Mathematics and Statistics Faculty Publications

For very general two-point boundary value problems we show that any positive solution satisfies a certain integral relation. As a consequence we obtain some new uniqueness and multiplicity results.

Recounting Fibonacci And Lucas Identities, Arthur T. Benjamin, Jennifer J. Quinn

All HMC Faculty Publications and Research

No abstract provided in this article.

Peak-To-Mean Power Control In Ofdm, Golay Complementary Sequences, And Reed–Muller Codes, James A. Davis, Jonathan Jedwab

Department of Math & Statistics Faculty Publications

We present a range of coding schemes for OFDM transmission using binary, quaternary, octary, and higher order modulation that give high code rates for moderate numbers of carriers. These schemes have tightly bounded peak-to-mean envelope power ratio (PMEPR) and simultaneously have good error correction capability. The key theoretical result is a previously unrecognized connection between Golay complementary sequences and second-order Reed–Muller codes over alphabets ℤ_2^h. We obtain additional flexibility in trading off code rate, PMEPR, and error correction capability by partitioning the second-order Reed–Muller code into cosets such that codewords with large values of PMEPR are isolated. …

Intersecting Chains In Finite Vector Spaces, Eva Czabarka

Faculty Publications

We prove an Erdős–Ko–Rado-type theorem for intersecting k-chains of subspaces of a finite vector space. This is the q-generalization of earlier results of Erdős, Seress and Székely for intersecting k-chains of subsets of an underlying set. The proof hinges on the author's proper generalization of the shift technique from extremal set theory to finite vector spaces, which uses a linear map to define the generalized shift operation. The theorem is the following.

For c = 0, 1, consider k-chains of subspaces of an n-dimensional vector space over GF(q), such that the smallest subspace in any chain has dimension at least …

Random Fluctuations Of Convex Domains And Lattice Points, Alex Iosevich, Kimberly Kinateder

Mathematics and Statistics Faculty Publications

In this paper, we examine a random version of the lattice point problem.

Axiomatic Approach For Quantification Of Image Resolution, Ge Wang, Yi Li

Mathematics and Statistics Faculty Publications

Image resolution is the primary parameter for performance characterization of any imaging system. In this work, we present an axiomatic approach for quantification of image resolution, and demonstrate that a good image resolution measure should be proportional to the standard deviation of the point spread function of an imaging system.

Splitting Tiled Surfaces With Abelian Conformal Tiling Group, Sean A. Broughton

Mathematical Sciences Technical Reports (MSTR)

Let p be a reflection on a closed Riemann Surface S, i.e., an anti-conformal involutary isometry of S with a non-empty fixed point subset. Let Sp denote the fixed point subset of p, which is also called the mirror of p. If S −Sp has two components, then p is called separating and we say that S splits at the mirror Sp. Otherwise p is called non-separating. We assume that the system of mirrors, Sq, as q varies over all reflections in the isometry group Aut*(S) defines a tiling of the surface, consisting of triangles. In turn, the tiling determines …

Dini-Campanato Spaces And Applications To Nonlinear Elliptic Equations, Jay Kovats

Mathematics and System Engineering Faculty Publications

We generalize a result due to Campanato [C] and use this to obtain regularity results for classical solutions of fully nonlinear elliptic equations. We demonstrate this technique in two settings. First, in the simplest setting of Poisson's equation Delta u=f in B, where f is Dini continuous in B, we obtain known estimates on the modulus of continuity of second derivatives D2u in a way that does not depend on either differentiating the equation or appealing to integral representations of solutions. Second, we use this result in the concave, fully nonlinear setting F(D^2u,x)=f(x) to obtain estimates on the modulus of …

Stochastic Functional Differential Equations On Manifolds (Conference On Probability And Geometry), Salah-Eldin A. Mohammed

Miscellaneous (presentations, translations, interviews, etc)

We prove an existence theorem for solutions of stochastic functional differential equations under smooth constraints in Euclidean space. The initial states are semimartingales on a compact Riemannian manifold. It is shown that, under suitable regularity hypotheses on the coefficients, and given an initial semimartingale, a sfde on a compact manifold admits a unique solution living on the manifold for all time. We also discuss the Chen-Souriau regularity of the solution of the sfde in the initial process. The results are joint work with Remi Leandre.

Stability Of Self-Similar Solutions For Van Der Waals Driven Thin Film Rupture, Thomas P. Witelski, Andrew J. Bernoff

All HMC Faculty Publications and Research

Recent studies of pinch-off of filaments and rupture in thin films have found infinite sets of first-type similarity solutions. Of these, the dynamically stable similarity solutions produce observable rupture behavior as localized, finite-time singularities in the models of the flow. In this letter we describe a systematic technique for calculating such solutions and determining their linear stability. For the problem of axisymmetric van der Waals driven rupture (recently studied by Zhang and Lister), we identify the unique stable similarity solution for point rupture of a thin film and an alternative mode of singularity formation corresponding to annular “ring rupture.”

On The Semivalues And The Power Core Of Cooperative Tu Games, Juan-Enrique Martínez-Legaz, Irinel C. Dragan

Mathematics Technical Papers - Archive

The Semivalues were introduced axiomatically by P.Dubey, A.Neyman and R.J.Weber (1981) as an important class of values for cooperative TU games. This class contains the Shapley value, the Banzhaf value, and many other values. For the Shapley value characterizations of games for which the Shapley value is coalitionally rational are due to Inarra and Usategui (1993), Izawa and Takahashi (1998), and Marin-Solano and Rafels (1999). In this paper the same problem of coalitional rationality is discussed for Semivalues, by using special formulas for the computation of Semivalues. The characterization shows that this is a prosperity property as defined by Van …

Codes Over Rings From Curves Of Higher Genus, José Felipe Voloch, Judy L. Walker

Department of Mathematics: Faculty Publications

We construct certain error-correcting codes over finite rings and estimate their parameters. These codes are constructed using plane curves and the estimates for their parameters rely on constructing “lifts” of these curves and then estimating the size of certain exponential sums.

THE purpose of this paper is to construct certain error-correcting codes over finite rings and estimate their parameters. For this purpose, we need to develop some tools; notably, an estimate for the dimension of trace codes over rings (generalizing work of van der Vlugt over fields and some results on lifts of affin curves from field of characteristic p …

A New Family Of Relative Difference Sets In 2-Groups, James A. Davis, Jonathan Jedwab

Department of Math & Statistics Faculty Publications

We recursively construct a new family of (2^6d+4, 8, 2^6d+4, 2^6d+1) semi-regular relative difference sets in abelian groups G relative to an elementary abelian subgroup U. The initial case d = 0 of the recursion comprises examples of (16, 8, 16, 2) relative difference sets for four distinct pairs (G, U).

Variational Principles For Average Exit Time Moments For Diffusions In Euclidean Space, Kimberly Kinateder, Patrick Mcdonald

Mathematics and Statistics Faculty Publications

Let D be a smoothly bounded domain in Euclidean space and let X_t be a diffusion in Euclidean space. For a class of diffusions, we develop variational principles which characterize the average of the moments of the exit time from D of a particle driven by X_t, where the average is taken overall starting points in D.

Almost Periodic Factorization Of Certain Block Triangular Matrix Functions, Ilya M. Spitkovsky, Darryl H. Yong

All HMC Faculty Publications and Research

Let

where , and . For rational such matrices are periodic, and their Wiener-Hopf factorization with respect to the real line always exists and can be constructed explicitly. For irrational , a certain modification (called an almost periodic factorization) can be considered instead. The case of invertible and commuting , was disposed of earlier-it was discovered that an almost periodic factorization of such matrices does not always exist, and a necessary and sufficient condition for its existence was found. This paper is devoted mostly to the situation when is not invertible but the commute pairwise (). The complete description is …

Divisible Tilings In The Hyperbolic Plane, Sean A. Broughton, Dawn M. Haney, Lori T. Mckeough, Brandy M. Smith

Mathematical Sciences Technical Reports (MSTR)

We consider triangle-quadrilateral pairs in the hyperbolic plane which "kaleidoscopically" tile the plane simultaneously. In this case the tiling by quadrilaterals is called a divisible tiling. All possible such divisible tilings are classified. There are a finite number of 1,2, and 3 parameter families as well as a finite number of exceptional cases.

Mathematical Modelling Of Extracellular Matrix Dynamics Using Discrete Cells: Fiber Orientation And Tissue Regeneration, J. C. Dallon, J. A. Sherratt, P. K. Maini

Faculty Publications

Matrix orientation plays a crucial role in determining the severity of scar tissue after dermal wounding. We present a model framework which allows us to examine the interaction of many of the factors involved in orientation and alignment. Within this framework, cells are considered as discrete objects, while the matrix is modeled as a continuum. Using numerical simulations, we investigate the effect on alignment of changing cell properties and of varying cell interactions with collagen and fibrin.

Openness And Monotoneity Of Induced Mappings, W. J. Charatonik

Mathematics and Statistics Faculty Research & Creative Works

It is shown that for locally connected continuum X if the induced mapping C(f) : C(X) ->C(Y) is open, then f is monotone. As a corollary it follows that if the continuum X is hereditarily locally connected and C(f) is open, then f is a homeomorphism. An example is given to show that local connectedness is essential in the result.

Dynamics And Asymptotic Behavior Of The Solutions Of A Nonlinear Differential Equation, Robert Palmer

Masters Theses & Specialist Projects

Initial value problems of the form dx/dt= t xP, x(a) = β are examined, first when p = 2. Applying Euler's method, a numerical approximation technique, when p = 2 for certain initial conditions produces a numerical solution which resembles a bifurcation diagram very similar to that produced by the logistic map. Comparisons of such numerical solutions to the logistic map are made, and a partial explanation of such numerical solutions is given. Then, the exact solution of the initial value problem with p = 2, for which the software package Mathematica 3.0 determines an explicit formula, is analyzed to …

A Stochastic Analog To The Richardson's Arms Race Model, John Fricks

Masters Theses & Specialist Projects

In this thesis, a stochastic version of the Richardson's arms race model is developed through the method of birth-death processes. The expected value of the model is explored and shown to be analogous to the original deterministic arms race model. The numerical method of randomization is then expanded and applied to the stochastic model. A comparison is then made between outcomes of the deterministic and stochastic models.

Positive Solutions To Semilinear Problems With Coefficient That Changes Sign, Nguyen Phuong Cac, Juan A. Gatica, Yi Li

Mathematics and Statistics Faculty Publications

No abstract provided.

Essays In Mechanism Design., Suresh Mutuswami Dr.

Doctoral Theses

The theory of implementation or mechanism design had its origins in the debates in the 1930s between Hayek, Lange and Lerner on the informational efficiency of the market economy. However, it was the work of Hurwicz in the 1950s and the 1960s which formalised the insights of Hayek, Lange and Lerner and paved the way for the body of work that followed his pioneering effort.In addition to the considerable theoretical literature on mechanism de- sign'. there also exists a body of literature which uses the mechanism design approach to address specific problems. Some examples of work in this vein include …

An Ellam Scheme For Advection-Diffusion Equations In Two Dimensions, Hong Wang, Helge K. Dahle, Richard E. Ewing, Magne S. Espedal, Robert Sharpley, Shushuang Man

Faculty Publications

We develop an Eulerian--Lagrangian localized adjoint method (ELLAM) to solve two-dimensional advection-diffusion equations with all combinations of inflow and outflow Dirichlet, Neumann, and flux boundary conditions. The ELLAM formalism provides a systematic framework for implementation of general boundary conditions, leading to mass-conservative numerical schemes. The computational advantages of the ELLAM approximation have been demonstrated for a number of one-dimensional transport systems; practical implementations of ELLAM schemes in multiple spatial dimensions that require careful algorithm development are discussed in detail in this paper. Extensive numerical results are presented to compare the ELLAM scheme with many widely used numerical methods and to …

Math Is A Subject That Is The Best, Alexandra Holliday

Humanistic Mathematics Network Journal

No abstract provided.

Middle School Math Teaching And How It Harms Our Children, Jerome Dancis

Humanistic Mathematics Network Journal

No abstract provided.

Changing The Subject, Or, Would You Hire A Good Clarinet Teacher To Teach Your Child The Violin?, Jerome Dancis

Humanistic Mathematics Network Journal

No abstract provided.

Is Mathematics Education Taking A Step Backward?, Frances Kuwahara Lang

Humanistic Mathematics Network Journal

This article addresses the current political, socioeconomic, and educational state of mathematics education in California. The “back-to-basics” movement in mathematics mirrors the “back-to-phonics” movement in language arts. At a time when ethnic minorities have become the majority, the dominant culture has chosen to revert back to practices that are inequitable and empower the elite. Critical educators must carry on the dialogue necessary to empower the disenfranchised mathematically and undermine the social injustice and economic inequality that will result if this movement is embraced.

A Window Into The Life Of Ramanujan, Marlene Neff

Humanistic Mathematics Network Journal

No abstract provided.

Numbers, Shea Ybarra

Humanistic Mathematics Network Journal

No abstract provided.