Harmonic Analysis and Representation Commons^™

Unions Of Lebesgue Spaces And A1 Majorants, Greg Knese, John E. Mccarthy, Kabe Moen

Mathematics Faculty Publications

We study two questions. When does a function belong to the union of Lebesgue spaces, and when does a function have an A1 majorant? We provide a systematic study of these questions and show that they are fundamentally related. We show that the union ofLwp(ℝn)spaces withw∈Apis equal to the union of all Banach function spaces for which the Hardy–Littlewood maximal function is bounded on the space itself and its associate space.

Tessellations: An Artistic And Mathematical Look At The Work Of Maurits Cornelis Escher, Emily E. Bachmeier

Honors Program Theses

The purpose of this study was to learn more about the mathematics of tessellations and their artistic potential. Whenever I have seen tessellations, I always admire them. It is baffling how such complicated shapes can be repeated infinitely on the plane. This research aimed to increase the amount of tessellation information and activities available to secondary mathematics teachers by connecting the tessellations of Maurits Cornelis Escher with their underlying mathematics in order to use them for teaching secondary mathematics. The overarching premise of this study was to create something that would also be beneficial in my career as a secondary …

Filters And Matrix Factorization, Myung-Sin Song, Palle E. T. Jorgensen

SIUE Faculty Research, Scholarship, and Creative Activity

We give a number of explicit matrix-algorithms for analysis/synthesis

in multi-phase filtering; i.e., the operation on discrete-time signals which

allow a separation into frequency-band components, one for each of the

ranges of bands, say N , starting with low-pass, and then corresponding

filtering in the other band-ranges. If there are N bands, the individual

filters will be combined into a single matrix action; so a representation of

the combined operation on all N bands by an N x N matrix, where the

corresponding matrix-entries are periodic functions; or their extensions to

functions of a complex variable. Hence our setting entails …

Reproducing Kernel Hilbert Space Vs. Frame Estimates, Palle E. T. Jorgensen, Myung-Sin Song

SIUE Faculty Research, Scholarship, and Creative Activity

We consider conditions on a given system F of vectors in Hilbert space H, forming a frame, which turn Hinto a reproducing kernel Hilbert space. It is assumed that the vectors in F are functions on some set Ω . We then identify conditions on these functions which automatically give H the structure of a reproducing kernel Hilbert space of functions on Ω. We further give an explicit formula for the kernel, and for the corresponding isometric isomorphism. Applications are given to Hilbert spaces associated to families of Gaussian processes.

The Structure And Unitary Representations Of Su(2,1), Andrew J. Pryhuber

Honors Projects

No abstract provided.

Numerical Solution For The Systems Of Variable-Coefficient Coupled Burgers’ Equation By Two-Dimensional Legendre Wavelets Method, Hossein Aminikhah, Sakineh Moradian

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, a numerical method for solving the systems of variable-coefficient coupled Burgers’ equation is proposed. The method is based on two-dimensional Legendre wavelets. Two-dimensional operational matrices of integration are introduced and then employed to find a solution to the systems of variable-coefficient coupled Burgers’ equation. Two examples are presented to illustrate the capability of the method. It is shown that the numerical results are in good agreement with the exact solutions for each problem.

A New Subgroup Chain For The Finite Affine Group, David Alan Lingenbrink Jr.

HMC Senior Theses

The finite affine group is a matrix group whose entries come from a finite field. A natural subgroup consists of those matrices whose entries all come from a subfield instead. In this paper, I will introduce intermediate sub- groups with entries from both the field and a subfield. I will also examine the representations of these intermediate subgroups as well as the branch- ing diagram for the resulting subgroup chain. This will allow us to create a fast Fourier transform for the group that uses asymptotically fewer opera- tions than the brute force algorithm.

Fast Algorithms For Analyzing Partially Ranked Data, Matthew Mcdermott

HMC Senior Theses

Imagine your local creamery administers a survey asking their patrons to choose their five favorite ice cream flavors. Any data collected by this survey would be an example of partially ranked data, as the set of all possible flavors is only ranked into subsets of the chosen flavors and the non-chosen flavors. If the creamery asks you to help analyze this data, what approaches could you take? One approach is to use the natural symmetries of the underlying data space to decompose any data set into smaller parts that can be more easily understood. In this work, I describe …

Eccentricity, Space Bending, Dimension, Florentin Smarandache, Marian Nitu, Mircea Eugen Selariu

Branch Mathematics and Statistics Faculty and Staff Publications

The main goal of this paper is to present new transformations, previously non-existent in traditional mathematics, that we call centric mathematics (CM) but that became possible due to the new born eccentric mathematics, and, implicitly, to the supermathematics (SM).

As shown in this work, the new geometric transformations, namely conversion or transfiguration, wipe the boundaries between discrete and continuous geometric forms, showing that the first ones are also continuous, being just apparently discontinuous.

On Local Fractional Continuous Wavelet Transform, Yang Xiaojun

Xiao-Jun Yang

We introduce a new wavelet transform within the framework of the local fractional calculus. An illustrative example of local fractional wavelet transform is also presented.

Local Fractional Discrete Wavelet Transform For Solving Signals On Cantor Sets, Yang Xiaojun

Xiao-Jun Yang

The discrete wavelet transform via local fractional operators is structured and applied to process the signals on Cantor sets. An illustrative example of the local fractional discretewavelet transformis given.

Characterization Of The Drilling Via The Vibration Augmenter Of Rotary-Drills And Sound Signal Processing Of Impacted Pipe As A Potential Water Height Assessment Tool, Nicholas Morris

STAR Program Research Presentations

The focus of the internship has been on two topics: a) Characterize the drilling performance of a novel percussive augmenter – this drill was developed by the JPL’s Advanced Technologies Group and its performance was characterized; and b) Examine the feasibility of striking a pipe as a means of assessing the water height inside the pipe. The purpose of this investigation is to examine the possibility of using a simple method of applying impacts to a pipe wall and determining the water height from the sonic characteristic differences including damping, resonance frequencies, etc. Due to multiple variables that are relevant …

Stability Of Multiwavelet Frames With Different Matrix Dilations And Matrix Translations, F. A. Shah, Sunita Goyal

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we study the stability of multiwavelet frames with different matrix dilations and matrix translations by means of operator theory and show that these frames remain stable over some kinds of perturbations of the basic generators.

Mathematical Aspects Of Heisenberg Uncertainty Principle Within Local Fractional Fourier Analysis, Yang Xiaojun

Xiao-Jun Yang

In this paper, we discuss the mathematical aspects of the Heisenberg uncertainty principle within local fractional Fourier analysis. The Schrödinger equation and Heisenberg uncertainty principles are structured within local fractional operators.

Early Investigations In Conformal And Differential Geometry, Raymond T. Walter

Raymond Walter

Shannon-Like Wavelet Frames On A Class Of Nilpotent Lie Groups, Vignon Oussa

Vignon Oussa

We construct Shannon-like Parseval frame wavelets on a class of non commutative two-step nilpotent Lie groups. Our work was inspired by a construction given by Azita Mayeli on the Heisenberg group. The tools used here are representation theoretic. However, a great deal of Gabor theory is used for the construction of the wavelets. The construction obtained here is very explicit, and we are even able to compute an upper bound for the L2 norm for these Parseval frame wavelets.

On The Lp-Error Of Approximation Of Bivariate Functions By Harmonic Splines, Yuliya Babenko, Tatyana Leskevich

Faculty and Research Publications

Interpolation by various types of splines is the standard procedure in many applications. In this paper we discuss harmonic spline “interpolation” (on the lines of a grid) as an alternative to polynomial spline interpolation (at vertices of a grid). We will discuss some advantages and drawbacks of this approach and present the asymptotics of the L_p-error for adaptive approximation by harmonic splines.

Linear Independence Of A Finite Set Of Dilations By A One-Parameter Matrix Lie Group, Vignon Oussa

Vignon Oussa

No abstract provided.

Supercharacters, Exponential Sums, And The Uncertainty Principle, J.L. Brumbaugh '13, Madeleine Bulkow '14, Patrick S. Fleming, Luis Alberto Garcia '14, Stephan Ramon Garcia, Gizem Karaali, Matt Michal '15, Andrew P. Turner '14

Pomona Faculty Publications and Research

The theory of supercharacters, which generalizes classical character theory, was recently introduced by P. Diaconis and I.M. Isaacs, building upon earlier work of C. Andre. We study supercharacter theories on $(Z/nZ)^d$ induced by the actions of certain matrix groups, demonstrating that a variety of exponential sums of interest in number theory (e.g., Gauss, Ramanujan, and Kloosterman sums) arise in this manner. We develop a generalization of the discrete Fourier transform, in which supercharacters play the role of the Fourier exponential basis. We provide a corresponding uncertainty principle and compute the associated constants in several cases.

Ramanujan Sums As Supercharacters, Christopher F. Fowler '12, Stephan Ramon Garcia, Gizem Karaali

Pomona Faculty Publications and Research

The theory of supercharacters, recently developed by Diaconis-Isaacs and Andre, can be used to derive the fundamental algebraic properties of Ramanujan sums. This machinery frequently yields one-line proofs of difficult identities and provides many novel formulas. In addition to exhibiting a new application of supercharacter theory, this article also serves as a blueprint for future work since some of the abstract results we develop are applicable in much greater generality.