Physical Sciences and Mathematics Commons^™

Normal Forms For Two-Inputs Nonlinear Control Systems, Issa Amadou Tall, Witold Respondek

Miscellaneous (presentations, translations, interviews, etc)

We study the feedback group action on two-inputs non-linear control systems. We follow an approach proposed by Kang and Krener which consists of analyzing the system and the feedback group step by step. We construct a normal form which generalizes that obtained in the single-input case. We also give homogeneous m-invariants of the action of the group of homogeneous transformations on the homogeneous systems of the same degree. We illustrate our results by analyzing the normal form and invariants of homogeneous systems of degree two.

Resolvability In Graphs, Varaporn Saenpholphat

Dissertations

The distance d (u, v ) between two vertices u and v in a connected graph G is the length of a shortest u - v path in G . For an ordered set W = { w1 , w2 , [Special characters omitted.] &cdots; , wk } of vertices in G and a vertex v of G , the code of v with respect to W is the k -vector cW (v ) = (d ( v, w1 ),d (v, w 2 ), [Special characters omitted.] &cdots; , d …

Detectability Of Excitatory Versus Inhibitory Drive In An Integrate-And-Fire-Or-Burst Thalamocortical Relay Neuron Model, Gregory D. Smith, S. M. Sherman

Arts & Sciences Articles

Although inhibitory inputs are often viewed as equal but opposite to excitatory inputs, excitatory inputs may alter the firing of postsynaptic cells more effectively than inhibitory inputs. This is because spike cancellation produced by an inhibitory input requires coincidence in time, whereas an excitatory input can add spikes with less temporal constraint. To test for such potential differences, especially in the context of the function of thalamocortical (TC) relay nuclei, we used a stochastic “integrate-and-fire-or-burst” TC neuron model to quantify the detectability of excitatory and inhibitory drive in the presence and absence of the low-threshold Ca 2+ current, IT, and …

Asymptotic Solitons For A Higher-Order Modified Korteweg–De Vries Equation, T. Marchant

Tim Marchant

Solitary wave interaction for a higher-order modified Korteweg–de Vries (mKdV) equation is examined. The higher-order mKdV equation can be asymptotically transformed to the mKdV equation, if the higher-order coefficients satisfy a certain algebraic relationship. The transformation is used to derive the higher-order two-soliton solution and it is shown that the interaction is asymptotically elastic. Moreover, the higher-order phase shifts are derived using the asymptotic theory. Numerical simulations of the interaction of two higher-order solitary waves are also performed. Two examples are considered, one satisfies the algebraic relationship derived from the asymptotic theory, and the other does not. For the example …

An Empirical Study Of Marginal Structural Models For Time-Independent Treatment, Tanya A. Henneman, Mark J. Van Der Laan

U.C. Berkeley Division of Biostatistics Working Paper Series

In non-randomized treatment studies a significant problem for statisticians is determining how best to adjust for confounders. Marginal structural models (MSMs) and inverse probability of treatment weighted (IPTW) estimators are useful in analyzing the causal effect of treatment in observational studies. Given an IPTW estimator a doubly robust augmented IPTW (AIPTW) estimator orthogonalizes it resulting in a more e±cient estimator than the IPTW estimator. One purpose of this paper is to make a practical comparison between the IPTW estimator and the doubly robust AIPTW estimator via a series of Monte- Carlo simulations. We also consider the selection of the optimal …

Fixed Point And Two-Cycles Of The Discrete Logarithm, Joshua Holden

Mathematical Sciences Technical Reports (MSTR)

We explore some questions related to one of Brizolis: does every prime p have a pair (g, h) such that h is a fixed point for the discrete logarithm with base g? We extend this question to ask about not only fixed points but also two-cycles. Campbell and Pomerance have not only answered the fixed point question for sufficiently large p but have also rigorously estimated the number of such pairs given certain conditions on g and h. We attempt to give heuristics for similar estimates given other conditions on g and h and also in the case …

A Review Of Selected Works On Crack Indentification, Kurt M. Bryan

Mathematical Sciences Technical Reports (MSTR)

We give a short survey of some of the results obtained within the last 10 years or so concerning crack identification using impedance imaging techniques. We touch upon uniqueness results, continuous dependence results, and computational algorithms.

Pattern Recognition For Electric Power System Protection, Yong Sheng

Doctoral Dissertations

The objective of this research is to demonstrate pattern recognition tools such as decision trees (DTs) and neural networks that will improve and automate the design of relay protection functions in electric power systems. Protection functions that will benefit from the research include relay algorithms for high voltage transformer protection (TP) and for high impedance fault (HIF) detection. A methodology, which uses DTs and wavelet analysis to distinguish transformer internal faults from other conditions that are easily mistaken for internal faults, has been developed. Also, a DT based solution is proposed to discriminate HIFs from normal operations that may confuse …

A Mathematical Model Of Bacterial Aerotaxis, Borbala Mazzag, Alex Mogilner

Borbala Mazzag

No abstract provided.

Characterizing A Defect In A One-Dimensional Bar, Cynthia Gangi, Sameer Shah

Mathematical Sciences Technical Reports (MSTR)

We examine the inverse problem of locating and describing an internal point defect in a one­ dimensional rod W by controlling the heat inputs and measuring the subsequent temperatures at the boundary of W. We use a variation of the forward heat equation to model heat flow through W, then propose algorithms for locating an internal defect and quantifying the effect the defect has on the heat flow. We implement these algorithms, analyze the stability of the procedures, and provide several computational examples.

Reconstruction Of Cracks With Unknown Transmission Condition From Boundary Data, F Ronald Ogborne Iii, Melissa E. Vellela

Mathematical Sciences Technical Reports (MSTR)

We examine the problem of Identifying both the location and constitutive law governing electrical current flow across a one-dimensional linear crack in a two dimensional region when the crack only partially blocks the flow of current. We develop a a constructive numerical procedure for solving the inverse problem and provide computational examples.

Bivariate Current Status Data, Mark J. Van Der Laan, Nicholas P. Jewell

U.C. Berkeley Division of Biostatistics Working Paper Series

In many applications, it is often of interest to estimate a bivariate distribution of two survival random variables. Complete observation of such random variables is often incomplete. If one only observes whether or not each of the individual survival times exceeds a common observed monitoring time C, then the data structure is referred to as bivariate current status data (Wang and Ding, 2000). For such data, we show that the identifiable part of the joint distribution is represented by three univariate cumulative distribution functions, namely the two marginal cumulative distribution functions, and the bivariate cumulative distribution function evaluated on the …

A Posteriori Error Estimates Based On Polynomial Preserving Recovery, Zhimin Zhang, Ahmed Naga

Mathematics Research Reports

Superconvergence of order O(h^1+rho), for some rho is greater than 0, is established for gradients recovered using Polynomial Preserving Recovery technique when the mesh is mildly structured. Consequently this technique can be used in building a posteriori error estimator that is asymptotically exact.

Matroid Duality From Topological Duality In Surfaces Of Nonnegative Euler Characteristic, Dan Slilaty

Mathematics and Statistics Faculty Publications

Let G be a connected graph that is 2-cell embedded in a surface S, and let G* be its topological dual graph. We will define and discuss several matroids whose element set is E(G), for S homeomorphic to the plane, projective plane, or torus. We will also state and prove old and new results of the type that the dual matroid of G is the matroid of the topological dual G*.

The Liouville-Bratu-Gelfand Problem For Radial Operators, Jon T. Jacobsen, Klaus Schmitt

All HMC Faculty Publications and Research

We determine precise existence and multiplicity results for radial solutions of the Liouville–Bratu–Gelfand problem associated with a class of quasilinear radial operators, which includes perturbations of k-Hessian and p-Laplace operators.

Analysis Of The N-Card Version Of The Game Le Her, Arthur T. Benjamin, Alan J. Goldman

All HMC Faculty Publications and Research

We present a complete solution to a card game with historical origins. Our analysis exploits the convexity properties in the payoff matrix, allowing this discrete game to be resolved by continuous methods.

Finite Extinction Time For Non-Linear Absorption-Diffusion Equations, Jacob B. Goldberg

Theses and Dissertations

In this thesis, we develop a numerical method in order to approximate the solutions of one-dimensional, non-linear absorption-diffusion equations. We test our method for accuracy against a linear diffusion equation with a solution that can be written in closed form. We then test various types of diffusion and absorption terms to determine which ones produce extinction in finite time. We also develop a numerical method to computationally solve diffusion-free equations. We compare the numerical solutions of the one-dimensional, non-linear absorption-diffusion equation and the diffusion-free equation and we find that for the cases tested, the numerical absorption-diffusion solutions are always less …

Comparing Clustering Algorithms For Use With Genomic And Proteomic Data, Rebecca A. Olson

Theses and Dissertations

The Human Genome Project and related projects have resulted in the development of a number of new experimental and analytic tools for use in genomic and proteomic research. In the area of toxicogenomics, researchers are concerned with how genes react to exposure to certain chemicals. The United States Air Force is interested in the effect of exposure to mission-essential chemicals. Although military personnel may come into contact with chemicals such as hydrazine, risk assessment is usually very limited. On the genomic level, risk assessment is a multi-stop and multi-disciplinary process. The process begins with an experiment that exposes cells to …

Gradient Recovery And A Posteriori Estimate For Bilinear Element On Irregular Quadrilateral Meshes, Zhimin Zhang

Mathematics Research Reports

A polynomial preserving gradient recovery method is proposed and analyzed for bilinear element under general quadrilateral meshes. It has been proven that the recovered gradient converges at a rate O(h^{1+rho) for rho = min(alpha, 1) when the mesh is distorted O(h1+alpha) (alpha > 0) from a regular one. Consequently, the a posteriori error estimator based on the recovered gradient is asymptotically exact.}

Fast Reconstruction Of Cracks Using Boundary Measurements, Nicholas A. Trainor, Rachel M. Krieger

Mathematical Sciences Technical Reports (MSTR)

This paper develops a fast algorithm for locating one or more perfectly insulating, pair-wise disjoint, linear cracks in a homogeneous two-dimensional electrical conductor, using boundary measurements.

Analysis Of Recovery Type A Posteriori Error Estimators For Mildly Structured Grids, Jinchao Xu, Zhimin Zhang

Mathematics Research Reports

Some recovery type error estimators for linear finite element method are analyzed under O(h^{1+alpha) (alpha greater than 0) regular grids. Superconvergence is established for recovered gradients by three different methods when solving general non-self-adjoint second-order elliptic equations. As a consequence, a posteriori error estimators based on those recovery methods are asymptotically exact.}

The Order At 2 Of The Odd-Partition Function, Chris Mehelich

Mathematical Sciences Technical Reports (MSTR)

Abstract. We evaluate the odd-partition function p₂(n) modulo 4 by elementary methods and analyze the asymptotic distribution of p₂(n) modulo 4. We use the theory of modular forms to obtain necessary and sufficient conditions for the order at 2 of p₂(n) to equal any given value between 0 and 4 inclusive.

Statistical Properties Of Composite Distortions In Hfc Systems And Their Effects On Digital Channels, Ron D. Katznelson

Ron D. Katznelson

The statistical properties of Composite Triple Beat (CTB) and Composite Second Order (CSO) distortion terms associated with the analog channels carried on Hybrid Fiber Coax (HFC) are presented. Both simulation and measurement results show that such distortion components falling on individual channels have amplitude Probability Density Function that is nearly Rayleigh distributed (having a Standard Deviation of 5.7 dB). It is shown that both CTB and CSO components have peak envelope power fluctuations that exceed their average (measured) power levels by more than 15 dB. The temporal statistical properties of these distortion components demonstrate peak envelope power fluctuations with characteristic …

On The Existence Of Multiple Positive Solutions For A Semilinear Problem In Exterior Domains, Yinbin Deng, Yi Li

Yi Li

In this paper, we study the existence and nonexistence of multiple positive solutions for problem where Ω=^N\ω is an exterior domain in ^N, ω⊂^N is a bounded domain with smooth boundary, and N>2. μ⩾0, p>1 are some given constants. K(x) satisfies: K(x)∈C^α_loc(Ω) and ∃C, ϵ, M>0 such that |K(x)|⩽C |x|^l for any |x|⩾M, with l⩽ −2−ϵ. Some existence and …

On The Existence Of Multiple Positive Solutions For A Semilinear Problem In Exterior Domains, Yinbin Deng, Yi Li

Mathematics and Statistics Faculty Publications

In this paper, we study the existence and nonexistence of multiple positive solutions for problem where Ω=^N\ω is an exterior domain in ^N, ω⊂^N is a bounded domain with smooth boundary, and N>2. μ⩾0, p>1 are some given constants. K(x) satisfies: K(x)∈C^α_loc(Ω) and ∃C, ϵ, M>0 such that |K(x)|⩽C |x|^l for any |x|⩾M, with l⩽ −2−ϵ. Some existence and …

Nonlinear Dynamics Of Mode-Locking Optical Fiber Ring Lasers, Kristin M. Spaulding, Darryl H. Yong, Arnold D. Kim, J Nathan Kutz

All HMC Faculty Publications and Research

We consider a model of a mode-locked fiber ring laser for which the evolution of a propagating pulse in a birefringent optical fiber is periodically perturbed by rotation of the polarization state owing to the presence of a passive polarizer. The stable modes of operation of this laser that correspond to pulse trains with uniform amplitudes are fully classified. Four parameters, i.e., polarization, phase, amplitude, and chirp, are essential for an understanding of the resultant pulse-train uniformity. A reduced set of four coupled nonlinear differential equations that describe the leading-order pulse dynamics is found by use of the variational nature …

Nonlinear Schrödinger Type Equations With Asymptotically Linear Terms, Francois A. Van Heerden

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

We study the nonlinear Schrödinger type equation

–Δu + (λg(x) + 1)u = f(u)

on the whole space R^N. The nonlinearity f is assumed to be asymptotically linear and g(x) ≥ 0 has a potential well. We do not assume a limit for g(x) as |x| → ∞ . Using variational techniques, we prove the existence of a positive solution for λ large. In the case where f is odd we obtain multiple pairs of solutions. The limiting behavior of solutions as λ → ∞ is also considered.

Tournament Matrices An Overview, Russel O. Carlson

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

The results of a round robin tournament can be represented as a matrix of zeros and ones, by ordering the players and placing a one in the (i,j) position if player i beat player j, and zeros otherwise. These matrices, called tournament matrices, can be represented by graphs, called tournament graphs. They have been the subject of much research and study, yet there have been few attempts to give a wide exposition on the subject. Those that have been done tend to focus on the graph theoretical aspects of tournaments. S. Ree and Y. Koh did write a brief …

Mixed Partial Derivatives And Fubini's Theorem, Asuman Güven Aksoy, Mario Martelli

CMC Faculty Publications and Research

A most fascinating aspect of calculus is its power to surprise even an experienced mathemat ician. Just when it appears that all ideas, results and connections have been discovered and thorough ly analyzed, the horizon suddenly broadens and somebody cries the familiar "eureka". The reason could be either a new result, a simpler way to prove an existing theorem, or a previously missed connection between different ideas. This potential for enrichment is second to none, and it reaffirms the unparalleled educational value of this area of mathematics.

An Analysis Of Mutually Dispersive Brown Symbols For Non-Linear Ambiguity Suppression, Matthew J. Papaphotis

Theses and Dissertations

This thesis significantly advances research towards the implementation of optimal Non-linear Ambiguity Suppression (NLS) waveforms by analyzing the Brown theorem. The Brown theorem is reintroduced with the use of simplified linear algebraic notation. A methodology for Brown symbol design and digitization is provided, and the concept of dispersive gain is introduced. Numerical methods are utilized to design, synthesize, and analyze Brown symbol performance. The theoretical performance in compression and dispersion of Brown symbols is demonstrated and is shown to exhibit significant improvement compared to discrete codes. As a result of this research a process is derived for the design of …