Physical Sciences and Mathematics Commons^™

Computer Assistance In Discovering Formulas And Theorems In System Engineering Ii, J. W. Helton, Mark Stankus, Kurt Schneider

Mathematics

[HSWcdc94] focused on procedures for simplifying complicated expressions automatically. [HScdc95] turned to the adventurous pursuit of developing a highly computer assisted method for “discovering” certain types of formulas and theorems.

It is often the case that some variables in the formulation of a problem are not the natural “coordinates” for solution of the problem. Gröbner Basis Algorithms, which lie at the core of our method, are very good at eliminating unknowns, but have no way of finding good changes of variables. This paper gives a way of incorporating changes of variables into our method.

As an example, we “discover” the …

M-Isometric Transformations Of Hilbert Space, Iii, Jim Alger, Mark Stankus

Mathematics

No abstract provided.

A Note On Distributions Of True Versus Fabricated Data, Theodore P. Hill

Research Scholars in Residence

New empirical evidence and statistical derivations of Benford’s Law have led to successful goodness-of fit tests to detect fraud in accounting data. Several recent case studies support the hypothesis that fabricated data does not conform to expected true digital frequencies.

A Stronger Triangle Inequality, Herb Bailey

Mathematical Sciences Technical Reports (MSTR)

The triangle inequality is basic for many results in real and complex analysis. The geometric form states that the sum of any two sides of a triangle is greater than the third. This was included as Proposition XX in the first book of Euclid's Elements. Many geometric triangle inequalities involving sides, angles, altitudes, inscribed circles and circumscribed circles have been found. Hundreds of these inequalities are summarized in [l] and [2]. A nice geometric proof of the triangle inequality is given in [3].

Note: On The Degree Of Lp Approximation With Positive Linear Operators, J. J. Swetits, B. Wood

Mathematics & Statistics Faculty Publications

The degree of approximation in L_p-spaces by positive linear operators is estimated in terms of the integral modulus of smoothness. It is shown that the conjectured optimal degree of approximation is not attained in the class of functions having a second derivative belonging to L_p.

Some Global Bifurcation Results For Variational Inequalities, Vy Khoi Le

Mathematics and Statistics Faculty Research & Creative Works

No abstract provided.

On The Homology Spectral Sequence For Topological Hochschild Homology, Thomas J. Hunter

Mathematics & Statistics Faculty Works

Marcel Bokstedt has computed the homotopy type of the topological Hochschild homology of Z/p using his definition of topological Hochschild homology for a functor with smash product. Here we show that easy conceptual proofs of his main technical result of are possible in the context of the homotopy theory of S-algebras as introduced by Elmendorf, Kriz, Mandell and May. We give algebraic arguments based on naturality properties of the topological Hochschild homology spectral sequence. In the process we demonstrate the utility of the unstable ''lower'' notation for the Dyer-Lashof algebra.

Automatic Realizability Of Galois Groups Of Order 16, Helen G. Grundman, Tara L. Smith

Mathematics Faculty Research and Scholarship

This article examines the realizability of small groups of order 2(k), k less than or equal to 4, as Galois groups over arbitrary fields of characteristic not 2. In particular we consider automatic realizability of certain groups given the realizability of others.

Existence And Bifurcation Of The Positive Solutions For A Semilinear Equation With Critical Exponent, Yinbin Deng, Yi Li

Mathematics and Statistics Faculty Publications

In this paper, we consider the semilinear elliptic equation[formula]Forp=2N/(N−2), we show that there exists a positive constantμ*>0 such that (∗)_μpossesses at least one solution ifμ∈(0, μ*) and no solutions ifμ>μ*. Furthermore, (∗)_μpossesses a unique solution whenμ=μ*, and at least two solutions whenμ∈(0, μ*) and 2<NN⩾6, under some monotonicity conditions onf((1.6)) we show that there exist two constants 0<μ_**⩽μ**<μ* such that problem (∗)_{μ …}

Computational Algebra Applications In Reliability, G Hartless, Lawrence Leemis

Arts & Sciences Articles

Reliability analysts are typically forced to choose between using an 'algorithmic programming language' or a 'reliability package' for analyzing their models and lifetime data. This paper shows that computational languages can be used to bridge the gap to combine the flexibility of a programming language with the ease of use of a package. Computational languages facilitate the development of new statistical techniques and are excellent teaching tools. This paper considers three diverse reliability problems that are handled easily with a computational algebra language: system reliability bounds; lifetime data analysis; and model selection.

Strongly-Consistent, Distribution-Free Confidence Intervals For Quantiles, David Gilat, Theodore P. Hill

Research Scholars in Residence

Strongly-consistent, distribution-free confidence intervals are derived to estimate the fixed quantiles of an arbitrary unknown distribution, based on order statistics of an iid sequence from that distribution. This new method, unlike classical estimates, works for totally arbitrary (including discontinuous) distributions, and is based on recent one-sided strong laws of large numbers.

Some Applications Of Sophisticated Mathematics To Randomized Computing, Ronald I. Greenberg

Computer Science: Faculty Publications and Other Works

No abstract provided.

Self-Consistency: A Fundamental Concept In Statistics, Thaddeus Tarpey, Bernard Flury

Mathematics and Statistics Faculty Publications

The term ''self-consistency'' was introduced in 1989 by Hastie and Stuetzle to describe the property that each point on a smooth curve or surface is the mean of all points that project orthogonally onto it. We generalize this concept to self-consistent random vectors: a random vector Y is self-consistent for X if E[X|Y] = Y almost surely. This allows us to construct a unified theoretical basis for principal components, principal curves and surfaces, principal points, principal variables, principal modes of variation and other statistical methods. We provide some general results on self-consistent random variables, give …

Positive Solutions For A Semilinear Elliptic Problem With Critical Exponent, Ismail Ali, Alfonso Castro

All HMC Faculty Publications and Research

No abstract provided in article.

Strong Laws For L- And U-Statistics, J. Aaronson, R. Burton, H. Dehling, D. Gilat, Theodore P. Hill, B. Weiss

Research Scholars in Residence

Strong laws of large numbers are given for L-statistics (linear combinations of order statistics) and for U-statistics (averages of kernels of random samples) for ergodic stationary processes, extending classical theorems of Hoeffding and of Helmers for iid sequences. Examples are given to show that strong and even weak convergence may fail if the given sufficient conditions are not satisfied, and an application is given to estimation of correlation dimension of invariant measures.

A Combinatorial Interpretation Of Lommel Polynomials And Their Derivatives, Philip J. Feinsilver, John P. Mcsorley, René Schott

Articles and Preprints

In this paper we present interpretations of Lommel polynomials and their derivatives. A combinatorial interpretation uses matchings in graphs. This gives an interpretation for the derivatives as well. Then Lommel polynomials are considered from the point of view of operator calculus. A step-3 nilpotent Lie algebra and finite-difference operators arise in the analysis.

A Mathematician Among The Molasses Barrels: Maclaurin's Unpublished Memoir On Volumes, Judith V. Grabiner

Pitzer Faculty Publications and Research

Suppose we are given a solid of revolution generated by a conic section. Slice out a frustum of the solid [14, diagrams pp. 77, 80]. Then, construct a cylinder, with the same height as the frustum, whose diameter coincides with the diameter of the frustum at the midpoint of its height. What is the difference between the volume of the frustum and the volume of this cylinder? Does this difference depend on where in the solid the frustum is taken?

The beautiful theorems which answer these questions first appear in a 1735 manuscript by Colin Maclaurin (1698–1746). This …

Wheels On Wheels On Wheels-Surprising Symmetry, Frank A. Farris

Mathematics and Computer Science

While designing a computer laboratory exercise for my calculus students, I happened to sketch the curve defined by this vector equation: (x, y) = (cos(t), sin(t)) + 1/2(cos(7t), sin(7t)) + 1/3(sin(17t), cos(17t)). I was thinking of the curve traced by a particle on a wheel mounted on a wheel mounted on a wheel, each turning at a different rate. The first term represents the largest wheel, of radius 1, turning counter-clockwise at one radian per second. The second term represents a smaller wheel centered at the edge of the first, turning 7 times as fast. The third term is for …

Cwatsets: Weights, Cardinalities, And Generalizations, Richard Mohr

Mathematical Sciences Technical Reports (MSTR)

This report provides an upper bound on the average weight of an element in a cwatset and discusses the ratio of the cardinality of a cwatset to the cardinality of the group containing the cwatset. The concept of a generalized cwatset is also introduced.

Σary, Moorhead State University, Mathematics Department

Math Department Newsletters

No abstract provided.

On Quaternionic Pseudo-Random Number Generators, Gary R. Greenfield

Department of Math & Statistics Technical Report Series

There is no dearth of published literature on the design, implementation, analysis, or use of pseudo-random number generators or PRNGs. For example, [6] [7] [14] and the references therein, provide a broad overview and firm grounding for the subject. This report complements and elaborates upon the work of McKeever [9], who investigated PRNGs constructed in a non-commutative setting with the target application being so-called cryptographically secure PRNGs as discussed in [12] or [13]. Novel "solutions" to the problem of designing cryptographically secure PRNGS continue to be proposed [1] [2] [10] [15], so despite the caution and skepticism required, the area …

The Exponential Stability Of A Coupled Hyperbolic/Parabolic System Arising In Structural Acoustics, George Avalos

Department of Mathematics: Faculty Publications

We show here the uniform stabilization of a coupled system of hyperbolic and parabolic PDE’s which describes a particular fluid/structure interaction system. This system has the wave equation, which is satisfied on the interior of a bounded domain Ω, coupled to a “parabolic–like” beam equation holding on ∂Ω, and wherein the coupling is accomplished through velocity terms on the boundary. Our result is an analog of a recent result by Lasiecka and Triggiani which shows the exponential stability of the wave equation via Neumann feedback control, and like that work, depends upon a trace regularity estimate for solutions of hyperbolic …

Droplet Evaporation And Deformations In An Amplitude Modulated Ultrasonic Field, Nihad E. Daidzic, Rene Stadler, Adrian Melling

Aviation Department Publications

The aim of the report presented is the measurements of droplet oscillations.

Imsa Math Journal: A Resource Notebook For High School Mathematics, Illinois Mathematics And Science Academy

IMSA Math Journal

We are proud to present the fourth issue of the IMSA Math Journal. In this edition we have included work by IMSA faculty, support staff, students, and alumni as well as articles by a faculty member of a sister school, The North Carolina School of Science and Mathematics and a colleague at Eastern Illinois University.

Excerpt: Letter from the Editors

Stability Analysis Of A Model For The Defect Structure Of Yba2cu3ox, Gregory Kozlowski, Tom Svobodny

Physics Faculty Publications

Unusual microstructures of YBa₂Cu₃O_x (123) crystals have been observed. These structures have been shown to pass very high transport currents. A model of the solidification of 123 from a melt with Y₂BaCuO₅ (211) inclusions indicates that the stability of the 123 interface can depend on the sizes of the 211 inclusions. The observed formations are interpreted in the light of this instability.

Determinants Of The Tournaments, Clifford A. Mccarthy '94, Arthur T. Benjamin

All HMC Faculty Publications and Research

No abstract provided in this article.

An Efficient Spectral Method For Ordinary Differential Equations With Rational Function Coefficients, Evangelos A. Coutsias, Thomas Hagstrom, David Torres

Branch Mathematics and Statistics Faculty and Staff Publications

We present some relations that allow the efficient approximate inversion of linear differential operators with rational function coefficients. We employ expansions in terms of a large class of orthogonal polynomial families, including all the classical orthogonal polynomials. These families obey a simple 3-term recurrence relation for differentiation, which implies that on an appropriately restricted domain the differentiation operator has a unique banded inverse. The inverse is an integration operator for the family, and it is simply the tridiagonal coefficient matrix for the recurrence. Since in these families convolution operators (i.e., matrix representations of multiplication by a function) are banded for …

Function Approximation Using A Sinc Neural Network, Wael R. Elwasif, Laurene V. Fausett

Mathematics and System Engineering Faculty Publications

Neural networks for function approximation are the basis of many applications. Such networks often use a sigmoidal activation function (e.g. tanh) or a radial basis function (e.g. gaussian). Networks have also been developed using wavelets. In this paper, we present a neural network approximation of functions of a single variable, using sinc functions for the activation functions on the hidden units. Performance of the sinc network is compared with that of a tanh network with the same number of hidden units. The sinc network generally learns the desired input-output mapping in significantly fewer epochs, and achieves a much lower total …

Divergence Diagrams: More Than Cantor Dust Lies At The Edge Of Feigenbaum Diagrams, John H. Rickert, Aaron Klebanoff

Mathematical Sciences Technical Reports (MSTR)

The dynamical system analysis of the logistic map f(x)=ax(1-x) is studied for values of a greater than 4.

Combinatorics Of Open Covers (I): Ramsey Theory, Marion Scheepers

Mathematics Faculty Publications and Presentations

We study several schemas for generating from one sort of open cover of a topological space a second sort of open cover. Some of these schemas come from classical literature, others are borrowed from the theory of ultrafilters on the set of positive integers. We show that the fact that such a schema actually succeeds in producing a cover imposes strong combinatorial structure on the family of open covers of a certain sort. In particular, we show that certain analogues of Ramsey’s theorem characterize some of these circumstances.