Physical Sciences and Mathematics Commons^™

Note: On Summability And Positive Linear Operators, J. J. Swetits

Mathematics & Statistics Faculty Publications

Quantitative estimates for approximation by positive linear operators are obtained with the use of a summability method which includes both convergence and almost convergence.

A Semilinear Dirichlet Problem, Alfonso Castro

All HMC Faculty Publications and Research

Let Ω be a bounded region in R^n. In this note we discuss the existence of weak solutions (see [4, Section 2]) of the Dirichlet problem:

Δu(x) + g(x, u(x)) + f(x, u(x), ∇u(x)) = 0 ; x є Ω

u(x) = 0 ; x є ∂Ω

where Δ is the Laplacian operator, g : Ω x R → R and f : Ω x Rⁿ⁺¹ → R are functions satisfying the Caratheodory condition (see [2, Section 3]), and ∇ is the gradient operator.

On The Application Of Coding Theory To Hashing, Nicholas Pippenger

All HMC Faculty Publications and Research

Quick proofs are given for the characterization (due to Schay, Raver, Hanan, and Palermo) of the collision distance of a linear hashing function and for a dual function (called the restriction distance), which relates to the accessibility of addresses by sets of keys and the uniform distribution of sets of keys over addresses.

A Summary Of Results On Order-Cauchy Completions Of Rings And Vector Lattices Of Continuous Functions, Melvin Henriksen

All HMC Faculty Publications and Research

This paper is a summary of joint research by F. Dashiell, A. Hager and the present author. Proofs are largely omitted. A complete version will appear in the Canadian Journal of Mathematics. It is devoted to a study of sequential order-Cauchy convergence and the associated completion in vector lattices of continuous functions. Such a completion for lattices C(X) is related to certain topological properties of the space X and to ring properties of C(X). The appropriate topological condition on the space X equivalent to this type of completeness for the lattice C(X) was first identified for compact spaces X in …

Optimal 2,3-Trees, Nicholas J. Pippenger, Raymond E. Miller, Arnold L. Rosenberg, Lawrence Snyder

All HMC Faculty Publications and Research

The 2,3-trees that are optimal in the sense of having minimal expected number of nodes visited per access are characterized in terms of their “profiles”. The characterization leads directly to a linear-time algorithm for constructing a K-key optimal 2,3-tree for a sorted list of K keys. A number of results are derived that demonstrate how different in structure these optimal 2,3-trees are from their “average” cousins.

On The Sampling Distribution Of Entropy Function, Pan Fu-Charo

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

Since the introduction of Shannon's Entropy Function in a non-probabilistic setting [l], there has been an expectancy that it will provide a useful measure for the fuzzy cluster validity problem. We hope to demonstrate here (empirically) by Monte Carlo simulation that the entropy Hc(U) of fuzzy c-partitions (U) defined below has expectaction

E(H_c(U)) = ∑^c_k=2 1/k

and variance

Var(H_c(U)) = 1/n(∑^c_k=2 1/k₂ - (c-1)/(c+1)(π²/6) - 1).

Furthermore, for sufficiently large n, that H is approximately normal.

In this report we propose a method which generates the required fuzzy matrix …

An Arithmetic Computer Approach To Gas Dynamical Modeling, Donald Greenspan, Aspi Rustom Wadia

Mathematics Technical Papers

Unsteady, two dimensional internal and external flows are analyzed using an arithmetic n-body formulation. A Lagrangian approach is used to study the internal shock formation in a shock tube and the external flows over a circular cyclinder, a flat plate and an airfoil. Viable physical mechanisms of the shock and boundary layer formation and of flow separation are developed. This approach results in a computer simulation of a wind tunnel in which arbitrarily shaped bodies can be studied. Extensive computer examples are described for which agreement with experimental data is good for all cases in which the damping of the …

Parameter Estimation In Illness-Death Processes: Preinfaractional Angina, Jerome Eisenfeld, B. Canada

Mathematics Technical Papers

An illness-death process is an absorbing Markov process where the illness states from the transient states of the process. This paper deals with the estimation of illness-death parameters from survivor data. The parameters of interest include the probabilities of recovery, relapse and death and mean residence times. The estimation problem is considered from the point of view of the theory of structural identifiability in compartmental analysis. As a prototypical example, we consider published data on preinfarctional angina.

Laminations, Or How To Build A Quantum-Logic-Valued Model Of Set Theory, Lawrence Stout

Scholarship

An explicit construction of the colimit of a filtered diagram in the category of topoi and logical morphisms is given and then used to construct a family of topoi with a fixed Boolean algebra of truth values but with varying amounts of cocompleteness. This same construction, when applied to the diagram of complete Boolean algebras in a quantum logic Q gives a partial topos, a noncategory which is a close to being a model of set theory with algebra of truth values Q as a noncategory can be.

The Total Negation Of A Topological Property, Paul Bankston

Mathematics, Statistics and Computer Science Faculty Research and Publications

No abstract provided.

On The Comparison Of Estimators In A Rectangular Distribution, Danny D. Dyer

Mathematics Technical Papers

Given two estimators ^1 and ^2 of an unknown parameter ^, Pitman (1937) has suggested calling ^1 a "closer" estimator of ^ than is ^2 if [see pdf for notations] for all values of ^. Using this "closeness" criterion, Johnson (1950) compared several well known estimators of ^ based on a random sample from the rectangular distribution [see pdf for notations], otherwise. In addition, each estimator's mean squared error was given since Pitman's criterion does not take into account the degree of closeness. Unfortunately when two such estimators of ^ are compared, one of them is often preferred under the …

An Alternative Semantics For Knowledge, Gregory Mellema

University Faculty Publications and Creative Works

No abstract provided.

Laminations, Or How To Build A Quantum-Logic-Valued Model Of Set Theory, Lawrence Stout

Lawrence N. Stout

An explicit construction of the colimit of a filtered diagram in the category of topoi and logical morphisms is given and then used to construct a family of topoi with a fixed Boolean algebra of truth values but with varying amounts of cocompleteness. This same construction, when applied to the diagram of complete Boolean algebras in a quantum logic Q gives a partial topos, a noncategory which is a close to being a model of set theory with algebra of truth values Q as a noncategory can be.

Band Matrices With Toeplitz Inverses, William F. Trench, T N. E. Greville

William F. Trench

No abstract provided.

An Investigation Of Montmort's "Probleme De Recontres" And Generalizations, Ronald I. Greenberg

Computer Science: Faculty Publications and Other Works

I have investigated a problem which may be phrased in many ways, such as finding the probability of answering a given number of questions correctly on a randomly-completed matching test which may have a number of extra "dud" answers. I have determined such probabilities, the average number of correct answers, and other allied results. I have also investigated a related problem involving the number of ways of choosing a different element from each of a certain collection of sets.

Existence And Comparison Results For Differential Equations Of Sobolev Type, Randy Vaughn, A. S. Vatsala

Mathematics Technical Papers

Recently a new class of differential equations, called differential equations of Sobolev type was studied in [4] in which an existence theorem of Picards type was investigated as well as a variation of constants formula. In [5], existence and comparison results for a class of Volterra integral equation of Sobolev-type were discussed. In this paper we recall the Peano type existence result from [5] for Sobolev-differential equations and show that solutions can be extended to the entire square under consideration. This result extends results found in [1] for nonlinear Volterra integral equations. Our results include a comparison result in addition …

Note On Special Relativistic Calculations With Identical Laboratory And Rocket Frame Computers, Donald Greenspan

Mathematics Technical Papers

Dynamical relativistic problems invariably require the solution of nonlinear differential equations. These equations are rarely solvable in closed form and are now being solved numerically on modern digital computers. In this paper, a numerical method is developed which has the special property that computations on identical computers in both the laboratory and rocket frames yield numerical results which continue to be related by the Lorentz transformation.

Scs 49: A Review Of A Theorem Of Dixmier's, Karl Heinrich Hofmann, Fred Watkins

Seminar on Continuity in Semilattices

No abstract provided.

Scs 48: Projective Limits In Cl And Scott's Construction (Comp. Iii-3), Karl Heinrich Hofmann, Jaime Niño

Seminar on Continuity in Semilattices

No abstract provided.

An Illustrative Model Describing The Refraction Of Long Water Waves By A Circular Island, Gregory A. Kriegsmann

Department of Mathematics: Faculty Publications

The refraction of small shallow water waves by an idealized island is studied in this paper. The island's shoal is modeled by a quartic polynomial in the radial variable. This particular model allows the explicit construction of the rays (wave orthogonals) and the determination of several important features of the wave motion. The various shortcomings of the particular profile are discussed.

N-Body Modeling Of Nonlinear, Free Surface Liquid Flow, Donald Greenspan

Mathematics Technical Papers

In this paper a new, n-body approach is developed for the modeling of liquid phenomena. Both long and short range forces are included. Computer examples of several prototype, nonlinear, free surface problems are described and discussed.

Groups Expressed As The Set-Theoretic Union Of Proper Subgroups, Lana Barrett

Masters Theses & Specialist Projects

In their paper “Groups as Unions of Proper Subgroups,” published in 1959, Haber and Rosenfeld posed the problem of characterizing groups which can be expressed as the set theoretic union of a finite collection of their proper subgroups. Haber and Rosenfeld established that a group G is the union of three proper subgroups (such a group G shall be called a U₃-group) if and only if the Klein 4-group is a homomorphic image of G. Further results concerning U₃-groups were obtained in the paper “Groups which are the Union of Three Subgroups” by Bruckhemier, Bryan and …

Second Leray Spectral Sequence Of Relative Hypercohomology, Saul Lubkin, Goro C. Kato

Mathematics

A second Leray spectral sequence of relative hypercohomology is constructed. (This is skew in generality to an earlier one constructed by S. Lubkin [(1968) Ann. Math. 87, 105-255].) The Mayer-Vietoris sequence of relative hypercohomology [Lubkin, S. (1968) Ann. Math. 87, 105-255] is also generalized.

Non-Splitting Unitary Perfect Polynomials Over Gf(Q)T, Mickie Sue Harbin, Jacob T. B. Beard Jr.

Mathematics Technical Papers

It has been conjectured that there are infinitely many distinct pd-equivalence classes of non-splitting unitary perfect polynomials over GF(pd) for each prime p and each odd integer d > 1. The conjecture is proved in the affirmative in the cases i) p < 97, ii) 2 ^ GF(p) is not a square, iii) 2 ^ GF(p) is a square and all of the positive integer intervals determined by distinct odd powers of ^t contain a square, where GF*(p) = (^). In addition, it has been determined that iii) is satisfied by 314 primes p > 97.2

Mathematical Models Of Porous Flow, Donald Greenspan

Mathematics Technical Papers

In this paper a new, viable mathematical approach to the analysis of porous flow is developed. Liquids and solids are modeled as sets of particles which interact in accordance with both long range (gravity) and short range (molecular type) forces. The resulting nonlinear, n-body problem is resolved numerically on a digital computer. A variety of fundamental porous flow type problems, solved on the University of Texas at Arlington IBM 370-155, are described and analyzed.

Existence And Comparison Results For A Class Of Volterra Integral Equations Of Sobolev Type, A. S. Vatsala, V. Lakshmikantham, Randy Vaughn

Mathematics Technical Papers

Recently [2] a class of new differential equations called Sobolev type differential equations was introduced and Picard type existence theorem for such equations was proved among other results like variation of constants formula and Gronwall type inequalities for such equations. However Peano's type existence result was open for such equations since it was not clear how to proceed. Special cases of such equations occur in imbedding method for solving Fredholm integral equations [5]. In this paper we consider Volterra integral equations of Sobolev type which include the foregoing class of differential equations as special cases. We consider Peano's type existence …

A Systems-Theory Approach To The Analysis Of Multiexponential Fluorescence Decay, Corey C. Ford, Jerome Eisenfeld

Mathematics Technical Papers

A mathematical model of the fluorescence decay experiment based on linear systems theory is presented. The model suggests an experimental technique which increases the probability of correctly determining the decay constants of a multicomponent system. Using moment methods for data analysis,accuracy is improved by combining information obtained from several discrete experiments. Examples are presented to show that the analysis of a three component system composed of known standards is improved as the number of experimental determinations is increased from 1 to 4. The discrete measurements are made by changing the excitation and emission wavelengths.

On A System Of Nonlinear Ordinary Differential Equations With An Irregular Type Singularity: A Degenerate Case, Jerome John Przybylski

Dissertations

No abstract provided.

Existence And Comparison Theorems For Differential Equations In Banach Spaces, V. Lakshmikantham, K. Deimling

Mathematics Technical Papers

In our recent paper [3] we have studied the existence of maximal and minimal solutions to the IVP in a Banach space [see pdf for notation]. (1) [see pdf for notation] where [see pdf for notation] maps [see pdf for notation] into [see pdf for notation], with [see pdf for notation] and [see pdf for notation] a cone. The essential hypotheses have been that [see pdf for notation]f is quasimonotone with respect to [see pdf for notation] and that [see pdf for notation] and [see pdf for notation] have some natural properties. If such extremal solutions exist then it is …

On Existence Of Extremal Solutions Of Differential Equations In Banach Spaces, V. Lakshmikantham, K. Deimling

Mathematics Technical Papers

Let X be a real Banach space, [see pdf for notation] a cone, [see pdf for notation] and [see pdf for notation] continuous. We look for conditions on X, K and f such that the IVP (1) [see pdf for notation] has a maximal solution [see pdf for notation] and a minimal solution u with respect to the partial ordering induced by K. Contrary to known results, [5,6], we shall not assume that K has interior points, since the standard cones of many infinite dimensional spaces have empty interior. The second essential new feature is that f is supposed to …