Digital Commons Network^™

Multivalue Methods, Clayton V. Henrie

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

Methods of solving ordinary differential equations with initial conditions are of a great importance to engineers and scientists. Many of these equations can be solved by well-known analytical techniques, but a greater number of physically significant differential equations cannot be so solved. Thus, the solutions of these equations must be approximated numerically. It is the purpose of this paper to investigate the techniques used in solving differential equations with initial conditions by "multivalue methods."

An Evaluation Of Bartlett's Chi-Square Approximation For The Determinant Of A Matrix Of Sample Zero-Order Correlation Coefficients, Stephen M. Hattori

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

The single equation least-squares regression model has been extensively studied by economists and statisticians alike in order to determine the problems which arise when particular assumptions are violated. Much literature is available in terms of the properties and limitations of the model. However, on the multicollinearity problem, there has been little research, and consequently, limited literature is available when the problem is encountered. Farrar & Glauber (1967) present a collection of techniques to use in order to detect or diagnose the occurrence of multicollinearity within a regression analysis. They attempt to define multicollinearity in terms of departures from a hypothesized …

Distribution-Free Interval Estimation Of The Largest Α-Quantile, William E. Plouff

Masters Theses

No abstract provided.

Models For Estimating Psychiatric Bed Needs, Patricia A. Brenneman

Loma Linda University Electronic Theses, Dissertations & Projects

Inland Counties Comprehensive Health Planning Council wished to estimate the psychiatric bed needs by type (such as state hospitals, board and care homes, etc.) in San Bernardino County from the distributions of patient arrivals and of lengths of stay. A statistical description of the system during 1973 was considered the first step toward estimating future bed needs. A computer simulation model using IBM's programming language General Purpose Simulation System (GPSS) was developed and was found to be unsatisfactory.

Theorems were then developed for a statistical model, seeking to predict future bed needs. The most convenient of these theorems states that …

Matrix Fields Over The Integers Modulo M, Robert M. Mcconnel, Jacob T. B. Beard Jr.

Mathematics Technical Papers - Archive

Let Zm denote the ring of integers modulo m and let [see pdf for notation] denote the complete ring of all [see pdf for notation] matrices over Zm under the usual matrix addition and multiplication. The primary purposes of this paper are to characterize and count all subfields of the ring [see pdf for notation] . Related results are given on field extensions in [see pdf for notation] of subfields of [see pdf for notation] . Partial results on subfields of [see pdf for notation] when R is an arbitrary finite commutative ring with identity are also given. Having basic …

Measurable Choice And The Invariant Subspace Problem, Edward Azoff, Frank Gilfeather

Department of Mathematics: Faculty Publications

In [1], J. Dyer, A. Pedersen and P. Porcelli announced that an affirmative answer to the invariant subspace problem would imply that every reductive operator is normal. Their argument, outlined in [1], provides a striking application of direct integral theory. Moreover, this method leads to a general decomposition theory for reductive algebras which in turn illuminates the close relationship between the transitive and reductive algebra problems.
The main purpose of the present note is to provide a short proof of the technical portion of [1] : that invariant subspaces for the direct integrands of a decomposable operator can be assembled …

A Polynomial Dual Of Partitions, Jacob T. B. Beard, Ann D. Dorris

Mathematics Technical Papers - Archive

Let [see pdf for notation] be a non-negative integral polynomial. The polynomial P(x) is m-graphical, and a multi-graph G a realization of P(x), provided there exists a multi-graph G containing exactly P(1) points where [see pdf for notation] of these points have degree i for [see pdf for notation]. For multi-graphs G,H having polynomials P(x), Q(x) and number-theoretic partitions (degree sequences) [see pdf for notation], the usual product P(x)Q(x) is shown to be the polynomial of the Cartesian product [see pdf for notation], thus inducing a natural product [see pdf for notation] which extends that of juxtaposing integral multiple copies …

On The Method Of Vector Lyapunov Functions, V. Lakshmikantham

Mathematics Technical Papers - Archive

The method of vector Lyapunov functions consists of employing several Lyapunov-like functions and the theory of vectorial differential inequalities. This general comparison technic leads to a more flexible mechanism to study the qualitative properties of nonlinear differential systems. This method is also effective in discussing interconnected dynamical systems where a construction of a single Lyapunov function is difficult. However, an unpleasant fact in this approach is the requirement of quasimonotone property of the comparison system. An idea to get around this will be indicated which might be useful in applications. Also, an extension of this technic to abstract cones will …

On The Theory Of Hamiltonian Graphs, Linda M. Lesniak

Dissertations

No abstract provided.

The Automorphism Group Of The Wreath Product Of Finite Groups, Kenneth G. Hummel

Dissertations

No abstract provided.

Levi Structures For Polynomial Ideals., Richard Michael Grassl

Mathematics & Statistics ETDs

Let R be the ring of polynomials in a denumerable set of independent indeterminates over a field F and let I be an ideal (Xo,x1,x2, ... ) in R. A Levi structure for I provides bases for I and R as vector spaces over F and an associated algorithm for determining whether an element of R is in I.

Such structures have been developed previously for certain principal differential ideals [i.e., ideals (Xc),x1 , ... ) with xj the j-th derivative of Xol in which the generator is homogeneous and isobaric and for a family of related ordinary ideals. The …

Branching Processes With Cataclysmic Environmental Changes., Juan F. Corona-BurgueñO

Mathematics & Statistics ETDs

In this dissertation we consider a continuous time branching process with a random environment in which the environment changes according to a continuous time Markov chain. The extinction problem for this model is posed and solved by two distinct methods: by the method of random evolutions of Griego and Hersh and by the results of Athreya and Karlin on branching processes with random environments. Limit theorems for the population size as well as a system of partial differential equations for the expected number of particles (as functions of time) are obtained.

Finite Topologies And Boolean Matrices, Jon Michael Kelley

Morehead State Theses and Dissertations

A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in the Department of Mathematics at Morehead State University by Jon Michael Kelley in July of 1974.

Existence And Comparison Results For Differential Equations In A Banach Space, V. Lakshmikantham

Mathematics Technical Papers - Archive

The study of the Cauchy problem for differential equations in a Banach space has taken two different directions. One direction is to find compactness type conditions that guarantee only existence of solutions and the corresponding results are extensions of the classical Peano's theorem. The other approach is to employ monotonicity (accretive or dissipative) type conditions that assure existence as well as uniqueness of solutions. In fact, this latter method shows that uniqueness conditions imply existence of solutions also [37] and therefore may be regarded as extensions of the classical Picard's theorem.

Differential Equations On Closed Subsets Of A Banach Space, V. Lakshmikantham

Mathematics Technical Papers - Archive

The problem of existence of solutions to the initial value problem [see pdf for notations], where [see pdf for notations], F is a locally closed subset of a Banach space E. Nonlinear comparison functions and dissipative type conditions in terms of Lyapunovlike functions are employed. A new comparison theorem is established which helps in surmounting the difficulties that arise in this general set-up.

A Technic In Perturbation Theory, G. S. Ladde, S. Leela, V. Lakshmikantham

Mathematics Technical Papers - Archive

A study of the effect of perturbations of differential equations depends on the method employed and on the nature of perturbations. One of the most used technics is that of Lyapunov method and the other is the nonlinear variation of parameters formula [3]. These methods dictate that we measure the perturbations by means of a norm and thus destroy the ideal nature, if any, of the perturbing terms. Recently an effort was made to correct this unpleasant situation [1,2]. In this paper, we wish to develop a new comparison theorem that connects the solutions of perturbed and unperturbed differential systems …

On The Redundancy Of Monotony Assumption, V. Lakshmikantham

Mathematics Technical Papers - Archive

It is well known that all the results in integral inequalities of Bellman-Gronwall-Reid type demand an assumption of monotony on the functions involved. Since the corresponding theory of differential inequalities does not require the monotonic assumption, it is believed that this extra condition is due to the technics employed rather than the necessity. It is also well known that in proving the convergence of successive approximations, it becomes crucial to suppose an additional restriction of monotony on the functions satisfying uniqueness criteria. The question whether this additional assumption is really needed has been open for many years. This problem was …

The Numerical Solution Of The Generalized Eigenvalue Problem For Rectangular Matrices, Charles Henry Burris Jr.

Mathematics & Statistics ETDs

This paper studies the Generalized Eigenvalue Problem Ax=λBx for real, rectangular matrices A and B. Several current algorithms for solving this problem are examined, most of which involve a determination of the rank of B, or one of its submatrices. An example is given in which such a decision cannot be made without introducing unnecessary error into the problem. The QZR Algorithm is then introduced as an algorithm which uses unitary transformations to reduce the matrices to a prescribed canonical form. This canonical form provides the information necessary for making decisions about rank. Since A and B can be non …

Independence, Essential Independence And Zero Correlation Of Two Random Variables Conditioned On A Third., Chester Raymond Crain Jr.

Mathematics & Statistics ETDs

In a recent paper Stoughton Bell and William J. Zimmer defined three distinct types of conditional independence; which they call pointwise, intervalwise and strong; of two random variables given a third. They considered these and four closely related independence properties, and they examined the relationships among the 128 (=23+4) Boolean combinations. Bell and Zimmer also considered the kinds of assumptions in applied research that would give rise to each type of conditional independence.

I have searched published texts and papers in probabil­ity and statistics for the three types of conditional inde­pendence. The type considered by Alfred Renyi [Foundations of Probability …

Is Mathematical Truth Time-Dependent?, Judith V. Grabiner

Pitzer Faculty Publications and Research

Another such mathematical revolution occurred between the eighteenth and nineteenth centuries, and was focused primarily on the calculus. This change was a rejection of the mathematics of powerful techniques and novel results in favor of the mathematics of clear definitions and rigorous proofs. Because this change, however important it may have been for mathematicians themselves, is not often discussed by historians and philosophers, its revolutionary character is not widely understood. In this paper, I shall first try to show that this major change did occur. Then, I shall investigate what brought it about. Once we have done this, we can …

Maximal And Minimal Solutions And Comparison Results For Differential Equations In Abstract Cones, V. Lakshmikantham, Roger W. Mitchell, A. Richard Mitchell

Mathematics Technical Papers - Archive

As is well known, an important technique in the theory of differential equations is concerned with estimating a function satisfying a differential inequality by means of the extremal solutions of the corresponding differential equation. This comparison principle has been widely employed in studying the qualitative theory of differential equations (see [3]). If we desire to develop a similar comparison result in abstract spaces we must consider cones. These results could be of great value in applications to the theory of differential equations in abstract spaces. First, we must consider existence results for maximal and minimal solutions in cones which can …

Lyapunov-Like Vector Functions Using Pointwise Degenerate Systems As Comparison Functions, Bernard A. Asner, V. Lakshmikantham

Mathematics Technical Papers - Archive

The use of Lyapunov-like vector functions is recognized as an important tool for estimating the behavior of a dynamical system. In applications, one needs to determine a suitable comparison function which contains information or properties that can be used to obtain some information about the behavior of the dynamical system. In recent years, the remarkable property of pointwise degeneracy has been discovered for linear, autonomous, delay-differential equations. For this class of equations one knows that all solutions will reach a subspace in finite time and remain on the subspace thereafter. As this class of equations is particularly simple, it becomes …

On The Existence Of Solutions Of Differential Equations In A Banach Space, Jerome Eisenfeld, V. Lakshmikantham

Mathematics Technical Papers - Archive

The study of Cauchy problem for differential equations in a Banach space has taken two different directions. One approach is to find compactness type conditions [1,2,4,9,14] to guarantee existence of solutions only and the corresponding results are extensions of the classical Peano's Theorem. The other approach is to employ accretive type conditions [9,10,11,12,15] which assure existence as well as uniqueness of solutions. In fact, this latter technic shows that uniqueness conditions imply existence of solutions [16]. In this paper we follow the first direction. Employing Lyapunov-like functions and the notion of the measure of noncompactness, we prove a local existence …

Asymptotic Equilibrium Of Ordinary Differential Systems In A Banach Space, Roger W. Mitchell, A. Richard Mitchell

Mathematics Technical Papers - Archive

A differential system [see pdf for notation] where [see pdf for notation] has asymptotic equilibrium if 1) for any initial condition [see pdf for notation] the system has a solution [see pdf for notation] existing on and such that [see pdf for notation] exists and is finite, and 2) for any v e B there exists [see pdf for notation] and a solution x(t) of (1)-(2) with [see pdf for notation] Several papers have appeared dealing with asymptotic equilibrium of (1)-(2) when [see pdf for notation], and f is majorized by a scalar function g(t,u) which is monotone in u …

Fixed Point Theorems Through Abstract Cones, Jerome Eisenfeld, V. Lakshmikantham

Mathematics Technical Papers - Archive

A well-known theorem of Banach states that if T is a mapping on a complete metric space [see pdf for notation] such that for some number [see pdf for notation], the inequality (1.1) [see pdf for notation] holds, then T has a unique fixed point (i.e., a point u such that Tu = u). Extensions of this theorem [1,2] continue to require that T is a contraction i.e., (1.2) [see pdf for notation] This condition is essential if p is a metric but if p takes values in a partially ordered set k, then the condition (1.2) is avoidable. In …

Studies In Probability Theory., Ramanathan Subramaniam Dr.

Doctoral Theses

This thesis consists of four chapters.The inpetus for the work in Chapter 1 comes from the concept of 'conditional atom' introduced by ileveu (191. Here, using conditional atoms we generalize the concept of nenatomicity of measures. (We confine ourselves to probability measures). We obtain generalizations of results on non atomic measures in [1), [3] and of Liapounoff's theorem. The results in Chapter 2 have their origins in a paper by Boylan [7]. To study 'e quiconvergence of martingales' Boylan introduced in [7] a metric on the space of complete sub d-algebras of a probability space. A little later, Never showed …

Studies In The Theory Of Measurable Maltifunctions., Sheshi Mohan Srivastava Dr.

Doctoral Theses

During the last fifteen years a large number of papers have been devoted to the study of elosed set valued multifunetions. The studies were notivated by both the theoretical and spplica- tional interests that such multifunetions have. From the appli- cational point of view it is worth noting that such multifune- tions arise in varlous problems of control theory, dynanie progra- aning etc. The theoretieal'aspects of these studies belong properly to classieal deseriptive set theory. Classionl deserip- tive set theory asks que stions about how sets are constructed and about other definability properties of sets. The results on closed set …

Invariant Subspaces Of Vector Valued Function Spaces On Bohr Groups., Somesh Chandra Bagchi Dr.

Doctoral Theses

The theory of invariant subspaces of various Cunclion-spaces of complox-valued and vector-valued functions on the circle group 13 well known through the 'Loctures on Invariant Subspaces' by Helson ([4]), Replacin; Line circle croup by a Bohr group B (that is, a compact ahelian group whose dual is a subgroup of tho ronl. line R, dense in the topology of R), Helson and Lowdenalager initiated the study of invariant subspaces of L2,(B) in (6]. They discovered that after suitable normalisation, the simply invariant subspaces of L2(B) arcinonc-to-one correspondence with a certain class of functions on R x B, which are called …

On Perturbing Lyapunov Functions, S. Leela, V. Lakshmikantham

Mathematics Technical Papers - Archive

It is known [2,3] that in proving uniform boundedness of a differential system by means of Lyapunov functions, it is sufficient to impose conditions in the complement of a compact set in [see pdf for notation], whereas, in the case of equiboundedness, the proofs demand that the assumptions hold everywhere in [see pdf for notation]. We wish to present, in this paper, a new idea which permits us to discuss nonuniform properties of solutions of differential equations under weaker assumptions. Our results will show that the equiboundedness can be proved without assuming conditions everywhere in [see pdf for notation] (as …

The Order Of An Entire Function With Some Derivatives Univalent, S. M. Shah, S. Y. Trimble

Mathematics and Statistics Faculty Research & Creative Works

No abstract provided.