Physical Sciences and Mathematics Commons^™

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

Mathematics Technical Papers

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 …

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

Mathematics Technical Papers

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 …

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

Mathematics Technical Papers

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

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.

A Note On T, Topologies, Wilson R. Crisler, Troy L. Hicks

Mathematics and Statistics Faculty Research & Creative Works

Let t be a T. topology for a set X. The problem of representing t as the lattice product (intersection) of stronger topologies is considered. © 1974 American Mathematical Society.

Lipschitz Spaces On The Surface Of Unit Sphere In Euclidean N-Space, Harvey Greenwald

Mathematics

This paper is concerned with defining Lipschitz spaces on Σn-1 the surface of the unit sphere in Rn. The importance of this example is that Σn-1 is not a group but a symmetric space. One begins with functions in Lp(Σn-1),1≤p≤∞. Σn-1 is a symmetric space and is related in a natural way to the rotation group SO(n). One can then use the group SO(n) to define first and second differences for functions in Lp(Σn-1). Such a function is the boundary value of its Poisson integral. This enables one to work with functions which are harmonic. Differences can then be replaced …

Stability And Transversality, Robert D. May

Mathematics & Computer Science Faculty Publications

No abstract provided.

Maintaining Topological Properties On The Brink Of Destruction, Kay J. Hamm

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

Throughout the paper we consider the setting where f is a continuous function (a mapping) whose domain X and range Y are both Hausdorff spaces. Our object is to determine conditions on the map f which insure that when X has a certain topological property Q, then Y will also have property Q. For example, if X is metrizable, then it does not necessarily follow that Y is a metric space; but if f is a perfect map, then metrizability is preserved. Chapter III is devoted to the study of this metrizability problem. In particular, we present Frink's [ 2] …

Functional Analysis Techniques In Numerical Analysis, Kenneth D. Schoenfeld

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

In this paper we will consider the problem of selecting the best, or optimal, numerical method of solution to a given mathematical problem. The admissible numerical methods will be a clearly defined set for each problem. Obviously, in order to find the best method in this set, we must have a clear mathematical formulation of just what "best" means; this will be the intent of Theorem 0.1. Intuitively, the best method will be understood to be the one which minimizes the maximum possible error where this error will be measured in terms of the norm of a given Hilbert space. …

Solvability Of Differential Systems Near Singular Points, Leon M. Hall

Doctoral Dissertations

"Functional analysis techniques are used to prove a theorem, analogous to the Harris-Sibuya-Weinberg theorem for ordinary differential equations, which yields as corollaries a number of existence theorems for holomorphic solutions of linear functional differential systems of the form z^Dy'(z) = A(z)y(z) + B(z)y(αz) + C(z)y'(αz) in the neighborhood of the singularity at z = 0"--Abstract, page 2.

Faces, Edges, Vertices Of Some Polyhedra, Charles H. Harbison

Journal of the Arkansas Academy of Science

A proof that: for any given polyhedron so shaped that every closed non-self intersecting broken line composed of edges of the polyhedron divides the surface of the polyhedron into precisely two disjoint regions each of which is bounded by the closed broken line, v - e + f = 2, where v is the number of vertices of the polyhedron, e the number of edges and f the number of faces.

Fields Medals And Nevanlinna Prize Presented At Icm-94 In Zurich, Nicholas Pippenger, J. Lindenstrauss, L.C. Evans, A. Douady, A. Shalev

All HMC Faculty Publications and Research

The Notices solicited the following five articles describing the work of the Fields Medalists and Nevanlinna Prize winner.

On Interaction In The Two-Way Aov Model Without Replication Including Power Comparisons Of Two Tests For Non-Additivity, Victor John Hegemann

Doctoral Dissertations

"Data analysts have long been concerned with the problem of giving a complete analysis of the two-way crossed classification design with one observation per cell when interaction is present in the data. Conventional tests of hypotheses and statistical inferences are not applicable in this case since an estimate of the error variance is not available. Transformation of the data to additive data has been one solution to this problem; however, interpretation of the physical units of the transformed data is difficult. Recent contributions to the analysis of this design include a test for interaction, a test for testing differences in …

Intuitive Concepts In Elementary Topology, Gary Rothwell

Honors Theses

My hour special study in intuitive topology originated in a curiosity of what exactly topology was and how it might be related to physics, my field of interest. The book I used was, Intuitive Concepts in Elementary Topology, by B.H. Arnold. This book is designed as a sophomore-junior level three hour course. Needless to say, I didn't quite cover the whole book in an hour a week. I mainly stuck to the intuitive concepts. Intuitive topology is dealing with more physical objects where the point set topology involves set theory; their unions, intersections and subsets.

On Integrability And L¹ Convergence Of Certain Cosine Sums, John William Garrett

Doctoral Dissertations

"Rees and Stanojevic altered the standard cosine series and obtained a necessary and sufficient condition for integrability of the altered sum. Here these sums are generalized, and it is shown that such sums converge in L¹-norm"--Abstract, page iv.

Linear Geometry, Phyllis L. Thomas

Masters Theses

"This paper contains material suitable for a one semester course in linear geometry where the student has had a one semester course in linear algebra previously. It compiles information in the areas of affine sets, affine geometry, convex sets, and a few applications. Also included are Caratheodory's theorem and the well-known theorem by Helly as proved by Radon"--Abstract, page ii.

Methods And Applications Of The Mean-Variance Portfolio Selection Model, John W. Marsh

Doctoral Dissertations

"The Mean-Variance portfolio selection model, or Efficient Market model, is examined in terms of the small investor. The performance is first tested on the small sample space of the thirty Dow Jones Industrials. The results show that it is possible to outperform the market by investing in the minimum-variance, or safest, portfolio. The Critical-Line algorithm as developed by Markowitz and modified by Sharpe is used in this analysis.

Since the Critical-Line algorithm is very time-consuming and does not always converge to a solution, an alternate algorithm is developed. This algorithm, referred to as the “Simplified Algorithm”, is designed to find …

Graphs With 1-Factors, David Sumner

Faculty Publications

In this paper it is shown that if G is a connected graph of order 2n (n > 1) not containing a 1-factor, then for each k, 1

The Completions Of Local Rings And Their Modules., Christopher Scott Taber

Masters Theses

"This paper gives a generally self-contained introduction to the completions of local rings and their modules. The '.Fbrmal Power Series, the Ideal-adic, and the Projective Limit completions are established, and the conditions under which they are equivalent are also given. In addition, two methods of obtaining the completion of an R-module from the completion of the ring Rare supplied. As an ideal of R may also be taken as an R-module, a discussion of obtaining its completion from the completed ring is also provided. Two well-known examples are furnished in this paper"--- Abstract, p. ii

Canonical Forms And Principal Systems For General Disconjugate Equations, William F. Trench

William F. Trench

No abstract provided.

Truncated Circular Normal Distribution With Applications In Ballistics And Meteorology, Danny D. Dyer

Mathematics Technical Papers

The use of the circular normal distribution (CND) to describe the behavior of random phenomena of a geophysical nature has been discussed by Crutcher [7, p. 9]. Based on the Mauchly [17] - Hsu [12] test, the harmonic dial points (the Fourier coefficients obtained from harmonic analysis of observations of periodic phenomena) representing i.) lunar semidiurnal atmospheric tides (Chapman and Lindzen [5, p. 66]), ii.) the westerly component of wind in the study of tidal oscillations in the upper atmosphere (Haurwitz [10]), and iii.) terrestrial-magnetic activity relative to solar activity (Bartels [2]) may be treated as observations from a CND. …

Existence Md Estimates For Solutions Of Nonlinear Equations Near A Branch Point, Jerome Eisenfeld, V. Lakshmikantham

Mathematics Technical Papers

Consider the equation (1.1) [see pdf for notation] on a Hilbert space H. Here n is a scalar and [see pdf for notation] is a linear Fredholm operator. That is: (a) L is closed; (b) The domain, D(L) is dense in H; (c) The range, R(L) is closed in H; (d) The dimension of the null space, dim n(L) <= (e) The dimension of the null space of the adjoint dim n(L*) <= The operator N, which may be nonlinear, is defined for sufficiently small and appropriately restricted [see pdf for notation], and [see pdf for notation] Using the method of Lyapunov-Schmidt (cf., e.g. [4] or [5]) we express w in the form (1.2) [see pdf for notation] where [see pdf for notation] denotes the orthogonal complement of n(L) in H. Suppose (u,v) satisfy the simultaneous equations (1.3) [see pdf for notation] (1.4) [see pdf for notation] where P is the orthogonal projection operator of H onto R(L), I is the identity operator on H, and J is a right inverse of L on R(L), i.e.

An Analytic Study Of A System Of Nonlinear Ordinary Differential Equations At An Irregular Type Singularity, James M. Lamb

Masters Theses

No abstract provided.

A Study Of Vector And Matrix Norms, Alvena Weiskircher

Theses & Honors Papers

A vector or matrix can be associated with a single nonnegative scalar . Basically this is the concept of a vector or matrix norm. While the study of these norms may be classified as numerical analysis , a limited background in linear algebra , matrix theory , and advanced calculus is sufficient to pursue the study of norms.

In this paper many of the properties and theorems concerning norms have been proven . It is thought that the most important results deal with the use of vector and matrix norms in testing for convergence of sequences and series of matrices …

Decompositions Of Modules And Matrices, Thomas Shores, Roger Wiegand

Department of Mathematics: Faculty Publications

A canonical form for a module M over a commutative ring R is a decomposition M ≈ R/I₁ Ο … Ο R/I_n, where the I_j are ideals of R and 1₁ ≤ . . . ≤ I_n. A complete structure theory is developed for those rings for which every finitely generated module has a canonical form. The (possibly larger) class of rings, for which every finitely generated module is a direct sum of cyclics, is also considered, and partial results are obtained for rings with fewer than 2^{c …}

On A Boundary Value Problem For A Class Of Differential Equations With A Deviating Argument, V. Lakshmikantham, Jerome Eisenfeld

Mathematics Technical Papers

Recently, J. Chandra [1] obtained comparison estimates for differential equations with deviating argument (1) [see pdf for notation] on the interval I: to [see pdf for notation] , where [see pdf for notation] is a given continuous function defined on a suitable interval (containing I ), together with the boundary conditions (2) [see pdf for notation] for [see pdf for notation] The class of BVP (1) - (2) is incorporated in a larger class discussed in [3]. The introduction in [1] of a maximal solution has furnished comparison results and an iterative procedure for obtaining solutions. The use of the …