Physical Sciences and Mathematics Commons^™

Estimation Of Spectral Variation., Rajendra Bhatia Dr.

Doctoral Theses

The study of spectra occupies a central place in the theory of linear operators. One part of this study consists of obtaining a detailed and complete knowledge of the spectrum of a given lincar operator. The other part - perturbation theory - consists of using this knowledge to obtain information about the spectra of nearby operators.Apart from the intrinsic mathematical interest it has, perturbation theory is of great importance in the study of several physical problems. In fact, the theory came into existence with the work of Rayleigh on sound waves and that of Schrodinger on quantum mechanics. Later, their …

Studies In Multipartite Self-Complementary Graphs., T. Gangopadhyay Dr.

Doctoral Theses

The class of self-complenentary egraphs has been extensively etudied by nany people, among others by C.R.J. Clephan, S.B, Rao, G, Ringel and H. Sache, and nany problems have been solved for this class, such as the Haniltonian problen and the characterisati on of potentially and forcibly self-complenentary dogree sequences (see [1], [2], [12), [13), [14, [15)). Thus self-complenentary graphs form an interesting class and this has been generalised by Hebbare [8] into the class of multipartite self-conplementary graphs.An r-partite self-complenentary graph is an r-partite graph G which is isomorphic to its r-partite complement where H has the sane vertex set …

Contribution To Theories Of Repetitive Sampling Strategies., Raghunath Arnab Dr.

Doctoral Theses

A common practical problem to whlch a survey - sampler has frequently to address himself 1s one of sampling a given finite population on successive occasions. One of the relevant issues requiring one's attention then 1s to adopt a suitable sampling strategy to estimate the population total of a variate of interest an the current occasion in an optimal manner. Here one has of necessity to take care to utilize the accumalated data on that variate procured in course of the survey along with other auxdliary information on one or more additional variables that may also incidentally be available, Several …

Determination Of Probability Measures Through Group Actions., Inder Kumar Rana Dr.

Doctoral Theses

One of the fundamental problems in Measure Theory is the following: given a measurable space (x, B,), to find subclasses D of B, such that whenever for two probability measures u and v on (X, B,), u(B) = v(3) for every B c D, then u(B) = v(B) for every Be B,. The first basic theorem of Measure Theory, viz., the Caratheodory Extension Theorem says that any sub-algebra D of B, which generates B, has the above mentioned property.Let (X, B, ) be a given measurable space. A subclass n of 3, is called a determining class for a class …

Statistical Analysis Of Nonestimable Functionals., Dibyen Majumdar Dr.

Doctoral Theses

Our interent will be centred around the Gaunn Markov model (Y,X6,2A), where Y ls a randon variable asnuning values in R" with expectation and dispersion natrix given byE(Y) = XA (1.1) (1.2) * X in Rnxm and A In the subset of nonnegative definite (n.n.d.) natrices of RAre known. Unless apecifled to the contrary, A will be assuted to be positive definite (p.d.). The unknown paraneter vector B varies in Ag, a subset of Rand o? in, will alvays be the positive half of the real line, unless otherwine specifled, and 1ikewfse satiafien the minimum requirement dim (n,) = m.Historically, …

Transformational Geometry Unit, Elizabeth Ann O'Neill

All Graduate Projects

The study included the development and writing of a unit on transformational geometry which involved a holistic approach including the cognitive, psychomotor, and affective domains. This unit was taught to the eighth grade class in the Oakville School District in Oakville, Washington. The results showed support that the teaching of this unit was effective.

An Octic Reciprocity Law Of Scholz Type, Duncan A. Buell, Kenneth S. Williams

Faculty Publications

No abstract provided.

Study Of Indian Nuptiality., Chitta R. Malaker Dr.

Doctoral Theses

DEFINITION OF NUPTIALITYAccordi ng to the latest International Demographic Terminelogy (Grebeni k and ill, 1974), t he term 'nuptiality' is used to denote the whole subject of the formation and dissolutien of sexual uni ons, includi ng the study of marriage, consensual uni ons, separation, di vorce, widowhood etc. It is also used, in a narrower sense, as a synonym for marriage in such terms as nuptiality rates;and nuptiality probabilities, The study of nuptiality deals with the frequency of marriage, 1.e., unions between persons of opposite sexes which involve rights and obligations fixed by law and custom; with the characteriatics …

Theory Of Estimation In Algebraic And Analytic Exponential Families With Applications To Variance Components Models., Krishnan Unni Dr.

Doctoral Theses

There are many important statistical problems of the following kind. The family of probability measures O is parametrized by a vector parame ter n varying in a q-dimen- sional domain. P can be represented as an exponential family of probability distributions with k canonical para- me ters where k is greater than q. The canonical parameters do not vary in a domain in R, but are restricted by polyno- mial or analytic equations. They vary on a curved surface defined by the polynomial or analytic equations within the natural parame ter space of the exponential family. The present work is …

Choice Of Strategies In Survey Sampling., Geetha Ramachandran Dr.

Doctoral Theses

The main problam of sampling from finite populations consi sts of devi sing an appropriate proceduro for sel ecting a sample from a given population and developing an appropriate procedure for estimating the population parameter of interest in order to maximize the precision of the estimator subject to certain restrictions on the cost for the survey or alternatively minimi ze the cost for achieving a given l evel of precision. During the thirties and forties several contributions were made to develop the theoretical background of sample survey techni ques in order to solve this problem, The most significant works of …

Maximal Residue Difference Sets Modulo P, Duncan A. Buell, Kenneth S. Williams

Faculty Publications

Let p ≡ 1 (mod 4) be a prime. A residue difference set modulo p is a set S = {a_i} of integers a_i such that (a_i/p) = +1 and ((a_i - a_j)/p) = +1 for all i and j with i ≠ j, where (n/p) is the Legendre symbol modulo p. Let m_p be the cardinality of a maximal such set S. The authors estimate the size of m_p.

Some Contribution To Descriptive Set Theory., Haimanti Sarbadhikari Dr.

Doctoral Theses

Current interest in the descriptive theory of sc ts stems largely from the many applications that the classical theory has found in probability theory, functional analysis, dynanic programming etc. As exarples, we cite the fundament al paper of Blackwell 15J wherein he has shown that many of the pathologies of probability theory c an be avoided if one takes the basic probability space to be an analytic set. As another example, we mention that several vritere [7), L23] have shown that in Blackwell's model of dynamic programming 131 the existence of optimal strategies ig related to the cxistence of measurable …

Fragments Of 1st-Order Logic, I: Universal Horn Logic, George F. Mcnulty

Faculty Publications

No abstract provided.

Mathematics In America: The First Hundred Years, Judith V. Grabiner

Pitzer Faculty Publications and Research

There are two main questions I shall discuss in this paper. First, why was American mathematics so weak from 1776 to 1876? Second, and much more important, how did what happened from 1776-1876 produce an American mathematics respectable by international standards by the end of the nineteenth century? We will see that the "weakness" -at least as measured by the paucity of great names- co-existed with the active building both of mathematics education and of a mathematical community which reached maturity in the 1890's.

Undecidable Properties Of Finite Sets Of Equations, George F. Mcnulty

Faculty Publications

No abstract provided.

Application Of Theorems Of Schur And Albert, Thomas L. Markham

Faculty Publications

No abstract provided.

System Of Imprimitivity For Ergodic Actions Of Locally Compact Abelian Groups., Joseph Mathew Dr.

Doctoral Theses

Mackeys theorem on inducud representation gives all the systems of imprimitivity for a locally compact second countable group G, acting in a separable Hilbert space and based on a transitive G-space X. These systems of imprimitivity are obtained from the unitary representationg of the closed subgroup H of G defining the transitive G-space X. The main tool for this study is a class of functions called cocycles. Maçkey showed that the (G, x) systems of imprimitivity are connected in a one-one way to unitary operator valued cocycles on GXX: This cocycle in turn gives a representation of the closed subgroup …

Multiplicatively Periodic Rings, Ted Chinburg, Melvin Henriksen

All HMC Faculty Publications and Research

We prove a generalization of Luh's result without using Dirichlet's Theorem. We then use Theorem 1 to show that the J-subrings of a periodic ring form a lattice with respect to join and intersection (the join of two subrings is the smallest subring containing both of them). After noting that every J-ring has nonzero characteristic, we determine for which positive integers n and m there exist J-rings of period n and characteristic m. This generalizes a problem posed by G. Wene.

Group Of Point Transformations Of Time Dependent Harmonic Oscillators, Jose Ricardo Bernal

University of the Pacific Theses and Dissertations

In general, a physical system has invariant quantities which are very often related to its symmetry and to the invariance of the equation that describe it. A detailed study of the invariance property of the differential equation will be helpful in understanding this relation.

The work is concerned with a preliminary investigation of the Lie-group which leaves invariant the Newtonian and Lagrangian equation of motion for a one-dimensional harmonic oscillator. A brief review of Ehrenfest's adiabatic principle and the later treatments on exact and adiabatic invariants will be presented.

Generalized Inverses Of Special Types Of Matrices., Rao Vara Prasada Pullepu Sree Satya Narayana Dr.

Doctoral Theses

Inverses, in the regular sense of the term, do not exist for singular square matrices and rectangular matrices. however for such matrices there exist matrices which satisfy many important properties similar to those of inverses of nonsingular matrices and for many purposes, can be used in the same way as regular inverses. These matrices are named generalised inverses (g-inverses) to distinguish then from the inverses of nonsingular matricos. Only since 1955 this field of study af generalized inverse was: invostigated systenatically and was expiorcd for nany beautiful and interesting results and applications though the concept of generalizod inverse was first …

Application Of Separation Theorem For Hermitian Matrices, Thomas L. Markham

Faculty Publications

No abstract provided.

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 …

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 …

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

Nonlinear Stochastic Integral-Equation Of Hammerstein Type, William J. Padgett

Faculty Publications

No abstract provided.

Maximal And Minimal Topologies., Asit Baran Raha Dr.

Doctoral Theses

Given a topological property Ï€ and a pon-void set X, let Ï€(X) denote the set of topologies on X with property Ï€. 7 (X) is obviously partially ordered by inclusion, A topologi- cal space (X,T) is minimal if T is a minimal element in T (X). (X, T) is said to be maximal Ï€ if T1 1s a maximalelement in Ï€(X), The study of maximal and minimal topological space is, as a matter of fact, a study of maximal Ï€ and minimal Ï€ spaces. Topological spaces closely related to minimal T sp aces are T-closed and KatetoV T spaces, (X,T) …

Studies In Boolean Algebras And Measure Theory., K. P. S. Bhaskara Rao Dr.

Doctoral Theses

This Thesis is divided into six chapters the first three chapters constituting studies in Boolean algebras and the last three chapters constituting studies in measure theory. We give below a sketch of the main problems treated in this Thesis.A De composition theorem due to Sobczyk and Hammer (28] implies that strongly continuous charges on Boolean algebras play a role similar to that of nonatomic measures on Boolean g-algebras. CHAPTER 1. Rudin [25] and Knowles [14] gave necessary and sufficient conditions for the Boral o-ficld of a compact Hausdorff space to admit a nonatomic measure. But there are no necessary and …

Nonnegative Matrices Whose Inverses Are M-Matrices, Thomas L. Markham

Faculty Publications

A characterization of a class of totally nonnegative matrices whose inverses are M-matrices is given. It is then shown that if A is nonnegative of order n and A^-1 is an M-matrix, then the almost principal minors of A of all orders are nonnegative.