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Measurable Sets In Product Spaces And Their Parametrizations., V. V. Srivatsa Dr.

Doctoral Theses

No abstract provided.

The Alfsen-Errors Structure Topology In The Theory Of Complex L1-Preduals., T. S.S.R.K. Rao Dr.

Doctoral Theses

A complex Banach space X is said to be an L1-predual if X* is isometric to L1() for some non-negative measure Well known examples of L1-preduals include the space C(X) of complex-valued continuous functions on a compact Hausdorff space and the abstract M-spaces of Kakutani. In [19], Grothendieck introduced a class of L'-preduals, now known as G-spaces, and conjectured that those are all the L'-preduals. In his 1964 memoir [35], Lândenstrauss settled this conjecture by exhibiting a wide class of Banach spaces, other than G-spaces, which are L1-preduals. He also gave several characterizations and interest- ing properties of L1-preduals in …

Pathwise Stochastic Calculus Of Continuous Semimartingales., Rajeeva L. Karandikar Dr.

Doctoral Theses

Stochastic integration with respect to Brownian motion was introduced by Ito. Stochastic integration with respect to martingales (and seminartingales) was developed by Kunita-Watanable [24 Fisk [9), Courrege D] and Meyer [33]. In this thesis, we study the path wise stochastic calculus restricting ourselves to continuous semimartingales. Here is a brief summary of our results.In Chapter I, we obtain a pathwise formula for the quadratic variation process < M > of a continuous local martingale M. Recall theat < M > is the natural increasing process in the Doob-Meyer decomposition of M. By a part wice formula for M> we mean a formula describing cxplicitly a w-path …

Spectral And Scattering Theory For Schrodinger Operator With A Class Of Momentum Dependent Long Range Potentials., P. L. Muthuramalingam Dr.

Doctoral Theses

No abstract provided.

Eigenvalues Of Nuclear Operators Of Diagonal Type., Raymond J. Kaiser

LSU Historical Dissertations and Theses

A nuclear operator T on a Banach space is an operator admitting a representation T = (SIGMA) T(,n) where each operator T(,n) has rank one and (SIGMA) (VERT.BAR)(VERT.BAR)(TAU)(,n)(VBAR)(VBAR) < (INFIN). A. Grothendieck proved that the sequence of eigenvalues, repeated according to their multiplicities, of such a nuclear operator must be square summable and that, in fact, two is the best possible exponent of summability. This work uses operators of diagonal type, i.e., operators which admit a diagonal representation with respect to a biorthogonal system, to construct examples of eigenvalue behavior. In particular we show that all of the known results concerning eigenvalue summability of nuclear operators can be obtained using operators of diagonal type. In Chapter 1, we present the terminology and notation to be used in this work. Chapter 2 gives a summary of results concerning nuclear (trace class) operators on Hilbert space. In Chapter 3, some of the important known results concerning eigenvalue behavior of nuclear operators on general Banach spaces are presented. The convolution operators on L(,p)(0,1), 1 < p < (INFIN), p (NOT=) 2, are diagonal operators since the trigonometric system (e('2(pi)inx)) is a conditional basis of these spaces. In Chapter 4 we use these important operators to exhibit extremal behavior of eigenvalue summability of nuclear operators on the L(,p)-spaces. In Chapter 5, we study nuclear cyclic diagonal operators. If X(,1),... ,X(,n) are sequence spaces, an operator D on (CRPLUS)X(,i) is a cyclic diagonal operator if its restriction to each X(,i) is a diagonal operator into X(,i+1) (if i = n into X(,1)). Certain eigenspaces of these operators have important applications to Banach space theory. Finally, in Chapter 6, we use a recently discovered space of G. Pisier to answer affirmatively a long outstanding question of. Pe l cynski and Saphar which is converse to Grothendieck's result: Given a (nonzero) sequence ((lamda)(,n)) in l(,2), is there a Banach space X and a nuclear operator on X whose eigenvalue sequence is ((lamda)(,n))?

On Valuation Rings As Homomorphic Images Of Valuation Domains., J. Paul Vicknair

LSU Historical Dissertations and Theses

A number of years ago, I. Kaplansky raised informally the question of whether every valuation ring could be expressed as a homomorphic image of a valuation domain. By a valuation ring, we mean a commutative ring with identity whose ideals are linearly ordered by inclusion. The classical notion of a valuation ring included the assumption that the ring is a domain. For two cases an affirmative answer to Kaplansky's question is known: for 0-dimensional valuation rings; and, for valuation rings which are monoid rings. In the early 40's, Kaplansky obtained structure theorems for a large class of (maximally complete) valuation …

On A Conjugate Class Of Subgroups Determined By A Formation, Mark Challis Hofmann

Doctoral Dissertations

This thesis is an investigation of the interrelationships between a formation f, a finite solvable group G, and G(,f) the residual of f in G. This study is developed by introducing the f-subgroups. It is proven that the f-subgroups of G form a characteristic conjugacy class of CAR-subgroups of G. Moreover these subgroups generate G(,f). As a result, G is an element of the formation f if and only if an f-subgroup is equal to the identity subgroup.

It is established that an f-subgroup is a product of known subgroups of the f-residual. The covering and avoidance properties of f-subgroups …

Ab?Ba : An Investigation Into The Historical Roots Of Noncommutative Algebra, Keith Chavey

Honors Capstones

Abstract Algebra is a branch of mathematics in which a large amount of research is currently taking place. This research includes the investigation into different types of algebraic structures such as fields, rings, groups, and their properties. The history of algebra is as rich as the science itself. It is my intention to investigate a crucial step in the development of algebra: the beginning of noncommutative algebra.

A Natural L(,P)-Metric For Spaces Composed Of Probability Measures With P-Th Moment., Mark William Scott

LSU Historical Dissertations and Theses

In the dissertation, it is seen that every probability measure with p-th moment on a complete, separable metric space can be viewed as a distribution of a metric space valued random variable. Between such random variables there exists an L(,p)-distance, and by finding the infimum of the L(,p)-distances between two types of random variables, it is possible to define a distance between distributions. It is seen that this distance can serve as a complete metric on the space of probability measures with p-th moment. The topology produced is shown to be equivalent to the topology of weak convergence of measures …

Generalized Hausdorff Matrices With Applications To Strong And Absolute Summability, John Edward Sayre

Digitized Theses

Hausdorff summability methods include such well known methods as those of Cesaro, Holder, and Euler. Consequently, the properties of Hausdorff matrices have been extensively investigated since 1921 when the German mathematician Flex von Hausdorff first published his paper entitled "Summationsmethoden und Momentfolgen I" {lcub}11{rcub}.;In 1922, Hausdorff published "Summationsmethoden und Momentfolgen II" {lcub}12{rcub} in which he defined what are now known as generalized Hausdorff matrices. The question then arose of finding appropriate conditions so that results about ordinary Hausdorff matrices would be true more generally. In this thesis, questions concerning strong and absolute summability are addressed.;In order to deal with the …