USF Mathematics -- Colloquia, Spring 2000

Ranked set sampling has attracted considerable attention as an efficient sampling design, particularly for environmental and ecological studies. A number of authors have noted a gain in efficiency over ordinary random sampling when specific estimators and tests of hypotheses are applied to rank set sample data. We generalize such results by deriving the asymptotic distribution for random sample U-statistics when applied to ranked set sample data. Our results show that the ranked set sample procedure is asymptotically at least as efficient as the random sample procedure, regardless of the accuracy of judgement ranking. Some errors in the ranked set sampling literature are also revealed, and counterexamples provided. Finally, application of majorization theory to these results shows when perfect ranking can be expected to yield greater efficiency than imperfect ranking. (Note: This work is joint with Lora L. Bohn.)

Friday, April 21, 2000

A quandle is an algebraic structure that imitates the moves in classical knot theory. We examine the structure in terms of examples of knotted arcs and circles. From several examples, we hope to develop some intuition about quandle homology. The talk will be elementary with lots of pictures, and the author hopes to make it accessible to beginning graduate students.

Friday, April 14, 2000

In this talk I present a procedure to compute all the multilinear identities and the multilinear central identities of the m × m matrices over a field of characteristic zero or large enough prime. The method uses group representation theory and relies heavily on computational techniques. Some knowledge of linear algebra (multiplication of matrices, nullspace of a matrix, etc.) is needed to follow the presentation.

Friday, April 7, 2000

We begin by defining the notion of bounded variation and explain various directions in which it can be generalized. Bounded variation arose in the context of Fourier series and made its first appearance in the Dirichlet-Jordan theorem on the convergence of Fourier series. Most (one-dimensional) generalizations of bounded variation came into being in attempts to improve this theorem. We will also discuss a recent failed attempt to generalize this notion. A more complete description of this talk can be found here.

Friday, March 31, 2000

This talk will center on the refinement equation for multi-wavelets and subdivision schemes. This is the relationship connecting the wavelets on adjacent levels of a multiresolution analysis. In particular we will focus on the information contained in the “refinement mask,” the finitely suppported sequence of coefficients appearing in this equation for vector subdivision schemes. How can one decode information about the existence of a solution to the functional equations defined by the equation, about the convergence to that solution, about the polynomials that can be reproduced with the scheme, and about the smoothness of the underlying solution? Most of the information is found from linear algebra applied to certain finite matrices formed from the refinement mask and should be accessible to non-experts.

Friday, March 24, 2000

We consider second order 3-point boundary value problems involving superlinear and sublinear terms. We show that there exist solutions with arbitrarily large number of oscillations. The methods involve applications of the Leray-Schauder degree theory. This is joint work with Professor Chaitan Gupta.

Monday, March 20, 2000

The fast systolic computation was designed by the author to achieve implementations that use fewer processors to execute the algorithm in less time then the conventional systolic algorithms. In 1978 H.T.Kung and C.S.Leiserson proposed systolic algorithms realized on a bidirectional linear array where two data streams flow in opposite directions. The data flow introduced for this solution requires data elements to appear in the data stream at each second time step, which is the only way to meet all the elements from the other data stream.

One way to speedup the solution is to use the regular folding procedure. This is achieved by a procedure called regular folding that consists of determination of symmetry planes and vector of interlocking properties and implementation of the translated folding. These solutions use up to 40% less processors and have double efficiency. However, these solutions do not execute the algorithms in less time and use a little bit more complex processors. To execute the algorithms in less time the author invented fast systolic designs.

The main idea to achieve fast systolic designs is based on algorithm partitioning, remapping, interlocking translation and composition of relevant parts. An additive splitting is obtained if partitioning is performed on input data. This is possible only if the operations used in the algorithm are associative and commutative. However we do not use partitioning to map large size algorithms onto small processor arrays, but use it organize the computation execution in a more efficient way. We partition the data stream into odd and even parts similar to the idea of partitioning the output data stream result in better solutions, since partitioning the input data stream requires an additional summation cell.

The best improvement achieved by the fast systolic designs is 33% less processors and 33% less time using the same type of processors and communication. Actually the same processor array used to solve a problem of dimension N can be used to solve a problem of dimension 3N/2 in the same array and the same time.

If further partitioning is implemented then the solutions are double pipelines. The algorithm can now be solved in half the time and the speed up is increased by a factor of 2. Two versions are identified for all four types of linear pipelines whether retiming or relocation is used. The fastest systolic design is a derivation of the last double pipeline solution and it uses 50% less processors and 50% less time. To achieve this solution an extra memory cell is required per cell. No increase in communications and processor design is required.

Friday, March 10, 2000

This lecture surveys a number of recent results in the analysis of the Newmark algorithm and related issues for nonlinear systems, both conservative and nonconservative. In particular, it is shown that the conservative Newmark algorithm is variational and hence symplectic. This analysis is believed to underly the reason that the second order accurate Newmark algorithm has excellent energy behavior for both conservative and dissipative systems. The variational approach is also studied in the context of collision algorithms using techniques from nonsmooth analysis, avoiding the use of gap functions. We will also briefly discuss some related issues, such as the fact that finite element elasticity using the Newmark algorithm is both variational and multisymplectic in a spacetime sense. The theory as well as simulations will be discussed. (Based on joint work with Michael Ortiz, Couro Kane, Anna Pandolfi, and Matt West).

Thursday, March 9, 2000

The talk will review some of the most common models used for the analysis of continuous and discrete ordinal data. Some drawbacks of these models will be highlighted in order to motivate the need for fully nonparametric approaches. Such nonparametric models and procedures will be presented and illustrated with the analysis of several data sets. The material for the talk is taken from Akritas, Arnold and Brunner (1997, JASA) and Akritas, Arnold and Du (2000, Biometrika, in press).

Friday, February 11, 2000

The Mislin genus classifies nilpotent groups of a certain type. By studying the Mislin genus of a group N and of its k-fold direct power, non-cancellation results for direct products are obtained.

Friday, January 28, 2000

A double-torus knot is a knot in a 3-sphere that can be embedded in the standard orientable closed surface of genus 2. The class of these knots is not new, but the systematic study of these knots started quite recently. In this talk, I explain the background and main objectives of this research, and discuss the connections with other prpblems in knot theory.

Wednesday, January 26, 2000

Friday, January 7, 2000

Title	Domination in Cartesian Products and Vizing’s Conjecture, Part I
Speaker	Professor Douglas F. Rall Furman University
Time	1:00-2:00 p.m.
Place	PHY 120
Sponsor	Professor Edwin Clark

An Introduction: A well-known problem in domination theory is the long standing conjecture of V.G. Vizing from 1963 that the domination number of the Cartesian product of two graphs is at least as large as the product of the domination numbers of the individual graphs. Although limited progress has been made this problem essentially remains open. In this introductory talk, a brief survey of progress will be outlined.

Title	Domination in Cartesian Products and Vizing’s Conjecture, Part II
Speaker	Professor Bert Hartnell Saint Mary’s University
Time	2:30-3:30 p.m.
Place	PHY 120
Sponsor	Professor Stephen Suen

Some recent results: As explained in the previous talk on this topic, 2-packings have long been recognized as playing a useful role in obtaining a lower bound for the domination number of the Cartesian product of two graphs. Here we will outline more recent attempts to exploit the nature of 2-packings more fully to establish improved lower bounds.

Title	A Survey of Double-Torus Knots
Speaker	Professor Kunio Murasugi (Professor Emeritus) University of Toronto
Time	3:00-4:00 p.m.
Place	PHY 108
Sponsor	Professor Masahiko Saito
Acknowledgement	Supported by CAS Faculty Development Program

Colloquia -- Spring 2000

Friday, April 28, 2000