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Mathematics & Statistics

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\(U\)-Statistics and Imperfect Ranking in Ranked Set Sampling

Brett Presnell

University of Florida

3:00pm-4:00pm

PHY 108

Shanti Gomatam

**Abstract**

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.)

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Diagrams of Knotted Arcs and Quandles

J. Scott Carter

University of S. Alabama

3:00pm-4:00pm

PHY 108

Masahiko Saito

**Abstract**

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.

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Constructing the Identities and the Central Identities of the Square Matrix Rings

Siamack Bondari

Saint Leo's University

3:00pm-4:00pm

PHY 108

Masahiko Saito

**Abstract**

In this talk I present a procedure to compute all the multilinear identities and the multilinear central identities of the \(m\times m\) 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.

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Bounded Variation and its Generalizations

Dan Waterman

Syracuse University

3:00pm-4:00pm

PHY 108

J. S. Ratti

**Abstract**

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.

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The Refinement Equation: Multi-wavelets and Subdivision

Sherman D. Riemenschneider

West Virginia University

3:00pm-4:00pm

PHY 108

Mourad Ismail

**Abstract**

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.

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Superlinear \(3\) point boundary value problems

Bruce Calvert

Auckland University

New Zealand

3:00pm-4:00pm

PHY 108

Athanassios Kartsatos

**Abstract**

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.

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The Fastest Systolic Designs

Marjan Gusev

University “St. Cyril and Methodius”

Skopje, Macedonia

3:00pm-4:00pm

LIF 260

Nataša Jonoska

**Abstract**

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.

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Variational integrators, the Newmark Algorithm, and Collisions

Jerry Marsden

California Institute of Technology

3:00pm-4:00pm

PHY 108

Nagle Lecture Committee

**Abstract**

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).

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Nonparametric Models and Methods for ANOVA and ANCOVA Designs

Michael G. Akritas

Penn State University

3:00pm-4:00pm

CHE 204

Chris Tsokos

**Abstract**

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).

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A new classification of nilpotent groups, and some non-cancellation results

Peter Hilton

SUNY Binghamton/University of Central Florida

3:00pm-4:00pm

PHY 108

Nataša Jonoska

**Abstract**

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.

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**Note**

A Survey of Double-Torus Knots

Kunio Murasugi (Professor Emeritus)

University of Toronto

3:00pm-4:00pm

PHY 108

Masahiko Saito

Supported by CAS Faculty Development Program

**Abstract**

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.

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Bifurcations, Chaos, and the Ultimate Fate of the Galaxy

Antonio Elipe

Department of Astronomy

University of Zaragoza, Spain

4:00pm-5:00pm

PHY 130

Carol Williams

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Domination in Cartesian Products and Vizing’s Conjecture, Part I

Douglas F. Rall

Furman University

1:00pm-2:00pm

PHY 120

W. Edwin Clark

**Abstract**

**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.

**Refreshments will be served from 2:00-2:30 p.m. in PHY 209 (Faculty Lounge)
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Domination in Cartesian Products and Vizing’s Conjecture, Part II

Bert Hartnell

Saint Mary’s University

2:30pm-3:30pm

PHY 120

Stephen Suen

**Abstract**

**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.