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Mathematics & Statistics

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**Speaker**

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Quasicrystals: from board games to complexity

Marjorie Senechal

Louise Wolff Kahn Professor Emerita of Mathematics and History of Science

Smith College

4:10pm-5:10pm

PHY 130

Mathmatics & Physics Departments

**Abstract**

Tessellations, ornamental art, tiling games, crystal structures: our fascination with patterns dates from prehistory. So does our urge to classify patterns. A quarter century ago the discovery of Penrose tilings and icosahedral crystals turned our received notion of “crystal” (and the rules of the tiling game) upside down. Our first, immediate, response was to extend the mathematical model of “crystal” beyond translation symmetry. But how? Matching rules? Self-similarity? Projections from high dimensions? Discrete diffraction spectra? It seems that none of these properties characterize “crystal”. But who cares? The complexity this research has uncovered is much more interesting.

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Topological Degree Theories at USF

Athanassios Kartsatos

3:00pm-4:00pm

PHY 118

Nataša Jonoska

**Abstract**

After a short introduction to the Leray-Schauder topological degree theory, I will go over around 10 years of research involving topological degree theories at USF. This research was done by Dr. Skrypnik (Ukrainian Academy of Sciences) and myself, and by several of my Ph.D. students and myself. These theories are for perturbations of nonlinear maximal monotone and \(m\)-accretive operators in real Banach spaces \(X\). \(m\)-accretive operators map \(X\) into \(X\), while maximal monotone operators map \(X\) into its dual space \(X^*\). Applications of such degree theories include results on invariance of domain, existence of eigenvalues/eigenvectors, ranges of sums, and other existence problems of Nonlinear Analysis.

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An integrated geo-spatial energy — air quality impact assessment tool

Daniel S. Zachary

Project LEAQ – Esch sur Alzette, Luxembourg

3:00pm-4:00pm

PHY 118

Marcus McWaters

**Abstract**

A geo-spatially distributed and coupled energy — air quality model is described in this paper. Our approach is to build upon the large scale systems analytic method from a non-geo-spatial (prototype). We focus on new developments of the geo-spatial distribution of emission and energy technologies in the energy model. Our approach is the building of a meta-modeling to perform a full geo-spatial analysis of energy sources, devices, and their emissions for the four broad sectors of the economy: transport, industrial, residential, commercial. An impact parameter is developed on the bases of projected energy infrastructure demand and population density. The application is designed for Luxembourg where two regions are designated (populated and rural) to test the principle of this experiment. A fifteen year (\(3\times 5\)) energy expansion scenario is simulated for Luxembourg in this example. This application is applicable to other European cities and regions.

Our approach in the search for an optimal energy/environmental solution employing a coupled meta-model and guided by a convex optimizer. We build upon the recent work in optimal control [ref. Carlson, Zachary et al, 2004]. Recent analysis including optimal spatial development is also presented. Cost and environmental impact (projected air pollution on population density with convolution techniques) are presented. A novel spatial distribution pro ject model is used in conjunction with the geo-spatially model. Here, we present a distribution tool that uses the concept of maximum entropy. In this model, we use fundamental distribution-of-state properties that are used to guide urban development projects.

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Rack shadows and their invariants

Sam Nelson

Claremont McKenna College

Claremont, CA

2:00pm-3:00pm

PHY 209 (Lounge)

Mohamed Elhamdadi

**Abstract**

A rack shadow is a set with an action by a rack, a set with a self-distributive right-invertible operation. We can use rack shadows to deﬁne invariants of knots and links by colorings of diagrams. In this talk we will see how shadow polynomials can be used to deﬁne enhancements of the rack counting invariant.

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On the long-time stability of the implicit Euler scheme for the two-dimensional Navier-Stokes equations

Florentina Tone

University of West Florida

3:00pm-4:00pm

PHY 118

Wen-Xu Ma

**Abstract**

In the first part of the talk I will introduce the Navier–Stokes equations of incompressible fluids and mention some of the basic results in the area. In the second part, I will discretize the Navier-Stokes equations in time using the implicit Euler scheme and I will prove that the fully implicit Euler scheme is unconditionally stable (uniformly in time).

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\(2\)-Groups and Crossed Modules

Alissa Crans

Loyola Marymount University

Los Angeles

3:30pm-4:00pm

PHY 118

Mohamed Elhamdadi and Masahiko Saito

**Abstract**

A \(2\)-group is a categorized version of a group in which the underlying set \(G\) has been replaced by a category and the multiplication, inverse, and identity maps have been replaced by functors. If we then allow the usual group axioms to hold on the nose as equations, we arrive at the notion of a strict \(2\)-group. There are many equivalent ways to precisely define a strict \(2\)-group, some dating back to the 1950's. We will discuss these equivalent definitions, focusing most of our attention on the relationship to the concept of a crossed module, formulated by J. H. C. Whitehead.

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Non-Involutory Connected Quandles with Good Involutions

J. Scott Carter

University of South Alabama

3:00pm-3:30pm

PHY 118

Mohamed Elhamdadi and Masahiko Saito

**Abstract**

There is a family of quandles that is not involutory connected, but which possess good involutions. So the purpose of the talk will be to define the words in the title, exemplify the concepts, and give the examples. In the process, I will indicate the notion of quandle extension and especially try to indicate why quandles are interesting. With luck, this will help Professor Crans by giving background information for her talk.