**COLLOQUIA 2012 -- 2013**

Refreshments are served in Parker Hall, Room 244, beginning at 3:30.

**Friday, April 26, 2013**

Title: A hyper-spherical sparse grid approach for high-dimensional discontinuity detection

Abstract: High-dimensional discontinuity detection is significantly important to several areas of science and engineering. For example, in uncertainty quantification (UQ), it is directly related to risk assessment and predicting rare events. More generally, in an N-dimensional Euclidean space, the location of the discontinuity is generally an N-1-dimensional manifold, and therefore, the most difficult challenge is to accurately and efficiently represent such manifold. Conventional adaptive sparse-grid hierarchical interpolation has been employed to characterize such manifolds, however, the mesh refinement densely places grid points around the discontinuity, and fails to "sparsely" represent the N-1 dimensional manifold.We propose a novel method for identifying jump discontinuities in high dimensional spaces by incorporating a hyper-spherical coordinate system (HSCS) into the sparse-grid approximation framework. The basic idea is to transform the Cartesian coordinate system to an N-dimensional HSCS and treat the manifold as the N-1 dimensional function in the subspace constituted by the N-1 angle coordinates. Then a sparse-grid approximation can be constructed in the subspace where the function value at each grid point is estimated by solving a nonlinear equation with use of the bisection or Newton method. This novel technique identifies the discontinuity with a reduced number of sparse grid points compared to existing methods. Several numerical examples will be presented that show the increased efficiency and accuracy of our approach for detecting the discontinuities.

Faculty host: Yanzhao Cao**Friday, April 19, 2013**

Speaker: Qinqin Hu, Department of Mathematics and Statistics, Shandong University

Title: The Dual and Degrees of Freedom of Linearly Constrained Generalized Lasso

Abstract: Lasso is a popular model building technique for simultaneous estimation and variable selection. As a way to incorporate prior knowledge in applications, we propose linearly constrained generalized lasso, an extension of Lasso. We derived the dual of the linearly constrained generalized lasso and proposed an efficient coordinate descent algorithm to estimate the dual. We derived an unbiased estimator of degrees of freedom of the linearly constrained generalized lasso and used it to select tuning parameters by AIC or BIC criteria. Simulation studies demonstrate the excellent performance.

Faculty host: Peng Zeng

**Friday, April 12, 2013**

Speaker: Professor Ron Smith, Department of Mathematics,

University of Tennessee at Chattanooga

Title: Recent Results on Normal Matrices

Abstract: High-dimensional discontinuity detection is significantly important to several areas of science and engineering. For example, in uncertainty quantification (UQ), it is directly related to risk assessment and predicting rare events. More generally, in an N-dimensional Euclidean space, the location of the discontinuity is generally an N-1-dimensional manifold, and therefore, the most difficult challenge is to accurately and efficiently represent such manifold. Conventional adaptive sparse-grid hierarchical interpolation has been employed to characterize such manifolds, however, the mesh refinement densely places grid points around the discontinuity, and fails to "sparsely" represent the N-1 dimensional manifold.We propose a novel method for identifying jump discontinuities in high dimensional spaces by incorporating a hyper-spherical coordinate system (HSCS) into the sparse-grid approximation framework. The basic idea is to transform the Cartesian coordinate system to an N-dimensional HSCS and treat the manifold as the N-1 dimensional function in the subspace constituted by the N-1 angle coordinates. Then a sparse-grid approximation can be constructed in the subspace where the function value at each grid point is estimated by solving a nonlinear equation with use of the bisection or Newton method. This novel technique identifies the discontinuity with a reduced number of sparse grid points compared to existing methods. Several numerical examples will be presented that show the increased efficiency and accuracy of our approach for detecting the discontinuities.

Faculty host: T.-Y. Tam

**Friday, April 5, 2013 **

Speakers: Dr. Gary Martin, Emily R. and Gerald S. Leischuck Endowed Professor

Department of Curriculum and Teaching, AU

Dr. Marilyn Strutchens, Mildred Cheshire Fraley Distinguished Professor

Department of Curriculum and Teaching, AU

Title: The Common Core State Standards for Mathematics: Implications for Mathematicians and Statisticians

Joint presentation on Math Education

Faculty host: Narendra Govil**Friday, March 29, 2013**

Speaker: Dr. Xing Liang, University of Science and Technology of China

Title: Spreading speed of the noncompact integer-difference models

Abstract: In this talk I will introduce our work on the spreading speed of the noncompact integer-difference models. In this work, we give an abstract framework of the spreading speed of the noncompact operator which has principal eigenvalue. Furthermore, we will investigate the existence of the principal eigenvalues of a class of positive integral operators which are not compact. Then we will show an application of our theory to such operators.

Faculty host: Wenxian Shen

**Monday****, March 18, 2013**

Speaker: Dr. Yi Wang, University of Science and Technology of China

Title: Floquet bundles for tridiagonal competitive-cooperative systems and applications

Abstract: For a general time-dependent linear competitive-cooperative tridiagonal system of differential equations, we obtain canonical Floquet invariant bundles which are exponentially separated in the framework of skew-product flows. The obtained Floquet theory is applied to study the dynamics on the hyperbolic omega-limit sets for the nonlinear competitive-cooperative tridiagonal systems in time-recurrent structures including almost periodicity and almost automorphy.

Faculty host: Wenxian Shen

Friday, August 23, 2013

Speaker: Dr. Nam-Kiu Tsing, Department of Mathematics, University of Hong Kong

Faculty host: T.-Y. Tam

**Friday, February 1, 2013**

Speaker: Dr. Minh Nguyen (Columbus State University)

Title: Asymptotic Behavior of Linear Almost Periodic Differential Equations

Abstract: The talk is concerned with strong stability of solutions of non-autonomous equations of the form $\dot u(t)=A(t)u(t)$, where $A(t)$ is an unbounded operator in a Banach space depending almost periodically on $t$. A general condition on strong stability is given in terms of Perron conditions on the solvability of the associated inhomogeneous equation.

This is a joint work with D.X. Bui, D. Luu, S. Siegmund.

Faculty host: Wenxian Shen

**Friday, January 25, 2013**

Speaker: Jose Ignacio Tello, Universidad Politécnica de Madrid

(Polytechnic University of Madrid), visiting Rutgers University

Title: Stability of steady states of the Cauchy problem for the exponential reaction-diffusion equation

Abstract available here

Faculty host: Georg Hetzer

**Friday, November 16, 2012**

Speaker: Anthony Hilton (University of Reading, UK,

and Queen Mary University of London, UK)

Title: Bounds on the simple graph and multigraph (*r; s; a; t*)-threshold numbers

Abstract available here

Faculty host: Pete Johnson

**Friday, November 9, 2012**

Speaker: Aklilu Zeleke

(Michigan State University, Department of Statistics and Probability;

and Lyman Briggs College)

Title: Fibonacci-Like Polynomials and Combinatorial Identities

Abstract available here

Faculty host: Erkan Nane

**Friday, November 2, 2012**

Speaker: Tin-Yau Tam

Title: Aluthge iteration and its generalizations

Abstract: We will discuss the notions Aluthge transform and Aluthge iteration, its convergence conjecture and its development towards the final solution. The ultimate solution of J. Antezana, E. Pujals, and D. Stojanoff requires the tools from dynamical system and differential geometry. We will also consider the Banach algebra extension of Yamazaki's result and Lie group extension of APS's result. The contributions of the Auburn group will be discussed. The Banach extension requires the notion unitarily invariant norm. The Lie group extension uses a well-known topological concept called covering map.

**Friday, October 12, 2012 **

Speaker: Denis Bell (University of North Florida, Jacksonville)

Title: Quasi-invariant Measures on Path Space

Abstract: Brownian motion was discovered in the early 1800’s and given and a statistical treatment by Einstein in 1905. Wiener constructed a probability measure on the space of continuous paths representing the law of Brownian motion, thereby giving a rigorous mathematical foundation to Einstein’s work. Wiener’s measure has an important translation property that goes under the name of the Cameron-Martin theorem. We discuss this property and describe a method to generalize it to a wider class of measures. Along the way we develop a relationship between quasi-invariant measures and admissible vector fields, both of which will be defined in the talk.

Faculty host: Ming Liao

**Friday, October 5, 2012**

Speaker: Geraldo De Souza

Title: Atomic Decomposition of Some Banach Spaces and Applications

Abstract available here

**MONDAY, September 17, 2012**

Speaker: Onur Alp Ilhan

(Erciyes University, Faculty of Education, 38039 Melikgazi, Kayseri, Turkey)

Title: Solvability of Some Integral Equations in Banach Space and Their Applications to the Theory of Viscoelasticity

Abstract available here

Faculty host: Erkan Nane

**Thursday, September 13, 2012, at 3:30 (refreshments at 3:00)
PLEASE NOTE DATE AND TIME**

Speaker: Ratnasingham Shivaji,

H. Barton Excellence Professor (The University of North Carolina at Greensboro)

Title: Semipositone Problems on Exterior Domains

Abstract available here

Faculty host: Georg Hetzer

** **

**Friday, September 7, 2012**

Speaker: Danyal Soybaş

(Erciyes University, Faculty of Education, Department of Mathematics, Melikgazi, 38500, Kayseri, Turkey)

Title: Representations of Some Geometric Banach Space Properties with Weakly Compact Operators

Abstract available here

Faculty host: Erkan Nane

**Friday, August 31, 2012**

Speaker: Dr Fatih Koyuncu (Yildirim Beyazit University, Ankara, Turkey)

Title: Absolute irreducibility of polynomials by the polytope method

Abstract available here

Faculty host: Erkan Nane