Dr. Yongchen Qi, University of Minnesota - Duluth
Title: Maximum likelihood estimation for the endpoint and exponent of a distribution
Time: Thursday, March 18.  2:30-3:30.  in RB-3024
  Consider a random sample from a regularly varying distribution function with a finite right endpoint θ and an exponent α of regular variation. The primary interest of the paper is to estimate both the endpoint and the exponent. Since the distribution is semiparametric and the endpoint and the exponent reveal asymptotic properties of the right tail for the distribution, estimates for both parameters involve only a few largest
observations from the sample. The conventional maximum likelihood method can be used  to estimate both α and θ, see e.g., Hall (1982) for regular cases when α ‥ 2, and Smith (1987), Dress, Ferreira and de Haan (2004) and Peng and Qi (2009) for nonregular cases when α ∈ (1, 2). The global maximum of the likelihood function doesn’t exist if one allows α ∈ (0, 1], and a local maximum exists with probability tending to one only if α > 1. Therefore, the maximum likelihood method fails when α ∈ (0, 1]. In this paper we propose a new likelihood method to estimate both parameters. The use of the new method requires no prior information on the exponent, the likelihood function is always bounded, and the estimates from this new likelihood exist in all cases. We present the asymptotic distributions for the estimates from the new method. Our simulation study shows that the proposed method has better finite sample properties than the conventional maximum likelihood method in regular cases.

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Dr. Brian Harbourne, University of Nebraska
Title:  The Spherical Tetrahedron: A Tale of a Texan, a Toilet Tank Float and  the Theorem of Gauss-Bonnet
Time:  Friday 05 February 2010, 1:30 pm â€" 2:30 p.m. in RB-1022  
 
Abstract:
Computing areas on spheres goes back to the ancient Egyptians. I'll discuss various aspects of spherical (as opposed to plane) geometry (such as how to find the area of a spherical triangle), culminating in the Theorem of Gauss-Bonnet. As an application, we'll determine the area of a spherical Reuleaux triangle and we'll use that to answer a problem posed by a Depression era Texas engineer (what is the volume of the spherical tetrahedron?). 

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In his thesis F. Hivert introduced a faithful action of the symmetric group S_n on the ring of polynomials C[x_1; x_2; : : : ; x_n], which does not preserves the multiplication, but still leads to interesting results. The invariants of the action are the quasi-symmetric functions.  Unfortunately, when we extend this action to the group algebra C[S_n], the action we obtain is not faithful anymore. However, by taking the quotient of C[S_n] by the kernel of this new action we obtain a faithful action of the Temperley-Lieb TL_n(2) on the ring of polynomials in n variables. Since the Temperley-Lieb algebra can be embedded in the Fuss Catalan algebra on two colours FC_n(a; b) with ab = 2, a natural question to ask is if the action of the Temperley-Lieb algebra can be extended to a faithful action of the Fuss-Catalan algebra, and if the extension is unique. We will show how we use the theory of subfactors to answer both these questions in the affirmative.  This is joint work with R. Burstein.

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Thursday, January 28, 2010, 2:30-3:30 in RB-1045
Title:  Specialised Mathematics for School Teachers:  who Needs It?

Abstract:
While the popular view remains that for most teachers, the mathematics they need is simply a 'remedial' examination of mathematics, an increasing body of research is arguing that there are specialised understandings of mathematics needed by school teachers, which are not developed during typical undergraduate mathematics courses. This talk will provide examples of what such specialised mathematics understanding might be. Aspects of a large data base of information on teacher-candidates' mathematical performance collected over the last five years will be shared. Results include levels of specialised understandings of elementary and early secondary mathematics concepts as needed for teaching of  teacher candidates in the Bachelor of Education professional certification program, broken down by number and types of secondary and undergraduate mathematics courses taken. This talk will be particularly of interest to undergraduate students considering a mathematics teaching career, as well to faculty members involved in preparing such students mathematically.

Title:    Polynomials Decomposition and Geometry

Abstract:  
We will see how to relate the algebraic problem of writing a polynomial in a specific way with some geometric constructions.  Everyone  knows an issue of this kind of problem, i.e. the canonical form for quadratic polynomials. We will begin with this very basic situation in two variables and show how to relate this with conics in the plane. Then we will use the twisted cubic curve to treat the degree three case. Eventually  rational normal curves will eventually appear to deal the general two variables case.

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Abstract:  
I will study the existence of multi-vortex solutions of the Ginzburg-Landau equations with an external potential in two dimensions. These equations model the equilibrium states of superconductors: the external potential represents doped impurities or defects of the superconductor. I will show that if the critical points of the potential are spaced widely enough and if the potential, W is "strong enough," then there exists a multi-vortex (perturbed) solution with each vortex centred near each critical point of W.