Primes Of The Form X² + Ny² In Function Fields,
2010
Louisiana State University and Agricultural and Mechanical College
Primes Of The Form X² + Ny² In Function Fields, Piotr Maciak
LSU Doctoral Dissertations
Let n be a square-free polynomial over F_q, where q is an odd prime power. In this work, we determine which irreducible polynomials p in F_q[x] can be represented in the form X^2+nY^2 with X, Y in F_q[x]. We restrict ourselves to the case where X^2+nY^2 is anisotropic at infinity. As in the classical case over Z, the representability of p by the quadratic form X^2+nY^2 is governed by conditions coming from class field theory. A necessary and almost sufficient condition is that the ideal generated by p splits completely in the Hilbert class field H of K=F_q(x,sqrt(-n)) for the …
Subgroups Of The Torelli Group,
2010
Louisiana State University and Agricultural and Mechanical College
Subgroups Of The Torelli Group, Leah R. Childers
LSU Doctoral Dissertations
Let Mod(Sg) be the mapping class group of an orientable surface of genus g, Sg. The action of Mod(Sg) on the homology of Sg induces the well-known symplectic representation:
Mod(Sg) ---> Sp(2g, Z).
The kernel of this representation is called the Torelli group, I(Sg).
We will study two subgroups of I(Sg). First we will look at the subgroup generated by all SIP-maps, SIP(Sg). We will show SIP(Sg) is not I(Sg) and is in fact an infinite index subgroup of I(Sg). We will also classify which SIP-maps are in the kernel of the Johnson homomorphism and Birman-Craggs-Johnson homomorphism.
Then we will …
A Class Of Discontinuous Petrov–Galerkin Methods. Ii. Optimal Test Functions,
2010
University of Texas at Austin
A Class Of Discontinuous Petrov–Galerkin Methods. Ii. Optimal Test Functions, Leszek Demkowicz, Jay Gopalakrishnan
Mathematics and Statistics Faculty Publications and Presentations
We lay out a program for constructing discontinuous Petrov–Galerkin (DPG) schemes having test function spaces that are automatically computable to guarantee stability. Given a trial space, a DPG discretization using its optimal test space counterpart inherits stability from the well posedness of the undiscretized problem. Although the question of stable test space choice had attracted the attention of many previous authors, the novelty in our approach lies in the fact we identify a discontinuous Galerkin (DG) framework wherein test functions, arbitrarily close to the optimal ones, can be locally computed. The idea is presented abstractly and its feasibility illustrated through …
Multigrid In A Weighted Space Arising From Axisymmetric Electromagnetics,
2010
Portland State University
Multigrid In A Weighted Space Arising From Axisymmetric Electromagnetics, Dylan M. Copeland, Jay Gopalakrishnan, Minah Oh
Mathematics and Statistics Faculty Publications and Presentations
Consider the space of two-dimensional vector functions whose components and curl are square integrable with respect to the degenerate weight given by the radial variable. This space arises naturally when modeling electromagnetic problems under axial symmetry and performing a dimension reduction via cylindrical coordinates. We prove that if the original three-dimensional domain is convex then the multigrid Vcycle applied to the inner product in this space converges, provided certain modern smoothers are used. For the convergence analysis, we first prove several intermediate results, e.g., the approximation properties of a commuting projector in weighted norms, and a superconvergence estimate for a …
Copula And Default Correlation,
2010
Louisiana State University and Agricultural and Mechanical College
Copula And Default Correlation, Dongxiang Yan
LSU Master's Theses
This work presents a study of copulas, with special focus on the Gaussian copula model and its behavior under a certain conditioning process. Simulations are carried out to examine the behavior of the moments on conditional copula model, as measured by the behavior of Wick identities which hold for multivariate Gaussians.
