Seminars this week

Abstract

> One goal of local helioseismology is to elicit three-dimensional information about the subsurface structure of sunspots. The physical quantities include sound-speed perturbations and magnetic fields. This information can be used to constrain sunspot models. Helioseismology involves solving both the forward and inverse problem. Traditionally, the inverse problem is solved assuming a linear relationship between the magnitude of the subsurface perturbation and the effect on the waves. We use three-dimensional numerical MHD simulations of wave propagation to explore the seismic effect of various perturbations to a reference sunspot model. These perturbations include modifications to the Wilson depression, subsurface sound-speed enhancements, and subsurface magnetic field changes. We comment on the possibility of measuring such perturbations on the Sun, and on the validity of using linear inversions.

Abstract

The determination of invariants of Lie algebras is motivated by the important role played by these elements in Physics and in representation theory. For semisimple Lie algebras, the invariants were determined long ago. But for non-semisimple Lie algebras, invariants have only been determined when the dimensions are low. Let g be a finite dimensional Lie algebra. Our interest is the centre of the enveloping algebra U(g), which has a close relation with the invariant subalgebra of the symmetric algebra S(g).

In this talk I will give the definition of Lie algebras, enveloping algebras, and the invariants. Then we recall the classical results for semisimple Lie algebras, and introduce the Duflo iosmorphism, giving a relationship between the invariants of S(g) and U(g). Finally I will introduce a method to compute the invariants by giving examples, and give some comments and criteria on the polynomiality of the centre of U(g).

Arrangements

Abstract

The Calculus of Variations was initiated in the 17th Century and forms a basic foundation of modern optimal (maximising or minimising) variational problems, nowadays often called optimal control. An introduction to the Calculus of Variations with some sample examples will be presented. This will include the Euler-Lagrange and Hamiltonian formulation together with the associated final boundary value conditions. A numerical shooting method can be used to solve the resulting Two Point Boundary Value Problem (TPBVP), a set of differential equations. There are many interesting applications including the optimal spending of capital, reservoir control, maintenance and replacement policy of vehicles and machinery, optimal delivery of medicines, drug bust strategies, study for examinations and optimal presentation of a lecture like this one. A new non-classical class of variational problems has been motivated by research on the non-linear revenue problem in the field of economics. This class of problem can be set up as a maximising problem in the Calculus of Variations (CoV) or Optimal Control. However, the state value at the final fixed time, $y(T)$, is {em a priori} unknown and the integrand to be maximised is also a function of the unknown $y(T)$. This is a non-standard CoV problem that has not been studied before. New final value costate boundary conditions will be presented for this CoV problem and some results will be shown.