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On symmetric invariants of centralisers in reductive Lie algebras
No full textLet g be a finite-dimensional simple Lie algebra of rank l over an algebraically closed field of characteristic
0. Let e be a nilpotent element of g and let q be the centraliser of e in g. In this paper we study the
algebra S(q)^q of symmetric invariants of q. We prove that if g is of type A or C, then S(q)^q is always
a graded polynomial algebra in l variables, and we show that this continues to hold for some nilpotent elements
in the Lie algebras of other types. In type A we prove that the invariant algebra S(q)q is freely
generated by a regular sequence in S(q) and describe the tangent cone at e to the nilpotent variety of g
Structure theorems over polynomial rings
No full textGiven a polynomial ring R over a field k and a finite group G, we consider a finitely generated graded RG-module S. We regard S as a kG-module and show that various conditions on S are equivalent, such as only containing finitely many isomorphism classes of indecomposable summands or satisfying a structure theorem in the sense of [D. Karagueuzian, P. Symonds, The module structure of a group action on a polynomial ring: A finiteness theorem, preprint, http://www.ma.umist.ac.uk/pas/preprints]
Effective wave propagation in a prestressed nonlinear elastic composite bar
Get PDFThe problem of determining the effective incremental response of nonlinearly elastic composite materials given some initial pre-stress, is of interest in numerous application areas. In particular the case when small amplitude elastic waves pass through a pre-stressed inhomogeneous structure is of great importance. Of specific interest is how the initial finite deformation affects the microstructure and thus the subsequent response of the structure. Modelling this effect is in general extremely difficult.
In this article we consider the simplest problem of this type where the material is a one dimensional composite bar consisting of two distinct phases, periodically distributed. Neglecting lateral contractions, the initial deformation is thus piecewise homogeneous and we can therefore determine the incremental behaviour semi-analytically, given the constitutive behaviour (strain energy function) of the phases in question. We apply asymptotic homogenization theory in the deformed configuration in order to find the effective response of the deformed material in the low frequency limit where the wavelength of the propagating waves is much longer than the characteristic lengthscale of the microstructure. We close by considering the arbitrary frequency case and illustrate how the initial deformation affects the location of stop and pass bands of the material. Work is underway to confirm these results experimentally
Pfaffians, the G-Signature Theorem and Galois Hodge Discriminants
No full textLet be a finite group acting freely on a smooth projective scheme over a locally compact field of characteristic 0. We show that the -constants associated to symplectic representations of and the action of on may be determined from Pfaffian invariants associated to duality pairings on Hodge cohomology. We also use such Pfaffian invariants, along with equivariant Arakelov Euler characteristics, to determine hermitian Euler characteristics associated to tame actions of finite groups on regular projective schemes over
Universal connection and curvature for statistical manifold geometry
No full textStatistical manifolds are representations of smooth families of
probability density functions
that allow differential geometric methods to be applied to
problems in stochastic processes, mathematical statistics and
information theory. It is common to have to consider a number of
linear connections on a given statistical manifold and so it is
important to know the corresponding universal connection and
curvature; then all linear connections and their curvatures are
pullbacks. An important class of statistical manifolds is that
arising from the exponential families and one particular family is
that of gamma distributions, which we showed recently to have
important uniqueness properties in stochastic processes. Here we
provide formulae for universal connections and curvatures on
exponential families and give an explicit example for the manifold
of gamma distributions
Premixed flames modelled with thermally sensitive intermediate branching kinetics
No full textThe foundations of a relatively simple two-step kinetic scheme for flame chemistry are outlined, involving a model chain branching process that should adopt the activation temperature of a rate-limiting branching reaction in order to offer a broad approximation for hydrocarbon flames. A model energetic intermediate reactant then acts as a buffer between fuel consumption and the release of heat, as the intermediate is converted into products through a completion reaction step. By taking the rate of the latter reaction to be linear in the concentration of the intermediate, which is consistent with the final state being an equilibrium in a broader chemical system, a form of the model is arrived at which admits asymptotic solutions in a thermodiffusive context with constant coefficients. These are developed to second order for large values of the activation energy of the branching reaction and are found to involve the same trends that are seen for lean methane and hydrogen flames calculated using detailed chemical and transport models. Linear stability analysis identifies the ranges of Lewis numbers in which cellular or oscillatory instability can arise, with the latter form of instability disappearing above a threshold heat of reaction. These and the underlying flame solutions themselves depend on the heat of reaction and the degree of heat loss but not on the activation temperature of the branching reaction, to leading order. Near the limit of flammability a direct parallel arises with one-step kinetic models for premixed flames
Scaling, Sensitivity and Stability in the Numerical Solution of Quadratic Eigenvalue Problems
No full textThe most common way of solving the quadratic eigenvalue problem (QEP)
(\l^2 M + \l D + K)x=0 is to convert it into a linear problem
(\l X + Y)z=0 of twice the dimension and solve the linear problem
by the QZ algorithm or a Krylov method.
In doing so, it is important to understand the influence of the linearization
process on the accuracy and stability of the computed solution.
We discuss these issues for three particular linearizations:
the standard companion linearization
and two linearizations that preserve symmetry in the problem.
For illustration we employ
a model QEP describing the motion of
a beam simply supported at both ends and damped at the midpoint.
We show that the above linearizations lead to poor numerical results
for the beam problem,
but that a two-parameter scaling proposed by Fan, Lin and Van Dooren
cures the instabilities.
We also show that half of the eigenvalues of the beam QEP are pure imaginary
and are eigenvalues of the undamped problem.
Our analysis makes use of recently developed theory explaining the sensitivity
and stability of linearizations,
the main conclusions of which are summarized.
As well as arguing that
scaling should routinely be used,
we give guidance on how to choose a linearization
and illustrate the practical value of condition numbers and backward errors
Enveloping algebras of Slodowy slices and the Joseph ideal
Get PDFLet be a simple algebraic group over an algebraically closed field \k of characteristic , and \g=\text{Lie}\,G. Let be an \sl_2-triple in \g with being a long root vector in \g. Let be the -invariant bilinear form on \g with and let \chi\in\g^* be such that for all x\in\g. Let be the Slodowy slice at through the adjoint orbit of and let be the enveloping algebra of ; see [\cite{P02}]. In this note we give an explicit presentation of by generators and relations. As a consequence we deduce that contains an ideal of codimension which is unique if \g is not of type . Applying Skryabin's equivalence of categories we then construct an explicit Whittaker model for the Joseph ideal of U(\g). Inspired by Joseph's Preparation Theorem we prove that there exists a homeomorphism between the primitive spectrum of and the spectrum of all primitive ideals of infinite codimension in U(\g) which respects Goldie rank and Gelfand--Kirillov dimension. We study highest weight modules for the algebra and apply earlier results of Mili{\v c}i{\'c}--Soergel and Backelin to express the composition multiplicities of the Verma modules for in terms of some inverse parabolic Kazhdan--Lusztig polynomials. Our results confirm in the minimal nilpotent case the de Vos--van Driel conjecture on composition multiplicities of Verma modules for finite -algebras. We also obtain some general results on the enveloping algebras of Slodowy slices and determine the associated varieties of related primitive ideals of U(\g). A sequel to this paper will treat modular aspects of this theory
Algorithm 866: IFISS, a Matlab toolbox for modelling incompressible flow
Get PDFIFISS is a graphical Matlab package for the interactive numerical study
of incompressible flow problems. It includes algorithms for discretisation by
mixed finite element methods and a posteriori error estimation of the
computed solutions. The package can also be used as a computational laboratory for experimenting with state-of-the-art preconditioned iterative solvers for the discrete linear equation systems that arise in incompressible flow modelling. A unique feature of the package is its comprehensive nature; for each problem addressed, it enables the study of both discretisation and iterative solution algorithms as well as the interaction between the two and the resulting effect on overall efficiency
Computing , and Related Matrix Functions by Contour Integrals
Get PDFNew methods are proposed for the numerical evaluation of
f(\A) or f(\A) b, where f(\A) is a function such as \sqrt
\A or \log (\A) with singularities in and
\A is a matrix with eigenvalues on or near .
The methods are based on combining contour integrals
evaluated by the periodic trapezoid rule with conformal
maps involving Jacobi elliptic functions. The convergence
is geometric, so that the computation of f(\A)b is
typically reduced to one or two dozen linear system solves