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Linear methods in the study of automorphisms
Get PDFMainly about automorphisms of finite groups, but also some infinite (especially nilpotent).
1. Survey: Results on fixed-point-free and almost fixed-point-free automorphisms.
Open problems
2. Some methods of representation theory: Automorphisms as linear transformations.
Clifford's theorem. Hall--Higman--type theorems.
Automorphism of prime order with fixed-point subgroup of given rank.
3. Lie ring methods: Automorphisms of Lie rings.
Associated Lie rings.
Method of graded centralizers.
Automorphism of order acting on a finite -group.
Frobenius groups of automorphisms with fixed-point-free kernel.
Lazard Lie algebra.
4. Baker--Campbell--Hausdorff formula: Mal'cev correspondence.
Lazard correspondence.
Automorphism of order acting on a finite -group.
5. Elimination of operators by nilpotency
On -soluble groups with a generalized -central or powerful Sylow -subgroup
Get PDFLet be a finite -soluble group, and a Sylow -sub\-group of . It is proved
that if all elements of of order (or of order for ) are
contained in the -th term of the upper central series of , then the -length of
is at most , where is the greatest integer such that
, and the exponent of the image of in is at most
. It is also proved that if is a powerful
-group, then the -length of is equal to 1
Vector spaces of linearizations for matrix polynomials: a bivariate polynomial approach
Get PDFWe revisit the important paper [D. S. Mackey, N. Mackey, C. Mehl, and V. Mehrmann, {SIAM J. Matrix Anal. Appl.}, 28 (2006), pp.~971--1004] and, by viewing matrices as coefficients for bivariate polynomials, we provide concise proofs for key properties of linearizations for matrix polynomials. We also show that every pencil in the double ansatz space is intrinsically connected to a Bezout matrix, which we use to prove the eigenvalue exclusion theorem.
In addition our exposition allows for any degree-graded basis, the monomials being a special case. Matlab code is given to construct the pencils in the double ansatz space for matrix polynomials expressed in any orthogonal basis
A spatially adaptive iterative method for a class of nonlinear operator eigenproblems
No full textWe present a new algorithm for the iterative solution of nonlinear operator eigenvalue problems arising from partial differential equations. This algorithm combines automatic spatial resolution of linear operators with the infinite Arnoldi method for nonlinear matrix eigenproblems proposed in [E. JARLEBRING, W. MICHIELS, AND K. MEERBERGEN, A linear eigenvalue algorithm for the nonlinear eigenvalue problem, Numer. Math., 122 (2012)]. The iterates in this infinite Arnoldi method are functions, and each iteration requires the solution of an inhomogeneous differential equation. This formulation is independent of the spatial representation of the functions, which allows us to employ a dynamic representation with an accuracy of about the level of machine precision at each iteration, similar to what is done in the Chebfun system [Z. BATTLES AND L. N. TREFETHEN, An extension of MATLAB to continuous functions and operators, SIAM J. Sci. Comput., 25 (2004)] with its chebop functionality [T. A. DRISCOLL, F. BORNEMANN, AND L. N. TREFETHEN, The chebop system for automatic solution of differential equations, BIT, 48 (2008)], although our function representation is entirely based on coefficients instead of function values. Our approach also allows for nonlinearities in the boundary conditions of the PDE. The algorithm is illustrated with several examples, e.g., the study of eigenvalues of a vibrating string with delayed boundary feedback control
Variational data assimilation using targetted random walks
Get PDFThe variational approach to data assimilation is a widely used methodology for both online prediction and for reanalysis. In either of these scenarios, it can be important to assess uncertainties in the assimilated state. Ideally, it is desirable to have complete information concerning the Bayesian posterior distribution for unknown state given data. We show that complete computational probing of this posterior distribution is now within the reach in the offline situation. We introduce a Markov chain�Monte Carlo (MCMC) method which enables us to directly sample from the Bayesian posterior distribution on the unknown functions of interest given observations. Since we are aware that these methods are currently too computationally expensive to consider using in an online filtering scenario, we frame this in the context of offline reanalysis. Using a simple random walk-type MCMC method, we are able to characterize the posterior distribution using only evaluations of the forward model of the problem, and of the model and data mismatch. No adjoint model is required for the method we use; however, more sophisticated MCMC methods are available which exploit derivative information. For simplicity of exposition, we consider the problem of assimilating data, either Eulerian or Lagrangian, into a low Reynolds number flow in a two-dimensional periodic geometry. We will show that in many cases it is possible to recover the initial condition and model error (which we describe as unknown forcing to the model) from data, and that with increasing amounts of informative data, the uncertainty in our estimations reduces
On linear equations in free Lie algebras
No full textWe investigate equations of the form over a free Lie algebra
. In the case where the coefficients are free generators of , we generalize a number of earlier results on equations with two variables to equations with an arbitrary number of indeterminates. Our main results refer to the case where the coefficients coincide with the free generators of . We give a detailed description of the solution space and we obtain an explicit basis for its multilinear fine homogeneous component
Generalized Dirichlet to Neumann operator on invariant differential forms and equivariant cohomology
Get PDFIn recent work, Belishev and Sharafutdinov show that the generalized Dirichlet to Neumann
(DN) operator � on a compact Riemannian manifold M with boundary �M determines
de Rham cohomology groups of M. In this paper, we suppose G is a torus acting by
isometries on M. Given X in the Lie algebra of G and the corresponding vector field XM
on M, Witten defines an inhomogeneous coboundary operator dXM
= d + ιXM on invariant
forms on M. The main purpose is to adapt Belishev�Sharafutdinov�s boundary data to
invariant forms in terms of the operator dXM in order to investigate to what extent the
equivariant topology of a manifold is determined by the corresponding variant of the DN
map. We define an operator �XM on invariant forms on the boundary which we call the
XM-DN map and using this we recover the XM-cohomology groups from the generalized
boundary data (�M,�XM ). This shows that for a Zariski-open subset of the Lie algebra,
�XM determines the free part of the relative and absolute equivariant cohomology groups
of M. In addition, we partially determine the ring structure of XM-cohomology groups
from �XM . These results explain to what extent the equivariant topology of the manifold
in question is determined by �XM
Cell Death: Linear Control Analysis of Eissing's Model
Get PDFWe deconstruct Eissing's intrinsic apoptosis model using linear control theory. In the life steady state the linearised dynamics are shown to be a tightly coupled but unstable multi- variable system. The life steady state is stabilized biochemically by decentralised XIAP and CARP acting as lead controllers. The small gain theorem is used to analyse stability and to give insight into how the inhibitors naturally modulate cell death and highlighting the role played by positive and negative feedback. Finally we use simulations to examine the extent to which recovery is possible once apoptosis has been initiated
Triangularizing Quadratic Matrix Polynomials
Get PDFWe show that any regular quadratic matrix polynomial can be reduced to an upper
triangular quadratic matrix polynomial over the complex numbers
preserving the finite and infinite elementary divisors.
We characterize the real quadratic matrix polynomials that are
triangularizable over the real numbers and show that those that are
not triangularizable over the real numbers are quasi-triangularizable
with diagonal blocks of sizes and .
We also derive complex and real Schur-like theorems for linearizations of
quadratic matrix polynomials with nonsingular leading coefficients.
In particular, we show that for any monic linearization \l I+A of an
quadratic matrix polynomial,
there exists a nonsingular matrix defined in terms of
orthonormal vectors that transforms to a companion
linearization of a (quasi)-triangular quadratic matrix polynomial.
This provides the foundation for designing numerical algorithms for the
reduction of quadratic matrix polynomials to upper (quasi)-triangular form
Preconditioning steady-state Navier-Stokes equations with random data
Get PDFWe consider the numerical solution of the steady-state
Navier--Stokes equations with uncertain data. Specifically, we treat the case of uncertain viscosity, which results in a flow with an uncertain Reynolds number. After linearization, we apply a stochastic Galerkin finite element method, combining standard inf-sup stable Taylor--Hood approximation on the spatial domain (on highly stretched grids), with orthogonal polynomials in the stochastic parameter. This yields a sequence of non-symmetric saddle-point problems with Kronecker product structure. The novel contribution of this study lies in the construction of efficient block triangular preconditioners for these discrete systems, for use with GMRES. Crucially, the preconditioners are robust with respect to the discretization and statistical parameters, and we exploit existing deterministic solvers based on the so-called Pressure Convection-Diffusion and Least-Squares Commutator approximations