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The cohomology of pro-p groups with a powerfully embedded sub-group
No full textWe calculate the cohomology of a pro-p group with an extendable and almost powerfully embedded subgroup
An alternative form of the Helmholtz criterion in the inverse problem of the calculus of variations
No full textWe give a necessary and sufficient condition for the existence of a local solution of the inverse problem of calculus of variations in terms of the identical vanishing of the variation of a functional on an extended space (with the number of independent variables increased by one), and explain its relation with the classical Helmholtz criterion using the de Rham complex on an infinite-dimensional space of fields
Algorithms for Cholesky and QR Factorizations, and the Semidefinite Generalized Eigenvalue Problem
Get PDFWe consider algorithms for three problems in numerical
linear algebra: computing the pivoted Cholesky factorization, solving the semidefinite generalized eigenvalue problem and updating the QR
factorization.
Fortran 77 codes exist in LAPACK for computing the
Cholesky
factorization (without pivoting) of a symmetric
positive definite
matrix using Level 2 and 3 BLAS. In LINPACK there is
a Level 1
BLAS routine for computing the Cholesky factorization
with complete
pivoting of a symmetric positive semidefinite
matrix. We
present two new algorithms and Fortran 77 LAPACK-style
codes for
computing this pivoted factorization: one using Level
2 BLAS and one
using Level 3 BLAS. We show that on modern machines
the new codes
can be many times faster than the LINPACK code. Also,
with a new
stopping criterion they provide more reliable rank
detection and can
have a smaller normwise backward error.
The generalized eigenvalue problem Ax=lambda Bx in
the case where
A and B are real and symmetric and B is positive
semidefinite
is considered. We present an algorithm for solving
this problem that
has a potentially smaller operation count than
existing methods and
requires no further restrictions on A and B. The
eigenvalues of
the problem are classified as finite or infinite.
Nonregular matrix
pencils, where the eigenvalues can take any value, are
also
discussed and a deflation strategy is given. We
include a MATLAB
code for our algorithm and give some numerical
experiments.
We also treat the problem of updating the QR
factorization, with
applications to the least squares problem. Algorithms
are presented
that compute the factorization A-tilde = Q-tilde R-tilde where
A-tilde is the
matrix A=QR after it has had a number of rows or
columns added or
deleted. This is achieved by updating the factors Q
and R, and
we show this can be much faster than computing the
factorization of
A-tilde from scratch. We consider algorithms that
exploit the Level 3
BLAS where possible and place no restriction on the
dimensions of
A or the number of rows and columns added or
deleted. For some of
our algorithms we present Fortran 77 LAPACK-style code
and show the
backward error of our updated factors is comparable to
the error
bounds of the QR factorization of A-tilde
w-function of the KDV hierarchy
No full textIn the present paper we construct a family of differential commuting
multidimensional operators of order 3, which is closely related to the KdV
hierarchy. We find a common eigenfunction of this family and an algebraic
relation between these operators. Using these operators we associate a hyperelliptic
curve to any solution of stationary KdV equation. A basic generating
function of the solution of stationary KdV equation is introduced as special
polarization of the hyperelliptic curve equation. The w-function is defined as
a unique function with the following property: the second logarithmic derivatives
of w are given by the basic generating function of the stationary g-KdV
equation solution, in particular, the second logarithmic derivative of w with
respect to x gives this solution
Blocks with trivial intersection defect groups
No full textWe show that each block whose defect groups intersect pairwise trivially either has cyclic or generalised quaternion defect groups, or is Morita equivalent to one of a given list of blocks of central extensions of automorphism groups of non-abelian simple groups. In particular we classify all blocks of automorphism groups of non-abelian simple groups whose defect groups are non-cyclic and intersect pairwise trivially. A consequence is that Donovans conjecture holds for blocks whose defect groups intersect pairwise trivially
Colimits, Stanley-Reisner algebras, and loop spaces
Get PDFThe book consists of articles at the frontier of current research in Algebraic Topology. It presents recent results by top notch experts, and is intended primarily for researchers and graduate students working in the field of algebraic topology. Included is an important article by Cohen, Johnes and Yan on the the homology of the space of smooth loops on a manifold M, endowed with the Chas-Sullivan intersection product, as well as an article by Goerss, Henn and Mahowald on stable homotopy groups of spheres, which uses the cutting edge technology of "topological modular forms"
Electromagnetic inverse problems for nematic liquid crystals and capacitance imaging
Get PDFThe aim of this study is to formulate and solve the high-frequency electromagnetic problem of wave propagation through nematic liquid crystal cells of arbitrary shape, and subsequently approach the inverse problem of reconstructing the orientational order by means of recovering the dielectric tensors in the interior from a finite set of boundary polarization measurements. The numerical solution of the forward electromagnetic problem is achieved by hybridizing the conventional vector finite elements with a boundary integral method, so that to preserve the necessary continuity conditions for the electromagnetic fields at the boundary of the domain. Combining the finite element equations with a magnetic field integral boundary equation yields surface integrals involving Green s function and its gradient. These integrals have kernels that become asymptotically singular as the distance between the observation and integration points reduces to zero, essentially making the numerical integration process problematic. For their computation a new basis of functions are introduced for the surface current density, the so-called Rao-Wilton-Glisson functions, which e®ectively substitute the tangential components of the magnetic field in the original boundary integrals. The transformed integrals are then treated with the singularity extraction method, essentially separating the smooth from the singular components of the kernels, the former of which are computed using conventional numerical integration and the later using closed form expressions derived for the RWG functions. The forward problem is then linearized with the aid of the Fr´echet derivative of the forward Maxwell operator and subsequently regularized using a Tikhonov type regularization. In regularization we construct a penalty term based on Frank s distortion energy functional, which is known to have a minimum in the neighborhood of stable liquid crystal director configurations. The inverse problem of reconstructing the orientation of the director vector in a uniaxial nematic liquid crystal using a finite set of noise infused boundary polarization measurements is approached as a special case of the inverse permittivity tensor problem, where the dielectric tensors are symmetric and expected to vary most significantly along the directions of their two biggest eigenvalues, which correspond to the associated Euler angles of the director vector
Computing the Polar Decomposition and the Matrix Sign Decomposition in Matrix Groups
Get PDFFor any matrix automorphism group \G associated with a
bilinear or sesquilinear form,
Mackey, Mackey, and Tisseur have recently shown that the
matrix sign decomposition factors of A\in\G also lie in \G;
moreover, the polar factors of lie in \G if the matrix of the underlying
form is unitary.
Groups satisfying the latter condition
include the complex orthogonal,
real and complex symplectic, and pseudo-orthogonal groups.
This work is concerned with exploiting the structure of \G when computing the
polar and matrix sign decompositions of matrices in \G.
We give sufficient conditions for a matrix iteration to preserve the
group structure and show that a family of globally convergent
rational Pad\'e-based iterations of Kenney and Laub satisfy these conditions.
The well-known scaled Newton iteration for computing the unitary polar
factor does not preserve group structure,
but we show that the approach of the iterates to the group
is precisely tethered to the approach to unitarity,
and that this forces a different and exploitable structure in the iterates.
A similar relation holds for the Newton iteration for the matrix sign function.
We also prove that the number of iterations needed for convergence of the
structure-preserving methods
can be precisely predicted by running an associated scalar iteration.
Numerical experiments are given to compare the cubically and quintically
converging iterations with Newton's method and to test stopping criteria.
The overall conclusion is that the structure-preserving iterations
and the scaled Newton iteration are all of practical interest,
and which iteration is to be preferred is problem-dependent
A 11,707,448,673,375 vertex graph related to the baby monster II
No full textContinuing the work in [7] we complete the combinatorial description of the point-line
collinearity graph of the characteristic 2 minimal parabolic geometry for the Baby Monster
simple group
