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On the volume of tubular neighbourhoods of real algebraic varieties
Get PDFThe problem of determining the volume of a tubular neighbourhood has a long and rich history. Bounds on the volume of neighbourhoods of algebraic sets have turned out to play an important role in the probabilistic analysis of condition numbers in numerical analysis. We present a self-contained derivation of bounds on the probability that a random point, chosen uniformly from a ball, lies within a given distance of a real algebraic variety of any codimension. The bounds are given in terms of the degrees of the defining polynomials, and contain as special case an unpublished result by Ocneanu
Mechanisms of intermittent state transitions in a coupled heterogeneous oscillator model of epilepsy
Get PDFWe investigate the dynamic mechanisms underlying intermittent state transitions in a recently proposed neural
mass model of epilepsy. We hypothesise the properties of the neural mass model contributing to the observed state
switching dynamics are i) coupling between oscillators and ii) heterogeneous proximity of these oscillators to a
bifurcation between distinct limit cycles. In order to test this hypothesis we construct a low dimensional, abstract
model preserving only these features and demonstrate that state transitions due to intermittency occur. This
suggests that there is a general bifurcation mechanism responsible for this behaviour and that this is independent
of the precise form of the evolution equations. Such abstractions of neural mass models allow a deeper insight
into the underlying dynamic and physiological mechanisms and also allow the more ecient exploration of large
scale brain dynamics in disease
Linear methods in the study of automorphisms
Get PDFMotivation: Studying automorphisms and their fixed points is one of the main
areas of research in Group Theory. For soluble and nilpotent groups, it is
natural to use the advantages of linear methods of representation theory and
Lie ring theory. For finite groups in particular, in view of the classification of
finite simple groups, many questions are now largely reduced to soluble and
nilpotent groups.
List of topics:
1. Automorphisms as linear transformations.
2. Clifford and Hall--Higman--type theorems.
3. Bounding Fitting height.
4. Powerful -groups.
5. Lie ring methods.
6. Using EXP and LOG functors.
7. Method of generalized centralizers.
8. Elimination of operators by nilpotenc
Connectivity of Local Fusion Graphs for Finite Simple Groups
Get PDFThe main result proved here is that for a finite simple group and a -conjugacy class of involutions the local fusion graph is a connected graph
Blocked Schur Algorithms for Computing the Matrix Square Root
Get PDFThe Schur method for computing a matrix square root reduces the matrix to the Schur triangular form and then computes a square root of the triangular matrix. We show that by using either standard blocking or recursive blocking the computation of the square root of the triangular matrix can be made rich in matrix multiplication. Numerical experiments making appropriate use of level 3 BLAS show significant speedups over the point algorithm, both in the square root phase and in the algorithm as a whole. In parallel implementations, recursive blocking is found to provide better performance than standard blocking when the parallelism comes only from threaded BLAS, but the reverse is true when parallelism is explicitly expressed using OpenMP. The excellent numerical stability of the point algorithm is shown to be preserved by blocking. These results are extended to the real Schur method. Blocking is also shown to be effective for multiplying triangular matrices
Covariance Structure Regularization via Entropy Loss Function
Get PDFThe need to estimate structured covariance matrices arises in a variety of applications and the problem is widely studied in statistics. We propose a new method for regularizing the covariance structure of a given covariance
matrix, in which the underlying structure is usually blurred due to random noises particularly when the dimension of the covariance matrix is high. The regularization is made
by choosing an optimal structure from an available class of covariance structures in terms of minimizing the discrepancy, defined via the entropy loss function, between the given matrix and the class. A range of potential candidate structures such as tridiagonal, compound symmetry,
AR(1), and Toeplitz are considered. Simulation studies are conducted, showing that the proposed new approach is reliable in regularization of covariance structures. The approach is also applied to real data analysis, demonstrating the usefulness of the proposed approach in practice
Minimal indices and minimal bases via filtrations
No full textIn this note we develop a new way of formulating
the notions of minimal basis and minimal indices,
based on the concept of a filtration of a vector space.
The goal is to provide useful new tools
for working with these important concepts,
as well as to gain deeper insight
into their fundamental nature.
This approach also readily reveals a strong minimality property of minimal indices, from which follows
a characterization of the vector polynomial bases
in rational vector spaces.
The effectiveness of this new formulation is further illustrated by proving two fundamental properties:
the invariance of the minimal indices
of a matrix polynomial under field extension,
and the direct sum property of minimal indices
Witten-Hodge theory for manifolds with boundary and equivariant cohomology
No full textWe consider a compact, oriented, smooth Riemannian
manifold (with or without boundary) and we suppose is a
torus acting by isometries on . Given in the Lie algebra and
corresponding vector field on , one defines Witten's
inhomogeneous coboundary operator \d_{X_M} = \d+\iota_{X_M}:
\Omega_G^\pm \to\Omega_G^\mp (even/odd invariant forms on ) and
its adjoint . Witten \cite{Witten} showed that the
resulting cohomology classes have -harmonic representatives
(forms in the null space of \Delta_{X_M} =
(\d_{X_M}+\delta_{X_M})^2), and the cohomology groups are
isomorphic to the ordinary de Rham cohomology groups of the set
of zeros of . Our principal purpose is to extend these
results to manifolds with boundary. In particular, we define
relative (to the boundary) and absolute versions of the
-cohomology and show the classes have representative
-harmonic fields with appropriate boundary conditions. To do
this we present the relevant version of the Hodge-Morrey-Friedrichs
decomposition theorem for invariant forms in terms of the operators
\d_{X_M} and . We also elucidate the connection
between the -cohomology groups and the relative and absolute
equivariant cohomology, following work of Atiyah and Bott. This
connection is then exploited to show that every harmonic field with
appropriate boundary conditions on has a unique
-harmonic field on , with corresponding boundary conditions.
Finally, we define the -Poincar\'{e} duality angles
between the interior subspaces of -harmonic fields on with
appropriate boundary conditions, following recent work of DeTurck and Gluck
A priori error analysis of stochastic Galerkin mixed approximations of elliptic PDEs with random data
Get PDFWe construct stochastic Galerkin approximations to the solution of a first order system of PDEs with random coefficients. Under the standard finite-dimensional noise assumption, we transform the variational saddle point problem to a parametric deterministic one. Approximations are constructed by combining mixed finite elements on the computational domain with -variate tensor product polynomials. We study the inf-sup stability and well-posedness of the continuous and finite-dimensional problems, the regularity of solutions with respect to the parameters describing the random coefficients, and establish a priori error estimates for stochastic Galerkin finite element approximations
Hybrid intelligent parameter estimation based on grey case-based reasoning for laminar cooling process
Get PDFIn this paper, a hybrid intelligent parameter estimation algorithm is proposed for predicting the strip
temperature during laminar cooling process. The algorithm combines a hybrid genetic algorithm (HGA) with
grey case-based reasoning (GCBR) in order to improve the precision of the strip temperature prediction. In
this context, the hybrid genetic algorithm is formed by combining the genetic algorithm with an annealing
and a local multidimensional search algorithm based on deterministic inverse parabolic interpolation. Firstly,
the weight vectors of retrieval features in case-based reasoning are optimised using hybrid genetic algorithm
in of�ine mode, and then they are used in grey case-based reasoning to accurately estimate the model
parameters online. The hybrid intelligent parameter estimation algorithm is validated using a set of
operational data gathered from a hot-rolled strip laminar cooling process in a steel plant. Experiment results
show the effectiveness of the proposed method in improving the precision of the strip temperature
prediction. The proposed method can be used in real-time temperature control of hot-rolled strip and has
potential for parameter estimation ofdifferenttypesofcoolingprocess