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The homotopy type of the complement coordinate subspace arrangement
Get PDFThe homotopy type of the complement of a complex coordinate subspace arrangement is studied by utilising some connections between its topological and combinatorial structures. A family of arrangements for which the complement is homotopy equivalent to a wedge of spheres is described. One consequence is an application in commutative algebra: certain local rings are proved to be Golod, that is, all Massey products in their homology vanish
Global flows for stochastic differential equations without global Lipschitz conditions
No full textWe consider stochastic differential equations driven by Wiener processes. The vector fields are supposed to satisfy only local Lipschitz conditions. The Lipschitz constants of the drift vector field, valid on balls of radius R, are supposed to grow not faster than log R, while those of the diffusion vector fields are supposed to grow not faster than . We regularize the stochastic differential equations by associating with them approximating ordinary differential equations obtained by discretization of the increments of the Wiener process on small intervals. By showing that the flow associated with a regularized equation converges uniformly to the solution of the stochastic differential equation, we simultaneously establish the existence of a global flow for the stochastic equation under local Lipschitz conditions
Stability and Convergence of the Method of Fundamental Solutions for Helmholtz problems on analytic domains
Get PDFThe Method of Fundamental Solutions (MFS) is a popular tool to solve Laplace
and Helmholtz boundary value problems. Its main drawback is that it often leads
to ill-conditioned systems of equations. In this paper we investigate for the
interior Helmholtz problem
on analytic domains how the singularities (charge points) of the MFS basis
functions have to be chosen such that approximate solutions can be represented
by the MFS basis in a numerically stable way. For Helmholtz problems on the
unit disc we give a full analysis
which includes the high frequency (short wavelength) limit.
For more difficult and nonconvex
domains such as crescents we demonstrate how the
right choice of charge points is connected to how far
into the complex plane the solution of the boundary value problem can be
analytically continued, which in turn depends on both domain shape
and boundary data.
Using this we develop a recipe for locating charge points which
allows us to reach error norms of typically on a wide variety
of analytic domains.
At high frequencies of order only 3 points per wavelength are
needed, which compares very favorably to boundary integral methods
A Newton Algorithm for the Nearest Correlation Matrix
Get PDFFirstly, we describe and investigate the algorithm of Qi and Sun which solves the problem of finding the nearest correlation matrix to a symmetric matrix. This algorithm claims a quadratic convergence. We discuss improving this algorithm's efficiency and reliability and detect a problem when we are aiming at a nearest correlation matrix with a high accuracy, using small error tolerences. As a consequence, we suggest a modified version, based on the algorithm of Qi and Sun, which is also a quadratically convergent algorithm, has improved efficiency and is modified so that the algorithm can return the nearest correlation matrix to high accuracy showing a robust and reliable behaviour.
Secondly, we investigate the general alternating projections method and also Higham's alternating projections method for the nearest correlation matrix. We discuss variations of the latter and include a further projection which allows more constraints to be added to the problem. We introduce a new algorithm and compare its convergence behaviour with Higham's alternating projections method
On the distance between a teacher and a class
No full textThe main aim of this article is to explore models of a teacher addressing a class
sitting in a rectangular classroom or lecture theatre and derive central measures of
the distance between the teacher and the class or between a lecturer and an audience.
This involves an interesting range of mathematics and two metrics. A secondary
aim is to discuss according to several criteria the optimization of the position chosen
by a teacher when addressing a class. Thus, in this article, mathematics is applied
to its teaching
Review of ``Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators'', by Lloyd N. Trefethen and Mark Embree. Princeton University Press, Princeton, NJ, USA, 2005.
Get PDFLAPACK-Style Codes for Pivoted Cholesky and QR Updating
Get PDFRoutines exist in LAPACK for computing the Cholesky
factorization of a symmetric positive definite
matrix and in LINPACK there is a pivoted routine for positive semidefinite matrices. We present new higher level BLAS LAPACK-style codes for computing this pivoted factorization. We show that these can be many times faster than the LINPACK code. Also, with a new stopping criterion, there is more reliable rank detection and smaller normwise backward error. We also present algorithms that update the QR factorization of a matrix after it has had a block of rows or columns added or a block of columns deleted.
This is achieved by updating the factors Q and R of the original matrix. We present some LAPACK-style codes and show these can be much faster than computing the factorization from scratch
Solving Systems of Linear Equations on the CELL Processor Using Cholesky Factorization
Get PDFThe STI CELL processor introduces
pioneering solutions in processor architecture. At the
same time it presents new challenges for the development
of numerical algorithms. One is effective exploitation
of the differential between the speed of single
and double precision arithmetic; the other is efficient
parallelization between the short vector SIMD
cores. In this work, the first challenge is addressed
by utilizing a mixed-precision algorithm for the solution
of a dense symmetric positive definite system of
linear equations, which delivers double precision accuracy,
while performing the bulk of the work in single
precision. The second challenge is approached by
introducing much finer granularity of parallelization
than has been used for other architectures and using
a lightweight decentralized synchronization. The
implementation of the computationally intensive sections
gets within 90 percent of peak floating point
performance, while the implementation of the memory
intensive sections reaches within 90 percent of
peak memory bandwidth. On a single CELL processor,
the algorithm achieves over 170 Gflop/s when
solving a symmetric positive definite system of linear
equation in single precision and over 150 Gflop/s
when delivering the result in double precision accuracy
Reduction groups and automorphic Lie algebras
Get PDFWe study a new class of infinite dimensional Lie algebras, which has important applications to the theory of integrable equations. The construction of these algebras is very similar to the one for automorphic functions and this motivates the name automorphic Lie algebras. For automorphic Lie algebras we present bases in which they are quasigraded and all structure constants can be written out explicitly. These algebras have useful factorisations on two subalgebras similar to the factorisation of the current algebra on the positive and negative parts
Multicomponent integrable wave equations: I. Darboux-dressing transformation
Get PDFThe Darboux-dressing transformations are applied to the Lax pair associated with systems of coupled nonlinear wave equations in the case of boundary values which are appropriate to both 'bright' and 'dark' soliton solutions. The general formalism is set up and the relevant equations are explicitly solved. Several instances of multicomponent wave equations of applicative interest, such as vector nonlinear Schrödinger-type equations and three resonant wave equations, are considered