University of Oxford

We treat the first n terms of general orthogonal series evolving with n as the partial sum process, and proved that under Menshov-Rademacher condition, the partial sum process can be enhanced into a geometric 2-rough process. For Fourier series, the condition can be improved, with an equivalent condition on limit function identified.

 In the recent series of papers Kleban, Simmons, and Ziff gave a non-rigorous computation  (base on Conformal Field Theory) of probabilities of several connectivity events for critical percolation. In particular they showed that the probability that there is a percolation cluster connecting two points on the boundary and a point inside the domain can be factorized in therms of pairwise connection probabilities. We are going to use SLE techniques to rigorously compute probabilities of several connectivity events and prove the factorization formula.

Boundary layers are often studied with no pressure gradient

or with an imposed pressure gradient. Either of these assumptions

can lead to difficulty in obtaining solutions. A major advance in fluid

dynamics last century (1969) was the development of a triple deck

formulation for boundary layers where the pressure is not

specified but emerges through an interaction between

boundary layer and the inviscid outer flow. This has given rise to

new computational problems and computations have in turn

fed ideas back into theoretical developments. In this survey talk

based on my new book, I will look at three problems:

flow past a plate, flow separation and flow in channels

and discuss the interaction between theory and computation

in advancing boundary layer theory.

The general importance of preconditioning in combination with an

appropriate iterative technique for solving large scale linear(ised)

systems is widely appreciated. For definite problems (where the

eigenvalues lie in a half-plane) there are a number of preconditioning

techniques with a range of applicability, though there remain many

difficult problems. For indefinite systems (where there are eigenvalues

in both half-planes), techniques are generally not so well developed.

Constraints arise in many physical and mathematical problems and

invariably give rise to indefinite linear(ised) systems: the incompressible

Navier-Stokes equations describe conservation of momentum in the

presence of viscous dissipation subject to the constraint of

conservation of mass, for transmission problems the solution on an

interior domain is often solved subject to a boundary integral which

imposes the exterior field, in optimisation the appearance of

constraints is ubiquitous...

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We will describe two approaches to preconditioning such constrained

systems and will present analysis and numerical results for each. In

particular, we will describe the applicability of these techniques to

approximations of incompressible Navier-Stokes problems using mixed

finite element approximation.

(Joint work with Felipe Cucker and Dennis Cheung, City University of Hong Kong.)

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Condition numbers are important complexity-theoretic tools to capture

a "distillation" of the input aspects of a computational problem that

determine the running time of algorithms for its solution and the

sensitivity of the computed output. The motivation for our work is the

desire to understand the average case behaviour of linear programming

algorithms for a large class of randomly generated input data in the

computational model of a machine that computes with real numbers. In

this model it is not known whether linear programming is polynomial

time solvable, or so-called "strongly polynomial". Closely related to

linear programming is the problem of either proving non-existence of

or finding an explicit example of a point in a polyhedral cone defined

in terms of certain input data. A natural condition number for this

computational problem was developed by Cheung and Cucker, and we analyse

its distributions under a rather general family of input distributions.

We distinguish random sampling of primal and dual constraints

respectively, two cases that necessitate completely different techniques

of analysis. We derive the exact exponents of the decay rates of the

distribution tails and prove various limit theorems of complexity

theoretic importance. An interesting result is that the existence of

the k-th moment of Cheung-Cucker's condition number depends only very

mildly on the distribution of the input data. Our results also form

the basis for a second paper in which we analyse the distributions of

Renegar's condition number for the randomly generated linear programming

problem.

Toeplitz matrices enjoy the dual virtues of ubiquity and beauty. We begin this talk by surveying some of the interesting spectral properties of such matrices, emphasizing the distinctions between infinite-dimensional Toeplitz matrices and the large-dimensional limit of the corresponding finite matrices. These basic results utilize the algebraic Toeplitz structure, but in many applications, one is forced to spoil this structure with some perturbations (e.g., by imposing boundary conditions upon a finite difference discretization of an initial-boundary value problem). How do such

perturbations affect the eigenvalues? This talk will address this question for "localized" perturbations, by which we mean perturbations that are restricted to a single entry, or a block of entries whose size remains fixed as the matrix dimension grows. One finds, for a broad class of matrices, that sufficiently small perturbations fail to alter the spectrum, though the spectrum is exponentially sensitive to other perturbations. For larger real single-entry

perturbations, one observes the perturbed eigenvalues trace out curves in the complex plane. We'll show a number of illustrations of this phenomenon for tridiagonal Toeplitz matrices.

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This talk describes collaborative work with Albrecht Boettcher, Marko Lindner, and Viatcheslav Sokolov of TU Chemnitz.