The Hanna Neumann Conjecture provides a bound on the rank of the intersection of finitely generated subgroups of a free group. We will follow Mineyev's recent elementary and beautiful proof of this longstanding conjecture.
Several stochastic simulation algorithms (SSAs) have been recently
proposed for modelling reaction-diffusion processes in cellular and molecular biology. In this talk, two commonly used SSAs will be studied. The first SSA is an on-lattice model described by the reaction-diffusion master equation. The second SSA is an off-lattice model based on the simulation of Brownian motion of individual molecules and their reactive collisions. The connections between SSAs and the deterministic models (based on reaction- diffusion PDEs) will be presented. I will consider chemical reactions both at a surface and in the bulk. I will show how the "microscopic" parameters should be chosen to achieve the correct "macroscopic" reaction rate. This choice is found to depend on which SSA is used. I will also present multiscale algorithms which use models with a different level of detail in different parts of the computational domain
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.