Anomalous ( non local) diffusion processes appear in many subjects: phase transition, fracture dynamics, game theory I will describe some of the issues involved, and in particular, existence and regularity for some non local versions of the p Laplacian, of non variational nature, that appear in non local tug of war.
The talk will be devoted to criteria of absence of arbitrage opportunities under small transaction costs for a family of multi-asset models of financial market.
Please note the earlier than usual start-time!
The Mathematical Institute invites you to attend the Inaugural Lecture of Professor Gui-Qiang G. Chen. Professor in the Analysis of Partial Differential Equations. Examination Schools, 75-81 High Street, Oxford, OX 4BG.
There is no charge to attend but registration is required. Please register your attendance by sending an email to @email specifying the number of people in your party. Admission will only be allowed with prior registration.
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ABSTRACT
While calculus is a mathematical theory concerned with change, differential equations are the mathematician's foremost aid for describing change. In the simplest case, a process depends on one variable alone, for example time. More complex phenomena depend on several variables – perhaps time and, in addition, one, two or three space variables. Such processes require the use of partial differential equations. The behaviour of every material object in nature, with timescales ranging from picoseconds to millennia and length scales ranging from sub-atomic to astronomical, can be modelled by nonlinear partial differential equations or by equations with similar features. The roles of partial differential equations within mathematics and in the other sciences become increasingly significant. The mathematical theory of partial differential equations has a long history. In the recent decades, the subject has experienced a vigorous growth, and research is marching on at a brisk pace.
In this lecture, Professor Gui-Qiang G. Chen will present several examples to illustrate the origins, developments, and roles of partial differential equations in our changing world.
Efficient linear and infinitely smooth approximation of functions from equidistant samples is a fascinating problem, at least since Runge showed in 1901 that it is not delivered by the interpolating polynomial.
In 1988, I suggested to substitute linear rational for polynomial interpolation by replacing the denominator 1 with a polynomial depending on the nodes, though not on the interpolated function. Unfortunately the so-obtained interpolant converges merely as the square of the mesh size. In 2007, Floater and Hormann have given for every integer a denominator that yields convergence of that prescribed order.
In the present talk I shall present the corresponding interpolant as well as some of its applications to differentiation, integration and the solution of boundary value problems. This is joint work with Georges Klein and Michael Floater.
The maximum principle is one of the main tools use to understand the behaviour of solutions to the Ricci flow. It is a very powerful tool that can be used to show that pointwise inequalities on the initial data of parabolic PDE are preserved by the evolution. A particular weak maximum principle for vector bundles will be discussed with references to Hamilton's seminal work [J. Differential Geom. 17 (1982), no. 2, 255–306; MR664497] on 3-manifolds with positive Ricci curvature and his follow up paper [J. Differential Geom. 24 (1986), no. 2, 153–179; MR0862046] that extends to 4-manifolds with various curvature assumptions.
This talk highlights the role of Gaussian Process models in sequential
data analysis. Issues of active data selection, global optimisation,
sensor selection and prediction in the presence of changepoints are
discussed.
Abstract. In this talk, I will present joint work with Uri Onn, Mark Berman, and Pirita Paajanen.
Let G be a linear algebraic group defined over the integers. Let O be a compact discrete valuation ring with a finite residue field of cardinality q and characteristic p. The group
G(O) has a filtration by congruence subgroups
G_m(O) (which is by definition the kernel of reduction map modulo P^m where P is the maximal ideal of O).
Let c_m=c_m(G(O)) denote the number of conjugacy classes in the finite quotient group G(O)/G_m(O) (which is called the mth congruence quotient of G(O)). The conjugacy class zeta function of
G(O) is defined to be the Dirichlet series Z_{G(O)}(s)=\sum_{m=0,1,...} c_m q^_{-ms}, where s is a complex number with Re(s)>0. This zeta function was defined by du Sautoy when G is a p-adic analytic group and O=Z_p, the ring of p-adic integers, and he proved that in this case it is a rational function in p^{-s}. We consider the question of dependence of this zeta function on p and more generally on the ring O.
We prove that for certain algebraic groups, for all compact discrete valuation rings with finite residue field of cardinality q and sufficiently large residue characteristic p, the conjugacy class zeta function is a rational function in q^{-s} which depends only on q and not on the structure of the ring. Note that this applies also to positive characteristic local fields.
A key in the proof is a transfer principle. Let \psi(x) and f(x) be resp.
definable sets and functions in Denef-Pas language.
For a local field K, consider the local integral Z(K,s)=\int_\psi(K)
|f(x)|^s dx, where | | is norm on K and dx normalized absolute value
giving the integers O of K volume 1. Then there is some constant
c=c(f,\psi) such that for all local fields K of residue characteristic larger than c and residue field of cardinality q, the integral Z(K,s) gives the same rational function in q^{-s} and takes the same value as a complex function of s.
This transfer principle is more general than the specialization to local fields of the special case when there is no additive characters of the motivic transfer principle of Cluckers and Loeser since their result is the case when the integral is zero.
The conjugacy class zeta function is related to the representation zeta function which counts number of irreducible complex representations in each degree (provided there are finitely many or finitely many natural classes) as was shown in the work of Lubotzky and Larsen, and gives information on analytic properties of latter zeta function.
We will investigate what one can detect about a discrete group from its profinite completion, with an emphasis on considering geometric properties.
Cryopreservation (using temperatures down to that of liquid nitrogen at
–196 °C) is the only way to preserve viability and function of mammalian cells for research and transplantation and is integral to the quickly evolving field of regenerative medicine. To cryopreserve tissues, cryoprotective agents (CPAs) must be loaded into the tissue. The loading is critical because of the high concentrations required and the toxicity of the CPAs. Our mathematical model of CPA transport in cartilage describes multi-component, multi-directional, non-dilute transport coupled to mechanics of elastic porous media in a shrinking and swelling domain.
Parameters are obtained by fitting experimental data. We show that predictions agree with independent spatially and temporally resolved MRI experimental measurements. This research has contributed significantly to our interdisciplinary group’s ability to cryopreserve human articular cartilage.
A family of graphs F on a fixed set of n vertices is said to be triangle-intersecting if for any two graphs G,H in F, the intersection of G and H contains a triangle. Simonovits and Sos conjectured that such a family has size at most (1/8)2^{\binom{n}{2}}, and that equality holds only if F
consists of all graphs containing some fixed triangle. Recently, the author, Yuval Filmus and Ehud Friedgut proved a strengthening of this conjecture, namely that if F is an odd-cycle-intersecting family of graphs, then |F| \leq (1/8) 2^{\binom{n}{2}}. Equality holds only if F consists of all graphs containing some fixed triangle. A stability result also holds: an odd-cycle-intersecting family with size close to the maximum must be close to a family of the above form. We will outline proofs of these results, which use Fourier analysis, together with an analysis of the properties of random cuts in graphs, and some results in the theory of Boolean functions. We will then discuss some related open questions.
All will be based on joint work with Yuval Filmus (University of Toronto) and Ehud Friedgut (Hebrew University of Jerusalem).
We study the nonhomogeneous boundary value problem for the
Navier--Stokes equations
\[
\left\{ \begin{array}{rcl}
-\nu \Delta{\bf u}+\big({\bf u}\cdot \nabla\big){\bf u} +\nabla p&=&{0}\qquad \hbox{\rm in }\;\;\Omega,\\[4pt]
{\rm div}\,{\bf u}&=&0 \qquad \hbox{\rm in }\;\;\Omega,\\[4pt]
{\bf u}&=&{\bf a} \qquad \hbox{\rm on }\;\;\partial\Omega
\end{array}\right
\eqno(1)
\]
in a bounded multiply connected domain
$\Omega\subset\mathbb{R}^n$ with the boundary $\partial\Omega$,
consisting of $N$ disjoint components $\Gamma_j$.
Starting from the famous J. Leray's paper published in 1933,
problem (1) was a subject of investigation in many papers. The
continuity equation in (1) implies the necessary solvability
condition
$$
\int\limits_{\partial\Omega}{\bf a}\cdot{\bf
n}\,dS=\sum\limits_{j=1}^N\int\limits_{\Gamma_j}{\bf a}\cdot{\bf
n}\,dS=0,\eqno(2)
$$
where ${\bf n}$ is a unit vector of the outward (with respect to
$\Omega$) normal to $\partial\Omega$. However, for a long time
the existence of a weak solution ${\bf u}\in W^{1,2}(\Omega)$ to
problem (1) was proved only under the stronger condition
$$
{\cal F}_j=\int\limits_{\Gamma_j}{\bf a}\cdot{\bf n}\,dS=0,\qquad
j=1,2,\ldots,N. \eqno(3)
$$
During the last 30 years many partial results concerning the
solvability of problem (1) under condition (2) were obtained. A
short overview of these results and the detailed study of problem
(1) in a two--dimensional bounded multiply connected domain
$\Omega=\Omega_1\setminus\Omega_2, \;\overline\Omega_2\subset
\Omega_1$ will be presented in the talk. It will be proved that
this problem has a solution, if the flux ${\cal
F}=\int\limits_{\partial\Omega_2}{\bf a}\cdot{\bf n}\,dS$ of the
boundary datum through $\partial\Omega_2$ is nonnegative (outflow
condition).
In this talk I will introduce some of the basic ideas linking the theory of complex multiplication for elliptic curves and class field theory. Time permitting, I'll mention Shimura and Taniyama's work on the case of abelian varieties.
Let $G $ be a compact Lie group, and consider the variety $\text {Hom} (\bb Z^k,G)$
of representations of the rank $k$ abelian free group $\bb Z^k$ into $G$. We prove
that the fundamental group of $\text {Hom} (\bb Z^k,G) $ is naturally isomorphic to direct
product of $k$ copies of the fundamental group of $G$. This is joint work with
Jose Manuel Gomez and Juan Souto.
In this talk, we will look at the diffusive scaling limit of a class of
one-dimensional random walks in a random space-time environment. In the
scaling limit, this gives rise to a so-called stochastic flow of kernels as
introduced by Le Jan and Raimond and generalized by Howitt and Warren. We will
prove several new results about these stochastic flows of kernels by making
use of the theory of the Brownian web and net. This is joint work with R. Sun
and E. Schertzer.
Abstract: An instance of the Potts model is defined by a graph of interactions and a number, q, of different ``spins''. A configuration in this model is an assignment of spins to vertices. Each configuration has a weight, which in the ferromagnetic case is greater when more pairs of adjacent spins are alike. The classical Ising model is the special case
of q=2 spins. We consider the problem of computing an approximation to the partition function, i.e., weighted sum of configurations, of
an instance of the Potts model. Through the random cluster formulation it is possible to make sense of the partition function also for non-integer q. Yet another equivalent formulation is as the Tutte polynomial in the positive quadrant.
About twenty years ago, Jerrum and Sinclair gave an efficient (i.e., polynomial-time) algorithm for approximating the partition function of a ferromagnetic Ising system. Attempts to extend this result to q≠2 have been unsuccessful. At the same time, no convincing evidence has been presented to indicate that such an extension is impossible. An interesting feature of the random cluster model when q>2 is that it exhibits a first-order phase transition, while for 1≤q≤2 only a second-order phase transition is apparent. The idea I want to convey in this talk is that this first-order phase transition can be exploited in order to encode apparently hard computational problems within the model. This provides the first evidence that the partition function of the ferromagnetic Potts model may be hard to compute when q>2.
This
is joint work with Leslie Ann Goldberg, University of Liverpool.
In this talk a number of identities will be discussed that relate to the event of level crossing of certain types of stochastic processes. Some of these identities are surprisingly simple and have interpretations in surplus modelling of insurance portfolios, the design of taxation schemes, optimal dividend strategies and the pricing of barrier options.
PLEASE NOTE THAT THIS WORKSHOP IS TO BE HELD IN 21 BANBURY ROAD BEGINNING AT 9AM! \\
We will give three short presentations of current work here on small scale mechanics :
1) micron-scale cantilever testing and nanoindentation - Dave Armstrong
2) micron-scale pillar compression – Ele Grieveson
3) Dislocation loop shapes – Steve Fitzgerald
These should all provide fuel for discussion, and I hope ideas for future collaborative work.\\
The meeting will be in the committee room in 21 Banbury Rd (1st floor, West end).
Vopenka's Principle is a very strong large cardinal axiom which can be used to extend ZFC set theory. It was used quite recently to resolve an important open question in algebraic topology: assuming Vopenka's Principle, localisation functors exist for all generalised cohomology theories. After describing the axiom and sketching this application, I will talk about some recent results showing that Vopenka's Principle is relatively consistent with a wide range of other statements known to be independent of ZFC. The proof is by showing that forcing over a universe satisfying Vopenka's Principle will frequently give an extension universe also satisfying Vopenka's Principle.
The presence of additives, which may or may not be surface-active, can have a dramatic influence on interfacial flows. The presence of surfactants alters the interfacial tension and drives Marangoni flow that leads to fingering instabilities in drops spreading on ultra-thin films. Surfactants also play a major role in coating flows, foam drainage, jet breakup and may be responsible for the so-called ``super-spreading" of drops on hydrophobic substrates. The addition of surface-inactive nano-particles to thin films and drops also influences the interfacial dynamics and has recently been shown to accelerate spreading and to modify the boiling characteristics of nanofluids. These findings have been attributed to the structural component of the disjoining pressure resulting from the ordered layering of nanoparticles in the region near the contact line. In this talk, we present a collection of results which demonstrate that the above-mentioned effects of surfactants and nano-particles can be captured using long-wave models.
The Bi-Conjugate Gradient method (BiCG) is a well-known iterative solver (Krylov method) for linear systems of equations, proposed about 35 years ago, and the basis for some of the most successful iterative methods today, like BiCGSTAB. Nevertheless, the convergence behavior is poorly understood. The method satisfies a Petrov-Galerkin property, and hence its residual is constrained to a space of decreasing dimension (decreasing one per iteration). However, that does not explain why, for many problems, the method converges in, say, a hundred or a few hundred iterations for problems involving a hundred thousand or a million unknowns. For many problems, BiCG converges not much slower than an optimal method, like GMRES, even though the method does not satisfy any optimality properties. In fact, Anne Greenbaum showed that every three-term recurrence, for the first (n/2)+1 iterations (for a system of dimension n), is BiCG for some initial 'left' starting vector. So, why does the method work so well in most cases? We will introduce Krylov methods, discuss the convergence of optimal methods, describe the BiCG method, and provide an analysis of its convergence behavior.
In the first part of this talk we introduce hypersymplectic manifolds and compare various aspects of their geometry with related notions in hyperkähler geometry. In particular, we explain the hypersymplectic quotient construction. Since many examples of hyperkähler structures arise from Yang-Mills moduli spaces via the hyperkähler quotient construction, we discuss the gauge theoretic equations for a (twisted) harmonic map from a Riemann surface into a compact Lie group. They can be viewed as the zero condition for a hypersymplectic moment map in an infinite-dimensional setup.
An agent is presented with an N Bandit (Fruit) machines. It is assumed that each machine produces successes or failures according to some fixed, but unknown Bernoulli distribution. If the agent plays for ever, how can he/she choose a strategy that ensures the average successes observed tend to the parameter of the "best" arm?
Alternatively suppose that the agent recieves a reward of a^n at the nth button press for a success, and 0 for a failure; now how can the agent choose a strategy to optimise his/her total expected rewards over all time? These are two examples of classic Bandit Problems.
We analyse the behaviour of two strategies, the Narendra Algorithm and the Gittins Index Strategy. The Narendra Algorithm is a "learning"
strategy, in that it answers the first question in the above paragraph, and we demonstrate this remains true when the sequences of success and failures observed on the machines are no longer i.i.d., but merely satisfy an ergodic condition. The Gittins Index Strategy optimises the reward stream given above. We demonstrate that this strategy does not "learn" and give some new explicit bounds on the Gittins Indices themselves.
In this talk, I will present joint work with Uri Onn, Mark Berman, and Pirita Paajanen.
Let G be a linear algebraic group defined over the integers. Let O be a compact discrete valuation ring with a finite residue field of cardinality q and characteristic p. The group
G(O) has a filtration by congruence subgroups
G_m(O) (which is by definition the kernel of reduction map modulo P^m where P is the maximal ideal of O).
Let c_m=c_m(G(O)) denote the number of conjugacy classes in the finite quotient group G(O)/G_m(O) (which is called the mth congruence quotient of G(O)). The conjugacy class zeta function of
G(O) is defined to be the Dirichlet series Z_{G(O)}(s)=\sum_{m=0,1,...} c_m q^_{-ms}, where s is a complex number with Re(s)>0. This zeta function was defined by du Sautoy when G is a p-adic analytic group and O=Z_p, the ring of p-adic integers, and he proved that in this case it is a rational function in p^{-s}. We consider the question of dependence of this zeta function on p and more generally on the ring O.
We prove that for certain algebraic groups, for all compact discrete valuation rings with finite residue field of cardinality q and sufficiently large residue characteristic p, the conjugacy class zeta function is a rational function in q^{-s} which depends only on q and not on the structure of the ring. Note that this applies also to positive characteristic local fields.
A key in the proof is a transfer principle. Let \psi(x) and f(x) be resp.
definable sets and functions in Denef-Pas language.
For a local field K, consider the local integral Z(K,s)=\int_\psi(K)
|f(x)|^s dx, where | | is norm on K and dx normalized absolute value
giving the integers O of K volume 1. Then there is some constant
c=c(f,\psi) such that for all local fields K of residue characteristic larger than c and residue field of cardinality q, the integral Z(K,s) gives the same rational function in q^{-s} and takes the same value as a complex function of s.
This transfer principle is more general than the specialization to local fields of the special case when there is no additive characters of the motivic transfer principle of Cluckers and Loeser since their result is the case when the integral is zero.
The conjugacy class zeta function is related to the representation zeta function which counts number of irreducible complex representations in each degree (provided there are finitely many or finitely many natural classes) as was shown in the work of Lubotzky and Larsen, and gives information on analytic properties of latter zeta function.
This talk will be an introduction to property (T). It was originally introduced by Kazhdan as a method of showing that certain discrete subgroups of Lie groups are finitely generated, but has expanded to become a widely used tool in group theory. We will take a short tour of some of its uses.
We derive solutions of the Kirchhoff equations for a knot tied on an infinitely long elastic rod subjected to combined tension and twist. We consider the case of simple (trefoil) and double (cinquefoil) knots; other knot topologies can be investigated similarly. The rod model is based on Hookean elasticity but is geometrically non-linear. The problem is formulated as a non-linear self-contact problem with unknown contact regions. It is solved by means of matched asymptotic expansions in the limit of a loose knot. Without any a priori assumption, we derive the topology of the contact set, which consists of an interval of contact flanked by two isolated points of contacts. We study the influence of the applied twist on the equilibrium and find an instability for a threshold value of the twist.
Geoghegan's stack construction is a tool for analysing groups
that act on simply connected CW complexes, by providing a topological
description in terms of cell stabilisers and the quotient complex,
similar to what Bass-Serre theory does for group actions on trees. We
will introduce this construction and see how it can be used to give
results on finiteness properties of groups.
I will describe recent work on motivic DT invariants for 3-manifolds, which are expected to be a refinement of Chern-Simons theory. The conclusion will be that these should be possible to define and work with, but there will be some interesting problems along the way. There will be a discussion of the problem of upgrading the description of the moduli space of flat connections as a critical locus to the problem of describing the fundamental group algebra of a 3-fold as a "noncommutative critical locus," including a recent topological result on obstructions for this problem. I will also address the question of how a motivic DT invariant may be expected to pick up a finer invariant of 3-manifolds than just the fundamental group.
This work demonstrates how we can extract clinically useful patterns
extracted
from time series data (speech signals) using nonlinear signal
processing and how to exploit those patterns using robust statistical
machine learning tools, in order to estimate remotely and accurately
average Parkinson's disease symptom severity.
String field theory is a candidate for a full non-perturbative definition
of string theory. We aim to define string field theory on a space-time
lattice to investigate its behaviour at the quantum level. Specifically, we
look at string field theory in a one dimensional linear dilaton background,
using level truncation to restrict the theory to a finite number of fields.
I will report on our preliminary results at level-0 and level-1.