We present an O(N logN) algorithm for the calculation of the first N coefficients in an expansion of an analytic function in Legendre polynomials. In essence, the algorithm consists of an integration of a suitably weighted function along an ellipse, a task which can be accomplished with Fast Fourier Transform, followed by some post-processing.
Hilbert Sixth Problem of Axiomatization of Physics is a problem of general nature and not of specific problem. We will concentrate on the kinetic theory; the relations between the Newtonian particle systems, the Boltzmann equation and the fluid dynamics. This is a rich area of applied mathematics and mathematical physics. We will illustrate the richness with some examples, survey recent progresses and raise open research directions.
Presentations by:
09.30 am Bernhard Langwallner Continuum limits of atomistic energies and new computational models of fracture
09.50 am Yasemin Sengul Well-posedness of dynamics
10.10 am Kostas Koumatos X-interfaces and nonclassical austenite-martensite interfaces
10.30 am Tim Squires Models for breast cancer and heart tissue
Abstract: Quantales are ordered algebras which can be thought of as pointfree noncommutative topologies. In recent years, their connections have been studied with fundamental notions in noncommutative geometry such as groupoids and C*-algebras. In particular, the setting of quantales corresponding to étale groupoids has been very well understood: a bijective correspondence has been defined between localic étale groupoids and inverse quantale frames. We present an equivalent but independent way of defining this correspondence for topological étale groupoids and we extend this correspondence to a non-étale setting.
Wound coils or rolls accumulate essentially flat strip compactly without folding or cutting and typically, strip is wound and unwound a number of times before its end use. The variety of material that is wound into coils or rolls is very extensive and includes magnetic tape, paper, cellophane, plastics, fabric and metals such as aluminium and steel.
Stresses wound into a coil provide its structural integrity via the frictional forces between the wraps. For a coil with inadequate inter-wrap pressure, the wraps may slip or telescope (causing surface scuffing) or the coil may slump and collapse. On the other hand, large internal stresses can cause increased creep and stress relaxation, collapse at the bore, stress wrinkling and rupture of the material in the coil.
Given the range of applications, it is not surprising that the literature on calculating stresses in wound coils is large and has a long history, which goes back at least to the wire winding of gun barrels. However the basic approach of the resulting accretion models, where the residual stress is recalculated each time a layer is added, has remained essentially the same. In this talk, we take a radically different approach in analysing the winding stresses in coils. Instead of the traditional method, we seek to deduce a winding policy that will achieve a target distribution of residual stresses within a coil. In this way, optimising the coiling tension profile is much more straight-forward, by
* Specifying the residue stresses required to avoid operational problems, tight-bore collapses, and other issues such as scuffing, then
* Determining the winding tension profile to produce the required residue stresses.
Strong explosive volcanic eruptions could inject in the lower stratosphere million tons of SO2, which being converted to sulfate aerosols, affect radiative balance of the planet for a few years. During this period the volcanic radiative forcing dominates other forcings producing distinct detectable climate responses. Therefore volcanic impacts provide invaluable natural test of climate nonlinearities and feedback mechanisms. In this talk I will overview volcanic impacts on tropospheric and strsatospheric temperature, ozone, high-latitude circulation, stratosphere-troposphere dynamic interaction, and focus on the long-term volcanic effect on ocean heat content and sea level.
This workshop is half-seminar, half-workshop. \\ \\ HSBC have an on-going problem and they submitted a proposal for an MSc in Applied Stats project on this topic. Unfortunately, the project was submitted too late for this cohort of students. Eurico will talk about "the first approach at the problem" but please be aware that it is an open problem which requires further work. Eurico's abstract is as follows. \\ \\
This article examines modelling yield curves through chaotic dynamical systems whose dynamics can be unfolded using non-linear embeddings in higher dimensions. We then refine recent techniques used in the state space reconstruction of spatially extended time series in order to forecast the dynamics of yield curves.
We use daily LIBOR GBP data (January 2007-June 2008) in order to perform forecasts over a 1-month horizon. Our method seems to outperform random walk and other benchmark models on the basis of mean square forecast error criteria.
In this talk I will discuss "motivic" Donaldson-Thomas invariants, following the now not-so-recent paper of Kontsevich and Soibelman on this subject. I will, in particular, present some understanding of the mysterious notion of "orientation data," and present some recent work. I will of course do my best to make this talk "accessible," though if you don't know what a scheme or a category is it will probably make you cry.
The second talk will present conjectural motivic generalizations
of ADHM sheaf invariants as well as their wallcrossing formulas.
It will be shown that these conjectures yield recursive formulas
for Poincare and Hodge polynomials of moduli spaces of Hitchin
pairs. It will be checked in many concrete examples that this recursion relation is in agreement with previous results of Hitchin, Gothen, Hausel and Rodriguez-Villegas.
The first talk will present a construction of equivariant
virtual counting invariants for certain quiver sheaves on a curve, called ADHM sheaves. It will be shown that these invariants are related to the stable pair theory of Pandharipande and Thomas in a specific stability chamber. Wallcrossing formulas will be derived using the theory of generalized Donaldson-Thomas invariants of Joyce and Song.
This talk will begin with an introduction to calibrations and calibrated submanifolds. Calibrated geometry generalizes Wirtinger's inequality in Kahler geometry by considering k-forms which are analogous to the Kahler form. A famous one-line proof shows that calibrated submanifolds are volume minimizing in their homology class. Our examples of manifolds with a calibration will come from complex geometry and from manifolds with special holonomy.
We will then discuss the deformation theory of the calibrated submanifolds in each of our examples and see how they differ from the theory of complex submanifolds of Kahler manifolds.
The presence and flow of fluid inside a crack within a solid causes deformation of the solid which in turn influences the flow of the fluid.
This coupled fluid-solid problem will be discussed in the context of dyke propagation and hydrofracture. The background material will be discussed in detail and some applications to specific geometries presented.
"We investigate a construction of a Skorokhod embedding due to Root (1969), which has been the subject of recent interest for applications in Mathematical Finance (Dupire, Carr & Lee), where the construction has applications for model-free pricing and hedging of variance derivatives. In this context, there are two related questions: firstly of the construction of the stopping time, which is related to a free boundary problem, and in this direction, we expand on work of Dupire and Carr & Lee; secondly of the optimality of the construction, which is originally due to Rost (1976). In the financial context, optimality is connected to the construction of hedging strategies, and by giving a novel proof of the optimality of the Root construction, we are able to identify model-free hedging strategies for variance derivatives. Finally, we will present some evidence on the numerical performance of such hedges. (Joint work with Jiajie Wang)"
This talk will be based on the article arXiv:1006.2788.
It is well-known that Morrey's quasiconvexity is closely related to gradient Young measures,
i.e., Young measures generated by sequences of gradients in
$L^p(\Omega;\mathbb{R}^{m\times n})$. Concentration effects,
however, cannot be treated by Young measures. One way how to describe both oscillation and
concentration effects in a fair generality are the so-called DiPerna-Majda measures.
DiPerna and Majda showed that having a sequence $\{y_k\}$ bounded in $L^p(\Omega;\mathbb{R}^{m\times n})$,$1\le p$ $0$.
Puck Rombach;
"Weighted Generalization of the Chromatic Number in Networks with Community Structure",
Christopher Lustri;
"Exponential Asymptotics for Time-Varying Flows,
Alex Shabala
"Mathematical Modelling of Oncolytic Virotherapy",
Martin Gould;
"Foreign Exchange Trading and The Limit Order Book"
The n-amalgamation property has recently been explored in connection with generalised imaginaries (groupoid imaginaries) by Hrushovski. This property is useful when studying models of a stable theory together with a generic automorphism, e.g.
elimination of imaginaries (e.i.) in ACFA may be seen as a consequence of 4-amalgamation (and e.i.) in ACF.
The talk is centered around 4-amalgamation of stably dominated types in algebraically closed valued fields. I will show that 4-amalgamation holds in equicharacteristic 0, even for systems with 1 vertex non stably dominated. This is proved using a reduction to the stable part, where 4-amalgamation holds by a result of Hrushovski. On the other hand, I will exhibit an NIP (even metastable) theory with 4-amalgamation for stable types but in which stably dominated types may not be 4-amalgamated.
The question in the title seems to be neglected in the studies of open dynamical systems. It occurred though that the features of dynamics may play a role comparable to the one played by the size of a hole. For instance, the escape through the smaller hole could be faster than through the larger one.
These studies revealed as well a new role of the periodic orbits in the dynamics which could be exactly quantified in some cases. Moreover, this new approach allows to characterize the elements of networks by their dynamical properties (rather than by static ones like centrality, betweenness, etc.)
Not only does the definition of an (abstract) saturated fusion system provide us with an interesting way to think about finite groups, it also permits the construction of exotic examples, i.e. objects that are non-realisable by any finite group. After recalling the relevant definitions of fusion systems and saturation, we construct an exotic fusion system at the prime 3 as the fusion system of the completion of a tree of finite groups. We then sketch a proof that it is indeed exotic by appealing to The Classification of Finite Simple Groups.
An important problem in algebra is the study of algebraic objects
defined over fields and how they behave under field extensions,
for example the Brauer group of a field, Galois cohomology groups
over fields, Milnor K-theory of a field, or the Witt ring of bilinear
forms over
a field. Of particular interest is the determination
of the kernel of the restriction map when passing to a field extension.
We will give an overview over some known results concerning the
kernel of the restriction map from the Witt ring of a field to the
Witt ring of an extension field. Over fields of characteristic
not two, general results are rather sparse. In characteristic two,
we have a much more complete picture. In this talk, I will
explain the full solution to this problem for extensions that are
given by function fields of hypersurfaces over fields of
characteristic two. An important tool is the study of the
behaviour of differential forms over fields of positive
characteristic under field extensions. The result for
Witt rings in characteristic two then follows by applying earlier
results by Kato, Aravire-Baeza, and Laghribi. This is joint
work with Andrew Dolphin.
Following work done by the 'Oxford Spies' we uncover more secrets of 'surface-active Agents'. In modern-day applications we refer to these agents as surfactants, which are now extensively used in industrial, chemical, biological and domestic applications. Our work focuses on the dynamic behaviour of surfactant and polymer-surfactant mixtures.
In this talk we propose a mathematical model that incorporates the effects of diffusion, advection and reactions to describe the dynamic behaviour of such systems and apply the model to the over-flowing-cylinder experiment (OFC). We solve the governing equations of the model numerically and, by exploiting large parameters in the model, obtain analytical asymptotic solutions for the concentrations of the bulk species in the system. Thus, these solutions uncover secrets of the 'surface-active Agents' and provide an important insight into the system behaviour, predicting the regimes under which we observe phase transitions of the species in the system. Finally, we suggest how our models can be extended to uncover the secrets of more complex systems in the field.
I will define and discuss the properties of the analytic torsion of
twisted cohomology and briefly of Z_2-graded elliptic complexes
in general, as an element in the graded determinant line of the
cohomology of the complex, generalizing most of the variants of Ray-
Singer analytic torsion in the literature. IThe definition uses pseudo-
differential operators and residue traces. Time permitting, I will
also give a couple of applications of this generalized torsion to
mathematical physics. This is joint work with Siye Wu.
We shall explain what is the weighted fundamental lemma and how it is related to the truncated Hitchin fibration.
We consider thin films of a cholesteric liquid-crystal material subject to an applied electric field. In such materials, the liquid-crystal "director" (local average orientation of the long axis of the molecules) has an intrinsic tendency to rotate in space; while the substrates that confine the film tend to coerce a uniform orientation.
The electric field encourages certain preferred orientations of the director as well, and these competing influences give rise to several different stable equilibrium states of the director field, including spatially uniform, translation invariant (functions only of position across the cell gap) and periodic (with 1-D or 2-D periodicity in the plane of the film). These structures depend on two principal control parameters: the ratio of the cell gap to the intrinsic "pitch" (spatial period of rotation) of the cholesteric and the magnitude of the applied voltage.
We report on numerical work (not complete) on the bifurcation and phase behavior of this system. The study was motivated by potential applications involving switchable gratings and eyewear with tunable transparency. We compare our results with experiments conducted in the Liquid Crystal Institute at Kent State University.
We shall explain what is the weighted fundamental lemma and how it is related to the truncated Hitchin fibration.
Let L be a positive definite lattice. There are only finitely many positive definite lattices
L' which are isomorphic to L modulo N for every N > 0: in fact, there is a formula for the number of such lattices, called the Siegel mass formula. In this talk, I'll review the Siegel mass formula and how it can be deduced from a conjecture of Weil on volumes of adelic points of algebraic groups. This conjecture was proven for number fields by Kottwitz, building on earlier work of Langlands and Lai. I will conclude by sketching joint work (in progress) with Dennis Gaitsgory, which uses topological ideas to attack Weil's conjecture in the case of function fields.
