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Recovering Riemannian metrics in monotone families from boundary data
Get PDFWe discuss the inverse problem of determining the anisotropic conductivity of a body described by a compact, orientable, Riemannian manifold M with
boundary bdy M, when measurements of electric voltages and currents are taken on all of bdy M. Specifically we consider a one parameter family of conductivity tensors,
extending results obtained in [AG] where the simpler Euclidean case is considered.
Our problem is equivalent to the geometric one of determining a Riemannian metric
in monotone one parameter family of metrics from its Dirichlet to Neumann map on bdy M
Forward and Inverse Problems in Towed Cable Hydrodynamics
Get PDFThis paper addresses the problem of reconstructing the velocities of t
he ocean currents impinging on a towed streamer cable during an offshore seismic
survey. This study considers a two-dimensional model describing the motion of a
flexible, inextensible cable in the presence of hydrodynamic drag forces in an in
compressible fluid. In the first part the forward model is introduced and then sol
ved to yield the cable’s velocity, curvature and tension in the knowledge of the
towing vessel motion and the hydrodynamic loads applied. In sequence, we formul
ate the inverse problem of inferring the ocean current velocities from discrete
samples of the cable’s shape and tension and show that this is rank deficient and
ill-posed. In approaching the inverse problem a numerically stable algorithm is
adopted based on generalized Tikhonov regularization, in the context of robust
differentiation of discrete noisy signals. In order to demonstrate the practical
performance of the scheme, some examples of ocean current reconstructions obtain
ed using simulated noisy data are presented
Ownership and Control: A Small-World Analysis
No full textIn this paper we investigate the ownership and control of British firms using recent techniques from computational graph theory. In particular, we analyze the `small-world' properties of UK company ownership and the corporate elite. A `small-world' is characterized by short `path-lengths' (actors are linked by a short chain of acquaintances) and high `clustering' (one's friends tend to be friends in their own right). We find that the network of both ownership and control can be characterized as a small-world. We simulate a set of corporate worlds using newly introduced random-graph models of Chung and Lu. In general, we reject the hypothesis that the corporate world of ownership and control is generated by a random graph model. The network structure of ownership and boards is decidedly more `clubby' than would be expected by chance, suggesting the presence of additional social structure not captured by the random graph model. In addition, we find that financial institutions are important and give rise to different network topologies
Heirs of box types in polynomially bounded structures
Get PDFWe characterize heirs of so called box types of a polynomially bounded o-minimal structure M. A box type is an n-type of M which is uniquely determined by the projections to the coordinate axes. From this, we deduce various structure theorems for subsets of Mk, definable in the expansion M* of M by all convex subsets of the line. Moreover we obtain a model completeness result for M*
Notes on lifting group actions
Get PDFGiven an action of a Lie group G on a manifold M, and a cover N of M (such as the universal cover M-tilde) the natural question arises of whether the action lifts to a cover of M. In these notes, we address this question and determine whether the G-action itself lifts or whether it is necessary to pass to a cover of G (there is always a lift of the action of the universal cover G-tilde). The results are presumably well-known to experts, but do not seem to be available in print.
These notes started life as the first section of a paper on symplectic (but non-hamiltonian) group actions, though it was felt that they were too detailed for a paper primarily on symplectic reduction. We are therefore making them available as a MIMS preprint
Renormalization for the boundary of chaos in piecewise monotonic maps with a single discontinuity
Get PDFMonotonic maps with a single discontinuity arise in a variety of situations. We describe the infinite sets of periods for such maps on the boundary of chaos; this gives a sense of the routes to chaos in such maps. The description involves an explicit subshift of finite type which describes the sequences of different renormalizations possible in these maps
IMPROVED NUMERICAL TECHNIQUES FOR OCCUPATION-TIME DERIVATIVES AND OTHER COMPLEX FINANCIAL INSTRUMENTS
Get PDFOccupation-time derivatives are complex barrier-type options where valuation
depends on the time spent beyond the barrier by the underlying asset. This thesis
presents a model for corporate bonds using an occupation-time derivative, the
ParAsian option, the features of which can capture bankruptcy resolution and complex
capital structure with violations of the absolute priority rule. It investigates the
numerics of the problem, and proposes appropriate numerical techniques to enable
accurate and rapid solutions. The model is extended to include bond conversion in
a two-tier structure, which presents its own numerical problems. A new occupationtime
derivative that takes into account the distance of deviations beyond the barrier
is presented and solved.
Using existing knowledge on the asymptotic structure, new fast and efficient techniques
are created for pricing American options. A second new occupation-time
derivative is proposed, combining elements of early exercise with the ParAsian option
to produce the American delayed-exercise option.
The numerical methods employed in this thesis are based on accurate finitedifference
schemes, specifically developed and enhanced to treat the various classes
of problem considered
On -definability without equality over the real numbers
Get PDFIn Delzell (1982) it has been shown that for first-order definability over the reals there exists an effective procedure which by a finite formula with equality defining an open set produces a finite formula without equality that defines the same set. In this paper we prove that there exists no such procedure for -definability over the reals. We also show that there exists even no uniform effective transformation of the definitions of -definable sets (i. e., -formulas) into new definitions of -definable sets in such a way that the results will define open sets, and if a definition defines an open set, then the result of this transformation will define the same set. These results highlight the important differences between -definability with equality and -definability without equality
Forays into Sequential Composition and Concatenation in EAGLE
Get PDFThe run-time verification logic Eagle is equipped with two forms of binary cut operator, sequential composition (;) and concatenation (·). Essentially, a concatenation formula F 1 ·F 2 holds on a trace if that trace can be cut into two non-overlapping traces such that F 1 holds on the first and F 2 on the second. Sequential composition differs from concatenation in that the two traces must overlap by one state. Both cut operators are non-deterministic in the sense that the cutting point is not uniquely defined. In this paper we establish that sequential composition and concatenation are equally expressive. We then extend Eagle with deterministic variants of sequential composition and concatenation. These variants impose a restriction on either the left or right operand so that the cut point defines either the shortest or longest possible satisfiable cut trace. Whilst it is possible to define such deterministic operators recursively within Eagle, such definitions based on the non-deterministic cut operators impose a complexity penalty. By augmenting Eagle’s evaluation calculus for the deterministic variants, we establish that the asymptotic time and space complexity of on-line monitoring for the variants with deterministic restrictions applied to the left operand is no worse than the asymptotic time and space complexity of the sub-formulæ
Linear groups of finite Morley rank
Get PDFThis paper is a brief survey of recent results and some open
problems related to linear groups of finite Morley rank, an area of research where Bruno Poizat's impact is very prominent. As a sign of respect to his strongly expressed views that mathematics has to be done, written and published only in the native tongue of the immediate author---the scribe, in effect---of the text, I insist on writing my paper in Russian, even if the results presented belong to a small but multilingual community of researchers of American, British, French, German, Kazakh, Russian, Turkish origin. To emphasise even further the linguistic subtleties involved, I use British spelling in the English fragments of my text