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Passage times of random walks and Lévy processes across power law boundaries
Get PDFWe establish an integral test involving only the distribution of the increments of a random walk S which determines whether limsup n→∞(S_n/n^κ) is almost surely zero, finite or infinite when 1/2 < κ < 1 and a typical step in the random walk has zero mean. This completes the results of Kesten and Maller [9] concerning finiteness of one-sided passage times over power law boundaries, so that we now have quite explicit criteria for all values of κ≥0. The results, and those of [9], are also extended to Lévy processes
Bifurcation and forced symmetry breaking in Hamiltonian systems
No full textWe consider the phenomenon of forced symmetry breaking in a symmetric Hamiltonian system on a symplectic manifold. In particular we study the persistence of an initial relative equilibrium subjected to this forced symmetry breaking. We see that, under certain nondegeneracy conditions, an estimate can be made on the number of bifurcating relative equilibria
Convexity of the optimal stopping boundary for the American put option
No full textWe show that the optimal stopping boundary for the American put option is convex in the standard Black-Scholes model. The methods are adapted from ice-melting problems and rely upon studying the behavior of level curves of solutions to certain parabolic differential equations
Spatio-temporal oscillations and rheochaos in a simple model of shear banding
No full textWe study a simple model of shear banding in which the flow-induced phase is destabilized by coupling between flow and microstructure (wormlike micellar length). By varying the strength of instability and the applied shear rate, we find a rich variety of oscillatory and chaotic shear banded flows. At low shear and weak instability, the induced phase pulsates next to one wall of the flow cell. For stronger instability, high shear pulses ricochet across the cell. At high shear we see oscillating bands on either side of central defects. We discuss our results in the context of recent experiments
A stochastic model for competing growth on Rd
Get PDFA stochastic model, describing the growth of two competing infections
on Rd, is introduced. The growth is driven by outbursts in the infected
region, an outburst in the type 1 (2) infected region transmitting the type
1 (2) infection to the previously uninfected parts of a ball with stochastic
radius around the outburst point. The main result is that with the growth
rate for one of the infection types ¯xed, mutual unbounded growth has
probability zero for all but at most countably many values of the other
infection rate. This is a continuum analog of a result of HÄaggstrÄom and
Pemantle. We also extend a shape theorem of Deijfen for the correspond-
ing model with just one type of infection
The Mathematics of Motion Camouflage
Get PDFMotion camouflage is a strategy whereby an aggressor moves towards a target whilst appearing stationary to the target
except for the inevitable perceived change in size of the aggressor as it approaches. The strategy has been observed in insects, and mathematical models using discrete time or neural network control have been used to simulate the behaviour. Here the differential equations for motion camouflage are derived and some simple cases are analysed. These equations are easy to simulate numerically, and simulations indicate that motion camouflage is more efficient than the classical pursuit strategy (‘move directly towards the target’)
The nonsmooth pitchfork bifurcation
Get PDFThe bifurcations of strange nonchaotic attractors in quasi-periodically forced systems are poorly understood. A simple two-parameter example is introduced which unifies previous observations of the non-smooth pitchfork bifurcation. There are two types of generalized pitchfork bifurcation which occur in this example, and the corresponding bifurcation curves can be calculated analytically. The example shows how these bifurcations are organized around a codimension two point in parameter space
A NON-CLASSICAL APPROACH TO MAXIMUM ENTROPY IN UNCERTAIN REASONING
Get PDFThis thesis is concerned with the question “Given a set of knowledge about propositional
variables, what is the ‘best’ way to assign probability values to those variables?”
I present here an approach to this question based upon a philosophical
concept of negation and its role in perception. This concept is discussed in detail
before a mathematical analysis of it is presented, in the form of structures in
propositional logic which, it is claimed, embody the principles of the underlying
philosophy. There follows the definition and mathematical characterisation of an
inference process which utilises these logical structures and also adheres closely
to the principles of Maximum Entropy. The properties of this inference process
are analysed and discussed.
Another inference process is then described based upon a modified version of
the philosophical principles defined earlier. A class of graphs is found which are
intimately connected with this inference process, and two attempts at characterising
this class are presented.
Cross Orbits
No full textThis paper contains a variety of results about the action of Conway's largest simple group upon the crosses in the Leech lattice. These results are tailor-made for use in 'A Monster Graph, I' (Proc. London Math. Soc. (3) 90 (2005) 42–60), where a graph related to the Monster simple group is studied