Nonlocal Operational Calculi for Dunkl Operators

The one-dimensional Dunkl operator $D_k$ with a non-negative parameter $k$, is considered under an arbitrary nonlocal boundary value condition. The right inverse operator of $D_k$, satisfying this condition is studied. An operational calculus of Mikusinski type is developed. In the frames of this operational calculi an extension of the Heaviside algorithm for solution of nonlocal Cauchy boundary value problems for Dunkl functional-differential equations $P(D_k)u=f$ with a given polynomial $P$ is proposed. The solution of these equations in mean-periodic functions reduces to such problems. Necessary and sufficient condition for existence of unique solution in mean-periodic functions is found

Projection Methods: Swiss Army Knives for Solving Feasibility and Best Approximation Problems with Halfspaces

We model a problem motivated by road design as a feasibility problem. Projections onto the constraint sets are obtained, and projection methods for solving the feasibility problem are studied. We present results of numerical experiments which demonstrate the efficacy of projection methods even for challenging nonconvex problems

Linear differential operators on contact manifolds

We consider differential operators between sections of arbitrary powers of the determinant line bundle over a contact manifold. We extend the standard notions of the Heisenberg calculus: noncommutative symbolic calculus, the principal symbol, and the contact order to such differential operators. Our first main result is an intrinsically defined "subsymbol" of a differential operator, which is a differential invariant of degree one lower than that of the principal symbol. In particular, this subsymbol associates a contact vector field to an arbitrary second order linear differential operator. Our second main result is the construction of a filtration that strengthens the well-known contact order filtration of the Heisenberg calculus

Relaxation of a Colloidal Particle into a Nonequilibrium Steady State

We study the relaxation of a single colloidal sphere which is periodically driven between two nonequilibrium steady states. Experimentally, this is achieved by driving the particle along a toroidal trap imposed by scanned optical tweezers. We find that the relaxation time after which the probability distributions have been relaxed is identical to that obtained by a steady state measurement. In quantitative agreement with theoretical calculations the relaxation time strongly increases when driving the system further away from thermal equilibrium

Using Gaussian Processes to Optimise Concession in Complex Negotiations against Unknown Opponents

In multi-issue automated negotiation against unknown opponents, a key part of effective negotiation is the choice of concession strategy. In this paper, we develop a principled concession strategy, based on Gaussian processes predicting the opponent's future behaviour. We then use this to set the agent's concession rate dynamically during a single negotiation session. We analyse the performance of our strategy and show that it outperforms the state-of-the-art negotiating agents from the 2010 Automated Negotiating Agents Competition, in both a tournament setting and in self-play, across a variety of negotiation domains

Negotiating Concurrently with Unknown Opponents in Complex, Real-Time Domains

We propose a novel strategy to enable autonomous agents to negotiate concurrently with multiple, unknown opponents in real-time, over complex multi-issue domains. We formalise our strategy as an optimisation problem, in which decisions are based on probabilistic information about the opponents' strategies acquired during negotiation. In doing so, we develop the first principled approach that enables the coordination of multiple, concurrent negotiation threads for practical negotiation settings. Furthermore, we validate our strategy using the agents and domains developed for the International Automated Negotiating Agents Competition (ANAC), and we benchmark our strategy against the state-of-the-art. We find that our approach significantly outperforms existing approaches, and this difference improves even further as the number of available negotiation opponents and the complexity of the negotiation domain increases