Best Response Games on Regular Graphs

With the growth of the internet it is becoming increasingly important to understand how the behaviour of players is affected by the topology of the network interconnecting them. Many models which involve networks of interacting players have been proposed and best response games are amongst the simplest. In best response games each vertex simultaneously updates to employ the best response to their current surroundings. We concentrate upon trying to understand the dynamics of best response games on regular graphs with many strategies. When more than two strategies are present highly complex dynamics can ensue. We focus upon trying to understand exactly how best response games on regular graphs sample from the space of possible cellular automata. To understand this issue we investigate convex divisions in high dimensional space and we prove that almost every division of $k-1$ dimensional space into $k$ convex regions includes a single point where all regions meet. We then find connections between the convex geometry of best response games and the theory of alternating circuits on graphs. Exploiting these unexpected connections allows us to gain an interesting answer to our question of when cellular automata are best response games

Optimal rates of decay for operator semigroups on Hilbert spaces

We investigate rates of decay for $C_0$-semigroups on Hilbert spaces under assumptions on the resolvent growth of the semigroup generator. Our main results show that one obtains the best possible estimate on the rate of decay, that is to say an upper bound which is also known to be a lower bound, under a comparatively mild assumption on the growth behaviour. This extends several statements obtained by Batty, Chill and Tomilov (J. Eur. Math. Soc., vol. 18(4), pp. 853-929, 2016). In fact, for a large class of semigroups our condition is not only sufficient but also necessary for this optimal estimate to hold. Even without this assumption we obtain a new quantified asymptotic result which in many cases of interest gives a sharper estimate for the rate of decay than was previously available, and for semigroups of normal operators we are able to describe the asymptotic behaviour exactly. We illustrate the strength of our theoretical results by using them to obtain sharp estimates on the rate of energy decay for a wave equation subject to viscoelastic damping at the boundary.Comment: 25 pages. To appear in Advances in Mathematic

Sector skills insights : construction

The UK Commission for Employment and Skills is a social partnership, led by Commissioners from large and small employers, trade unions and the voluntary sector. Our mission is to raise skill levels to help drive enterprise, create more and better jobs and promote economic growth. Our strategic objectives are to: • Provide outstanding labour market intelligence which helps businesses and people make the best choices for them; • Work with businesses to develop the best market solutions which leverage greater investment in skills; • Maximise the impact of employment and skills policies and employer behaviour to support jobs and growth and secure an internationally competitive skills base. These strategic objectives are supported by a research programme that provides a robust evidence base for our insights and actions and which draws on good practice and the most innovative thinking. The research programme is underpinned by a number of core principles including the importance of: ensuring ‘relevance ’ to our most pressing strategic priorities; ‘salience ’ and effectively translating and sharing the key insights we find; internationa

Searching and Stopping: An Analysis of Stopping Rules and Strategies

Searching naturally involves stopping points, both at a query level (how far down the ranked list should I go?) and at a session level (how many queries should I issue?). Understanding when searchers stop has been of much interest to the community because it is fundamental to how we evaluate search behaviour and performance. Research has shown that searchers find it difficult to formalise stopping criteria, and typically resort to their intuition of what is "good enough". While various heuristics and stopping criteria have been proposed, little work has investigated how well they perform, and whether searchers actually conform to any of these rules. In this paper, we undertake the first large scale study of stopping rules, investigating how they influence overall session performance, and which rules best match actual stopping behaviour. Our work is focused on stopping at the query level in the context of ad-hoc topic retrieval, where searchers undertake search tasks within a fixed time period. We show that stopping strategies based upon the disgust or frustration point rules - both of which capture a searcher's tolerance to non-relevance - typically result in (i) the best overall performance, and (ii) provide the closest approximation to actual searcher behaviour, although a fixed depth approach also performs remarkably well. Findings from this study have implications regarding how we build measures, and how we conduct simulations of search behaviours

Multiphoton detachment from negative ions: new theory vs experiment

In this paper we compare the results of our adiabatic theory (Gribakin and Kuchiev, Phys. Rev. A, accepted for publication) with other theoretical and experimental results, mostly for halogen negative ions. The theory is based on the Keldysh approach. It shows that the multiphoton detachment rates and the corresponding n-photon detachment cross sections depend only on the asymptotic behaviour of the bound state radial wave function. The dependence on the exponent is very strong. This is the main reason for the disagreement with some previous calculations, which employed bound state wave functions with incorrect asymptotic forms. In a number of cases our theoretical results produces best agreement with absolute and relative experimental data.Comment: 9 pages, Latex, IOP style, and 3 figures fig1.ps, fig2.ps, fig3.ps, submitted to J. Phys.

Asymptotic tensor rank of graph tensors: beyond matrix multiplication

We present an upper bound on the exponent of the asymptotic behaviour of the tensor rank of a family of tensors defined by the complete graph on $k$ vertices. For $k\geq4$, we show that the exponent per edge is at most 0.77, outperforming the best known upper bound on the exponent per edge for matrix multiplication ($k=3$), which is approximately 0.79. We raise the question whether for some $k$ the exponent per edge can be below $2/3$, i.e. can outperform matrix multiplication even if the matrix multiplication exponent equals 2. In order to obtain our results, we generalise to higher order tensors a result by Strassen on the asymptotic subrank of tight tensors and a result by Coppersmith and Winograd on the asymptotic rank of matrix multiplication. Our results have applications in entanglement theory and communication complexity

Physical simulation for monocular 3D model based tracking

The problem of model-based object tracking in three dimensions is addressed. Most previous work on tracking assumes simple motion models, and consequently tracking typically fails in a variety of situations. Our insight is that incorporating physics models of object behaviour improves tracking performance in these cases. In particular it allows us to handle tracking in the face of rigid body interactions where there is also occlusion and fast object motion. We show how to incorporate rigid body physics simulation into a particle filter. We present two methods for this based on pose and force noise. The improvements are tested on four videos of a robot pushing an object, and results indicate that our approach performs considerably better than a plain particle filter tracker, with the force noise method producing the best results over the range of test videos

Revisiting Norm Optimization for Multi-Objective Black-Box Problems: A Finite-Time Analysis

The complexity of Pareto fronts imposes a great challenge on the convergence analysis of multi-objective optimization methods. While most theoretical convergence studies have addressed finite-set and/or discrete problems, others have provided probabilistic guarantees, assumed a total order on the solutions, or studied their asymptotic behaviour. In this paper, we revisit the Tchebycheff weighted method in a hierarchical bandits setting and provide a finite-time bound on the Pareto-compliant additive $\epsilon$-indicator. To the best of our knowledge, this paper is one of few that establish a link between weighted sum methods and quality indicators in finite time.Comment: submitted to Journal of Global Optimization. This article's notation and terminology is based on arXiv:1612.0841

Adaptive estimation of the density matrix in quantum homodyne tomography with noisy data

In the framework of noisy quantum homodyne tomography with efficiency parameter $1/2 < \eta \leq 1$, we propose a novel estimator of a quantum state whose density matrix elements $\rho_{m,n}$ decrease like $Ce^{-B(m+n)^{r/ 2}}$, for fixed $C\geq 1$, $B>0$ and $0<r\leq 2$. On the contrary to previous works, we focus on the case where $r$, $C$ and $B$ are unknown. The procedure estimates the matrix coefficients by a projection method on the pattern functions, and then by soft-thresholding the estimated coefficients. We prove that under the $\mathbb{L}_2$ -loss our procedure is adaptive rate-optimal, in the sense that it achieves the same rate of conversgence as the best possible procedure relying on the knowledge of $(r,B,C)$. Finite sample behaviour of our adaptive procedure are explored through numerical experiments