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Multifractal Structure of Bernoulli Convolutions
Get PDFLet be the distribution of the random series , where is a sequence of i.i.d. random variables taking the values 0,1 with probabilities . These measures are the well-known (biased) Bernoulli convolutions.
In this paper we study the multifractal spectrum of for typical . Namely, we investigate the size of the sets
Our main results highlight the fact that for almost all, and in some cases all, in an appropriate range, is nonempty and, moreover, has positive Hausdorff dimension, for many values of . This happens even in parameter regions for which is typically absolutely continuous
Boundary-equilibrium bifurcations in piecewise-smooth slow-fast systems
No full textIn the paper we study the qualitative dynamics of piecewise-smooth slow-fast systems (singularly perturbed
systems). We consider phase space topology of systems with 1-dimensional slow dynamics and 1-dimensional
fast dynamics. The slow manifold of the reduced system is formed by a piecewise-continuous curve, and
the di®erentiability is lost across the switching surface. In the full system the slow manifold is no longer
continuous, and there is an O(") discontinuity across the switching manifold, but the discontinuity cannot
qualitatively alter system dynamics. Revealed phase space topology is used to unfold qualitative dynamics of
planar slow-fast systems with an equilibrium point on the switching surface. In this case the local dynamics
corresponds to so-called boundary-equilibrium bifurcations, and four qualitative phase portraits are uncov-
ered. Our results are then used to investigate the dynamics of a box model of a thermohaline circulation, and
the presence of a boundary-equilibrium bifurcation of a fold type is shown
Fast Implicit Solvers using Stabilized Mixed Approximation
No full textThis paper concerns a new class of robust and efficient methods for solving the Navier-Stokes equations for unsteady incompressible flow. In previous work (Kay et al. SIAM J. Sci. Comput. 2010; 32:111-128) we established the effectiveness of an implicit time integrator using a stabilized trapezoid rule with an explicit Adams-Bashforth method for error control. The role of the stability of the spatial approximation on the overall accuracy of the implicit solution algorithm is the primary focus here. In particular, the relationship between spatial stabilization and temporal solution accuracy is assessed computationally for the case of the lowest order conforming mixed approximation
Smoothing non-smooth systems with the moving average transformation
Get PDFWe present a novel, systematic procedure for smoothing non-smooth dynamical systems. In particular we introduce the Moving Average Transformation which can be thought of as a change of variables which transforms discontinuous systems into \emph{dynamically equivalent} continuous systems and continuous but non-differentiable systems into dynamically equivalent differentiable systems. This smoothing gives us a new way to compute the stability properties of a non-smooth systems and provides a new theoretical link between smooth and non-smooth systems. The dynamics and algebraic structure of systems obtained by transforming typical non-smooth systems are investigated
A note on the algebraic structure of conditionals
Get PDFWe investigate the idea of representing conditional measures as simple measures (possibly with further properties) on conditional objects. To formalise this intuition we introduce a logico-algebraic framework based on the concept of conditional algebras. These are structures which allow us to distinguish between the logical properties of conditionals and those of conditional measures, a distinction which, we argue, helps us clarifying both concepts.
We illustrate the applicability of our framework by considering three popular conditional measures in the uncertain reasoning literature, namely, plausibility, probability and possibility measures
Regulating Industries under Exogenous Uncertainty
Get PDFWe present a quantitative method to find jointly optimal strategies for an industry regulator and a firm, who operate under exogenous uncertainty. The firm controls its operating policy in order to maximize its expected future profits, whilst taking account of regulatory fines. The regulator aims to control the probability that the firm enters a given undesirable state, such as ceasing production, by imposing a fine which is as low as possible, while achieving the required reduction in probability.
The exogenous uncertainty is modeled using a stochastic differential equation, and we show this implies that the firm's behavior can be solved via the Hamilton-Jacobi-Bellman equation, and the regulatory fine can be obtained via the Feynman-Kac formula. We discuss both analytic and numerical solution methods. Our results
are illustrated for a security of supply problem for vaccine production where future production costs are uncertain and, using empirical data, for an abandonment problem in a gold mining operation where future commodity prices are uncertain. The method determines the level of fine which establishes a Nash equilibrium in these nonzero-sum games, under uncertainty
GPU-based solution of Continuous Time Markov Chains using CUSP
Get PDFThis technical report describes the parallelisation of the
response-time analyser HYDRA using CUSP and the results of executing it on HECToR's GPGPU testbed. We achieved good speed-ups in execution time, but these were outweighed by increased setup time
GPU-based solution of Continuous Time Markov Chains using CUSP
Get PDFThis technical report describes the parallelisation of the
response-time analyser HYDRA using CUSP and the results of executing it on HECToR's GPGPU testbed. We achieved good speed-ups in execution time, but these were outweighed by increased setup time
Investigating the Performance of Asynchronous Jacobi's Method for Solving Systems of Linear Equations
Get PDFEver-increasing core counts create the need to develop parallel algorithms that avoid closely-coupled execution across cores. In this paper we present two case studies investigating the performance of several parallel asynchronous implementations of Jacobi's method for solving systems of linear equations. Although conditions for the convergence of asynchronous Jacobi are well known, what drives its rate of convergence is less well understood. The first case study investigates the algorithm's performance when executed on large numbers of processors on a Cray XE6, while the second explores the effect of varying the number of synchronous and asynchronous processors. We observe that the performance of parallel asynchronous Jacobi is highly implementation, problem and architecture-dependent
Emergence of hierarchical networks and polysynchronous behaviour in simple adaptive systems
Get PDFWe describe the dynamics of a simple adaptive network. The network architecture evolves to a number of disconnected components on which the dynamics is characterized by the possibility of differently synchronized nodes within the same network (polysynchronous states). These systems may have implications for the evolutionary emergence of polysynchrony and hierarchical networks in physical or biological systems modeled by adaptive networks