On dependence of dynamical structure of numerical solutions of fluid simulations on forcibly added randomness

In the present paper, the dependencies of the numerical results of fluid simulations on forcibly added randomness are discussed. The incompressible Navier-Stokes equations and the continuity equation are solved numerically by using the MAC (Maker-And-Cell) method and implicit temporal scheme. The model adopted in the present study is a flow around a two-dimensional circular cylinder and the Reynolds number is 1500. The randomness which is given by using the pseudo-random number is forcibly added in the time marching step of the discretized Navier-Stokes equations. Dependencies of the averaged structure of asymptotic numerical solutions on the randomness are discussed. Furthermore, the dependence of the qualitative structure of the asymptotic solution of each sample calculation on the amplitude of randomness is also studied. It is clarified that forcibly added random errors may cover the nonlinear errors which make the system unstable.     

Simple ingredients leading to very efficient heuristics for the maximum clique problem

Starting from an algorithm recently proposed by Pullan and Hoos, we formulate and analyze iterated local search algorithms for the maximum clique problem. The basic components of such algorithms are a fast neighbourhood search (not based on node evaluation but on completely random selection) and simple, yet very effective, diversification techniques and restart rules. A detailed computational study is performed in order to identify strengths and weaknesses of the proposed algorithms and the role of the different components on several classes of instances. The tested algorithms are very fast and reliable: most of the DIMACS benchmark instances are solved within very short CPU times. For one of the hardest tests, a new putative optimum was discovered by one of our algorithms. Very good performances were also shown on recently proposed and more difficult instances. It is important to remark that the heuristics tested in this paper are basically parameter free (the appropriate value for the unique parameter is easily identified and was, in fact, the same value for all problem instances used in this paper).     

模糊数学,随机数学与精确数学的逻辑比较

本文揭示模糊数学，随机数学与精确数学有相似的形式公理化描述，它们的区别只在语义上。     

Quantum Mechanics and Imprecise Probability

An extension of the Born rule, the quantum typicality rule, has recently been proposed [B. Galvan in Found. Phys. 37:1540–1562 (2007)]. Roughly speaking, this rule states that if the wave function of a particle is split into non-overlapping wave packets, the particle stays approximately inside the support of one of the wave packets, without jumping to the others. In this paper a formal definition of this rule is given in terms of imprecise probability. An imprecise probability space is a measurable space endowed with a set of probability measures ℘. The quantum formalism and the quantum typicality rule allow us to define a set of probabilities on (X ^T ,ℱ), where X is the configuration space of a quantum system, T is a time interval and ℱ is the σ-algebra generated by the cylinder sets. Thus, it is proposed that a quantum system can be represented as the imprecise stochastic process , which is a canonical stochastic process in which the single probability measure is replaced by a set of measures. It is argued that this mathematical model, when used to represent macroscopic systems, has sufficient predictive power to explain both the results of the statistical experiments and the quasi-classical structure of the macroscopic evolution.     

The Interplay Between Fairness and Randomness in a Spatial Computer Game

This article describes how children use an expressive microworld to articulate ideas about how to make a game seem fair with the use of randomness. Our aim in this study is to disentangle different flavours of fairness and to find out how children used each flavour to make sense of potentially complex behaviour. In order to achieve this, a spatial computer game was designed to enable children to examine the consequences of their attempts to make the game fair. The study investigates how 23 children, aged between 5.5 and 8 years, engaged in constructing a crucial part of a mechanism for a fair spatial lottery machine (microworld). In particular, the children tried to construct a fair game given a situation in which the key elements happened randomly. The children could select objects, determine their properties, and arrange their spatial layout in the machine. The study is based on task-based interviewing of children who were interacting with the computer game. The study shows that children have various cognitive resources for constructing a random fair environment. The spatial arrangement, the visualisation and the manipulations in the lottery machine allow us gain a view into the children’s thinking of the two central concepts, fairness and randomness. The paper reports on two main strategies by which the children attempted to achieve a balance in the lottery machine. One involves arranging the balls symmetrically and the other randomly. We characterize the nature of the thinking in these two strategies: the first we see as deterministic and the latter as stochastic, exploiting the random collisions of the ball. In this article we trace how the children’s thinking moved between these two perspectives.

Comparison of the Hurst and DEA exponents between the catalogue and its clusters: The California case

In this work, we analyse the long-run correlations of the seismic catalogue and of its clusters, by means of the Diffusion Entropy Analysis (DEA) and the value of the Hurst exponent. First we calculate the values for the whole catalogue, for the rate and the inter-event times and distance distribution over time, and subsequently we calculate the values for the declustered catalogue and the main clusters. We find a wide variety of behaviours, which depart from the Poisson statistics.     

On the Dealer's Randomness Required in Secret Sharing Schemes

In this paper we provide upper and lower bounds on the randomness required by the dealer to set up a secret sharing scheme for infinite classes of access structures. Lower bounds are obtained using entropy arguments. Upper bounds derive from a decomposition construction based on combinatorial designs (in particular, t-(v,k,) designs). We prove a general result on the randomness needed to construct a scheme for the cycle Cn; when n is odd our bound is tight. We study the access structures on at most four participants and the connected graphs on five vertices, obtaining exact values for the randomness for all them. Also, we analyze the number of random bits required to construct anonymous threshold schemes, giving upper bounds. (Informally, anonymous threshold schemes are schemes in which the secret can be reconstructed without knowledge of which participants hold which shares.)     

Strong disorder renormalization group primer and the superfluid–insulator transition

This brief review introduces the method and application of real-space renormalization group to strongly disordered quantum systems. The focus is on recent applications of the strong disorder renormalization group to the physics of disordered-boson systems and the superfluid–insulator transition in one dimension. The fact that there is also a well-understood weak disorder theory for this problem allows us to illustrate what aspects of the physics change at strong disorder. In particular, the strong disorder RG analysis suggests that the transitions at weak disorder and strong disorder belong to distinct universality classes, but this question remains under debate and is not fully resolved to date. Further applications of the strong disorder renormalization group to higher-dimensional Bose systems and to bosons coupled to dissipation are also briefly reviewed.     

Measuring randomness with periodic media

We show that the sensitivity to disorder of certain physical properties of periodic media depends on whether the disorder is truly random or not. This allows to utilize periodic media for testing sequences of random numbers and to quantify their departure from true randomness via simple transmission and/or reflection data analysis. This physics-based model shows promises for device applications to test random data.     

The hierarchy of correlations in random binary sequences

The meaning of randomness is studied for the simple case of binary sequences. Ensemble theory is used, together with correlation coefficients of any order. Conservation laws for the total amount of correlation are obtained. They imply that true randomness is an ensemble property and can never be achieved in a single sequence. The relation with entropy is discussed for different ensembles. Well-tempered pseudorandom sequences turn out to be suitable sources of random numbers, and practical recipes to generate them for use in large-scale Monte Carlo simulations are found.