Showing papers on "Randomness published in 1986"

How to construct random functions

Abstract: A constructive theory of randomness for functions, based on computational complexity, is developed, and a pseudorandom function generator is presented. This generator is a deterministic polynomial-time algorithm that transforms pairs (g, r), where g is any one-way function and r is a random k-bit string, to polynomial-time computable functions ƒr: {1, … , 2k} → {1, … , 2k}. These ƒr's cannot be distinguished from random functions by any probabilistic polynomial-time algorithm that asks and receives the value of a function at arguments of its choice. The result has applications in cryptography, random constructions, and complexity theory.

Observation of weak localization of light in a random medium

Abstract: The concept of Anderson localization, that is, localization due to randomness, is one of the most interesting new concepts of contemporary physics.1 This strong type of localization pertains to the absence of diffusion in random materials as a result of interference of all the scattered waves. A precursor of Anderson localization, sometimes referred to as weak localization, concerning enhanced backseattering in a strongly scattering random medium, was first pointed out for electrons by Abrahams et al.2

An Exhaustive Analysis of Multiplicative Congruential Random Number Generators with Modulus $2^{31} - 1$

Abstract: This paper presents the results of an exhaustive search to find optimal full period multipliers for the multiplicative congruential random number generator with prime modulus $2^{31} - 1$. Here a multiplier is said to be optimal if the distance between adjacent parallel hyperplanes on which k-tuples lie does not exceed the minimal achievable distance by more than 25 percent for $k = 2, \cdots ,6$. This criterion is considerably more stringent than prevailing standards of acceptability and leads to a total of only 414 multipliers among the more than 534 million candidate multipliers.Section 1 reviews the basic properties of linear congruential generators and § 2 describes worst case performance measures. These include the maximal distance between adjacent parallel hyperplanes, the minimal number of parallel hyperplanes, the minimal distance between k-tuples, the lattice ratio and the discrepancy. Section 3 presents the five best multipliers and compares their performances with those of three commonly employed multipliers for all measures but the lattice test. Comparisons using packing measures in the space of k-tuples and in the dual space are also made. Section 4 presents the results of applying a battery of statistical tests to the best five to detect local departures from randomness. None were found. The Appendix contains a list of all optimal multipliers.

The statistical error of green's function Monte Carlo

Abstract: The statistical error in the ground state energy as calculated by Green's Function Monte Carlo (GFMC) is analyzed and a simple approximate formula is derived which relates the error to the number of steps of the random walk, the variational energy of the trial function, and the time step of the random walk. Using this formula it is argued that as the thermodynamic limit is approached withN identical molecules, the computer time needed to reach a given error per molecule increases asN h where 0.5

Generalized Portmanteau Statistics and Tests of Randomness

Abstract: This paper considers the problem of testing the randomness of Gaussian and non–Gaussian time series. A general class of parametric portmanteau statistics, which include the Box–Pierce and the Ljung–Box statistics, is introduced. Using the exact first and second moments of the sample autocorrelations when the observations are i.i.d. normal with unknown mean, the exact expected value of any portmanteau statistics is obtained for this case. Two new portmanteau statistics, which exploit the exact moments of the sample autocorrelations, are studied. For the nonparametric case, a rank portmanteau statistic is introduced. The latter has the same distribution for any series of exchangeable random variables and uses the exact moments of the rank autocorrelations. We show that its asymptotic distribution is chi–squate. Simulation results indicate that the new portmanteau statistics are better approximated by the chi–square asymptotic distribution than the Ljung–Box statistics. Several analytical results presented i...

Linear complexity and random sequences

Abstract: The problem of characterizing the randomness of finite sequences arises in cryptographic applications. The idea of randomness clearly reflects the difficulty of predicting the next digit of a sequence from all the previous ones. The approach taken in this paper is to measure the (linear) unpredictability of a sequence (finite or periodic) by the length of the shortest linear feedback shift register (LFSR) that is able to generate the given sequence. This length is often referred to in the literature as the linear complexity of the sequence. It is shown that the expected linear complexity of a sequence of n independent and uniformly distributed binary random variables is very close to n/2 and, that the variance of the linear complexity is virtually independent of the sequence length, i.e. is virtually a constant! For the practically interesting case of periodically repeating a finite truly random sequence of length 2m or 2m-1, it is shown that the linear complexity is close to the period length.

Relaxation behaviour in ultrametric spaces

Abstract: The authors model transport in random media through random walks on ultrametric spaces, which allow them to account for energetic randomness They monitor the relaxation patterns for the trapping and target annihilation problems and show that the relaxation depends qualitatively on temperature

Flux quantization on quasicrystalline networks

Abstract: We have measured the superconducting transition temperature (Tc(H)) as a function of magnetic field for a network of thin aluminum wires arranged in patterns from periodic to random and including the intermediate configurations of quasi-crystalline and incommensurate. We find sharp cusp-like dips in the Tc(H) curve for periodic, incommensurate (quasi-periodic) and quasi-crystalline networks reflecting the lock-in of the flux lattice with the underlying network. We find no similar fine structure for random arrays. The experiments therefore strongly suggest that there is a sense of commensurability with incommensurate structures. Can commensurate states exist on incommensurate lattices? To investigate this problem theoretically we introduce a simple model related to both the flux quantization problem as well as the problem of charged particles on a substrate and study the ground states by Monte Carlo simulated annealing. The model is one dimensional but allows for a wide variety of different patterns to be investigated including periodic systems with incommensurate areas, quasi-periodic systems with commensurate areas and various degrees of randomness corresponding to the patterns which can be readily produced by electron beam lithography. The Monte Carlo studies, as well as the experimental observations suggest that there are two types of energetically favorable ground states μ those which reflect the periodicity of the substrate (the configuration of the wire network) and those which reflect the inflation symmetry of the substrate, its self-similarity with change of length scale. This allows us to generalize the concept of commensurability to include incommensurate structures. In particular the commensurate states are those which have a Fourier spectrum consisting of Bragg spots which completely contain the Fourier spectrum of the substrate. However, for the quasi-crystalline substrates, we find that the states reflecting the inflation symmetry are more stable or robust than the periodic states.

How to measure self-generated complexity

Abstract: In an increasing number of simple dynamical systems, patterns arise which are judged as “complex” in some naive sense. In this talk, quantities are discussed which can serve as measures of this complexity. They are measure-theoretic constructs. In contrast to the Kolmogorov complexity, they are small both for completely ordered and for completely random patterns. Some of the most interesting patterns have indeed zero randomness but infinite complexity in the present sense.

Exactly soluble random field Ising models in one dimension

Abstract: The authors solve exactly the one-dimensional random field Ising model for two classes of magnetic field distributions: symmetric exponential (model I) and non-symmetric exponential (model II). For both models, expressions for the free-energy at all finite temperatures are presented. The low-temperature region is examined in more detail; they obtain the zero-temperature energy and entropy in closed form; it is shown that the free energy of both models has an expansion in integer powers of temperature. Model I has a non-vanishing zero-point entropy for all values of the parameters as soon as randomness is diluted. In model II the zero-point entropy is zero except for a discrete sequence of values of one parameter. In some cases the zero-temperature magnetisation is positive whereas the average magnetic field is negative; the magnetisation may also change sign as a function of temperature.

Chaos: solving the unsolvable, predicting the unpredictable!

Abstract: Publisher Summary This chapter focuses on chaos. Chaos means exponentially sensitive dependence of final state upon initial state, positive Liapunov exponents, positive topological or metric entropy, and fractal attractors. Chaos means deterministic randomness —deterministic because of existence–uniqueness and randomness. Chaos is a synonym for randomness. Almost all infinite binary sequences, having positive complexity, pass every computable test for randomness, where a computable test is one expressible as a finite algorithm. A-integrable systems can approximate randomness in the same sense that rationals can approximate irrationals. The virtue of plane billiards, as a class, is that they are the simplest systems that exhibit almost all possible dynamical behavior.

Randomness of a true coin toss

Abstract: A discussion is given of what it means for a coin toss to be random. To aid in that discussion, we produce and solve numerically a physically realistic model of such a toss. The ideas we develop should apply in a general way to other commonly used mechanical randomizers. The coin's randomness is determined by the nature of the basins of attraction of heads and tails. Although mechanical systems are known which are intrinsically random, we conclude that the coin flip is not among them. Rather, the effective randomness in practice is based on the magnitude of the scale of the variation of the basins of attraction relative to the precision of the flipping mechanism.

Moment‐equation and path‐integral techniques for wave propagation in random media

Abstract: Differential equations for all moments of the field of a wave propagating through a random medium are derived under the parabolic approximation and the Markov approximation, but including anisotropy in the random medium and a deterministic background refractive index. Mathematical equivalence is demonstrated between these moment equations and path‐integral expressions for the moments obtained under the same approximations. A discussion of approximations that are weaker than Markov is given.

Variance of the range of a random walk

Abstract: Tn, the expectation of the square of the number of distinct sites occupied by a random walk in steps 1 throughn, is obtained from its relation to the dual first occupancy probabilityFij(x, x′), and the latter quantity is obtained from a recursion with the first occupancy probabilityF k (x″). The varianceVn of the number of distinct sites occupied is calculated directly from Tn; the procedure is illustrated by the calculation ofVn (4096 /⩾n) and the derivation of asymptotic expansions forVn for a particular random walk in dimensions 1 through 3.

Anderson localization and breakdown of hydrodynamics in random ferromagnets.

Abstract: The dynamic structure factor of Heisenberg magnets with weak randomness is computed. Under circumstances which are explained in detail, we find failure of hydrodynamic theory in the longitudinal structure factor due to localization of spin waves. Localization induces a power-law dependence on q and \ensuremath{\omega} for the neutron scattering line shape near magnetic Bragg spots. The exponent describing the power law is related to the correlation-length exponent of Anderson localization. Random anisotropy magnets appear to be promising candidates for experimental investigations.

On Strict-Sense Forms of the Hida-Cramer Representation

Abstract: This paper is an attempt to delineate more clearly than before the role of the standard Brownian motion and Poisson processes in generating general stochastic processes (Ω δt,Xt, P). In discrete time, the analog would be to use a sequence of independent Bernoulli random walks. This is a very different setting, and one about which we have nothing to contribute. Evidently such a sequence does not go far toward generating a general discrete parameter process, at least in the sense we have in mind here. The situation in continuous time, however, is antithetical. One can obtain representation theorems of considerable generality, which to some extent crystalize an important aspect of all continuous time processes. We consider this as the aspect which involves randomness of time without randomness of place. In some sense, there is a natural dichotomy of the two kinds of randomness, and our aim is to isolate and study the case in which the role of randomness of place can be eliminated. A basic tool in our investigation is the “prediction process” Zt of [11], but it is no more than that. Theoretically, one could attempt to define respresentations of Zt itself, analogous to those obtained below but valid for all P. In the present paper, however, each P is treated separately.

Analyzing Catch–Effort Data Allowing for Randomness in the Catching Process

Abstract: For many fisheries the only reliable data is a (bivariate) time series of catches and efforts. Most existing methods of analyzing such data implicitly assume that the main source of randomness is in the dynamics of the population, while ignoring randomness in the catching process. The assumption of a deterministic catch production function (usually of the Schaefer form C = qEX) must be contrary to the experience of almost everyone who has ever gone fishing. In this paper a stochastic catch model coupled with a deterministic dynamic model is used in the analysis of catch–effort data and shown to give very plausible results. Estimates (with confidence intervais) of catchability, maximum sustainable yield, and other dynamic model parameters are obtained numerically by the method of maximum likelihood. The incorporation of stochastic dynamics with the stochastic catch model is difficult.

Random-site spin-glass models

Abstract: A simple general method is presented for solving mean-field spin-glass models where the bond randomness is expressible in terms of an underlying site randomness. Both separable and non-separable models can be solved.

Pattern formation of dendritic fractals in fracture and electric breakdown

Abstract: By introducing a simple network model, growth process of cracks and electric breakdown patterns is analysed numerically. In both cases, random dendritic fractals are created by a deterministic evolution rule with randomness added only initially. Some resemblance and difference are clarified between these two phenomena.

Estimation and Verification of a Stochastic Neuron Model

Abstract: The treatment of a neuron as an information processor is complicated by the nonlinear, time-varying, and distributed parameter attributes of the classical Hodgkin-Huxley neuron model. In this paper, we fit data from experiments on spontaneously firing snail neurons to a much simpler integrate and fire model featuring random process descriptions of the input current density, threshold, and reset potentials. The method of generalized least squares is used to show that the integrate and fire model explains 99.6 percent of the variation in the data used to describe the population behavior of neurons on the visceral ganglion of the Helix Aspersa snail. Experimental histograms suggest that most of the random variation in the interspike interval is caused by the randomness in the input current density and comparatively little by the random fluctuations in threshold and reset potentials, although the latter are still significant from an information processing viewpoint. Residuals from the regression are used to estimate the range of the input current density random process. The residuals also show that slight, but significant, autocorrelation exists in the input current densities. This suggests that information in addition to the mean input current density is being transmitted in the interspike interval code.

Some observations about the randomness of hard problems

Abstract: In this note we investigate some connections between hard languages and random languages. We show that there exist languages that are both hard and random. We also show that every EXPTIME-hard language is polynomial-time weakly random.

High Performance Random Pulser Based on Photon Counting

Abstract: A random pulser featuring high stability, wide range, and good randomness is described. Change of the mean rate per 50°C change of the ambient temperature is less than 0.01%. The mean rate is variable from 10 pps to more than 10 Mpps with good randomness.

Strategic behaviour and a notion of ex ante efficiency in a voting model

Abstract: A person is said to prefer in the stochastic dominance sense one lottery-over-outcomes over another lottery-over-outcomes if the probability of his (at least) first choice being selected in the first lottery is greater than or equal to the analogous probability in the second lottery, the probability of his at least second choice being selected in the first lottery is greater than or equal to the analogous probability in the second lottery, and so on, with at least one strict inequality. This (partial) preference relation is used to define straightforwardness of a social choice function that maps profiles of ordinal preferences into lotteries over outcomes. Given a prior probability distribution on profiles this partial preference ordering (taking into account the additional randomness) is used to induce a partial preference ordering over social choice functions for each individual. These are used in turn to define ex ante Pareto undominated (efficient) social choice functions. The main result is that it is impossible for a social choice function to be both ex ante efficient and straightforward. We also extend the result to cardinal preferences and expected utility evaluations.

Theory of light detection in the presence of feedback

Abstract: The usual formulations of the quantum and semiclassical theories of photodetection presume open-loop configurations, i.e., that there are no feedback paths leading from the output of the photodetector to the light beam impinging on that detector. In such configurations, the qualitative and quantitative distinctions between the quantum and semiclassical theories are well understood. In the quantum theory, photocurrent and photocount randomness arises from the quantum noise in the illumination beam, whereas in the semiclassical theory the fundamental source of randomness is associated with the excitations of the atoms forming the detector. Nevertheless, the quantum theory subsumes the semiclassical theory in a natural way.1

Criteria for randomness of randomised networks

Abstract: A method for the generation of random network models for the structure of amorphous semiconductors is critically re-examined in terms of the structure factor of the models in reciprocal space. It is found that, while the authors' previous models show a feature that is retained from the original diamond cubic structure (as pointed out by Guttman, Thorpe and de Leeuw), this may be suppressed simply by taking the randomisation process further.