Showing papers on "Randomness published in 1970"
TL;DR: For generating the numbers 1, 2, … M in random sequence, an electronic modulo‐M counter is used, driven by the pulses from a high‐frequency pulse generator, producing a random‐number modulo M if the pulse train is interrupted at a random time.
Abstract: For generating the numbers 1, 2, … M in random sequence, an electronic modulo‐M counter is used, driven by the pulses from a high‐frequency pulse generator. If the pulse train is interrupted at a random time, then the counter stops at random in one of its M possible states, producing thus a random‐number modulo M. The random time is the time at which a Geiger‐Mueller tube registers an electron from a 90Sr source. The electronic circuitry is designed such that variations in the characteristics of the components do not impair the randomness. Due to the simplicity of the circuit, the degree of randomness obtainable can be discussed in detail. The randomness of the primarily generated ``basic sequence'' is limited by the finite value of the quotient (pulse‐generator frequency) / (number‐generation frequency). An extremely high degree of randomness can be realized by ``contracting'' the basic sequence, i.e., by adding (modulo M) strings of k (k=2, 3, 4, or even higher) successive numbers of the basic sequence, to form one number of the final sequence. This contraction can be achieved very easily by interrupting the pulse train only after exactly k electrons have been recorded by the Geiger‐Mueller tube. The performance of a generator was tested by recording a basic sequence of generated numbers on paper tape. For probing the long‐time reliability of the generator these recordings were made over an 18 months period, and the randomness tests were designed to discover also temporary malfunctions. In accord with the theoretical expectation, these tests did not indicate any deviation from randomness.
130 citations
TL;DR: This paper showed that there was no significant effect of response rate on first-and second-order nonrandomness, and that there appeared to be substantial individual differences for first and secondorder non-randomness.
73 citations
TL;DR: In this paper, it was shown that higher-order terms in the Wiener-Hermite expansion are capable of representing shocks, which dissipate the energy, even for a nearly Gaussian field of evolving three-dimensional turbulence.
Abstract: Meecham and his co-workers have developed a theory of turbulence involving a truncated Wiener–Hermite expansion of the velocity field. The randomness is taken up by a white-noise function associated, in the original version of the theory, with the initial state of the flow. The mechanical problem then reduces to a set of coupled integro-differential equations for deterministic kernels. We have solved numerically an analogous set for Burgers's model equation and have computed, for the sake of comparison, actual random solutions of the Burgers equation. We find that the theory based on the first two terms of the Wiener–Hermite expansion predicts an insufficient rate of energy decay for Reynolds numbers larger than two, because the equations for the kernels contain no convolution integrals in wave-number space and therefore permit no cascade of energy. An energy cascade in wave-number space corresponds to a cascade up through successive terms of the Wiener-Hermite expansion. Pictures of the Gaussian and non-Gaussian components of an actual solution of the Burgers equation show directly that only higher-order terms in the Wiener–Hermite expansion are capable of representing shocks, which dissipate the energy. Higher-order terms would be needed even for a nearly Gaussian field of evolving three-dimensional turbulence. ‘Gaussianity’, in the experimentalist's sense, has no bearing on the rate of convergence of a Wiener–Hermite expansion whose white-noise function is associated with the initial state. Such an expansion would converge only if the velocity field and its initial state were joint-normally distributed. The question whether a time-varying white-noise function can speed the convergence is treated in the paper following this one.
71 citations
TL;DR: In this article, the authors describe an experiment in which 203 Ss judged binary sequences of white and black dots with respect to randomness, and the results showed that sequences with conditional probabilities around 0.4 were judged as most random.
57 citations
TL;DR: In this article, the notion of randomness was introduced to avoid the paradox, born of the classical conception of the totality of all sets of probability one, to properties expressible in the constructive infinitary propositional calculus.
Abstract: Publisher Summary This chapter focusses on the notion of randomness. The sequences satisfying the definition of randomness form a set of probability one with respect to the measure that makes all coordinates independently take the values 0 and 1 with probability ½. A recursive sequence is necessarily non random. It is proposed to avoid the paradox, born of the classical conception of the totality of all sets of probability one, to properties expressible in the constructive infinitary propositional calculus. The specific Borel sets considered are always obtained by applying the Borelian operations to recursive sequences of the defined sets, which means that they are hyperarithmetical. The chapter proves the theorem, which states that the intersection of all hyperarithmetical sets of measure one is a Σ 1 1 set of measure one.
55 citations
TL;DR: For stochastic differential equations arising in physical problems, the objectives, limitations, and restrictive assumptions of the various methods are studied and some promising new methods are derived which eliminate various limitations and allow treatment of a wide class of applications in physics as discussed by the authors.
Abstract: For stochastic differential equations arising in physical problems, the objectives, limitations, and restrictive assumptions of the various methods are studied and some promising new methods are derived which eliminate various limitations and allow treatment of a wide class of applications in physics. (Among these, will be an adequate treatment of the propagation of an electromagnetic wave in a random continuum or a random d'Alembertian operator without assumptions of ``small randomness'' and other restrictions.)
53 citations
TL;DR: In this paper, the effect of randomness in the coupling constants on the low-temperature thermodynamics was studied and it was found that a phase transition which occurs at zero temperature in the non-random system disappears in the random system.
Abstract: The one-dimensional X-Y model in a z axis magnetic field with random coupling constants is studied. The free energy is calculated for the case in which the squares of the coupling constants have a generalized Poisson distribution. The effect of randomness in the coupling constants on the low-temperature thermodynamics is studied. In particular, it is found that a phase transition which occurs at zero temperature in the non-random system disappears in the random system. The thermodynamic quantities which are singular in the non- random system have derivatives of all orders in the random system.
37 citations
01 Feb 1970
32 citations
TL;DR: In this article, the authors considered two ways of treatment of ties, and the distribution of the respective test statistics is derived under the hypothesis of randomness and under the contiguous alternative.
Abstract: This paper is devoted to problems of rank tests when samples are drawn from purely discrete distributions. There are considered two ways of treatment of ties, and the distribution of the respective test statistics is derived under the hypothesis of randomness and under the contiguous alternative. Furthermore, their asymptotic power and efficiency are established.
26 citations
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TL;DR: In the GEOLOGIC PHENOMENA, the concept of "RANDOMENESS" is more common than previously believed as discussed by the authors, and it has been recognized more and more frequently in the last few decades.
Abstract: MATHEMATICAL PROBABILITY THEORY AND ENSUING FIELDS OF STATISTICS, STOCHASTIC METHODS, AND OPERATIONAL RESEARCH HAVE FREED SCIENTISTS FROM THE RESTRAINT OF DETERMINISTIC METHODS AND RELATED CONCEPTS OF STRICT CASUALITY IN THE ANALYSIS OF NATURAL PHENOMENA. RECENT RECOGNITION OF RANDOMNESS AND APPARENT RANDOMNESS IN FLUVIAL PROCESSES, GEOMORPHIC EVOLUTION, HYDROLOGY, LANDMASS DISTRIBUTION, GEOGRAPHICAL SHAPES, SEISMIC PHENOMENA, STRATIFICATION, AND LITHOLOGY, TOGETHER WITH KNOWN RANDOMNESS IN SUCH BASIC PROCESSES AS RADIOACTIVE DECAY, ORGANIC EVOLUTION, AND GALACTIC EVOLUTION SUGGEST STRONGLY THAT RANDOMNESS IS INHERENT IN THE NATURAL PROCESS. FURTHERMORE, VARYING DEGREES OF RANDOMNESS ARE INDICATED BY THE EVIDENCE IN THE GEOLOGIC PHENOMENA. RECOGNITION THAT RANDOMNESS IS MORE COMMON THAN FORMERLY WAS ADMITTED POSSIBLE, AND RECOGNITION THAT FORMERLY WAS ADMITTED POSSIBLE, AND RECOGNITION THAT RANDOMNESS MAY BE A PROFOUND CONCEPT OF NATURE, DEMAND THAT GEOLOGISTS EXAMINE CLOSELY THE EXTENT AND CHARACTER OF RANDOMNESS IS INHERENT OR SIMPLY APPARENT. /AUTHOR/
TL;DR: In this article, Anderson's theory of localization is critically reviewed and extended, with particular emphasis on some controversial aspects, and it is shown that when the randomness exceeds a certain critical value, all the eigenstates become localized in agreement with Anderson's original result.
TL;DR: In this paper, a definition of agreement between a physical theory and experiment, proposed in earlier work, is extended to be relative to τ where τ is Zermelo-Fraenkel set theory, and the main aim of this definition is to include relations between limit properties of empirical outcome sequences and expectation values obtained from the physical theory.
Abstract: In this work, a definition of agreement between a physical theory and experiment, proposed in earlier work, is extended to be relative to τ where τ is Zermelo‐Fraenkel set theory. The main aim of this work is to show that this definition, unlike that of earlier work, is sufficiently powerful to include relations between limit properties of empirical outcome sequences and expectation values obtained from the physical theory. We also extend, to the more powerful τ, some earlier results on randomness and the empirical determinability of the probability measure which a physical theory assigns to the outcome set of an infinite sequence of experiments.
TL;DR: In this article, a general survey of minimizing criteria and methods suitable for the determination of the constants of the Wilson equation is described, and a method awarding a statistical weight to each experimental point is developed.
TL;DR: In this paper, an analysis is made of the motion of a spherical Brownian particle whose surface can diffusely reflect the molecules of an equilibrium host gas. But the analysis is restricted to the case for which the Knudsen number of the particle is large compared to unity.
Abstract: An analysis is made of the motion of a spherical Brownian particle whose surface can diffusely reflect the molecules of an equilibrium host gas. The analysis is based on Newton's second law and a limiting form of Markov's method. It is shown, both for specular and diffuse reflections, that equipartition of energy is a consequence of the dynamics and randomness of the motion. In addition, it is demonstrated that the diffusion coefficient can depend on the temperature of the particle. The entire analysis is restricted to the case for which the Knudsen number of the particle is large compared to unity.
TL;DR: In this article, a method based on a modification of the "nearest neighbor" relationship, as used in ecology, is developed for evaluating the extent of mixing of nonhomogeneous fibers in a blended yarn.
Abstract: A method based on a modification of the "nearest neighbor" relationship, as used in ecology, is developed for evaluating the extent of mixing of nonhomogeneous fibers in a blended yarn The evaluation parameter is the degree of randomness in the radial fiber dispersion of a yarn Simulated cross sections are created for study in such a manner that they feature either randomness or some imposed form of nonrandomness The procedure for applying the method and the manner of evaluating its results are explained in detail The method of computation has been programmed so that it can be per formed on a computer
TL;DR: In this paper, a model for a satellite under the influence of aerodynamic and gravitational torques is developed as a canonical diffusion process on a cylinder; the atmospheric density uncertainty introduces the randomness into the system.
Abstract: A diffusion process on a multi-dimensional cylinder is developed that can be used to study stochastic processes defining states in physical systems having angular measurements. Stochastic Liapunov functions and supermartingales are used to investigate asymptotic stability and related properties of the process. A model for a satellite under the influence of aerodynamic and gravitational torques is developed as a canonical diffusion process on a cylinder; the atmospheric density uncertainty introduces the randomness into the system. Stochastic stability of this system, along with general pendulum systems, is investigated.
01 Jun 1970
TL;DR: A series expansion method and a rejection method for Monte Carlo radiation transport calculations described by Planck`s Radiation Law are discussed.
Abstract: In some Monte Carlo radiation transport calculations one must sample a random variable which is described by Planck`s Radiation Law. We discuss a series expansion method and a rejection method in this note.
TL;DR: In this paper, the authors used the maximum likelihood method to find a set of parameters which maximizes the likelihood function, which gives the minimum values of variances of estimated parameters, and determined, in the case of Ge(Li) detectors for a random shape and noise, the expression of the likelihood functions.
Abstract: Ge(Li) detectors used for gamma rays detection deliver signals with random parameters in noise. Timing studies with such pulses must take into account the two sources of errors; random shape and noise. This has not been made until now. In this paper, we use the "maximum likelihood method" which has been widely studied in communication theory. It mainly consists in the search of a set of parameters which maximizes the "likelihood function". This condition gives the minimum values of variances of estimated parameters. With the likelihood method we have determined, in the case of Ge(Li) detectors for a random shape (but sinqle interaction) and noise, the expression of the likelihood functions. If an electronic system could give this function as response to Ge(Li) pulses it would be the best pulse Forocessor, but it is obviously impossible to realize. However the method qives the bounds of accuracy of measurements theoretically possible : For planar pulses, this bound is found to be proportional to the inverse of the expectation of the energy of the delivered current. For coaxial pulses, one derives bounds in low and high electric field regions. Results are not so simple as in the previous case and additive terms appear in the formulas with the expectation of the energy of the current. Rather surprising results are obtained : timing nearly as good for planar pulses could be approached.
TL;DR: A "Random Number Generator" is a program that produces a number each time it is invoked that is unpredictable and has the same chance of being selected as any other number among those the generator can produce.
Abstract: A "Random Number Generator" is a program that produces a number each time it is invoked. This number is, for a broad range of uses, "random." More exactly, it is unpredictable and has the same chance of being selected as any other number among those the generator can produce. A random generator is useful in the same way as flipping a coin can be useful: randomness is independent of human motives and desires, it is impartial and fair. Its applications include generating test data, Monte Carlo procedures, testing grammars, and making choices in programs for creating music, poetry, and other art. The definition of randomness is a philosophical and mathematical question, beyond the scope of this discussion. It would not be out of place, however, to explain that there is no such thing as a "random number." While a die may I and five-spot uppermost we would not say that therefore the number 5 is random. Put in a relatively precise way, we would say that a sequence of numbers is random if there is no possible program that could be written to predict what the next member of the sequence was going to be given all previous members of the sequence. We would also require that the sequence had certain statistical properties, for example, that each possible output was equally likely. A random generator is a device (such as an honest roulette wheel) that produces such a sequence. It follows that a computer program cannot generate a random sequence (if it could, then a faster computer with the same program could 'predict' the outputted numbers). This is generally true. The usual "random number generator" or "random function" provided at most installations is more properly a pseudorandom generator. It is, in fact, a program that produces a string of numbers that for most practical purposes appears random. It has good statistical properties and while it will eventually repeat itself and is certainly predictable, it is unlikely that you (or your program) would ever detect these "faults." There is a large and still growing literature on methods of programming pseudorandom functions, and one can probably be written in relatively short order (see for example, the Algorithms department of the Communications of the Association for Computing Machinery ). The random generator at your installation may have any of a number of forms. Usually it is a function or procedure whose name looks like RND, RAN, RANDOM, RANF, IRANF, etc. In APL it is simply a
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TL;DR: The one-sample runs test as mentioned in this paper enables one to determine the randomness of the sequence of events in a sample, which can be used to demonstrate that statistics are fallible.
Abstract: I t is difficult to conceptualize for students in elementary statistics, the meaning and consequences of a Type I or a Type I1 error; that in fact statistics are fallible One method of demonstrating a Type I1 error, that is, accepting the null hypothesis when in fact it is false, can be illustrated by the non-parametric one-sample Runs Test (Siegel, 1956) The one-sample Runs Test enables one to determine the randomness of the sequence of events in a sample For example, one might be interested in whether a group of rats show any systematic preference for one side over another in a simple T-maze W e would expect on the basis of chance that our animals would turn right as often as they would turn left (50/50), but we would also be interested in the randomness of patterning The following patterns are non-random even though the animals turn right 50% of the time and left 50% of the time
TL;DR: In this paper, the problem of scattering a plane elastic wave by a rigid circular cylindrical-inclusion imbedded in a random medium is analyzed on the assumptions that the medium differs slightly from a homogeneous medium, is independent of axial space variable and is statistically homogeneous and isotropic in other space variables.
TL;DR: In this article, the authors compared the potential energy of clusters of stars in which the distribution of matter is continuous with that of static model clusters, and the comparison was made from the point of view of applying the virial theorem to estimate the masses of the clusters.
Abstract: The potential energy of clusters of stars in which the distribution of matter is taken to be continuous is compared with that of static model clusters in which the distribution of matter is discrete, the comparison being made from the point of view of applying the virial theorem to estimate the masses of the clusters. There is good agreement on the average between the two cases as long as the stellar distribution is random. Systematic differences occur whenever there is any departure from randomness. However, reduction of the mass of a cluster as estimated by means of the virial theorem by even as much as a factor of 2 on the average would seem to require even greater departures from randomness in the stellar distribution than are considered here. As might be expected there are sometimes very large fluctuations in the potential energy from one cluster to the next in the discrete case.