Perfect Randomness Research Articles

An Algorithmic Complexity Interpretation of Lin's Third Law of Information Theory

Instead of static entropy we assert that the Kolmogorov complexity of a static structure such as a solid is the proper measure of disorder (or chaoticity). A static structure in a surrounding perfectly-random universe acts as an interfering entity which introduces local disruption in randomness. This is modeled by a selection rule R which selects a subsequence of the random input sequence that hits the structure. Through the inequality that relates stochasticity and chaoticity of random binary sequences we maintain that Lin’s notion of stability corresponds to the stability of the frequency of 1s in the selected subsequence. This explains why more complex static structures are less stable. Lin’s third law is represented as the inevitable change that static structure undergo towards conforming to the universe’s perfect randomness.

Spatial distributions of aerosol particles: Investigation of the Poisson assumption

Atmospheric particulates are known to be distributed non-uniformly in space due to spatially localized sources and incomplete mixing. Many studies have extensively examined the variability of aerosol particle concentrations on diurnal, seasonal, and annual timescales. Receiving much less attention, however, are aerosol particle spatial distributions on shorter ( ≲ 5 min ) temporal scales. Knowledge regarding the spatial distribution on these temporal scales is vital to fully understand and predict the properties of microphysical processes like activation, coagulation, and collision/coalescence, as well as some large-scale phenomena like radiation attenuation through a thick yet dilute layer of particles. Traditional treatments of microphysical phenomena often implicitly assume that particles are perfectly randomly distributed (obey Poisson statistics) on short temporal scales, even though this assumption may be inconsistent with observations made on longer scales. This study seeks to explicitly resolve the statistics of particle positions on a variety of temporal scales (over five orders of magnitude) to determine if and when the assumption of perfect spatial randomness can be justified in nature. Some implications of the results are discussed.

On the extinction of radiation by a homogeneous but spatially correlated random medium: reply to comment

In response to comments by Borovoi [J. Opt. Soc. Am. A19, 2517 (2002)] on my earlier work [J. Opt. Soc. Am. A18, 1929 (2001)], the kinetic approach to extinction is compared with the traditional radiative transfer formalism and advantages of the former are illustrated with concrete examples. It is pointed out that the basic differential equation dI(l)=-cσI(l)dl already implies perfect randomness (absence of correlations) on small scales. One of the consequences is that the extinction of radiation in a negatively correlated random medium cannot be treated within the traditional framework. This limits the usefulness of the Jensen inequality. Also, simple counterexamples to theorems given in the first reference above and in Dokl. Akad. Nauk SSSR, 276, 1374 (1984) are presented.

Metastable Coordination Dynamics of Brain and Behavior.

One of the biggest puzzles in trying to understand ourselves is how we (and our brains) make decisions or choices. What kind of thing is the brain that it can make choices? If choice is a function of the large-scale activity of the brain, what form do these large-scale spatiotemporal patterns of brain activity take and how are they related to what people do? How are localized regions of the brain coordinated so as to support large-scale network activity? Fundamentally speaking, why is one kind of pattern of behavior selected over another? What are the principles and mechanisms? Choice implies the creation (and annihilation) of information. How is information created so that it may modulate, and be modulated by, the dynamics of the brain? These are big questions, so let's treat them systematically, albeit briefly. There are good reasons to believe that the answers may lie within Coordination Dynamics, a line of scientific enquiry that aims to understand—through theory and experiment—how patterns of coordination form, persist and change in living things. Coordination is not just matter in motion. It refers to the functional ordering of parts and processes in space and time. Coordination Dynamics is unique because it contains two coexistent aspects, a self-organizing, or ‘undirected’ aspect and an informational, or ‘directed’ aspect. The former deals with collective or cooperative effects that arise spontaneously when ordinary matter takes on novel properties, as in lasers and superconductors, or when new forms of organization among water molecules arise as in the weather (Cross and Hohenberg, 19931); Haken, 19772); Nicolis and Prigogine, 19773)). The human brain and the behavior it produces has been demonstrated to exploit these self-organizing, cooperative effects, an intriguing finding in its own right (Sect. 2.0). The latter, ‘directed’ aspect of coordination dynamics reveals how information is created de novo and how information guides, directs, modulates and is modified by self-organizing dynamics. “Directed” terms like “plans,” “programs,” “intentions” and so forth, rather than reified are embraced by metastable coordination dynamics. Metastable Coordination Dynamics means that there is no such thing as perfect order or randomness in the way natural systems are coordinated. Only tendencies toward order and randomness exist in spacetime. These tendencies arise from the coexistence of intrinsic properties of individual parts and the essentially nonlinear coupling between them (Sect. 5.0).

Effect of packing method on the randomness of disc packings

The randomness of disc packings, generated by random sequential adsorption (RSA), random packing under gravity (RPG) and Mason packing (MP) which gives a packing density close to that of the RSA packing, has been analysed, based on the Delaunay tessellation, and is evaluated at two levels, i.e. the randomness at individual subunit level which relates to the construction of a triangle from a given edge length distribution and the randomness at network level which relates to the connection between triangles from a given triangle frequency distribution. The Delaunay tessellation itself is also analysed and its almost perfect randomness at the two levels is demonstrated, which verifies the proposed approach and provides a random reference system for the present analysis. It is found that (i) the construction of a triangle subunit is not random for the RSA, MP and RPG packings, with the degree of randomness decreasing from the RSA to MP and then to RPG packing; (ii) the connection of triangular subunits in the network is almost perfectly random for the RSA packing, acceptable for the MP packing and not good for the RPG packing. Packing method is an important factor governing the randomness of disc packings.

Randomized algorithms and pseudorandom numbers

Randomized algorithms are analyzed as if unlimited amounts of perfect randomness were available, while pseudorandom number generation is usually studied from the perspective of cryptographic security. Bach recently proposed studying the interaction between pseudorandom number generators and randomized algorithms. We follow Bach's lead; we assume that a (small) random seed is available to start up a simple pseudorandom number generator which is then used for the randomized algorithm. We study randomized algorithms for (1) sorting; (2) selection; and (3) oblivious routing in networks.

Isotopic lattice symmetry: An unexplored area of research

The existence of isotopes is a well-known and widely used fact of chemistry. Nevertheless, despite all the attention to the isotopes of individual elements, there have been very few, if any, attempts to look at the whole notion of isotopicity as a unique phenomenon in its own way. This article is an attempt to fill this gap. The general assumption of the perfect isotopic randomness in molecules and crystals might have a complimentary aspect in the possibility of positional correlations in isotope distribution. I introduce the notion of isotopic freedom to reflect positional flexibility of different isotopes within structural constraints of a given elemental sublattice. In this vein, the whole notion of isotopicity can be seen as an overlooked outfit for “Nature's quest for patterns,” complimentary to other known manifestations of spontaneous pattern formation. I also discuss the potential of artificial isotopic structuring for fundamental and applied research directions, an option that also has remained largely unused until the present time.

Local randomness in pseudorandom sequences

The concept of provable cryptographic security for pseudorandom number generators that was introduced by Schnorr is investigated and extended. The cryptanalyst is assumed to have infinite computational resources and hence the security of the generators does not rely on any unproved hypothesis about the difficulty of solving a certain problem, but rather relies on the assumption that the number of bits of the generated sequence the enemy can access is limited. The concept of perfect local randomness of a sequence generator is introduced and investigated using some results from coding theory. The theoretical and practical cryptographic implications of this concept are discussed. Possible extensions of the concept of local randomness as well as some applications are proposed.

Random Number Generation by Normal Subjects

Human facility for generating random numbers has been proposed as a brief simple test of psychological functioning (5). Evans (2) reported a standardised procedure and formulae for obtaining an index.of random number generation. A sample of 50 male and 71 female Christchurch residents was interviewed at home. In terms of age and socioeconomic status (1) the sample was representative of the general population. Subjects were instructed in the concept of randomness and then attempted to generate random numbers, each subject generating 100 numbers between 1 and 10 at a paced rate of l/sec. An index, based on the frequency with which each number was succeeded by another, was calculated for each subject. Index values can range from zero (indicating perfect randomness) to one (when each number is completely predictable from its predecessor, e.g., 7 might be always followed by 3, etc.). The mean index was 0.335 (SD = 0.056). This is somewhat higher than that found with volunteer college students (0.317) but lower than that found with some psychiatric inpatient groups (3). No significant (# < 0.05) effect of age, age on leaving school, history of concussion, or of serious concussion (defined by hospitalization for 24 hr. or more for concussion) was found on the index. There was a significant correlation (0.33) between the index and number of errors on a word-reading test known to correlate highly with intelligence (4). Facilicy at generating random numbers is then related .to general intelligence, a feature which should be considered when the index is used clinically. There was also a suggestive (P < 0.10) positive correlation bemeen the index and socioeconomic status which, however, disappeared when perform

Thermodynamic properties of liquids, including solutions. III. Solutions of polymers, especially poly(viny chloride)

AbstractA new theory of solution thermodynamics is outlined. The energy and enthalpy are related to the contacting areas of the chemically different kinds of molecular segments, the energy per unit contact area for each kind of contact, and an equilibrium constant. Departures of the entropy of mixing from the classical Raoult's law relationship result from differences in the volumes of solvent and solute molecules, from departures from perfect randomness of mixing, and from the concentration dependence of the orientational randomness of the bonds in the molecules. Simple equations for these relationships have been tested. Quantitative agreement is usually obtained.Examples are given of applications of the theory to polymer solutions. Application to poly(vinyl chloride) solutions has not yet been made, but the appropriate procedures and methods for determining the parameters are discussed.

On Some Statistical Tests for $M$th Order Markov Chains

Certain $\psi^2$ statistics were defined by Good in [8] and he stated that there is a "strong analogy" between certain functions of these statistics and some given likelihood ratio (LR) statistics appropriate for testing hypotheses concerning the order of a Markov chain. He also indicated that the analogy held when the hypothesis of "perfect randomness" is true. The present author has indicated in [12] that, for the $\psi^2$ statistics in [8], this analogy (i.e., the asymptotic equivalence of the corresponding statistics) does not hold under some more general conditions when the "perfect randomness" hypothesis is not true. It will be seen herein that certain functions of a modified form of the $\psi^2$ statistics are asymptotically equivalent to certain LR statistics in the more general case when the hypothesis $H(P_m)$ that the positively regular Markov chain (see [2]) is governed by a completely specified system $P_m$ of $m$th order transition probabilities is true. Also, certain functions of a different modified form of the $\psi^2$ statistics will be seen to be asymptotically equivalent to certain LR statistics in the case when the hypothesis $H_m$ that the positively regular Markov chain is of order $m$ is true. These results are helpful in determining the asymptotic distributions of various statistics and the null hypotheses that can be tested with a given statistic. For example, if a given statistic $G$ is asymptotically equivalent, under $H(P_1),$ to the LR statistic $L$ for testing the null hypothesis $H(P_1)$ within the alternate hypothesis $H_2$, then the asymptotic distribution, under $H(P_1),$ of $G$ will be $\chi^2$ with a known number of degrees of freedom (i.e., with a known expectation); $G$ can be used directly to test $H(P_1)$ within $H_2$ (if $G$ is sensitive to these hypotheses), although the asymptotic distribution, under $H_2,$ of $G$ may differ, in a certain sense, from that of $L$ (see Section 6 in [1]). However, if a given statistic $\Delta G$ is asymptotically equivalent, under $H(P_1),$ to the LR statistic $\Delta L$ for testing the null hypothesis $H_1$ within the alternate hypothesis $H_2$ (i.e., if there is an "ostensible analogy" between $\Delta G$ and $\Delta L$), but this asymptotic equivalence does not hold under some more general conditions (e.g., under $H_1$), then $\Delta G$ can not be used to test $H_1$ within $H_2$; the asymptotic distribution, under $H(P_1),$ of $\Delta G$ will be $\chi^2,$ but the asymptotic distribution, under the null hypothesis $H_1,$ will not be $\chi^2,$ and furthermore the expectation, under $H_1,$ of $G$ can approach infinity (see [15]). The present author has indicated in [15] that certain functions of a modified form of the $\psi^2$ statistics, which were investigated by Stepanow in [18] and which are computed for a specified $P_1$, are asymptotically equivalent, under $H(P_1),$ to certain LR statistics, but that they will not be equivalent under $H_1.$ Although it is stated in [18] that the results presented there can be applied to the solution of the problem of testing the null hypothesis $H_1,$ it is shown in [15] that none of the statistics in [18] can be used directly to test this composite hypothesis. In the present paper, it will be seen that a statistic based on a different modified form of $\psi^2,$ as well as certain other statistics described herein, will be asymptotically equivalent, under $H_1,$ to a certain LR statistic and can be used to test the null hypothesis $H_1.$ The $\psi^2$ statistics defined by Good in [10] are more general than those given in [3], [8], [18]. Besides studying the relation between these statistics and the LR statistics, we shall also discuss certain conjectures proposed in [10] concerning the asymptotic distributions of these statistics, which were investigated by Billingsley [4] for the cases $H_0$ and $H_1$ (the author mentions that a more general result for $H_m(m \geqq 0)$ can be obtained using similar methods) and independently, using different methods, by the present author [14] for the case $H_m(m \geqq 0)$ when the transition probabilities are all positive (this author also mentions that a more general result can be obtained by similar methods). The $\psi^2$ and LR statistics defined in [10] and [8], as well as some related statistics developed in the present paper, will be generalized further herein, and the asymptotic distributions of these generalized statistics will be investigated. This investigation leads to generalizations of the asymptotic distributions obtained by Good [8], Billingsley [3] [4], and the present author [13] [14], and it helps to clarify the relation between the various statistics. The different asymptotically equivalent forms of various statistics presented here make it possible for the statistician to choose whichever form he finds preferable both from the computational point of view and also from some other viewpoints (see [1], [5], [11]).

Asymptotic Distributions of "Psi-Squared" Goodness of Fit Criteria for $m$-th Order Markov Chains

Asymptotic Distributions of "Psi-Squared" Goodness of Fit Criteria for $m$-th Order Markov Chains