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Looking through Li Yans History of Calculation in China , and other works on the history of premodern Chinese mathematics, one is impressed by the virtual absence of anything explicitly relating to the theory of probability. Premodern China had an abundance of the availability of repeating patterns that can be expressed numerically (stochastically determined oracles, dice-casting and analogous phenomena from gaming) but since this abundance was not in itself a sufficient cause for the emergence of a calculus of probabilities sharpens the questions that must be asked about China, and also suggests the inadequacies of exclusively external-factor analyses of the rise of probabilistic thinking in Europe. Perhaps, in the last analysis, there was a lingering belief in China that there was no inherent equiprobability of outcomes, and personal luck ruled; and given that this was so, that calculating probabilities correctly was impossible, and so not worth bothering with. Keywords: cubic dice; personal luck; premodern China; probabilistic thinking; tortoise-shell oracles

USING THE EXPERIMENTAL AND THEORETICAL RESULTS OF PREVIOUS RESEARCH WORK, THE AUTHOR PRESENTS A PROBABILISTIC DESCRIPTION OF CONSTANT STRESS AMPLITUDE FATIGUE-TEST RESULTS. THIS DESCRIPTION INCLUDES THE S-N CURVES (OR EQUIPROBABILITY OF FRACTURE CURVES), P-S CURVES (OR STRESS-RESPONSE CURVES), P-N CURVES (CUMULATIVE DISTRIBUTION FUNCTIONS OF FATIGUE ENDURANCES), AND ACCOUNTS FOR THE OCCURRENCE OF RUNOUTS. A METHOD OF ESTIMATION OF THE MODEL COEFFICIENTS WHICH USES THE INFORMATION PROVIDED BY ALL THE SPECIMENS TESTED (BROKEN OR UNBROKEN) IS ALSO PRESENTED. THE APPLICATION OF THIS METHOD IS DEMONSTRATED THROUGH FIVE EXAMPLES FOR EACH OF WHICH SEVERAL HUNDRED TEST RESULTS ARE AVAILABLE. /ASTM/

A graph with no point of degree greater than f is called an f -graph. An f -graph to which no line can be added without introducing a point of degree greater than f is called a line maximal f-graph . We consider the following procedure. Starting with n labeled points and no lines, sequentially add lines one at a time to these points so as to obtain at each step a labeled f -graph. At each step the line to be added is selected with equiprobability from among those lines whose addition would not violate the f -degrees restriction. A terminal graph of this procedure is a random sequentially generated line maximal f -graph. Let P ( m = i ; n ; f ) denote the probability that a random sequentially generated line maximal f -graph has i points of degree less than f . The exact determination of P ( m = i ; n ; f ) is an open problem posed by P. Erdos. In this paper we obtain, for f = 2, 3 and 4, approximation functions for P ( m = i ; n ; f ) and conjecture values for lim P ( m = i ; n ; f ) as n → ∞. Heretofore, neither theoretical nor qualitative results concerning the asymptotic distribution of P ( m = i ; n ; f ) were available.

Uniformity, that is, equiprobability of all available options is central as a theoretical presupposition and as a computational tool in probability theory. It is justified only when applied to an appropriate sample space. In five studies, we posed diversified problems that called for unequal probabilities or weights to be assigned to the given units. The predominant response was choice of equal probabilities and weights. Many participants failed the task of partitioning the possibilities into elements that justify uniformity. The uniformity fallacy proved compelling and robust across varied content areas, tasks, and cases in which the correct weights should either have been directly or inversely proportional to their respective values. Debiasing measures included presenting individualized and visual data and asking for extreme comparisons. The preference of uniformity obtains across several contexts. It seems to serve as an anchor also in mathematical and social judgments. People’s pervasive partiality for uniformity is explained as a quest for fairness and symmetry, and possibly in terms of expediency.

A method of mining association rules from time series based on Rough Set is introduced. To clean the data, Fourier transformation is employed, and LPF operator is adopted. Partial and overall features of a time series are defined, and some innovative methods for extracting features from a time series or for segmenting a time series are proposed. Thereafter, a discretization technique that will produce symbols with equiprobability is adopted to discretize the features since Rough Set can only tackle discretized values. Traced time segments problem has already been a serious problem of data mining from a time series with Rough Set, so an innovative method to determine the traced time segments is proposed. Finally, two mining strategies are proposed to demonstrate the process of mining association rules in a time series with Rough Set, and an example is presented too.

A system for feature extraction and recognition of lip postures is presented, by constructing a shape model of the inner and outer contours of the lips, whose parameters are controlled by a canonical genetic algorithm. Features of each posture are the model parameters which better fit to the the analyzed image. A Bayesian Classifier is used to classify, under the assumption that there is equiprobability between the classes, obtaining results up to 82,4% of accurate classification.

Quantum electrodynamics is a time-symmetric theory that is part of the electroweak interaction, which is invariant under a generalized form of this symmetry, the PCT transformation. The thesis is defended that the arrow of time in electrodynamics is a consequence of the assumption of an initial state of high order, together with the quantum version of the equiprobability postulate. (c) 2006 Elsevier Ltd. All rights reserved.

The objective of this paper is the statistical characterization of natural hyperspectral backgrounds using themultivariate t-elliptically contoured distribution. Traditionally, hyperspectral backgrounds have been modeledusingmultivariateGaussiandistributions;howeveritiswellknownthatrealdataoftenexhibit"long-tail"behaviorthat cannot be accounted by normal distribution models. The proposed multivariate t-distribution model haselliptical equiprobability contours whose center and ellipticity is speciÞed by the mean vector and covariancematrixofthedata. Thedensityofthecontours,whichisresectedintothedistributionoftheMahalanobisdistance,is controlled by an extra parameter, the number of degrees of freedom. As the number of degrees of freedomincreases, the tails decrease and approach those of a normal distribution with the same mean and covariance. Inthisworkweinvestigatetheapplicationof t-ellipticallycontoureddistributionstothecharacterizationofdifferenthyperspectral background data obtained by visually interactive spatial segmentation ("physically" homogeneousclasses), automated clustering algorithms using spectral similarity metrics (spectrally homogeneous classes),and by Þtting normal mixture models (statistically homogeneous classes). These investigations are done usinghyperspectral data from theAVIRIS sensor.Keywords: Hyperspectral imaging, statistical modeling, background characterization, target detection.

Given an equidistribution for probabilities p(i)=1/N, i=1..N. What is the expected corresponding rank ordered frequency distribution f(i), i=1..N, if an ensemble of M events is drawn?, 4 pages, 4 figures

In this paper we extend to upper and lower probabilities our approach to independence, based solely on conditional probability (in a coherent setting). Most difficulties arise (when this notion is put forward in the classical framework) either from the introduction of marginals for upper and lower probabilities (often improperly called, in the relevant literature, "imprecise" probabilities) when trying to extend to them the "product rule", or from the different ways of introducing conditioning for upper and lower probabilities. Our approach to conditioning in the context of "imprecise" probabilities is instead the most natural: in fact its starting point refers to a direct definition (through coherence) of the "enveloping" conditional "precise" probabilities. The discussion of some critical examples seems to suggest that the intuitive aspects of independence are better captured by referring to just one (precise) probability than to a family (such as that one singling-out a lower or upper probability).

Summary. Identification of fragile sites is a way to investigate the genetic abnormalities that are the hallmark of cancer and play an important role in carcinogenesis. Manifestation of nonrandom breakage at a chromosome band is an essential criterion for determination of the fragility of the band. In this article, a new detection procedure is proposed. This new procedure takes the relationship of one site with the others into consideration and can be applied to tests of the randomness of breakpoints under either the proportional probability model (PPM) or the equiprobability model (EPM). The procedure can form a grouping structure that classifies all sites into several clusters. It is applied to identification of fragile sites for a real data set for Chinese patients with colorectal carcinoma for illustration of the proposed method.

Limiting dilution assays (LDA) are used to estimate an unknown cell fraction of interest within a sample. This paper discusses a method for designing an LDA using the distribution of the cell fraction of interest, examining three different design approaches: geometric progression, equally spaced log, and equiprobability. Two common estimation methods, minimum chi-square and maximum likelihood, also are investigated. These designs and estimation methods, coupled with varying numbers of wells per dilution and dilutions per design, are compared quantitatively through computer simulation. Performance measures computed were mean relative bias and mean squared error.

We introduce a nonextensive entropic measure $S_{\chi}$ that grows like $N^{\chi}$, where $N$ is the size of the system under consideration. This kind of nonextensivity arises in a natural way in some $N$-body systems endowed with long-range interactions described by $r^{-\alpha}$ interparticle potentials. The power law (weakly nonextensive) behavior exhibited by $S_{\chi}$ is intermediate between (1) the linear (extensive) regime characterizing the standard Boltzmann-Gibbs entropy and the (2) the exponential law (strongly nonextensive) behavior associated with the Tsallis generalized $q$-entropies. The functional $S_{\chi} $ is parametrized by the real number $\chi \in[1,2]$ in such a way that the standard logarithmic entropy is recovered when $\chi=1$ >. We study the mathematical properties of the new entropy, showing that the basic requirements for a well behaved entropy functional are verified, i.e., $S_{\chi}$ possesses the usual properties of positivity, equiprobability, concavity and irreversibility and verifies Khinchin axioms except the one related to additivity since $S_{\chi}$ is nonextensive. For $1, Comment: LaTeX file, 20 pages, 7 figures

International audience; We present a new string matching algorithm optimal on average (with equiprobability and independence of letters, in O(m + n log(\Sigma\) m/m), where n is the size of the text and m the size of the starched word, both taken on an alphabet Sigma) and linear in the worst case (in O(m + n)). Of all the algorithms that verify these two complexities. ours is the: simplest since it uses only a single structure, a suffix automaton, Moreover. its preprocessing phase is linearly dynamical; i.e., it is possible to search the words p(1), then p(1) p(2), p(1) p(2)p(3),,,., p(1) p(2)p(3).,. p(i) with O(Sigma\p(i)\) total preprocessing time, Among the algorithms that verify this property (for instance, the Knuth-Morris-Pratt algorithm) our algorithm is the only one to be optimal on average. (C) 2000 Academic Press.

The volume of the body enclosed by the n-dimensional Lame curve defined by Ʃni=1 xbi = E is computed.

International audience; The subset X-0,={AAC,AAT,ACC,ATC,ATT,CAG,CTC,CTG,GAA,GAC,GAG,GAT,GCC,GGC,GGT,GTA, GTC,GTT,TAC,TTC} of 20 trinucleotides has a preferential occurrence in the frame 0 (reading frame established by the ATG start trinucleotide) of protein (coding) genes of both prokaryotes and eukaryotes. This subset X-0 is a complementary maximal circular code with two permutated maximal circular codes X-1 and X-2 in the frames 1 and 2 respectively (frame 0 shifted by one and two nucleotides respectively in the 5'-3' direction). X-0 is called a C-3 code (Arquis and Michel, 1997, J. Biosyst 44, 107-134). A quantitative study of these three subsets, X-0, X-1, and X-2 in the three frames 0, 1 and 2 of eukaryotic protein genes shows that their occurrence frequencies are constant functions of the trinucleotide positions in the sequences. The frequencies of X-0, X-1 and X-2 in the frame 0 of eukaryotic protein genes are 48.5%, 29% and 22.5% respectively. These properties are not observed in the 5' and 3' regions of eukaryotes where X-0, X-1 and X-2 occur with variable frequencies around the random value (1/3). Several frequency asymmetries unexpectedly observed, e.g. the frequency difference between X-1 and X-2 in the frame 0, are related to a new property of the C-3 code X-0 involving substitutions. An evolutionary analytical model at three parameters (p, q, t) based on an independent mixing of the 20 codons (trinucleotides in the frame 0) of X-0 with equiprobability (1/20) followed by t approximate to 4 substitutions per codon according to the proportions p approximate to 0.1, q approximate to 0.1 and r = 1 - p - q approximate to 0.8 in the three codon sites respectively, retrieves the frequencies of X-0, X-1 and X-2 observed in the three frames of protein genes and explains these asymmetries. The complex behaviour of these analytical curves is totally unexpected and a priori difficult to imagine. Finally, the evolutionary analytical method developed could be applied to the phylogenetic tree reconstruction and the DNA. sequence alignment. (C) 1999 Elsevier Science Ireland Ltd. All rights reserved.

Through a simple computer simulation, we provide clear evidence that the base composition of nucleotide sequences should be taken into account to obtain accurate estimates of the genetic distances between homologous nucleotide sequences. We tested several stochastic models and demonstrated clearly that the inferred estimates of the number of substitutions are strongly dependent on the a priori conditions superimposed in the model, i.e., equiprobability in base composition and/or transition transversion ratio. In general, the more a priori assumptions there are, the more inaccurate the results. Whereas, rather accurate estimates are obtained with stochastic methods, which take into account base composition and do not superimpose unverified a priori conditions.

Individuals with a previous experience of dominance are likely to be dominants in further encounters. To test this effect, individuals with a previous experience of dominance are used for the experiments. One way to obtain such individuals is to let opponents 'self-select': encounters between pairs of more or less equivalent opponents are staged and one selects ex post facto the dominant and subordinate from the ensuing conflict. This paper formally shows that the selection of dominant and subordinate animals modifies the dominance probability functions of the two corresponding sub-samples of animals. As a result, the propensity of previous winners to win again and of previous losers to repeat their loss can be attributed to this artefact rather than to prior social experience. This result has serious methodological implications. When one relies solely on selection to obtain winners and losers, equiprobability is no longer the appropriate null hypothesis against which prior social experience effects have to be tested. To clearly demonstrate the effect of dominance experience, one must show that prior winners defeat neutral opponents in at least two-thirds of all cases; reciprocally, to show that prior subordinate experience plays a role, prior losers must win in less than one-third of all fights against neutral opponents; finally, to conclude that a combined effect of the two kinds of prior experience is in operation, one must obtain that prior winners defeat prior losers in more than 83% of all planned conflicts. The present result does not imply that experience effects are not at work when the selection procedure is used, but that the procedure used to show their effects is inadequate because effects of experience on a subsequent encounter are confounded with those introduced by statistical selection.

Many people believe the “Nugget Effect” is the result of rap: in-situ changes in the concentration of a given constituent interest, taking place on a very small scale. They are right however, there is much more to it. Indeed, the “Nugget Effect is the result of at least seven types of variability: 1 The true in-situ, small-scale, random variability; 2 The variability introduced by Constitution Heterogenei during all sampling and subsampling stages, which is function of fragment size and sample or subsample weight 3 The variability introduced by small-scale Distributiol Heterogeneity during all sampling and subsampling stages which is a function of transient segregation as soon a the material has been broken up; 4 The variability introduced by any deviation from an isotropic module of observation ensuring sampling equiprobability in all relevant dimensions, during al: sampling and subsampling stages; 5 The variability introduced by selectivity and poor recovery during all sampling and subsampling stages; 6 The variability introduced by contamination, losses, alteration of physical or chemical properties, and huma errors; and 7 The variability introduced by the analytical procedure

1. Applicability. Following previous studies, Tai et al. (1993) proposed use of the relationship between the binomial and F-distributions to test for nonrandomness of chromosomal breakpoints. All of these methods were developed based on the concept of the binomial model. Dahm and Greenbaum (1994) contended that appropriate models for statistical identification of fragile sites will require tests that establish site-specific chromosomal breakage as being nonrandom with respect to the number and distribution of total breaks. They thus questioned the statistical efficacy of Tai et al. (1993) with their own idea in this (cytogenetic) area. Even if their idea were correct, their question should be applicable to all methods under the binomial model, not only for Tai et al. (1993). We have little confidence in the biological and, specifically, cytogenetic reasoning put forth by Dahm and colleagues. They draw strong conclusions on the basis of very little biological evidence. Bohm et al. (1995) chose to present their method assuming an equiprobability model (EPM), although they noted their test procedures can be directly modified to scale the multinomial-homogeneity expectations to reflect band width or other relevant proportions. From our perspective, if Dahm and colleagues had been as clear in their published manuscript as they were in their Reader Reaction, readers might have more easily deduced the meaning of "directly," and we would not have misinterpreted them. We appreciate that this problem has now been clarified. Nevertheless, we would like to emphasize that Hou, Chiang, and Tai (1999) proposed using the highest standardized observed breakage (Ni np9)/{np?(1 9)11/2 as an exclusion statistic and the M test (Fuchs and Kenett, 1980) for resolving the problem. 2. Binomial, multinomial, or hierarchical. The iterative procedure proposed by Bohm et al. (1995) was established under a multistage changeable multinomial model. Because in each step the comparison base of sites changes, this method is a sequentially adjusted comparison base procedure rather than a fixed comparison base procedure. As a consequence, all the fragile sites identified by this procedure are obtained by different comparison bases. Instead of thinking that identification of fragile sites is a two-group classification procedure as Bohm et al. (1995) proposed, Hou, Chiang, and Tai (2001) considered that the classification procedure can be a multiple-group classification procedure. Obviously, the two-group setting is a special case of the multigroup setting. If Bohm and colleagues thought that the work of Bohm et al. (1995) extending the binomial model to the multinomial model is meaningful for identification of fragile sites, we do not see why they declared that it is not apparent that hierarchical clusters of fragile sites have any biological meaning. 3. Pooled or individual. Bohm et al. (1995) concluded from a numerical study that the analysis of chromosomal breakage data pooled over individuals causes a large number of sites to be inaccurately identified as fragile and insisted on the necessity of per-individual fragile sites analysis. In fact, very simple examples can be constructed in which just the opposite conclusion is reached: Failing to pool samples across individuals leads to the false exclusion of fragile sites. Thus, to our mind, the logic behind the claim of Dahm and colleagues is overly simplistic. 4. Sample size. Greenbaum et al. (1997) recommended a minimum of half as many breaks as bands in identifying

Knowledge is definedoperationally by the ability to make predictions. In multiple choice tests one has to predict which alternative will be scored as the correct one. In this context a testee is called a predictor. A prediction is defined as the assignment of probabilities to mutually exclusive events. The “true” probabilities are dependent on the knowledgeability of the predictor, on the Information he has: all probabilities are personal probabilities. Probabilities are called “true” if for each probability the proportion of occurrence of events approaches \the probability assigned to those events. A measure of “realism” indicating if reported probabilities are an overestimation or an underestimation of the true probabilities will be derived as will be the scoring rule to evaluate predictors.

A Lagrangian trajectory model, TRACMASS with the use of velocity fields calculated by the Rossby Centre (Swedish Hydrological and Meteorological Institute) circulation model, is employed to analyse trajectories of current-driven surface transport in the Gulf of Finland, the Baltic Sea, for the period of 1987-1991. Statistical analysis of trajectories is performed to calculate a map of probabilities for adverse impacts released in different sea areas to hit the coast. There is a clearly defined curve (equiprobability line) in the western part of the gulf from which the chances of the propagation of adverse impacts to either of the coasts are equal. The current-driven propagation of tracers from a wide area (of reduced risk) to the coast in the central and eastern parts of the gulf is unlikely within about three weeks. A safe fairway in terms of coastal protection goes over the equiprobability line and the area of reduced risk.

The concepts of the basic symmetric operator and complete symmetric operator of the equivalent-electron regular Young tableau T ［ λ ］ ig are presented, and the concepts of the root state and generative state generated by these symmetric operators acting on each Slater function  i are also given. Based on the establishment rules of the vertical permutation operator A ［ λ ］ ig of the orthogonal normalization Young tableau T ［ λ ］ ie , the symmetric operators in A ［ λ ］ ie and the equiprobability comparison method for solving T ［ λ ］ ie are presented, which can avoid the complicated algebra involving many operators. Finally, a new method for sloving the Young basis of the electron system with a large N is presented.

From an experimental study of two biases in probabilistic judgments — the representativeness bias in situations of inductive inference, and the equiprobability bias in "purely random" situations — it is shown that it is possible to characterize two types of cognitive aids which differ by their impact point: the object to which the probabilistic judgment pertains (for the representativeness bias), or the activated representation (for the equiprobability bias).

The evolution of certain pointer-based implementations of dictionaries, linear lists and priority queues is studied. Under the assumption of equiprobability of histories, i.e., of paths through the internal state space of the implementation, the n → ∞ asymptotics of the space and time costs of a sequence of n supported operations are computed.

For many applications one wishes to decide whether a certain set of numbers originates from an equiprobability distribution or whether they are unequally distributed. Distributions of relative frequencies may deviate significantly from the corresponding probability distributions due to finite sample effects. Hence, it is not trivial to discriminate between an equiprobability distribution and non-equally distributed probabilities when knowing only frequencies. Based on analytical results we provide a software tool which allows to decide whether data correspond to an equiprobability distribution. The tool is available at http://bioinf.charite.de/equifreq/ . Its application is demonstrated for the distribution of point mutations in coding genes.

The concepts of evenness, diversity and dominance have been traditionally regarded as basic parameters of community structure in interspecific competition, and during the past several decades there has been an impressive development of their measurements. It is quite depressing to realize, however, that there is still no consensus on how to measure and interrelate these parameters. In consequence, an increasing number of field ecologists are questioning the utility of evenness, dominance and diversity statistics to evaluate changes in the functional structure of biological communities with competing species. Recently, in this journal, Bulla (1994) has proposed a new index of species evenness and its associated diversity measurement to clarify the current confusion and, at the same time, reverse the loss of interest in the use of evenness and diversity statistics. Bulla properly justifies the development of a new evenness index because older ones, such as Pielou's (1969) J index, Hill's (1973) E index, Alatalo's (1981) F index and Molinari's (1989) G index, exhibit serious shortcomings. Nevertheless, it seems to me that Bulla's contribution only brings more confusion to the subject because it has both conceptual and practical drawbacks, overlooking the importance of the complementary relationship between evenness and dominance. Furthermore, Bulla's contribution is not entirely original since I had already proposed and utilized the same measurement of species diversity (Camargo 1992a, b, 1993a, b). To explain the derivation of his evenness index, Bulla begins with a graphical representation of the relative abundances of ten plant species in a real community and in a theoretical community (Fig. 1). Two different distributions are represented, the observed distribution and the expected distribution (equiprobability between species). Bulla correctly argues that the closer the observed distribution approaches the expected distribution, the 0.5

Five simple models of the implicit social process whereby a group combines individual preferences into a collective decision are investigated: plurality, majority, proportionality, equiprobability and highest expected value. For each model, an expression is developed which provides the probability that a group selects a given alternative, as a function of (i) group size, (ii) number of alternatives and (iii) the set of probabilities governing the likelihood with which a group member advocates each alternative. Predictions yielded by the probability expressions are used as the basis for a comparison of the models.

Expressions are developed for the probability that a group chooses a particular course of action in each of four complex social decision scheme models: majority with proportionality, majority with equiprobability, plurality with proportionality and plurality with equiprobability. A method of discriminating by experiment among a number of social decision scheme models is suggested involving, where feasible, experimenter manipulation of individual member preference probabilities.

The distribution of the chi-square goodness-of-fit statistic is studied in the equiprobable case. Tables of exact critical values are given for a = .1, .05, .01, .005; k = 2(1)4, N = 26(1)50; k = 5, N = 26(1)40; k = 6(1)10, N = 26(1)30, where a is the desired significance level, k is the number of cells and N is the sample size. Methods of fitting the true distribution are compared. If k> 3, it is found that a simple additive adjustment to the asymptotic chi-square fit leads to high accuracy even for N between 10 and 20. For k = 2, the Yates corrected chi-square statistic is very accurately fitted by the usual chi-square distribution.

Complementary color coding is used in conjunction with nontraditional legend arrangements to create two-variable maps. The logic of a complementary color-coding scheme is simpler and easier for the map user to comprehend than the spectrally encoded scheme used to produce most of the two-variable maps published to date. In addition to the complementary color-coding scheme, an unclassed legend based on a line of best fit and a legend arrangement based on the binormal distribution are used to represent more accurately the structure of the bivariate scatter diagram. These legends avoid many of the statistical distortions introduced by the rectangular legend currently being used to produce two-variable maps. Although the unclassed two-variable map presents a true graphic correlation, the large number of color tones contained in the map confounds the map user. The two-variable map based on the binormal distribution combines the rectangular legend and an equiprobability ellipse to represent the legend d...

Keynes introduces three different principles under the single title “the principle of indifference”. The first is Bernoulli's princple of non-sufficient reason.If there is no known reason for predicating of our subject one rather than another of several alternatives, then relatively to such knowledge the assertions of each of these alternatives have an equal probability. Thus equal probabilities must be assigned to each of several arguments, if there is an absence of positive ground for assigning unequal ones.

This paper presents a very simple approximation, having “graph” accuracy, for calculating bivariate normal probability contours. Such contours are useful in simultaneous inference and other applications. This method is then adapted to approximate roughly bivariate and multivariate normal union probabilities.

Boltzmann's hypothesis of the equiprobability of the various ways of combining microslates (giving rise to the different configurations of a system) is erroneous. This renders untenable the ergodic hypothesis according to which the configurations that prime the order in an isolated system are reached randomly and points to the need to postulate the existence of a hidden variable in order to explain the system's shift away from its state of maximum entropy.

The strength of a significance test is defined as a weighted average of the power especially if the weights are a prior density. The concept is a compromise between Neyman-Pearsonian and Bayesian methods and is applied to tests for multinomial equiprobability and for no association in contingency tables. The differences of the power functions of two criteria, X 2 and G, are illustrated for the trinomial, where G arose earlier from a partially Bayesian approach. Other methods for apprehending power functions of several variables are discussed. For 2 × 2 contingency tables with equal row or column totals the power and strength functions of several statistics are equal.

A study of the effect of excluded volume on the mean-square optical anisotropy 〈γ2〉 and the mean-square end-to-end distance 〈R2〉 has been carried out for polymer chains of up to 2000 bonds. The calculations have been made for models assuming (a) equiprobability of internal rotations and (b) interdependence of short-range intramolecular interactions. All the results indicate that 〈γ2〉 is practically insensitive to the excluded-volume effect. Concomitantly the behavior of 〈R2〉 has been analyzed.

The behaviour of a statistic proposed byMorgenstern [1979] for testing the hypothesis of equiprobability of the Lotto-numbers is studied under a suitable class of alternatives.

The geographical evolution of the economy requires reproduction of the existing structure as well as change. Both ordering within and variety outside the constraints are involved and are measured by entropy. Equiprobability, independence, compartmentalization and alphabet size are examined, sometimes in game theory terms. Evolution is a continual throwing up of possible innovations in spatial economic organization, those accepted increasing income. Entropy of inefficiency is thus maintained, freedom of choice in consumption replacing choice in activity. Innovations covarying in space-time with the existing structure are likely to be accepted. Communications theory can express these ideas.

This study presents an expository geometric derivation of the Pareto distribution with applications to conflict behavior. In contrast to typical stochastic models which assume equiprobability of access to scarce resources, the derivation here assumes sequential arrivals of colonists or other resource acquisitors. A log-exponential distribution is derived which turns out to be identical to the Pareto law of income distribution, the derivation here differing from others in its emphasis on sequential arrivals and resource scarcity. This model is then applied to the upper tail of land distribution in El Salvador and the distribution of colonial populations in 1914. In both cases, the log-exponential (Pareto) distribution can be accepted while the exponential itself as a plausible theoretical rival is rejected. Invariance properties of the Pareto distribution suggest reasons why political violence is likely if the model holds throughout the range of resource holdings, as was the case in 1914.

The theory of degenerate coding is presented in a way enabling further application to molecular biology. There are two kinds of redundancy of a degenerate code. The first is due to the excess in codon length and the second to the code degeneracy. If the code is asymmetrically degenerate, the second kind of redundancy can be profitable for control of error rate. This control can be performed just by selective synonymous codon usage. Utilisation of the genetic code is partially influenced by this theoretical possibility. In particular the degree of error protectivity is well correlated with deviation from equiprobability in synonymous codon usage. The biological significance of this fact is discussed.

Risk assessments done by the Ford Foundation-Mitre Corporation and the Union of Concerned Scientists agree on the hazard probabilities and consequences associated with commercial nuclear fission, but make opposed recommendations regarding using atomic energy to generate electricity. The UCS risk analysis decided against use of the technology; the Ford-Mitre study advised in favor of it (Union of Concerned Scientists 1977; Nuclear Energy Policy Study Group 1977; Cooke 1982, p. 334).The two studies agreed on the data but made different policy recommendations because they followed different methodological rules at the third, risk-evaluation, stage of assessment. (The first two stages are risk identification and risk estimation.) The Ford-Mitre research was based on the widely accepted Bayesian decision criterion that it is rational to choose the action with the best expected value or utility, where ‘expected value’ or ‘expected utility’ is defined as the weighted sum of all possible consequences of the action, and where the weights are given by the probability associated with each consequence.

An inference procedure is required to assign a density operator D to a system when the only available information is the set of expectation values a i of some observables A i , commuting or not. To do this, we consider as a single “supersystem” the Gibbsian ensemble consisting of N replicas of the system, on which thought experiments compatible with the data a i can be performed. In this supersystem, we identify the expectation values a i with mean values, and implement the principle that equal ignorance should be represented by equiprobability. New features are brought in through the non-commutativity of the observables A i , and through the operator nature of the quantum states. These generate difficulties, both conceptual and technical, which we discuss and resolve. The density operator D of the system considered is obtained from that of the supersystem by taking the partial trace over the N − 1 other systems. The rather involved calculations are accomplished by field theoretical techniques. They provide a generalized canonical distribution, with deviations which vanish as N → ∞, the same result as would be obtained by maximizing von Neumann's entropy with the constraints 〈 A i 〉 = a i . We succeed thus to justify the quantal maximum entropy principle and to construct simultaneously von Neumann's entropy, starting from the idea of equiprobability in the supersystem.

We argue that the evolution of order in living systems and certain nonliving physical systems obeys a common fundamental principle which we call the Darwinian dynamic. Such ordered systems deviate greatly from the thermodynamic equiprobability rule according to which, for example, all nucleotide sequences of comparable length should be found in roughly equal abundance. We formulate the Darwinian dynamic by first considering how macroscopic order is generated in a simple nonbiological system far from the thermodynamic equilibrium. We then extend our consideration to short, replicating RNA molecules, which we assume to be like the earliest forms of life, and show that the underlying order-generating process is basically similar. The equation we use as an example of the Darwinian dynamic for these simple replicators contains variables that express the basic conditions necessary for the process of natural selection as conceived by Darwin: variation of type, heritability, and competition for limited resources....

Two models are proposed to solve the sodium nitrate structure near the order-disorder phase transition (T c =276°C). The first model is built with the static disorder assumption between the two equilibrium positions of NO − 3 ions. In this case, disorder is characterized by an order parameter: η. For the second model, a one-dimensional orientational probability function is introduced to take disorder into account. In the latter formalism, a coupling between NO − 3 rotations around the three fold axis and translations along the same axis has been introduced. Comparison between the two models shows that the static model can be held to be valid while the square root of the rotational mean square amplitude is lower than 12°. Structure determination by the two models yields equivalent fits for temperatures below ∼250°C, fits with the second model become better for temperatures above 250°C (T c -26°C). Equiprobability curves display the NO − 3 ion jump mechanism between their two equilibrium positions: the rotation around the three-fold axis is made via a translation so that the distance between oxygen atoms and the nearest neighbour Na + ion stays approximately constant during the jump Proposition de deux modeles pour resoudre la structure pres de la transition ordre desordre correspondant a deux mecanismes de desordre differents. Proposition d'un modele de desordre statique entre les deux positions d'equilibre des ions NO − 3 et d'un second modele dans lequel le desordre est pris en compte par une fonction de probabilite orientationnelle unidimensionnelle