Multivariate rank statistics for testing randomness concerning some marginal distributions

In the present paper there is proposed a rank statistic for multivariate testing of randomness conceriing some marginal distributions. The asymptotic distribution of this statistic under hypothesis and “near” alternatives is treated.     

Māl, enunciations, and the prehistory of Arabic algebra

Medieval Arabic algebra books intended for practical training generally have in common a first “book” which is divided into two sections: one on the methods of solving simplified equations and manipulating expressions, followed by one consisting of worked-out problems. By paying close attention to the wording of the problems in the books of al-Khwārizmī, Abū Kāmil, and Ibn Badr, we reveal the different ways the word māl was used. In the enunciation of a problem it is a common noun meaning “quantity,” while in the solution it is the proper noun naming the square of “thing” (shay '). We then look into the differences between the wording of enunciations and equations, which clarify certain problems solved without “thing,” and help explain the development of algebra before the time of al-Khwārizmī.     

Passage to the limit in Proposition I, Book I of Newton's Principia

The purpose of this paper is to analyze the way in which Newton uses his polygon model and passes to the limit in Proposition I, Book I of his Principia. It will be evident from his method that the limit of the polygon is indeed the orbital arc of the body and that his approximation of the actual continuous force situation by a series of impulses passes correctly in the limit into the continuous centripetal force situation. The analysis of the polygon model is done in two ways: (1) using the modern concepts of force, linear momentum, linear impulse, and velocity, and (2) using Newton's concepts of motive force and quantity of motion. It should be clearly understood that the term “force” without the adjective “motive,” is used in the modern sense, which is that force is a vector which is the time rate of change of the linear momentum. Newton did not use the word “force” in this modern sense. The symbol F denotes modern force. For Newton “force” was “motive force,” which is measured by the change in the quantity of motion of a body. Newton's “quantity of motion” is proportional to the magnitude of the modern vector momentum. Motive force is a scalar and the symbol F_m is used for motive force.     

Multiway Dependence in Exponential Family Conditional Distributions

Conditionally specified statistical models are frequently constructed from one-parameter exponential family conditional distributions. One way to formulate such a model is to specify the dependence structure among random variables through the use of a Markov random field (MRF). A common assumption on the Gibbsian form of the MRF model is that dependence is expressed only through pairs of random variables, which we refer to as the “pairwise-only dependence” assumption. Based on this assumption, J. Besag (1974, J. Roy. Statist. Soc. Ser. B36, 192–225) formulated exponential family “auto-models” and showed the form that one-parameter exponential family conditional densities must take in such models. We extend these results by relaxing the pairwise-only dependence assumption, and we give a necessary form that one-parameter exponential family conditional densities must take under more general conditions of multiway dependence. Data on the spatial distribution of the European corn borer larvae are fitted using a model with Bernoulli conditional distributions and several dependence structures, including pairwise-only, three-way, and four-way dependencies.     

Exploiting structure in quantified formulas

We study the computational problem “find the value of the quantified formula obtained by quantifying the variables in a sum of terms.” The “sum” can be based on any commutative monoid, the “quantifiers” need only satisfy two simple conditions, and the variables can have any finite domain. This problem is a generalization of the problem “given a sum-of-products of terms, find the value of the sum” studied in [R.E. Stearns and H.B. Hunt III, SIAM J. Comput. 25 (1996) 448–476]. A data structure called a “structure tree” is defined which displays information about “subproblems” that can be solved independently during the process of evaluating the formula. Some formulas have “good” structure trees which enable certain generic algorithms to evaluate the formulas in significantly less time than by brute force evaluation. By “generic algorithm,” we mean an algorithm constructed from uninterpreted function symbols, quantifier symbols, and monoid operations. The algebraic nature of the model facilitates a formal treatment of “local reductions” based on the “local replacement” of terms. Such local reductions “preserve formula structure” in the sense that structure trees with nice properties transform into structure trees with similar properties. These local reductions can also be used to transform hierarchical specified problems with useful structure into hierarchically specified problems having similar structure.     

Analysis in distribution of two randomized algorithms for finding the maximum in a broadcast communication model

The limit laws of three cost measures are derived of two algorithms for finding the maximum in a single-channel broadcast communication model. Both algorithms use coin flips and comparisons. Besides the ubiquitous normal limit law, the Dickman distribution also appears in a natural way. The method of proof proceeds along the line via the method of moments and the “asymptotic transfers,” which roughly bridges the asymptotics of the “conquering cost of the subproblems” and that of the total cost. Such a general approach has proved very fruitful for a number of problems in the analysis of recursive algorithms.     

A comparison of procedures for estimating the parent probability distribution from a given set of fractiles

On one hand, eliciting subjective probabilities (fractiles) is a well-established procedure. On the other hand, knowledge of a subjective variable's central moments or distribution function is often assumed. However, the problem of converting elicited fractiles into the required moments or distribution function has been largely ignored. We show that this conversion problem is far from trivial, and that the most commonly used conversion procedures often produce huge errors. Alternative procedures are proposed; the “Tocher's curve” and “linear function of fractiles” methods are shown to be much more accurate than the commonly used procedures.     

CDS simulation and pattern formation of phase separation

Several properties of the generation and evolution of phase separating patterns for binary material studied by CDS model are proposed. The main conclusions are (1) for alloys spinodal decomposition, the conceptions of “macro-pattern” and “micropattern” are posed by “black-and- white graph” and “gray-scale graph” respectively. We find that though the four forms of map f that represent the self-evolution of order parameter in a cell (lattice) are similar to each other in “macro-pattern”, there are evident differences in their micro-pattern, e.g., some different fine netted structures in the black domain and the white domain are found by the micro-pattern, so that distinct mechanical and physical behaviors shall be obtained. (2) If the two constitutions of block copolymers are not symmetric (i.e. r ≠ 0.5), a pattern called “grain-strip cross pattern” is discovered, in the 0.43 <r <0.45.     

Family of multivariate generalized t distributions

In this paper, we introduce a new family of multivariate distributions as the scale mixture of the multivariate power exponential distribution introduced by Gómez et al. (Comm. Statist. Theory Methods 27(3) (1998) 589) and the inverse generalized gamma distribution. Since the resulting family includes the multivariate t distribution and the multivariate generalization of the univariate GT distribution introduced by McDonald and Newey (Econometric Theory 18 (11) (1988) 4039) we call this family as the “multivariate generalized t-distributions family”, or MGT for short. We show that this family of distributions belongs to the elliptically contoured distributions family, and investigate the properties. We give the stochastic representation of a random variable distributed as a multivariate generalized t distribution. We give the marginal distribution, the conditional distribution and the distribution of the quadratic forms. We also investigate the other properties, such as, asymmetry, kurtosis and the characteristic function.     

A note on the convergence of trivariate extreme order statistics and extension

Necessary and sufficient conditions, under which there exists (at least) a sequence of vectors of real numbers for which the distribution function (d.f.) of any vector of extreme order statistics converges to a nondegenerate limit, are derived. The interesting thing is that these conditions solely depend on the univariate marginals. Moreover, the limit splits into the product of the limit univariate marginals if all the bivariate marginals of the trivariate d.f., from which the sample is drawn, is of negative quadrant dependent random variables (r.v.'s). Finally, all these results are stated for the multivariate extremes with arbitrary dimensions.     

The Enumeration of Permutations with a Prescribed Number of “Forbidden” Patterns

We initiate a general approach for the fast enumeration of permutations with a prescribed number of occurrences of “forbidden” patterns that seems to indicate that the enumerating sequence is always P-recursive. We illustrate the method completely in terms of the patterns “abc,” “cab,” and “abcd.”     

Implication in fuzzy logic

Intermediate truth values and the order relation “as true as” are interpreted. The material implication A → B quantifies the degree by which “B is at least as true as A.” Axioms for the → operator lead to a representation of → by the pseudo-Lukasiewicz model. A canonical scale for the truth value of a fuzzy proposition is selected such that the → operator is the Lukasiewicz operator and the negation is the classical 1−. operator. The mathematical structure of some conjunction and disjunction operators related to → are derived.     

The difference in a problem of rations from the Rhind mathematical papyrus

In an ancient Egyptian problem of bread distribution from the Rhind mathematical papyrus (dated between 1794 and 1550 B.C.), a procedure of “false position” is used in the calculation of a series of five rations. The algorithm is only partially illustrated in the problem text, and last century's prevailing interpretations suggested a determination of the series by trial and error. The missing part of the computational procedure is reconstructed in this article as an application of the algorithm, exemplified in the preceding section of the papyrus, to calculate an unknown quantity by means of the method of “false position.”     

The rate of convergence for the cyclic projections algorithm III: Regularity of convex sets

The cyclic projections algorithm is an important method for determining a point in the intersection of a finite number of closed convex sets in a Hilbert space. That is, for determining a solution to the “convex feasibility” problem. This is the third paper in a series on a study of the rate of convergence for the cyclic projections algorithm. In the first of these papers, we showed that the rate could be described in terms of the “angles” between the convex sets involved. In the second, we showed that these angles often had a more tractable formulation in terms of the “norm” of the product of the (nonlinear) metric projections onto related convex sets.In this paper, we show that the rate of convergence of the cyclic projections algorithm is also intimately related to the “linear regularity property” of Bauschke and Borwein, the “normal property” of Jameson (as well as Bakan, Deutsch, and Li’s generalization of Jameson’s normal property), the “strong conical hull intersection property” of Deutsch, Li, and Ward, and the rate of convergence of iterated parallel projections. Such properties have already been shown to be important in various other contexts as well.     

The Cost Distribution of Queue-Mergesort, Optimal Mergesorts, and Power-of-2 Rules

Queue-mergesort is introduced by Golin and Sedgewick as an optimal variant of mergesorts in the worst case. In this paper, we present a complete analysis of the cost distribution of queue-mergesort, including the best, average, and variance cases. The asymptotic normality of its cost is also established under the uniform permutation model. We address the corresponding optimality problems and we show that if we fix the merging scheme then the optimal mergesort as far as the average number of comparisons is concerned is to divide as evenly as possible at each recursive stage (top-down mergesort). On the other hand, the variance of queue-mergesort reaches asymptotically the minimum value. We also characterize a class of mergesorts with the latter property. A comparative discussion is given on the probabilistic behaviors of top-down mergesort, bottom-up mergesort, and queue-mergesort. We derive an “invariance principle” for asymptotic linearity of divide-and-conquer recurrences based on general “power-of-2” rules of which the underlying dividing rule of queue-mergesort is a special case. These analyses reveal an interesting algorithmic feature for general power-of-2 rules.     

On the Average Distribution of Inversive Pseudorandom Numbers

The inversive congruential method is an attractive alternative to the classical linear congruential method for pseudorandom number generation. The authors have recently introduced a new method for obtaining nontrivial upper bounds on the multidimensional discrepancy of inversive congruential pseudorandom numbers in parts of the period. This method has also been used to study the multidimensional distribution of several other similar families of pseudorandom numbers. Here we apply this method to show that, “on average” over all initial values, much stronger results than those known for “individual” sequences can be obtained.     

The perception of randomness

Psychologists have studied people's intuitive notions of randomness by two kinds of tasks: judgment tasks (e.g., “is this series like a coin?” or “which of these series is most like a coin?”), and production tasks (e.g., “produce a series like a coin”). People's notion of randomness is biased in that they see clumps or streaks in truly random series and expect more alternation, or shorter runs, than are there. Similarly, they produce series with higher than expected alternation rates. Production tasks are subject to other biases as well, resulting from various functional limitations. The subjectively ideal random sequence obeys “local representativeness”; namely, in short segments of it, it represents both the relative frequencies (e.g., for a coin, 50%–50%) and the irregularity (avoidance of runs and other patterns). The extent to which this bias is a handicap in the real world is addressed.     

Bicolored graph partitioning, or: gerrymandering at its worst

This study is motivated by an electoral application where we look into the following question: how much biased can the assignment of parliament seats be in a majority system under the effect of vicious gerrymandering when the two competing parties have the same electoral strength? To give a first theoretical answer to this question, we introduce a stylized combinatorial model, where the territory is represented by a rectangular grid graph, the vote outcome by a “balanced” red/blue node bicoloring and a district map by a connected partition of the grid whose components all have the same size. We constructively prove the existence in cycles and grid graphs of a balanced bicoloring and of two antagonist “partisan” district maps such that the discrepancy between their number of “red” (or “blue”) districts for that bicoloring is extremely large, in fact as large as allowed by color balance.     

Routing Problems on the Mesh of Buses

The mesh of buses (MBUSs) is a parallel computation model which consists ofn×nprocessors,nrow buses, andncolumn buses, but no local connections between neighboring processors. Annlower bound for the permutation routing on this model is shown. The proof does not depend on common predetermined assumptions such as “if a packet has to move horizontally then it has to ride on a horizontal bus at least once.” As for upper bounds, a 1.5nalgorithm is shown.     

Regression models for multivariate ordered responses via the Plackett distribution

We investigate the properties of a class of discrete multivariate distributions whose univariate marginals have ordered categories, all the bivariate marginals, like in the Plackett distribution, have log-odds ratios which do not depend on cut points and all higher-order interactions are constrained to 0. We show that this class of distributions may be interpreted as a discretized version of a multivariate continuous distribution having univariate logistic marginals. Convenient features of this class relative to the class of ordered probit models (the discretized version of the multivariate normal) are highlighted. Relevant properties of this distribution like quadratic log-linear expansion, invariance to collapsing of adjacent categories, properties related to positive dependence, marginalization and conditioning are discussed briefly. When continuous explanatory variables are available, regression models may be fitted to relate the univariate logits (as in a proportional odds model) and the log-odds ratios to covariates.