On medieval Islamic multiplication tables

Multiplication tables for sexagesimal numbers were used by the astronomers of the Islamic Middle Ages in their arithmetical calculations and in the preparation of more sophisticated tables for finding planetary positions and for reckoning time by the sun and stars. Most known examples of Islamic multiplication tables contain about 3,600 entries, although one has been found containing 216,000 entries.     

On Markov chain Monte Carlo Algorithms for Computing Conditional Expectations Based on Sufficient Statistics

Much work has focused on developing exact tests for the analysis of discrete data using log linear or logistic regression models. A parametric model is tested for a dataset by conditioning on the value of a sufficient statistic and determining the probability of obtaining another dataset as extreme or more extreme relative to the general model, where extremeness is determined by the value of a test statistic such as the chi-square or the log-likelihood ratio. Exact determination of these probabilities can be infeasible for high dimensional problems, and asymptotic approximations to them are often inaccurate when there are small data entries and/or there are many nuisance parameters. In these cases Monte Carlo methods can be used to estimate exact probabilities by randomly generating datasets (tables) that match the sufficient statistic of the original table. However, naive Monte Carlo methods produce tables that are usually far from matching the sufficient statistic. The Markov chain Monte Carlo method used in this work (the regression/attraction approach) uses attraction to concentrate the distribution around the set of tables that match the sufficient statistic, and uses regression to take advantage of information in tables that “almost” match. It is also more general than others in that it does not require the sufficient statistic to be linear, and it can be adapted to problems involving continuous variables. The method is applied to several high dimensional settings including four-way tables with a model of no four-way interaction, and a table of continuous data based on beta distributions. It is powerful enough to deal with the difficult problem of four-way tables and flexible enough to handle continuous data with a nonlinear sufficient statistic.     

医学随访研究设计表及其使用方法

本文由过去提出的医学随访研究基础方法和应用模型设计出３２０个常用临床试验方案，制成设计供临床应用．与以往设计相比，其优点是１．便于限定失访水平，２．没有来自信息性终检的偏性，３．便于限定终末终检率在其阈值之下，避免设计错误，保证结果的可靠性，４．设计灵活，无Ｍｏｒｇａｎ误差．利用这些表可简化设计过程．附工作实例描述其使用方法．     

Dynamic Markov Bases

This article presents a computational approach for generating Markov bases for multiway contingency tables whose cell counts might be constrained by fixed marginals and by lower and upper bounds. Our framework includes tables with structural zeros as a particular case. Instead of computing the entire Markov bases in an initial step, our framework finds sets of local moves that connect each table in the reference set with a set of neighbor tables. We construct a Markov chain on the reference set of tables that requires only a set of local moves at each iteration. The union of these sets of local moves forms a dynamic Markov basis. We illustrate the practicality of our algorithms in the estimation of exact p-values for a three-way table with structural zeros and a sparse eight-way table. This article has online supplementary materials.     

Sphérique et sphéropée en Grèce ancienne

In ancient Greece, astronomy and geography were closely connected. Geography was based on the teachings of astronomy, but it determined them as well: for example the needs of scientific geography forced astronomers to use geocentric hypotheses. To study the relationship between a motionless earth situated in the central part of the cosmos and a sky moving in a circle round the axis of the world, two methods were available, either the geometrical method, used more particularly in the selection called Small Astronomy (Autolycos, Euclid, Theodose of Bithynia) or the experimental approach, dealing with reduced representations of the world, more or less precise (celestial globes or armillary spheres). The first method was supported by Spheric, or Geometry of the sphere; the second resorted to Spheropoeia, a branch of Mechanics. There was mutual interaction between these two methods.     

Visualizing parameters from loglinear models

Summary  This paper presents a graphical display for the parameters resulting from loglinear models. Loglinear models provide a method for analyzing associations between two or several categorical variables and have become widely accepted as a tool for researchers during the last two decades. An important part of the output of any computer program focused on loglinear models is that devoted to estimation of parameters in the model. Traditionally, this output has been presented using tables that indicate the values of the coefficients, the associated standard errors and other related information. Evaluation of these tables can be rather tedious because of the number of values shown as well as their rather complicated structure, mainly when the analyst needs to consider several models before reaching a model with a good fit. Therefore, a graphical display summarizing tables of parameters could be of great help in this situation. In this paper we put forward an interactive dynamic graphical display that could be used in such fashion.     

Forecasting and simulating mortality tables

In this paper we suggest solutions to the actuaries, facing the problem of estimating future mortality tables, especially in cases where there is a lack of relevant data and where the tendencies are not easy to estimate directly. We propose the utilization of external sources of information in the form of other, published mortality tables and use formal statistical tests to decide among these possible candidates. The procedure can also be applied for checking e.g. the goodness of mortality selection factors. We suggest the use of parametric families in modelling; for example the simple 2-parameter Azbel model. We conclude the paper by a simulation study which allows for the quantification of the possible risks related to unforeseen changes in the mortality tables in the future. To calibrate the variances of these models, initial estimates are needed, which we get by the Lee–Carter method.     

A Markov chain Monte Carlo Algorithm for Approximating Exact Conditional Probabilities

Conditional inference eliminates nuisance parameters by conditioning on their sufficient statistics. For contingency tables conditional inference entails enumerating all tables with the same sufficient statistics as the observed data. For moderately sized tables and/or complex models, the computing time to enumerate these tables is often prohibitive. Monte Carlo approximations offer a viable alternative provided it is possible to obtain samples from the correct conditional distribution. This article presents an MCMC extension of the importance sampling algorithm, using a rounded normal candidate to update randomly chosen cells while leaving the remainder of the table fixed. This local approximation can greatly increase the efficiency of the rounded normal candidate. By choosing the number of cells to be updated at random, a balance is struck between dependency in the Markov chain and accuracy of the candidate.     

A finiteness theorem for Markov bases of hierarchical models

We show that the complexity of the Markov bases of multidimensional tables stabilizes eventually if a single table dimension is allowed to vary. In particular, if this table dimension is greater than a computable bound, the Markov bases consist of elements from Markov bases of smaller tables. We give an explicit formula for this bound in terms of Graver bases. We also compute these Markov and Graver complexities for all K×2×2×2 tables.     

Markov bases and subbases for bounded contingency tables

In this paper we study the computation of Markov bases for contingency tables whose cell entries have an upper bound. It is known that in this case one has to compute universal Gröbner bases, and this is often infeasible also in small- and medium-sized problems. Here we focus on bounded two-way contingency tables under independence model. We show that when these bounds on cells are positive the set of basic moves of all 2 × 2 minors connects all tables with given margins. We also give some results about bounded incomplete table and we conclude with an open problem on the necessary and sufficient condition on the set of structural zeros so that the set of basic moves of all 2 × 2 minors connects all incomplete contingency tables with given positive margins.     

Tail Behavior, Modes and other Characteristics of Stable Distributions

Stable distributions have heavy tails that are asymptotically Paretian. Accurate computations of stable densities and distribution functions are used to analyze when the Paretian tail actually appears. Implications for estimation procedures are discussed. In addition to numerically locating the mode of a general stable distribution, analytic and numeric results are given for the mode. Extensive tables of stable percentiles have been computed; aspects of these tables and the appropriateness of infinite variance stable models are discussed.     

Sampling for Conditional Inference on Contingency Tables

We propose new sequential importance sampling methods for sampling contingency tables with given margins. The proposal for each method is based on asymptotic approximations to the number of tables with fixed margins. These methods generate tables that are very close to the uniform distribution. The tables, along with their importance weights, can be used to approximate the null distribution of test statistics and calculate the total number of tables. We apply the methods to a number of examples and demonstrate an improvement over other methods in a variety of real problems. Supplementary materials are available online.     

Solar and lunar velocities in the Alfonsine Tables

Although not previously noticed, the solar and lunar velocity tables in the 1483 edition of the Alfonsine Tables differ from the corresponding tables in the 1518 edition (Venice) which are identical to those that appear in the Toledan Tables and the z$\text{i}$j of al-Batt$\text{a}$n$\text{i}$. The velocity tables in the unpublished Hebrew version of the Alfonsine Tables, composed in 1460 (Avignon), agree with the 1483 edition, and the Hebrew translator tells us that these tables (together with some others) were added to the Alfonsine Tables by Christian scholars after the time of Alfonso. A discussion of the underlying parameters is also presented.     

Group theory for tetraammineplatinum(II) withC 2v andC 4v point group in the non-rigid system

The non-rigid molecule group theory (NRG) in which the dynamical symmetry operations are defined as physical operations is a new field of chemistry. Smeyers in a series of papers applied this notion to determine the character table of restricted NRG of some molecules. In this work, a simple method is described, by means of which it is possible to calculate character tables for the symmetry group of molecules consisting of a number of NH3 groups attached to a rigid framework. We study the full non-rigid group (f-NRG) of tetraammineplatinum (II) with two separate symmetry groups C2v and C4v. We prove that they are groups of order 216 and 5184 with 27 and 45 conjugacy classes, respectively. Also, we will compute the character tables of these groups.     

Decision tables in Algol 60

This paper defines an extension to Algol 60, which allows the programmer to write decision tables in his Algol program. A pre-processor converts the decision tables to Algol, and its output is used as input to the ordinary Algol compiler. The generated Algol program uses a straight-forward and efficient algorithm for choosing the appropriate decision rule.     

Tensor product of C-injective modules

The purpose of this paper is to calculate all the character tables of Hecke algebras associated with finite Chevalley groups of exceptional type and their maximal parabolic subgroups when they are commutative. In the case when the groups are of classical type, the character values of Hecke algebras are expressed by using the q-Krawtchouk polynomials and the q-Hahn polynomials (See [10] and [15]). On the other hand, the character tables of commutative Hecke algebras associated with exceptional Weyl groups and their maximal parabolic subgroups are given in [12]. In §1, we discuss the structure of Hecke algebras and in §2, we calculate all the character tables of these commutative Hecke algebras associated with finite Chevalley groups of exceptional type. Although some of them are well known, we include them for completeness     

Morse inequalities for the Koszul complex of multi-persistence

In this paper, we define the homological Morse numbers of a filtered cell complex in terms of relative homology of nested filtration pieces, and derive inequalities relating these numbers to the Betti tables of the multi-parameter persistence modules of the considered filtration. Using the Mayer-Vietoris spectral sequence we first obtain strong and weak Morse inequalities involving the above quantities, and then we improve the weak inequalities achieving a sharp lower bound for homological Morse numbers. Furthermore, we prove a sharp upper bound for homological Morse numbers, expressed again in terms of the Betti tables.     

Exact and heuristic methods for cell suppression in multi-dimensional linked tables

The increasing demand for information, coupled with the increasing capability of computer systems, has compelled information providers to reassess their procedures for preventing disclosure of confidential information. This paper considers the problem of protecting an unpublished, sensitive table by suppressing cells in related, published tables. A conventional integer programming technique for two-dimensional tables is extended to find an optimal suppression set for the public tables. This can be used to protect the confidentiality of sensitive data in three- and higher-dimensional tables. More importantly, heuristics that are intimately related to the structure of the problem are also presented to mitigate the computational difficulty of the integer program. An example is drawn from healthcare management. Data tables are randomly generated to assess the computational time/space restrictions of the IP model, and to evaluate the heuristics.     

Analyzing multiset data by the Power STATIS-ACT method

The STATIS-ACT method is a generalization of principal component analysis used to study simultaneously several data tables measured on the same observation units or variables. The goal of this method is to analyze the relationship between these data tables and to combine them into a compromise matrix corresponding to an optimal agreement between the data. In this paper, we propose a new approach to this method, referred to as the Power STATIS-ACT method, where the compromise matrix is derived from a general s-power based criterion ${(s\geqslant 1)}$ and investigate some of its theoretical and practical properties. Special attention is devoted to the 1-power case which makes the introduction of low rank versions of the compromise possible. We also examine the effect of varying the power parameter s on the compromise solutions. All results are illustrated with a number of real data tables.