On affine motions and universal rigidity of tensegrity frameworks

Recently, Alfakih and Ye (2013) [4] proved that if an r  -dimensional bar framework (G,p)$(G,p)$ on n?r+2$n?r+2$ nodes in general position in R^r${\mathbb{R}}^r$ admits a positive semidefinite stress matrix with rank n−r−1$n-r-1$, then (G,p)$(G,p)$ is universally rigid. In this paper, we generalize this result in two directions. First, we extend this result to tensegrity frameworks. Second, we replace the general position assumption by the weaker assumption that in configuration p, each point and its neighbors in G   affinely span R^r${\mathbb{R}}^r$.     

A parametric approach to correspondence analysis

We compare correspondence analysis (CA) and the alternative approach using Hellinger distance (HD), for representing categorical data in a contingency table. As both methods may be appropriate, we introduce a parameter and define a generalized version of correspondence analysis (GCA) which contains CA and HD as particular cases. Comparison with alternative approaches are performed. We propose a coefficient which globally measures the similarity between CA and GCA, which can be decomposed into several components, one component for each principal dimension, indicating the contribution of the dimensions on the difference between both representations. Two criteria for choosing the best value of the parameter are proposed.     

A one-to-one correspondence between colorings and stable sets

Given a graph G, we construct an auxiliary graph with vertices such that the set of all stable sets of is in one-to-one correspondence with the set of all colorings of G. Then, we show that the Max-Coloring problem in G reduces to the Maximum Weighted Stable set problem in .     

A correspondence between Hopf ideals and sub-Hopf algebras

In this paper we show a bijective correspondence between normal Hopf ideals and sub-Hopf algebras of a commutative Hopf algebra over a field k. This gives a purely algebraic proof of the fundamental theorem [2, III, §3, n^o7] of the theory of affine k-groups.     

A correspondence between the AKNS hierarchy and the JM hierarchy

Reference [1] presented a gauge transformation between thex parts of the AKNS eigenvalue problem and those of the JM (Jaulent-Miodek) eigenvalue problem. In this paper we discuss the correspondence between thet parts of the AKNS eigenvalue problem and thet parts of the JM eigenvalue prohlem under the gauge transformation, and give a correspondence between the AKNS hierarchy and the JM hierarchy and also three types of Darboux transformation for the JM hierarchy.Project supported by the Science Fund of the Ministry of Education.     

Interpreting multiple correspondence analysis

The practical aspects of interpreting multiple correspondence analysis are reviewed. The geometric concepts associated with simple correspondence analysis are shown to be inadequate for multiple correspondence analysis. An alternative approach, called joint correspondence analysis, is shown to be a more natural generalization of the simple case. The practical interpretation of homogeneity analysis is also discussed and a compromise is proposed between classical multiple correspondence analysis and joint correspondence analysis, preserving the optimal scaling properties of homogeneity analysis. A simple example is used throughout the discussion to illustrate the issues involved.     

Multiple taxicab correspondence analysis

We compare the statistical analysis of multidimensional contingency tables by multiple correspondence analysis (MCA) and multiple taxicab correspondence analysis (MTCA). We will show in this paper: First, MTCA and MCA can produce different results. Second, taxicab correspondence analysis of a Burt table is equivalent to centroid correspondence analysis of the indicator matrix. Third, along the first principal axis, the projected response patterns in MTCA will be clustered and the number of cluster points is less than or equal to 1+ the number of variables. Fourth, visual maps produced by MTCA seem to be clearer and more readable in the presence of rarely occurring categories of the variables than the graphical displays produced by MCA. Two well known data sets are analyzed.     

A correspondence between a class of monoids and self-similar group actions I

We describe a correspondence between a class of left cancellative monoids and self-similar group actions in the sense of Nekrashevych et al. This correspondence originated in Perrot’s 1972 thesis, and developed the ideas to be found in Rees’ 1948 paper.     

Littlewood's correspondence between Schur polynomials and immanants

Littlewood's correspondence between Schur polynomials and immanants is discussed in the context of character theory. Some of the identities for immanants that result from the correspondence are used to prove inequalities involving immanants of positive semidefinite matrices.     

Bootstrap tests in correspondence analysis

A bootstrap test is developed for testing models specified by Goodman in 1985 and 1986 for the correspondence analysis of two-way contingency tables. It enables testing goodness-of-fit in relation with the usual matrix decomposition method. An approximate table of critical values for the proposed test statistic is presented. Bootstrap confidence interval construction is also included. The behaviour of the test statistic and the confidence intervals is studied using Monte Carlo simulation.     

The correspondence between Moritz Pasch and Felix Klein

The extant correspondence, consisting of ten letters from the period from 1882 to 1902, from Moritz Pasch to Felix Klein is presented together with an English translation and a short introduction. These letters provide insights into the views of Pasch and Klein regarding the role of intuition and axioms in mathematics, and also into the hiring practices of mathematics professors in the 1880s.     

S-PLUS code for ordinal correspondence analysis

Summary  Correspondence analysis is a popular graphical tool used to analyse contingency tables. In the past, it has commonly been performed by applying a singular value decomposition to a transformation of the data in the contingency table. A recent advance in its theory is to perform a bivariate moment decomposition instead. This approach is especially useful for the detection of linear and non-linear associations between ordinal variables; a feature not readily available using singular value decomposition. This paper outlines S-PLUS code that will perform correspondence analysis using bivariate moment decomposition. It also includes a simple plotting function that will enable the graphical interpretation of the different levels of association.     

S-PLUS code for simple and multiple correspondence analysis

An exploration of the application of S-PLUS code designed to perform classical correspondence analysis is made in this paper. This code allows for the “classical” analysis to be performed on categorical data consisting of two or more variables. For multi-way contingency tables, correspondence analysis can be performed by considering either the indicator matrix or the Burt matrix. The code also allows the user to incorporate into the analysis confidence circles (to identify categorical responses that deviate from the hypothesis of independence) and for an asymmetrical analysis to be performed. The function includes various warnings and stoppages to help the user properly analyse their data.     

Tests of ignoring and eliminating in nonsymmetric correspondence analysis

Nonsymmetric correspondence analysis (NSCA) aims to examine predictive relationships between rows and columns of a contingency table. The predictor categories of such tables are often accompanied by some auxiliary information. Constrained NSCA (CNSCA) incorporates such information as linear constraints on the predictor categories. However, imposing constraints also means that part of the predictive relationship is left unaccounted for by the constraints. A method of NSCA is proposed for analyzing the residual part along with the part accounted for by the constraints. The CATANOVA test may be invoked to test the significance of each part. The two tests parallel the distinction between tests of ignoring and eliminating, and help gain some insight into what is known as Simpson’s paradox in the analysis of contingency tables. Two examples are given to illustrate the distinction.     

On a correspondence between probabilistic and fuzzy choice functions

Probabilistic and fuzzy choice functions are used to describe decision situations in which some degree of uncertainty or imprecision is involved. We propose a way to equate these two formalisms by means of residual implication operations. Furthermore, a set of new rationality conditions for probabilistic choice functions is proposed and proved to be sufficient to ensure that the associated fuzzy choice function is rational.     

Stochastic analysis, rough path analysis and fractional Brownian motions

 In this paper we show, by using dyadic approximations, the existence of a geometric rough path associated with a fractional Brownian motion with Hurst parameter greater than 1/4. Using the integral representation of fractional Brownian motions, we furthermore obtain a Skohorod integral representation of the geometric rough path we constructed. By the results in [Ly1], a stochastic integration theory may be established for fractional Brownian motions, and strong solutions and a Wong-Zakai type limit theorem for stochastic differential equations driven by fractional Brownian motions can be deduced accordingly. The method can actually be applied to a larger class of Gaussian processes with covariance functions satisfying a simple decay condition. Received: 11 May 2000 / Revised version: 20 March 2001 / Published online: 11 December 2001