Subpermanents of doubly stochastic matrices †

It is shown that if all subpermaneats of order k of an n × n doubly stochastic matrix are equal for some k ≤ n ? 2, then all the entries of the matrix must be equal to 1/n.     

Set inertia-preserving matrices∗ †

The inertia-preservers of several sets of matrices are identified. The sets include: all real matrices, all complex matrices, triangular matrices, real symmetric matrices and Hermitian matrices.     

The irreducible highest weight representations of c∞

An explicit construction is given for the level one irreducible highest weight representations of completed infinite rank affine Lie algebra c∞.     

On convertible nonnegative matrices * †

An n× nmatrix Ais called convertible if there is an n× n(1, -1)-matrix Hsuch that per A= det(H°A) where H ° Adenotes the Hadamard product of Hand A. A convertible (0,l)-matrix is called extremal if replacing any zero entry with a 1 breaks the convertibility. In this paper some properties of

nonnegative convertible matrices are investigated and some classes of extremal convertible (0,1)-matrices are obtained.     

Minimum permanents of doubly stochastic matrices with prescribed zero entries †

A study of properties of matrices with minimum permanent in a face of the polyhedron of doubly stochastic n × n matrices. The minima are determined for certain faces.     

Minimum permanents of doubly stochastic matrices with at least one zero entry †

It is shown that the minimum value of the permanent on the n× ndoubly stochastic matrices which contain at least one zero entry is achieved at those matrices nearest to J_n in Euclidean norm, where J_n is the n× nmatrix each of whose entries is n^-1 . In case n ≠ 3 the minimum permanent is achieved only at those matrices nearest J_n ; for n= 3 it is achieved at other matrices containing one or more zero entries as well.     

The structure of a class of Z-local rings

A local ring R is called Z-local if J(R) = Z(R) and J(R)2 = 0. In this paper the structure of a class of Z-local rings is determined.     

The use of simulation and case study†

Although statistics is a rational subject, it does not follow that students can or do learn statistics in a rational way. Students learn statistics in the pattern‐forming mode more than in the rational mode. The paper describes and compares the two modes, and identifies three conditions that prevent or discourage learning in the rational mode. The diagnosis of pattern‐forming de‐emphasizes mathematical derivations and proofs as a means of allowing students to understand statistical concepts. The primary recommendation is that the teacher should deliberately manipulate the pattern‐forming process. The underlying idea is for the teacher to view the process for what it means to the student.     

The minimal eigenvalue of the Lyapunov transform †

An (n m) hypergraph is a coupleH=(N E), where the vertex set N is {1,…n} and the edge set E is an m-element multiset of nonempty subsets of N. In this paper, we count nonisomorphic hypergraphs where isomorphism of hypergraphs is the natural extension of that of graphs. A main result is an explicit formula for the cycle index of the permutation representation of any permutation group P with object set N acting on the k-element subsets of N. By making a simple substitution in these cycle indices for P the symmetric group S_N and k=1,…,n, we obtain generating functions which enumerate various types of hypergraphs. Using the technique developed, we extend Snapper's results on characteristic polynomials of permutation representations and group characters from the case where the group has odd order to the general case.     

The structure of L*-inverse semigroups

The concepts of L*-inverse semigroups and left wreath products of semigroups are introduced. It is shown that the L*-inverse semigroup can be described as the left wreath product of a type A semigroupΓand a left regular band B together with a mapping which maps the semigroupΓinto the endomorphism semigroup End(B). This result generalizes the structure theorem of Yamada for the left inverse semigroups in the class of regular semigroups. We shall also provide a constructed example for the L*-inverse semigroups by using the left wreath products.     

The cohomology group of weak entwining structure

In this paper, we reveal that a weak entwining structure admits a rich cohomology theory. As an application we compute the cohomology of a weak entwining structure associated to a weak coalgebra-Galois extension.     

The structural implications of measurement error in sociometry†

Measurement error, an inherent quality of any empirical data collection technique, is discussed in the context of sociometric data. These data have long been assumed to possess face validity and to be the data of choice in any study of the sentiment structure of small scale social systems. However, it is argued that while methods of sociometric analysis have become increasingly more sophisticated they have failed to yield unequivocal results because they do not distinguish structural complexity from measurement error. Through a discussion of increasingly more complex examples the distortion laden character of most sociometric data is illustrated. This distortion is introduced by the formalities of the sociometric test and it will not be removed by developing increasingly more sophisticated structural models or throwing out some of the data. Instead, when issues concerning the nature of specific relational networks are raised data of much higher quality than those which are commonly available are required. A technique for generating high quality sociometric data is briefly discussed. On the other hand, it is suggested that the extant body of sociometric data ought to be adequate when sizeable aggregations are examined for evidence of statistical tendencies in structure.     

A distance model for sociometric structure†

The conceptual and mathematical framework of a general model for distance within sociometric structure is described. The model characterizes “balance” in terms of the triangle inequality, in which the distance between two people (A and C) should be less than or equal to the sum of the distances to a third person (B), i.e., d (A,C) ≤ d (A,B) + d (B,C). The notion of addition of distances is developed. Different ways of adding distances result in different models of sociometric structure. Two families of models for symmetric graphs are discussed. The general model is extended to asymmetric graphs by generalizing the notion of transitivity. The model's potential for resolving a problem of the transitivity’ model is then discussed. The general model provides a means of examining the relationship between stratification and clustering in the structure of groups.