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In the 1960s, Atkinson introduced an abstract algebraic setting for multiparameter eigenvalue problems. He showed that a nonsingular multiparameter eigenvalue problem is equivalent to the associated system of generalized eigenvalue problems, which is a key relation for many theoretical results and numerical methods for nonsingular multiparameter eigenvalue problems. In 2009, Muhič and Plestenjak extended the above relation to a class of singular two-parameter eigenvalue problems with coprime characteristic polynomials and such that all finite eigenvalues are algebraically simple. They introduced a way to solve a singular two-parameter eigenvalue problem by computing the common regular eigenvalues of the associated system of two singular generalized eigenvalue problems. Using new tools, in particular the stratification theory, we extend this connection to singular two-parameter eigenvalue problems with possibly multiple eigenvalues and such that characteristic polynomials can have a nontrivial common factor. [ABSTRACT FROM AUTHOR]

• We introduce extreme copulas in each class of shock induced copulas (s.a. Marshall-Olkin). • Extreme copulas are dense in each of these classes, while shuffles of M are everywhere dense. • Statistical interpretation of an extreme copula is that the shocks inducing it have disjoint supports. In a recent paper, extreme points (in the Krein-Milman sense) of the class of semilinear copulas were introduced, motivated by the lack of known extreme copulas such as shuffles of M. We propose an extension of this concept to the class of all bivariate shock induced copulas, the most well-known part of them being the Marshall-Olkin copulas. This class properly contains semilinear copulas. Our technique coincides with the existing notion on them and has some advantages. First, it is defined on a wider family of copulas, which is helpful in finding more examples of extreme copulas. Second, we show that they are dense in each class they belong to (including the class of semilinear copulas) in a stronger sense than in the Krein-Milman approach; actually, they are dense in a similar way as shuffles of M are dense in the set of all copulas. Third, this definition enables practitioners to give stochastic interpretation of extremality. Roughly speaking, a shock induced copula is extreme whenever the inducing shocks have pairwise disjoint supports. [ABSTRACT FROM AUTHOR]

We review some of the current research in multiparameter spectral theory. We prove a version of the Cayley–Hamilton theorem for multiparameter systems and list a few inverse problems for such systems. Some consequences of results on determinantal representations proved by Dixon, Dickson and Vinnikov for the inverse problems are discussed. [Copyright &y& Elsevier]