Polynomial relations among principal minors of a -matrix

The image of the principal minor map for n×n-matrices is shown to be closed. In the 19th century, Nanson and Muir studied the implicitization problem of finding all relations among principal minors when n=4. We complete their partial results by constructing explicit polynomials of degree 12 that scheme-theoretically define this affine variety and also its projective closure in . The latter is the main component in the singular locus of the 2×2×2×2-hyperdeterminant.     

Commutation relations and Markov chains

It is shown that the combinatorics of commutation relations is well suited for analyzing the convergence rate of certain Markov chains. Examples studied include random walk on irreducible representations, a local random walk on partitions whose stationary distribution is the Ewens distribution, and some birth–death chains.     

Commutation relations and hypercyclic operators

In this paper we establish hypercyclicity of continuous linear operators on \({H(\mathbb{C})}\) that satisfy certain commutation relations.     

Commutation relations for Schramm‐Loewner evolutions

Schramm‐Loewner evolutions (SLEs) describe a one‐parameter family of growth processes in the plane that have particular conformal invariance properties. For instance, SLE can define simple random curves in a simply connected domain. In this paper we are interested in questions pertaining to the definition of several SLEs in a domain (i.e., several random curves). In particular, we derive infinitesimal commutation conditions, discuss some elementary solutions, study integrability conditions following from commutation, and show how to lift these infinitesimal relations to global relations in simple cases. The situation in multiply connected domains is also discussed. © 2007 Wiley Periodicals, Inc.     

Commutation relations,homomorphisms, and differential equations

We present an algebraic approach to the theory of ordinary differential equations and indicate a method for constructing first integrals of such equations.     

Cut-off lengths and uncertainty relations in quantum general relativity

The object of this paper is to discuss the status of the inaccuracy relations arising in quantum general relativity and to give a short review of some of their physical implications.     

Double dispersion relations in quantum Statistical Mechanics—II

Double dispersion relations are given for the functions $$\gamma (12;3) = \frac{1}{i}\langle T\left( {\psi (1)\psi ^\dag (2)B (3)} \right)\rangle$$ common in many electron problems. Here ψ and ψ^T refer to electron destruction and creation operators, and B (3) is a boson or boson-like operator such as particle density, ion displacement, local spin, particle current density, or electron spin density. Consequences of Hermiticity properties and time reversal invariance as well as sum rules are established for the spectral density functions entering the double dispersion relations. A subtracted double dispersion relation is suggested for the corresponding vertex function. Details of the derivation and examples are given which were not included in an earlier brief report of this work.     

How to distinguish quantum covariances using uncertainty relations

To each operator monotone function it is possible to associate a quantum version of the classical covariance. We show that: i) only for regular quantum covariances one can prove non-trivial uncertainty relations; ii) the usual quantum covariance gives the best inequalities in this setting.     

Geometric relations among Voronoi diagrams

Two general classes of Voronoi diagrams are introduced and, along with their modifications to higher order, are shown to be geometrically related. This geometric background, on the one hand, serves to analyse the size and combinatorial structure and, on the other, implies general and efficient methods of construction for various important types of Voronoi diagrams considered in the literature.Research supported by the Austrian Fond zur Foerderung der wissenschaftlichen Forschung.     

Topological minors in bipartite graphs

For a bipartite graph G on m and n vertices, respectively, in its vertices classes, and for integers s and t such that 2 ≤ s ≤ t, 0 ≤ m − s ≤ n − t, and m + n ≤ 2s + t − 1, we prove that if G has at least mn − (2(m − s) + n − t) edges then it contains a subdivision of the complete bipartite K _(s,t) with s vertices in the m-class and t vertices in the n-class. Furthermore, we characterize the corresponding extremal bipartite graphs with mn − (2(m − s) + n − t + 1) edges for this topological Turan type problem.     

Compact topological minors in graphs

Let ε be a real number such that and t a positive integer. Let n be a sufficiently large positive integer as a function of t and ε. We show that every n‐vertex graph with at least n^1+ε edges contains a subdivision of K_t in which each edge of K_t is subdivided less than 10/ε times. This refines the main result in [A. Kostochka and Pyber, Combinatorica 8 (1988), 83–86] and resolves an open question raised there. We also pose some questions. © 2010 Wiley Periodicals, Inc. J Graph Theory 67:139‐152, 2011     

Complete minors in pseudorandom graphs

Random graphs have the smallest complete minors among all graphs of given order and density. We show that, unlike random graphs, many well‐known examples of pseudorandom graphs (such as Paley graphs) have very large minors. © 2000 John Wiley & Sons, Inc. Random Struct. Alg., 17: 26–28, 2000