Generalisation : graphs and colourings

The interaction between practice and theory in mathematics is a central theme. Many mathematical structures and theories result from the formalisation of a real problem. Graph Theory is rich with such examples. The graph structure itself was formalised by Leonard Euler in the quest to solve the problem of the Bridges of Königsberg. Once a structure is formalised, and results are proven, the mathematician seeks to generalise. This can be considered as one of the main praxis in mathematics. The idea of generalisation will be illustrated through graph colouring. This idea also results from a classic problem, in which it was well known by topographers that four colours suffice to colour any map such that no countries sharing a border receive the same colour. The proof of this theorem eluded mathematicians for centuries and was proven in 1976. Generalisation of graphs to hypergraphs, and variations on the colouring theme will be discussed, as well as applications in other disciplines.peer-reviewe

Direct solutions to tropical optimization problems with nonlinear objective functions and boundary constraints

We examine two multidimensional optimization problems that are formulated in terms of tropical mathematics. The problems are to minimize nonlinear objective functions, which are defined through the multiplicative conjugate vector transposition on vectors of a finite-dimensional semimodule over an idempotent semifield, and subject to boundary constraints. The solution approach is implemented, which involves the derivation of the sharp bounds on the objective functions, followed by determination of vectors that yield the bound. Based on the approach, direct solutions to the problems are obtained in a compact vector form. To illustrate, we apply the results to solving constrained Chebyshev approximation and location problems, and give numerical examples.Comment: Mathematical Methods and Optimization Techniques in Engineering: Proc. 1st Intern. Conf. on Optimization Techniques in Engineering (OTENG '13), Antalya, Turkey, October 8-10, 2013, WSEAS Press, 2013, pp. 86-91. ISBN 978-960-474-339-

Hilbert C*-modules and related subjects - a guided reference overview I

The overview contains 450 references of books, chapters of monographs, papers, preprints and Ph.~D.~thesises which are concerned with the theory and/or various applications of Hilbert C*-modules. To show a way through this amount of literature a four pages guide is added clustering sources around major research problems and research fields, and giving information on the historical background. Two smaller separate parts list references treating Hilbert modules over Hilbert*-algebras and Hilbert modules over (non-self-adjoint) operator algebras. Any additions, corrections and forthcoming information are welcome.Comment: LaTeX 2.09, 23 page

On the algorithmic complexity of twelve covering and independence parameters of graphs

The definitions of four previously studied parameters related to total coverings and total matchings of graphs can be restricted, thereby obtaining eight parameters related to covering and independence, each of which has been studied previously in some form. Here we survey briefly results concerning total coverings and total matchings of graphs, and consider the aforementioned 12 covering and independence parameters with regard to algorithmic complexity. We survey briefly known results for several graph classes, and obtain new NP-completeness results for the minimum total cover and maximum minimal total cover problems in planar graphs, the minimum maximal total matching problem in bipartite and chordal graphs, and the minimum independent dominating set problem in planar cubic graphs