Related papers: Six topics on inscribable polytopes
Questions such as whether we live in a spatially finite universe, and what its shape and size may be, are among the fundamental open problems that high precision modern cosmology needs to resolve. These questions go beyond the scope of…
This is a review (including some background material) of the author's work and related activity on certain exactly solvable statistical models in two dimensions, including the six-vertex model, loop models and lozenge tilings. Applications…
We present five open problems in the theory of vertex rings. They cover a variety of different areas of research where vertex rings have been, or are threatening to be, relevant. They have also been chosen because I personally find them…
An age-old controversy in mathematics concerns the necessity and the possibility of constructive proofs. The controversy has been rekindled by recent advances which demonstrate the feasibility of a fully constructive mathematics. This…
We approach several themes of classical geometry of the circle and complete them with some original results, showing that not everything in traditional math is revealed, and that it still has an open character. The topics were chosen…
Creative telescoping is the method of choice for obtaining information about definite sums or integrals. It has been intensively studied since the early 1990s, and can now be considered as a classical technique in computer algebra. At the…
Throughout this book, we discuss some open problems in various branches of science, including mathematics, theoretical physics, astro-physics, geophysics etc. It is of our hope that some of the problems discussed in this book will find…
Some particular examples of classical and quantum systems on the lattice are solved with the help of orthogonal polynomials and its connection to continuous models are explored.
This is the first part of a small-scale explorative study in an effort to start assessing reproducibility issues specific to scientometrics research. This effort is motivated by the desire to generate empirical data to inform debates about…
This is a straightforward introduction to the properties of polynomials in many variables that do not vanish in the open upper half plane. Such polynomials generalize many of the well-known properties of polynomials with all real roots.
An alternative approach towards quantum theory is described, and tentative attempts to connect his approach to special and general relativity are discussed. Important concepts are gauge groups and information/entropy connected to some…
We consider unimodality and related properties of f-vectors of polytopes in various dimensions. By a result of Kalai (1988), f-vectors of 5-polytopes are unimodal. In higher dimensions much less can be said; we give an overview on current…
We prove that any finite, abstract n-polytope is covered by a finite, abstract regular n-polytope.
The paper is a survey of notions and results related to classical and new generalizations of the notion of a periodic sequence. The topics related to almost periodicity in combinatorics on words, symbolic dynamics, expressibility in logical…
We characterize all the strongly monotypic polytopes. Hadwiger's conjecture for this class of polytopes is deduced from the characterization.
We present and discuss a number of known results and open problems abelian squares in words on small alphabets.
Rapid advances in computing technology over the past few decades have spurred two extraordinary phenomena in science: large-scale and high-throughput data collection coupled with the creation and implementation of complex statistical…
A triangulation of a surface with fixed topological type is called irreducible if no edge can be contracted to a vertex while remaining in the category of simplicial complexes and preserving the topology of the surface. A complete list of…
This is a survey on tropical polytopes from the combinatorial point of view and with a focus on algorithms. Tropical convexity is interesting because it relates a number of combinatorial concepts including ordinary convexity, monomial…
We study multiple sampling, interpolation and uniqueness for the classical Fock space in the case of unbounded mul-tiplicities.