Journal of Prime Research in Mathematics

Abstract: Inui and Katori introduced Fermi walk configurations which are non-crossing subsets of the directed random walks between opposite corners of a rectangular \(l × w\) grid. They related them to Bose configurations which are similarly defined except that they include multisets. Bose configurations biject to vicious walker watermelon configurations. It is found that the maximum number of walks in a Fermi configuration is \(lw + 1\) and the number of configurations corresponding to this number of walks is a w-dimensional Catalan number \(C_{l,w}\). Product formulae for the numbers of Fermi configurations with \(lw\) and \(lw − 1\) walks are derived. We also consider generating functions for the numbers of \(n−\)walk configurations as a function of \(l\) and \(w\). The Bose generating function is rational with denominator \((1-z)^{lw+1}\). The Fermi generating function is found to have a factor \((1+z)^{lw+1}\) and the complementary factor , \( Q_{l,w}^{frmi}(z)\)is related to the numerator of the Bose generating function which is a generalized Naryana polynomial introduced by Sulanke. Recurrence relations for the numbers of Fermi walks and for the coefficients of the polynomial \( Q_{l,w}^{frmi}(z)\) are obtained.

Abstract: Here we discuss the application of control theory to dynamical non-linear systems in the form of speed gradient method. This approach can be practically used for the space focusing of classical or quantum particles, for instance, in nanolithography with the beams of cooled atoms. Standard speed gradient method works here not very efficient because it allows to achieve the selected level of energy (the eigenfunction of Hamiltonian for dynamical differential equation). For the practical purposes we re-formulate the mathematical task and introduce principally new type of controlled systems. We demand achievement of the space distribution of the dynamical particles. Additionally we investigate the statistical properties of the particle dynamics but not the behavior of the single particle. Efficiency of the approach was verified by means of computer simulations.

Abstract: We use arithmetical tools (algebraic numbers, valuations, Galois groups) in order to describe the structure of all compact subsets of the \(p-\)adic complex field, which are invariant with respect to the absolute Galois group of the \(p-\)adic number field

Abstract: In a given graph \(G = (V;E)\), a set of vertices S with an assignment of colors to them is said to be a defining set of the vertex coloring of G, if there exists a unique extension of the colors of S to a \(c \geq \chi(G)\) coloring of the vertices of G. A defining set with minimum cardinality is called a minimum defining set and its cardinality is the defining number, denoted by \((d(G; c)\). If F is a family of graphs then \( Spec_{c}(F)=\{d| \exists G, G \epsilon F, d(G,C)=d \} \). Here we study the cases where \(F\) is the family of \(k-\)regular (connected and disconnected) graphs on n vertices and \(c = k-1\). Also the \(Spec_{k-1}(F)\) defining spectrum of all \(k-\)regular (connected and disconnected) graph on n vertices are verified for \(k = 3, 4\) and \(5\).

Abstract: A clique of a graph G is a complete subgraph of \(G\) which is maximal relatively to set inclusion and a covering C of G consisting of s cliques is an irreducible covering if the union of any \(s − 1\) cliques from C is a proper subset of the vertex-set of \(G\). Some discrete optimization problems involve irreducible coverings of graphs: minimization of Boolean functions, minimization of incompletely specified finite automata, finding the chromatic number of a graph. This paper surveys some recent results by the author on the irreducible coverings of graphs by cliques: the recurrence relation and the exponential generating function of the number of irreducible coverings for bipartite graphs, asymptotic behavior of these numbers and of the maximum number of irreducible coverings by cliques of an n-vertex graph as n tends to infinity, extremal graphs of order n for irreducible coverings by \(n − 2\) and \(n − 3\) cliques and the structure of irreducible coverings for bipartite and nonbipartite cases. Some conjectures and open problems are proposed.