Journal of Prime Research in Mathematics

Abstract: By a convenient parameter change we give a necessary and sufficient condition for the
conversion of a degree \(n\) rational triangular Bézier surfaces to a degree \(n × n\) rectangular surface whose one of the four edges is reduced to a single multiple point. Using then the method of base point we obtain the explicit expression of multi-sided patches in Bézier form.

Abstract: Let K be an arbitrary commutative field and let \(R = K[[X]]\) be the ring of formal power series in one variable. Let \(G_{R}\) be the set of all power series of the form \(u = Xv\), where \(v\) is a unity in \(R\). Relative to the usual composition \(G_{R}\) becomes a topological group with respect to the \(X-\)adic topology of \(R\). We also study an equivalence relation on \(R\). Let \(R = K[[X]]\) be the ring of formal power series in one variable over a fixed commutative field \(K\). We denote by \(ord f = min \{i: a_{i} ≠ 0\} \) for any \(  f ∈ R\) . It is well known that \(ord f\) is a valuation on \(R\) and \(R\) becomes a complete topological ring relative to the topology induced by this valuation. Let  \(G_{G}=\{u∈ R: ord u=1\}\) and for \(u,v ∈ R_{G}\) we denote \((uov)(X)=v(u(X))\), a new element of \(R_{G}\).

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 \(5\).