I. Naci Cangul

Analysis approach to finite monoids

In a previous paper by the authors, a new approach between algebra and analysis has been recently developed. In detail, it has been generally described how one can express some algebraic properties in terms of special generating functions. To continue the study of this approach, in here, we state and prove that the presentation which has the minimal number of generators of the split extension of two finite monogenic monoids has different sets of generating functions (such that the number of these functions is equal to the number of generators) that represent the exponent sums of the generating pictures of this presentation. This study can be thought of as a mixture of pure analysis, topology and geometry within the purposes of this journal.AMS Subject Classification:11B68, 11S40, 12D10, 20M05, 20M50, 26C05, 26C10.

On the first Zagreb index and multiplicative Zagreb coindices of graphs

Abstract For a (molecular) graph G with vertex set V (G) and edge set E(G), the first Zagreb index of G is defined as , where dG(vi) is the degree of vertex vi in G. Recently Xu et al. introduced two graphical invariants and named as first multiplicative Zagreb coindex and second multiplicative Zagreb coindex, respectively. The Narumi-Katayama index of a graph G, denoted by NK(G), is equal to the product of the degrees of the vertices of G, that is, NK(G) = . The irregularity index t(G) of G is defined as the number of distinct terms in the degree sequence of G. In this paper, we give some lower and upper bounds on the first Zagreb index M1(G) of graphs and trees in terms of number of vertices, irregularity index, maxi- mum degree, and characterize the extremal graphs. Moreover, we obtain some lower and upper bounds on the (first and second) multiplicative Zagreb coindices of graphs and characterize the extremal graphs. Finally, we present some relations between first Zagreb index and Narumi-Katayama index, and (first and second) multiplicative Zagreb index and coindices of graphs.

Minimality over free monoid presentations

As a continues study of the paper [4], in here, we first state and prove thep-Cockcroft property (or, equivalently, eciency) for a presentation, say PE, of the semi-direct product of a free abelian monoid rank two by a finite cyclic monoid. Then, in a separate section, we present sucient conditions on a special case for PE to be minimal whilst it is inecient. 2000 AMS Classification: 20L05, 20M05, 20M15, 20M50.

Properties of n‐th Degree Bernstein Polynomials

In this paper, derivatives of the product of Bernstein polynomials of the same and different degrees are obtained. Also a recurrence formula for those polynomials together with some new properties are given.

On the Norms of Toeplitz and Hankel Matrices with Pell Numbers

Let us define A = [a ij ] i,i = 0 n-1 and B = [b ij ] i,i = 0 n-1 as n×n Toeplitz and Hankel matrices, respectively, such that a ij = P i−j and b ij = P i+j , where P denotes the Pell number. We present upper and lower bounds for the spectral norms of these matrices.

Power subgroups of Hecke groups H(n)

Results in discrete group theory are applied to some Hecke groups to determine the group theoretical structure of power subgroups.

The graph based on Gröbner-Shirshov bases of groups

Let us consider groups G1=Zk∗(Zm∗Zn), G2=Zk×(Zm∗Zn), G3=Zk∗(Zm×Zn), G4=(Zk∗Zl)∗(Zm∗Zn) and G5=(Zk∗Zl)×(Zm∗Zn), where k,l,m,n≥2. In this paper, by defining a new graph Γ(Gi) based on the Gröbner-Shirshov bases over groups Gi, where 1≤i≤5, we calculate the diameter, maximum and minimum degrees, girth, chromatic number, clique number, domination number, degree sequence and irregularity index of Γ(Gi). Since graph theoretical studies (including such above graph parameters) consist of some fixed point techniques, they have been applied in such fields as chemistry (in the meaning of atoms, molecules, energy etc.) and engineering (in the meaning of signal processing etc.), game theory and physics. In addition, the Gröbner-Shirshov basis and the presentations of algebraic structures contain a mixture of algebra, topology and geometry within the purposes of this journal.MSC:05C25, 13P10, 20M05, 20E06, 26C10.

Some Properties of the Minimal Polynomials of 2cos(π/q) for odd q

The number λq = 2 cos π/q, q∈N, q≥3,, appears in the study of Hecke groups which are Fuchsian groups, and in the study of regular polyhedra. There are many results about the minimal polynomial of this algebraic number. Here we obtain the minimal polynomial of this number by means of the better known Chebycheff polynomials for odd q and give some of their properties.

The Minimal Polynomials of 2cos(π/2k)over the Rationals

The number λq = 2 cos π/q, q∈N, q≥3, appears in the study of Hecke groups which are Fuchsian groups of the first kind, and in the study of regular polyhedra. Here we obtained the minimal polynomial of this number by means of the better known Chebycheff polynomials and the set of roots on the extension Q(λq). We follow some kind of inductive method on the number q. The minimal polynomial is obtained for even q.

Some Special Cases of the Minimal Polynomial of 2cos(π/q) over Q

The number λq = 2 cos π/q, q∈N, q≥3, appears in the study of Hecke groups which are Fuchsian groups of the first kind, and in the study of regular polyhedra. Here we obtained some results on the values of the minimal polynomial of this number in modulo prime p. This results help in the calculation of the congruence subgroups of the Hecke groups which is an important problem in discrete group theory.