A new generalization of Moulton affine planes

We consider a variation on W. A. Pierce's construction of Moulton planes. For any pseudo-ordered fieldF, the pairs of elements ofF are taken as points, and straight lines are given by the equationsx=c,y=mx+n withm≥0 andg(y=mf(x)+n withm < 0, wheref andg are mappings ofF into itself which have to satisfy a number of conditions.     

On a generalization of starlike functions

In this paper, we mainly study the order of $q$-starlikeness of the well-known basic hypergeometric function. In addition, we discuss the Bieberbach-type problem and the second order Hankel determinant for a generalized class of starlike functions.     

On a generalization of linecritical graphs

Let β(G) be the maximal β such that for any edge xy of G there is an independent β-set that contains no neighbours of x and y. Then $\text{0}\text{}\text{?}\text{β(G)}\text{}\text{?}\text{α(G)?1}$ and G is linecritical iff β(G) = α(G)?1. We determine the minimal connected graphs for any given β(G) or for any given β(G) and α(G). We study the case when β(G)??2 and give upper bounds for the minimal valencies. We generalize some results on linecritical graphs of [1] and [4].     

On a generalization of Hermite polynomials

We consider a new generalization of Hermite polynomials to the case of several variables. Our construction is based on an analysis of the generalized eigenvalue problem for the operator _∂Ax+D${\partial }_{A\mathbf{x}}+D$, acting on a linear space of polynomials of N variables, where A   is an endomorphism of the Euclidean space R^N${\mathbb{R}}^N$ and D is a second order differential operator. Our main results describe a basis for the space of Hermite–Jordan polynomials.     

Hyperplane Sections of Kantor's Unitary Ovoids

The projective plane is embedded as a variety of projective points in , where M is a nine dimensional -module for the groupG=GL(3,q ²). The hyperplane sections of thisvariety and their stabilizers in the group G aredetermined. When q 2 (mod 3) one such hyperplanesection is a member of the family of Kantor's unitary ovoids.We furtherdetermine all sections whereD has codimension two in M and demonstratethat these are never empty. Consequences are drawn for Kantor'sovoids.     

On a generalization of cyclic semifields

A new construction is given of cyclic semifields of orders q ²ⁿ , n odd, with kernel (left nucleus) and right and middle nuclei isomorphic to , and the isotopism classes are determined. Furthermore, this construction is generalized to produce potentially new semifields of the same general type that are not isotopic to cyclic semifields. In particular, a new semifield plane of order 4⁵ and new semifield planes of order 16⁵ are constructed by this method.     

On the generalization of a retrocirculant

We study the properties of matrices of the form P(σ)A where σ is induced by an automorphism of an abelian group G and A is a group matrix. P(σ)A is a generalization of a retrocirculant. We also determine the eigenvalues of P(σ)A.     

On a generalization of Rubin's theorem

The work is devoted to the calculation of asymptotic value of the choice number of the complete r‐partite graph K_{m* r} = K_{m, …, m} with equal part size m. We obtained the asymptotics in the case lnr = o(lnm). The proof generalizes the classical result of A.L. Rubin for the case r = 2. © 2010 Wiley Periodicals, Inc. J Graph Theory 67: 226–234, 2011 67: 226‐234, 2011     

On a generalization of duality triads

Some aspects of duality triads introduced recently are discussed. In particular, the general solution for the triad polynomials is given. Furthermore, a generalization of the notion of duality triad is proposed and some simple properties of these generalized duality triads are derived.     

On a generalization of semisimple modules

Let R be a ring with identity. A module \(M_R\) is called an r-semisimple module if for any right ideal I of R, MI is a direct summand of \(M_R\) which is a generalization of semisimple and second modules. We investigate when an r-semisimple ring is semisimple and prove that a ring R with the number of nonzero proper ideals \(\le \)4 and \(J(R)=0\) is r-semisimple. Moreover, we prove that R is an r-semisimple ring if and only if it is a direct sum of simple rings and we investigate the structure of module whenever R is an r-semisimple ring.     

On a generalization of distance sets

A subset X in the d-dimensional Euclidean space is called a k-distance set if there are exactly k distinct distances between two distinct points in X and a subset X is called a locally k-distance set if for any point x in X, there are at most k distinct distances between x and other points in X.Delsarte, Goethals, and Seidel gave the Fisher type upper bound for the cardinalities of k-distance sets on a sphere in 1977. In the same way, we are able to give the same bound for locally k-distance sets on a sphere. In the first part of this paper, we prove that if X is a locally k-distance set attaining the Fisher type upper bound, then determining a weight function w, (X,w) is a tight weighted spherical 2k-design. This result implies that locally k-distance sets attaining the Fisher type upper bound are k-distance sets. In the second part, we give a new absolute bound for the cardinalities of k-distance sets on a sphere. This upper bound is useful for k-distance sets for which the linear programming bound is not applicable. In the third part, we discuss about locally two-distance sets in Euclidean spaces. We give an upper bound for the cardinalities of locally two-distance sets in Euclidean spaces. Moreover, we prove that the existence of a spherical two-distance set in (d−1)-space which attains the Fisher type upper bound is equivalent to the existence of a locally two-distance set but not a two-distance set in d-space with more than d(d+1)/2 points. We also classify optimal (largest possible) locally two-distance sets for dimensions less than eight. In addition, we determine the maximum cardinalities of locally two-distance sets on a sphere for dimensions less than forty.