Abelian actions on compact nonorientable Riemann surfaces

Given an integer $g>2$ , we state necessary and sufficient conditions for a finite Abelian group to act as a group of automorphisms of some compact nonorientable Riemann surface of genus g. This result provides a new method to obtain the symmetric cross-cap number of Abelian groups. We also compute the least symmetric cross-cap number of Abelian groups of a given order and solve the maximum order problem for Abelian groups acting on nonorientable Riemann surfaces.

Order Statistics of Additive Uniform Exponential Distribution

In this paper we investigated the order statistics by using Additive Uniform Exponential Distribution (AUED) proposed by Venkata Subbarao Uppu (2010).The probability density functions of rth order Statistics, lth moment of the rth order Statistic, minimum, maximum order statistics, mean of the maximum and minimum order statistics, the joint density function of two order statistics were calculated and discussed in detailed . Applications and several aspects were discussed Keywords: Additive Uniform Exponential Distribution, Moments, Minimum order statistic, Maximum order statistic, Joint density of the order Statistics, complete length of service.

Transitivity of trees

For a graph [Formula: see text], a partition [Formula: see text] of the vertex set [Formula: see text] is a transitive partition if [Formula: see text] dominates [Formula: see text] whenever [Formula: see text]. The transitivity [Formula: see text] of a graph [Formula: see text] is the maximum order of a transitive partition of [Formula: see text]. For any positive integer [Formula: see text], we characterize the smallest tree with transitivity [Formula: see text] and obtain an algorithm to determine the transitivity of any tree of finite order.

Maximum order complexity of the sum of digits function in Zeckendorf base and polynomial subsequences

Automatic sequences are not suitable sequences for cryptographic applications since both their subword complexity and their expansion complexity are small, and their correlation measure of order 2 is large. These sequences are highly predictable despite having a large maximum order complexity. However, recent results show that polynomial subsequences of automatic sequences, such as the Thue–Morse sequence, are better candidates for pseudorandom sequences. A natural generalization of automatic sequences are morphic sequences, given by a fixed point of a prolongeable morphism that is not necessarily uniform. In this paper we prove a lower bound for the maximum order complexity of the sum of digits function in Zeckendorf base which is an example of a morphic sequence. We also prove that the polynomial subsequences of this sequence keep large maximum order complexity, such as the Thue–Morse sequence.

On the k -Component Independence Number of a Tree

Let G be a graph and k ≥ 1 be an integer. A subset S of vertices in a graph G is called a k -component independent set of G if each component of G S has order at most k . The k -component independence number, denoted by α c k G , is the maximum order of a vertex subset that induces a subgraph with maximum component order at most k . We prove that if a tree T is of order n , then α k T ≥ k / k + 1 n . The bound is sharp. In addition, we give a linear-time algorithm for finding a maximum k -component independent set of a tree.