We use generating functions over group rings to count polynomials finite fields with the first few coefficients and a factorization pattern prescribed. In particular, we obtain different exact formulas for number of monic n-smooth degree m field, as well prescribed trace coefficient.

In this paper, we propose a new public key cryptosystem in non-commutative group over ring, using hard problem, Factorization with Discrete Logarithm Problem (FDLP). The security analysis of the proposed scheme is discussed and it shown that system secure.

A 1-factorization of a graph is a decomposition of the graph into edge disjoint perfect matchings. There is a well-known method, which we call the K-construction, for building a 1-factorization of Kn;n from a 1-factorization of Knþ1. The 1-factorization of Kn;n can be written as a latin square of order n. The K-construction has been used, among other things, to make perfect 1-factorizations, su...

In this paper, we study the problem of nonnegative graph embedding, originally investigated in [14] for reaping the benefits from both nonnegative data factorization and the specific purpose characterized by the intrinsic and penalty graphs [13]. Our contributions are two-fold. On the one hand, we present a multiplicative iterative procedure for nonnegative graph embedding, which significantly ...

Data clustering is the task to group the data samples into certain clusters based on the relationships of samples and structures hidden in data, and it is a fundamental and important topic in data mining and machine learning areas. In the literature, the spectral clustering is one of the most popular approaches and has many variants in recent years. However, the performance of spectral clusteri...

Abstract We show that for the family of complex reflection groups G = ( m , p ,2) appearing in Shephard–Todd classification, endomorphism ring reduced hyperplane arrangement A ) is a non-commutative resolution coordinate discriminant Δ . This furthers work Buchweitz, Faber and Ingalls who showed this result holds any true group. In particular, we construct matrix factorization from decompose it...

This paper is the second in a series of three papers concerning the surface T×T , where T is a complex torus. We compute the fundamental group of the branch curve of the surface in C, using the van Kampen Theorem and the braid monodromy factorization of the curve. 1. Background The concentration on the fundamental group of a complement of a branch curve of an algebraic surfaceX with respect to ...

In this paper, we present several methods for construction of elliptic curves with large torsion group and positive rank over number fields of small degree. We also discuss potential applications of such curves in the elliptic curve factorization method (ECM).