Cycles of polynomial mappings in two variables over rings of integers in quadratic fields

We find all possible cycle-lengths for polynomial mappings in two variables over rings of integers in quadratic extensions of rationals.     

A finite algorithm for the Drazin inverse of a polynomial matrix

Based on Greville's finite algorithm for Drazin inverse of a constant matrix we propose a finite numerical algorithm for the Drazin inverse of polynomial matrices. We also present a new proof for Decell's finite algorithm through Greville's finite algorithm.     

An asymptotic expansion for the error in a linear map that reproduces polynomials of a certain order

Han's ‘multinode higher-order expansion’ in [H] is shown to be a special case of an asymptotic error expansion available for any bounded linear map on C([a..b]) that reproduces polynomials of a certain order. The key is the formula for the divided difference at a sequence containing just two distinct points.     

Parallel Itoh–Tsujii multiplicative inversion algorithm for a special class of trinomials

In this contribution, we derive a novel parallel formulation of the standard Itoh–Tsujii algorithm for multiplicative inverse computation over the field GF(2 ^m ). The main building blocks used by our algorithm are: field multiplication, field squaring and field square root operators. It achieves its best performance when using a special class of irreducible trinomials, namely, P(x) = x ^m  + x ^k  + 1, with m and k odd numbers and when implemented in hardware platforms. Under these conditions, our experimental results show that our parallel version of the Itoh–Tsujii algorithm yields a speedup of about 30% when compared with the standard version of it. Implemented in a Virtex 3200E FPGA device, our design is able to compute multiplicative inversion over GF(2¹⁹³) after 20 clock cycles in about 0.94 μS.      

An algorithm for source location in directed graphs

A polynomial time solution algorithm is described to find a smallest subset R of nodes of a directed graph D=(V,A) such that, for every node v∈V-R, there are k edge-disjoint paths from R to v and there are l edge-disjoint paths from v to R.     

On the period function of x″+f(x)x′+g(x)=0

We study the period function T of a center O of the title's equation. A sufficient condition for the monotonicity of T, or for the isochronicity of O, is given. Such a condition is also necessary, when f and g are odd and analytic. In this case a characterization of isochronous centers is given. Some classes of plane systems equivalent to such equation are considered, including some Kukles’ systems.     

The stable crews problem

In this paper the classical stable roommates problem is generalized to situations when the two partners in a pair perform different roles. We propose an efficient algorithm to decide the existence of a stable matching in this problem.     

Eventual Positivity of Hermitian Polynomials and Integral Operators

Quillen proved that if a Hermitian bihomogeneous polynomial is strictly positive on the unit sphere, then repeated multiplication of the standard sesquilinear form to this polynomial eventually results in a sum of Hermitian squares. Catlin-D'Angelo and Varolin deduced this positivstellensatz of Quillen from the eventual positive-definiteness of an associated integral operator. Their arguments involve asymptotic expansions of the Bergman kernel. The goal of this article is to give an elementary proof of the positive-definiteness of this integral operator.     

Some Inequalities for the Polynomial with S-Fold Zeros at the Origin

Let p(z) be a polynomial of degree n, which has no zeros in |z|1,Dewan et al.[K.K.Dewan and Sunil Hans, Generalization of certain well known polynomial inequalities, J. Math. Anal. Appl.,363(2010),pp.38–41] established |zp'(z)+nβ/2p(z)≤n/2{(|β/2|+|1+β/2|)max|z|=1|p(z)|-(|1+β/2|-|β/2|)min|z|=1|p(z)|},for any|β|≤1 and |z|=1.In this paper we improve the above inequality for the polynomial which has no zeros in |z|k,k≥1,except s-fold zeros at the origin. Our results generalize certain well known polynomial inequalities.