The Critical Values of a Polynomial

It has been known for a long time that any real sequence y ₁ , . . . ,y _n-1 is the sequence of critical values of some real polynomial. Here we show that any complex sequence w ₁ , . . . ,w _n-1 is the sequence of critical values of some complex polynomial.     

A remark on analytic continuation

A simple example is given to show that the space of germs obtained by analytic continuation of a given germ need not be a covering space in the topological sense.

     

The Schwarz-Pick Lemma for derivatives

The Schwarz-Pick Lemma states that any analytic function of the unit disc into itself is a contraction with respect to the hyperbolic metric. In this note a related result is proved for the derivative of an analytic function.

     

Sphere-preserving maps in inversive geometry

We give an extensive discussion of sphere-preserving maps defined on subdomains of Euclidean -space, and their relationship to Möbius maps and to the preservation of cross-ratios. In the case (the complex plane) we also relate these ideas to the solutions of certain functional equations.

     

On Ritt's Factorization of Polynomials

Ritt has shown that any complex polynomial p can be writtenas the composition of polynomials p₁,...,p_m, where each p_j isprime in the sense that it cannot be written as a non-trivialcomposition of polynomials. The factors p_j are not unique butthe number m of them is, as is the set of the degrees of thep_j. The paper extends Ritt's theory and, in particular, a thirdinvariant of the decomposition is introduced.     

Models for capacitated lot-sizing problem with backlogging,setup carryover and crossover

Setup operations are significant in some production environments. It is mandatory that their production plans consider some features, as setup state conservation across periods through setup carryover and crossover. The modelling of setup crossover allows more flexible decisions and is essential for problems with long setup times. This paper proposes two models for the capacitated lot-sizing problem with backlogging and setup carryover and crossover. The first is in line with other models from the literature, whereas the second considers a disaggregated setup variable, which tracks the starting and completion times of the setup operation. This innovative approach permits a more compact formulation. Computational results show that the proposed models have outperformed other state-of-the-art formulation.