Integer Points in Backward Orbits

A theorem of J. Silverman states that a forward orbit of a rational map $\phi(z)$ on $\mathbb P^1(K)$ contains finitely many $S$-integers in the number field $K$ when $(\phi\circ\phi)(z)$ is not a polynomial. We state an analogous conjecture for the backward orbits using a general $S$-integrality notion based on the Galois conjugates of points. This conjecture is proven for the map $\phi(z)=z^d$, and consequently Chebyshev polynomials, by uniformly bounding the number of Galois orbits for $z^n-\beta$ when $\beta\not =0$ is a non-root of unity. In general, our conjecture is true provided that the number of Galois orbits for $\phi^n(z)-\beta$ is bounded independently of $n$.Comment: 13 page

Is project management the new management 2.0?

This paper considers the evolving nature of project management (PM) and offers a comparison with the evolving nature of management generally. Specifically, we identify a number of management trends that are drawn from a paper that documents a proposed ‘Management 2.0’ model, and we compare those trends to the way in which PM is maturing to embrace the challenges of modern organizational progress.Some theoretical frameworks are offered that assist in explaining the shift from the historically accepted ‘tools and techniques’ model to a more nuanced and behaviorally driven paradigm that is arguably more appropriate to manage change in today’s flexible and progressive organizations, and which provide a more coherent response, both in PM and traditional management, to McDonald’s forces. In addition, we offer a number of examples to robustly support our assertions, based around the development of innovative products from Apple Inc. In using this metaphor to demonstrate the evolution of project-based work, we link PM with innovation and new product development.