The HAL 9000 computer, the inimitable star of the classic Kubrick and Clarke film "2001: A Space Odyssey," displayed image understanding capabilities vastly beyond today’s computer systems. HAL could not only instantly recognize who he was interacting with, but also he could lip read, judge aesthetics of visual sketches, recognize emotions subtly expressed by scientists on board the ship, and r...

In the movie 2001: A Space Odyssey (Kubrick 1968), crewmembers of a spaceship interact with the ship’s computer by talking to it. The computer, named HAL9000, has no keyboard or mouse; the only way to communicate with it is conversationally. HAL9000 behaves the same way people do in conversation, using subtle strategies such as hedging, indirect requests, and politeness to accomplish its goals....

A set is said to be 3-free if no three elements form an arithmetic progression. Given a 3-free set A of integers 0 = a0 < a1 < · · · < at, the Stanley sequence S(A) = {an} is defined using the greedy algorithm: For each successive n > t, we pick the smallest possible an so that {a0, a1, . . . , an} is 3-free and increasing. Work by Odlyzko and Stanley indicates that Stanley sequences may be div...

We define a new family F̃w(X) of generating functions for w ∈ S̃n which are affine analogues of Stanley symmetric functions. We establish basic properties of these functions such as their symmetry and conjecture certain positivity properties. As an application, we relate these functions to the k-Schur functions of Lapointe, Lascoux and Morse as well as the cylindric Schur functions of Postnikov. ...

Following a construction of Stanley we consider toric face rings associated to rational pointed fans. This class of rings is a common generalization of the concepts of Stanley–Reisner and affine monoid algebras. The main goal of this article is to unify parts of the theories of Stanley–Reisnerand affine monoid algebras. We consider (nonpure) shellable fan’s and the Cohen–Macaulay property. More...