Hypercube orientations with only two in-degrees

We consider the problem of orienting the edges of the $n$-dimensional hypercube so only two different in-degrees $a$ and $b$ occur. We show that this can be done, for two specified in-degrees, if and only if an obvious necessary condition holds. Namely, there exist non-negative integers $s$ and $t$ so that $s+t=2^n$ and $as+bt=n2^{n-1}$. This is connected to a question arising from constructing a strategy for a "hat puzzle."Comment: 9 pages, 4 figure

Computing optimal strategies for a cooperative hat game

We consider a `hat problem' in which each player has a randomly placed stack of black and white hats on their heads, visible to the other player, but not the wearer. Each player must guess a hat position on their head with the goal of both players guessing a white hat. We address the question of finding the optimal strategy, i.e., the one with the highest probability of winning, for this game. We provide an overview of prior work on this question, and describe several strategies that give the best known lower bound on the probability of winning. Upper bounds are also considered here

Note A Note on the Binomial Drop Polynomial of a Poset

Suppose (P,-<) is a poset of size n and n: P- ~ P is a permutation. We say that n has a drop at x if n(x)~x. Let fie(k) denote the number of n having k drops, 0 < ~ k < n, and define the drop polynomial A p(2) by Further, define the incomparability graph I(P) to have vertex set P and edges 0" whenever i and j are incomparable in P, i.e., neither i-<j nor j< i holds. In this note we give a short proof that Ae(2) is equal to the chromatic polynomial of](P). © 1994 Academic Press, Inc. 1

Cycles, L-functions and triple products of elliptic curves

A variant of a conjecture of Beilinson and Bloch relates the rank of the Griffiths group of a smooth projective variety over a number field to the order of vanishing of an L-function at the center of the critical strip. Presently, there is little evidence to support the conjecture, especially when the L-function vanishes to order greater than 1. We study 1-cycles on E^3 for various elliptic curves E/Q. In each of the 76 cases considered we find that the empirical order of vanishing of the L-function is at least as large as our best lower bound on the rank of the Griffiths group. In 11 cases this lower bound is two.

Fast and Precise Computations of Discrete Fourier Transforms using Cyclotomic Integers

Many applications of fast fourier transforms (FFT's), such as computer-tomography, geophysical signal processing, high resolution imaging radars, and prediction filters, require high precision output. The usual method of fixed point computation of FFT's of vectors of length 2 ` leads to an average loss of `=2 bits of precision. This phenomenon, often referred to as computational noise, causes major problems for arithmetic units with limited precision which are often used for real time applications. Several researchers have noted that calculation of FFT's with algebraic integers avoids computational noise entirely, see, e.g., [3]. We will show that complex numbers can be approximated accurately by cyclotomic integers, and combine this idea with Chinese remaindering strategies in the cyclotomic integers to, roughly, give a O(b 1+ffl L log(L)) algorithm to compute b-bit precision FFT's of length L. The first part of the paper will describe the FFT strategy, assuming good approximation..