Coins of Three Different Weights

We discuss several coin-weighing problems in which coins are known to be of three different weights and only a balance scale can be used. We start with the task of sorting coins when the pans of the scale can fit only one coin. We prove that the optimal number of weighings for $n$ coins is $\lceil 3n/2\rceil -2$. When the pans have an unlimited capacity, we can sort the coins in $n+1$ weighings. We also discuss variations of this problem, when there is exactly one coin of the middle weight.Comment: 18 page

On the Metric Dimension of Cartesian Products of Graphs

A set S of vertices in a graph G resolves G if every vertex is uniquely determined by its vector of distances to the vertices in S. The metric dimension of G is the minimum cardinality of a resolving set of G. This paper studies the metric dimension of cartesian products G*H. We prove that the metric dimension of G*G is tied in a strong sense to the minimum order of a so-called doubly resolving set in G. Using bounds on the order of doubly resolving sets, we establish bounds on G*H for many examples of G and H. One of our main results is a family of graphs G with bounded metric dimension for which the metric dimension of G*G is unbounded

Fast Decoder for Overloaded Uniquely Decodable Synchronous Optical CDMA

In this paper, we propose a fast decoder algorithm for uniquely decodable (errorless) code sets for overloaded synchronous optical code-division multiple-access (O-CDMA) systems. The proposed decoder is designed in a such a way that the users can uniquely recover the information bits with a very simple decoder, which uses only a few comparisons. Compared to maximum-likelihood (ML) decoder, which has a high computational complexity for even moderate code lengths, the proposed decoder has much lower computational complexity. Simulation results in terms of bit error rate (BER) demonstrate that the performance of the proposed decoder for a given BER requires only 1-2 dB higher signal-to-noise ratio (SNR) than the ML decoder.Comment: arXiv admin note: substantial text overlap with arXiv:1806.0395

Quantum computing classical physics

In the past decade quantum algorithms have been found which outperform the best classical solutions known for certain classical problems as well as the best classical methods known for simulation of certain quantum systems. This suggests that they may also speed up the simulation of some classical systems. I describe one class of discrete quantum algorithms which do so--quantum lattice gas automata--and show how to implement them efficiently on standard quantum computers.Comment: 13 pages, plain TeX, 10 PostScript figures included with epsf.tex; for related work see http://math.ucsd.edu/~dmeyer/research.htm