The complexity of normal form rewrite sequences for Associativity

The complexity of a particular term-rewrite system is considered: the rule of associativity (x*y)*z --> x*(y*z). Algorithms and exact calculations are given for the longest and shortest sequences of applications of --> that result in normal form (NF). The shortest NF sequence for a term x is always n-drm(x), where n is the number of occurrences of * in x and drm(x) is the depth of the rightmost leaf of x. The longest NF sequence for any term is of length n(n-1)/2.Comment: 5 page

Efficient recovering of operation tables of black box groups and rings

People have been studying the following problem: Given a finite set S with a hidden (black box) binary operation * on S which might come from a group law, and suppose you have access to an oracle that you can ask for the operation x*y of single pairs (x,y) you choose. What is the minimal number of queries to the oracle until the whole binary operation is recovered, i.e. you know x*y for all x,y in S? This problem can trivially be solved by using |S|^2 queries to the oracle, so the question arises under which circumstances you can succeed with a significantly smaller number of queries. In this presentation we give a lower bound on the number of queries needed for general binary operations. On the other hand, we present algorithms solving this problem by using |S| queries, provided that * is an abelian group operation. We also investigate black box rings and give lower and upper bounds for the number of queries needed to solve product recovering in this case.Comment: 5 page

The Hilbert scheme of a plane curve singularity and the HOMFLY polynomial of its link

The intersection of a complex plane curve with a small three-sphere surrounding one of its singularities is a non-trivial link. The refined punctual Hilbert schemes of the singularity parameterize subschemes supported at the singular point of fixed length and whose defining ideals have a fixed number of generators. We conjecture that the generating function of Euler characteristics of refined punctual Hilbert schemes is the HOMFLY polynomial of the link. The conjecture is verified for irreducible singularities y^k = x^n, whose links are the k,n torus knots, and for the singularity y^4 = x^7 - x^6 + 4 x^5 y + 2 x^3 y^2, whose link is the 2,13 cable of the trefoil.Comment: 20 pages; some improvements are mad

An experimental investigation of criteria for continuous variable entanglement

We generate a pair of entangled beams from the interference of two amplitude squeezed beams. The entanglement is quantified in terms of EPR-paradox [Reid88] and inseparability [Duan00] criteria, with observed results of $\Delta^{2} X_{x|y}^{+} \Delta^{2} X_{x|y}^{-} = 0.58 \pm 0.02$ and $\sqrt{\Delta^{2} X_{x \pm y}^{+} \Delta^{2} X_{x \pm y}^{-}} = 0.44 \pm 0.01$, respectively. Both results clearly beat the standard quantum limit of unity. We experimentally analyze the effect of decoherence on each criterion and demonstrate qualitative differences. We also characterize the number of required and excess photons present in the entangled beams and provide contour plots of the efficacy of quantum information protocols in terms of these variables.Comment: 4 pages, 5 figure

On the nature of chaos

Based on newly discovered properties of the shift map (Theorem 1), we believe that chaos should involve not only nearby points can diverge apart but also faraway points can get close to each other. Therefore, we propose to call a continuous map $f$ from an infinite compact metric space $(X, d)$ to itself chaotic if there exists a positive number $\delta$ such that for any point $x$ and any nonempty open set $V$ (not necessarily an open neighborhood of $x$) in $X$ there is a point $y$ in $V$ such that $\limsup_{n \to \infty} d(f^n(x), f^n(y)) \ge \delta$ and $\liminf_{n \to \infty} d(f^n(x), f^n(y)) = 0$.Comment: 5 page

On the number of limit cycles of the Lienard equation

In this paper, we study a Lienard system of the form dot{x}=y-F(x), dot{y}=-x, where F(x) is an odd polynomial. We introduce a method that gives a sequence of algebraic approximations to the equation of each limit cycle of the system. This sequence seems to converge to the exact equation of each limit cycle. We obtain also a sequence of polynomials R_n(x) whose roots of odd multiplicity are related to the number and location of the limit cycles of the system.Comment: 10 pages, 5 figures. Submitted to Physical Review