Cellular automata are a discrete, synchronous, and uniform dynamical system that give rise to a wide range of dynamical behaviours. In this paper, we investigate whether this system can achieve synchronisation. We study the cases of classical bi-infinite configurations, periodic configurations, and periodic configurations of prime period. In the two former cases, we prove that only a “degenerat...

Let a and b be two polynomials having numerical coeecients. We consider the question: when are a and b relatively prime? Since the coeecients of a and b are approximant, the question is the same as: when are two polynomials relatively prime, even after small perturbations of the coeecients? In this paper we provide a numeric parameter for determining that two polynomials are prime, even under s...

The King’s Problem [1] is the following: A physicist is trapped on an island ruled by a mean king who promises to set her free if she can give him the answer to the following puzzle. The physicist is asked to prepare a d-state quantum system in any state of her choosing and give it to the king, who measures one of several sets of mutually unbiased observables (this term will be defined below) o...

A long-standing difficulty for connectionism has been to implement compositionality, the idea of building a knowledge representation out of components such that the meaning arises from the meanings of the individual components and how they are combined. Here we show how a neural-learning algorithm, knowledge-based cascade-correlation (KBCC), creates a compositional representation of the prime-n...

Linear congruential generators (LCGs) remain the most popular method of pseudorandom number generation on digital computers. Ease of implementation has favored implementing LCGs with power-of-two moduli. However, prime modulus LCGs are superior in quality to power-of-two modulus LCGs, and the use of a Mersenne prime minimizes the computational cost of generation. When implemented for parallel c...

The tunnel number of a PL knot K is the minimum number of PL one-cells which must be attached in order that the regular neighborhood of the resulting complex has complement a handlebody [2]. It is easy to see that n-bridge knots have tunnel numbers (n 1) and torus knots have tunnel number one. More difficult is finding knots which have higher tunnel number. If a knot has tunnel number one, it w...