iv : m at h . L O / 9 81 20 14 v 1 2 D ec 1 99 8 Fuzzy L languages

ar X iv : m at h - ph / 9 81 20 09 v 1 1 3 D ec 1 99 8 Quantum dots A survey of rigorous results

X iv : m at h - ph / 9 81 20 19 v 1 1 8 D ec 1 99 8 ON ANGLES WHOSE SQUARED TRIGONOMETRIC FUNCTIONS ARE RATIONAL

We consider the rational linear relations between real numbers whose squared trigonometric functions have rational values, angles we call " geodetic ". We construct a convenient basis for the vector space over Q generated by these angles. Geodetic angles and rational linear combinations of geodetic angles appear naturally in Euclidean geometry; for illustration we apply our results to equidecom...

ar X iv : m at h - ph / 9 81 20 28 v 1 2 8 D ec 1 99 8 ( q , h ) – analogue of Newton ’ s binomial formula

In this letter, the (q, h)–analogue of Newton’s binomial formula is obtained in the (q, h)–deformed quantum plane which reduces for h = 0 to the q–analogue. For (q = 1, h = 0), this is just the usual one as it should be. Moreover, the h–analogue is recovered for q = 1. Some properties of the (q, h)–binomial coefficients are also given. This result will contribute to an introduction of the (q, h...

ar X iv : m at h - ph / 9 81 20 16 v 1 1 8 D ec 1 99 8 ISOMORPHISM OF HIERARCHICAL STRUCTURES

We consider hierarchical structures such as Fibonacci sequences and Penrose tilings, and examine the consequences of different choices for the definition of isomorphism. In particular we discuss the role such a choice plays with regard to matching rules for such structures.

ar X iv : m at h / 99 07 08 5 v 2 [ m at h . G R ] 1 7 D ec 1 99 9 LOOPS AND SEMIDIRECT PRODUCTS