A crisis in chaotic dynamical systems is characterized by the conversion of a nonattracting, Cantorset-like chaotic saddle into a chaotic attractor. The gaps in between various pieces of the chaotic saddle are densely filled after the crisis. We give a quantitative scaling theory for the growth of the topological entropy for a major class of crises, the interior crisis. The theory is confirmed ...

Generalizing the results of Thue (for n = 2) and of Klepinin and Sukhanov (for n = 3), we prove that for all n ≥ 2, the critical exponent of the Arshon word of order n is given by (3n− 2)/(2n− 2), and this exponent is attained at position 1.

A set of positive integers is primitive (or 1-primitive) if no member divides another. Erd\H{o}s proved in 1935 that the weighted sum $\sum1/(n \log n)$ for $n$ ranging over a $A$ universally bounded all choices $A$. In 1988 he asked this universal bound attained by prime numbers. One source difficulty conjecture $\sum n^{-\lambda}$ maximized primes and only $\lambda$ at least critical exponent...