Lens opacities, or cataract(s), may be inherited as a classic Mendelian disorder usually with early-onset or, more commonly, acquired with age as a multi-factorial or complex trait. Many genetic forms of cataract have been described in mice and other animal models. Considerable progress has been made in mapping and identifying the genes and mutations responsible for inherited forms of cataract,...

Since the 1990s chaotic cat maps are widely used in data encryption, for their very complicated dynamics within a simple model and desired characteristics related to requirements of cryptography. The number of cat map parameters and the map period length after discretization are two major concerns in many applications for security reasons. In this paper, we propose a new family of 36 distinctiv...

We introduce an alternative approach to the third order helicity of a volume preserving vector field X. The proposed approach exploits correspondence between the Milnor ¯ µ 123-invariant for 3-component links and the homotopy invariants of maps to configuration spaces, and we provide a simple geometric proof of this fact in the case of borromean links. Based on these connections we develop a fo...

The notion of A-graded algebras was introduced by V.I. Arnold, who made a complete classification for the case of 3 generators [l]. Unfortunately he did not present proofs of his statements. When our group together with him started further investigations of A-graded algebras, he suggested to me first to construct and publish the proof of his classification in the way that we need for our work.

We answer a question of V.I. Arnold concerning the growth rate of the number of Morse functions on the two sphere.

We consider several Hamiltonian systems for which the existence of Arnold's mechanism for diiusion (whiskered tori, transition ladder, etc.) has been proven. By means of Mather theory we show that the diiusion time may be bounded by a power of the homoclinic splitting.