1 . Introduction and statement of results Let (3(n) = Y_, In p and B (n) = y_p . j jn ap denote the sum of distinct prime divisors of n and the sum of all prime divisors of n respectively . Both (3(n) and B(n) are additive functions which are in a certain sense large (the average order of B(n) is Tr'n/(6log n), [1]) . For a fixed integer m the number of solutions of B(n) = m, is the number of p...

1. Under suitable smoothness assumptions, quasiperiodicity, and hence the absence of any kind of hyperbolicity, is non-negligible in a measure-theoretical sense [Kol] (under suitable smoothness assumptions). 2. In low regularity (C), failure of non-uniform hyperbolicity (here nonuniform hyperbolicity can be understood as positivity of the metric entropy) is a fairly robust phenomenon in the top...

In the present paper, the Sawada–Kotera equation is considered by Lie symmetry analysis. All of the geometric vector fields to the Sawada–Kotera equation are obtained, and then the symmetry reductions and exact solutions of the Sawada–Kotera equation are investigated. Our results show that symmetry analysis is a very efficient and powerful technique in finding the solution of the proposed equat...

2. Theorem. Let £ be a reflexive B-space and let (S, 2, m) be a a-finite measure space. Let there be defined on S a strongly measurable function Ts with values in the B-space B(H) of bounded linear operators on H. Suppose that || 7\|| ^=1 for all sES. Let h be a measure-preserving transformation (m.p.t.) in (S, 2, m). Then for each XELi(S, £) there is an XELi(S, X) such that limn<„ «_1E"=i T*Th...

We consider a smooth bracket generating control-affine system in Rd and show that any orientation preserving diffeomorphism of Rd can be approximated, in the very strong sense, by a diffeomorphism included in the flow generated by a time-varying feedback control which is polynomial with respect to the state variables and trigonometricpolynomial with respect to the time variable .

We consider the effect of noise on the dynamics generated by volume-preserving maps on a d-dimensional torus. The quantity we use to measure the irreversibility of the dynamics is the dissipation time. We focus on the asymptotic behaviour of this time in the limit of small noise. We derive universal lower and upper bounds for the dissipation time in terms of various properties of the map and it...

nonstandard finite difference schemes for the black-scholes partial differential equation preserving the positivity property are proposed. computationally simple schemes are derived by using a nonlocal approximation in the reaction term of the black-scholes equation. unlike the standard methods, the solutions of new proposed schemes are positive and free of the spurious oscillations.

Unifying approaches by amongst others Archimedes, Kepler, Goldberg, Caspar and Klug, Coxeter, Conway, extending on a previous formalization of the concept local symmetry preserving (lsp) operations, we introduce formal definition operations plane graphs that preserve orientation-preserving symmetries, but not necessarily orientation-reversing symmetries. This include, e.g., chiral Goldberg Conw...