We give a general framework incorporating the most important multifractal spectra using the thermodynamic approach to establish the multifractal formalism for weighted topological entropy.
We investigate the ideal generated by Jacobian determinants. We generalize Freudenburg’s Lemma to the case of r polynomials in n variables over an arbitrary unique factorization domain. This lemma is useful in reformulations of the Jacobian Conjecture.
Let 1 < p < infinity be a fixed exponent. Suppose that (f(n))(n >= 0), (g(n))(n >= 0) are real-valued martingales satisfying f(0) equivalent to x, g(0 )equivalent to( )y, parallel to(f(n))(n >= 0)parallel to(p )= F and g(n)-g(n-1 )= v(n)(f(n)-f(n-1)), n = 1, 2, ... , for some predictable sequence (v(n))(n >= 1) taking values in [-1, 1]. The purpose of this paper is to determine the optimal (i.e. the smallest) constant B= B-1,B-p(x, y, F) for which parallel to(g(n))(n >= 0)parallel to(1) <= B-1,B-p(x, y, F).
Suppose that G is a finite solvable group and V is a finite, faithful and completely reducible G-module. Let H be a nilpotent subgroup of G. Then there exists v is an element of V such that |CH(v)| <= (|H|/p)1/p, where CH(v) is the centralizer of v in H and p is the smallest prime divisor of |H|.
Let f be an arithmetic function satisfying some simple conditions. The aim of this paper is to establish some asymptotic estimates for the quantities X psi f(x) := n <= x fx]) X Lambda(n)f , pi f(x) := n p <= x (fx]) f p for x -> infinity, where Lambda(n) is the von Mangoldt function and [t] is the integer part of t is an element of R. These generalise or sharpen some recent results of Saito-Suzuki-Takeda-Yoshida. As an application, we show that X 1 similar to x ->infinity p <= x, [x/p] is prime (X p 1 p(p + 1) x log x
