作者：Ford, K
摘要：An old conjecture of Sierpinski asserts that for every integer k greater than or equal to 2, there is a number m for which the equation phi(x) = m has exactly k solutions. Here phi is Euler's totient function. In 1961, Schinzel deduced this conjecture from his Hypothesis H. The purpose of this paper is to present an unconditional proof of Sierpinski's conjecture. The proof uses many results from sieve theory, in particular the famous theorem of Chen.
作者：Lyubich, M
摘要：We prove the Feigenbaum-Coullet-Tresser conjecture on the hyperbolicity of the renormalization transformation of bounded type. This gives the first computer-free proof of the original Feigenbaum observation of the universal parameter scaling laws. We use the Hyperbolicity Theorem to prove Milnor's conjectures on self-similarity and hairiness of the Mandelbrot set near the corresponding parameter values. We also conclude that the set of real infinitely renormalizable quadratics of type bounded ...
作者：Trudinger, NS; Wang, XJ
摘要：In our previous paper on this topic, we introduced the notion of k-Hessian measure associated with a continuous k-convex function in a domain Omega in Euclidean n-space, k = 1,, n, and proved a weak continuity result with respect to local uniform convergence. In this paper, we consider L-convex functions, not necessarily continuous, and prove the weak continuity of the associated k-Hessian measure with respect to convergence in measure. The proof depends upon local integral estimates for the g...
作者：Hilsum, M
摘要：We construct analytically the signature operator for a new family of topological manifolds. This family contains the quasi-conformal manifolds and the topological manifolds modeled on germs of homeomorphisms of R-n possessing a derivative which is in L-p, with p > 1/2n(n + 1). We obtain an unbounded Fredholm module which defines a class in the K-homology of the manifold, the Chern character of which is the Hirzebruch polynomial in the Pontrjagin classes of the manifold. This generalizes previo...
作者：Safonov, MV; Yuan, Y
摘要：We prove the doubling property of L-caloric measure corresponding to the second order parabolic equation in the whole space and in Lipschitz domains. For parabolic equations in the divergence form, a weaker form of the doubling property follows easily from a recent result, the backward Harnack inequality, and known estimates of Green's function. Our method works for both the divergence and nondivergence cases. Moreover, the backward Harnack inequality and estimates of Green's function are not ...
