In his report at ICM,Smale posed a problem whether the inequality α(t,ψd)≤1 holds for allt∈(0,1),where .This paper gives a negative answer.In addition,elementary proofs are given for the theorems on the tractability of random algorithm and on the average area of approximate zeros.Meanwhile,discussion about the algorithm of the global Newton's method is devoted in this paper.
In this paper,we study the uniformly asymptotic stability of nonautonomous retarded difference-differential equations by constructing Lyapunov functionals.We conclude that the retarded difference-differential equation is uniformly asymptotically stable under the strong diagonal dominance.
Let f:(X,A)→(X,A) be an extension of a given map ψ:A→A,where (X,A) is a pair of compact polyhedra.We shall introduce a special Nielsen number,SN(f|ψ),which is a lower bound for the number of fixed points on X-A for all extensions in the homotopy class of f.It is shown that for many space pairs this lower bound is the best possible one,and that it can be realized without the by-passing condition.
We prove in this paper the C∞ regularity for a "very strict" local minimum of class Clocρ,ρ>3,of functionals with genuine degenerate quasiconvex integrand depending on a vector-valued function u.Such a minimum satisfies the condition: for all x∈Ω,there exists a neighbourhood K(x) of x in Ω and C1(x)>0,C2(x)>0,1≥ε(x)>0,such that for all real Ψ∈c0∞ (K).
In this paper we discuss the concept ‘generalized exponential dichotomy' and give the existence of Ck invariant manifolds for abstract nonautonomous differential equations in Banach or Hilbert spaces.Also we give a classification of invariant manifolds and an estimate of the locality radius of invariant manifolds.
Let A,B be unital C*-algebras,DA1 the set of all completely positive maps ψ from A to Mn(C),with Tr ψI)≤1(n≥3).If Ψ is an α-invariant affine homeomorphism between DA1 and DB1 with Ψ (0)=0,then A is *-isomorphic to B.Obtained results can be viewed as non-commutative Kadison-Shultz theorems.
This paper provides two kinds of forbidden configurations for the rectilinear O-embeddability of triangle free planar graphs.Meanwhile,the characterizations of the O-embeddability for outerplanar graphs and Halin graphs are found.
In this paper we apply the Malliavin calculus for two-parameter Wiener functionals to show that the solutions of stochastic differential equations in plane have a smooth density under the restricted Hörmander's condition.This answers a question mentioned by Nualart and Sanz in [3].