* Correspondence: [email protected]. cn Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University, Jiangsu Nanjing 210046, China Full list of author information is available at the end of the article Abstract The existence of at least three weak solutions is established for a class of quasilinear elliptic systems involving the p(x)-Laplace operator with Neumann boundary ...

This paper deals with the critical point of three-dimensional bootstrap percolation-like cellular automata. Some general sufficient or necessary conditions for pc = 0 are obtained. In the case of pc > 0, some explicit upper and lower bounds are provided in terms of the critical value of oriented site percolation.

In this work we study the critical behavior of a three-state (+1, −1, 0) opinion model with independence. Each agent has a probability q to act as independent, i.e., he/she can choose his/her opinion independently of the opinions of the other agents. On the other hand, with the complementary probability 1 − q the agent interacts with a randomly chosen individual through a kinetic exchange. Our ...

Theorem 1. Convergence to local optimum (a) The sequence of iterates Ut of Algorithm 4 satisfies: limt→∞ ||∇f(Ut)|| = 0. This means that the accumulation points of the sequence {Ut}t=1 are critical points of f . (b) Assume the critical points of f are isolated. Let U∗ be a critical point of f . Then U∗ is a local minimum of f if and only if it is asymptotically stable with regard to the sequenc...

in order to present a quantitative indicator for the onset of instability, in this paper, the critical points of a stratified gravitational flow on a slope are found and analyzed. these points are obtained by means of the solution of the two-dimensional navier-stokes equations via the standard arakawa-c finite-difference method. results show that in the marginal richardson numbers, the critical...

We investigate the density of critical factorizations of infinte sequences of words. The density of critical factorizations of a word is the ratio between the number of positions that permit a critical factorization, and the number of all positions of a word. We give a short proof of the Critical Factorization Theorem and show that the maximal number of noncritical positions of a word between t...

Here we say that a real polynomial is hyperbolic or Axiom A, if the real line is the union of a repelling hyperbolic set, the basin of hyperbolic attracting periodic points and the basin of infinity. We call a C1 endomorphism of the compact interval (or the circle) hyperbolic if it has finitely many hyperbolic attracting periodic points and the complement of the basin of attraction of these poi...