Abstract We provide an elementary proof of the Brouwer’s fixed point theorem based solely on introductory topological concepts such as compactness and connectedness both covered in a first undergraduate course point-set topology.

In this paper, we establish a weighted version of the well-known Fréchet–Kolmogorov theorem, which holds for weights beyond \(A_\infty \). This theory extends previous known unsatisfactory results in terms relaxing index to natural range. As applications, obtain compactness commutators multilinear vector-valued Calderón–Zygmund type operators, including Littlewood–Paley operators. It is worthy ...

According to a result of one of the authors [7] there are at most a countable number of inequivalent differentiable actions of a compact Lie group on a compact differentiable manifold. The results above show that both the compactness of the manifold and the differentiability of the action are necessary assumptions. In the course of our proof we also prove the following theorem, which is an elem...

In this paper, locally Lipschitz functions acting between infinite dimensional normed spaces are considered. When the range is a dual space and satisfies the Radon–Nikod´ym property, Clarke's generalized Jacobian will be extended to this setting. Characterization and fundamental properties of the extended generalized Jacobian are established including the nonemptiness, the β-compactness, the β-...

We present a sufficient condition for weak continuity in the method of compensated compactness. The condition links weak continuity to the structure of the wave cone and the characteristic set for polynomials of degree greater than two. The condition applies to all the classical examples of weakly continuous functions and generalizes the Quadratic Theorem and the Wedge Product Theorem. In fact,...

This paper addresses a number of fundamental problems in logic and the philosophy of mathematics by considering some more technical problems in model theory and set theory. The interplay between syntax and semantics is usually considered the hallmark of model theory. At first sight, Shelah’s notion of abstract elementary class shatters that icon. As in the beginnings of the modern theory of str...

The stochastic 2D primitive equations with anisotropic viscosity are studied in this paper. existence of the martingale solutions and pathwise uniqueness obtained. proof is based on estimates, compactness method, tightness criteria Jakubowski version Skorokhod Theorem for nonmetric spaces.