Paracompact-type mappings

Recently a new direction of uniform topology called the uniform topology of uniformly continuous mappings has begun to develop intensively. This direction is devoted, first of all, to the extension to uniformly continuous mappings of the basic concepts and statements concerning uniform spaces. In this case a uniform space is understood as the simplest uniformly continuous mapping of this uniform space into a one-point space. The investigations carried out have revealed large uniform analogs of continuous mappings and made it possible to transfer to uniformly continuous mappings many of the main statements of the uniform topology of spaces. The method of transferring results from spaces to mappings makes it possible to generalize many results. Therefore, the problem of extending some concepts and statements concerning uniform spaces to uniformly continuous mappings is urgent. In this article, we introduce and study uniformly R-paracompact, strongly uniformly R-paracompact, and uniformly R-superparacompact mappings. In particular, we solve the problem of preserving R-paracompact (respectively, strongly uniformly R-paracompact, uniformly R-superparacompact) spaces towards the preimage under uniformly R-paracompact (respectively, strongly uniformly R-paracompact, uniformly R-superparacompact) mappings.

The Navier–Stokes–Vlasov–Fokker–Planck System as a Scaling Limit of Particles in a Fluid

Convergence of a system of particles, interacting with a fluid, to Navier–Stokes–Vlasov–Fokker–Planck system is studied. The interaction between particles and fluid is described by Stokes drag force. The empirical measure of particles is proved to converge to the Vlasov–Fokker–Planck component of the system and the velocity of the fluid coupled with the particles converges in the uniform topology to the the Navier–Stokes component. A new uniqueness result for the PDE system is added.

Generic homeomorphisms have full metric mean dimension

We prove that for $C^0$ -generic homeomorphisms, acting on a compact smooth boundaryless manifold with dimension greater than one, the upper metric mean dimension with respect to the smooth metric coincides with the dimension of the manifold. As an application, we show that the upper box dimension of the set of periodic points of a $C^0$ -generic homeomorphism is equal to the dimension of the manifold. In the case of continuous interval maps, we prove that each level set for the metric mean dimension with respect to the Euclidean distance is $C^0$ -dense in the space of continuous endomorphisms of $[0,1]$ with the uniform topology. Moreover, the maximum value is attained at a $C^0$ -generic subset of continuous interval maps and a dense subset of metrics topologically equivalent to the Euclidean distance.

On B-ideals in a Topological B-Algebra

This paper presents the characterizations and properties of B-ideals in a topological B-algebra and introduces the uniform topology on a B-algebra in terms of its B- ideals. Moreover, this paper shows that a uniform B-topological space is a topological B-algebra.

New topology in residuated lattices

In this paper, by using the notion of upsets in residuated lattices and defining the operator Da(X), for an upset X of a residuated lattice L we construct a new topology denoted by τa and (L, τa) becomes a topological space. We obtain some of the topological aspects of these structures such as connectivity and compactness. We study the properties of upsets in residuated lattices and we establish the relationship between them and filters. O. Zahiri and R. A. Borzooei studied upsets in the case of BL-algebras, their results become particular cases of our theory, many of them work in residuated lattices and for that we offer complete proofs. Moreover, we investigate some properties of the quotient topology on residuated lattices and some classes of semitopological residuated lattices. We give the relationship between two types of quotient topologies τa/F and $\begin{array}{} \displaystyle \mathop {{\tau _a}}\limits^ - \end{array}$. Finally, we study the uniform topology $\begin{array}{} \displaystyle {\tau _{\bar \Lambda }} \end{array}$ and we obtain some conditions under which $\begin{array}{} \displaystyle (L/J,{\tau _{\bar \Lambda }}) \end{array}$ is a Hausdorff space, a discrete space or a regular space ralative to the uniform topology. We discuss briefly the applications of our results on classes of residuated lattices such as divisible residuated lattices, MV-algebras and involutive residuated lattices and we find that any of this subclasses of residuated lattices with respect to these topologies form semitopological algebras.

An optimal transportation approach to the decay of correlations for non-uniformly expanding maps

We consider the transfer operators of non-uniformly expanding maps for potentials of various regularity, and show that a specific property of potentials (‘flatness’) implies a Ruelle–Perron–Frobenius theorem and a decay of the transfer operator of the same speed as that entailed by the constant potential. The method relies neither on Markov partitions nor on inducing, but on functional analysis and duality, through the simplest principles of optimal transportation. As an application, we notably show that for any map of the circle which is expanding outside an arbitrarily flat neutral point, the set of Hölder potentials exhibiting a spectral gap is dense in the uniform topology. The method applies in a variety of situations, including Pomeau–Manneville maps with regular enough potentials, or uniformly expanding maps of low regularity with their natural potential; we also recover in a united fashion variants of several previous results.

Uniform topology on EQ-algebras

In this paper, we use filters of an EQ-algebra E to induce a uniform structure (E, 𝓚), and then the part 𝓚 induce a uniform topology 𝒯 in E. We prove that the pair (E, 𝒯) is a topological EQ-algebra, and some properties of (E, 𝒯) are investigated. In particular, we show that (E, 𝒯) is a first-countable, zero-dimensional, disconnected and completely regular space. Finally, by using convergence of nets, the convergence of topological EQ-algebras is obtained.