In 1966, Michael [11] introduced the concept of compact-covering maps. Since many important kinds of maps are compact-covering, such as closed maps on paracompact spaces, much work has been done to seek the characterizations of metric spaces under various compact-covering maps, for example, compact-covering (open) s-maps, pseudosequence-covering (quotient) s-maps, sequence-covering (quotient) s...

We prove that the obstacle problem for a non-uniformly parabolic operator of Kolmogorov type, with Cauchy (or Cauchy-Dirichlet) boundary conditions, has a unique strong solution u. We also show that u is a solution in the viscosity sense.

Ateya and Madhagi (2011) introduced a multivariate form of truncated generalized Cauchy distribution (TGCD), which introduced by Ateya and Al-Hussaini (2007). The multivariate version of (TGCD) is denoted by (MVTGCD). Among the features of this form are that subvectors and conditional subvectors of random vectors, distributed according to this distribution, have the same form of distribution ...

In this paper, we study the concepts of Wijsman statistical convergence, Hausdorff statistical convergence and  Wijsman statistical Cauchy double sequences of sets and investigate the relationship between them.

This paper is devoted to the study of solutions with critical regularity for two-dimensional Muskat equation. We prove that Cauchy problem well-posed on endpoint Sobolev space $$L^2$$ functions three-half derivative in . result optimal respect scaling One well-known difficulty one cannot define a flow map such lifespan bounded from below subsets this space. To overcome this, we estimate norm wh...

In this paper, we consider the Cauchy problem for generalized Camassa–Holm equation that containing, as its members, three integrable equations: equation, Degasperis–Procesi and Novikov equation. We present a new unified method to prove sharp ill-posedness in $$B^s_{p,\infty }$$ with $$s>\max \{1+1/p, 3/2\}$$ $$1\le p\le \infty $$ sense solution map starting from $$u_0$$ is discontinuous at $$t...

The Cauchy problem for a quasilinear system of hyperbolic-parabolic equations is addressed with the method linearization and fixed point. Coupling between hyperbolic parabolic variables allowed in we do not assume Friedrich's symmetrizability system. This coupling results linear energy estimates that prevent application Banach's contraction principle. A metric point theorem developed order to c...

The lower bounds for the explosion and implosion breakdown points of the simultaneous Cauchy M-estimator (Cauchy MLE) of the regression and scale parameters are derived. For appropriate tuning constants, the breakdown point attains the maximum possible value.