Let A and B be nonempty subsets of a finite group G in which the order of the smallest nonzero subgroup is not smaller than d = |A| + |B| − 1. Then at least d different elements of G has a representation in the form ab, where a ∈ A and b ∈ B. This extends a classical theorem of Cauchy and Davenport to noncommutative groups. We also generalize Vosper’s inverse theorem in the same spirit, giving ...

In this paper the problem of the geodesic connectedness and convexity of incomplete Riemannian manifolds is analyzed. To this aim, a detailed study of the notion of convexity for the associated Cauchy boundary is carried out. In particular, under widely discussed hypotheses, we prove the convexity of open domains (whose boundaries may be nondifferentiable) of a complete Riemannian manifold. Var...

We introduce and study the notions of a strongly completable and of a strongly complete quasi-uniform space. A quasi-uniform space (X,U) is said to be strongly complete if every Cauchy filter (in the sense of Sieber and Pervin) clusters in the uniform space (X,U ∨ U−1). An interesting motivation for the study of this notion of completeness is the fact, proved here, that the quasi-uniformity ind...

This paper studies the uniqueness of bounded solution to a new Cauchy problem fractional nonlinear partial integro-differential equation based on multivariate Mittag–Leffler function as well Banach’s contractive principle in complete space. Applying Babenko’s approach, we convert with variable coefficients an implicit integral equation, which is powerful method studying solutions. Furthermore, ...

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.

We reduce the construction of a weak solution of the Cauchy problem for the Navier-Stokes system to the construction of a stochastic problem solution. Under suitable conditions we solve the stochastic problem and prove that simultaneously we obtain a weak (generalized) solution to the Cauchy problem for the Navier-Stokes system. RESUMEN Nosotros reducimos la construcción de una solución débil d...

We investigate the interior of regular axisymmetric and stationary black holes surrounded by matter and find that for non-vanishing angular momentum of the black hole the spacetime can always be extended regularly up to and including an inner Cauchy horizon. We provide an explicit relation for the regular metric at the inner Cauchy horizon in terms of that at the event horizon. As a consequence...

The construction of analogues of the Cauchy kernel is crucial for the solution of Riemann–Hilbert problems on compact Riemann surfaces. A formula for the Cauchy kernel can be given as an infinite sum over the elements of a Schottky group, and this sum is often used for the explicit evaluation of the kernel. In this paper a new formula for a quasi-automorphic analogue of the Cauchy kernel in ter...

We show that linear scalar waves are bounded and continuous up to the Cauchy horizon of Reissner–Nordström–de Sitter and Kerr–de Sitter spacetimes, and in fact decay exponentially fast to a constant along the Cauchy horizon. We obtain our results by modifying the spacetime beyond the Cauchy horizon in a suitable manner, which puts the wave equation into a framework in which a number of standard...

We examine the evidence for the widespread belief that heavy tail distributions enhance the search for minima on multimodal objective functions. We analyze isotropic and anisotropic heavy-tail Cauchy distributions and investigate the probability to sample a better solution, depending on the step length and the dimensionality of the search space. The probability decreases fast with increasing st...