Non-Existence Results for Stable Solutions to Weighted Elliptic Systems Including Advection Terms

In this paper, we study a non-linear weighted Grushin system including advection terms. We prove some Liouville-type theorems for stable solutions of the system, based on the comparison property and the bootstrap iteration. Our results generalise and improve upon some previous works.

On the extremal solutions of superlinear Helmholtz problems

In this note, we deal with the Helmholtz equation −∆u+cu = λf(u) with Dirichlet boundary condition in a smooth bounded domain Ω of R n , n > 1. The nonlinearity is superlinear that is limt−→∞ f(t) t = ∞ and f is a positive, convexe and C 2 function defined on [0,∞). We establish existence of regular solutions for λ small enough and the bifurcation phenomena. We prove the existence of critical value λ ∗ such that the problem does not have solution for λ > λ∗ even in the weak sense. We also prove the existence of a type of stable solutions u ∗ called extremal solutions. We prove that for f(t) = e t , Ω = B1 and n ≤ 9, u ∗ is regular.

Modulo Periodic Poisson Stable Solutions of Quasilinear Differential Equations

In this paper, modulo periodic Poisson stable functions have been newly introduced. Quasilinear differential equations with modulo periodic Poisson stable coefficients are under investigation. The existence and uniqueness of asymptotically stable modulo periodic Poisson stable solutions have been proved. Numerical simulations, which illustrate the theoretical results are provided.

On the structure of the Levinson center for monotone non-autonomous dynamical systems with a first integral

In this paper we give a description of the structure of compact global attractor (Levinson center) for monotone Bohr/Levitan almost periodic dynamical system $x'=f(t,x)$ (*) with the strictly monotone first integral. It is shown that Levinson center of equation (*) consists of the Bohr/Levitan almost periodic (respectively, almost automorphic, recurrent or Poisson stable) solutions. We establish the main results in the framework of general non-autonomous (cocycle) dynamical systems. We also give some applications of theses results to different classes of differential/difference equations.

Application of Machine Learning for Lithology-on-Bit Prediction using Drilling Data in Real-Time

SUMMARY Researchers base their analysis on basic drilling parameters obtained during mud logging and demonstrate impressive results. However, due to limitations imposed by data quality often present during drilling, those solutions often tend to lose their stability and high levels of predictivity. In this work, the concept of hybrid modeling was introduced which allows to integrate the analytical correlations with algorithms of machine learning for obtaining stable solutions consistent from one data set to another.

The unbearable lightness of charged gravitini

We prove that charged gravitini cannot have parametrically small or vanishing Lagrangian mass in de Sitter vacua of extended supergravity while respecting the magnetic weak gravity conjecture. This places large classes of de Sitter solutions of gauged supergravity in the swampland, including all known stable solutions of the N=2 theory. We illustrate this result by analyzing a variety of de Sitter critical points of N=2 matter-coupled supergravity that also include new stable de Sitter solutions. Our results provide concrete evidence that (quasi) de Sitter with charged light gravitini should belong to the swampland, which also strongly resonates with the “festina lente” bound.

Universal Hardy–Sobolev inequalities on hypersurfaces of Euclidean space

In this paper, we study Hardy–Sobolev inequalities on hypersurfaces of [Formula: see text], all of them involving a mean curvature term and having universal constants independent of the hypersurface. We first consider the celebrated Sobolev inequality of Michael–Simon and Allard, in our codimension one framework. Using their ideas, but simplifying their presentations, we give a quick and easy-to-read proof of the inequality. Next, we establish two new Hardy inequalities on hypersurfaces. One of them originates from an application to the regularity theory of stable solutions to semilinear elliptic equations. The other one, which we prove by exploiting a “ground state” substitution, improves the Hardy inequality of Carron. With this same method, we also obtain an improved Hardy or Hardy–Poincaré inequality.