Brazilian Society of Computational and Applied Mathematics

We show that one can construct a germ of a local solution of systems of semi-linear second order partial differential equations (PDEs) from any given n -jet that satisfies the system up to the ( n − 2 ) -jet level. We treat in more details the case of harmonic map-germs to the unit sphere. Our results allow, for instance, to determine which local singularities a harmonic map to the sphere can have.

ABSTRACT We investigate some chemostat models incorporating wall growth, competition, random fluctuations on the dilution rate, and different consumption functions (Monod and Haldane). We analyze the asymptotic behavior of the solutions of the corresponding random differential systems to establish conditions on the model parameters under which the microbes persist in the gut or disappear. Moreover, several numerical simulations are presented to support the theoretical results and illustrate their biological interpretation.

Abstract In this paper we prove the gradient structure of solutions for a nonautonomous cascade system defined on Banach spaces, where the x –variable evolves independently via $$\dot{x} = Ax+f(t,x)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mover> <mml:mi>x</mml:mi> <mml:mo>˙</mml:mo> </mml:mover> <mml:mo>=</mml:mo> <mml:mi>A</mml:mi> <mml:mi>x</mml:mi> <mml:mo>+</mml:mo> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> and influences the y –variable through $$\dot{y} = By+g(x,y)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mover> <mml:mi>y</mml:mi> <mml:mo>˙</mml:mo> </mml:mover> <mml:mo>=</mml:mo> <mml:mi>B</mml:mi> <mml:mi>y</mml:mi> <mml:mo>+</mml:mo> <mml:mi>g</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . By first analyzing the long–time dynamics of the nonautonomous x –equation and then examining the resulting y –dynamics for each asymptotic state of x , we provide a complete description of the system’s gradient structure in two levels: a more abstract and general, with less hypotheses on f , and a deeper level of description, when the term f ( t , x ) is asymptotically autonomous. Finally, we present a description when the term f ( t , x ) is a small nonautonomous perturbation of an autonomous term.

Abstract We consider the empirical neighborhood distribution of marked sparse Erdős-Rényi random graphs, obtained by decorating edges and vertices of a sparse Erdős-Rényi random graph with i.i.d. random elements taking values on Polish spaces. We prove that the empirical neighborhood distribution of this model satisfies a large deviation principle in the framework of local weak convergence. We rely on the concept of BC-entropy introduced by Delgosha and Anantharam (2019) which is inspired on the previous work by Bordenave and Caputo (2015). Our main technical contribution is an approximation result that allows one to pass from graph with marks in discrete spaces to marks in general Polish spaces. As an application of the results developed here, we prove a large deviation principle for interacting diffusions driven by gradient evolution and defined on top of sparse Erdős-Rényi random graphs. In particular, our results apply for the stochastic Kuramoto model. We obtain analogous results for the sparse uniform random graph with given number of edges.

Let \(E\) be a Banach space (or a Banach lattice), let \(\tau\) be a vector topology on \(E\) and let \({\bf x}\) be a sequence (or a positive sequence) in \(E\) not converging to zero with respect to \(\tau\). We show how to construct infinite dimensional Banach spaces (or Banach lattices) consisting, up to the origin, of sequences in \(E\) not converging to zero with respect to \(\tau\) and containing a subsequence of \({\bf x}\). Plenty of applications to Banach space theory and to Banach lattice theory are provided.