Search results for "singular"

A singular (p,q)-equation with convection and a locally defined perturbation

2021

Abstract We consider a parametric Dirichlet problem driven by the ( p , q ) -Laplacian and a reaction which is gradient dependent (convection) and the competing effects of two more terms, one a parametric singular term and a locally defined perturbation. We show that for all small values of the parameter the problem has a positive smooth solution.

p −1-Linear Maps in Algebra and Geometry

2012

At least since Habousch’s proof of Kempf’s vanishing theorem, Frobenius splitting techniques have played a crucial role in geometric representation theory and algebraic geometry over a field of positive characteristic. In this article we survey some recent developments which grew out of the confluence of Frobenius splitting techniques and tight closure theory and which provide a framework for higher dimension geometry in positive characteristic. We focus on local properties, i.e. singularities, test ideals, and local cohomology on the one hand and global geometric applicatioms to vanishing theorems and lifting of sections on the other.

Adjacency matrices of random digraphs: singularity and anti-concentration

2017

Let ${\mathcal D}_{n,d}$ be the set of all $d$-regular directed graphs on $n$ vertices. Let $G$ be a graph chosen uniformly at random from ${\mathcal D}_{n,d}$ and $M$ be its adjacency matrix. We show that $M$ is invertible with probability at least $1-C\ln^{3} d/\sqrt{d}$ for $C\leq d\leq cn/\ln^2 n$, where $c, C$ are positive absolute constants. To this end, we establish a few properties of $d$-regular directed graphs. One of them, a Littlewood-Offord type anti-concentration property, is of independent interest. Let $J$ be a subset of vertices of $G$ with $|J|\approx n/d$. Let $\delta_i$ be the indicator of the event that the vertex $i$ is connected to $J$ and define $\delta = (\delta_1, …

Settlement dynamics and recruitment responses of Mediterranean gorgonians larvae to different crustose coralline algae species

2020

International audience; Sessile marine species such as Anthozoans act as ecosystem engineers due to their three-dimensional structure. Gorgonians, in particular, can form dense underwater forests that give shelter to other species increasing local biodiversity. In the last decades, several Mediterranean gorgonian populations have been affected by natural and anthropogenic impacts which drastically reduced their size. However, some species showed unexpected resilience, mainly due to the supply of new individuals. To understand the mechanisms underlying recovery processes, studies on the first life history stages (i.e. larval dispersal, settlement and recruitment) are needed. In tropical cora…

A singular elliptic equation and a related functional

2021

We study a class of Dirichlet boundary value problems whose prototype is [see formula in PDF] where 0 < p < 1 and f belongs to a suitable Lebesgue space. The main features of this problem are the presence of a singular term |u|p−2u and a datum f which possibly changes its sign. We introduce a notion of solution in this singular setting and we prove an existence result for such a solution. The motivation of our notion of solution to problem above is due to a minimization problem for a non–differentiable functional on [see formula in PDF] whose formal Euler–Lagrange equation is an equation of that type. For nonnegative solutions a uniqueness result is obtained.

Do Randomized Algorithms Improve the Efficiency of Minimal Learning Machine?

2020

Minimal Learning Machine (MLM) is a recently popularized supervised learning method, which is composed of distance-regression and multilateration steps. The computational complexity of MLM is dominated by the solution of an ordinary least-squares problem. Several different solvers can be applied to the resulting linear problem. In this paper, a thorough comparison of possible and recently proposed, especially randomized, algorithms is carried out for this problem with a representative set of regression datasets. In addition, we compare MLM with shallow and deep feedforward neural network models and study the effects of the number of observations and the number of features with a special dat…

Jaume Vicent: un escultor singular en la encrucijada de dos siglos

2019

Sustainable growth and environmental catastrophes

2017

Abstract In the standard AK growth model we introduce the threat of an ecological catastrophe and study the consequences for the economic variables in the long-run. We extend the basic framework by considering two environmental externalities: the first one is local and gives account of the marginal damage from emissions flow; the second one is aggregate, or global, and relates to the extreme damage which may happen if the accumulated stock of pollutants is on the threshold of a worldwide catastrophe. In this context dominated by market failures, we focus on the socially optimal solution and the search of conditions for sustainability. We identify the efficient balanced growth path, which ma…

On the continuous and discontinuous maximal operators

2018

Abstract In the first part of this paper we study the regularity properties of a wide class of maximal operators. These results are used to show that the spherical maximal operator is continuous W 1 , p ( R n ) ↦ W 1 , p ( R n ) , when p > n n − 1 . Other given applications include fractional maximal operators and maximal singular integrals. On the other hand, we show that the restricted Hardy–Littlewood maximal operator M λ , where the supremum is taken over the cubes with radii greater than λ > 0 , is bounded from L p ( R n ) to W 1 , p ( R n ) but discontinuous.

Dynamical Features of the MAP Kinase Cascade

2017

The MAP kinase cascade is an important signal transduction system in molecular biology for which a lot of mathematical modelling has been done. This paper surveys what has been proved mathematically about the qualitative properties of solutions of the ordinary differential equations arising as models for this biological system. It focuses, in particular, on the issues of multistability and the existence of sustained oscillations. It also gives a concise introduction to the mathematical techniques used in this context, bifurcation theory and geometric singular perturbation theory, as they relate to these specific examples. In addition further directions are presented in which the application…