We solve the initial value problem for the diffusion induced by dyadic fractional derivative s in \mathbb{R}^{+}. First we obtain the spectral analysis of the dyadic fractional derivative operator in terms of the Haar system, which unveils a structure for the underlying “heat kernel”. We show that this kernel admits an integrable and decreasing majorant that involves the dyadic distance. This allows us to provide an estimate of the maximal operator of the diffusion by the Hardy-Littlewood dyadic maximal operator. As a consequence we obtain the pointwise convergence to the initial data., Marcelo Actis, Hugo Aimar., and Obsahuje seznam literatury
Given a distribution $T$ on the sphere we define, in analogy to the work of Łojasiewicz, the value of $T$ at a point $\xi$ of the sphere and we show that if $T$ has the value $\tau$ at $\xi$, then the Fourier-Laplace series of $T$ at $\xi$ is Abel-summable to $\tau$., Francisco Javier González Vieli., and Obsahuje bibliografii
In this paper, we study the structure of polycyclic groups admitting an automorphism of order four on the basis of Neumann’s result, and prove that if α is an automorphism of order four of a polycyclic group G and the map φ: G → G defined by gφ = [g,α] is surjective, then G contains a characteristic subgroup H of finite index such that the second derived subgroup H″ is included in the centre of H and CH(α2) is abelian, both CG(α2) and G/[G, α2] are abelian-by-finite. These results extend recent and classical results in the literature., Tao Xu, Fang Zhou, Heguo Liu., and Obsahuje seznam literatury
We describe the relation between quasi-minuscule representations, polytopes and Weyl group orbits in Picard lattices of rational surfaces. As an application, to each quasi-minuscule representation we attach a class of rational surfaces, and realize such a representation as an associated vector bundle of a principal bundle over these surfaces. Moreover, any quasi-minuscule representation can be defined by rational curves, or their disjoint unions in a rational surface, satisfying certain natural numerical conditions., Jae-Hyouk Lee, Mang Xu, Jiajin Zhang., and Seznam literatury
We consider a single-species stochastic logistic model with the population's nonlinear diffusion between two patches. We prove the system is stochastically permanent and persistent in mean, and then we obtain sufficient conditions for stationary distribution and extinction. Finally, we illustrate our conclusions through numerical simulation., Li Zu, Daqing Jiang, Donal O'Regan., and Obsahuje bibliografii
We show that the Porous Medium Equation and the Fast Diffusion Equation, \dot u - \Delta {u^m} = f with m\in (0, \infty ), can be modeled as a gradient system in the Hilbert space H^{-1}(\Omega ), and we obtain existence and uniqueness of solutions in this framework. We deal with bounded and certain unbounded open sets \Omega \subset \mathbb{R}^{n} and do not require any boundary regularity. Moreover, the approach is used to discuss the asymptotic behaviour and order preservation of solutions., Samuel Littig, Jürgen Voigt., and Obsahuje seznam literatury
Let G be a finite group and k a field of characteristic p > 0. In this paper, we obtain several equivalent conditions to determine whether the principal block B0 of a finite p-solvable group G is p-radical, which means that B0 has the property that e0(kP)G is semisimple as a kG-module, where P is a Sylow p-subgroup of G, kP is the trivial kP-module, (kP)G is the induced module, and e0 is the block idempotent of B0. We also give the complete classification of a finite p-solvable group G which has not more than three simple B0-modules where B0 is p-radical., Xiaohan Hu, Jiwen Zeng., and Obsahuje seznam literatury
In this paper we analyse a definition of a product of Banach spaces that is naturally associated by duality with a space of operators that can be considered as a generalization of the notion of space of multiplication operators. This dual relation allows to understand several constructions coming from different fields of functional analysis that can be seen as instances of the abstract one when a particular product is considered. Some relevant examples and applications are shown, regarding pointwise products of Banach function spaces, spaces of integrable functions with respect to vector measures, spaces of operators, multipliers on Banach spaces of analytic functions and spaces of Lipschitz functions., Enrique A. Sánchez Pérez., and Obsahuje seznam literatury
In this paper, for the impulsive fractional integro-differential equations involving Caputo fractional derivative in Banach space, we investigate the existence and uniqueness of a pseudo almost periodic PC-mild solution. The working tools are based on the fixed point theorems, the fractional powers of operators and fractional calculus. Some known results are improved and generalized. Finally, existence and uniqueness of a pseudo almost periodic PC-mild solution of a two-dimensional impulsive fractional predator-prey system with diffusion are investigated., Zhinan Xia., and Obsahuje bibliografii
We discuss the representability almost everywhere (a.e.) in C of an irreducible algebraic function as the Cauchy transform of a signed measure supported on a finite number of compact semi-analytic curves and a finite number of isolated points. This brings us to the study of trajectories of the particular family of quadratic differentials A(z − a)(z − b)×(z − c)−2 dz2. More precisely, we give a necessary and sufficient condition on the complex numbers a and b for these quadratic differentials to have finite critical trajectories. We also discuss all possible configurations of critical graphs., Mohamed Jalel Atia, Faouzi Thabet., and Obsahuje seznam literatury