Differentiable spaces that are subcartesian

We show that the differential structure of the orbit space of a proper action of a Lie group on a smooth manifold is weakly reflexive. This implies that the orbit space is a differentiable space in the sense of Smith, which ensures that the orbit space has an exterior algebra of differential forms, that satisfies Smith’s version of de Rham’s theorem. Because the orbit space is a locally closed subcartesian space, it has vector fields and their flows.     

Strain effects on multiferroic BiFeO3 films

The field of multiferroics has experienced a rapid progress resulting in the discovery of many new physical phenomena. BiFeO₃ (BFO) compound, which is one of the few room-temperature single-phase multiferroics, has contributed subsequently to this progress. As a result, significant review articles have been devoted specifically to this famous system. This chapter is dedicated to the strain effects on the structure stability and property changes of BFO thin films. It is a short and non-exhaustive topical overview that may be seen as an invitation for interested readers to go beyond. There is a very active and prolific research in this field and we apologize to the authors whose relevant work is not cited here. After a short introduction, we will thus review the effect of strain on BFO films by describing the consequences on the structure and the phase transitions as well as on polar, magnetic and magnetoelectric properties.     

Pattern universes

In this essay we explore analogies between macroscopic patterns, which result from a sequence of phase transitions/instabilities starting from a homogeneous state, and similar phenomena in cosmology, where a sequence of phase transitions in the early universe is believed to have separated the fundamental forces from each other, and also shaped the structure and distribution of matter in the universe. We discuss three distinct aspects of this analogy: (i) Defects and topological charges in macroscopic patterns are analogous to spins and charges of quarks and leptons; (ii) Defects in generic 3+1 stripe patterns carry an energy density that accounts for phenomena that are currently attributed to dark matter; (iii) Space-time patterns of interacting nonlinear waves display behaviors reminiscent of quantum phenomena including inflation, entanglement and dark energy.     

Convergence of solutions for the fractional Cahn–Hilliard system

This paper deals with the Cauchy–Dirichlet problem for the fractional Cahn–Hilliard equation. The main results consist of global (in time) existence of weak solutions, characterization of parabolic smoothing effects (implying under proper condition eventual boundedness of trajectories), and convergence of each solution to a (single) equilibrium. In particular, to prove the convergence result, a variant of the so-called ?ojasiewicz–Simon inequality is provided for the fractional Dirichlet Laplacian and (possibly) non-analytic (but $C^1$) nonlinearities.     

A short historical account of period doublings in the pre-renormalization era

I will shortly review the history of experimental and theoretical findings on period doubling until the discovery of the quantitative universal properties of the infinite period-doubling cascade.     

Heisenberg群上Laplace算子的离散谱

本文证明了Heisenberg群上Laplace算子的Dirichlet特征值的存在性，给出了特征值的估计     

弱(L_1,L_2)-BLD映射的正则性

本文首先将文［１］中的ＢＬＤ映射推广为弱（Ｌ１，Ｌ２）－ＢＬＤ映射，并证明了如下正则性结果：存在两个可积指数 Ｐ１＝Ｐ１（ｎ，Ｌ１，Ｌ２）＜ｎ＜ｑ１＝ｑ１（ｎ，Ｌ１，Ｌ２），使得对任意弱（Ｌ１，Ｌ２）－ＢＬＤ映射ｆ∈（Ω，Ｒｎ），都有ｆ∈（Ω，Ｒｎ），即ｆ为（Ｌ１，Ｌ２）－ＢＬＤ映射．     

Fractions continues de matrices,convergence et applications

The aim of this article is to give some property of continued fraction with matrices arguments, about their convergence and others applications. At the end of this work, we present a resolution of the Algebraic Riccati Equation by giving an explicit continued fraction development of its solution. This revised version was published online in August 2006 with corrections to the Cover Date.