Simplicity and Commutative Bases of Derivations in Polynomial and Power Series Rings

The first part of the paper will describe a recent result of K. Retert in (\cite{Ret}) for $k[x_1,\ldots,x_n]$ and $k[[x_1,\ldots,x_n]]$. This result states that if $\mathfrak{D}$ is a set of commute $k$-derivations of $k[x,y]$ such that both $\partial_x \in \mathfrak{D}$ and the ring is $\mathfrak{D}$-simple, then there is $d \in \mathfrak{D}$ such that $k[x,y]$ is $\{\partial_x,d\}$-simple. As applications, we obtain relationships with known results of A. Nowicki on commutative bases of derivations

On Critical Point Equation of Compact Manifolds with Zero radial Weyl Curvature

Let $\mathcal{C}$ be the space of smooth metrics $g$ on a given compact manifold $M^{n}$ ($n\geq3$) with constant scalar curvature and unitary volume. The goal of this paper is to study the critical point of the total scalar curvature functional restricted to the space $\mathcal{C}$ (we shall refer to this critical point as CPE metrics) under assumption that $(M,g)$ has zero radial Weyl curvature. Among the results obtained, we emphasize that in 3-dimension we will be able to prove that a CPE metric with nonnegative sectional curvature must be isometric to a standard $3$-sphere. We will also prove that a $n$-dimensional, $4\leq n\leq10,$ CPE metric satisfying a $L^{n/2}$-pinching condition will be isometric to a standard sphere. In addition, we shall conclude that such critical metrics are isometrics to a standard sphere under fourth-order vanishing condition on the Weyl tensor.Comment: 20 page

Remarks on critical metrics of the scalar curvature and volume functionals on compact manifolds with boundary

We provide a general B\"ochner type formula which enables us to prove some rigidity results for $V$-static spaces. In particular, we show that an $n$-dimensional positive static triple with connected boundary and positive scalar curvature must be isometric to the standard hemisphere, provided that the metric has zero radial Weyl curvature and satisfies a suitable pinching condition. Moreover, we classify $V$-static spaces with non-negative sectional curvature.Comment: Fixed typo

Restrictions of harmonic functions and Dirichlet eigenfunctions of the Hata set to the interval

In this paper we study the harmonic functions and the Dirichlet eigenfunctions of the Hata set, and their restrictions to the interval $[0,1]$, its main edge. We prove that these restrictions of the harmonic functions are singular, \ie monotone and with zero derivatives almost everywhere, and provide numerical evidence that this is also the case for the eigenfunctions.Comment: 7 figure

On the classification of noncompact steady quasi-Einstein manifold with vanishing condition on the Weyl tensor

The aim of this paper is to study complete (noncompact) steady $m$-quasi-Einstein manifolds satisfying a fourth-order vanishing condition on the Weyl tensor. In this case, we are able to prove that a steady $m$-quasi-Einstein manifold ($m>1$) on a simply connected $n$-dimensional manifold $(M^{n},g)$, $(n\geq4),$ with nonnegative Ricci curvature and zero radial Weyl curvature must be a warped product with $(n-1)-$dimensional Einstein fiber, provided that $M$ has fourth order divergence-free Weyl tensor (i.e., ${\rm div}^{4}W=0$).Comment: 12 page

On the volume functional of compact manifolds with boundary with harmonic Weyl tensor

One of the main aims of this article is to give the complete classification of critical metrics of the volume functional on a compact manifold $M$ with boundary $\partial M$ and with harmonic Weyl tensor, which improves the corresponding classification for complete locally conformally flat case, due to Miao and Tam [18]. In particular, we prove that a critical metric with harmonic Weyl tensor on a simply connected compact manifold with boundary isometric to a standard sphere $\mathbb{S}^{n-1}$ must be isometric to a geodesic ball in a simply connected space form $\Bbb{R}^n,$ $\Bbb{H}^n$ and $\Bbb{S}^n.$ In order to achieve our goal, firstly we shall conclude the classification of such critical metrics under the Bach-flat assumption and then we will prove that both geometric conditions are indeed equivalent.Comment: 20 page

Volume functional of compact 44-manifolds with a prescribed boundary metric

We prove that a critical metric of the volume functional on a $4$-dimensional compact manifold with boundary satisfying a second-order vanishing condition on the Weyl tensor must be isometric to a geodesic ball in a simply connected space form $\mathbb{R}^{4}$, $\mathbb{H}^{4}$ or $\mathbb{S}^{4}.$ Moreover, we provide an integral curvature estimate involving the Yamabe constant for critical metrics of the volume functional, which allows us to get a rigidity result for such critical metrics.Comment: To appear in The Journal of Geometric Analysi

Geometric Inequalities for Critical Metrics of the Volume Functional

The goal of this article is to investigate the geometry of critical metrics of the volume functional on an $n$-dimensional compact manifold with (possibly disconnected) boundary. We establish sharp estimates to the mean curvature and area of the boundary components of critical metrics of the volume functional on a compact manifold. In addition, localized version estimates to the mean curvature and area of the boundary of critical metrics are also obtained.Comment: Fixed typo

A Model for Interactive Scores with Temporal Constraints and Conditional Branching

Interactive Scores (IS) are a formalism for the design and performance of interactive multimedia scenarios. IS provide temporal relations (TR), but they cannot represent conditional branching and TRs simultaneously. We propose an extension to Allombert et al.'s IS model by including a condition on the TRs. We found out that in order to have a coherent model in all possible scenarios, durations must be flexible; however, sometimes it is possible to have fixed durations. To show the relevance of our model, we modeled an existing multimedia installation called Mariona. In Mariona there is choice, random durations and loops. Whether we can represent all the TRs available in Allombert et al.'s model into ours, or we have to choose between a timed conditional branching model and a pure temporal model before writing a scenario, still remains as an open question.Comment: 14 pages, extended version of conference paper on Journ\'ees de INformatique Musicale 201

Weakly Einstein critical metrics of the volume functional on compact manifolds with boundary

The goal of this paper is to study weakly Einstein critical metrics of the volume functional on a compact manifold $M$ with smooth boundary $\partial M$. Here, we will give the complete classification for an $n$-dimensional, $n=3$ or $4,$ weakly Einstein critical metric of the volume functional with nonnegative scalar curvature. Moreover, in the higher dimensional case ($n\geq5$), we will established a similar result for weakly Einstein critical metric under a suitable constraint on the Weyl tensor.Comment: 11 page