7,635 research outputs found
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 and . This result
states that if is a set of commute -derivations of
such that both and the ring is
-simple, then there is such that is
-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 be the space of smooth metrics on a given compact
manifold () 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 (we shall refer to
this critical point as CPE metrics) under assumption that 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 -sphere. We will also
prove that a -dimensional, CPE metric satisfying a
-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 -static spaces. In particular, we show that an
-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 -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 ,
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
-quasi-Einstein manifolds satisfying a fourth-order vanishing condition on
the Weyl tensor. In this case, we are able to prove that a steady
-quasi-Einstein manifold () on a simply connected -dimensional
manifold , with nonnegative Ricci curvature and zero
radial Weyl curvature must be a warped product with dimensional
Einstein fiber, provided that has fourth order divergence-free Weyl tensor
(i.e., ).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 with
boundary 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 must be isometric to a geodesic ball in a
simply connected space form and 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 -manifolds with a prescribed boundary metric
We prove that a critical metric of the volume functional on a -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 , or 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 -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 with smooth boundary .
Here, we will give the complete classification for an -dimensional, or
weakly Einstein critical metric of the volume functional with nonnegative
scalar curvature. Moreover, in the higher dimensional case (), we will
established a similar result for weakly Einstein critical metric under a
suitable constraint on the Weyl tensor.Comment: 11 page
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