800 research outputs found
Bifurcations of piecewise smooth flows:perspectives, methodologies and open problems
In this paper, the theory of bifurcations in piecewise smooth flows is critically surveyed. The focus is on results that hold in arbitrarily (but finitely) many dimensions, highlighting significant areas where a detailed understanding is presently lacking. The clearest results to date concern equilibria undergoing bifurcations at switching boundaries, and limit cycles undergoing grazing and sliding bifurcations. After discussing fundamental concepts, such as topological equivalence of two piecewise smooth systems, discontinuity-induced bifurcations are defined for equilibria and limit cycles. Conditions for equilibria to exist in n-dimensions are given, followed by the conditions under which they generically undergo codimension-one bifurcations. The extent of knowledge of their unfoldings is also summarized. Codimension-one bifurcations of limit cycles and boundary-intersection crossing are described together with techniques for their classification. Codimension-two bifurcations are discussed with suggestions for further study
A characterization of the central shell-focusing singularity in spherical gravitational collapse
We give a characterization of the central shell-focusing curvature
singularity that can form in the spherical gravitational collapse of a bounded
matter distribution obeying the dominant energy condition. This
characterization is based on the limiting behaviour of the mass function in the
neighbourhood of the singularity. Depending on the rate of growth of the mass
as a function of the area radius R, the singularity may be either covered or
naked. The singularity is naked if this growth rate is slower than R, covered
if it is faster than R, and either naked or covered if the growth rate is same
as R.Comment: 12 pages, Latex, significantly revised version, including change of
title. Revised version to appear in Classical and Quantum Gravit
Invariant-based approach to symmetry class detection
In this paper, the problem of the identification of the symmetry class of a
given tensor is asked. Contrary to classical approaches which are based on the
spectral properties of the linear operator describing the elasticity, our
setting is based on the invariants of the irreducible tensors appearing in the
harmonic decomposition of the elasticity tensor [Forte-Vianello, 1996]. To that
aim we first introduce a geometrical description of the space of elasticity
tensors. This framework is used to derive invariant-based conditions that
characterize symmetry classes. For low order symmetry classes, such conditions
are given on a triplet of quadratic forms extracted from the harmonic
decomposition of the elasticity tensor , meanwhile for higher-order classes
conditions are provided in terms of elements of , the higher irreducible
space in the decomposition of . Proceeding in such a way some well known
conditions appearing in the Mehrabadi-Cowin theorem for the existence of a
symmetry plane are retrieved, and a set of algebraic relations on polynomial
invariants characterizing the orthotropic, trigonal, tetragonal, transverse
isotropic and cubic symmetry classes are provided. Using a genericity
assumption on the elasticity tensor under study, an algorithm to identify the
symmetry class of a large set of tensors is finally provided.Comment: 32 page
Some open questions in "wave chaos"
The subject area referred to as "wave chaos", "quantum chaos" or "quantum
chaology" has been investigated mostly by the theoretical physics community in
the last 30 years. The questions it raises have more recently also attracted
the attention of mathematicians and mathematical physicists, due to connections
with number theory, graph theory, Riemannian, hyperbolic or complex geometry,
classical dynamical systems, probability etc. After giving a rough account on
"what is quantum chaos?", I intend to list some pending questions, some of them
having been raised a long time ago, some others more recent
A model of host response to a multi-stage pathogen
We model the immune surveillance of a pathogen which passes through
immunologically distinct stages. The biological parameters of this system
induce a partial order on the stages, and this, in turn, determines which
stages will be subject to immune regulation. This corresponds to the system's
unique asymptotically stable fixed point.Comment: 22 pages, no figure
Naked singularities in Tolman-Bondi-de Sitter collapse
We study the formation of central naked singularities in spherical dust
collapse with a cosmological constant. We find that the central curvature
singularity is locally naked, Tipler strong, and generic, in the sense that it
forms from a non-zero-measure set of regular initial data. We also find that
the Weyl and Ricci curvature scalars diverge at the singularity, with the
former dominating over the latter, thereby signaling the non-local origin of
the singularity.Comment: 8 pages, RevTeX, 1 eps figure; accepted for publication in Phys. Rev.
Generalized Strong Curvature Singularities and Cosmic Censorship
A new definition of a strong curvature singularity is proposed. This
definition is motivated by the definitions given by Tipler and Krolak, but is
significantly different and more general. All causal geodesics terminating at
these new singularities, which we call generalized strong curvature
singularities, are classified into three possible types; the classification is
based on certain relations between the curvature strength of the singularities
and the causal structure in their neighborhood. A cosmic censorship theorem is
formulated and proved which shows that only one class of generalized strong
curvature singularities, corresponding to a single type of geodesics according
to our classification, can be naked. Implications of this result for the cosmic
censorship hypothesis are indicated.Comment: LaTeX, 11 pages, no figures, to appear in Mod. Phys. Lett.
- …