307 research outputs found
Heat flow method to Lichnerowicz type equation on closed manifolds
In this paper, we establish existence results for positive solutions to the
Lichnerowicz equation of the following type in closed manifolds -\Delta
u=A(x)u^{-p}-B(x)u^{q},\quad in\quad M, where , and ,
are given smooth functions. Our analysis is based on the global
existence of positive solutions to the following heat equation {ll} u_t-\Delta
u=A(x)u^{-p}-B(x)u^{q},\quad in\quad M\times\mathbb{R}^{+}, u(x,0)=u_0,\quad
in\quad M with the positive smooth initial data .Comment: 10 page
Photon mediated interaction between distant quantum dot circuits
Engineering the interaction between light and matter is an important goal in
the emerging field of quantum opto-electronics. Thanks to the use of cavity
quantum electrodynamics architectures, one can envision a fully hybrid
multiplexing of quantum conductors. Here, we use such an architecture to couple
two quantum dot circuits . Our quantum dots are separated by 200 times their
own size, with no direct tunnel and electrostatic couplings between them. We
demonstrate their interaction, mediated by the cavity photons. This could be
used to scale up quantum bit architectures based on quantum dot circuits or
simulate on-chip phonon-mediated interactions between strongly correlated
electrons
On the local existence of maximal slicings in spherically symmetric spacetimes
In this talk we show that any spherically symmetric spacetime admits locally
a maximal spacelike slicing. The above condition is reduced to solve a
decoupled system of first order quasi-linear partial differential equations.
The solution may be accomplished analytical or numerically. We provide a
general procedure to construct such maximal slicings.Comment: 4 pages. Accepted for publication in Journal of Physics: Conference
Series, Proceedings of the Spanish Relativity Meeting ERE200
First order hyperbolic formalism for Numerical Relativity
The causal structure of Einstein's evolution equations is considered. We show
that in general they can be written as a first order system of balance laws for
any choice of slicing or shift. We also show how certain terms in the evolution
equations, that can lead to numerical inaccuracies, can be eliminated by using
the Hamiltonian constraint. Furthermore, we show that the entire system is
hyperbolic when the time coordinate is chosen in an invariant algebraic way,
and for any fixed choice of the shift. This is achieved by using the momentum
constraints in such as way that no additional space or time derivatives of the
equations need to be computed. The slicings that allow hyperbolicity in this
formulation belong to a large class, including harmonic, maximal, and many
others that have been commonly used in numerical relativity. We provide details
of some of the advanced numerical methods that this formulation of the
equations allows, and we also discuss certain advantages that a hyperbolic
formulation provides when treating boundary conditions.Comment: To appear in Phys. Rev.
Gluing construction of initial data with Kerr-de Sitter ends
We construct initial data sets which satisfy the vacuum constraint equa-
tions of General Relativity with positive cosmologigal constant. More pre-
silely, we deform initial data with ends asymptotic to Schwarzschild-de Sitter
to obtain non-trivial initial data with exactly Kerr-de Sitter ends. The method
is inspired from Corvino's gluing method. We obtain here a extension of a
previous result for the time-symmetric case by Chru\'sciel and Pollack.Comment: 27 pages, 3 figure
The Cauchy problem on a characteristic cone for the Einstein equations in arbitrary dimensions
We derive explicit formulae for a set of constraints for the Einstein
equations on a null hypersurface, in arbitrary dimensions. We solve these
constraints and show that they provide necessary and sufficient conditions so
that a spacetime solution of the Cauchy problem on a characteristic cone for
the hyperbolic system of the reduced Einstein equations in wave-map gauge also
satisfies the full Einstein equations. We prove a geometric uniqueness theorem
for this Cauchy problem in the vacuum case.Comment: 83 pages, 1 figur
Discrete Dynamical Systems Embedded in Cantor Sets
While the notion of chaos is well established for dynamical systems on
manifolds, it is not so for dynamical systems over discrete spaces with
variables, as binary neural networks and cellular automata. The main difficulty
is the choice of a suitable topology to study the limit . By
embedding the discrete phase space into a Cantor set we provided a natural
setting to define topological entropy and Lyapunov exponents through the
concept of error-profile. We made explicit calculations both numerical and
analytic for well known discrete dynamical models.Comment: 36 pages, 13 figures: minor text amendments in places, time running
top to bottom in figures, to appear in J. Math. Phy
The Cauchy Problem for the Einstein Equations
Various aspects of the Cauchy problem for the Einstein equations are
surveyed, with the emphasis on local solutions of the evolution equations.
Particular attention is payed to giving a clear explanation of conceptual
issues which arise in this context. The question of producing reduced systems
of equations which are hyperbolic is examined in detail and some new results on
that subject are presented. Relevant background from the theory of partial
differential equations is also explained at some lengthComment: 98 page
Dynamically generated embeddings of spacetime
We discuss how embeddings in connection with the Campbell-Magaard (CM)
theorem can have a physical interpretation. We show that any embedding whose
local existence is guaranteed by the CM theorem can be viewed as a result of
the dynamical evolution of initial data given in a four-dimensional spacelike
hypersurface. By using the CM theorem, we establish that for any analytic
spacetime, there exist appropriate initial data whose Cauchy development is a
five-dimensional vacuum space into which the spacetime is locally embedded. We
shall see also that the spacetime embedded is Cauchy stable with respect these
the initial data.Comment: (8 pages, 1 figure). A section on Cauchy Stability of the embedding
was added. (To appear in Class. Quant. Grav.
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