Unique continuation property with partial information for two-dimensional anisotropic elasticity systems

In this paper, we establish a novel unique continuation property for two-dimensional anisotropic elasticity systems with partial information. More precisely, given a homogeneous elasticity system in a domain, we investigate the unique continuation by assuming only the vanishing of one component of the solution in a subdomain. Using the corresponding Riemann function, we prove that the solution vanishes in the whole domain provided that the other component vanishes at one point up to its second derivatives. Further, we construct several examples showing the possibility of further reducing the additional information of the other component. This result possesses remarkable significance in both theoretical and practical aspects because the required data is almost halved for the unique determination of the whole solution.Comment: 14 pages, 1 figur

The Use of Dispersion Relations in the ππ\pi\pi and KKˉK\bar K Coupled-Channel System

Systematic and careful studies are made on the properties of the IJ=00 $\pi\pi$ and $K\bar K$ coupled-channel system, using newly derived dispersion relations between the phase shifts and poles and cuts. The effects of nearby branch point singularities to the determination of the $f_0(980)$ resonance are estimated and and discussed.Comment: 22 pages with 5 eps figures. A numerical bug in previous version is fixed, discussions slightly expanded. No major conclusion is change

Dimensional regularization of the gravitational interaction of point masses

We show how to use dimensional regularization to determine, within the Arnowitt-Deser-Misner canonical formalism, the reduced Hamiltonian describing the dynamics of two gravitationally interacting point masses. Implementing, at the third post-Newtonian (3PN) accuracy, our procedure we find that dimensional continuation yields a finite, unambiguous (no pole part) 3PN Hamiltonian which uniquely determines the heretofore ambiguous ``static'' parameter: namely, $\omega_s=0$. Our work also provides a remarkable check of the perturbative consistency (compatibility with gauge symmetry) of dimensional continuation through a direct calculation of the ``kinetic'' parameter $\omega_k$, giving the unique answer compatible with global Poincar\'e invariance ($\omega_k={41/24}$) by summing $\sim50$ different dimensionally continued contributions.Comment: REVTeX, 8 pages, 1 figure; submitted to Phys. Lett.

Linearised Higher Variational Equations

This work explores the tensor and combinatorial constructs underlying the linearised higher-order variational equations of a generic autonomous system along a particular solution. The main result of this paper is a compact yet explicit and computationally amenable form for said variational systems and their monodromy matrices. Alternatively, the same methods are useful to retrieve, and sometimes simplify, systems satisfied by the coefficients of the Taylor expansion of a formal first integral for a given dynamical system. This is done in preparation for further results within Ziglin-Morales-Ramis theory, specifically those of a constructive nature.Comment: Minor changes with respect to previous versio

Finite-size and correlation-induced effects in Mean-field Dynamics

The brain's activity is characterized by the interaction of a very large number of neurons that are strongly affected by noise. However, signals often arise at macroscopic scales integrating the effect of many neurons into a reliable pattern of activity. In order to study such large neuronal assemblies, one is often led to derive mean-field limits summarizing the effect of the interaction of a large number of neurons into an effective signal. Classical mean-field approaches consider the evolution of a deterministic variable, the mean activity, thus neglecting the stochastic nature of neural behavior. In this article, we build upon two recent approaches that include correlations and higher order moments in mean-field equations, and study how these stochastic effects influence the solutions of the mean-field equations, both in the limit of an infinite number of neurons and for large yet finite networks. We introduce a new model, the infinite model, which arises from both equations by a rescaling of the variables and, which is invertible for finite-size networks, and hence, provides equivalent equations to those previously derived models. The study of this model allows us to understand qualitative behavior of such large-scale networks. We show that, though the solutions of the deterministic mean-field equation constitute uncorrelated solutions of the new mean-field equations, the stability properties of limit cycles are modified by the presence of correlations, and additional non-trivial behaviors including periodic orbits appear when there were none in the mean field. The origin of all these behaviors is then explored in finite-size networks where interesting mesoscopic scale effects appear. This study leads us to show that the infinite-size system appears as a singular limit of the network equations, and for any finite network, the system will differ from the infinite system

Normal form for travelling kinks in discrete Klein-Gordon lattices

We study travelling kinks in the spatial discretizations of the nonlinear Klein--Gordon equation, which include the discrete $\phi^4$ lattice and the discrete sine--Gordon lattice. The differential advance-delay equation for travelling kinks is reduced to the normal form, a scalar fourth-order differential equation, near the quadruple zero eigenvalue. We show numerically non-existence of monotonic kinks (heteroclinic orbits between adjacent equilibrium points) in the fourth-order equation. Making generic assumptions on the reduced fourth-order equation, we prove the persistence of bounded solutions (heteroclinic connections between periodic solutions near adjacent equilibrium points) in the full differential advanced-delay equation with the technique of center manifold reduction. Existence and persistence of multiple kinks in the discrete sine--Gordon equation are discussed in connection to recent numerical results of \cite{ACR03} and results of our normal form analysis