Limit cycles and invariant cylinders for a class of continuous and discontinuous vector field in dimension 2n

Agraïments: The first author is partially supported by CNPq grand number 200293/2010-9. Both authors are also supported by the joint project CAPES-MECD grant PHB-2009-0025-PC.The subject of this paper concerns with the bifurcation of limit cycles and invariant cylinders from a global center of a linear differential system in dimension 2n perturbed inside a class of continuous and discontinuous piecewise linear differential systems. Our main results show that at most one limit cycle and at most one invariant cylinder can bifurcate using the expansion of the displacement function up to first order with respect to a small parameter. This upper bound is reached. For proving these results we use the averaging theory in a form where the differentiability of the system is not needed

From K.A.M. Tori to Isospectral Invariants and Spectral Rigidity of Billiard Tables

This article is a part of a project investigating the relationship between the dynamics of completely integrable or close to completely integrable billiard tables, the integral geometry on them, and the spectrum of the corresponding Laplace-Beltrami operators. It is concerned with new isospectral invariants and with the spectral rigidity problem for families of Laplace-Beltrami operators with Dirichlet, Neumann or Robin boundary conditions, associated with C^1 families of billiard tables. We introduce a notion of weak isospectrality for such deformations. The main dynamical assumption on the initial billiard table is that the corresponding billiard ball map or an iterate of it has a Kronecker invariant torus with a Diophantine frequency and that the corresponding Birkhoff Normal Form is nondegenerate in Kolmogorov sense. Then we obtain C^1 families of Kronecker tori with Diophantine frequencies. If the family of the Laplace-Beltrami operators satisfies the weak isospectral condition, we prove that the average action on the tori and the Birkhoff Normal Form of the billiard ball maps remain the same along the perturbation. As an application we obtain infinitesimal spectral rigidity for Liouville billiard tables in dimensions two and three. Applications are obtained also for strictly convex billiard tables of dimension two as well as in the case when the initial billiard table admits an elliptic periodic billiard trajectory. Spectral rigidity of billard tables close elliptical billiard tables is obtained. The results are based on a construction of C^1 families of quasi-modes associated with the Kronecker tori and on suitable KAM theorems for C^1 families of Hamiltonians.Comment: 170 pages; new results about the spectral rigidity of elliptical billiard tables; new Modified Iterative Lemma in the proof of KAM theorem with parameter

Manufacturing Metrology

Metrology is the science of measurement, which can be divided into three overlapping activities: (1) the definition of units of measurement, (2) the realization of units of measurement, and (3) the traceability of measurement units. Manufacturing metrology originally implicates the measurement of components and inputs for a manufacturing process to assure they are within specification requirements. It can also be extended to indicate the performance measurement of manufacturing equipment. This Special Issue covers papers revealing novel measurement methodologies and instrumentations for manufacturing metrology from the conventional industry to the frontier of the advanced hi-tech industry. Twenty-five papers are included in this Special Issue. These published papers can be categorized into four main groups, as follows: Length measurement: covering new designs, from micro/nanogap measurement with laser triangulation sensors and laser interferometers to very-long-distance, newly developed mode-locked femtosecond lasers. Surface profile and form measurements: covering technologies with new confocal sensors and imagine sensors: in situ and on-machine measurements. Angle measurements: these include a new 2D precision level design, a review of angle measurement with mode-locked femtosecond lasers, and multi-axis machine tool squareness measurement. Other laboratory systems: these include a water cooling temperature control system and a computer-aided inspection framework for CMM performance evaluation