4,984 research outputs found
Unified formalism for higher-order non-autonomous dynamical systems
This work is devoted to giving a geometric framework for describing
higher-order non-autonomous mechanical systems. The starting point is to extend
the Lagrangian-Hamiltonian unified formalism of Skinner and Rusk for these
kinds of systems, generalizing previous developments for higher-order
autonomous mechanical systems and first-order non-autonomous mechanical
systems. Then, we use this unified formulation to derive the standard
Lagrangian and Hamiltonian formalisms, including the Legendre-Ostrogradsky map
and the Euler-Lagrange and the Hamilton equations, both for regular and
singular systems. As applications of our model, two examples of regular and
singular physical systems are studied.Comment: 43 pp. We have corrected and clarified the statement of Propositions
2 and 3. A remark is added after Proposition
Non-local fractional derivatives. Discrete and continuous
We prove maximum and comparison principles for fractional discrete
derivatives in the integers. Regularity results when the space is a mesh of
length , and approximation theorems to the continuous fractional derivatives
are shown. When the functions are good enough, these approximation procedures
give a measure of the order of approximation. These results also allows us to
prove the coincidence, for good enough functions, of the Marchaud and
Gr\"unwald-Letnikov derivatives in every point and the speed of convergence to
the Gr\"unwald-Letnikov derivative. The fractional discrete derivative will be
also described as a Neumann-Dirichlet operator defined by a semi-discrete
extension problem. Some operators related to the Harmonic Analysis associated
to the discrete derivative will be also considered, in particular their
behavior in the Lebesgue spaces $\ell^p(\mathbb{Z}).
Mechanisms for photon sorting based on slit-groove arrays
Mechanisms for one-dimensional photon sorting are theoretically studied in
the framework of a couple mode method. The considered system is a nanopatterned
structure composed of two different pixels drilled on the surface of a thin
gold layer. Each pixel consists of a slit-groove array designed to squeeze a
large fraction of the incident light into the central slit. The Double-Pixel is
optimized to resolve two different frequencies in the near infrared. This
system shows a high transmission efficiency and a small crosstalk. Its response
is found to strongly depend on the effective area shared by overlapping pixels.
Three different regimes for the process of photon sorting are identified and
the main physical trends underneath in such regimes are unveiled. Optimal
efficiencies for the photon sorting are obtained for a moderate number of
grooves that overlap with grooves of the neighbor pixel. Results could be
applied to optical and infrared detectors.Comment: 12 pages, 4 figure
Nonholonomic constraints in -symplectic Classical Field Theories
A -symplectic framework for classical field theories subject to
nonholonomic constraints is presented. If the constrained problem is regular
one can construct a projection operator such that the solutions of the
constrained problem are obtained by projecting the solutions of the free
problem. Symmetries for the nonholonomic system are introduced and we show that
for every such symmetry, there exist a nonholonomic momentum equation. The
proposed formalism permits to introduce in a simple way many tools of
nonholonomic mechanics to nonholonomic field theories.Comment: 27 page
Transmittance of a subwavelength aperture flanked by a finite groove array \\ placed near the focus of a conventional lens
One-dimensional light harvesting structures illuminated by a conventional
lens are studied in this paper. Our theoretical study shows that high
transmission efficiencies are obtained when the structure is placed near the
focal plane of the lens. The considered structure is a finite slit-groove array
(SGA) with a given number of grooves that are symmetrically distributed with
respect to a central slit. The SGA is nano-patterned on an opaque metallic
film. It is found that a total transmittance of 80% is achieved even for a
single slit when (i) Fabry-Perot like modes are excited inside the slit and
(ii) the effective cross section of the aperture becomes of the order of the
full width at half maximum of the incident beam. A further enhancement of 8% is
produced by the groove array. The optimal geometry for the groove array
consists of a moderate number of grooves () at either side of the
slit, separated by a distance of half the incident wavelength .
Grooves should be deeper (with depth ) than those typically
reported for plane wave illumination in order to increase their individual
scattering cross section.Comment: 7 pages, 6 figure
Higher-order Mechanics: Variational Principles and other topics
After reviewing the Lagrangian-Hamiltonian unified formalism (i.e, the
Skinner-Rusk formalism) for higher-order (non-autonomous) dynamical systems, we
state a unified geometrical version of the Variational Principles which allows
us to derive the Lagrangian and Hamiltonian equations for these kinds of
systems. Then, the standard Lagrangian and Hamiltonian formulations of these
principles and the corresponding dynamical equations are recovered from this
unified framework.Comment: New version of the paper "Variational principles for higher-order
dynamical systems", which was presented in the "III Iberoamerican Meeting on
Geometry, Mechanics and Control" (Salamanca, 2012). The title is changed. A
detailed review is added. Sections containing results about variational
principles are enlarged with additional comments, diagrams and summarizing
results. Bibliography is update
Lipschitz spaces adapted to Schrödinger operators and regularity properties
Consider the Schrödinger operator L= - Δ + V in Rn, n≥ 3 , where V is a nonnegative potential satisfying a reverse Hölder condition of the type (1|B|∫BV(y)qdy)1/q ≤C|B|∫BV(y)dy, for some q > n/2. We define ΛαL, 0 0‖f(·+z) + f(·-z)-2f(·)‖∞|z|α 0 , we denote by Λ Wα/2 the set of functions f which satisfy ‖ρ(·)-αf(·)‖∞ 0. We prove that for 0 < α ≤ 2 - n/q, ΛαL = Λ Wα/2. As application, we obtain regularity properties of fractional powers (positive and negative) of the operator L, Schrödinger Riesz transforms, Bessel potentials and multipliers of Laplace transforms type. The proofs of these results need in an essential way the language of semigroups. Parallel results are obtained for the classes defined through the Poisson semigroup, Py f = e-y √LfMarta De León-Contreras was partially supported by grant EPSRC Research Grant EP/S029486/1. José L. Torrea was partially supported by Grant PGC2018-099124-B-I00 (MINECO/FEDER
Symmetries in Classical Field Theory
The multisymplectic description of Classical Field Theories is revisited,
including its relation with the presymplectic formalism on the space of Cauchy
data. Both descriptions allow us to give a complete scheme of classification of
infinitesimal symmetries, and to obtain the corresponding conservation laws.Comment: 70S05; 70H33; 55R10; 58A2
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