Search results for "Differential operator"
showing 10 items of 70 documents
Nambu-Poisson manifolds and associated n-ary Lie algebroids
2001
We introduce an n-ary Lie algebroid canonically associated with a Nambu-Poisson manifold. We also prove that every Nambu-Poisson bracket defined on functions is induced by some differential operator on the exterior algebra, and characterize such operators. Some physical examples are presented.
Explicit polynomial solutions of fourth order linear elliptic Partial Differential Equations for boundary based smooth surface generation
2011
We present an explicit polynomial solution method for surface generation. In this case the surface in question is characterized by some boundary configuration whereby the resulting surface conforms to a fourth order linear elliptic Partial Differential Equation, the Euler–Lagrange equation of a quadratic functional defined by a norm. In particular, the paper deals with surfaces generated as explicit Bézier polynomial solutions for the chosen Partial Differential Equation. To present the explicit solution methodologies adopted here we divide the Partial Differential Equations into two groups namely the orthogonal and the non-orthogonal cases. In order to demonstrate our methodology we discus…
Pseudodifferential operators on manifolds with a Lie structure at infinity
2003
to appear in Anal. Math.; Several examples of non-compact manifolds $M_0$ whose geometry at infinity is described by Lie algebras of vector fields $V \subset \Gamma(TM)$ (on a compactification of $M_0$ to a manifold with corners $M$) were studied by Melrose and his collaborators. In math.DG/0201202 and math.OA/0211305, the geometry of manifolds described by Lie algebras of vector fields -- baptised "manifolds with a Lie structure at infinity" there -- was studied from an axiomatic point of view. In this paper, we define and study the algebra $\Psi_{1,0,\VV}^\infty(M_0)$, which is an algebra of pseudodifferential operators canonically associated to a manifold $M_0$ with the Lie structure at …
SPECTRAL INVARIANCE FOR CERTAIN ALGEBRAS OF PSEUDODIFFERENTIAL OPERATORS
2001
We construct algebras of pseudodifferential operators on a continuous family groupoid G that are closed under holomorphic functional calculus, contain the algebra of all pseudodifferential operators of order 0 on G as a dense subalgebra, and reflect the smooth structure of the groupoid G, when G is smooth. As an application, we get a better understanding on the structure of inverses of elliptic pseudodifferential operators on classes of non-compact manifolds. For the construction of these algebras closed under holomorphic functional calculus, we develop three methods: one using two-sided semi-ideals, one using commutators, and one based on Schwartz spaces on the groupoid.
Compactness of Fourier integral operators on weighted modulation spaces
2019
Using the matrix representation of Fourier integral operators with respect to a Gabor frame, we study their compactness on weighted modulation spaces. As a consequence, we recover and improve some compactness results for pseudodifferential operators.
Pairs of nontrivial smooth solutions for nonlinear Neumann problems
2020
Abstract We consider a nonlinear Neumann problem driven by a nonhomogeneous differential operator with a reaction term that exhibits strong resonance at infinity. Using variational tools based on the critical point theory, we prove the existence of two nontrivial smooth solutions.
Spectral properties of random non-self-adjoint operators
2015
In this thesis we are interested in the spectral properties of random non-self-adjoint operators. Weare going to consider primarily the case of small random perturbations of the following two types of operators: 1. a class of non-self-adjoint h-differential operators Ph, introduced by M. Hager [32], in the semiclassical limit (h→0); 2. large Jordan block matrices as the dimension of the matrix gets large (N→∞). In case 1 we are going to consider the operator Ph subject to small Gaussian random perturbations. We let the perturbation coupling constant δ be e (-1/Ch) ≤ δ ⩽ h(k), for constants C, k > 0 suitably large. Let ∑ be the closure of the range of the principal symbol. Previous results o…
Differential operator formalism for axial optical vortex beam and the double-phase-ramp converter
2019
A systematic study of the properties of the output dark rays or singular skeleton for the Laguerre-Gaussian beam LG 01 passed through double-phase-ramp converter is presented. When the DOE is discontinuous at the origin, as is the case here, the transfer function is not analytical, so that a special theoretical approach is needed. The previously reported formalism of scattering modes, which permitted the analytical calculation of arbitrary multisingular Gaussian beams, requires analyticity everywhere. We present here an adaption of this formalism that overcomes this limitation. The procedure is based on the differential operator algebra used in the previous construction. We give an example …
A physically based connection between fractional calculus and fractal geometry
2014
We show a relation between fractional calculus and fractals, based only on physical and geometrical considerations. The link has been found in the physical origins of the power-laws, ruling the evolution of many natural phenomena, whose long memory and hereditary properties are mathematically modelled by differential operators of non integer order. Dealing with the relevant example of a viscous fluid seeping through a fractal shaped porous medium, we show that, once a physical phenomenon or process takes place on an underlying fractal geometry, then a power-law naturally comes up in ruling its evolution, whose order is related to the anomalous dimension of such geometry, as well as to the m…
Return to Equilibrium, Non-self-adjointness and Symmetries, Recent Results with M. Hitrik and F. Hérau
2014
In this talk we review some old and new results about the use of supersymmetric structures in semi-classical problems. Necessary and sufficient conditions are obtained for a real semiclassical partial differential operator of order two to possess a supersymmetric structure. For operators coming from a chain of oscillators, coupled to two heat baths, we show the non-existence of a smooth supersymmetric structure. The recent and new results all come from joint works with Michael Hitrik and Frederic Herau.