Suitable Geometric Conditions Research Articles

Nonflatness of certain contact and CR manifolds

We prove that, under a suitable geometric condition, a Riemannian manifold of dimension at least 7 endowed with a contact distribution cannot be flat. This result yields nonflatness of some classes of almost contact metric manifolds, contact sub-Riemannian symmetric spaces, locally symmetric C R spaces and C R submanifolds of Kähler manifolds. As an application, we prove that a compact flat Riemannian manifold of odd dimension at least 7 cannot be isometrically immersed as a hypersurface of a simply connected, complete Kähler manifold of nonpositive curvature.

POLICRYPS composite structures: realization, characterization and exploitation for electro-optical and all-optical applications

The acronym POLICRYPS indicates a nano/micro composite structure made of films of well aligned nematic liquid crystal (NLC) alternated to slices of almost pure polymer. Structures are fabricated by curing a homogeneous mixture of mesogenic material, monomer and a curing agent with UV radiation, under suitable physical and geometrical conditions. In particular, geometrical conditions determine (in the range 0.2 ÷ 15 μ m) the spatial periodicity of the realized sample, which can be utilized for transmitting, diffracting or reflecting an impinging light beam, with negligible scattering losses. The spatial modulation of the refractive index (from polymer to NLC) can be switched ON and OFF both by applying an electric field of few V/μ m or, in some cases, by irradiating the sample with a light beam of suitable wavelength. This electrical/optical tuneability is responsible for a series of distinctive characteristics of the optical effects produced by the structure, and determines the range of possible applications. In fact, in different geometries, the POLICRYPS can be exploited as a switchable holographic grating, a switchable optical phase modulator, a switchable beam splitter, a tuneable Bragg filter or it can be exploited as an electro-optical edge filter in an optical interrogation system.

Concentration of solutions for a singularly perturbed mixed problem in non-smooth domains

We consider a singularly perturbed problem with mixed Dirichlet and Neumann boundary conditions in a bounded domain Ω⊂Rn whose boundary has an (n−2)-dimensional singularity. Assuming 1<p<n+2n−2, we prove that, under suitable geometric conditions on the boundary of the domain, there exist solutions which approach the intersection of the Neumann and the Dirichlet parts as the singular perturbation parameter tends to zero.

Measurements of the Thermal Neutron Macroscopic Absorption Cross Section for Neutron Absorbing Layers

Objective of this study is measuring the macroscopic cross section of a neutron absorbing layer for thermal neutrons. For this purpose a neutron source and BF 3 detector have been applied. For measuring macroscopic cross section of thermal neutrons by the $$ I = BI_{0} e^{{ - \sum\nolimits_{tot} t}} $$ Formula, it is necessary to provide suitable geometric conditions in order to assume the production and build-up coefficient to be the unit value (=1). To fulfill required conditions for this assumption, surface of the detector is covered with a 2 mm thick layer of cadmium. Radiation window of the detector has a 3 cm diameter, situated directly in front of the source. By placing the cadmium cover over the detector, variation of $$ Ln{\frac{{I_{0}^{{}} }}{I}} $$ values verses thickness of absorbent layer, renders linear function behavior, making it possible to measure the macroscopic cross section. The next stage is applying the MCNP code by simulating F1 tally and cosine-cards for calculating Total Macroscopic Cross-Section. Validation of this study is achieved through comparison of simulation by the MCNP code and results rendered by experiment measurements.

Asymptotic stability of finite energy in Navier Stokes-elastic wave interaction

We consider a model of fluid-structure interaction in a bounded domain Ω∈ℝ2 where Ω is comprised of two open adjacent sub-domains occupied, respectively, by the solid and the fluid. This leads to a study of Navier Stokes equation coupled on the interface to the dynamic system of elasticity. The characteristic feature of this coupled model is that the resolvent is not compact and the energy function characterizing balance of the total energy is weakly degenerated. These combined with the lack of mechanical dissipation and intrinsic nonlinearity of the dynamics render the problem of asymptotic stability rather delicate. Indeed, the only source of dissipation is the viscosity effect propagated from the fluid via interface. It will be shown that under suitable geometric conditions imposed on the geometry of the interface, finite energy function associated with weak solutions converges to zero when the time t converges to infinity. The required geometric conditions result from the presence of the pressure acting upon the solid.

Polynomial decay for coupled Schrödinger equations with variable coefficients and damped by one Dirichlet boundary feedback

The main purpose of this paper is to study, under a suitable geometric conditions, the indirect boundary stabilization for coupled Schrödinger equations with variable coefficients and one Dirichlet boundary feedback. The polynomial energy decay rate for smooth solutions is obtained by the combination of the Riemannian geometry method in $\left[ Ya1\right] $ and the ideas of I. Lasiecka and R. Triggiani in $\left[ LT3\right].

Asymptotics for Some Proximal-like Method Involving Inertia and Memory Aspects

Given a Hilbert space H and a closed convex function Φ: $H\rightarrow{\mathbb{R}} \cup \{+\infty\}$ , we consider the inertial proximal algorithm $$ x_{n+1}-x_n-\alpha_n(x_n-x_{n-1})+\beta_n\partial\Phi(x_{n+1})\ni 0, \qquad \qquad \qquad \qquad({\mathcal{A}}) $$ where (α n ) and (β n ) are nonnegative sequences. The notation $\partial \Phi$ stands for the subdifferential of Φ in the sense of convex analysis. This algorithm can be viewed as the implicit discretization of a continuous gradient system involving a memory term. We give conditions that ensure that a suitable discrete energy decreases to $\inf\Phi$ as n→ + ∞. When Φ has a unique minimum, the question of the convergence of (x n ) is solved. In the case of multiple minima, it is proved that if $\left(\prod_{k=1}^n \alpha_k\right)\not \in l^1$ and if a suitable geometric condition on the set argmin Φ is fulfilled, then non stationary sequences of (A) cannot converge.

Formula omitted]-estimates and existence theorems for the [formula omitted]-operator on complete Kähler manifolds

In this paper we establish some L p -estimates and existence theorems for the ∂ ¯ -operator on a holomorphic Hermitian vector bundle over a complete Kähler manifold. As applications, we prove some vanishing theorems of the L p -Dolbeault cohomology on complete Kähler manifolds with suitable geometric conditions.

Sobolev Inequalities on Forms and L p,q -Cohomology on Complete Riemannian Manifolds

We prove some Sobolev inequalities on differential forms over a class of complete non-compact Riemannian manifolds with suitable geometric conditions. Moreover, we establish some Lp,q-estimates and existence theorems of the Cartan-De Rham equation and the Hodge systems. As applications, we prove some vanishing theorems of the Lp,q-cohomology and prove the Lq-solvability of the nonlinear p-Laplace equation on forms on complete non-compact Riemannian manifolds with suitable geometric conditions.

Some geometric constraints and the problem of global regularity for the Navier–Stokes equations**Dedicated to the memory of Fausto Saleri.

We consider the Cauchy problem for the 3D Navier–Stokes equations and show that weak solutions satisfying suitable geometric conditions are smooth. The first condition we consider requires the vorticity's direction at the point y (near to x) to be within a circular cone of a given small amplitude, with vertex at x and axis along the vorticity's direction at x. The amplitude of the angle at the vertex of the cone depends mainly on the viscosity and on the size of the initial datum. Hence, the vorticity's direction need not be (Hölder) continuous to ensure regularity, as in the results proved in previous studies. Our new results show that a possible singularity deriving from non-trivial collision of strong vortex tubes is depleted if the angle formed by the tubes is small enough.Other conditions regarding the direction of the curl of the vorticity are considered: it is shown that if the direction of curl of the vorticity at point x is ‘nearly parallel’ to the same quantity at neighbouring points, then weak solutions are smooth.

Concentration of Solutions for Some Singularly Perturbed Mixed Problems: Existence Results

In this paper,we study the asymptotic behavior of some solutions to a singularly perturbed problem with mixed Dirichlet and Neumann boundary conditions. We prove that, under suitable geometric conditions on the boundary of the domain, there exist solutions which approach the intersection of the Neumann and the Dirichlet parts as the singular perturbation parameter tends to zero.

On the Strong [formula omitted]-Hodge decomposition over complete Riemannian manifolds

We establish the Strong L p -Hodge decomposition theorem and the L p -Poincaré inequalities on differential forms over complete non-compact Riemannian manifolds. As applications, we prove some L p -estimates and existence theorems for the de Rham operator as well as some vanishing theorems for the L p -cohomology and the L p -torsion over complete non-compact Riemannian manifolds with suitable geometric conditions.

Hardy Inequalities, Observability, and Control for the Wave and Schrödinger Equations with Singular Potentials

We address the question of exact controllability of the wave and Schrödinger equations perturbed by a singular inverse-square potential. Exact boundary controllability is proved in the range of subcritical coefficients of the singular potential and under suitable geometric conditions. The proof relies on the method of multipliers. The key point in the proof of the observability inequality is a suitable Hardy-type inequality with sharp constants. On the contrary, in the supercritical case, we prove that exact controllability is false.

In-line holography and coherent diffractive imaging with x-ray waveguides

A Fresnel coherent diffraction imaging experiment with hard x rays is here presented, using two planar crossed waveguides as optical elements, leading to a virtual pointlike source. The coherent wave field obtained with this setup is used to illuminate a micrometric single object having the shape of a butterfly. A digital two-dimensional in-line holographic reconstruction of the unknown object at low resolution (200 nm) has been obtained directly via fast Fourier transform (FFT) of the raw data. The object and its twin image are well separated because suitable geometrical conditions are satisfied. A good estimate of the incident wave field phase has been extracted directly from the FFT of the raw data. A partial object reconstruction with 50 nm spatial resolution was achieved by fast iterative phase retrieval, the major limitation for a full reconstruction being the nonideal structure of the guided beam. The method offers a route for fast and reliable phase retrieval in x-ray coherent diffraction.

Interaction between Dynamics and Cloud Microphysics in Orographic Precipitation Enhancement: A High-Resolution Modeling Study of Two North Alpine Heavy-Precipitation Events

Abstract Interactions of atmospheric dynamics and cloud microphysics with the Alpine orography are investigated for two north Alpine heavy-precipitation cases (20–22 May 1999 and 22–23 August 2005). Both cases were related to a deep cyclone propagating slowly eastward along the Alps, advecting moist air of Mediterranean origin toward the northern side of the Alps. A validation against high-resolution rain gauge data reveals that the average model bias is below 15% in the region of interest, but there is a tendency to systematically underestimate very heavy precipitation. A scale decomposition of the discrepancies between model and observations reveals that errors on the meso-β-scale contribute at least as much to the total model error as discrepancies on the meso-γ-scale. On the scale of single mountain ridges and valleys, the formation of precipitation maxima at valley locations is investigated, with particular emphasis on a region in which a valley receives systematically more precipitation than the adjacent mountain ridges. It is found that the downstream advection of precipitation hydrometeors generated in an orographic feeder cloud is essential for the development of valley maxima. Strong ambient winds and (due to the fall speed difference between snow/graupel and rain) a low freezing level favor a large distance of the precipitation maximum from the upstream mountain ridge. Under suitable geometrical conditions, downstream advection of hydrometeors can even lead to systematically more precipitation in the valley than over the adjacent ridges. Another mechanism capable of generating a systematic rainfall maximum at valley locations requires a freezing level between valley bottom and crest height and a mountain wave flow penetrating into the valley. Under such conditions, the increase in the fall velocity related to melting of snow or graupel into rain leads to a locally intensified fallout of hydrometeors and thus to a maximum in the precipitation rate.

The Poisson problem with mixed boundary conditions in Sobolev and Besov spaces in non-smooth domains

We introduce certain Sobolev-Besov spaces which are particularly well adapted for measuring the smoothness of data and solutions of mixed boundary value problems in Lipschitz domains. In particular, these are used to obtain sharp well-posedness results for the Poisson problem for the Laplacian with mixed boundary conditions on bounded Lipschitz domains which satisfy a suitable geometric condition introduced by R. Brown in (1994). In this context, we obtain results which generalize those by D. Jerison and C. Kenig (1995) as well as E. Fabes, O. Mendez and M. Mitrea (1998). Applications to Hodge theory and the regularity of Green operators are also presented.

Simultaneous Exact Control of Quasi-Electrostatic Piezoelectric Systems in Multilayered Media

We consider two quasi-electrostatic models in a multilayered piezoelectric structure. Both models only differ on the boundary conditions. For each one of them we prove a boundary observability inequality which together with the Hilbert Uniqueness Method introduced by J.L. Lions allow us to drive both systems to rest at time T with the same control function. We require suitable geometric conditions on the region and the interfaces as well as monotonicity assumptions on the coefficients of the model.

Polarization of Light from Warm Clouds above an Accretion Disk: Effects of Strong Gravity near a Black Hole

We consider polarization from the scattering of light on a cloud in radial motion along the symmetry axis of an accretion disk. Radiation drag from the disk and gravitational attraction of the central black hole are taken into account, as well as the effect of the cloud cooling in the radiation field. This provides us with a self-consistent toy-model for predicted lightcurves, including the linear polarization that arises from the scattering. Strong gravitational lensing creates indirect images; these are formed by photons that originate from the disk, become backscattered onto the photon circular orbit and eventually redirected toward an observer. Under suitable geometrical conditions the indirect photons may visibly influence the resulting magnitude of the polarization and light-curve profiles. Relevant targets are black holes in active galactic nuclei and stellar-mass Galactic black-holes exhibiting episodic accretion/ejection events.

Boundary Stabilization and a Problem of Transmission for a System of Propagation of Sound

We consider a transmission problem for a model describing the evolution of sound in a compressible fluid. Assuming that dissipative mechanisms of memory type are effective in part of the boundary, the coefficients are piecewise constants and suitable geometric conditions on the domain and the interfaces, we prove that the total energy associated with the model decays exponentially fast as t → + ∞.

Polynomial decay for a hyperbolic–parabolic coupled system

This paper analyzes the long time behavior of a linearized model for fluid-structure interaction. The space domain consists of two parts in which the evolution is governed by the heat equation and the wave equation respectively, with transmission conditions at the interface. Based on the construction of ray-like solutions by means of Geometric Optics expansions and a careful analysis of the transfer of the energy at the interface, we show the lack of uniform decay of solutions in general domains. Also, we prove the polynomial decay result for smooth solutions under a suitable Geometric Control Condition. This condition requires that all rays propagating in the wave domain reach the interface in a uniform time after, possibly, bouncing in the exterior boundary.