227 research outputs found
Two-Scale Kirchhoff Theory: Comparison of Experimental Observations With Theoretical Prediction
We introduce a non-perturbative two scale Kirchhoff theory, in the context of
light scattering by a rough surface. This is a two scale theory which considers
the roughness both in the wavelength scale (small scale) and in the scales much
larger than the wavelength of the incident light (large scale). The theory can
precisely explain the small peaks which appear at certain scattering angles.
These peaks can not be explained by one scale theories. The theory was assessed
by calculating the light scattering profiles using the Atomic Force Microscope
(AFM) images, as well as surface profilometer scans of a rough surface, and
comparing the results with experiments. The theory is in good agreement with
the experimental results.Comment: 6 pages, 8 figure
Local Isometric immersions of pseudo-spherical surfaces and evolution equations
The class of differential equations describing pseudo-spherical surfaces,
first introduced by Chern and Tenenblat [3], is characterized by the property
that to each solution of a differential equation, within the class, there
corresponds a 2-dimensional Riemannian metric of curvature equal to . The
class of differential equations describing pseudo-spherical surfaces carries
close ties to the property of complete integrability, as manifested by the
existence of infinite hierarchies of conservation laws and associated linear
problems. As such, it contains many important known examples of integrable
equations, like the sine-Gordon, Liouville and KdV equations. It also gives
rise to many new families of integrable equations. The question we address in
this paper concerns the local isometric immersion of pseudo-spherical surfaces
in from the perspective of the differential equations that give
rise to the metrics. Indeed, a classical theorem in the differential geometry
of surfaces states that any pseudo-spherical surface can be locally
isometrically immersed in . In the case of the sine-Gordon
equation, one can derive an expression for the second fundamental form of the
immersion that depends only on a jet of finite order of the solution of the
pde. A natural question is to know if this remarkable property extends to
equations other than the sine-Gordon equation within the class of differential
equations describing pseudo-spherical surfaces. In an earlier paper [11], we
have shown that this property fails to hold for all other second order
equations, except for those belonging to a very special class of evolution
equations. In the present paper, we consider a class of evolution equations for
of order describing pseudo-spherical surfaces. We show that
whenever an isometric immersion in exists, depending on a jet of
finite order of , then the coefficients of the second fundamental forms are
functions of the independent variables and only.Comment: Fields Institute Communications, 2015, Hamiltonian PDEs and
Applications, pp.N
On C-smooth Surfaces of Constant Width
A number of results for C-smooth surfaces of constant width in Euclidean
3-space are obtained. In particular, an integral inequality
for constant width surfaces is established. This is used to prove that the
ratio of volume to cubed width of a constant width surface is reduced by
shrinking it along its normal lines. We also give a characterization of
surfaces of constant width that have rational support function.
Our techniques, which are complex differential geometric in nature, allow us
to construct explicit smooth surfaces of constant width in ,
and their focal sets. They also allow for easy construction of tetrahedrally
symmetric surfaces of constant width.Comment: 14 pages AMS-LATEX, 5 figure
Extension of geodesic algebras to continuous genus
Using the Penner--Fock parameterization for Teichmuller spaces of Riemann
surfaces with holes, we construct the string-like free-field representation of
the Poisson and quantum algebras of geodesic functions in the continuous-genus
limit. The mapping class group acts naturally in the obtained representation.Comment: 16 pages, submitted to Lett.Math.Phy
Bayesian Estimation of Intensity Surfaces on the Sphere via Needlet Shrinkage and Selection
This paper describes an approach for Bayesian modeling in spherical datasets. Our method is based upon a recent construction called the needlet, which is a particular form of spherical wavelet with many favorable statistical and computational properties. We perform shrinkage and selection of needlet coefficients, focusing on two main alternatives: empirical-Bayes thresholding, and Bayesian local shrinkage rules. We study the performance of the proposed methodology both on simulated data and on two real data sets: one involving the cosmic microwave background radiation, and one involving the reconstruction of a global news intensity surface inferred from published Reuters articles in August, 1996. The fully Bayesian approach based on robust, sparse shrinkage priors seems to outperform other alternatives.Business Administratio
Discrete asymptotic nets and W-congruences in Plucker line geometry
The asymptotic lattices and their transformations are studied within the line
geometry approach. It is shown that the discrete asymptotic nets are
represented by isotropic congruences in the Plucker quadric. On the basis of
the Lelieuvre-type representation of asymptotic lattices and of the discrete
analog of the Moutard transformation, it is constructed the discrete analog of
the W-congruences, which provide the Darboux-Backlund type transformation of
asymptotic lattices.The permutability theorems for the discrete Moutard
transformation and for the corresponding transformation of asymptotic lattices
are established as well. Moreover, it is proven that the discrete W-congruences
are represented by quadrilateral lattices in the quadric of Plucker. These
results generalize to a discrete level the classical line-geometric approach to
asymptotic nets and W-congruences, and incorporate the theory of asymptotic
lattices into more general theory of quadrilateral lattices and their
reductions.Comment: 28 pages, 4 figures; expanded Introduction, new Section, added
reference
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