scholarly journals The singular terms in the fundamental matrix of crystal optics

2011 ◽  
Vol 467 (2133) ◽  
pp. 2663-2689 ◽  
Author(s):  
Peter Wagner
Keyword(s):  
Explicit Formula ◽  
Fundamental Matrix ◽  
Singular Part ◽  
Wave Surface ◽  
Crystal Optics ◽  
Cauchy Principal ◽  
Singular Terms ◽  
Negatively Curved ◽  
Principal Value ◽  
Positively Curved

We derive an explicit formula for the singular part of the fundamental matrix of crystal optics. It consists of a singularity remaining fixed at the origin x =0, of delta terms located on the positively curved parts of the wave surface, the well-known Fresnel surface and of a Cauchy principal value distribution on the negatively curved part of the wave surface.

On Flag Curvature of Homogeneous Randers Spaces

2013 ◽  
Vol 65 (1) ◽  
pp. 66-81 ◽  
Author(s):  
Shaoqiang Deng ◽  
Zhiguang Hu
Keyword(s):  
Explicit Formula ◽  
Lie Group ◽  
Flag Curvature ◽  
Nilpotent Lie Group ◽  
Finsler Spaces ◽  
Rigidity Result ◽  
Randers Space ◽  
Negatively Curved

AbstractIn this paper we give an explicit formula for the flag curvature of homogeneous Randers spaces of Douglas type and apply this formula to obtain some interesting results. We first deduce an explicit formula for the flag curvature of an arbitrary left invariant Randersmetric on a two-step nilpotent Lie group. Then we obtain a classification of negatively curved homogeneous Randers spaces of Douglas type. This results, in particular, in many examples of homogeneous non-Riemannian Finsler spaces with negative flag curvature. Finally, we prove a rigidity result that a homogeneous Randers space of Berwald type whose flag curvature is everywhere nonzero must be Riemannian.

scholarly journals An Algorithm for the Numerical Evaluation of Certain Finite Part Integrals

2011 ◽  
Vol 2011 ◽  
pp. 1-21
Author(s):  
Samir A. Ashour ◽  
Hany M. Ahmed
Keyword(s):  
Numerical Evaluation ◽  
Gaussian Quadrature ◽  
Quadrature Rules ◽  
Finite Part ◽  
Cauchy Principal ◽  
Cauchy Principal Value ◽  
Cauchy Principal Value Integrals ◽  
Principal Value

Many algorithms that have been proposed for the numerical evaluation of Cauchy principal value integrals are numerically unstable. In this work we present some formulae to evaluate the known Gaussian quadrature rules for finite part integrals , and extend Clenshow's algorithm to evaluate these integrals in a stable way.

The Spectra of Semi-Normal Singular Integral Operators

1970 ◽  
Vol 22 (1) ◽  
pp. 134-150 ◽  
Author(s):  
C. R. Putnam
Keyword(s):  
Hilbert Space ◽  
Singular Integral ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Cauchy Principal ◽  
Cauchy Principal Value ◽  
Principal Value ◽  
Image Position ◽  
Adjoint Operators

Suppose that(1.1)and define the bounded self-adjoint operators H and J on the Hilbert space L2(0, 1) by(1.2)the integral being a Cauchy principal valueIt is seen that(1.3)or, equivalently,(1.4)Since (Cƒ, ƒ) = π–1|(ƒ, ϕ)|2 ≧ 0, A is semi-normal. (For a discussion of such operators, see [4].)

The Hilbert transform of Schwartz distributions. II

1987 ◽  
Vol 102 (3) ◽  
pp. 553-559 ◽  
Author(s):  
M. Aslam Chaudhry ◽  
J. N. Pandey
Keyword(s):  
Hilbert Transform ◽  
Open Problem ◽  
Compact Support ◽  
Schwartz Space ◽  
Cauchy Principal ◽  
Cauchy Principal Value ◽  
Schwartz Distributions ◽  
Principal Value ◽  
Image Position ◽  
The Hilbert Transform

AbstractLet D(R) be the Schwartz space of C∞ functions with compact support on R and let H(D) be the space of all C∞ functions defined on R for which every element is the Hilbert transform of an element in D(R), i.e.where the integral is defined in the Cauchy principal-value sense. Introducing an appropriate topology in H(D), Pandey [3] defined the Hilbert transform Hf of f ∈ (D(R))′ as an element of (H(D))′ by the relationand then with an appropriate interpretation he proved that.In this paper we give an intrinsic description of the space H(D) and its topology, thereby providing a solution to an open problem posed by Pandey ([4], p. 90).

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