scholarly journals POLAR DECOMPOSITION OF THE k-FOLD PRODUCT OF LEBESGUE MEASURE ON ℝn

2012 ◽  
Vol 85 (2) ◽  
pp. 315-324 ◽  
Author(s):  
S. REZA MOGHADASI
Keyword(s):  
Lebesgue Measure ◽  
Polar Decomposition ◽  
Rotational Symmetry ◽  
Orthogonal Transformation ◽  
Geometric Measure ◽  
Measure Decomposition

AbstractThe Blaschke–Petkantschin formula is a geometric measure decomposition of the q-fold product of Lebesgue measure on ℝn. Here we discuss another decomposition called polar decomposition by considering ℝn×⋯×ℝn as ℳn×k and using its polar decomposition. This is a generalisation of the Blaschke–Petkantschin formula and may be useful when one needs to integrate a function g:ℝn×⋯×ℝn→ℝ with rotational symmetry, that is, for each orthogonal transformation O,g(O(x1),…,O(xk))=g(x1,…xk). As an application we compute the moments of a Gaussian determinant.

An induction proof of the generalized Blaschke-Petkantschin formula

1996 ◽  
Vol 28 (2) ◽  
pp. 334-334
Author(s):  
E. B. Vedel Jensen
Keyword(s):  
Lebesgue Measure ◽  
Hausdorff Measure ◽  
Integral Geometry ◽  
Geometric Measure ◽  
Measure Decomposition

The classical Blaschke-Petkanschin formula is a formula in integral geometry givmg a geometric measure decomposition of the q-fold product of Lebesgue measure. The original versions are due to Blaschke and Petkanschin in the 1930s. In Zähle (1990) and Jensen and Kiêu (1992), generalized versions have been derived, where Lebesgue measure is replaced by Hausdorff measure.

An induction proof of the generalized Blaschke-Petkantschin formula

1996 ◽  
Vol 28 (02) ◽  
pp. 334
Author(s):  
E. B. Vedel Jensen
Keyword(s):  
Lebesgue Measure ◽  
Hausdorff Measure ◽  
Integral Geometry ◽  
Geometric Measure ◽  
Measure Decomposition

The classical Blaschke-Petkanschin formula is a formula in integral geometry givmg a geometric measure decomposition of the q-fold product of Lebesgue measure. The original versions are due to Blaschke and Petkanschin in the 1930s. In Zähle (1990) and Jensen and Kiêu (1992), generalized versions have been derived, where Lebesgue measure is replaced by Hausdorff measure.

scholarly journals On Perimeters and Volumes of Fattened Sets

2019 ◽  
Vol 2019 ◽  
pp. 1-12
Author(s):  
Andrea C. G. Mennucci
Keyword(s):  
Lebesgue Measure ◽  
Measure Theory ◽  
Geometric Measure Theory ◽  
Coarea Formula ◽  
Compact Set ◽  
Geometric Measure ◽  
General Bound

In this paper we analyze the shape of fattened sets; given a compact set C⊂RN let Cr be its r-fattened set; we prove a general bound rP(Cr)≤NL({Cr∖C}) between the perimeter of Cr and the Lebesgue measure of Cr∖C. We provide two proofs: one elementary and one based on Geometric Measure Theory. Note that, by the Flemin–Rishel coarea formula, P(Cr) is integrable for r∈(0,a). We further show that for any integrable continuous decreasing function ψ:(0,1/2)→(0,∞) there exists a compact set C⊂RN such that P(Cr)≥ψ(r). These results solve a conjecture left open in (Mennucci and Duci, 2015) and provide new insight in applications where the fattened set plays an important role.

A note on the Blaschke-Petkantschin formula, Riesz distributions, and Drury’s identity

2018 ◽  
Vol 21 (6) ◽  
pp. 1641-1650
Author(s):  
Boris Rubin
Keyword(s):  
Lebesgue Measure ◽  
Laplace Operator ◽  
Elementary Proof ◽  
Polar Decomposition ◽  
Fractional Powers ◽  
Linear Subspaces ◽  
Precise Meaning ◽  
Matrix Spaces ◽  
Admissible Functions

Abstract The Blaschke-Petkantschin formula is a variant of the polar decomposition of the k-fold Lebesgue measure on ℝn in terms of the corresponding measures on k-dimensional linear subspaces of ℝn. We suggest a new elementary proof of this famous formula and discuss its connection with Riesz distributions associated with fractional powers of the Cayley-Laplace operator on matrix spaces. Another application of our proof is the celebrated Drury identity that plays a key role in the study of mapping properties of the Radon-John k-plane transforms. Our proof gives precise meaning to the constants in Drury’s identity and to the class of admissible functions.

Comparison of Calculated and Recorded Electron Energy-Loss Spectra of Aluminium and Carbon

1990 ◽  
Vol 48 (2) ◽  
pp. 64-65
Author(s):  
L. Reimer ◽  
R. Oelgeklaus
Keyword(s):  
Fourier Transform ◽  
Energy Loss ◽  
Electron Energy ◽  
Rotational Symmetry ◽  
Mean Free Path ◽  
Single Scattering ◽  
Specimen Thickness ◽  
Two Dimensional ◽  
Image Position ◽  
Electron Energy Loss

Quantitative electron energy-loss spectroscopy (EELS) needs a correction for the limited collection aperture α and a deconvolution of recorded spectra for eliminating the influence of multiple inelastic scattering. Reversely, it is of interest to calculate the influence of multiple scattering on EELS. The distribution f(w,θ,z) of scattered electrons as a function of energy loss w, scattering angle θ and reduced specimen thickness z=t/Λ (Λ=total mean-free-path) can either be recorded by angular-resolved EELS or calculated by a convolution of a normalized single-scattering function ϕ(w,θ). For rotational symmetry in angle (amorphous or polycrystalline specimens) this can be realised by the following sequence of operations :(1)where the two-dimensional distribution in angle is reduced to a one-dimensional function by a projection P, T is a two-dimensional Fourier transform in angle θ and energy loss w and the exponent -1 indicates a deprojection and inverse Fourier transform, respectively.

LEBESGUE MEASURE AND GAMBLING

1993 ◽  
Vol 19 (1) ◽  
pp. 40
Author(s):  
Kanovei ◽  
Linton
Keyword(s):  
Lebesgue Measure

Polar Decomposition of Wiener Measure and Schwarzian Integrals

2020 ◽  
Vol 41 (4) ◽  
pp. 709-713
Author(s):  
E. T. Shavgulidze ◽  
N. E. Shavgulidze
Keyword(s):  
Polar Decomposition ◽  
Wiener Measure

On a Choquet-Stieltjes type integral on intervals

2019 ◽  
Vol 69 (4) ◽  
pp. 801-814 ◽  
Author(s):  
Sorin G. Gal
Keyword(s):  
Lebesgue Measure ◽  
Choquet Integral ◽  
Stieltjes Integral ◽  
Line Integral ◽  
Type Integral ◽  
Choquet Integrals ◽  
Lebesgue Measures ◽  
Stieltjes Type

Abstract In this paper we introduce a new concept of Choquet-Stieltjes integral of f with respect to g on intervals, as a limit of Choquet integrals with respect to a capacity μ. For g(t) = t, one reduces to the usual Choquet integral and unlike the old known concept of Choquet-Stieltjes integral, for μ the Lebesgue measure, one reduces to the usual Riemann-Stieltjes integral. In the case of distorted Lebesgue measures, several properties of this new integral are obtained. As an application, the concept of Choquet line integral of second kind is introduced and some of its properties are obtained.

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