Mixed curvature measures for sets of positive reach and a translative integral formula

1995 ◽  
Vol 57 (3) ◽  
pp. 259-283 ◽  
Author(s):  
J. Rataj ◽  
M. Z�hle
Keyword(s):  
Integral Formula ◽  
Positive Reach ◽  
Curvature Measures ◽  
Sets Of Positive Reach

Mixed curvature measures and a translative integral formula

1996 ◽  
Vol 28 (2) ◽  
pp. 341-341 ◽  
Author(s):  
Jan Rataj
Keyword(s):  
Integral Formula ◽  
Convex Bodies ◽  
Curvature Measure ◽  
Positive Reach ◽  
Rectifiable Currents ◽  
Curvature Measures ◽  
Dimension 2 ◽  
Image Position ◽  
Almost All ◽  
Sets Of Positive Reach

Let X, Y be two sets of positive reach in ℝd. The translative integral formula says that, for 0 ≦ k ≦ d − 1 and bounded Borel subsets A, B ε ℝd, where is the curvature measure (of order k) of X and is the mixed curvature measure of the sets X, Y and order r, S [1]. The mixed curvature measures are introduced by means of rectifiable currents, which leads to a relatively simple proof of (1). The proof needs an additional assumption on X, Y assuring that also reach (X ∩ Yz) > 0 for almost all z. This assumption is satisfied automatically for convex bodies, in dimension 2, or for sets with a sufficiently smooth boundary. Using the additivity of mixed curvature measures, (1) can be extended to unions of sets of positive reach.

Mixed curvature measures and a translative integral formula

1996 ◽  
Vol 28 (02) ◽  
pp. 341
Author(s):  
Jan Rataj
Keyword(s):  
Integral Formula ◽  
Convex Bodies ◽  
Curvature Measure ◽  
Positive Reach ◽  
Rectifiable Currents ◽  
Curvature Measures ◽  
Dimension 2 ◽  
Image Position ◽  
Almost All ◽  
Sets Of Positive Reach

Let X, Y be two sets of positive reach in ℝ d . The translative integral formula says that, for 0 ≦ k ≦ d − 1 and bounded Borel subsets A, B ε ℝ d , where is the curvature measure (of order k) of X and is the mixed curvature measure of the sets X, Y and order r, S [1]. The mixed curvature measures are introduced by means of rectifiable currents, which leads to a relatively simple proof of (1). The proof needs an additional assumption on X, Y assuring that also reach (X ∩ Yz ) > 0 for almost all z. This assumption is satisfied automatically for convex bodies, in dimension 2, or for sets with a sufficiently smooth boundary. Using the additivity of mixed curvature measures, (1) can be extended to unions of sets of positive reach.

An integral formula for the even part of the texture function or « the apparition of the fπ and fω ghost distributions »

1982 ◽  
Vol 43 (2) ◽  
pp. 189-195 ◽  
Author(s):  
Claude Esling ◽  
Jacques Muller ◽  
Hans-Joachim Bunge
Keyword(s):  
Integral Formula

scholarly journals ON p – q DUALITY AND EXPLICIT SOLUTIONS IN c ≤ 1 2D GRAVITY MODELS

1995 ◽  
Vol 10 (08) ◽  
pp. 1219-1236 ◽  
Author(s):  
S. KHARCHEV ◽  
A. MARSHAKOV
Keyword(s):  
String Theory ◽  
Exact Solutions ◽  
2D Gravity ◽  
Integral Formula ◽  
Integral Representations ◽  
Gravity Models ◽  
Explicit Solutions ◽  
String Equation ◽  
Duality Transformation

We study the role of integral representations in the description of nonperturbative solutions to c ≤ 1 string theory. A generic solution is determined by two functions, W(x) and Q(x), which behave at infinity like xp and xq respectively. The integral formula for arbitrary (p, q) models is derived, which explicitly realizes a duality transformation between (p, q) and (q, p) 2D gravity solutions. We also discuss the exact solutions to the string equation and reduction condition and present several explicit examples.

A new approach to curvature measures in linear shell theories

2020 ◽  
pp. 108128652097275
Author(s):  
Miroslav Šilhavý
Keyword(s):  
Strain Tensor ◽  
Middle Surface ◽  
Surface Strain ◽  
Affine Function ◽  
Shear Deformations ◽  
Strain Measure ◽  
Bending Strain ◽  
Curvature Measures ◽  
Tensor Formula ◽  
Strain Measures

The paper presents a coordinate-free analysis of deformation measures for shells modeled as 2D surfaces. These measures are represented by second-order tensors. As is well-known, two types are needed in general: the surface strain measure (deformations in tangential directions), and the bending strain measure (warping). Our approach first determines the 3D strain tensor E of a shear deformation of a 3D shell-like body and then linearizes E in two smallness parameters: the displacement and the distance of a point from the middle surface. The linearized expression is an affine function of the signed distance from the middle surface: the absolute term is the surface strain measure and the coefficient of the linear term is the bending strain measure. The main result of the paper determines these two tensors explicitly for general shear deformations and for the subcase of Kirchhoff-Love deformations. The derived surface strain measures are the classical ones: Naghdi’s surface strain measure generally and its well-known particular case for the Kirchhoff-Love deformations. With the bending strain measures comes a surprise: they are different from the traditional ones. For shear deformations our analysis provides a new tensor [Formula: see text], which is different from the widely used Naghdi’s bending strain tensor [Formula: see text]. In the particular case of Kirchhoff–Love deformations, the tensor [Formula: see text] reduces to a tensor [Formula: see text] introduced earlier by Anicic and Léger (Formulation bidimensionnelle exacte du modéle de coque 3D de Kirchhoff–Love. C R Acad Sci Paris I 1999; 329: 741–746). Again, [Formula: see text] is different from Koiter’s bending strain tensor [Formula: see text] (frequently used in this context). AMS 2010 classification: 74B99

scholarly journals The surface layer integral method for the modelling of the radial gravity gradient component over sea surface

2019 ◽  
Vol 9 (1) ◽  
pp. 127-132
Author(s):  
D. Zhao ◽  
Z. Gong ◽  
J. Feng
Keyword(s):  
Surface Layer ◽  
Integral Method ◽  
Integral Formula ◽  
Radial Component ◽  
Second Order ◽  
Sea Surface ◽  
Gravity Potential ◽  
Gravity Gradient ◽  
Disturbing Potential ◽  
Gradient Component

Abstract For the modelling and determination of the Earth’s external gravity potential as well as its second-order radial derivatives in the space near sea surface, the surface layer integral method was discussed in the paper. The reasons for the applicability of the method over sea surface were discussed. From the original integral formula of disturbing potential based on the surface layer method, the expression of the radial component of the gravity gradient tensor was derived. Furthermore, an identity relation was introduced to modify the formula in order to reduce the singularity problem. Numerical experiments carried out over the marine area of China show that, the modi-fied surface layer integral method effectively improves the accuracy and reliability of the calculation of the second-order radial gradient component of the disturbing potential near sea surface.

scholarly journals Remarks on Manifolds with Two-Sided Curvature Bounds

2021 ◽  
Vol 9 (1) ◽  
pp. 53-64
Author(s):  
Vitali Kapovitch ◽  
Alexander Lytchak
Keyword(s):  
Distance Functions ◽  
Cut Locus ◽  
Bounded Curvature ◽  
Curvature Bounds ◽  
Positive Reach

Abstract We discuss folklore statements about distance functions in manifolds with two-sided bounded curvature. The topics include regularity, subsets of positive reach and the cut locus.

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