Euler–Poincaré formula in equal characteristic¶under ordinariness assumptions

2000 ◽  
Vol 102 (1) ◽  
pp. 1-24 ◽  
Author(s):  
Richard Pink
Keyword(s):  
Poincaré Formula

3D Image-Based Stereology

2012 ◽  
Vol 706-709 ◽  
pp. 2687-2692 ◽  
Author(s):  
Yoshitaka Adachi ◽  
Mayumi Ojima ◽  
Naoko Sato ◽  
Yuan Tsung Wang
Keyword(s):  
3D Structure ◽  
Line Length ◽  
Differential Geometry And Topology ◽  
Dual Phase ◽  
Topological Properties ◽  
Serial Sectioning ◽  
Metric Properties ◽  
Topological Features ◽  
Bonnet Theorem ◽  
Poincaré Formula

The features present in 3D structure have geometric properties that fall into two broad categories: topological and metric. Metric properties are generally the more familiar; these include volume, surface area, line length and curvature. Equally or even more important in some applications are the topological properties of features. The two principal topological properties are number per unit volume and connectivity. In the present study, a change in morphology of pearlite and dual phase microstructures was examined from differential geometry and topology viewpoint. 3D images of eutectoid pearlite and dual phase steels were obtained by reconstructing serial sectioning images. Their metric and topological features were evaluated using The Euler Poincare formula and The Gauss-Bonnet Theorem. In addition, newly developed fully-automated serial sectioning 3D microscope “Genus_3D” will be also introduced.

scholarly journals An Elementary Proof of the General Poincaré Formula for λ-additive Measures

2019 ◽  
Vol 24 (2) ◽  
pp. 173-185 ◽  
Author(s):  
József Dombi ◽  
Tamás Jónás
Keyword(s):  
Probability Theory ◽  
General Formula ◽  
Elementary Proof ◽  
Additive Measure ◽  
Limit Case ◽  
Additive Measures ◽  
Poincaré Formula

In a previous paper of ours [4], we presented the general formula for lambda-additive measure of union of n sets and gave a proof of it. That proof is based on the fact that the lambda-additive measure is representable. In this study, a novel and elementary proof of the formula for lambda-additive measure of the union of n sets is presented. Here, it is also demonstrated that, using elementary techniques, the well-known Poincare formula of probability theory is just a limit case of our general formula.

The general Poincaré formula for λ-additive measures

2019 ◽  
Vol 490 ◽  
pp. 285-291 ◽  
Author(s):  
József Dombi ◽  
Tamás Jónás
Keyword(s):  
Additive Measures ◽  
Poincaré Formula

The Case of SL(2); the Examples of Evans and Rudnick

2012 ◽  
Author(s):  
Nicholas M. Katz
Keyword(s):  
To Come ◽  
Poincaré Formula

This chapter treats both of these examples, as well as all the examples to come, using the Euler–Poincaré formula, cf. [Ray, Thm. 1] or [Ka-GKM, 2.3.1] or [Ka-SE, 4.6, (v) atop p. 113] or [De-ST, 3.2.1], to compute the “dimension” of the object N in question.

scholarly journals An Extension of Poincaré Formula in Integral Geometry

1951 ◽  
Vol 2 ◽  
pp. 55-61 ◽  
Author(s):  
Minoru Kurita
Keyword(s):  
Finite Length ◽  
Integral Geometry ◽  
Euclidean Plane ◽  
Kinematic Measure ◽  
Fixed Curve ◽  
Poincaré Formula

A curve c2 of finite length L2 moves on a euclidean plane. Let the number of points of intersection of c2 with the fixed, curve C1 of length Ls1 be n, and the element of kinematic measure of the position of c2 be dK.

On the Poincare formula and the Riemann singularity theorem over nodal curves

2008 ◽  
Vol 342 (4) ◽  
pp. 885-902 ◽  
Author(s):  
Usha N. Bhosle ◽  
A. J. Parameswaran
Keyword(s):  
Singularity Theorem ◽  
Nodal Curves ◽  
Poincaré Formula
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