scholarly journals Vector fields on some class of complete symmetric varieties

1986 ◽  
Vol 103 ◽  
pp. 85-94
Author(s):  
Yoshifumi Kato
Keyword(s):  
Coordinate System ◽  
Homogeneous Space ◽  
Weyl Group ◽  
Maximal Torus ◽  
Vector Fields ◽  
Local Coordinate System ◽  
Torus Action ◽  
Characteristic Classes ◽  
Open Set ◽  
Symmetric Varieties

In the previous papers [6], [7], we show that the set of an algebraic homogeneous space G/P fixed under the action of a maximal torus T can be canonically identified with the coset W1 = W/W1 of Weyl group W. We find a T invariant Zariski open set near each element w ∊ W1 and introduce a very nice local coordinate system such that we can express the maximal torus action explicitly. As a result, we become able to apply the study of J. B. Carrell and D. Lieberman [2], [3] to the space G/P and investigate the numerical properties of its characteristic classes and cycles.

BIFURCATION OF ROUGH HETEROCLINIC LOOP WITH ORBIT FLIPS

2012 ◽  
Vol 22 (11) ◽  
pp. 1250278 ◽  
Author(s):  
XINGBO LIU ◽  
ZHENZHEN WANG ◽  
DEMING ZHU
Keyword(s):  
Periodic Orbit ◽  
Coordinate System ◽  
Vector Fields ◽  
Poincaré Map ◽  
Local Coordinate System ◽  
Heteroclinic Orbit ◽  
Dimensional Vector ◽  
Homoclinic Loop ◽  
Heteroclinic Loop ◽  
Approximate Expressions

In this paper, heteroclinic loop bifurcations with double orbit flips are investigated in four-dimensional vector fields. We obtain the bifurcation equations by setting up a local coordinate system near the rough heteroclinic orbit and establishing the Poincaré map. By means of the bifurcation equations, we investigate the existence, coexistence and noncoexistence of periodic orbit, homoclinic loop and heteroclinic loop under some nongeneric conditions. The approximate expressions of corresponding bifurcation curves (or surfaces) are also given. An example of application is also given to demonstrate the existence of the heteroclinic loop with double orbit flips.

Establishment of local coordinate systems in the transition to the geodetic system of coordinates of GSK-2011

2017 ◽  
Vol 929 (11) ◽  
pp. 2-10
Author(s):  
A.V. Vinogradov
Keyword(s):  
Coordinate System ◽  
Small Businesses ◽  
Transition Period ◽  
Local Coordinate System ◽  
Initial Point ◽  
Central Meridian ◽  
Administrative District ◽  
Local Networks ◽  
Small Areas ◽  
Geodetic System

Pretty before long there will be transition to the geodetic system of coordinates of GSK-2011. For the transition period it is necessary to develop a method of recalculating coordinates from one system to another. The existing methods of recalculating coordinates are designed for recalculating coordinate points of state geodetic networks (GGS) and geodetic local networks (GSS). For small areas (administrative districts, populated areas) simplified methods are more acceptable. You need to choose the resampling methods that can be applied in small businesses, performing surveying works. The article presents the the results of calculations of changes of coordinates of the same point in GSK-2011 and SC-95 in six-degree zones of Gauss projection. It was found that in each region values of the shifts changed to small ones. Therefore, it is possible to convert the coordinates of the points by the simplified formulae. For recalculation from the coordinates of GSK-2011 in SK-95 or local coordinate system (WCS) of the administrative district it is necessary to find the origin of coordinates, scale value and rotation of the coordinate axes. The error of the conversion shall not exceed 0,001 m. The coordinates of the initial point of the local coordinate system relative to the central meridian of the local coordinate system shall be added in the list of parameters of the transition from local coordinate system to the state one.

Consistent co-rotational framework for Euler-Bernoulli and Timoshenko beam-column elements under distributed member loads

2021 ◽  
pp. 136943322098663
Author(s):  
Yi-Qun Tang ◽  
Wen-Feng Chen ◽  
Yao-Peng Liu ◽  
Siu-Lai Chan
Keyword(s):  
Coordinate System ◽  
Stiffness Matrix ◽  
Traditional Method ◽  
Local Coordinate System ◽  
Frame Structures ◽  
Global Coordinate System ◽  
Single Element ◽  
Geometrically Nonlinear ◽  
Geometrically Nonlinear Analysis ◽  
Geometric Stiffness

Conventional co-rotational formulations for geometrically nonlinear analysis are based on the assumption that the finite element is only subjected to nodal loads and as a result, they are not accurate for the elements under distributed member loads. The magnitude and direction of member loads are treated as constant in the global coordinate system, but they are essentially varying in the local coordinate system for the element undergoing a large rigid body rotation, leading to the change of nodal moments at element ends. Thus, there is a need to improve the co-rotational formulations to allow for the effect. This paper proposes a new consistent co-rotational formulation for both Euler-Bernoulli and Timoshenko two-dimensional beam-column elements subjected to distributed member loads. It is found that the equivalent nodal moments are affected by the element geometric change and consequently contribute to a part of geometric stiffness matrix. From this study, the results of both eigenvalue buckling and second-order direct analyses will be significantly improved. Several examples are used to verify the proposed formulation with comparison of the traditional method, which demonstrate the accuracy and reliability of the proposed method in buckling analysis of frame structures under distributed member loads using a single element per member.

scholarly journals Springer’s Weyl Group Representation via Localization

2017 ◽  
Vol 60 (3) ◽  
pp. 478-483 ◽  
Author(s):  
Jim Carrell ◽  
Kiumars Kaveh
Keyword(s):  
Weyl Group ◽  
Equivariant Cohomology ◽  
Group Representation ◽  
General Theorem ◽  
Torus Action ◽  
Cohomology Algebra ◽  
Reductive Algebraic Group ◽  
Point Set ◽  
Closed Subvariety ◽  
Springer Variety

AbstractLet G denote a reductive algebraic group over C and x a nilpotent element of its Lie algebra 𝔤. The Springer variety Bx is the closed subvariety of the flag variety B of G parameterizing the Borel subalgebras of 𝔤 containing x. It has the remarkable property that the Weyl group W of G admits a representation on the cohomology of Bx even though W rarely acts on Bx itself. Well-known constructions of this action due to Springer and others use technical machinery from algebraic geometry. The purpose of this note is to describe an elementary approach that gives this action when x is what we call parabolic-surjective. The idea is to use localization to construct an action of W on the equivariant cohomology algebra H*S (Bx), where S is a certain algebraic subtorus of G. This action descends to H*(Bx) via the forgetful map and gives the desired representation. The parabolic-surjective case includes all nilpotents of type A and, more generally, all nilpotents for which it is known that W acts on H*S (Bx) for some torus S. Our result is deduced from a general theorem describing when a group action on the cohomology of the ûxed point set of a torus action on a space lifts to the full cohomology algebra of the space.

scholarly journals MEASUREMENT OF VERTICALITY CONSTRUCTION OF BUSHINGS FOR LABORATORY CONSTRUCTION MATERIALS

2014 ◽  
Vol 40 (4) ◽  
pp. 171-174 ◽  
Author(s):  
Petr Jadviščok ◽  
Rostislav Dandoš ◽  
Tomaš Jiroušek
Keyword(s):  
Coordinate System ◽  
Building Materials ◽  
Civil Engineering ◽  
Local Coordinate System ◽  
Construction Materials ◽  
Polar Method ◽  
Standard Equipment ◽  
Final Number ◽  
Loading Tests ◽  
Design And Manufacture

This contribution describes process which was used for verticality measurement of the bushings for laboratory construction materials in the pavilion of testing. This pavilion is newly built in VŠB-TU Ostrava, Faculty of Civil Engineering, as part of the Testing house of the building materials. The requirement of the building investor was to determine the verticality of the bushings placed between the first aboveground and the first underground floor. After the building finishing, the bushings with the diameter 70 mm will be used for loading tests of various building materials. The final number of bushings is 169, and they are placed lengthwise and crosswise in the step of 750 mm. The centres of the bushings were measured by polar method in pavilion local coordinate system. The precision of the bushing centres determination was }5 mm according to the investor´s requirement. The precision would not be followed if the standard equipment for reflector fixing was used. In that case, it was necessary to design and manufacture special tool in the shape of truncated cone. On the top part was placed central pivot for reflector with additional plate bubble.

scholarly journals Computation of electrostatic fields in anisotropic human tissues using the Finite Integration Technique (FIT)

2005 ◽  
Vol 2 ◽  
pp. 309-313 ◽  
Author(s):  
V. C. Motresc ◽  
U. van Rienen
Keyword(s):  
Coordinate System ◽  
Electromagnetic Fields ◽  
Human Body ◽  
Local Coordinate System ◽  
Anisotropic Materials ◽  
The Body ◽  
Integration Technique ◽  
Data Set ◽  
Computer Data ◽  
Finite Integration Technique

Abstract. The exposure of human body to electromagnetic fields has in the recent years become a matter of great interest for scientists working in the area of biology and biomedicine. Due to the difficulty of performing measurements, accurate models of the human body, in the form of a computer data set, are used for computations of the fields inside the body by employing numerical methods such as the method used for our calculations, namely the Finite Integration Technique (FIT). A fact that has to be taken into account when computing electromagnetic fields in the human body is that some tissue classes, i.e. cardiac and skeletal muscles, have higher electrical conductivity and permittivity along fibers rather than across them. This property leads to diagonal conductivity and permittivity tensors only when expressing them in a local coordinate system while in a global coordinate system they become full tensors. The Finite Integration Technique (FIT) in its classical form can handle diagonally anisotropic materials quite effectively but it needed an extension for handling fully anisotropic materials. New electric voltages were placed on the grid and a new averaging method of conductivity and permittivity on the grid was found. In this paper, we present results from electrostatic computations performed with the extended version of FIT for fully anisotropic materials.

scholarly journals On Formality of Some Homogeneous Spaces

Symmetry ◽  
2019 ◽  
Vol 11 (8) ◽  
pp. 1011
Author(s):  
Aleksy Tralle
Keyword(s):  
Homogeneous Space ◽  
Root System ◽  
Lie Algebra ◽  
Lie Group ◽  
Maximal Torus ◽  
Homogeneous Spaces ◽  
Direct Proof ◽  
Classical Type ◽  
Sasakian Manifolds ◽  
Full Analysis

Let G / H be a homogeneous space of a compact simple classical Lie group G. Assume that the maximal torus T H of H is conjugate to a torus T β whose Lie algebra t β is the kernel of the maximal root β of the root system of the complexified Lie algebra g c . We prove that such homogeneous space is formal. As an application, we give a short direct proof of the formality property of compact homogeneous 3-Sasakian spaces of classical type. This is a complement to the work of Fernández, Muñoz, and Sanchez which contains a full analysis of the formality property of S O ( 3 ) -bundles over the Wolf spaces and the proof of the formality property of homogeneous 3-Sasakian manifolds as a corollary.

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