scholarly journals Covariant Differential Operator under General Coordinate Transformation and Structure of Lagrangian Operator

1976 ◽  
Vol 55 (4) ◽  
pp. 1276-1287 ◽  
Author(s):  
T. Okabayashi
Keyword(s):  
Differential Operator ◽  
Coordinate Transformation ◽  
Lagrangian Operator ◽  
General Coordinate Transformation

Study of Large Scale Measurement Method Based on Leapfrog Principle

2011 ◽  
Vol 130-134 ◽  
pp. 1560-1563
Author(s):  
Long Jiang Zheng ◽  
Xue Li ◽  
Ling Ling Qin ◽  
Hong Bin Chen ◽  
Xue Gao ◽  
...  
Keyword(s):  
Coordinate Transformation ◽  
Large Scale ◽  
Manufacturing Industry ◽  
Coordinate Measuring Machine ◽  
Measuring Method ◽  
Measuring System ◽  
Wide Range ◽  
General Coordinate Transformation ◽  
Measuring Machine ◽  
The Mathematical Model

At present，large scale and space coordinates measuring system with wide-range and high-precision has been widely used in modern manufacturing industry. In this paper, large scale measuring method based on leapfrog principle of flexible three coordinate measuring machine is described. The mathematical model of coordinate transformation is built and the general coordinate transformation formula after number of times leapfrogging is derived. The best positioning and each step of leapfrog are given.

scholarly journals AN INVARIANT APPROACH TO DYNAMICAL FUZZY SPACES WITH A THREE-INDEX VARIABLE

2006 ◽  
Vol 21 (13) ◽  
pp. 1017-1028 ◽  
Author(s):  
NAOKI SASAKURA
Keyword(s):  
Field Theory ◽  
Scalar Field ◽  
Linear Transformation ◽  
Coordinate Transformation ◽  
General Linear ◽  
Variable Formula ◽  
Fuzzy Space ◽  
General Coordinate Transformation ◽  
Invariant Approach ◽  
Invariant Tensors

A dynamical fuzzy space might be described by a three-index variable [Formula: see text], which determines the algebraic relations [Formula: see text] among the functions fa on the fuzzy space. A fuzzy analogue of the general coordinate transformation would be given by the general linear transformation on fa. We study equations for the three-index variable invariant under the general linear transformation and show that the solutions can be generally constructed from the invariant tensors of Lie groups. As specific examples, we study SO(3) symmetric solutions and discuss the construction of a scalar field theory on a fuzzy two-sphere within this framework.

Sturm-Liouville Problem for Stationary Differential Operator with Nonlocal Two-Point Boundary Conditions

2006 ◽  
Vol 11 (1) ◽  
pp. 47-78 ◽  
Author(s):  
S. Pečiulytė ◽  
A. Štikonas
Keyword(s):  
Boundary Condition ◽  
Differential Operator ◽  
Boundary Conditions ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Liouville Problem ◽  
Complex Case ◽  
Qualitative Behavior ◽  
Point Boundary ◽  
Sturm Liouville

The Sturm-Liouville problem with various types of two-point boundary conditions is considered in this paper. In the first part of the paper, we investigate the Sturm-Liouville problem in three cases of nonlocal two-point boundary conditions. We prove general properties of the eigenfunctions and eigenvalues for such a problem in the complex case. In the second part, we investigate the case of real eigenvalues. It is analyzed how the spectrum of these problems depends on the boundary condition parameters. Qualitative behavior of all eigenvalues subject to the nonlocal boundary condition parameters is described.

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