Tensor algebra and the inertia tensor

2011 ◽  
pp. 492-521
Author(s):  
R. Douglas Gregory
2020 ◽  
Vol 4 (OOPSLA) ◽  
pp. 1-30 ◽  
Author(s):  
Ryan Senanayake ◽  
Changwan Hong ◽  
Ziheng Wang ◽  
Amalee Wilson ◽  
Stephen Chou ◽  
...  

Author(s):  
MÁTYÁS DOMOKOS ◽  
VESSELIN DRENSKY

AbstractThe problem of finding generators of the subalgebra of invariants under the action of a group of automorphisms of a finite-dimensional Lie algebra on its universal enveloping algebra is reduced to finding homogeneous generators of the same group acting on the symmetric tensor algebra of the Lie algebra. This process is applied to prove a constructive Hilbert–Nagata Theorem (including degree bounds) for the algebra of invariants in a Lie nilpotent relatively free associative algebra endowed with an action induced by a representation of a reductive group.


Sensors ◽  
2021 ◽  
Vol 21 (15) ◽  
pp. 5076
Author(s):  
Javier Martinez-Roman ◽  
Ruben Puche-Panadero ◽  
Angel Sapena-Bano ◽  
Carla Terron-Santiago ◽  
Jordi Burriel-Valencia ◽  
...  

Induction machines (IMs) are one of the main sources of mechanical power in many industrial processes, especially squirrel cage IMs (SCIMs), due to their robustness and reliability. Their sudden stoppage due to undetected faults may cause costly production breakdowns. One of the most frequent types of faults are cage faults (bar and end ring segment breakages), especially in motors that directly drive high-inertia loads (such as fans), in motors with frequent starts and stops, and in case of poorly manufactured cage windings. A continuous monitoring of IMs is needed to reduce this risk, integrated in plant-wide condition based maintenance (CBM) systems. Diverse diagnostic techniques have been proposed in the technical literature, either data-based, detecting fault-characteristic perturbations in the data collected from the IM, and model-based, observing the differences between the data collected from the actual IM and from its digital twin model. In both cases, fast and accurate IM models are needed to develop and optimize the fault diagnosis techniques. On the one hand, the finite elements approach can provide highly accurate models, but its computational cost and processing requirements are very high to be used in on-line fault diagnostic systems. On the other hand, analytical models can be much faster, but they can be very complex in case of highly asymmetrical machines, such as IMs with multiple cage faults. In this work, a new method is proposed for the analytical modelling of IMs with asymmetrical cage windings using a tensor based approach, which greatly reduces this complexity by applying routine tensor algebra to obtain the parameters of the faulty IM model from the healthy one. This winding tensor approach is explained theoretically and validated with the diagnosis of a commercial IM with multiple cage faults.


2001 ◽  
Vol 16 (25) ◽  
pp. 4207-4222 ◽  
Author(s):  
J. A. NIETO ◽  
L. N. ALEJO-ARMENTA

By using tensor analysis, we find a connection between normed algebras and the parallelizability of the spheres S 1, S 3 and S 7. In this process, we discovered the analog of Hurwitz theorem for curved spaces and a geometrical unified formalism for the metric and the torsion. In order to achieve these goals we first develop a proof of Hurwitz theorem based on tensor analysis. It turns out that in contrast to the doubling procedure and Clifford algebra mechanism, our proof is entirely based on tensor algebra applied to the normed algebra condition. From the tersor analysis point of view our proof is straightforward and short. We also discuss a possible connection between our formalism and the Cayley–Dickson algebras and Hopf maps.


2019 ◽  
Vol 19 (1) ◽  
pp. 147-163 ◽  
Author(s):  
Alwin Stegeman ◽  
Lieven De Lathauwer

AbstractThe problem is considered of approximately solving a system of univariate polynomials with one or more common roots and its coefficients corrupted by noise. The goal is to estimate the underlying common roots from the noisy system. Symbolic algebra methods are not suitable for this. New Rayleigh quotient methods are proposed and evaluated for estimating the common roots. Using tensor algebra, reasonable starting values for the Rayleigh quotient methods can be computed. The new methods are compared to Gauss–Newton, solving an eigenvalue problem obtained from the generalized Sylvester matrix, and finding a cluster among the roots of all polynomials. In a simulation study it is shown that Gauss–Newton and a new Rayleigh quotient method perform best, where the latter is more accurate when other roots than the true common roots are close together.


2021 ◽  
Author(s):  
Liancheng Jia ◽  
Zizhang Luo ◽  
Liqiang Lu ◽  
Yun Liang
Keyword(s):  

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