On the invariant scalar products and the UIR of SO  (n,1)

1980 ◽  
Vol 21 (4) ◽  
pp. 675-679 ◽  
Author(s):  
Takayoshi Maekawa
Keyword(s):  
Invariant Scalar ◽  
Scalar Products

scholarly journals Nilpotent symmetries as a mechanism for Grand Unification

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Lars Andersson ◽  
András László ◽  
Błażej Ruba
Keyword(s):  
Gauge Field ◽  
Scalar Product ◽  
Local Symmetry ◽  
Matter Field ◽  
Gauge Fields ◽  
Spacetime Symmetries ◽  
Invariant Scalar ◽  
Spacetime Manifold ◽  
Local Symmetries ◽  
Dilatation Group

Abstract In the classic Coleman-Mandula no-go theorem which prohibits the unification of internal and spacetime symmetries, the assumption of the existence of a positive definite invariant scalar product on the Lie algebra of the internal group is essential. If one instead allows the scalar product to be positive semi-definite, this opens new possibilities for unification of gauge and spacetime symmetries. It follows from theorems on the structure of Lie algebras, that in the case of unified symmetries, the degenerate directions of the positive semi-definite invariant scalar product have to correspond to local symmetries with nilpotent generators. In this paper we construct a workable minimal toy model making use of this mechanism: it admits unified local symmetries having a compact (U(1)) component, a Lorentz (SL(2, ℂ)) component, and a nilpotent component gluing these together. The construction is such that the full unified symmetry group acts locally and faithfully on the matter field sector, whereas the gauge fields which would correspond to the nilpotent generators can be transformed out from the theory, leaving gauge fields only with compact charges. It is shown that already the ordinary Dirac equation admits an extremely simple prototype example for the above gauge field elimination mechanism: it has a local symmetry with corresponding eliminable gauge field, related to the dilatation group. The outlined symmetry unification mechanism can be used to by-pass the Coleman-Mandula and related no-go theorems in a way that is fundamentally different from supersymmetry. In particular, the mechanism avoids invocation of super-coordinates or extra dimensions for the underlying spacetime manifold.

scholarly journals Cosmology of non-Hermitian (C)PT-invariant scalar matter

2009 ◽  
Vol 171 ◽  
pp. 012043 ◽  
Author(s):  
Alexander A Andrianov ◽  
Francesco Cannata ◽  
Alexander Y Kamenshchik ◽  
Daniele Regoli
Keyword(s):  
Invariant Scalar ◽  
Scalar Matter

Features of determining the deformed state of a particle material during hot stamping of porous moldings

2021 ◽  
pp. 21-30
Author(s):  
B. G. Gasanov ◽  
A. A. Aganov ◽  
P. V. Sirotin
Keyword(s):  
Displacement Vector ◽  
Hot Stamping ◽  
Strain Tensor ◽  
Particle Material ◽  
Software Algorithms ◽  
Deformed State ◽  
Metal Strain ◽  
Scalar Products ◽  
Analytical Expressions ◽  
Kinetics Of

The paper describes main methods for assessing the deformed state of porous body metal frames developed by different authors based on the analysis of yield conditions and governing equations, using the principle of equivalent strains and stresses, and studying the kinetics of metal strain during pressing. Formulas were derived to determine the components of the powder particle material strain tensor through dyads, as scalar products of the basis vectors of the convected coordinate system at each moment of porous molding strain. The expediency of using the analytical expressions developed to determine the deformed state of the particle material was experimentally substantiated subject to the known displacement vector parameters of representative elements (macrostrains) of porous billets. The applications of well-known analytical expressions were established, and the proposed formulas proved applicable for the deformed state assessment of particle metal during the pressure processing of powder products of different configurations and designing billets with a defined porosity and geometric parameters as a basis for compiling software algorithms for the computer simulation of porous molding hot stamping.

Sunbeam vectors and scalar products

1997 ◽  
Vol 35 (5) ◽  
pp. 310-311
Author(s):  
Ernest Zebrowski
Keyword(s):  
Scalar Products

A WAVE APPROACH TO PATTERN RECOGNITION (WITH APPLICATION TO OPTICAL CHARACTER RECOGNITION)

1994 ◽  
Vol 04 (01) ◽  
pp. 193-207 ◽  
Author(s):  
VADIM BIKTASHEV ◽  
VALENTIN KRINSKY ◽  
HERMANN HAKEN
Keyword(s):  
Pattern Recognition ◽  
Character Recognition ◽  
Optical Character Recognition ◽  
Dissipative Structures ◽  
Binary Images ◽  
Optical Character ◽  
Nonlinear Version ◽  
Medium Quality ◽  
Scalar Products ◽  
Machine Reading

The possibility of using nonlinear media as a highly parallel computation tool is discussed, specifically for image classification and recognition. Some approaches of this type are known, that are based on stationary dissipative structures which can “measure” scalar products of images. In this paper, we exploit the analogy between binary images and point sets, and use the Hausdorff metrics for comparing the images. It does not require the measure at all, and is based only on the metrics of the space whose subsets we consider. In addition to Hausdorff distance, we suggest a new “nonlinear” version of this distance for comparison of images, called “autowave” distance. This distance can be calculated very easily and yields some additional advantages for pattern recognition (e.g. noise tolerance). The method was illustrated for the problem of machine reading (Optical Character Recognition). It was compared with some famous OCR programs for PC. On a medium quality xerocopy of a journal page, in the same conditions of learning and recognition, the autowave approach resulted in much fewer mistakes. The method can be realized using only one chip with simple uniform connection of the elements. In this case, it yields an increase in computation speed of several orders of magnitude.

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