Section 62 The scalar product (dot product)

2004 ◽  
pp. 533-537
Keyword(s):  
Scalar Product ◽  
Dot Product

Black-white contrast of dislocation loops

1978 ◽  
Vol 36 (1) ◽  
pp. 328-329
Author(s):  
J. J. Hren ◽  
W. D. Cooper ◽  
L. J. Sykes
Keyword(s):  
Electron Microscopy ◽  
Transmission Electron Microscopy ◽  
Dislocation Loops ◽  
Major Premise ◽  
Burgers Vector ◽  
Transmission Electron ◽  
Primary Means ◽  
Contrast Analysis ◽  
Dot Product ◽  
Imaging Conditions

Small dislocation loops observed by transmission electron microscopy exhibit a characteristic black-white strain contrast when observed under dynamical imaging conditions. In many cases, the topography and orientation of the image may be used to determine the nature of the loop crystallography. Two distinct but somewhat overlapping procedures have been developed for the contrast analysis and identification of small dislocation loops. One group of investigators has emphasized the use of the topography of the image as the principle tool for analysis. The major premise of this method is that the characteristic details of the image topography are dependent only on the magnitude of the dot product between the loop Burgers vector and the diffracting vector. This technique is commonly referred to as the (g•b) analysis. A second group of investigators has emphasized the use of the orientation of the direction of black-white contrast as the primary means of analysis.

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 Lexicographic Orders of Intuitionistic Fuzzy Values and Their Relationships

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 166 ◽  
Author(s):  
Feng Feng ◽  
Meiqi Liang ◽  
Hamido Fujita ◽  
Ronald Yager ◽  
Xiaoyan Liu
Keyword(s):  
Decision Making ◽  
Comparative Analysis ◽  
Scalar Product ◽  
Multiple Attribute Decision Making ◽  
Benchmark Problem ◽  
Score Functions ◽  
Intuitionistic Fuzzy ◽  
Multiple Attribute ◽  
Risk Investment ◽  
Membership Score

Intuitionistic fuzzy multiple attribute decision making deals with the issue of ranking alternatives based on the decision information quantified in terms of intuitionistic fuzzy values. Lexicographic orders can serve as efficient and indispensable tools for comparing intuitionistic fuzzy values. This paper introduces a number of lexicographic orders by means of several measures such as the membership, non-membership, score, accuracy and expectation score functions. Some equivalent characterizations and illustrative examples are provided, from which the relationships among these lexicographic orders are ascertained. We also propose three different compatible properties of preorders with respect to the algebraic sum and scalar product operations of intuitionistic fuzzy values, and apply them to the investigation of compatible properties of various lexicographic orders. In addition, a benchmark problem regarding risk investment is further explored to give a comparative analysis of different lexicographic orders and highlight the practical value of the obtained results for solving real-world decision-making problems.

scholarly journals NON-HERMITIAN OSCILLATOR-LIKE HAMILTONIANS AND λ-COHERENT STATES REVISITED

2001 ◽  
Vol 16 (02) ◽  
pp. 91-98 ◽  
Author(s):  
JULES BECKERS ◽  
NATHALIE DEBERGH ◽  
JOSÉ F. CARIÑENA ◽  
GIUSEPPE MARMO
Keyword(s):  
Harmonic Oscillator ◽  
Hilbert Spaces ◽  
Coherent States ◽  
Scalar Product ◽  
Points Of View ◽  
Usual Scalar ◽  
Usual Scalar Product

Previous λ-deformed non-Hermitian Hamiltonians with respect to the usual scalar product of Hilbert spaces dealing with harmonic oscillator-like developments are (re)considered with respect to a new scalar product in order to take into account their property of self-adjointness. The corresponding deformed λ-states lead to new families of coherent states according to the DOCS, AOCS and MUCS points of view.

scholarly journals Sphere and Dot Product Representations of Graphs

2012 ◽  
Vol 47 (3) ◽  
pp. 548-568 ◽  
Author(s):  
Ross J. Kang ◽  
Tobias Müller
Keyword(s):  
Dot Product
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