scholarly journals Chirality of Dirac Spinors Revisited

Symmetry ◽  
2020 ◽  
Vol 12 (4) ◽  
pp. 616
Author(s):  
Michel Petitjean
Keyword(s):  
Group Theory ◽  
Clifford Algebras ◽  
Euclidean Spaces ◽  
Dirac Algebra ◽  
Mathematical Definition ◽  
Higher Dimensional ◽  
Symmetry Operators ◽  
Definition Of ◽  
Dirac Spinors

We emphasize the differences between the chirality concept applied to relativistic fermions and the ususal chirality concept in Euclidean spaces. We introduce the gamma groups and we use them to classify as direct or indirect the symmetry operators encountered in the context of Dirac algebra. Then we show how a recent general mathematical definition of chirality unifies the chirality concepts and resolve conflicting conclusions about symmetry operators, and particularly about the so-called chirality operator. The proofs are based on group theory rather than on Clifford algebras. The results are independent on the representations of Dirac gamma matrices, and stand for higher dimensional ones.

scholarly journals About Chirality in Minkowski Spacetime

Symmetry ◽  
2019 ◽  
Vol 11 (10) ◽  
pp. 1320 ◽  
Author(s):  
Michel Petitjean
Keyword(s):  
Metric Space ◽  
Time Reversal ◽  
Minkowski Spacetime ◽  
Euclidean Spaces ◽  
Mathematical Definition ◽  
Definition Of ◽  
Mathematical Definitions ◽  
Lorentz Boosts

In this paper, we show that Lorentz boosts are direct isometries according to the recent mathematical definitions of direct and indirect isometries and of chirality, working for any metric space. Here, these definitions are extended to the Minkowski spacetime. We also show that the composition of parity inversion and time reversal is an indirect isometry, which is the opposite of what could be expected in Euclidean spaces. It is expected that the extended mathematical definition of chirality presented here can contribute to the unification of several definitions of chirality in space and in spacetime, and that it helps clarify the ubiquitous concept of chirality.

A Hull Normal Based Approach for Cylindrical Size Assessment

2011 ◽  
Vol 133 (1) ◽  
Author(s):  
Steven Turek ◽  
Sam Anand
Keyword(s):  
New York ◽  
Least Squares ◽  
Measured Data ◽  
Size Determination ◽  
Mathematical Definition ◽  
Dimensioning And Tolerancing ◽  
Normal Method ◽  
Computational Metrology ◽  
American Society ◽  
Definition Of

Digital measurement devices, such as coordinate measuring machines, laser scanning devices, and digital imaging, can provide highly accurate and precise coordinate data representing the sampled surface. However, this discrete measurement process can only account for measured data points, not the entire continuous form, and is heavily influenced by the algorithm that interprets the measured data. The definition of cylindrical size for an external feature as specified by ASME Y14.5.1M-1994 [The American Society of Mechanical Engineers, 1995, Dimensioning and Tolerancing, ASME Standard Y14.5M-1994, ASME, New York, NY; The American Society of Mechanical Engineers, 1995, Mathematical Definition of Dimensioning and Tolerancing Principles, ASME Standard Y14.5.1M-1994, ASME, New York, NY] matches the analytical definition of a minimum circumscribing cylinder (MCC) when rule no. 1 [The American Society of Mechanical Engineers, 1995, Dimensioning and Tolerancing, ASME Standard Y14.5M-1994, ASME, New York, NY; The American Society of Mechanical Engineers, 1995, Mathematical Definition of Dimensioning and Tolerancing Principles, ASME Standard Y14.5.1M-1994, ASME, New York, NY] is applied to ensure a linear axis. Even though the MCC is a logical choice for size determination, it is highly sensitive to the sampling method and any uncertainties encountered in that process. Determining the least-sum-of-squares solution is an alternative method commonly utilized in size determination. However, the least-squares formulation seeks an optimal solution not based on the cylindrical size definition [The American Society of Mechanical Engineers, 1995, Dimensioning and Tolerancing, ASME Standard Y14.5M-1994, ASME, New York, NY; The American Society of Mechanical Engineers, 1995, Mathematical Definition of Dimensioning and Tolerancing Principles, ASME Standard Y14.5.1M-1994, ASME, New York, NY] and thus has been shown to be biased [Hopp, 1993, “Computational Metrology,” Manuf. Rev., 6(4), pp. 295–304; Nassef, and ElMaraghy, 1999, “Determination of Best Objective Function for Evaluating Geometric Deviations,” Int. J. Adv. Manuf. Technol., 15, pp. 90–95]. This work builds upon previous research in which the hull normal method was presented to determine the size of cylindrical bosses when rule no. 1 is applied [Turek, and Anand, 2007, “A Hull Normal Approach for Determining the Size of Cylindrical Features,” ASME, Atlanta, GA]. A thorough analysis of the hull normal method’s performance in various circumstances is presented here to validate it as a superior alternative to the least-squares and MCC solutions for size evaluation. The goal of the hull normal method is to recreate the sampled surface using computational geometry methods and to determine the cylinder’s axis and radius based upon it. Based on repetitive analyses of random samples of data from several measured parts and generated forms, it was concluded that the hull normal method outperformed all traditional solution methods. The hull normal method proved to be robust by having a lower bias and distributions that were skewed toward the true value of the radius, regardless of the amount of form error.

Mathematical definition of the curve of Spee in permanent healthy dentitions in man

1992 ◽  
Vol 37 (9) ◽  
pp. 691-694 ◽  
Author(s):  
V.F. Ferrario ◽  
C. Sforza ◽  
A. Miani ◽  
A. Colombo ◽  
G. Tartaglia
Keyword(s):  
Mathematical Definition ◽  
Curve Of Spee ◽  
Definition Of

scholarly journals From Euclidean Spaces to Metric Spaces

2019 ◽  
Vol 8 (1) ◽  
Author(s):  
Ryan Joseph Rogers ◽  
Ning Zhong
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Euclidean Spaces ◽  
Definition Of

In this note, we provide the definition of a metric space and establish that, while all Euclidean spaces are metric spaces, not all metric spaces are Euclidean spaces. It is then natural and interesting to ask which theorems that hold in Euclidean spaces can be extended to general metric spaces and which ones cannot be extended. We survey this topic by considering six well-known theorems which hold in Euclidean spaces and rigorously exploring their validities in general metric spaces.

scholarly journals Mathematical and Physical Continuity

2008 ◽  
Vol 6 ◽  
Author(s):  
Mark Colyvan ◽  
Kenny Easwaran
Keyword(s):  
Common Sense ◽  
Natural Question ◽  
Alternative Definition ◽  
Continuous Motion ◽  
Mathematical Definition ◽  
Definition Of ◽  
Point Of Departure

There is general agreement in mathematics about what continuity is. In this paper we examine how well the mathematical definition lines up with common sense notions. We use a recent paper by Hud Hudson as a point of departure. Hudson argues that two objects moving continuously can coincide for all but the last moment of their histories and yet be separated in space at the end of this last moment. It turns out that Hudson’s construction does not deliver mathematically continuous motion, but the natural question then is whether there is any merit in the alternative definition of continuity that he implicitly invokes.

A City is Not a Semilattice Either

1976 ◽  
Vol 8 (4) ◽  
pp. 375-384 ◽  
Author(s):  
F Harary ◽  
J Rockey
Keyword(s):  
Graphical Analysis ◽  
Theoretical Concept ◽  
Theoretical Terms ◽  
Natural Form ◽  
Mathematical Definition ◽  
Christopher Alexander ◽  
Structural Configurations ◽  
Definition Of ◽  
The Relationship ◽  
The City

In 1965 Christopher Alexander took the original step of analysing the city in graph theoretical terms and concluded that its historical or natural form is a semilattice and that urban planners of the future should adhere to this model. The idea was well received in architectural circles and has passed without serious challenge. In this paper, the value of such analysis is once again emphasized, although some of Alexander's arguments and his conclusions are refuted. Beginning with an exposition of the relationship between the graph theoretical concept of a tree, and the representation of a tree by a family of sets, we present a mathematical definition of a semilattice and discuss the ‘points’ and ‘lines’ of a graph in terms of a city, concluding that it is neither a tree nor a semilattice. This clears the ground for future graphical analysis. It seems that even general structural configurations, such as graphs or digraphs with certain specified properties, will fail to characterize a city, whose complexity, at this stage, may well continue to be understood more readily through negative rather than positive descriptions.

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