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.

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 Mathematics education undergraduates’ personal definitions of the notion of angle of contiguity in Kinematics

2021 ◽  
Vol 6 (2) ◽  
pp. 111-127
Author(s):  
Zakaria Ndemo ◽  
David Mtetwa
Keyword(s):  
Student Teachers ◽  
Mathematics Instruction ◽  
State University ◽  
Written Responses ◽  
Mathematical Definition ◽  
Concept Images ◽  
Abstraction In Context ◽  
Definition Of ◽  
Mathematical Definitions

The concept of a mathematical definition causes severe difficulties among students during problem solving and proving activities. Students’ difficulties with the use of mathematical definitions often arise from the fact that students are often given those definitions instead of constructing them. With the aim of developing an understanding of the kinds of student teachers evoked concept images of the notion of  angle of contiguity, a qualitative case study was conducted at one state university in Zimbabwe. Purposive sampling was used to select 28 mathematics undergraduate student teachers who responded to a test item. Qualitative data analysis was guided by ideas drawn from the theoretical framework of Abstraction in Context and idea of imperative features of a mathematical definition.  Student teachers written responses revealed that student teachers personal concept definitions consisted of ambiguous and irrelevant formulations that did not capture the essence of the idea of the angle of contiguity. In some cases their responses were not consistent with the definition of the angle of contiguity.  Although there were a few instances of adequate descriptions of the concept, (8 out of  32) these and the inadequate descriptors elicited can contribute significantly towards efforts intended to improve mathematics instruction.  Improved mathematics instruction will lead to enhanced conceptualizations of mathematics concepts.

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 Fixed Point of Interpolative Rus–Reich–Ćirić Contraction Mapping on Rectangular Quasi-Partial b-Metric Space

Symmetry ◽  
2020 ◽  
Vol 13 (1) ◽  
pp. 32
Author(s):  
Pragati Gautam ◽  
Luis Manuel Sánchez Ruiz ◽  
Swapnil Verma
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Real Number ◽  
Metric Space ◽  
Contraction Mapping ◽  
Metric Spaces ◽  
Rectangular Metric Space ◽  
New Type ◽  
Symmetry Requirement ◽  
Definition Of

The purpose of this study is to introduce a new type of extended metric space, i.e., the rectangular quasi-partial b-metric space, which means a relaxation of the symmetry requirement of metric spaces, by including a real number s in the definition of the rectangular metric space defined by Branciari. Here, we obtain a fixed point theorem for interpolative Rus–Reich–Ćirić contraction mappings in the realm of rectangular quasi-partial b-metric spaces. Furthermore, an example is also illustrated to present the applicability of our result.

scholarly journals Strong submeasures and applications to non-compact dynamical systems

2020 ◽  
pp. 1-23
Author(s):  
TUYEN TRUNG TRUONG
Keyword(s):  
Metric Space ◽  
Bounded Operator ◽  
Continuous Functions ◽  
Open Dense Subset ◽  
Compact Metric Space ◽  
Basic Properties ◽  
Banach Theorem ◽  
Complex Varieties ◽  
Definition Of ◽  
Meromorphic Maps

Abstract A strong submeasure on a compact metric space X is a sub-linear and bounded operator on the space of continuous functions on X. A strong submeasure is positive if it is non-decreasing. By the Hahn–Banach theorem, a positive strong submeasure is the supremum of a non-empty collection of measures whose masses are uniformly bounded from above. There are many natural examples of continuous maps of the form $f:U\rightarrow X$ , where X is a compact metric space and $U\subset X$ is an open-dense subset, where f cannot extend to a reasonable function on X. We can mention cases such as transcendental maps of $\mathbb {C}$ , meromorphic maps on compact complex varieties, or continuous self-maps $f:U\rightarrow U$ of a dense open subset $U\subset X$ where X is a compact metric space. For the aforementioned mentioned the use of measures is not sufficient to establish the basic properties of ergodic theory, such as the existence of invariant measures or a reasonable definition of measure-theoretic entropy and topological entropy. In this paper we show that strong submeasures can be used to completely resolve the issue and establish these basic properties. In another paper we apply strong submeasures to the intersection of positive closed $(1,1)$ currents on compact Kähler manifolds.

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 Fixed Points for a Pair of F-Dominated Contractive Mappings in Rectangular b-Metric Spaces with Graph

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 884 ◽  
Author(s):  
Tahair Rasham ◽  
Giuseppe Marino ◽  
Abdullah Shoaib
Keyword(s):  
Fixed Point ◽  
Metric Space ◽  
Metric Spaces ◽  
Partial Metric Space ◽  
Contractive Condition ◽  
Partial Metric ◽  
Contractive Mappings ◽  
Dislocated Metric Space ◽  
Definition Of ◽  
Dislocated Metric

Recently, George et al. (in Georgea, R.; Radenovicb, S.; Reshmac, K.P.; Shuklad, S. Rectangular b-metric space and contraction principles. J. Nonlinear Sci. Appl. 2015, 8, 1005–1013) furnished the notion of rectangular b-metric pace (RBMS) by taking the place of the binary sum of triangular inequality in the definition of a b-metric space ternary sum and proved some results for Banach and Kannan contractions in such space. In this paper, we achieved fixed-point results for a pair of F-dominated mappings fulfilling a generalized rational F-dominated contractive condition in the better framework of complete rectangular b-metric spaces complete rectangular b-metric spaces. Some new fixed-point results with graphic contractions for a pair of graph-dominated mappings on rectangular b-metric space have been obtained. Some examples are given to illustrate our conclusions. New results in ordered spaces, partial b-metric space, dislocated metric space, dislocated b-metric space, partial metric space, b-metric space, rectangular metric spaces, and metric space can be obtained as corollaries of our results.

INTERTWINED BASINS OF ATTRACTION

2012 ◽  
Vol 22 (06) ◽  
pp. 1250130
Author(s):  
CHANGMING DING
Keyword(s):  
Dynamical Systems ◽  
Metric Space ◽  
Basins Of Attraction ◽  
General Definition ◽  
Sufficient Condition ◽  
Topological Equivalence ◽  
Definition Of

This paper deals with intertwined basins of attraction for dynamical systems in a metric space. After giving a general definition of intertwining property, which is preserved by a topological equivalence between dynamical systems, we present a sufficient condition to guarantee the existence of intertwined basins for dynamical systems in ℝn.

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