scholarly journals Some Applications of Clifford Algebra in Geometry

2020 ◽  
Author(s):  
Ying-Qiu Gu
Keyword(s):  
Clifford Algebra ◽  
Simple Calculation ◽  
Riemann Geometry ◽  
Geometric Concepts ◽  
Vector Algebra ◽  
Complex Quaternion ◽  
Concise Representation ◽  
Definition Of ◽  
Mathematics And Physics ◽  
Area And Volume

In this paper, we provide some enlightening examples of the application of Clifford algebra in geometry, which show the concise representation, simple calculation and profound insight of this algebra. The definition of Clifford algebra implies geometric concepts such as vector, length, angle, area and volume, and unifies the calculus of scalar, spinor, vector and tensor, so that it is able to naturally describe all variables and calculus in geometry and physics. Clifford algebra unifies and generalizes real number, complex, quaternion and vector algebra, converts complicated relations and operations into intuitive matrix algebra independent of coordinate systems. By localizing the basis or frame of space-time and introducing differential and connection operators, Clifford algebra also contains Riemann geometry. Clifford algebra provides a unified, standard, elegant and open language and tools for numerous complicated mathematical and physical theories. Clifford algebra calculus is an arithmetic-like operation that can be well understood by everyone. This feature is very useful for teaching purposes, and popularizing Clifford algebra in high schools and universities will greatly improve the efficiency of students to learn fundamental knowledge of mathematics and physics. So Clifford algebra can be expected to complete a new big synthesis of scientific knowledge.

Kinematics of Meshing Surfaces Using Geometric Algebra

2003 ◽  
Author(s):  
Scott M. Miller
Keyword(s):  
Geometric Algebra ◽  
Screw Theory ◽  
Three Dimensional ◽  
Normal Vector ◽  
The Past ◽  
Geometric Concepts ◽  
Vector Algebra ◽  
The Common ◽  
Two Bodies ◽  
Theoretical Developments

As is well known, analysis of two surfaces in mesh plays a fundamental role in gear theory. In the past, special coordinate systems, vector algebra, or screw theory was used to analyze the kinematics of meshing. The approach here instead relies on geometric algebra, an extension of conventional vector algebra. The elegance of geometric algebra for theoretical developments is demonstrated by examining the so-called “equation of meshing,” which requires that the relative velocity of two bodies at a point of contact be perpendicular to the common surface normal vector. With surprisingly little effort, several alternative forms of the equation of meshing are generated and, subsequently, interpreted geometrically. Via straightforward algebraic manipulations, the results of screw theory and vector algebra are unified. Due to the simplicity with which complex geometric concepts are expressed and manipulated, the effort required to grasp the general three-dimensional meshing of surfaces is minimized.

An intuitive introduction to the Euclidean concept of betweenness

1968 ◽  
Vol 15 (8) ◽  
pp. 683-686
Author(s):  
Tom Denmark
Keyword(s):  
Mathematics Curriculum ◽  
Careful Consideration ◽  
Line Segments ◽  
Fundamental Notion ◽  
Geometric Concepts ◽  
Definition Of

If geometry is to be an integral part of the mathematics curriculum for Grades K–6, then careful consideration must be given to the provision of a sound basis for the development of geometric concepts. One fundamental notion which is often overlooked or neglected is the concept of a point B being between two points A and C. Since the Euclidean definition of a point lying between two other points is essential to the formulation of definitions for line segments, rays, and lines, this concept should be introduced early in the children's study of geometry

STABLE QUASIPARTICLE PICTURE IN THERMAL QUANTUM FIELD PHYSICS

1994 ◽  
Vol 09 (10) ◽  
pp. 1703-1729 ◽  
Author(s):  
H. CHU ◽  
H. UMEZAWA
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Degrees Of Freedom ◽  
Fock Space ◽  
Quantum Field ◽  
Self Energy ◽  
Vector Algebra ◽  
Physical Particle ◽  
Usual Quantum ◽  
Definition Of

It is well known that physical particles are thermally dissipative at finite temperature. In this paper we reformulate both the equilibrium and nonequilibrium thermal field theories in terms of stable quasiparticles. We will redefine the thermal doublets, the double tilde conjugation rules and the thermal Bogoliubov transformations so that our theory can be consistent for most general situations. All operators, including the dissipative physical particle operators, are realized in a Fock space defined by the stable quasiparticles. The propagators of the physical particles are expressed in terms of the operators of such stable quasiparticles, which is a simple diagonal matrix with the diagonal elements being the temporal step functions, same as the propagators in the usual quantum field theory without thermal degrees of freedom. The proper self-energies are also expressed in terms of these stable quasiparticle propagators. This formalism inherits the definition of on-shell self-energy in the usual quantum field theory. With this definition, a self-consistent renormalization is formulated which leads to quantum Boltzmann equation and the entropy law. With the aid of a doublet vector algebra we have an extremely simple recipe for computing Feynman diagrams. We apply this recipe to several examples of equilibrium and nonequilibrium two-point functions, and to the kinetic equation for the particle numbers.

scholarly journals Analyzing Educational Objectives that Include Critical Thinking: Dot Product Problems in Vector Algebra

2020 ◽  
Vol 2 (2) ◽  
pp. 108-118
Author(s):  
Aneta Gacovska-Barandovska ◽  
Vesna Celakoska-Jordanova ◽  
Emilija Celakoska
Keyword(s):  
First Year ◽  
Mathematical Activity ◽  
Solo Taxonomy ◽  
Algebra Students ◽  
Individual Interviews ◽  
Vector Algebra ◽  
Dot Product ◽  
Definition Of ◽  
The University ◽  
First Year University

The primary and secondary school educational system should be stable and any upgrading reforms should be made gradually and consistently. This is especially important in mathematics education, since the element of logical reasoning while learning is more prominent there. Inconsistencies in reforms generate deficiencies in the higher levels of young students’ reasoning skills and this situation continues on the university stage of education. We will report our findings about the reasoning of first-year university students on elements of geometry and associated algebra. We conducted an experiment where students’ understanding of the definition of dot product of two vectors, cosine function and linear (in)dependence of vectors is evaluated, and address their mathematical activity to provide insight into the key elements of the problem they are solving. We use Bloom’s and SOLO taxonomy as a tool for the assessment of our findings. We obtained the data from written exams given to vector algebra students and also from individual interviews.   

Clifford Algebra to Geometric Calculus. A Unified Language for Mathematics and Physics

1985 ◽  
Vol 53 (5) ◽  
pp. 510-511 ◽  
Author(s):  
David Hestenes ◽  
Garret Sobczyk ◽  
James S. Marsh
Keyword(s):  
Clifford Algebra ◽  
Geometric Calculus ◽  
Mathematics And Physics

Learning Geometric Concepts through Ceramic Tile Design

2003 ◽  
Vol 9 (3) ◽  
pp. 134-140
Author(s):  
Clare V. Bell
Keyword(s):  
Native American ◽  
Ceramic Tile ◽  
Cultural Representation ◽  
Religious Art ◽  
Geometric Patterns ◽  
Geometric Figures ◽  
Geometric Concepts ◽  
The Creation ◽  
Definition Of

Symmetry and geometric patterns are commonly used in the creation of designs that symbolize and contribute to the definition of culture. Native American weaving and pottery designs, Mexican tiles, and Islamic religious art are forms of cultural representation that rely heavily on a repetition of geometric figures and symmetry. These items are used as examples of geometric art for the lessons in this article (see fig. 1).

scholarly journals Global Dimensional Mathematics

2021 ◽  
pp. 1-10
Author(s):  
Olivier Denis
Keyword(s):  
Prime Numbers ◽  
Mathematical Framework ◽  
Total Change ◽  
Mathematical Concepts ◽  
Scientific Paradigm ◽  
Geometric Concepts ◽  
Definition Of ◽  
Fundamental Theorem Of Arithmetic ◽  
Point Line ◽  
Opening Up

Here we build the fundamentals of global dimensional mathematics in order to build the new basis of the new theoretical scientific paradigm. Research on the foundations of mathematics covering the definition of fundamental mathematical concepts such as point, line, direction, dimension, addition, multiplication, division, zeros, infinities, limits, factorization, integers and prime numbers, were carried out and further still the resolution of the Goldbach conjecture is now effective. New original fundamental mathematical notions are established building the core of global dimensional mathematics based on the decomposition of integers into an addition of prime numbers terms and on fundamental geometric concepts such as the concept dimension based on the notion of direction and point, as well as research on set theory and work on the notion of limits and infinity. The termization as a decomposition of integers into an addition of prime number terms by a python program, breaks the unique factorization theorem, the fundamental theorem of arithmetic while the geometric notions developed break Euclidean geometry leading to a new mathematical framework, geometry, topology and metrics leading to a total change of theoretical scientific paradigm. Global dimensional mathematics forms the basis for the construction of the new scientific paradigm of the 21st century and beyond, opening up a still unknown perspective on the world of science in general.

scholarly journals Ingenious Solution for the Rank Reversal Problem of TOPSIS Method

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Wenguang Yang
Keyword(s):  
Theoretical Analysis ◽  
Evaluation Process ◽  
Simple Calculation ◽  
Global Perspective ◽  
Ideal Solution ◽  
Topsis Method ◽  
Reversal Problem ◽  
Rank Reversal ◽  
Negative Ideal Solution ◽  
Definition Of

Although the classic TOPSIS method is very practical, there may be a problem of rank reversal in the addition, deletion, or replacement of the candidate set, which makes its credibility greatly compromised. Based on the understanding of the classical TOPSIS method, this paper establishes a new improved TOPSIS method called NR-TOPSIS. Firstly, the historical maximum and minimum values of all attribute indicators from a global perspective during the evaluation process are determined. Secondly, according to whether the attributes belong to the benefit attribute or cost attribute, standardization is carried out. And then, in the case where the historical values of attributes are determined, we re-fix the positive ideal solution and the negative ideal solution. At the same time, this paper gives the definition of ranking stable and proves that the NR-TOPSIS proposed satisfies ranking stable, which theoretically guarantees that the rank reversal phenomenon does not exist. Finally, in the verification of examples, the results are consistent with the theoretical analysis, which further support the theoretical analysis. The NR-TOPSIS method overcomes rank reversal, which is not only obviously superior to the classical TOPSIS method but also relatively superior to the R-TOPSIS method which has also overcome rank reversal. It is also superior to other reference methods due to its simple calculation.

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