Differentials of the basis in Clifford Geometric Algebra

Yingqiu Gu

doi:10.59400/jam.v2i4.1700

Differentials of the basis in Clifford Geometric Algebra

Yingqiu Gu School of Mathematical Science, Fudan University, Shanghai 200433, China

Article ID: 1700

446 Views, 89 PDF Downloads

DOI: https://doi.org/10.59400/jam.v2i4.1700

Keywords: Clifford algebra; connection; Dirac-γ; moving frame; variation

Abstract

In this paper we discuss the dynamic effects of the varying frames. The differential of frame or basis vectors is always equivalent to a linear transformation of the frame, and the linear transformation is not the same in different contexts. In differential geometry, the linear transformation is the connection operator. While in quantum mechanics, the operator algebra corresponds to the differentials of matrices. Corresponding to the variation of the metric, the variation of the frame contains a unusual fourth-order tensor. We also derive the Lie differential of the frame corresponding to the Lorentz transformation group. The definition of differential of the frame is different, so the corresponding linear transformation is also different. In this paper, the unified point of view to deal with the variation of frame or basis vectors will bring great convenience to the research and application of Clifford algebras.

References

[1]Clifford WK. Applications of Grassmann’s Extensive Algebra. American Journal of Mathematics. 1878; 1(4):350-358, doi: 10.2307/2369379

[2]Grassmann HG. Extension Theory, (Die Ausdehnungslehre von 1862) Lloyd C. Kannenberg, trans. American Mathematical Society, Rhode Island. London Mathematical Society Publishing; 2000.

[3]Hamilton WR. Elements of quaternions. Longmans, Green, & Company Publishing; 1866.

[4]Doran C, Lasenby A. Geometric algebra for physicists. Cambridge University Press Publishing; 2003.

[5]Macdonald A. Linear and geometric algebra. Nottingham: Alan Macdonald Publishing; 2010.

[6]Pozo JM, Sobczyk G. Geometric algebra in linear algebra and geometry. Acta Applicandae Mathematica. 2002; 71: 207-244. doi:10.1023/A:1015256913414

[7]Hestenes D. Observables, operators, and complex numbers in the Dirac theory. J. Math. Phys. 1975; 16:556-572. doi:10.1063/1.522554

[8]Keller J, Rodriguez-Romo S. Multivectorial representation of Lie groups. Int. J. Theor. Phys. 1991; 30: 185-196. doi:10.1007/BF00670711

[9]Shirokov DS. On inner automorphisms preserving subspaces of Clifford algebras. Adv. Appl. Clifford Algebras. 2021; 31: 30-52. doi:10.1007/s00006-021-01135-6

[10]Keller J. Spinors and Multivectors as a Unified Tool for Spacetime Geometry and for Elementary Particle Physics. Int. J. Theor. Phys. 1991; 30: 137-184. doi:10.1007/BF00670710

[11]Ablamowicz R, Sobczyk G. Lectures on Clifford (geometric) algebras and applications. Available online: https://link.springer.com/book/10.1007/978-0-8176-8190-6 (accessed on 20 June 2024)

[12]Ablamowicz R. Clifford Algebras: Applications to Mathematics, Physics, and Engineering. Available online: https://link.springer.com/book/10.1007/978-1-4612-2044-2 (accessed on 20 June 2024)

[13]Lounesto P. Clifford Algebras and Spinors. Cambridge Univ. Press; 2001.

[14]Breuils S, Tachibana K, Hitzer, E. New applications of Clifford’s geometric algebra. Adv. Appl. Clifford Algebras. 2022; 32:17-55. doi:10.1007/s00006-021-01196-7

[15]Gu Y-Q. A Note on the Representation of Clifford Algebras. J. Geom. Symmetry Phys. 2021; 62: 29-52. doi:10.7546/jgsp-62-2021-29-52

[16]Snygg J. Clifford Algebra-A Computational Tool For Physicists. Oxford University Press; 1997.

[17]Snygg J. A new approach to differential geometry using Clifford’s geometric algebra. Springer Science & Business Media Publishing; 2011.

[18]Hehl FW, Von der Heyde P, Kerlick GD, Nester JM. General relativity with spin and torsion: Foundations and prospects. Rev. Mod. Phys. 1976; 48: 393-416. doi:10.1103/RevModPhys.48.393.

[19]Nester JM. Special orthonormal frames. Journal of mathematical physics. 1992; 33: 910-913,

[20]Dimakis A, Muller-Hoissen F. Clifform calculus with applications to classical field theories. Classical and Quantum Gravity. 1991; 8: 2093. doi:10.1088/0264-9381/8/11/018

[21]Nester JM, Tung RS, Zhytnikov VV. Some Spinor-Curvature Identities. Classical and Quantum Gravity. 1994; 11: 983. doi:10.1088/0264-9381/11/4/014

[22]Gu Y-Q. Theory of Spinors in Curved Space-Time. Symmetry. 2021; 13: 1931, doi:10.3390/sym13101931

[23]Gu Y-Q. Clifford A. Hypercomplex Numbers and Nonlinear Equations in Physics. Geometry Integrability and Quantization. 2023; 25: 47-72. doi:10.7546/giq-25-2023-47-72

[24]Gu Y-Q. Clifford Algebra and Hypercomplex Number as well as Their Applications in Physics. J. Appl. Math. Phys. 2022; 10: 1375-1393. doi:10.4236/jamp.2022.104097

[25]Gu Y.-Q. Miraculous Hypercomplex Numbers. Mathematics and Systems Science. 2023; 1: 2258. doi:10.54517/mss.v1i1.2258

[26]Calvet RG. On Matrix Representations of Geometric (Clifford) Algebras. J. Geom. Symmetry Phys. 2017; 43: 1-36. doi:10.7546/jgsp-43-2017-1-36

[27]Cartan É. Riemannian Geometry in an Orthogonal Frame (translated into English by V. Goldberg). World Scientific Publishing; 2001.

[28]Hestenes D, Sobczyk G. Clifford algebra to geometric calculus: a unified language for mathematics and physics (Vol. 5). Springer Science & Business Media Publishing; 2012

[29]Morozov O. Moving coframes and symmetries of differential equations. J. Phys. Am. 2002; 35: 2965- 2977. doi: 10.1088/0305-4470/35/12/317

[30]Kogan IA, Olver PJ. Invariant Euler-Lagrange equations and the invariant variational bicomplex. Acta Appl. Math. 2003; 76: 137-193. doi:10.1023/A:1022993616247

[31]Arnaldsson Ö. Involutive moving frames. Diff. Geom. Appl. 2020; 69: 101603. doi:10.1016/j.difgeo.2020.101603

Published

2024-08-18

How to Cite

Gu, Y. (2024). Differentials of the basis in Clifford Geometric Algebra. Journal of AppliedMath, 2(4), 1700. https://doi.org/10.59400/jam.v2i4.1700

Download Citation

Issue

Vol. 2 No. 4 (2024)

Section

Article

This work is licensed under a Creative Commons Attribution 4.0 International License.

Editor-in-Chief

Assoc. Prof. Ahmad Zeeshan

International Islamic University, Pakistan

eISSN

2972-4805

Publication Frequency

Bi-monthly

About the Publisher

Academic Publishing insists on taking academic exchange and publication as the main line, carrying out comprehensive management based on science and technology, and fully exploring excellent international publishing resources. Within 5 years, it will form a strategic framework and scale with science (S), technology (T), medicine (M), education (E), and humanities and arts (H) as the main publishing fields. Academic Publishing is headquartered in Singapore and based in Malaysia, with the United States and China providing the main scientific and academic resources. At the same time, it has established long-term good cooperative relations with other publishing companies, scientific research communities, and academic organizations in more than a dozen countries and regions. Academic Publishing uses English and Chinese as its main publishing languages, mainly publishing books, journals, and conference papers in print and online. The vast majority of publications follow the international open access policy, providing stable and long-term quality and professional publications. With the joint efforts of the expert team and our professional editorial team, our publications will gradually be indexed by international databases in stages to provide convenient and professional retrieval for various scholars. At the same time, manuscripts we accept will be subject to the peer review principle, and cutting-edge and innovative research articles will be preferentially accepted for peer reference and discussion. All kinds of our publications are welcome for peer to contribute, access, and download.

more

Volume Arrangement

2024

2023

Featured Articles

Octagonal-square tessellation model for masting GSM network: A case study of MTN Kumasi-East, Ghana

Masting in GSM network design is one of the most challenging problems in cell planning. The effect of uniform design pattern has been proven geographically to be hexagonal using uniform cell range. In this paper, we present a new uniform greedy semi-regular tessellation model called the octagonal square tessellation model (OSTM) to address the problem of global minimum overlap difference and area. Data from MTN Kumasi-East Ghana was collected and analyzed using the developed model. The original layout for the 0.6 km cell range accounted for an overlap difference of 937.66 m and a total area of 21.41 km² for 50 GSM mosts whereas the OSTM model accounted for an overlap difference of 1316.95 m with an area of 34.23 km². This is a 59.87% reduction of the original total area. Our solution is shown to be optimal in overlap difference and area for non-uniform cell range.

Computation of topological indices of linear chains of perylene and coronene using M-polynomial

The M-polynomial yields degree based topological indices that anticipate different physical and chemical properties of material being scrutinized. In this work, M- polynomial of linear chains of perylene and coronene are acquired. From M-polynomial, some degree based topological dicriptors are determined. Some topological indices of these compounds are compared by plotting graphs.

H∞ hybrid control and MRD in a steel frame building subjected to excessivevibrations caused by the dynamic action of wind and earthquake

The dynamic loads from earthquakes and winds can destroy lives, cause collapse in civil structures, and interrupt basic services provided to the population. In this scenario, structural designs must be developed to decrease the damage induced by these actions. The objective of this work is to design a hybrid controller based on the H∞ optimization via state feedback and the magneto-rheological damper (MRD) to mitigate the excessive vibrations of a three-story steel frame building, represented through the shear building model, subjected to the simultaneous dynamic action of wind and earthquake. All research is based on computational simulation, experimental research and results will not be addressed. In the numerical analysis, digital computer and MATLAB® software are used, and implemented codes generate the expected results based on the mathematical modeling. With the application of the H∞ control technique via state feedback, the displacements were reduced by 77%. With MRD this reduction was 79%. With the hybrid controller, this reduction was 100%. Thus, the verifications in relation to maximum displacements were met for NBR 15421:2006, NBR 8800:2008 and NBR 6118:2014. From the results, it is concluded that the hybrid controller proved to be more efficient and achieved the proposed objective. The exogenous inputs had zero influence on the behavior of the system output.

Topological analysis of multiple tables

The paper proposes a topological approach in order to explore several data tables simultaneously. These data tables of quantitative and/or qualitative variables measured on different homogeneous themes, collected from the same individuals. This approach, called topological analysis of multiple tables (TAMT), is based on the notion of neighborhood graphs in the context of a joint analysis of several data tables. It allows the simultaneous study of possible links between several thematic tables. The structure of the correlations or associations of the variables in each thematic table is analyzed according to quantitative, qualitative or mixed variables considered. Like the multiple factorial analysis (MFA), the TAMT allows several tables of variables to be analyzed simultaneously, and to obtain results, in particular graphical representations, which make it possible to study the relationship between individuals, variables and tables of data. These can also be tables of temporal data, collected at different times on the same individuals. The proposed TAMT approach is illustrated using real data associated with several and different homogeneous themes. Its results are compared to those from the MFA method.

Development of a stacked hybrid Decision Tree model leveraging the NSL-KDD dataset

Intrusion detection in information technology as well as operational technology networks is highly required in modern day systems due to the increased spate of cyber-attacks in both number and complexity. Anomaly-based intrusion detection systems which have the capacity to detect novel or zero-day attacks are highly employed in this regard. One important component of anomaly-based intrusion detection systems which ensures their behaviour is artificial intelligence in general and machine learning in particular. The burden in modern day cybersecurity research is to investigate and develop models that can outperform existing ones. This paper is aimed at developing a hybrid decision tree model using the stacking ensemble approach. Performances were measured on the basis of recall, precision, accuracy, F1-score, receiver operating characteristics and confusion matrices. The hybrid model presented a precision of 97%, accuracy of 81%, F1-score of 80% and AUC score of 0.96, respectively.