Dr. Garret Sobczyk

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List of Publications:

2023

• G. Sobczyk,  Global and Local Duality in Grassmann Algebras

• G. Sobczyk,  Global and Local Nilpotent Basis of Matrices

• G. Sobczyk,  Itinerent Quantum Integers: The Language of Quantum Computers

• G. Sobczyk,  Calculus of Compatible Nilpotents

• G. Sobczyk,  Geometric Algebras of Light Cone Projective Graph Geometries

2022

• R. Santilli, G. Sobczyk,  Representation of nuclear magnetic moments via a Clifford algebra formulation of Bohm’s hidden variables

2021

• G. Sobczyk,  Spheroidal Domains and Geometric Analysis in Euclidean Space,  Contemporary Mathematics, Vol. 2 Issue 3, (2021).

2020

•  G. Sobczyk,   Periodic Table of Geometric Numbers, DOI: 10.13140/RG.2.2.16033.35681

•  G. Sobczyk,   Notes on Plucker's Relations in Geometric Algebra, Advances in Mathematics, 106959, 363, 25 March 2020.

•  G. Sobczyk,   Geometric Matrices of Dual Null Vectors

2019

• Mathematica Package: Geometric Matrices

• Matrix Gateway to Geometric Algebra, Spacetime and Spinors

• Scaffolding of Spacetime

• What's in Pauli Matrix?

• Clemson University Talk: "Matrix Gateway to Geometric Algebra, Spacetime and Spinors"

• Talk: Geometric Matrices, 52nd Congreso Nacional de La SMM, Oct. 21-25, 2019, Monterrey, NL.

2018

• Geometric Matrices and the Symmetric Group

• From Vectors to Geometric Algebra

• De Vectores al Algebra Geometrica

2017

• Spinors in Spacetime Algebra and Euclidean 4-Space

• Geometrization of the Real Number System

2016

• Group Manifolds in Geometric Algebra

2015

• Geometric Spinors, Relativity and the Hopf Fibration
• Spacetime Algebra of Dirac Spinors
• Vector Analysis of Spinors
• Geometric Number Systems and Spinors
• Geometry of Spin ½ Particles, Revista Mexicana de Fisica 61 (2015) pp. 211-223.

• Geometric Numbers: The Language of Quantum Mechanics, Invited Lecturer at the National Youth Science Foundation, West Virgina, June 26, 2015.

2014

• Vector Analysis of Spinors, Talk: Universidad de Las Americas – Puebla, October 2014.

• Geometry of Bivectors, November 4, 2014.

2013

• The Spectral Basis of a Linear Operator, Talk: AMS/MAA Joint Mathematical Meeting, San Diego, Calif., January 10, 2013.

• New Foundations in Mathematics: The Geometric Concept of Number, Springer 2013.

2012 

• Conformal Mappings in Geometric Algebra, February 2012 Issue of AMS Notices.

• Unitary Geometric Algebra, To appear in Special Issue of Advances in Applied Clifford Algebras in Memory of Professor Jaime Keller (2012).

2010 

2009

·         Garret Sobczyk, The Symmetric Group and Twisted Symmetric Product in Geometric Algebra, October 13, 2009.

• Do Black Holes Really Exist?, Talk presented at El Tercer Congreso Nacional Universitario de Fisica, Universidad de Las Americas – Cholula, Puebla, February 20, 2009. 

2008

·         Garret Sobczyk and Tolga Yarman, Unification of Space-Time-Matter-Energy, Appl. Comput. Math. 7 (2008), no.2, pp.255-268.

·         G. Sobczyk and Omar Leon Sanchez, Fundamental Theorem of Calculus,  to appear, Advances in Applied Clifford Algebras.

·         G. Sobczyk, Geometric Matrix Algebra, Linear Algebra and its Applications, 429 (2008) 1163-1173.

2007

• arXiv:0710.0143 [ps, pdf, other]
  Title: Generalized Taylor's Theorem
  Authors: Garret Sobczyk
  Comments: 3 pages
  Subjects: Numerical Analysis (math.NA)

• arXiv:0710.0092[ps, pdf, other]
  Title: Geometry of moving planes
  Authors: Garret Sobczyk
  Comments: 16 pages, 4 figures
  Subjects: Mathematical Physics (math-ph)

• arXiv:0710.0084[ps, pdf, other]
  Title: Special relativity in complex vector algebra
  Authors: Garret Sobczyk
  Comments: 11 pages, 3 figures, PIRT Moscow, July 2007
  Subjects: Mathematical Physics (math-ph)

• The Spectral Basis and Rational Interpolation, submitted to Proceedings of “Curves and Surfaces”, Avignon 2006.

2006

• 4. arXiv:math/0602405 [ps, pdf, other]

  • Title: The Spectral Basis and Rational Interpolation

  • Authors: Garret Sobczyk

  • Comments: 8 pages, 2 figures

  • Subjects: Numerical Analysis (math.NA); Rings and Algebras (math.RA)

• The Spectral Basis and Rational Interpolation, (accepted) Proceedings of “Curves and Surfaces”, Avignon 2006.

• Un Enfoque Modular a la Interpolación Racional, (with Omar León Sánchez), submitted to “Miscelánea Matemática” (May 2006).

• Geometric matrix algebra, (accepted) Proceedings of “International Linear Algebra Society”, Amsterdam, October

• Generalized Rolle’s Theorem and Extended Modular Polynomials, submitted to “NOTES” of “The American Mathematical Monthly”, November 2006.

• Geometric Algebra, Spectral Basis and Interpolation, Talk, Universidad de Las Americas – Puebla, Feb. 24, 2006.

2005

• “Hybrid Matrix Geometric Algebra”, with G. Erlebacher, Proceedings of International Workshop on Geometric Invariance and Applications in Engineering (IWMM&GIAE), Beijing, China, (to appear).

2004

• First Order Linear Ordinary Differential Equations in Associative Algebras, (with Gordon Erlebacher), Electronic Journal of Differential Equations, Vol. 2004(2004), No. 01, pp. 1-18. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or ftp ejde.math.txstate.edu (login: ftp)

• Chapter 7: Software for Clifford (geometric) algebras, (with R. Ablamowicz), in Lecture on Clifford Geometric Algebras and Applications, Editor with Rafal Ablamowicz, International Conference on Clifford Algebras and their Applications in Mathematical Physics, Birkhauser 2004. http://www.amazon.com

• Relativity in Clifford’s Geometric Algebras of Space and Spacetime, W. E. Baylis, G. E. Sobczyk, International Journal of Theoretical Physics, Vol. 43, No. 10, Oct. 2004, pp1386-1399.

• Clifford´s Geometric Algebras in Multilinear Algebra and Non-Euclidean Geometries, in Computacional Non-Commutative Algebras and Applications, Edited by Jim Byrnes, NATO Science Series, Kluwer Academic Publishers 2004.

• “Quantum Hermite Interpolation Polynomials”, in PARAMATRIC OPTIMIZATION AND RELATED TOPICS VII, Edited by J. Guddat, H. Jongen, J. Ruckmann, M. Todorov, Aportaciones Matemáticas 18 Investigación, Sociedad Matemática Mexicana, 2004, pp. 105 – 112.

• Lecture on Clifford Geometric Algebras and Applications, Editor with Rafal Ablamowicz, International Conference on Clifford Algebras and their Applications in Mathematical Physics, Birkhauser 2004. http://www.frontlist.com/detail/0817632573

2003

• Quantum Hermite Interpolation Polynomials, Proceedings of 7th International Conference on Parametric Optimization and Related Topics, Sept. 2002.

• Clifford Geometric Algebras in Multilinear Algebra and Non-Euclidean Geometries, NATO Advanced Studies Institute: Computational Non-Commutative Algebras and Applications (To Appear).

2002

• Generalized Vandermonde Determinants and Applications, Aportaciones Matematicas Vol. 30: 41-53, 2002. (Divulgación)

• Geometric Algebra in Linear Algebra and Geometry,(With Jose M‡ria Pozo) Acta Applicandae Mathematicae, Vol. 71: 207-244, 2002. (Investigación)

2001

• The Missing Spectral Basis in Algebra and Number Theory, The American Mathematical Monthly, Vol. 108, No. 4, April 2001, 289-387. (Investigación)

• Edited Book (with Eduardo Bayro-Corrochano) Geometric Algebra with Applications in Science and Engineering, Birkhauser 2001.

• Applications of Lie Algebras and the Algebra of Incidence, (with Eduardo Bayro - Corrochano), in Geometric Algebra with Applications in Science and Engineering, Chapter 13, pp. 252-277, Birkhauser 2001. (Investigación)

• Realizations of the Conformal Group (with Jose Maria Pozo) in Geometric Algebra with Applications in Science and Engineering, Chapter 3, pp. 42-60, Birkhauser 2001. (Investigación)

• Universal Geometric Algebra, in Geometric Algebra with Applications in Science and Engineering, Chapter 2, pp. 18-40, Birkhauser 2001. (Investigación)

1997

• Spectral Integral Domain in the Classroom, Aportaciones matematicas, Serie Comunicaciones 20 (1997), 169-188. DVI

• The Generalized Spectral Decomposition of a Linear Operator, The College Mathematics Journal, January (1997), 27-38.

1996

• A Unipodal Algebra Package for MATHEMATICA, Clifford Algebras with Numeric and Symbolic Computations, Edited by R. Ablamowicz and P. Lounesto, Birkhauser Pub., pp. 189-199, May 1996.

• Structure of Factor Algebras and Clifford Algebra, Linear Algebra and Its applications, 241-243:803-810, Elsevier Science Inc., 1996.

• A Universal Geometric Algebra, Advanced Mathematics: Computations and Applications, Edito/rs: A.S. Alekseev, N.S. Bakhvalov, NCC Publisher, Novosibirsk, pp. 447-456, 1996.

  • Chapter 3, Introduction to Geometric Algebras (pp. 37-44)

  • Chapter 4, Linear Transformations (pp. 45-52)

  • Chapter 5, Directed Integration (pp. 53-64)

• Clifford (Geometric) Algebras with Applications in Physics, Mathematics, and Engineering, Editor: W.E. Baylis, Birkhauser,1996.

1995

• The Hyperbolic Number Plane, The College Mathematics Journal, Vol. 26, No. 4, September (1995), 268-280. Pictures

• Book Review: Clifford Algebras and Spinor Structures: A Special Volume Dedicated to the Memory of Albert Crumeyrolle, Edited by Rafael Ablamowicz,Pertti Lounesto, Kluwer 194. Advances in Applied Clifford Algebras, Vol. 1, 1995.

1994

• Generalized Finite Fourier Transforms, Proceedings, 14th IMACS World Congress on Computational and Applied Mathematics, Georgia Institute of Technology, July 1994.

• Clifford Algebra Techniques in Linear Algebra, Collection of Papers

• Dedicated to the memory of Albert Crumeyrolle, Kluwer (1994).

• Derivations of Endomorphisms, Proceedings, Differential Geometry Methods in Theoretical Physics, Ixtapa, 1993, Advances In Applied Clifford

• Algebras (1994).

1991-1993

• Jordan Form in Clifford Algebra, Clifford Algebras and their Applications in Mathematical Physics, Editors: F. Brackx et al. (Kluwer, Dordrecht, 1993) p.33-41.

• Jordan Form in Associative Algebras, Spinors, Twistors, Clifford Algebras and Quantum Deformations, Editors: Z. Oziewicz et al. Kluwer, Dordrecht, 1993) p. 357-364.

• With Marius Mitrea, Conference Report, Advances in Applied Clifford Algebras, 3, No. 1, 99-104, Mexico 1993.

• Linear Transformations in Unitary Geometric Algebra, Foundations in Physics, Vol. 23, No. 10, 1375-1385, 1993.The Early Years, Foundations in Physics, 23, No. 10, 1291-1293, 1993.

• With J. Cruz-Guzman, Remarks on Circulant Matrices and Polynomial Number Systems, Advances in Applied Clifford Algebras 2, No. 2, 205-214 (1992).

• Noncommutative Extensions of Number: An Introduction to Clifford's Geometric Algebra, Aportaciones Matemáticas Comunicaciones 11, 207-218 (1992).

• Program: Spinor Analysis of Space and Time, Universidad Nacional Autonoma de Mexico, Facultad de Estudios Superiores, Cuautitlan, VI Foro Interno de Investigacion, January 1992.

• Report: Unipodal Algebra and Roots of Polynomials, with D. Hestenes and P. Reany, Advances in Applied Clifford Algebras, Vol. 1, No. 1, p.51-64, Dec. 1991.

• Unipotents, Idempotents, and a Spinor Basis for Matrices, Advances in Applied Clifford Algebras, Vol. 2, No. 1, p.53-64, July 1992.

• Geometry of Complex Vectors and Algebraic Roots, Differential Geometric Methods in Theoretical Physics, p.397-407, Edited by S. Catto, and A. Rocha, World Scientific Press, 1992.

• Simplicial Calculus with Geometric Algebra, in Clifford Algebras and Their Applications in Mathematical Physics, Edited by A. Micali, R. Boudet, and J. Helmstetter, (Kluwer Academic Publishers, Dordrecht 1992). Clifford's Geometric Algebra and Hermitian Operators, p. 112 in 13th International Conference on General Relativity and Gravitation, Abstracts of Contributed Papers, Edited by Pedro W. Lamberti and Omar E. Ortiz, Cordoba, Argentina, 1992.

1980-1990

• Graphing Rational Functions Without Plotting, Lander College Science Newsletter, No. 51, April 1990.

• Killing Vectors and Embedding of Exact Solutions in General Relativity, in Clifford Algebras and their Applications in Mathematical Physics, Edited by S.R. J. Chisholm, Kluwer, 227-244 (1986).

• Vector Calculus with Complex Variables, Lecture Notes, Spring Hill College, 1986 (First taught at Spring Hill College in Fall Semester 1984. Revised and taught as a two semester course in the Fall and Spring Semesters 1986-87.

• CLIFFORD ALGEBRA TO GEOMETRIC CALCULUS: A Unified Language for Mathematics and Physics, coauthored with D. Hestenes, Reidel Publishing Co, 1984, (314 pages).

• Clifford Analytical Techniques in Lie-Admissible Mechanics, (with K. Morawiecki), Proceedings of Conference on Non Potential Interactions, Orleans, France, 1982.

• Plebanski Classification of the Tensor of Matter, Acta Physica Polonica, Vol. B11, No.6, 1981.

• A Complex Gibbs-Heaviside Vector Algebra for Space-time, Acta Physica Polonica, Vol. B12, No.5, 407-418, 1981.

• Spacetime Vector Analysis, Physics Letters, 84A, 45-49, 1981.

• Algebraic Classification of Important Tensors in Physics, Physics Letters, 84A, 49-54, 1981. Spacetime Algebra Approach to Curvature, J. Mathematical Physics, Vol. 24, 1981.

• Conjugations and Hermitian Operators in Spacetime, Acta Physica Polonica, Vol.B12, No.6, 509-521, 1981.

Before 1980

• Computerized Pedigrees in Family Practice, Am J Human Genetics, Vol. 9-10, July 1976.

• Geometric Structures, Polish Academy of Science, Warsaw 1974.

• Dissertation: Mappings of Surfaces in Euclidean Space Using Geometric Algebra Thesis1, Thesis 2, page 108

• Fixing Seismograms with Polynomials, Technical Reports No. 45, United Electro-Dynamics, Teledyne Inc., 1965.