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Dr. Garret Sobczyk
local time: 2024-02-25 18:06 (-06:00 )
Dr. Garret Sobczyk (Abstracts)
Titles Abstracts Details
  • New Foundations in Mathematics: The Geometric Concept of Number (2011) [Updated 7 years ago]
    by Garret Sobczyk   read the paper:

    The development of the real number system represents both a milestone and a cornerstone in the foundation of modern mathematics. We go further and suggest that the real number system should be completed to include the concept of direction. Some of this work has already been done by the invention of the complex numbers, quaternions, and vectors. What has been lacking, however, is a general geometric number system. In 1878, William Kingdom Clifford invented his "geometric algebra", based upon the earlier work of Grassmann and Hamilton. Geometric algebra is the completion of the real number system to include new anticommuting square roots of plus and minus one, each such root representing an orthogonal direction in successively higher dimensions. All of the usual rules of the real number system remain valid, except that the commutative law of multiplication is no longer universally valid. The book, "New Foundations in Mathematics: The Geometric Concept of Number" by the author, represents an attempt to show how many ideas of modern mathematics can be developed within this new framework, including modular number systems, complex and hyperbolic numbers, geometric algebra of Euclidean and pseudo-Euclidean spaces, linear and multilinear algebra, Hermitian inner product spaces, the theory of special relativity, representations of the symmetric group, calculus and differential geometry of n-dimensional surfaces, Lie groups and Lie algebras, and other topics.


  • Unification of Space-Time-Matter-Energy (2008) [Updated 7 years ago]
    by Tolga Yarman, Garret Sobczyk   read the paper:

    Appl. Comput. Math. 7(2) (2008), pp. 255-268. A complete description of space-time, matter and energy is given in Einstein's special theory of relativity. We derive explicit equations of motion for two falling bodies, based upon the principle that each body must subtract the mass-equivalent for any change in its kinetic energy that is incurred during the fall. We find that there are no singularities and consequently no blackholes.

    A complete description of space-time, matter and energy is given in Einstein's special theory of relativity. We derive explicit equations of motion for two falling bodies, based upon the principle that each body must subtract the mass-equivalent for any change in its kinetic energy that is incurred during the fall. We find that there are no singularities and consequently no blackholes.


  • Geometry of Moving Planes (2008) [Updated 1 decade ago]
    by Garret Sobczyk   read the paper:

    The concept of number and its generalization has played a central role in the development of mathematics over many centuries and many civilizations. Noteworthy milestones in this long and arduous process were the developments of the real and complex numbers which have achieved universal acceptance. Serious attempts have been made at further extensions, such as Hamiltons quaternions, Grassmann's exterior algebra and Clifford's geometric algebra. By examining the geometry of moving planes, we show how new mathematics is within reach, if the will to learn these powerful methods can be found.