199,263 research outputs found
Geometric operations implemented by conformal geometric algebra neural nodes
Geometric algebra is an optimal frame work for calculating with vectors. The
geometric algebra of a space includes elements that represent all the its
subspaces (lines, planes, volumes, ...). Conformal geometric algebra expands
this approach to elementary representations of arbitrary points, point pairs,
lines, circles, planes and spheres. Apart from including curved objects,
conformal geometric algebra has an elegant unified quaternion like
representation for all proper and improper Euclidean transformations, including
reflections at spheres, general screw transformations and scaling. Expanding
the concepts of real and complex neurons we arrive at the new powerful concept
of conformal geometric algebra neurons. These neurons can easily take the above
mentioned geometric objects or sets of these objects as inputs and apply a wide
range of geometric transformations via the geometric algebra valued weights.Comment: 6 pages, 2 tables, 10 figure
Representation of Crystallographic Subperiodic Groups in Clifford's Geometric Algebra
This paper explains how, following the representation of 3D crystallographic
space groups in Clifford's geometric algebra, it is further possible to
similarly represent the 162 so called subperiodic groups of crystallography in
Clifford's geometric algebra. A new compact geometric algebra group
representation symbol is constructed, which allows to read off the complete set
of geometric algebra generators. For clarity moreover the chosen generators are
stated explicitly. The group symbols are based on the representation of point
groups in geometric algebra by versors (Clifford monomials, Lipschitz
elements).
Keywords: Subperiodic groups, Clifford's geometric algebra, versor
representation, frieze groups, rod groups, layer groups .Comment: 17 pages, 6 figures, 11 tables. arXiv admin note: substantial text
overlap with arXiv:1306.128
Representation of Crystallographic Subperiodic Groups by Geometric Algebra
We explain how following the representation of 3D crystallographic space
groups in geometric algebra it is further possible to similarly represent the
162 socalled subperiodic groups of crystallography in geometric algebra. We
construct a new compact geometric algebra group representation symbol, which
allows to read off the complete set of geometric algebra generators. For
clarity we moreover state explicitly what generators are chosen. The group
symbols are based on the representation of point groups in geometric algebra by
versors (Clifford group, Lipschitz elements).Comment: 11 pages, 5 figures, 9 table
Projective Geometric Algebra as a Subalgebra of Conformal Geometric algebra
We show that if PGA is understood as a subalgebra of CGA in mathematically
correct sense, then the flat objects share the same representation in PGA and
CGA. Particularly, we treat duality in PGA. This leads to unification of PGA
and CGA objects which is important especially for software implementation and
symbolic calculations
Part I: Vector Analysis of Spinors
Part I: The geometric algebra of space is derived by extending the real
number system to include three mutually anticommuting square roots of plus one.
The resulting geometric algebra is isomorphic to the algebra of complex 2x2
matrices, also known as the Pauli algebra. The so-called spinor algebra of
C(2), the language of the quantum mechanics, is formulated in terms of the
idempotents and nilpotents of the geometric algebra of space, including its
beautiful representation on the Riemann sphere, and a new proof of the
Heisenberg uncertainty principle. In "Part II: Spacetime Algebra of Dirac
Spinors", the ideas are generalized to apply to 4-component Dirac spinors, and
their geometric interpretation in spacetime.Comment: 20 pages, 4 figure
Spinors in Spacetime Algebra and Euclidean 4-Space
This article explores the geometric algebra of Minkowski spacetime, and its
relationship to the geometric algebra of Euclidean 4-space. Both of these
geometric algebras are algebraically isomorphic to the 2x2 matrix algebra over
Hamilton's famous quaternions, and provide a rich geometric framework for
various important topics in mathematics and physics, including stereographic
projection and spinors, and both spherical and hyperbolic geometry. In
addition, by identifying the time-like Minkowski unit vector with the extra
dimension of Euclidean 4-space, David Hestenes' Space-Time Algebra of Minkowski
spacetime is unified with William Baylis' Algebra of Physical Space.Comment: 16 pages, 3 figure
Reflections in Conics, Quadrics and Hyperquadrics via Clifford Algebra
In this article we present a new and not fully employed geometric algebra
model. With this model a generalization of the conformal model is achieved. We
discuss the geometric objects that can be represented. Furthermore, we show
that the Pin group of this geometric algebra corresponds to inversions with
respect to axis aligned quadrics. We discuss the construction for the two- and
three-dimensional case in detail and give the construction for arbitrary
dimension. Key Words: Clifford algebra, geometric algebra, generalized
inversion, conic, quadric, hyperquadri
Non-associativity in non-geometric string and M-theory backgrounds, the algebra of octonions, and missing momentum modes
We propose a non-associative phase space algebra for M-theory backgrounds
with locally non-geometric fluxes based on the non-associative algebra of
octonions. Our proposal is based on the observation that the non-associative
algebra of the non-geometric R-flux background in string theory can be obtained
by a proper contraction of the simple Malcev algebra generated by imaginary
octonions. Furthermore, by studying a toy model of a four-dimensional locally
non-geometric M-theory background which is dual to a twisted torus, we show
that the non-geometric background is "missing" a momentum mode. The resulting
seven-dimensional phase space can thus be naturally identified with the
imaginary octonions. This allows us to interpret the full uncontracted algebra
of imaginary octonions as the uplift of the string theory R-flux algebra to
M-theory, with the contraction parameter playing the role of the string
coupling constant .Comment: 27 page
Geometric algebra, qubits, geometric evolution, and all that
The earlier approach is used for description of qubits and geometric phase
parameters, the things critical in the area of topological quantum computing.
The used tool, Geometric (Clifford) Algebra is the most convenient formalism
for that case. Generalization of formal complex plane to an an arbitrary plane
in three dimensions and of usual Hopf fibration to the map generated by an
arbitrary unit value element of even sub-algebra of the three-dimensional
Geometric Algebra are resulting in more profound description of qubits compared
to quantum mechanical Hilbert space formalism
Notes on Geometric-Algebra Quantum-Like Algorithms
In these notes we present preliminary results on quantum-like algorithms
where tensor product is replaced by geometric product. Such algorithms possess
the essential properties typical of quantum computation (entanglement,
parallelism) but employ additional algebraic structures typical of geometric
algebra -- structures absent in standard quantum computation. As a test we
reformulate in Geometric Algebra terms the Deutsch-Jozsa problem.Comment: Preliminary version of note
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