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Carlos Castro - One of the best experts on this subject based on the ideXlab platform.
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Exceptional Jordan Matrix Models, Octonionic $p$-Branes and Star Product Deformations
viXra, 2020Co-Authors: Carlos CastroAbstract:A brief review of the essentials behind the construction of a Chern-Simons-like brane action from the Large $N$ limit of Exceptional Jordan Matrix Models paves the way to the construction of actions for membranes and $p$-branes moving in octonionic-spacetime backgrounds endowed with octonionic-valued metrics. The main result is that action of a membrane moving in spacetime backgrounds endowed with an octonionic-valued metric is not invariant under the usual diffeomorphisms of its world volume coordinates $ \sigma^a \rightarrow \sigma'^a (\sigma^b) $, but instead it is invariant under the rigid $ E_{ 6 ( - 26) } $ transformations which preserve the volume (cubic) form. The star-product deformations of octonionic $p$-branes follow. In particular, we focus on the octonionic membrane along with the phase space quantization methods developed by \cite{Szabo} within the context of Nonassociative Quantum Mechanics. We finalize with some concluding remarks on Double and Exceptional Field theories, Nonassociative Gravity and $A_\infty, L_\infty$ algebras.
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c x h x o valued gravity su 4 4 unification hermitian Matrix geometry and nonsymmetric kaluza klein theory
viXra, 2018Co-Authors: Carlos CastroAbstract:We review briefly how {\bf R} $\otimes$ {\bf C} $\otimes$ {\bf H} $\otimes$ {\bf O}-valued Gravity (real-complex-quaterno-octonionic Gravity) naturally can describe a grand unified field theory of Einstein's gravity with a Yang-Mills theory containing the Standard Model group $SU(3) \times SU(2) \times U(1)$. In particular, the $ C \otimes H \otimes O$ algebra is explored deeper. It is found that it can furnish the gauge group {\bf [SU(4)]}$^4$ revealing the possibility of extending the Standard Model by introducing additional gauge bosons, heavy quarks and leptons, and a $fourth$ family of fermions with profound physical implications. An analysis of $ C \otimes H \otimes O$-valued gravity reveals that it bears a connection to Nonsymmetric Kaluza-Klein theories and complex Hermitian Matrix Geometry. The key behind these connections is in finding the relation between $ C \otimes H \otimes O$-valued metrics in $two$ $complex$ dimensions with metrics in $higher$ dimensional $real$ manifolds ($ D = 32 $ real dimensions in particular). It is desirable to extend these results to hypercomplex, quaternionic manifolds and Exceptional Jordan Matrix Models.
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C x H x O-valued Gravity, [SU(4)]$^4$ Unification, Hermitian Matrix Geometry and Nonsymmetric Kaluza-Klein Theory
viXra, 2018Co-Authors: Carlos CastroAbstract:We review briefly how {\bf R} $\otimes$ {\bf C} $\otimes$ {\bf H} $\otimes$ {\bf O}-valued Gravity (real-complex-quaterno-octonionic Gravity) naturally can describe a grand unified field theory of Einstein's gravity with a Yang-Mills theory containing the Standard Model group $SU(3) \times SU(2) \times U(1)$. In particular, the $ C \otimes H \otimes O$ algebra is explored deeper. It is found that it can furnish the gauge group {\bf [SU(4)]}$^4$ revealing the possibility of extending the Standard Model by introducing additional gauge bosons, heavy quarks and leptons, and a $fourth$ family of fermions with profound physical implications. An analysis of $ C \otimes H \otimes O$-valued gravity reveals that it bears a connection to Nonsymmetric Kaluza-Klein theories and complex Hermitian Matrix Geometry. The key behind these connections is in finding the relation between $ C \otimes H \otimes O$-valued metrics in $two$ $complex$ dimensions with metrics in $higher$ dimensional $real$ manifolds ($ D = 32 $ real dimensions in particular). It is desirable to extend these results to hypercomplex, quaternionic manifolds and Exceptional Jordan Matrix Models.
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The Large N Limit of Exceptional Jordan Matrix Models and M, F Theory
viXra, 2009Co-Authors: Carlos CastroAbstract:The large N → ∞ limit of the Exceptional F4,E6 Jordan Matrix Models of Smolin-Ohwashi leads to novel Chern-Simons Membrane Lagrangians which are suitable candidates for a nonperturbative bosonic formulation of M Theory in D = 27 real, complex dimensions, respectively. Freudenthal algebras and triple Freudenthal products permits the construction of a novel E7 X SU(N) invariant Matrix model whose large N limit yields generalized nonlinear sigma models actions on 28 complex dimensional backgrounds associated with a 56 real-dim phase space realization of the Freudenthal algebra. We argue why the latter Matrix Model, in the large N limit, might be the proper arena for a bosonic formulation of F theory. To finalize we display generalized Dirac-Nambu-Goto membrane actions in terms of 3 X 3 X 3 cubic Matrix entries that match the number of degrees of freedom of the 27-dim exceptional Jordan algebra J3[0].
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The large N limit of exceptional Jordan Matrix models and M,F theory
Journal of Geometry and Physics, 2007Co-Authors: Carlos CastroAbstract:The large N→∞ limits of the exceptional F4,E6 Jordan Matrix models of Smolin and Ohwashi lead to novel Chern–Simons membrane Lagrangians which are suitable candidates for providing a nonperturbative bosonic formulation of M theory in D=27 real and complex dimensions, respectively. Freudenthal algebras and triple Freudenthal products permit the construction of a novel E7×SU(N) invariant Matrix model whose large N limit yields generalized nonlinear sigma model actions on 28-complex-dimensional backgrounds associated with a 56-real-dimensional phase space realization of the Freudenthal algebra. We argue as to why the latter Matrix model, in the large N limit, might be the proper arena for a bosonic formulation of F theory. Finally, we display generalized Dirac–Nambu–Goto membrane actions in terms of 3×3×3 cubic Matrix entries that match the numbers of degrees of freedom of the 27-dimensional exceptional Jordan algebra J3[0].
Guang-ren Duan - One of the best experts on this subject based on the ideXlab platform.
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Parametric solutions to rectangular high-order Sylvester equations—Case of F arbitrary
2014 Proceedings of the SICE Annual Conference (SICE), 2014Co-Authors: Guang-ren DuanAbstract:In an recent paper, we have proposed a general complete parametric solution in a simple and neat analytical closed form to a type of rectangular high-order Sylvester Matrix equations with the parameter Matrix F being an arbitrary Matrix. Based on this result, in this paper we are presenting, for a type of rectangular high-order Sylvester Matrix equations with the parameter Matrix F being in a Jordan Matrix form, a complete parametric solution which is expressed in terms of a free parameter Matrix Z representing the degrees of freedom. The primary feature of this solution is that the Matrix F, together with the parameter Matrix R, does not need to be even known a priori, and thus may be set undetermined and used as degrees of freedom beyond the completely free parameter Matrix Z. The results provide great convenience to the computation and analysis of the solutions to this class of equations, and can perform important functions in many control systems analysis and design problems involving high-order dynamical systems.
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On a type of generalized sylvester equations
2013 25th Chinese Control and Decision Conference (CCDC), 2013Co-Authors: Guang-ren DuanAbstract:In this paper a new type of generalized Sylvester equations (GSEs) associated with the generalized eigenstructure assignment in a type of descriptor linear systems are proposed. Based on the concept of F-coprimeness, degrees of freedom existing in the general solution to this type of equations are first given, and then a general complete parametric solution in explicit closed form is established based on generalized right factorization. The primary feature of this solution is that the parameter Matrix F, which corresponds to the finite closed-loop Jordan Matrix in the generalized eigenstructure assignment problem, does not need to be in any canonical form, and may be even unknown a priori. The results provide great convenience to the computation and analysis of the solutions to this class of equations, and can perform important functions in many control systems analysis and design problems.
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Explicit general solution to the Matrix equation AV 1 BW 5 EVF 1 R
IET Control Theory & Applications, 2008Co-Authors: Guang-ren DuanAbstract:A Matrix equation AV+BW = EVF + R is considered, where E, A, B and R are matrices with appropriate dimensions, F is an arbitrarily given Jordan Matrix and V and W are the matrices to be determined. A complete explicit general solution for this equation is established based on elementary transformations of polynomial matrices. The proposed solution does not involve the derivative of polynomial matrices and not require the eigenvalues of Matrix F to be known. When the eigenvalues of Matrix F are prescribed, the solution for matrices V and W can be obtained by carrying out singular value decompositions, thus possesses good numerical stability. An example is employed to illustrate the effect of the proposed approaches.
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An explicit solution to the Matrix equation AV + BW = EV J
Journal of Control Theory and Applications, 2007Co-Authors: Guang-ren Duan, Bin ZhouAbstract:In this note, the Matrix equation AV + BW = EV J is considered, where E, A and B are given matrices of appropriate dimensions, J is an arbitrarily given Jordan Matrix, V and W are the matrices to be determined. Firstly, a right factorization of (sE — A)−1 B is given based on the Leverriver algorithm for descriptor systems. Then based on this factorization and a proposed parametric solution, an alternative parametric solution to this Matrix equation is established in terms of the R-controllability Matrix of (E, A, B), the generalized symmetric operator and the observability Matrix associated with the Jordan Matrix J and a free parameter Matrix. The proposed results provide great convenience for many analysis and design problems. Moreover, some equivalent forms are proposed. A numerical example is employed to illustrate the effect of the proposed approach.
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The solution to the Matrix equationAV + BW = EVJ + R
Applied Mathematics Letters, 2004Co-Authors: Guang-ren DuanAbstract:Abstract This note considers the solution to the generalized Sylvester Matrix equation AV + BW = EVJ + R, where A, B, E, and R are given matrices of appropriate dimensions, J is an arbitrary given Jordan Matrix, while V and W are matrices to be determined. A general parametric solution for this equation is proposed, based on the Smith form reduction of the Matrix [A − sE B]. The solution possesses a very simple and neat form, and does not require the eigenvalues of Matrix J to be known. An example is presented to illustrate the proposed solution.
Carlos Castro Perelman - One of the best experts on this subject based on the ideXlab platform.
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on c varvec otimes h varvec otimes o valued gravity sedenions hermitian Matrix geometry and nonsymmetric kaluza klein theory
Advances in Applied Clifford Algebras, 2019Co-Authors: Carlos Castro PerelmanAbstract:We review briefly how R $$\otimes $$ C $$\otimes $$ H $$\otimes $$ O-valued gravity (real-complex-quaterno-octonionic gravity) naturally can describe a grand unified field theory of Einstein’s gravity with a Yang–Mills theory containing the Standard Model group $$SU(3) \times SU(2) \times U(1)$$ . The algebra of left actions associated with the composite algebras involving the Division algebras, and the Sedenions $$\mathbf{S}$$ , and acting on themselves, all lead to complex Clifford algebras (complex Matrix algebras). The complex Cl(16) algebra is the most appealing one since it is the one corresponding to the algebra of left actions of $$\mathbf{C} \otimes \mathbf{H} \otimes \mathbf{O} \otimes \mathbf{S}$$ acting on itself, and containing the $$\mathbf{e_8 \oplus e_8}$$ algebra of the anomaly free 10D Heterotic String. An analysis of $$ C \otimes H \otimes O$$ -valued gravity reveals that it bears a connection to Nonsymmetric Kaluza–Klein theories and complex Hermitian Matrix Geometry. The key behind these connections is in finding the relation between $$ C \otimes H \otimes O$$ -valued metrics in two complex dimensions with complex metrics in higher dimensional real manifolds ( $$ D = 32 $$ real dimensions in particular). It is desirable to extend these results to hypercomplex, quaternionic manifolds and Exceptional Jordan Matrix Models.
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On C $$\varvec{\otimes }$$ ⊗ H $$\varvec{\otimes }$$ ⊗ O-Valued Gravity, Sedenions, Hermitian Matrix Geometry and Nonsymmetric Kaluza–Klein Theory
Advances in Applied Clifford Algebras, 2019Co-Authors: Carlos Castro PerelmanAbstract:We review briefly how R $$\otimes $$ C $$\otimes $$ H $$\otimes $$ O-valued gravity (real-complex-quaterno-octonionic gravity) naturally can describe a grand unified field theory of Einstein’s gravity with a Yang–Mills theory containing the Standard Model group $$SU(3) \times SU(2) \times U(1)$$ . The algebra of left actions associated with the composite algebras involving the Division algebras, and the Sedenions $$\mathbf{S}$$ , and acting on themselves, all lead to complex Clifford algebras (complex Matrix algebras). The complex Cl(16) algebra is the most appealing one since it is the one corresponding to the algebra of left actions of $$\mathbf{C} \otimes \mathbf{H} \otimes \mathbf{O} \otimes \mathbf{S}$$ acting on itself, and containing the $$\mathbf{e_8 \oplus e_8}$$ algebra of the anomaly free 10D Heterotic String. An analysis of $$ C \otimes H \otimes O$$ -valued gravity reveals that it bears a connection to Nonsymmetric Kaluza–Klein theories and complex Hermitian Matrix Geometry. The key behind these connections is in finding the relation between $$ C \otimes H \otimes O$$ -valued metrics in two complex dimensions with complex metrics in higher dimensional real manifolds ( $$ D = 32 $$ real dimensions in particular). It is desirable to extend these results to hypercomplex, quaternionic manifolds and Exceptional Jordan Matrix Models.
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On C $$\varvec{\otimes }$$ ⊗
Advances in Applied Clifford Algebras, 2019Co-Authors: Carlos Castro PerelmanAbstract:We review briefly how R $$\otimes $$ ⊗ C $$\otimes $$ ⊗ H $$\otimes $$ ⊗ O -valued gravity (real-complex-quaterno-octonionic gravity) naturally can describe a grand unified field theory of Einstein’s gravity with a Yang–Mills theory containing the Standard Model group $$SU(3) \times SU(2) \times U(1)$$ S U ( 3 ) × S U ( 2 ) × U ( 1 ) . The algebra of left actions associated with the composite algebras involving the Division algebras, and the Sedenions $$\mathbf{S}$$ S , and acting on themselves, all lead to complex Clifford algebras (complex Matrix algebras). The complex Cl (16) algebra is the most appealing one since it is the one corresponding to the algebra of left actions of $$\mathbf{C} \otimes \mathbf{H} \otimes \mathbf{O} \otimes \mathbf{S}$$ C ⊗ H ⊗ O ⊗ S acting on itself, and containing the $$\mathbf{e_8 \oplus e_8}$$ e 8 ⊕ e 8 algebra of the anomaly free 10 D Heterotic String. An analysis of $$ C \otimes H \otimes O$$ C ⊗ H ⊗ O -valued gravity reveals that it bears a connection to Nonsymmetric Kaluza–Klein theories and complex Hermitian Matrix Geometry. The key behind these connections is in finding the relation between $$ C \otimes H \otimes O$$ C ⊗ H ⊗ O -valued metrics in two complex dimensions with complex metrics in higher dimensional real manifolds ( $$ D = 32 $$ D = 32 real dimensions in particular). It is desirable to extend these results to hypercomplex, quaternionic manifolds and Exceptional Jordan Matrix Models.
Liu Zhen - One of the best experts on this subject based on the ideXlab platform.
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Public verifiable algorithm of threshold secret sharing with short share
Journal of Computer Applications, 2009Co-Authors: Liu ZhenAbstract:To make up the limitation that the length of secret can not be too long and prevent the action of cheating,using the theory of Jordan Matrix,and combining with the formulary of Lagrange,the authors put forward an algorithm of threshold secret sharing with short share.It could effectively resist the statistical attack and the united attack of corrupt participants less than r.The length of secret share that each participator needed to conserve was very short.It had a very important application when the secret was a big privacy file,a big message transmitted in an insecure channel,a secret database shared by several participants or enormous data in distributed storage.
Jörg Schray - One of the best experts on this subject based on the ideXlab platform.
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The general classical solution of the superparticle
Classical and Quantum Gravity, 1996Co-Authors: Jörg SchrayAbstract:The theory of vectors and spinors in (9 + 1)-dimensional spacetime is introduced in a completely octonionic formalism based on an octonionic representation of the Clifford algebra . The general solution of the classical equations of motion of the CBS superparticle is given to all orders of the Grassmann hierarchy. A spinor and a vector are combined into a Grassmann, octonionic, Jordan Matrix in order to construct a superspace variable to describe the superparticle. The combined Lorentz and supersymmetry transformations of the fermionic and bosonic variables are expressed in terms of Jordan products.
