| 1. |
- Abramov, Viktor, et al.
(författare)
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3-Hom-Lie Algebras Based on σ-Derivation and Involution
- 2020
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Ingår i: Advances in Applied Clifford Algebras. - : Springer. - 0188-7009 .- 1661-4909. ; 30:3
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Tidskriftsartikel (refereegranskat)abstract
- We show that, having a Hom-Lie algebra and an element of its dual vector space that satisfies certain conditions, one can construct a ternary totally skew-symmetric bracket and prove that this ternary bracket satisfies the Hom-Filippov-Jacobi identity, i.e. this ternary bracket determines the structure of 3-Hom-Lie algebra on the vector space of a Hom-Lie algebra. Then we apply this construction to two Hom-Lie algebras constructed on an associative, commutative algebra using σ-derivation and involution, and we obtain two 3-Hom-Lie algebras.
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| 2. |
- Farhangdoost, Mohammed Reza, et al.
(författare)
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Simply Complete Hom-Lie Superalgebras and Decomposition of Complete Hom-Lie Superalgebras
- 2023
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Ingår i: Advances in Applied Clifford Algebras. - : Birkhauser. - 0188-7009 .- 1661-4909. ; 33:2
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Tidskriftsartikel (refereegranskat)abstract
- Complete hom-Lie superalgebras are considered and some equivalent conditions for a hom-Lie superalgebra to be a complete hom-Lie superalgebra are established. In particular, the relation between decomposition and completeness for a hom-Lie superalgebra is described. Moreover, some conditions that the linear space of αs-derivations of a hom-Lie superalgebra to be complete and simply complete are obtained.
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| 3. |
- Khrennikov, Andrei
(författare)
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Hyperbolic quantization
- 2008
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Ingår i: Advances in Applied Clifford Algebras. - : Springer Science and Business Media LLC. - 0188-7009 .- 1661-4909. ; 18:3-4, s. 843-852
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Tidskriftsartikel (refereegranskat)abstract
- We demonstrated that classical mechanics have, besides the well known quantum deformation, another deformation – so called hyperbolic quantum mechanics. The classical Poisson bracket can be obtained as the limit h → 0 not only of the ordinary Moyal bracket, but also hyperbolic analogue of the Moyal bracket. Thus there are two different deformations of classical phase-space: complex Hilbert space and hyperbolic Hilbert space (module over a so called hyperbolic algebra – the two-dimensional Clifford algebra). Ordinary (complex) and hyperbolic quantum mechanics are characterized by two types of interference perturbation of the classical formula of total probability: ordinary cos-interference and hyperbolic cosh-interference.
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| 4. |
- Khrennikov, Andrei, 1958-
(författare)
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Representation of probabilistic data by quantum-like hyperbolic amplitudes
- 2010
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Ingår i: Advances in Applied Clifford Algebras. - Berlin-Heidelberg : Springer. - 0188-7009 .- 1661-4909. ; 20:1, s. 43-56
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we study the problem of representation of statistical data of any origin by hyperbolic probabilistic amplitudes (normalized vectors in hyperbolic Hilbert space). It generalizes the conventional QM which is based on complex Hilbert space. We performed extended numerical simulation. Similar to the conventional quantum formalism for Bloch's sphere, we visualize results of simulation for a special class of statistical data on so called Bloch's hyperboloid. The notion of hyperbolic qubit is introduced. © 2008 Birkḧauser Verlag Basel/Switzerland.
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| 5. |
- Krasnov, Yakov, et al.
(författare)
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Idempotent Geometry in Generic Algebras
- 2018
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Ingår i: Advances in Applied Clifford Algebras. - Basel, Switzerland : Birkhaeuser Science. - 0188-7009 .- 1661-4909. ; 28:5
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Tidskriftsartikel (refereegranskat)abstract
- Using the syzygy method, established in our earlier paper (Krasnov and Tkachev, 2018), we characterize the combinatorial stratification of the variety of two-dimensional real generic algebras. We show that there exist exactly three different homotopic types of such algebras and relate this result to potential applications and known facts from qualitative theory of quadratic ODEs. The genericity condition is crucial. For example, the idempotent geometry in Clifford algebras or Jordan algebras of Clifford type is very different: such algebras always contain nontrivial submanifolds of idempotents.
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| 6. |
- Lankinen, Paul
(författare)
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New construction of casual tensors
- 2004
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Ingår i: Advances in Applied Clifford Algebras. - : Birkhauser. - 0188-7009 .- 1661-4909. ; 14:2, s. 191-205
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Tidskriftsartikel (refereegranskat)abstract
- We study the problem of constructing tensors satisfying the dominant property, a generalization of the dominant energy condition T ab u a v b ≥ 0 for all future directed causal vectors u, v. The construction is done on the paravector subspace of the r-fold Euclidean Clifford algebra ⊗rCℓp and is a generalization of the representation of superenergy tensors with complex 2-spinors. Especially, as with 2-spinors, we are able to construct causal tensors of arbitrary rank, contrary to earlier constructions using tensors or the r-fold Lorentzian Clifford algebra ⊗rCℓp,1 that only produce causal tensors of even rank. An advantage of the construction in ⊗rCℓp is that several algebraic properties become trivial due to the Euclidean norm on it.
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| 7. |
- Lian, Binbin
(författare)
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Geometric Error Modeling of Parallel Manipulators Based on Conformal Geometric Algebra
- 2018
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Ingår i: Advances in Applied Clifford Algebras. - : Birkhauser Verlag AG. - 0188-7009 .- 1661-4909. ; 28:1
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Tidskriftsartikel (refereegranskat)abstract
- An approach for geometric error modeling of parallel manipulators (PMs) based on the visual representation and direct calculation of conformal geometric algebra is introduced in this paper. In this method, the finite motion of an open-loop chain is firstly formulated. Through linearization of the finite motion, error propagation of the open-loop chain is analyzed. Then the error sources are separated in terms of joint perturbations and geometric errors. Next, motions and constraints of PMs are analyzed visually by their reciprocal property. Finally geometric error model of PMs are formulated considering the actuations and constraints. The merits of this new approach are twofold: (1) complete and continuous geometric error modeling can be achieved since finite motions are considered, (2) visual and analytical computation of motions and constraints are applied for transferring geometric errors from the open-loop chain to the PM. A 2-DoF rotational PM is applied to demonstrate the geometric error modeling process. Comparisons between simulation and analytical models show that this approach is highly effective.
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| 8. |
- Mabrouk, S., et al.
(författare)
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Generalized Derivations and Rota-Baxter Operators of n-ary Hom-Nambu Superalgebras
- 2021
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Ingår i: Advances in Applied Clifford Algebras. - : Birkhauser. - 0188-7009 .- 1661-4909. ; 31:3
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Tidskriftsartikel (refereegranskat)abstract
- The aim of this paper is to generalise the construction of n-ary Hom-Lie bracket by means of an (n-2)-cochain of given Hom-Lie algebra to super case inducing n-Hom-Lie superalgebras. We study the notion of generalized derivations and Rota-Baxter operators of n-ary Hom-Nambu and n-Hom-Lie superalgebras and their relation with generalized derivations and Rota-Baxter operators of Hom-Lie superalgebras. We also introduce the notion of 3-Hom-pre-Lie superalgebras which is the generalization of 3-Hom-pre-Lie algebras.
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| 9. |
- Nyman, Peter
(författare)
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On the consistency of the quantum-like representation algorithm for hyperbolic interference.
- 2011
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Ingår i: Advances in Applied Clifford Algebras. - : Springer Science and Business Media LLC. - 0188-7009 .- 1661-4909. ; 21:4, s. 799-811
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Tidskriftsartikel (refereegranskat)abstract
- Recently quantum-like representation algorithm (QLRA) wasintroduced by A. Khrennikov [20]–[28] to solve the so-called “inverseBorn’s rule problem”: to construct a representation of probabilistic databy a complex or hyperbolic probability amplitude or more general complextogether with hyperbolic which matches Born’s rule or its generalizations.The outcome from QLRA is coupled to the formula of totalprobability with an additional term corresponding to trigonometric, hyperbolicor hyper-trigonometric interference. The consistency of QLRAfor probabilistic data corresponding to trigonometric interference was recentlyproved [29].We complete the proof of the consistency of QLRA tocover hyperbolic interference as well. We will also discuss hyper trigonometricinterference. The problem of consistency of QLRA arises, becauseformally the output of QLRA depends on the order of conditioning. Fortwo observables (e.g., physical or biological) a and b, b|a- and a|b- conditionalprobabilities produce two representations, say in Hilbert spacesHb|a and Ha|b (in this paper over the hyperbolic algebra). We provethat under “natural assumptions” these two representations are unitaryequivalent (in the sense of hyperbolic Hilbert space).
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