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Fabien Besnard - One of the best experts on this subject based on the ideXlab platform.

  • algebraic backgrounds for nonCommutative kaluza klein theory ii the almost Commutative Case and the standard model
    Journal of Mathematical Physics, 2019
    Co-Authors: Fabien Besnard
    Abstract:

    We define almost-Commutative algebraic backgrounds and give conditions on them allowing us to compute their configuration space in terms of those of the continuous and finite parts. We apply these results to a background with finite algebra C⊕H⊕M3(C) and find that the configuration space is larger than the one obtained from the fluctuations of the metric and contains in addition to the Standard Model (SM) gauge fields, the ZB-L′-boson, a complex scalar field σ, and flavor changing fields. The latter can be removed similarly to centralizing fields in the gravity model studied in the first part. The remaining fields belong to a U(1)B-L-extension of the SM.We define almost-Commutative algebraic backgrounds and give conditions on them allowing us to compute their configuration space in terms of those of the continuous and finite parts. We apply these results to a background with finite algebra C⊕H⊕M3(C) and find that the configuration space is larger than the one obtained from the fluctuations of the metric and contains in addition to the Standard Model (SM) gauge fields, the ZB-L′-boson, a complex scalar field σ, and flavor changing fields. The latter can be removed similarly to centralizing fields in the gravity model studied in the first part. The remaining fields belong to a U(1)B-L-extension of the SM.

  • NonCommutative ordered spaces: examples and counterexamples
    Classical and Quantum Gravity, 2015
    Co-Authors: Fabien Besnard
    Abstract:

    In order to introduce the notion of causality in nonCommutative geometry, it is necessary to extend Gelfand theory to the context of ordered spaces. In a previous work we have already given an algebraic characterization of the set of non-decreasing continuous functions on a certain class of topological ordered spaces. Such a set is called an isocone, and there exist at least two versions of them (strong and weak) which coincide in the Commutative Case. In this paper, we introduce yet another breed of isocones, ultraweak isocones, which has a simpler definition with a clear physical meaning. We show that ultraweak and weak isocones are in fact the same, and completely classify those that live in a finite-dimensional -algebra, hence corresponding to finite nonCommutative ordered spaces. We also give some examples in infinite dimension.

Rodica D Costin - One of the best experts on this subject based on the ideXlab platform.

  • matrix valued polynomials generated by the scalar type rodrigues formulas
    Journal of Approximation Theory, 2009
    Co-Authors: Rodica D Costin
    Abstract:

    The properties of matrix-valued polynomials generated by the scalar-type Rodrigues' formulas are analyzed. A general representation of these polynomials is found in terms of products of simple differential operators. The recurrence relations, leading coefficients, completeness are established, as well as, in the Commutative Case, the second order equations for which these polynomials are eigenfunctions and the corresponding eigenvalues, and ladder operators. A new, direct proof is given to the conjecture of Duran and Grunbaum that if the weights are self-adjoint and positive semidefinite then they are necessarily of scalar type. Commutative classes of orthogonal polynomials (corresponding to weights that are self-adjoint but not positive semidefinite) are found, which satisfy all the properties usually associated to orthogonal polynomials, and are not of scalar type.

  • matrix valued polynomials generated by the scalar type rodrigues formulas
    arXiv: Classical Analysis and ODEs, 2008
    Co-Authors: Rodica D Costin
    Abstract:

    The properties of matrix valued polynomials generated by the scalar-type Rodrigues' formulas are analyzed. A general representation of these polynomials is found in terms of products of simple differential operators. The recurrence relations, leading coefficients, completeness are established, as well as, in the Commutative Case, the second order equations for which these polynomials are eigenfunctions and the corresponding eigenvalues, and ladder operators. The conjecture of Duran and Grunbaum that if the weights are self-adjoint and positive semidefinite then they are necessarily of scalar type is proved for Q(x)=x and Q(x)=x^2-1 in dimension two, and for any dimension under genericity assumptions. Commutative classes of quasi-orthogonal polynomials are found, which satisfy all the properties usually associated to orthogonal polynomials.

Rinat Kedem - One of the best experts on this subject based on the ideXlab platform.

  • The Solution of the Quantum A _1 T-System for Arbitrary Boundary
    Communications in Mathematical Physics, 2012
    Co-Authors: Philippe Di Francesco, Rinat Kedem
    Abstract:

    We solve the quantum version of the A _1 T -system by use of quantum networks. The system is interpreted as a particular set of mutations of a suitable (infinite-rank) quantum cluster algebra, and Laurent positivity follows from our solution. As an application we re-derive the corresponding quantum network solution to the quantum A _1 Q -system and generalize it to the fully non-Commutative Case. We give the relation between the quantum T -system and the quantum lattice Liouville equation, which is the quantized Y -system.

Renzo Sprugnoli - One of the best experts on this subject based on the ideXlab platform.

  • mixed succession rules the Commutative Case
    Journal of Combinatorial Theory Series A, 2010
    Co-Authors: Silvia Bacchelli, Luca Ferrari, Renzo Pinzani, Renzo Sprugnoli
    Abstract:

    We begin a systematic study of the enumerative combinatorics of mixed succession rules, i.e. succession rules such that, in the associated generating tree, nodes are allowed to produce sons at several different levels according to different production rules. Here we deal with a specific Case, namely that of two different production rules whose rule operators commute. In this situation, we are able to give a general formula expressing the sequence associated with the mixed succession rule in terms of the sequences associated with the component production rules. We end by providing examples illustrating our approach.

  • mixed succession rules the Commutative Case
    arXiv: Combinatorics, 2008
    Co-Authors: Silvia Bacchelli, Luca Ferrari, Renzo Pinzani, Renzo Sprugnoli
    Abstract:

    We begin a systematic study of the enumerative combinatorics of mixed succession rules, which are succession rules such that, in the associated generating tree, the nodes are allowed to produce their sons at several different levels according to different production rules. Here we deal with a specific Case, namely that of two different production rules whose rule operators commute. In this situation, we are able to give a general formula expressing the sequence associated with the mixed succession rules in terms of the sequences associated with the component production rules. We end by providing some examples illustrating our approach.

Mira A. Peterka - One of the best experts on this subject based on the ideXlab platform.

  • Finitely-Generated Projective Modules over the θ-Deformed 4-Sphere
    Communications in Mathematical Physics, 2013
    Co-Authors: Mira A. Peterka
    Abstract:

    We investigate the “ θ -deformed spheres” $${C(S^{3}_{\theta})}$$ and $${C(S^{4}_{\theta})}$$ , where θ is any real number. We show that all finitely-generated projective modules over $${C(S^{3}_{\theta})}$$ are free, and that $${C(S^{4}_{\theta})}$$ has the cancellation property. We classify and construct all finitely-generated projective modules over $${C(S^{4}_{\theta})}$$ up to isomorphism. An interesting feature is that if θ is irrational then there are nontrivial “rank-1” modules over $${C(S^{4}_{\theta})}$$ . In that Case, every finitely-generated projective module over $${C(S^{4}_{\theta})}$$ is a sum of a rank-1 module and a free module. If θ is rational, the situation mirrors that for the Commutative Case θ = 0.