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A M Semikhatov - One of the best experts on this subject based on the ideXlab platform.

  • higher string Functions higher level appell Functions and the logarithmic widehat s ell 2_k u 1 cft model
    Communications in Mathematical Physics, 2009
    Co-Authors: A M Semikhatov
    Abstract:

    We generalize the string Functions \({{\fancyscript {C}_{n,r}(\tau)}}\) associated with the coset \({{{\widehat{s\ell}2_k/u(1)}}}\) to higher string Functions \({{\fancyscript {A}_{n,r}(\tau)}}\) and \({{\fancyscript {B}_{n,r}(\tau)}}\) associated with the coset W(k)/u(1) of the W-algebra of the logarithmically extended \({{{\widehat{s\ell}2_k}}}\) conformal field model with positive integer k. The higher string Functions occur in decomposing W(k) characters with respect to level-k theta and Appell Functions and their derivatives (the characters are neither quasiperiodic nor holomorphic, and therefore cannot decompose with respect to only theta-Functions). The decomposition coefficients, to be considered “logarithmic parafermionic characters,” are given by \({{\fancyscript {A}_{n,r}(\tau)}}\) , \({{\fancyscript {B}_{n,r}(\tau)}}\) , \({{\fancyscript {C}_{n,r}(\tau)}}\) , and by the triplet \({{\fancyscript{W}(p)}}\) -algebra characters of the (p = k + 2, 1) logarithmic model. We study the properties of \({{\fancyscript {A}_{n,r}}}\) and \({{\fancyscript {B}_{n,r}}}\) , which nontrivially generalize those of the classic string Functions \({{\fancyscript {C}_{n,r}}}\) , and evaluate the modular group representation generated from \({{\fancyscript {A}_{n,r}(\tau)}}\) and \({{\fancyscript {B}_{n,r}(\tau)}}\) ; its structure inherits some features of modular transformations of the higher-level Appell Functions and the associated Transcendental Function Φ.

  • higher string Functions higher level appell Functions and the logarithmic sl 2 _k u 1 cft model
    arXiv: Quantum Algebra, 2007
    Co-Authors: A M Semikhatov
    Abstract:

    We generalize the string Functions C_{n,r}(tau) associated with the coset ^sl(2)_k/u(1) to higher string Functions A_{n,r}(tau) and B_{n,r}(tau) associated with the coset W(k)/u(1) of the W-algebra of the logarithmically extended ^sl(2)_k conformal field model with positive integer k. The higher string Functions occur in decomposing W(k) characters with respect to level-k theta and Appell Functions and their derivatives (the characters are neither quasiperiodic nor holomorphic, and therefore cannot decompose with respect to only theta-Functions). The decomposition coefficients, to be considered ``logarithmic parafermionic characters,'' are given by A_{n,r}(tau), B_{n,r}(tau), C_{n,r}(tau), and by the triplet \mathscr{W}(p)-algebra characters of the (p=k+2,1) logarithmic model. We study the properties of A_{n,r} and B_{n,r}, which nontrivially generalize those of the classic string Functions C_{n,r}, and evaluate the modular group representation generated from A_{n,r}(tau) and B_{n,r}(tau); its structure inherits some features of modular transformations of the higher-level Appell Functions and the associated Transcendental Function Phi.

Zhiyun Cheng - One of the best experts on this subject based on the ideXlab platform.

Stephan Stieberger - One of the best experts on this subject based on the ideXlab platform.

  • motivic multiple zeta values and superstring amplitudes
    Journal of Physics A, 2013
    Co-Authors: Oliver Schlotterer, Stephan Stieberger
    Abstract:

    The structure of tree-level open and closed superstring amplitudes is analyzed. For the open superstring amplitude we find a striking and elegant form, which allows one to disentangle its α′-expansion into several contributions accounting for different classes of multiple zeta values. This form is bolstered by the decomposition of motivic multiple zeta values, i.e. the latter encapsulate the α′-expansion of the superstring amplitude. Moreover, a morphism induced by the coproduct maps the α′-expansion onto a non-commutative Hopf algebra. This map represents a generalization of the symbol of a Transcendental Function. In terms of elements of this Hopf algebra the α′-expansion assumes a very simple and symmetric form, which carries all the relevant information. Equipped with these results we can also cast the closed superstring amplitude into a very elegant form.

  • motivic multiple zeta values and superstring amplitudes
    arXiv: High Energy Physics - Theory, 2012
    Co-Authors: Oliver Schlotterer, Stephan Stieberger
    Abstract:

    The structure of tree-level open and closed superstring amplitudes is analyzed. For the open superstring amplitude we find a striking and elegant form, which allows to disentangle its alpha'-expansion into several contributions accounting for different classes of multiple zeta values. This form is bolstered by the decomposition of motivic multiple zeta values, i.e. the latter encapsulate the alpha'-expansion of the superstring amplitude. Moreover, a morphism induced by the coproduct maps the alpha'-expansion onto a non-commutative Hopf algebra. This map represents a generalization of the symbol of a Transcendental Function. In terms of elements of this Hopf algebra the alpha'-expansion assumes a very simple and symmetric form, which carries all the relevant information. Equipped with these results we can also cast the closed superstring amplitude into a very elegant form.

Oliver Schlotterer - One of the best experts on this subject based on the ideXlab platform.

  • motivic multiple zeta values and superstring amplitudes
    Journal of Physics A, 2013
    Co-Authors: Oliver Schlotterer, Stephan Stieberger
    Abstract:

    The structure of tree-level open and closed superstring amplitudes is analyzed. For the open superstring amplitude we find a striking and elegant form, which allows one to disentangle its α′-expansion into several contributions accounting for different classes of multiple zeta values. This form is bolstered by the decomposition of motivic multiple zeta values, i.e. the latter encapsulate the α′-expansion of the superstring amplitude. Moreover, a morphism induced by the coproduct maps the α′-expansion onto a non-commutative Hopf algebra. This map represents a generalization of the symbol of a Transcendental Function. In terms of elements of this Hopf algebra the α′-expansion assumes a very simple and symmetric form, which carries all the relevant information. Equipped with these results we can also cast the closed superstring amplitude into a very elegant form.

  • motivic multiple zeta values and superstring amplitudes
    arXiv: High Energy Physics - Theory, 2012
    Co-Authors: Oliver Schlotterer, Stephan Stieberger
    Abstract:

    The structure of tree-level open and closed superstring amplitudes is analyzed. For the open superstring amplitude we find a striking and elegant form, which allows to disentangle its alpha'-expansion into several contributions accounting for different classes of multiple zeta values. This form is bolstered by the decomposition of motivic multiple zeta values, i.e. the latter encapsulate the alpha'-expansion of the superstring amplitude. Moreover, a morphism induced by the coproduct maps the alpha'-expansion onto a non-commutative Hopf algebra. This map represents a generalization of the symbol of a Transcendental Function. In terms of elements of this Hopf algebra the alpha'-expansion assumes a very simple and symmetric form, which carries all the relevant information. Equipped with these results we can also cast the closed superstring amplitude into a very elegant form.

Weinian Zhang - One of the best experts on this subject based on the ideXlab platform.

  • equilibria and their bifurcations in a recurrent neural network involving iterates of a Transcendental Function
    IEEE Transactions on Neural Networks, 2008
    Co-Authors: B Gao, Weinian Zhang
    Abstract:

    Some practical models contain so complicated mathematical expressions that it is hard to determine the number and distribution of all equilibria, not mentioning the qualitative properties and bifurcations of those equilibria. The three-node recurrent neural network system with two free weight parameters, originally introduced by Ruiz, Owens, and Townley in 1997, is such a system, for which the equation of equilibria involves Transcendental Function and its iterates. Not computing coordinates of its equilibria, in this paper, we display an effective technique to determine the number and distribution of its equilibria. Without full information about equilibria, our method enables to further study qualitative properties of those equilibria and discuss their saddle node, pitchfork, and Hopf bifurcations by approximating center manifolds.