The Integral Analog of the Leibniz Rule

We propose an extension of [Formula: see text]-ary Nambu–Poisson bracket to superspace [Formula: see text] and construct by means of superdeterminant a family of Nambu–Poisson algebras of even degree functions, where the parameter of this family is an invertible transformation of Grassmann coordinates in superspace [Formula: see text]. We prove in the case of the superspaces [Formula: see text] and [Formula: see text] that our [Formula: see text]-ary bracket, defined with the help of superdeterminant, satisfies the conditions for [Formula: see text]-ary Nambu–Poisson bracket, i.e. it is totally skew-symmetric and it satisfies the Leibniz rule and the Filippov–Jacobi identity (fundamental identity). We study the structure of [Formula: see text]-ary bracket defined with the help of superdeterminant in the case of superspace [Formula: see text] and show that it is the sum of usual [Formula: see text]-ary Nambu–Poisson bracket and a new [Formula: see text]-ary bracket, which we call [Formula: see text]-bracket, where [Formula: see text] is the product of two odd degree smooth functions.

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Certain Fractional Derivative Formulae Involving the Product of a General Class of Polynomials and the Multivariable Gimel-Function

Global Journal of Science Frontier Research ◽

10.34257/gjsfrfvol18is6pg1 ◽

2018 ◽

pp. 1-13

Author(s):

Frédéric Ayant

Keyword(s):

Fractional Derivative ◽

General Class ◽

Leibniz Rule ◽

General Character ◽

General Class Of Polynomials

In the present paper, we obtain three unified fractional derivative formulae. The first involves the product of a general class of polynomials and the multivariable Gimel-function. The second involves the product of a general class of polynomials and two multivariable Gimel-functions and has been obtained with the help of the generalized Leibniz rule for fractional derivatives.The last fractional derivative formulae also involves the product of a general class of polynomials and the multivariable Gimel-function but it is obtained by the application of the first fractional derivative formulae twice and, it involve two independents variables instead of one.The polynomials and the functions involved in all our fractional derivative formulae as well as their arguments which are of the type The formulae are the very general character and thus making them useful in applications. In the end, we shall give a particular case.

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No-go theorem of Leibniz rule and supersymmetry on the lattice

10.22323/1.066.0233 ◽

2009 ◽

Author(s):

Makoto Sakamoto ◽

Mitsuhiro Kato ◽

Hiroto So

Keyword(s):

Leibniz Rule

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On deformations of classical mechanics due to Planck-scale physics

International Journal of Modern Physics D ◽

10.1142/s0218271820500704 ◽

2020 ◽

Vol 29 (10) ◽

pp. 2050070

Author(s):

Olga I. Chashchina ◽

Abhijit Sen ◽

Zurab K. Silagadze

Keyword(s):

Quantum System ◽

Evolution Equations ◽

Hidden Variables ◽

Classical Mechanics ◽

Planck Scale ◽

Leibniz Rule ◽

Uncertainty Relations ◽

Interesting Possibility ◽

Heisenberg Uncertainty ◽

Space Formulation

Several quantum gravity and string theory thought experiments indicate that the Heisenberg uncertainty relations get modified at the Planck scale so that a minimal length do arises. This modification may imply a modification of the canonical commutation relations and hence quantum mechanics at the Planck scale. The corresponding modification of classical mechanics is usually considered by replacing modified quantum commutators by Poisson brackets suitably modified in such a way that they retain their main properties (antisymmetry, linearity, Leibniz rule and Jacobi identity). We indicate that there exists an alternative interesting possibility. Koopman–von Neumann’s Hilbert space formulation of classical mechanics allows, as Sudarshan remarked, to consider the classical mechanics as a hidden variable quantum system. Then, the Planck scale modification of this quantum system naturally induces the corresponding modification of dynamics in the classical substrate. Interestingly, it seems this induced modification in fact destroys the classicality: classical position and momentum operators cease to be commuting and hidden variables do appear in their evolution equations.

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Bifurcations and Exact Solutions for a Class of MKdV Equations with the Conformable Fractional Derivative via Dynamical System Method

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127420500042 ◽

2020 ◽

Vol 30 (01) ◽

pp. 2050004 ◽

Cited By ~ 4

Author(s):

Jianli Liang ◽

Longkun Tang ◽

Yonghui Xia ◽

Yi Zhang

Keyword(s):

Fractional Derivative ◽

Exact Solutions ◽

Fractional Order ◽

Traveling Wave ◽

Nonlinear Differential Equations ◽

Traveling Wave Solutions ◽

Leibniz Rule ◽

Wave Transformation ◽

Conformable Fractional Derivative ◽

Simplest Equation Method

In 2014, Khalil et al. [2014] proposed the conformable fractional derivative, which obeys chain rule and the Leibniz rule. In this paper, motivated by the monograph of Jibin Li [Li, 2013], we study the exact traveling wave solutions for a class of third-order MKdV equations with the conformable fractional derivative. Our approach is based on the bifurcation theory of planar dynamical systems, which is much different from the simplest equation method proposed in [Chen & Jiang, 2018]. By employing the traveling wave transformation [Formula: see text] [Formula: see text], we reduce the PDE to an ODE which depends on the fractional order [Formula: see text], then the analysis depends on the order [Formula: see text]. Moreover, as [Formula: see text], the exact solutions are consistent with the integer PDE. However, in all the existing papers, the reduced ODE is independent of the fractional order [Formula: see text]. It is believed that this method can be applicable to solve the other nonlinear differential equations with the conformable fractional derivative.

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On new applications of fractional calculus

Boletim da Sociedade Paranaense de Matemática ◽

10.5269/bspm.v37i3.18626 ◽

2017 ◽

Vol 37 (3) ◽

pp. 113-118 ◽

Cited By ~ 2

Author(s):

Shilpi Jain ◽

Praveen Agarwal

Keyword(s):

Fractional Calculus ◽

Special Functions ◽

Leibniz Rule ◽

The One ◽

New Applications ◽

Paper Author

In the present paper author derive a number of integrals concerning various special functions which are applications of the one of Osler result. Osler provided extensions to the familiar Leibniz rule for the nth derivative of product of two functions.

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Long-time behavior of numerical solutions to nonlinear fractional ODEs

ESAIM Mathematical Modelling and Numerical Analysis ◽

10.1051/m2an/2019055 ◽

2020 ◽

Vol 54 (1) ◽

pp. 335-358 ◽

Cited By ~ 4

Author(s):

Dongling Wang ◽

Aiguo Xiao ◽

Jun Zou

Keyword(s):

Numerical Methods ◽

Numerical Solutions ◽

Analytical Techniques ◽

Decay Rates ◽

Leibniz Rule ◽

Discrete Version ◽

Time Behavior ◽

Long Time Behavior ◽

Integration Schemes ◽

Long Time

In this work, we study the long time behavior, including asymptotic contractivity and dissipativity, of the solutions to several numerical methods for fractional ordinary differential equations (F-ODEs). The existing algebraic contractivity and dissipativity rates of the solutions to the scalar F-ODEs are first improved. In order to study the long time behavior of numerical solutions to fractional backward differential formulas (F-BDFs), two crucial analytical techniques are developed, with the first one for the discrete version of the fractional generalization of the traditional Leibniz rule, and the other for the algebraic decay rate of the solution to a linear Volterra difference equation. By means of these auxiliary tools and some natural conditions, the solutions to F-BDFs are shown to be contractive and dissipative, and also preserve the exact contractivity rate of the continuous solutions. Two typical F-BDFs, based on the Grünwald–Letnikov formula and L1 method respectively, are studied. For high order F-BDFs, including convolution quadrature schemes based on classical second order BDF and product integration schemes based on quadratic interpolation approximation, their numerical contractivity and dissipativity are also developed under some slightly stronger conditions. Numerical experiments are presented to validate the long time qualitative characteristics of the solutions to F-BDFs, revealing very different decay rates of the numerical solutions in terms of the the initial values between F-ODEs and integer ODEs and demonstrating the superiority of the structure-preserving numerical methods.

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