Characterizing the derivative and the entropy function by the Leibniz rule

König, Hermann; Milman, Vitali

doi:10.1016/j.jfa.2011.05.003

Cited by 21 publications

(17 citation statements)

References 5 publications

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“…All solution operators of the classical Leibniz product rule were determined by König and Milman. 16 Their method is extended to the operators with the Leibniz defect, and the uniqueness is confirmed.…”

Section: Introductionmentioning

confidence: 91%

“…The next addressed question is the uniqueness of the derivative definition, based on the product rule with the Leibniz defect. All solution operators of the classical Leibniz product rule were determined by König and Milman . The method they established could be immediately extended to the rule with the Leibniz defect.…”

Section: Generalized Differentiation Operatormentioning

confidence: 99%

“…The application of the method of theorem 2 in the work of König and Milman leads for the operator Ð to the functional equation:

F ((), fg) false/ ((), fg) = F ((), f) false/ f + F ((), g) false/ g - κ false/ 2 .

…”

Section: Generalized Differentiation Operatormentioning

confidence: 99%

“…The only difference between the functional equation and the functional equation that is considered in is the last term. The solution of the functional equation delivers

H ((), s) = s + κ false/ 2 .

…”

Section: Generalized Differentiation Operatormentioning

confidence: 99%

“…The considerations of König and Milman are repeated with some minor alterations for the product rule with a similar entropy function that contains the nonvanishing Leibniz defect κ . Thus, all solution operators of the product rule with the Leibniz defect are determined.…”

Section: Generalized Differentiation Operatormentioning

confidence: 99%

See 4 more Smart Citations

Non‐Leibniz Hamiltonian and Lagrangian formalisms for certain class of dissipative systems

Kobelev

2019

Comp and Math Methods

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The non‐Leibniz Hamiltonian and Lagrangian formalism for the certain class of dissipative systems is introduced in this article. The formalism is based on the generalized differentiation operator (κ‐operator) with a nonzero Leibniz defect. The Leibniz defect of the introduced operator linearly depends on one scaling parameter. In a special case, if the Leibniz defect vanishes, the generalized differentiation operator reduces to the common differentiation operator. The κ‐operator allows the formulation of the variational principles and corresponding equations of Lagrange and Hamiltonian types for dissipative systems. The solutions of some generalized dynamical equations are provided closed form.

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Section: Introductionmentioning

confidence: 91%

Section: Generalized Differentiation Operatormentioning

confidence: 99%

“…The application of the method of theorem 2 in the work of König and Milman leads for the operator Ð to the functional equation:

F ((), fg) false/ ((), fg) = F ((), f) false/ f + F ((), g) false/ g - κ false/ 2 .

…”

Section: Generalized Differentiation Operatormentioning

confidence: 99%

“…The only difference between the functional equation and the functional equation that is considered in is the last term. The solution of the functional equation delivers

H ((), s) = s + κ false/ 2 .

…”

Section: Generalized Differentiation Operatormentioning

confidence: 99%

Section: Generalized Differentiation Operatormentioning

confidence: 99%

See 3 more Smart Citations

Non‐Leibniz Hamiltonian and Lagrangian formalisms for certain class of dissipative systems

Kobelev

2019

Comp and Math Methods

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Mechanics with non‐Leibniz derivatives

Kobelev

2018

Proc Appl Math and Mech

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The non-Leibniz Hamiltonian and Lagrangian formalism for the certain class of dissipative systems is introduced in this article. The formalism is based on the generalized differentiation κ-operator with a non-zero Leibniz defect. The Leibniz defect of the introduced operator linearly depends on one scaling parameter. In a special case, if the Leibniz defect vanishes, the κ-operator reduces to the common differentiation operator. The new operator allows the formulation of the variational principles and corresponding equations of Lagrange and Hamiltonian types for dissipative systems. The solutions of some generalized dynamical equations are provided in closed form.

show abstract