On the Leibniz rule and Laplace transform for fractional derivatives

Taylor series is a useful mathematical tool when describing and constructing a function. With the series representation, some properties of fractional calculus can be revealed clearly. This paper investigates two typical applications: Lebiniz rule and Laplace transform. It is analytically shown that the commonly used Leibniz rule cannot be applied for Caputo derivative. Similarly, the well-known Laplace transform of Riemann-Liouville derivative is doubtful for n-th continuously differentiable function. By the aid of this series representation, the exact formula of Caputo Leibniz rule and the explanation of Riemann-Liouville Laplace transform are presented. Finally, three illustrative examples are revisited to confirm the obtained results

The Leibniz-De Volder Correspondence, with Selections from the Correspondence Between Leibniz and Johann Bernoulli, ed. P. Lodge

Paul Lodge’s excellent new contribution to the Yale Leibniz series collects together the entirety of the Leibniz-De Volder correspondence, totaling some thirty-three letters, together with a generous selection of relevant excerpts from Leibniz’s concurrent correspondence with Bernoulli, which Lodge has helpfully interspersed throughout. As with previous volumes in the series, the texts appear in the original language, in this case Latin, together with an English translation on opposing pages. Lodge’s transcriptions reflect his careful study of all the available manuscripts and represent a significant improvement over the existing versions in GP II (Leibniz-De Volder) and GM III (Leibniz-Bernoulli). Rounding out the volume are a long introduction (79 pp.), itself a valuable contribution to Leibniz scholarship, together with extensive notes on the texts, a bibliography, and indexes for names and subjects

Noncommutative Differential Forms on the kappa-deformed Space

We construct a differential algebra of forms on the kappa-deformed space. For a given realization of the noncommutative coordinates as formal power series in the Weyl algebra we find an infinite family of one-forms and nilpotent exterior derivatives. We derive explicit expressions for the exterior derivative and one-forms in covariant and noncovariant realizations. We also introduce higher-order forms and show that the exterior derivative satisfies the graded Leibniz rule. The differential forms are generally not graded-commutative, but they satisfy the graded Jacobi identity. We also consider the star-product of classical differential forms. The star-product is well-defined if the commutator between the noncommutative coordinates and one-forms is closed in the space of one-forms alone. In addition, we show that in certain realizations the exterior derivative acting on the star-product satisfies the undeformed Leibniz rule.Comment: to appear in J. Phys. A: Math. Theo

The Apokatastasis Essays in Context: Leibniz and Thomas Burnet on the Kingdom of Grace and the Stoic/Platonic Revolutions

One of Leibniz’s more unusual philosophical projects is his presentation (in a series of unpublished drafts) of an argument for the conclusion that a time will necessarily come when “nothing would happen that had not happened before." Leibniz’s presentations of the argument for such a cyclical cosmology are all too brief, and his discussion of its implications is obscure. Moreover, the conclusion itself seems to be at odds with the main thrust of Leibniz’s own metaphysics. Despite this, we can discern a serious and important point to Leibniz’s consideration of the doctrine, namely in what it suggests about the proper boundary between metaphysics and theology, on the one hand, and ordinary history (whether human or natural), on the other. And we can get a better sense of Leibniz purpose in the essays by considering them in the context of Leibniz's response to Thomas Burnet's "Telluris theoria sacra" (1681-89). Leibniz praises Burnet's history of earth for presenting a harmony between the principles of nature and grace, a harmony absent in the cosmogonies of Descartes and the Newtonians. But Leibniz also complains that Burnet misconceives the boundary between natural explanation and reflections on divine wisdom. And Leibniz's essays on cyclical cosmology suggest the alternative to Burnet's account: a natural history of the earth and its inhabitants should be radically autonomous from, even if ultimately harmonious with, theological principles

Lie-central derivations, Lie-centroids and Lie-stem Leibniz algebras

In this paper, we introduce the notion of a Lie-derivation. This concept generalizes derivations for non-Lie Leibniz algebras. We study these Lie-derivations in the case where their image is contained in the Lie-center, and call them Lie-central derivations. We provide a characterization of Lie-stem Leibniz algebras by their Lie-central derivations, and prove several properties of the Lie algebra of Lie-central derivations for Lie-nilpotent Leibniz algebras of class 2. We also introduce ID∗-Lie-derivations. An ID∗-Lie-derivation of a Leibniz algebra g is a Lie-derivation of g in which the image is contained in the second term of the lower Lie-central series of g, and which vanishes on Lie-central elements. We provide an upper bound for the dimension of the Lie algebra IDLie∗ (g) of ID∗-Lie-derivation of g, and prove that the sets IDLie∗ (g) and IDLie∗ (q) are isomorphic for any two Lie-isoclinic Leibniz algebras g and q

Lie-central derivations, Lie-centroids and Lie-stem Leibniz algebras

In this paper, we introduce the notion Lie-derivation. This concept generalizes derivations for non-Lie Leibniz algebras. We study these Lie-derivations in the case where their image is contained in the Lie-center, call them Lie-central derivations. We provide a characterization of Lie-stem Leibniz algebras by their Lie-central derivations, and prove several properties of the Lie algebra of Lie-central derivations for Lie-nilpotent Leibniz algebras of class 2. We also introduce ${\sf ID}_*-Lie$-derivations. A ${\sf ID}_*-Lie$-derivation of a Leibniz algebra G is a Lie-derivation of G in which the image is contained in the second term of the lower Lie-central series of G, and that vanishes on Lie-central elements. We provide an upperbound for the dimension of the Lie algebra $ID_*^{Lie}(G)$ of $ID_*Lie$-derivation of G, and prove that the sets $ID_*^{Lie}(G)$ and $ID_*^{Lie}(G)$ are isomorphic for any two Lie-isoclinic Leibniz algebras G and Q

Leibniz Series for π

In this article we prove the Leibniz series for π which states that π4=∑n=0∞(−1)n2⋅n+1. The formalization follows K. Knopp [8], [1] and [6]. Leibniz’s Series for Pi is item #26 from the “Formalizing 100 Theorems” list maintained by Freek Wiedijk at http://www.cs.ru.nl/F.Wiedijk/100/.Pąk Karol - Institute of Informatics, University of Białystok, Ciołkowskiego 1M, 15-245 Białystok, PolandGeorge E. Andrews, Richard Askey, and Ranjan Roy. Special Functions. Cambridge University Press, 1999.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507–513, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55–65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.Lokenath Debnath. The Legacy of Leonhard Euler: A Tricentennial Tribute. World Scientific, 2010.Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Definition of integrability for partial functions from ℝ to ℝ and integrability for continuous functions. Formalized Mathematics, 9(2):281–284, 2001.Konrad Knopp. Infinite Sequences and Series. Dover Publications, 1956. ISBN 978-0-486-60153-3.Jarosław Kotowicz. Partial functions from a domain to the set of real numbers. Formalized Mathematics, 1(4):703–709, 1990.Jarosław Kotowicz. Monotone real sequences. Subsequences. Formalized Mathematics, 1 (3):471–475, 1990.Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269–272, 1990.Jarosław Kotowicz. Convergent real sequences. Upper and lower bound of sets of real numbers. Formalized Mathematics, 1(3):477–481, 1990.Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887–890, 1990.Xiquan Liang and Bing Xie. Inverse trigonometric functions arctan and arccot. Formalized Mathematics, 16(2):147–158, 2008. doi:10.2478/v10037-008-0021-3.Akira Nishino and Yasunari Shidama. The Maclaurin expansions. Formalized Mathematics, 13(3):421–425, 2005.Chanapat Pacharapokin, Kanchun, and Hiroshi Yamazaki. Formulas and identities of trigonometric functions. Formalized Mathematics, 12(2):139–141, 2004.Konrad Raczkowski. Integer and rational exponents. Formalized Mathematics, 2(1):125–130, 1991.Konrad Raczkowski and Andrzej Nędzusiak. Real exponents and logarithms. Formalized Mathematics, 2(2):213–216, 1991.Piotr Rudnicki and Andrzej Trybulec. Abian’s fixed point theorem. Formalized Mathematics, 6(3):335–338, 1997.Yuguang Yang and Yasunari Shidama. Trigonometric functions and existence of circle ratio. Formalized Mathematics, 7(2):255–263, 1998