95,960 research outputs found
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 Lodgeâs 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 Leibnizâs 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. Lodgeâs 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 Leibnizâs 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 ânothing would happen that had not happened before." Leibnizâs 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 Leibnizâs own metaphysics. Despite this, we can discern a serious and important point to Leibnizâs 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 IDâ-Lie-derivations. An IDâ-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 IDLieâ (g) of IDâ-Lie-derivation of g, and prove that the sets IDLieâ (g) and IDLieâ (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 -derivations. A -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 of -derivation of G, and
prove that the sets and are isomorphic for any
two Lie-isoclinic Leibniz algebras G and Q
Leibniz Series for Ï
In this article we prove the Leibniz series for Ï which states that Ï4=ân=0â(â1)n2â
n+1. The formalization follows K. Knopp [8], [1] and [6]. Leibnizâs Series for Pi is item #26 from the âFormalizing 100 Theoremsâ list maintained by Freek Wiedijk at http://www.cs.ru.nl/F.Wiedijk/100/.PÄ
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