Classroom note: A Generalization of Leibniz rule for higher derivatives

Thaheem, A. B.; Laradji, A.

doi:10.1080/00207390310001595410

Cited by 17 publications

(13 citation statements)

References 1 publication

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“…+ α n . Leibniz considered the problem of determining the kth derivative of the product of n smooth functions v 1 (t)v 2 (t) • • • v n (t) and obtained the formula (e.g., [12])…”

Section: Auxiliariesmentioning

confidence: 99%

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Exponential and Hypoexponential Distributions: Some Characterizations

Yanev

2020

Mathematics

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The (general) hypoexponential distribution is the distribution of a sum of independent exponential random variables. We consider the particular case when the involved exponential variables have distinct rate parameters. We prove that the following converse result is true. If for some n≥2, X1,X2,…,Xn are independent copies of a random variable X with unknown distribution F and a specific linear combination of Xj’s has hypoexponential distribution, then F is exponential. Thus, we obtain new characterizations of the exponential distribution. As corollaries of the main results, we extend some previous characterizations established recently by Arnold and Villaseñor (2013) for a particular convolution of two random variables.

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“…+ α n . Leibniz considered the problem of determining the kth derivative of the product of n smooth functions v 1 (t)v 2 (t) • • • v n (t) and obtained the formula (e.g., [12])…”

Section: Auxiliariesmentioning

confidence: 99%

“…By (12) we have H(t) ≡ 1 and therefore h 0 = 1 and h k = 0 for all k ≥ 1. Equating h k 's to the corresponding coefficients of the series in the right-hand side of ( 12), we will obtain equations for {a k } ∞ k=0 .…”

mentioning

confidence: 98%

Exponential and Hypoexponential Distributions: Some Characterizations

Yanev

2020

Mathematics

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“…It is expected that the conventional Born series (3) will not converge in this case. We will apply instead the modified series (29) proposed in this work.…”

Section: A Examplementioning

confidence: 99%

Acceleration of Born Series by Change of Variables

Lopez-Menchon

Rius

Heldring

et al. 2021

IEEE Trans. Antennas Propagat.

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In this work, we propose a method to enhance the convergence of the Born series. The Born series is widely used in scattering theory, but its convergence is only guaranteed under certain restrictive conditions which limit the cases where this formulation can be applied. The proposed method, based on modifying the singularities of the resolvent operator by a change of variables, accelerates the convergence of the series or even achieves convergence in otherwise divergent cases. Due to its low computational cost and ease of implementation, the method preserves the advantages of the Born series while it extends its range of application. It can also be applied to a variety of methods relying on the same mathematical principles as the Born series. Numerical examples are provided to show that a significant improvement can be achieved with simple techniques and with a limited knowledge about the operator spectrum.

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“…for all F, G ∈ C α . The derivative of the product of more than two functions can be found according to [25] (2.2)…”

Section: Multiindexes Multinomial Theorem and Generalized Leibniz's mentioning

confidence: 99%

Cross-Moments Computation for Stochastic Context-Free Grammars

2019

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In this paper we consider the problem of efficient computation of cross-moments of a vector random variable represented by a stochastic context-free grammar. Two types of cross-moments are discussed. The sample space for the first one is the set of all derivations of the context-free grammar, and the sample space for the second one is the set of all derivations which generate a string belonging to the language of the grammar. In the past, this problem was widely studied, but mainly for the cross-moments of scalar variables and up to the second order. This paper presents new algorithms for computing the cross-moments of an arbitrary order, while the previously developed ones are derived as special cases.

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Classroom note: A Generalization of Leibniz rule for higher derivatives

Cited by 17 publications

References 1 publication

Exponential and Hypoexponential Distributions: Some Characterizations

Exponential and Hypoexponential Distributions: Some Characterizations

Acceleration of Born Series by Change of Variables

Cross-Moments Computation for Stochastic Context-Free Grammars

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