Probability and Statistics

Mathai, A. M.; Haubold, Hans J.

doi:10.1515/9783110562545

Cited by 20 publications

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“…Rewriting the relations for the charges q i using q i = N i e, and knowing that the Laplace transform of the sum of independent variables is the product of the Laplace transforms 48 , we obtain the Laplace transform of N t (t) = q t (t)/e as:…”

Section: Macroscopic Electrode Responsementioning

confidence: 99%

Power-law charge relaxation of inhomogeneous porous capacitive electrodes

Allagui¹,

Benaoum²

2022

Preprint

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Porous electrodes-made of hierarchically nanostructured materials-are omnipresent in various electrochemical energy technologies from batteries and supercapacitors to sensors and electrocatalysis. Modeling the system-level macroscopic transport and relaxation in such electrodes given their complex microscopic geometric structure is important to better understand the performance of the devices in which they are used. The discharge response of capacitive porous electrodes in particular do not necessarily follow the traditional exponential decay observed with flat electrodes, which is good enough for describing the general dynamics of processes in which the rate of a dynamic quantity (such as charge) is proportional to the quantity itself.Electric double-layer capacitors (EDLCs) and other similar systems exhibit instead power law-like discharge profiles that are best described with differential equations involving non-integer derivatives. Using the fractional-order integral in the Riemann-Liouville sense and superstatistics we present a treatment of the macroscopic response of such type of electrode systems starting from the mesoscopic behavior of sub-parts of it. The solutions can be in terms of the Mittag-Leffler (ML) function or a power law-like function depending on the underlying assumptions made on the physical parameters of initial charge and characteristic time response. The generalized threeparameter ML function is found to be the best suited to describe experimental results of a commercial EDLC at different time scales of discharge.

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Section: Macroscopic Electrode Responsementioning

confidence: 99%

Power-law charge relaxation of inhomogeneous porous capacitive electrodes

Allagui¹,

Benaoum²

2022

Preprint

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“…For tests that make use of the Student-t statistic, refer to Mathai and Haubold (2017b). Since the density given in (5.6.5) is an even function, when ρ = 0, all odd order moments are equal to zero and the even order moments can easily be evaluated from type-1 beta integrals.…”

Section: The Special Case ρ =mentioning

confidence: 99%

“…, x p ). Correlation does not measure general relationships between the variables; counterexamples are provided in Mathai and Haubold (2017b). Hence "maximum correlation" should be interpreted as maximum joint scale-free variation or joint scatter in the variables.…”

Section: Different Derivations Of ρ 1(2p)mentioning

confidence: 99%

“…For the next property, we will use the following two basic results on conditional expectations, referring also to Mathai and Haubold (2017b). Let x and y be two real scalar random variables having a joint distribution.…”

Section: Different Derivations Of ρ 1(2p)mentioning

confidence: 99%

“…where it is assumed that the expected value of the conditional variance and the variance of the conditional expectation exist. Situations where the results stated in (i) and (ii) are applicable or not applicable are described and illustrated in Mathai and Haubold (2017b). In result (i), x can be a scalar, vector or matrix variable.…”

Section: Different Derivations Of ρ 1(2p)mentioning

confidence: 99%

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Chapter 5: Matrix-Variate Gamma and Beta Distributions

Mathai

Provost

Haubold

2022

Multivariate Statistical Analysis in the Real and Complex Domains

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The notations introduced in the preceding chapters will still be followed in this one. Lower-case letters such as x, y will be utilized to represent real scalar variables, whether mathematical or random. Capital letters such as X, Y will be used to denote vector/matrix random or mathematical variables. A tilde placed on top of a letter will indicate that the variables are in the complex domain. However, the tilde will be omitted in the case of constant matrices such as A, B. The determinant of a square matrix A will be denoted as |A| or det(A) and, in the complex domain, the absolute value or modulus of the determinant of B will be denoted as |det(B)|. Square matrices appearing in this chapter will be assumed to be of dimension p × p unless otherwise specified.

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Chapter 1: Mathematical Preliminaries

Mathai

Provost

Haubold

2022

Multivariate Statistical Analysis in the Real and Complex Domains

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It is assumed that the reader has had adequate exposure to basic concepts in Probability, Statistics, Calculus and Linear Algebra. This chapter provides a brief review of the results that will be needed in the remainder of this book. No detailed discussion of these topics will be attempted. For essential materials in these areas, the reader is, for instance, referred to Mathai and Haubold (2017a, 2017b). Some properties of vectors, matrices, determinants, Jacobians and wedge product of differentials to be utilized later on, are included in the present chapter. For the sake of completeness, we initially provide some elementary definitions. First, the concepts of vectors, matrices and determinants are introduced.

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Probability and Statistics

Cited by 20 publications

References 0 publications

Power-law charge relaxation of inhomogeneous porous capacitive electrodes

Power-law charge relaxation of inhomogeneous porous capacitive electrodes

Chapter 5: Matrix-Variate Gamma and Beta Distributions

Chapter 1: Mathematical Preliminaries

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