Ordinary Exponential Function Research Articles

For Thine Is the Stretch, the Power, and the Glory: Bibliometrics as a Branch of Relaxation Kinetics

A stretched exponential function taking the form, f(x) = exp(-x^β), x ∈ [0, ∞), β ∈ (0, 1], characterizes decay, diffusion, and relaxation phenomena known as Kohlrausch-Williams-Watts (KWW) processes. Recent work, particularly on relaxation kinetics in metallic glasses, has described the conditions under the shape parameter β deviates from its usual value. Where β > 1, the corresponding exponential function is compressed rather than stretched. The β-generalized exponential function provides good parametric fits for two measures of influence in legal academia: law review impact factors and Social Science Research Network (SSRN) downloads per author. A stretched exponential function models impact factors from 2007 through 2019. A compressed exponential function describes SSRN downloads per author by law school, except the single outlier atop the rankings. A power law distribution fits the SSRN data, but only for the top 100 schools. The fact that the value of the exponent β straddles the value of 1, which characterizes an ordinary exponential function, is not a trivial artifact of fitting a model to observed bibliometric data. Treating the reciprocal of β as a rough measure of heterogeneity, h = 1/β, suggests that impact factors and SSRN downloads measure different aspects of academic influence in law. Law reviews, especially as the stigma of online publishing recedes, have become more heterogeneous. By contrast, the compressed rather than stretched exponential kinetics of SSRN data implies the presence of avalanche-like processes in the posting and sharing of preprints among legal scholars.

Newton’s cooling law in generalised statistical mechanics

The study of cooling materials is essential to analyse the thermal properties of substances, electronic devices, among many others. The physical law for describing the temporal temperature decrease has been dominated by Newton’s law of cooling (NLC), which assumes that the natural cooling occurs by following an exact exponential trend. However, several studies have questioned the broad validity of this law by arguing that cooling occurs following an approximate rather than an exact exponential trend. In this way, to mitigate the difficulties of a reliable-description of the natural cooling processes, we introduce in this work a new formulation of NLC in the context of the q- and κ-calculus, which are obtained from the Tsallis q- and Kaniadakis κ-generalised statistical mechanics. The NLC in the Tsallis and Kaniadakis frameworks, in which we call q-NLC and κ-NLC, respectively, are derived from the related deformed derivative operators. The q- and κ-NLC describes the cooling process through the deformation of the ordinary exponential function. To empirically validate our proposal, we consider two real data sets: in the first one, a water-cooling case-study; and then, in the second one, the cooling of a power battery pack. The results show that the NLC based on generalised statistics outperform the classical NLC, which demonstrates a new path to cooling-analyses.

Transient superdiffusion in random walks with a [formula omitted]-exponentially decaying memory profile

We propose a random walk model with q-exponentially decaying memory profile. The q-exponential function is a generalization of the ordinary exponential function. In the limit q→1, the q-exponential becomes the ordinary exponential function. This model presents a Markovian diffusive regime that is characterized by finite memory correlations. It is well known, that central limit theorems prohibit superdiffusion for Markovian walks with finite variance of step sizes. In this problem we report the outcome of a transient superdiffusion for finite sized walks.

The effects of nonextensivity on quantum dissipation

Nonextensive dynamics for a quantum dissipative system described by a Caldirola-Kanai (CK) Hamiltonian is investigated in SU(1,1) coherent states. To see the effects of nonextensivity, the system is generalized through a modification fulfilled by replacing the ordinary exponential function in the standard CK Hamiltonian with the q-exponential function. We confirmed that the time behavior of the system is somewhat different depending on the value of q which is the degree of nonextensivity. The effects of q on quantum energy dissipation and other parameters are illustrated and discussed in detail.

QUANTUM ANALYSIS OF A MODIFIED CALDIROLA-KANAI OSCILLATOR MODEL FOR ELECTROMAGNETIC FIELDS IN TIME-VARYING PLASMA

Quantum properties of a modifled Caldirola-Kanai oscillator model for propagating electromagnetic flelds in a plasma medium are investigated using invariant operator method. As a modiflcation, ordinary exponential function in the Hamiltonian is replaced with a modifled exponential function, so-called the q-exponential function. The system described in terms of q-exponential function exhibits nonextensivity. Characteristics of the quantized flelds, such as quantum electromagnetic energy, quadrature ∞uctuations, and uncertainty relations are analyzed in detail in the Fock state, regarding the q-exponential function. We conflrmed, from their illustrations, that these quantities oscillate with time in some cases. It is shown from the expectation value of energy operator that quantum energy of radiation flelds dissipates with time, like a classical energy, on account of the existence of non-negligible conductivity in media.

An Efficient High-Order Time Integration Method for Spectral-Element Discontinuous Galerkin Simulations in Electromagnetics

We investigate efficient algorithms and a practical implementation of an explicit-type high-order timestepping method based on Krylov subspace approximations, for possible application to large-scale engineering problems in electromagnetics. We consider a semi-discrete form of the Maxwell's equations resulting from a high-order spectral-element discontinuous Galerkin discretization in space whose solution can be expressed analytically by a large matrix exponential of dimension $$\kappa \times \kappa $$ ? × ? . We project the matrix exponential into a small Krylov subspace by the Arnoldi process based on the modified Gram---Schmidt algorithm and perform a matrix exponential operation with a much smaller matrix of dimension $$m\times m$$ m × m ( $$m\ll \kappa $$ m ? ? ). For computing the matrix exponential, we obtain eigenvalues of the $$m\times m$$ m × m matrix using available library packages and compute an ordinary exponential function for the eigenvalues. The scheme involves mainly matrix-vector multiplications, and its convergence rate is generally $$O(\Delta t^{m-1})$$ O ( Δ t m - 1 ) in time so that it allows taking a larger timestep size as $$m$$ m increases. We demonstrate CPU time reduction compared with results from the five-stage fourth-order Runge---Kutta method for a certain accuracy. We also demonstrate error behaviors for long-time simulations. Case studies are also presented, showing loss of orthogonality that can be recovered by adding a low-cost reorthogonalization technique.

THE MINIMAL κ-ENTROPY MARTINGALE MEASURE

We introduce the notion of κ-entropy (κ ∈ ℝ, |κ| ≤ 1), starting from Kaniadakis' (2001, 2002, 2005) one-parameter deformation of the ordinary exponential function. The κ-entropy is in duality with a new class of utility functions which are close to the exponential utility functions, for small values of wealth, and to the power law utility functions, for large values of wealth. We give conditions on the existence and on the equivalence to the basic measure of the minimal κ-entropy martingale measure. Moreover, we provide characterizations of its density as a κ-exponential function. We show that the minimal κ-entropy martingale measure is closely related to both the standard entropy martingale measure and the well known q-optimal martingale measures. We finally establish the convergence of the minimal κ-entropy martingale measure to the minimal entropy martingale measure as κ tends to 0.

New connection formulae for some q-orthogonal polynomials in q-Askey scheme

New nonlinear connection formulae of the q-orthogonal polynomials, such as continuous q-Laguerre, continuous big q-Hermite, q-Meixner–Pollaczek and q-Gegenbauer polynomials, in terms of their respective classical analogues are obtained using a special realization of the q-exponential function as infinite multiplicative series of ordinary exponential function.

New connection formulae for theq-orthogonal polynomials via a series expansion of theq-exponential

Using a realization of the q-exponential function as an infinite multiplicative sereis of the ordinary exponential functions we obtain new nonlinear connection formulae of the q-orthogonal polynomials such as q-Hermite, q-Laguerre and q-Gegenbauer polynomials in terms of their respective classical analogs.

Special deformed exponential functions leading to more consistent Klauder's coherent states

We give a general approach for the construction of deformed oscillators. These ones could be seen as describing deformed bosons. Basing on new definitions of certain quantum series, we demonstrate that they are nothing but the ordinary exponential functions in the limit when the deformation parameters goes to one. We also prove that these series converge to a complex function, in a given convergence radius that we calculate. Klauder's coherent states are explicitly found through these functions that we design by deformed exponential functions.

Is there a “Universal” Explanation for the “Universal” Dynamic Response?

AbstractStructurally disordered solid electrolytes, both crystalline and glassy, as well as ionic melts, exhibit a set of spectroscopic peculiarities that are at variance with the predictions of simple random‐hopping models. Typically, stretched exponential functions are encountered where ordinary exponential functions are expected. The effect has come to be called “universal dynamic response.” In this paper, it is traced back to a mean square displacement which, in a broad time regime, varies as time to the power of β, with 0 < β < 1. Computer simulations as well as model considerations show that this particular shape of the mean square displacement is caused by the frequent occurrence of correlated forward‐backward hopping sequences. The latter processes result from the mutual repulsive Coulomb interaction of the mobile ions.

Algebraic groups and small transcendence degree I

The purpose of this paper is to study the algebraic independence of numbers associated with one-parameter subgroups of commutative algebraic groups. Sufficient conditions are given for certain fields to have transcendence degree at least two over the field of rational numbers. As an application one may deduce generalizations of several classical results concerning the algebraic independence of values of the ordinary exponential function or the Weierstrass elliptic function.

An analysis of transient flow of heat and moisture during drying of granular beds

The transient process of drying granular materials in fixed packed beds is analysed, and equations expressing moisture contents and temperatures of material and drying medium as functions of time and location in the bed are derived. While moisture content of the granules may be expressed in terms of ordinary exponential functions of these variables, humidity and temperature of the drying medium additionally require numerical solutions. The theory is developed for materials drying in four stages—one stage at constant rate, one at increasing rate, and two at falling rates—and should be applicable to numerous products in the chemical and food industries, with wider applicability if suitably modified. Application of the theory is demonstrated for through-drying of a granular bed of wheat, using published data to evaluate transfer parameters appropriate to special drying conditions and employing a CDC3200 digital computer to calculate the moisture content of grains as a function of time and location. The Fortran programme written for this computation may be used to predict the progress of drying other materials within the category of the present analysis.