Complete Monotonicity Research Articles

A Comparative View of Becker, Lomnitz, and Lambert Linear Viscoelastic Models

We compare the classical viscoelastic models due to Becker and Lomnitz with respect to a recent viscoelastic model based on the Lambert W function. We take advantage of this comparison to derive new analytical expressions for the relaxation spectrum in the Becker and Lomnitz models, as well as novel integral representations for the retardation and relaxation spectra in the Lambert model. In order to derive these analytical expressions, we have used the analytical properties of the exponential integral and the Lambert W function, as well as the Titchmarsh’s inversion formula of the Stieltjes transform. In addition, we prove some interesting inequalities by comparing the different models considered, as well as the non-negativity of the retardation and relaxation spectral functions. This means that the complete monotonicity of the rate of creep and the relaxation functions is satisfied, as required by the classical theory of linear viscoelasticity.

Computable error bounds for quasi-Monte Carlo using points with non-negative local discrepancy

Abstract Let $f:[0,1]^{d}\to{\mathbb{R}}$ be a completely monotone integrand as defined by Aistleitner and Dick (2015, Acta Arithmetica, 167, 143–171) and let points $\boldsymbol{x}_{0},\dots ,\boldsymbol{x}_{n-1}\in [0,1]^{d}$ have a non-negative local discrepancy (NNLD) everywhere in $[0,1]^{d}$. We show how to use these properties to get a non-asymptotic and computable upper bound for the integral of $f$ over $[0,1]^{d}$. An analogous non-positive local discrepancy property provides a computable lower bound. It has been known since Gabai (1967, Illinois J. Math., 11, 1–12) that the two-dimensional Hammersley points in any base $b\geqslant 2$ have NNLD. Using the probabilistic notion of associated random variables, we generalize Gabai’s finding to digital nets in any base $b\geqslant 2$ and any dimension $d\geqslant 1$ when the generator matrices are permutation matrices. We show that permutation matrices cannot attain the best values of the digital net quality parameter when $d\geqslant 3$. As a consequence the computable absolutely sure bounds we provide come with less accurate estimates than the usual digital net estimates do in high dimensions. We are also able to construct high-dimensional rank one lattice rules that are NNLD. We show that those lattices do not have good discrepancy properties: any lattice rule with the NNLD property in dimension $d\geqslant 2$ either fails to be projection regular or has all its points on the main diagonal. Complete monotonicity is a very strict requirement that for some integrands can be mitigated via a control variate.

Complete monotonicity for a ratio of finitely many gamma functions

Abstract In this paper, we solve the question completely raised by Feng Qi and Dongkyu Lim in paper “Monotonicity properties for a ratio of finite many gamma functions” published in Advances in Difference Equations and get some properties about ratios of finitely many gamma functions such as complete monotonicity, logarithmically complete monotonicity, the Bernstein function property, null point and extreme value.

Decreasing and complete monotonicity of functions defined by derivatives of completely monotonic function involving trigamma function

Decreasing and complete monotonicity of functions defined by derivatives of completely monotonic function involving trigamma function

Logarithmically complete monotonicity of reciprocal ARCTAN function

We prove the conjecture stated in F. Qi and R. Agarwal, On complete monotonicity for several classes of functions related to ratios of gamma functions, J. Inequal. Appl. (2019), that the function 1∕ arctan is logarithmically completely monotonic on (0,∞), but not a Stieltjes transform.

Complete monotonicity of the remainder of an asymptotic expansion of the generalized Gurland's ratio

Complete monotonicity of the remainder of an asymptotic expansion of the generalized Gurland's ratio

Constructing models for spherical and elliptical densities

Abstract The article provides construction algorithms for consistent model classes of continuous spherical and elliptical distributions. The algorithms are based on characterization theorems for consistent families of generator functions. This characterization uses the term of complete monotonicity. The algorithms are applied to several examples of generators leading to consistent families of generators with explicit formulas for marginal densities of arbitrary dimension.

Several Functions Originating from Fisher–Rao Geometry of Dirichlet Distributions and Involving Polygamma Functions

In this paper, the authors review and survey some results published since 2020 about (complete) monotonicity, inequalities, and their necessary and sufficient conditions for several newly introduced functions involving polygamma functions and originating from the estimation of the sectional curvature of the Fisher–Rao geometry of the Dirichlet distributions in the two-dimensional case.

Monotonicity and Convexity Properties and Some Inequalities Involving $E_{n,p}(x)$

In this paper, we established some monotonicity and convexity properties of the p-analogue of the exponential integral function. The increasing and decreasing, positive and negative, and convexity and concavity properties of the function were established and proved. Complete monotonicity of the function was also considered.

On the monotonicity of the discrete Choquet-like operators

Recent research has highlighted the importance of studying the different forms of monotonicity for various discrete Choquet-like integrals in the field of data aggregation and its real-world applications. In this paper, we focus on the standard monotonicity condition for two operators encompassing many modifications of the discrete Choquet-like integral known in the literature, such as the CQO-integral, the CT-integral, the CC-integral, the CO-integral, or the n-ary discrete d-Choquet integral. Using a new approach, we provide complete monotonicity characterizations for both operators. This, in turn, allows us to give full characterizations of the operators as aggregation functions. Our results extend and complete some results in the literature.

Anomalous Kinetics of Photoluminescence from Degrading Organic Materials and Its Theoretical Rendition

The Photoluminescence (PL) degradation of thermally evaporated Alq3 thin films is described by four Kohlrausch-Williams-Watt (KWW) functions, which are solely mathematical expressions. The present contribution not only unfolds the physical meaning of the KWW function, but also reveals new mathematical tools. By introducing the concept of the material clock, the system has been described by a damped harmonic oscillator, which in certain conditions allows the expansion of the KWW function in the so-called Prony series. The terms of this series can be attributed to chemical and physical processes that really contribute to the decay, i.e., degradation of Alq3 thin films when interacting with internal and environmental agents. These insights unveiled the usefulness of proper mathematical procedures and properties, such as the monotonicity and the complete monotonicity, for investigating the PL of this ubiquitous organometallic molecule, which possesses one among the highest emission yield. Moreover, this method is also promising for describing the photoluminescent processes of similar organic molecules important both for basic research and optoelectronic applications.

New Monotonicity and Infinite Divisibility Properties for the Mittag-Leffler Function and for Stable Distributions

Hyperbolic complete monotonicity property (HCM) is a way to check if a distribution is a generalized gamma (GGC), hence is infinitely divisible. In this work, we illustrate to which extent the Mittag-Leffler functions Eα,α∈(0,2], enjoy the HCM property, and then intervene deeply in the probabilistic context. We prove that for suitable α and complex numbers z, the real and imaginary part of the functions x↦Eαzx, are tightly linked to the stable distributions and to the generalized Cauchy kernel.

Inner approximations of coherent lower probabilities and their application to decision making problems

AbstractWe consider a decision making problem under imprecision, where the probabilistic information is given in terms of a set of probability measures, and where finding the optimal alternative(s) may be difficult. To ease the computation, we propose to transform the initial model into another one that (1) belongs to some subclass with better mathematical properties, such as supermodularity or complete monotonicity; (2) is at least as informative as the original model, while being as close as possible to it. We show that the problem can be approached in terms of linear or quadratic programming and that it can be connected with the one of determining the incenter of a credal set. Finally, we compare the solutions of a decision making problem with the initial and the transformed models and illustrate how our approach can be applied in a decision making problem under severe uncertainty.

Consistent Prony’s Approximation for FDTD Modeling of Dispersive Media Characterized by Completely Monotone Susceptibilities

Consistent Prony’s Approximation for FDTD Modeling of Dispersive Media Characterized by Completely Monotone Susceptibilities

Non-Gaussian Measures in Infinite Dimensional Spaces: the Gamma-Grey Noise

In the context of non-Gaussian analysis, Schneider [29] introduced grey noise measures, built upon Mittag-Leffler functions; analogously, grey Brownian motion and its generalizations were constructed (see, for example, [6, 7, 9, 27]). In this paper, we construct and study a new non-Gaussian measure, by means of the incomplete-gamma function (exploiting its complete monotonicity). We label this measure Gamma-grey noise and we prove, for it, the existence of Appell system. The related generalized processes, in the infinite dimensional setting, are also defined and, through the use of the Riemann-Liouville fractional operators, the (possibly tempered) Gamma-grey Brownian motion is consequently introduced. A number of different characterizations of these processes are also provided, together with the integro-differential equation satisfied by their transition densities. They allow to model anomalous diffusions, mimicking the procedures of classical stochastic calculus.

Exponential bounds for the logarithmic derivative of Whittaker functions

Some well-known results of Grönwall on logarithmic derivative of modified Bessel functions of the first kind concerning exponential bounds are extended to Whittaker functions of the first and second kind M κ , μ M_{\kappa ,\mu } and W κ , μ W_{\kappa ,\mu } . Moreover, a complete monotonicity result is proved for the logarithmic derivative of the Whittaker function W κ , μ , W_{\kappa ,\mu }, and some monotonicity results with respect to the parameters and argument are shown for the logarithmic derivative of M κ , μ . M_{\kappa ,\mu }. The results extend and complement the known results in the literature about modified Bessel functions of the first and second kind.

Increasing property and logarithmic convexity of functions involving Dirichlet lambda function

Abstract In this article, with the help of an integral representation of the Dirichlet lambda function, by means of a monotonicity rule for the ratio of two integrals with a parameter, and by virtue of complete monotonicity and another property of an elementary function involving the exponential function, the authors find increasing property and logarithmic convexity of two functions containing the gamma function and the Dirichlet lambda function.

Delta Complete Monotonicity and Completely Monotonic Degree on Time Scales

Delta Complete Monotonicity and Completely Monotonic Degree on Time Scales

Sharp inequalities related to the volume of the unit ball in mathbb{R}^{n}

Let Omega _{n}=pi ^{n/2}/Gamma (frac{n}{2}+1) (n in mathbb{N}) denote the volume of the unit ball in mathbb{R}^{n}. In this paper, the logarithmically complete monotonicity of a function involving the ratio of two gamma functions is presented, which yields a sharp double inequality for the quantity Omega _{n}^{2}/(Omega _{n-1}Omega _{n+1}). Also, we establish new sharp inequalities for the quantity Omega _{n}^{2}/(Omega _{n-1}Omega _{n+1}).

Monotonicity and complete monotonicity of some functions involving the modified Bessel functions of the second kind

In this paper, we introduce some monotonicity rules for the ratio of integrals. Furthermore, we demonstrate that the function -T ν,α,β (s) is completely monotonic in s and absolutely monotonic in ν if and only if β≥1, where T ν,α,β (s)=K ν 2 (s)-βK ν-α (s)K ν+α (s) defined on s>0 and K ν (s) is the modified Bessel function of the second kind of order ν. Finally, we determine the necessary and sufficient conditions for the functions s↦T μ,α,1 (s)/T ν,α,1 (s), s↦(T μ,α,1 (s)+T ν,α,1 (s))/(2T (μ+ν)/2,α,1 (s)), and s↦d n 1 dν n 1 T ν,α,1 (s)/d n 2 dν n 2 T ν,α,1 (s) to be monotonic in s∈(0,∞) by employing the monotonicity rules.