Metric Spaces and Completely Monotone Functions

1988 ◽  
pp. 115-145
Author(s):  
I. J. Schoenberg
Keyword(s):  
Metric Spaces ◽  
Monotone Functions ◽  
Completely Monotone Functions ◽  
Completely Monotone

scholarly journals On the Correlation Structure of a Lévy-Driven Queue

2008 ◽  
Vol 45 (4) ◽  
pp. 940-952 ◽  
Author(s):  
Abdelghafour Es-Saghouani ◽  
Michel Mandjes
Keyword(s):  
Random Variable ◽  
Correlation Structure ◽  
Single Server ◽  
Monotone Functions ◽  
Single Server Queue ◽  
Completely Monotone Functions ◽  
Complementary Distribution ◽  
Completely Monotone ◽  
Server Queue ◽  
Heavy Tailed

In this paper we consider a single-server queue with Lévy input and, in particular, its workload process (Qt)t≥0, with a focus on the correlation structure. With the correlation function defined asr(t) := cov(Q0,Qt) / var(Q0) (assuming that the workload process is in stationarity at time 0), we first determine its transform ∫0∞r(t)e-ϑtdt. This expression allows us to prove thatr(·) is positive, decreasing, and convex, relying on the machinery of completely monotone functions. We also show thatr(·) can be represented as the complementary distribution function of a specific random variable. These results are used to compute the asymptotics ofr(t), for larget, for the cases of light-tailed and heavy-tailed Lévy inputs.

ON THE $ \langle \rho_j,W_j \rangle $ GENERALIZED COMPLETELY MONOTONE FUNCTIONS

2020 ◽  
Vol 54 (1 (251)) ◽  
pp. 35-43
Author(s):  
B.A. Sahakyan
Keyword(s):  
Monotone Functions ◽  
Completely Monotone Functions ◽  
Completely Monotone ◽  
Class Of Functions

We consider sequences $ {\lbrace \rho_j \rbrace}_{0}^{\infty} $ $ (\rho_0 \mathclose{=} 1, \rho_j \mathclose{\geq} 1) $, $ {\lbrace \alpha_j \rbrace}_{0}^{\infty} $ $ (\alpha_0 \mathclose{=} 1, \alpha_j \mathclose{=} 1 \mathclose{-} (1/\rho_j )) $, $ {\lbrace W_j (x) \rbrace}_{0}^{\infty} \mathclose{\in} W $, where $$ W \mathclose{=} \lbrace {\lbrace W_j (x) \rbrace}_{0}^{\infty} / W_0 (x) \mathclose{\equiv} 1, W_j (x) \mathclose{>} 0, {W}_{j}^{\prime} (x) \mathclose{\leq} 0, W_j (x) \mathclose{\in} C^\infty [0,a] \rbrace, $$ $ C^\infty [0,a] $ is the class of functions of infinitely differentiable. For such sequences we introduce systems of operators $ {\lbrace {A}_{a,n}^{\ast} f \rbrace}_{0}^{\infty} $, $ {\lbrace \tilde{A}_{a,n}^{\ast} f \rbrace}_{0}^{\infty} $ and functions $ {\lbrace {U}_{a,n} (x) \rbrace}_{0}^{\infty} $, $ {\lbrace {\Phi}_{n} (x,t) \rbrace}_{0}^{\infty} $. For a certain class of functions a generalization of Taylor–Maclaurin type formulae was obtained. We also introduce the concept of $ \langle \rho_j,W_j \rangle $ generalized completely monotone functions and establish a theorem on their representation.

scholarly journals Models of dielectric relaxation based on completely monotone functions

2016 ◽  
Vol 19 (5) ◽  
Author(s):  
Roberto Garrappa ◽  
Francesco Mainardi ◽  
Maione Guido
Keyword(s):  
Time Domain ◽  
Differential Operators ◽  
Dielectric Materials ◽  
Monotone Functions ◽  
Mathematical Functions ◽  
Completely Monotone Functions ◽  
Wing Model ◽  
Completely Monotone ◽  
The Time Domain

AbstractThe relaxation properties of dielectric materials are described, in the frequency domain, according to one of the several models proposed over the years: Kohlrausch-Williams-Watts, Cole-Cole, Cole-Davidson, Havriliak-Negami (with its modified version) and Excess wing model are among the most famous. Their description in the time domain involves some mathematical functions whose knowledge is of fundamental importance for a full understanding of the models. In this work, we survey the main dielectric models and we illustrate the corresponding time-domain functions. In particular, we stress the attention on the completely monotone character of the relaxation and response functions. We also provide a characterization of the models in terms of differential operators of fractional order.

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