Search results for "Calculus"
showing 10 items of 617 documents
On Finite Translation Structures with Proper Dilatations
1986
Recently, Biliotti and the author obtained a certain number of results on translation structures with proper dilatations including structure-and characterisation-theorems, which here will be reformulated in a different manner, throwing a new light on some of the regarded questions.
A Lagrangian method for deriving new indefinite integrals of special functions
2015
A new method is presented for obtaining indefinite integrals of common special functions. The approach is based on a Lagrangian formulation of the general homogeneous linear ordinary differential equation of second order. A general integral is derived which involves an arbitrary function, and therefore yields an infinite number of indefinite integrals for any special function which obeys such a differential equation. Techniques are presented to obtain the more interesting integrals generated by such an approach, and many integrals, both previously known and completely new are derived using the method. Sample results are given for Bessel functions, Airy functions, Legendre functions and hype…
Is there an absolutely continuous random variable with equal probability density and cumulative distribution functions in its support? Is it unique? …
2014
This paper inquires about the existence and uniqueness of a univariate continuous random variable for which both cumulative distribution and density functions are equal and asks about the conditions under which a possible extrapolation of the solution to the discrete case is possible. The issue is presented and solved as a problem and allows to obtain a new family of probability distributions. The different approaches followed to reach the solution could also serve to warn about some properties of density and cumulative functions that usually go unnoticed, helping to deepen the understanding of some of the weapons of the mathematical statistician’s arsenal.
On the use of fractional calculus for the probabilistic characterization of random variables
2009
In this paper, the classical problem of the probabilistic characterization of a random variable is re-examined. A random variable is usually described by the probability density function (PDF) or by its Fourier transform, namely the characteristic function (CF). The CF can be further expressed by a Taylor series involving the moments of the random variable. However, in some circumstances, the moments do not exist and the Taylor expansion of the CF is useless. This happens for example in the case of $\alpha$--stable random variables. Here, the problem of representing the CF or the PDF of random variables (r.vs) is examined by introducing fractional calculus. Two very remarkable results are o…
A method for the probabilistic analysis of nonlinear systems
1995
Abstract The probabilistic description of the response of a nonlinear system driven by stochastic processes is usually treated by means of evaluation of statistical moments and cumulants of the response. A different kind of approach, by means of new quantities here called Taylor moments, is proposed. The latter are the coefficients of the Taylor expansion of the probability density function and the moments of the characteristic function too. Dual quantities with respect to the statistical cumulants, here called Taylor cumulants, are also introduced. Along with the basic scheme of the method some illustrative examples are analysed in detail. The examples show that the proposed method is an a…
Experimental evidence for fractional time evolution in glass forming materials
2002
The infinitesimal generator of time evolution in the standard equation for exponential (Debye) relaxation is replaced with the infinitesimal generator of composite fractional translations. Composite fractional translations are defined as a combination of translation and the fractional time evolution introduced in [Physica A, 221 (1995) 89]. The fractional differential equation for composite fractional relaxation is solved. The resulting dynamical susceptibility is used to fit broad band dielectric spectroscopy data of glycerol. The composite fractional susceptibility function can exhibit an asymmetric relaxation peak and an excess wing at high frequencies in the imaginary part. Nevertheless…
Dynamics of Two Picophytoplankton Groups in Mediterranean Sea: Analysis of the Deep Chlorophyll Maximum by a Stochastic Advection-Reaction-Diffusion …
2013
A stochastic advection-reaction-diffusion model with terms of multiplicative white Gaussian noise, valid for weakly mixed waters, is studied to obtain the vertical stationary spatial distributions of two groups of picophytoplankton, i.e., picoeukaryotes and Prochlorococcus, which account about for 60% of total chlorophyll on average in Mediterranean Sea. By numerically solving the equations of the model, we analyze the one-dimensional spatio-temporal dynamics of the total picophytoplankton biomass and nutrient concentration along the water column at different depths. In particular, we integrate the equations over a time interval long enough, obtaining the steady spatial distributions for th…
Identification of stiffness, dissipation and input parameters of multi degree of freedom civil systems under unmeasured base excitations
2009
A time domain dynamic identification technique based on a statistical moment approach has been formulated for civil systems under base random excitations in the linear state. This technique is based on the use of classically damped models characterized by a mass proportional damping. By applying the Itô stochastic calculus, special algebraic equations that depend on the statistical moments of the response can be obtained. These equations can be used for the dynamic identification of the mechanical parameters that define the structural model, in the case of unmeasured input as well, and the identification of the input itself. Furthermore, the above equations demonstrate the possibility of id…
An output-only stochastic parametric approach for the identification of linear and nonlinear structures under random base excitations: Advances and c…
2014
In this paper a time domain output-only Dynamic Identification approach for Civil Structures (DICS) first formulated some years ago is reviewed and presented in a more generalized form. The approach in question, suitable for multi- and single-degrees-of-freedom systems, is based on the statistical moments and on the correlation functions of the response to base random excitations. The solving equations are obtained by applying the Itô differential stochastic calculus to some functions of the response. In the previous version ([21] Cavaleri, 2006; [22] Benfratello et al., 2009), the DICS method was based on the use of two classes of models (Restricted Potential Models and Linear Mass Proport…
Oscillatory Behavior of Second-Order Nonlinear Neutral Differential Equations
2014
Published version of an article in the journal: Abstract and Applied Analysis. Also available from the publisher at: http://dx.doi.org/10.1155/2014/143614 Open Access We study oscillatory behavior of solutions to a class of second-order nonlinear neutral differential equations under the assumptions that allow applications to differential equations with delayed and advanced arguments. New theorems do not need several restrictive assumptions required in related results reported in the literature. Several examples are provided to show that the results obtained are sharp even for second-order ordinary differential equations and improve related contributions to the subject.