The solution of an n-dimensional
stochastic differential equation driven by Gaussian white noises is a Markov vector.
In this way, the transition joint probability density function (JPDF) of this
vector is given by a deterministic parabolic partial differential
equation, the so-called Fokker-Planck-Kolmogorov (FPK) equation. There exist few
exact solutions of this equation so that the analyst must resort to approximate
or numerical procedures. The finite element method (FE) is among the latter,
and is reviewed in this paper. Suitable computer codes are written for the two
fundamental versions of the FE method, the Bubnov-Galerkin and the
Petrov-Galerkin method. In order to reduce the computational effort, which is
to reduce the number of nodal points, the following refinements to the method
are proposed: 1) exponential (Gaussian) weighting functions different from the
shape functions are tested; 2) quadratic and cubic splines are used to
interpolate the nodal values that are known in a limited number of points. In
the applications, the transient state is studied for first order systems only,
while for second order systems, the steady-state JPDF is determined, and it is
compared with exact solutions or with simulative solutions: a very good
agreement is found.

Abstract:
A second order oscillator with
nonlinear restoring force and nonlinear damping is considered: it is subject to
both external and internal (parametric) excitations of Gaussian white noise
type. The nonlinearities are chosen in such a way that the associated Fokker-Planck-Kolmogorov
equation is solvable in the steady state. Different choices of some system
parameters give rise to different and interesting shapes of the joint
probability density function of the response, which in some cases appears to be
multimodal. The problem of the determination of the power spectral density of
the response is also addressed by using the true statistical linearization
method.

Abstract:
This paper analyzes the stochastic stability of a damped Mathieu oscillator subjected to a parametric excitation of the form of a stationary Gaussian process, which may be both white and coloured. By applying deterministic and stochastic averaging, two Itô’s differential equations are retrieved. Reference is made to stochastic stability in moments. The differential equations ruling the response statistical moment evolution are written by means of Itô’s differential rule. A necessary and sufficient condition of stability in the moments of order r is that the matrix of the coefficients of the ODE system ruling them has negative real eigenvalues and complex eigenvalues with negative real parts. Because of the linearity of the system the stability of the first two moments is the strongest condition of stability. In the case of the first moments (averages), critical values of the parameters are expressed analytically, while for the second moments the search for the critical values is made numerically. Some graphs are presented for representative cases.

Abstract:
This paper addresses the problem of the interpretation of the stochastic differential equations (SDE). Even if from a theoretical point of view, there are infinite ways of interpreting them, in practice only Stratonovich’s and Itô’s interpretations and the kinetic form are important. Restricting the attention to the first two, they give rise to two different Fokker-Planck-Kolmogorov equations for the transition probability density function (PDF) of the solution. According to Stratonovich’s interpretation, there is one more term in the drift, which is not present in the physical equation, the so-called spurious drift. This term is not present in Itô’s interpretation so that the transition PDF’s of the two interpretations are different. Several examples are shown in which the two solutions are strongly different. Thus, caution is needed when a physical phenomenon is modelled by a SDE. However, the meaning of the spurious drift remains unclear.

Abstract:
Palmar intertriradial ridge counts (a-b, b-c, c-d, a-d) in 260 males and 260 females of Sardinian origin were considered. Bilateral and sex differences, correlation coefficients, skewness and kurtosis of the four distributions and an index of asymmetry were calculated. There are no sex differences, ridge counts show positive and almost always significant correlations. The a-d ridge count shows a normal distribution in both sexes.

Abstract:
We report on five
dermatoglyphic traits recorded on palm prints in sample of Sardinians and
Corsicans. The two populations are very similar but not identical.

Abstract:
This paper presents a brief review of the interpretation of measurements of hadron yields in hadronic interactions within the framework of thermal models, over a broad energy range (from SIS to LHC energies, $\sqrt{s_{NN}} \simeq$ 2.5 GeV -- 5 TeV). Recent experimental results and theoretical developments are reported, with an emphasis on topics discussed during the Quark Matter 2014 conference.

Abstract:
Let X be a Fano manifold of dimension n and index n-3. Kawamata proved the non vanishing of the global sections of the fundamental divisor in the case n=4. Moreover he proved that if Y is a general element of the fundamental system then Y has at most canonical singularities. We prove a generalization of this result in arbitrary dimension.