where for some kernels \(k(\mathfrak {u},v)\), $$\begin{aligned} \nu (v)=\int _{\mathbb {R}^3}\int _{\mathbb {S}^2}q(\omega ,\left| \mathfrak {u}-v\right| )\mu (\mathfrak {u})\textrm{d}{\omega }\textrm{d}{\mathfrak {u}},\qquad K[f](v)=\int _{\mathb...
However, the peak of the α-stable density curve in Figure 9.28 lies far to the left of this point. The mean deviates significantly from the mode, because the concentration curve has a heavy tail. Consider the histogram in Figure 9.32. For the N = 10, 000 particles simulated, the sample...
Goldbach’s parametrisation, rather than the use of the Lambert function, that is best suited to uniformise the complex equationz^w = w^z. The complex logarithm and complex exponents are, of course familiar ideas; nevertheless, as we depend on the exact details of, and the notation for, ...
Then, we considered the nonlinear fractional Cauchy reaction–diffusion equation (CRDE), which is used to describe the evolution of a system over time with reaction and diffusion processes in various fields. For instance, in biology and medicine, these equations are used to model the spreading of...
However, since the integrand of the stochastic part is a stochastic process, this part cannot be solved using Riemann integration theory. Instead, Itô’s lemma for stochastic integration is used for solving the stochastic part.5 The stochastic differential equation of the Black-Scholes model for...
The paper is devoted to the study of one class of problems with nonlocal conditions for a mixed diffusion-wave equation with two independent variables. The main results of the work are the proof of regular and strong solvability, as well as the Volterra
The Mellin transform of the square of Riemann's zeta-function and Atkinson's formula This leads to a decomposition of Z 1 (s) as a sum of four functions, of which the most interesting one is a series with d(n). After a transformation formula, based on the functional equation for ...
We present a new Riemann-Hilbert problem formalism for the initial value problem for the derivative nonlinear Schr"odinger (DNLS) equation on the line. We show that the solution of this initial value problem can be obtained from the solution of some associated Riemann-Hilbert problem. This new...
In the case of two-dimensional gradient index cavities designed by the conformal transformation optics, we propose a boundary integral equation method for the calculation of resonant mode functions by employing a fictitious space which is reciprocally equivalent to the physical space. Using the Green’...
The (hardest) problem is clearly the derivation of the fluctuation-dissipation relation and this is the point from which our analysis departs from that in [KLO12]. Even though, as we will see, it is enough to determine (1.11) approximately (i.e. that the difference of left and right han...