We prove a number of results concerning the large $N$ asymptotics of the free energy of a random matrix model with a polynomial potential. Our approach is based on a deformation of potential and on the use of the underlying integrable structures of the matrix model. The main results include the existence of a full asymptotic expansion in even powers of $N$ of the recurrence coefficients of the related orthogonal polynomials for a one-cut regular potential and the double scaling asymptotics of the free...

The uniform approach to calculation of MISE for histogram and density box-spline estimators gives us a possibility to obtain estimators of derivatives of densities and the asymptotic constant.

The calculation of sound generation and propagation in low Mach number flows requires serious reflections on the characteristics of the underlying equations. Although the compressible Euler/Navier-Stokes equations cover all effects, an approximation via standard compressible solvers does not have the ability to represent acoustic waves correctly. Therefore, different methods have been developed to deal with the problem. In this paper, three of them are considered and compared to each other. They...

The calculation of sound generation and propagation in low Mach number flows requires serious reflections on the characteristics of the underlying equations. Although the compressible Euler/Navier-Stokes equations cover all effects, an approximation via standard compressible solvers does not have the ability to represent acoustic waves correctly. Therefore, different methods have been developed to deal with the problem. In this paper, three of them are considered and compared to each other....

En este trabajo discutimos la resolución de la ecuación de Besseld2x/dx2 + (1/x)(dy/dx) - (1 - s2/x2)y = 0.Las funciones de Bessel modificadas Kv(x) e Iv(x) son las soluciones a la ecuación anterior cuando v = is. El valor de la función Kis(x) es real y el de la función Iis(x) es complejo, por ello definimos en su lugar una función real Mis(x). La función Iis(x) resultará ser una combinación de las funciones Kis(x) y Mis(x). Daremos algunos desarrollos en serie de Mis(x) y Kis(x) junto con sus derivadas...

A general method of deriving canonical functions for ray field singularities involving caustics, shadow boundaries and their intersections is presented. It is shown that many time-domain canonical functions can be expressed in terms of elementary functions and elliptic integrals.

The commutative neutrix convolution product of the functions $x^r{e}_{-}^{\lambda x}$ and $x^s{e}_{+}^{\mu x}$ is evaluated for $r,s=0,1,2,...$ and all $\lambda ,\mu$. Further commutative neutrix convolution products are then deduced.

Let $p,q\in ℝ$ with $p-q\ge 0$, $\sigma =\frac{1}{2}(p+q-1)$ and $s=\frac{1}{2}(1-p+q)$, and let $${𝒟}_m(x;p,q)={𝒟}_0(x;p,q)+\sum _{k=1}^m\frac{B_{2k}\left(s\right)}{2k{(x+\sigma )}^{2k}},$$ where $${𝒟}_0(x;p,q)=\frac{\psi (x+p)+\psi (x+q)}{2}-ln(x+\sigma ).$$ We establish the asymptotic expansion $${𝒟}_0(x;p,q)\sim -\sum _{n=1}^{\infty }\frac{B_{2n}\left(s\right)}{2n{(x+\sigma )}^{2n}}\text{as}x\to \infty ,$$ where $B_{2n}\left(s\right)$ stands for the Bernoulli polynomials. Further, we prove that the functions ${(-1)}^m{𝒟}_m(x;p,q)$ and ${(-1)}^{m+1}{𝒟}_m(x;p,q)$ are completely monotonic in $x$ on $(-\sigma ,\infty )$ for every $m\in {\mathbb{N}}_0$ if and only if $p-q\in [0,\frac{1}{2}]$ and $p-q=1$, respectively. This not only unifies the two known results but also yields some new results.

This work deals with the study of the bounds of the asymptotic expansions of complex functions in a normed space. Some inequalities are obtained similar to the Gorny-Cartan for the bounds of asymptotic expansions in an angle of the complex space.