相关论文: Love--Lieb integral equations: applications, theor…
We present a method for comparing the classical and quantum calculations of the electric quadrupolar Love number $k_2$ and show that our previous derivation of the quantum Love number of a quantum blackhole matches exactly the classical…
These lectures present a survey of recent developments in the area of random matrices (finite and infinite) and random permutations. These probabilistic problems suggest matrix integrals (or Fredholm determinants), which arise very…
The g-function is a measure of degrees of freedom associated to a boundary of two-dimensional quantum field theories. In integrable theories, it can be computed exactly in a form of the Fredholm determinant, but it is often hard to evaluate…
We obtain the full set of tidal Love numbers of non-rotating black holes in an effective field theory extension of general relativity. We achieve our results using a recently introduced modified Teukolsky equation that describes the…
Conical functions appear in a large number of applications in physics and engineering. In this paper we describe an extension of our module CONICAL for the computation of conical functions. Specifically, the module includes now a routine…
In this paper, positive solutions to the Laplace equation with 1-dimensional circular singularities are investigated. First, we establish $L^p$ integrability estimates for such solutions $u$ near the singularities, in comparison with…
In this paper, the following critical biharmonic elliptic problem \begin{eqnarray*} \begin{cases} \Delta^2u= \lambda u+\mu u\ln u^2+|u|^{2^{**}-2}u, &x\in\Omega,\\ u=\dfrac{\partial u}{\partial \nu}=0, &x\in\partial\Omega \end{cases}…
We prove that the auxiliary function $\mathop{\mathcal R}(s)$ has the integral representation \[\mathop{\mathcal R}(s)=-\frac{2^s \pi^{s}e^{\pi i s/4}}{\Gamma(s)}\int_0^\infty y^{s}\frac{1-e^{-\pi y^2+\pi \omega y}}{1-e^{2\pi \omega…
For real numbers $p,q>1$ we consider the following family of integrals: \begin{equation*} \int_{0}^{1}\frac{(x^{q-2}+1)\log\left(x^{mq}+1\right)}{x^q+1}{\rm d}x \quad \mbox{and}\quad…
We consider fractional differential equations of order $\alpha \in (0,1)$ for functions of one independent variable $t\in (0,\infty)$ with the Riemann-Liouville and Caputo-Dzhrbashyan fractional derivatives. A precise estimate for the order…
We offer in this article some modification of Monte-Carlo method for solving of a linear integral Fredholm's equation of a second kind (Fredholm's well posed problem). We prove that the rate of convergence of offered method is optimal under…
Integral relations and transformation rules are used to obtain, out of an asymptotic solution, a new group of four pairs of solutions to the double-confluent Heun equation. Each pair presents the same series coefficients but has solutions…
Hypersingular integrals of the type $I_{\alpha}(T_n,m,r) = \int_{-1}^{1} \hpsngAbs \frac{T_n(s)(1-s^2)^{m-{1/2}}}{(s-r)^\alpha}ds |r|<1$ and $I_{\alpha}(U_n,m,r) = \int_{-1}^{1} \hpsngAbs \frac{U_n(s)(1-s^2)^{m-{1/2}}}{(s-r)^\alpha}ds…
We come back to the Cauchy integral equations occurring in radiative transfer problems posed in finite, plane-parallel media with light scattering taken as monochromatic and isotropic. Their solution is calculated following the classical…
In this pedagogical note I will discuss one-loop integrals, where (i) different regions of the integration region lead to divergences and (ii) where these divergences cancel in the sum over all regions. These integrals cannot be calculated…
In this paper, we are concerned with a Liouville-type result of the nonlinear integral equation \begin{equation*} u(x)=\int_{\mathbb{R}^{n}}\frac{u(1-|u|^{2})}{|x-y|^{n-\alpha}}dy, \end{equation*} where $u: \mathbb{R}^{n} \to…
We study existence of a weak solution for one-dimensional problems as \begin{equation}\label{intro}\tag{1} \begin{cases} \displaystyle -\frac{d}{dx}\left(a(x) \frac{d u}{dx}\right) = - \frac{d \phi (u) }{dx}- \frac{d g(x) }{dx}&…
The theory of normal variance mixture distributions is used to provide elementary derivations of closed-form expressions for the definite integrals $\int_0^\infty x^{-2\nu}\cos(bx)\gamma(\nu,\alpha x^2)\,\mathrm{d}x$ (for $\nu>1/2$, $b>0$…
The Fredholm integral equations of the first kind is a typical ill-posed problem, so that it is usually difficult to obtain its analytical minimal-norm solution. This paper gives a closed-form minimal-norm solution for the degenerate kernel…
We study the following semilinear biharmonic equation $$ \left\{\begin{array}{lllllll} \Delta^{2}u=\frac{\lambda}{1-u}, &\quad \mbox{in}\quad \B, u=\frac{\partial u}{\partial n}=0, &\quad \mbox{on}\quad \partial\B, \end{array} \right.…