Related papers: Scaling limits of discrete holomorphic functions

We prove that the Coleff-Herrera residue current, corresponding to a pair of holomorphic functions defining a complete intersection, can be obtained as the unrestricted weak limit of a natural smooth $(0,2)$-form depending on two…

We study in the present article the Kardar-Parisi-Zhang (KPZ) equation $$ \partial_t h(t,x)=\nu\Delta h(t,x)+\lambda |\nabla h(t,x)|^2 +\sqrt{D}\, \eta(t,x), \qquad (t,x)\in\mathbb{R}_+\times\mathbb{R}^d $$ in $d\ge 3$ dimensions in the…

It has been shown recently [10] that Cauchy transforms of orthogonal polynomials appear naturally in general correlation functions containing ratios of characteristic polynomials of random NxN Hermitian matrices. Our main goal is to…

The Cauchy problem for fractional derivatives linear systems of ordinary differential equations with constant coefficients is considered, where at first the analytic expressions are given through the matrix exponent of its corresponding…

We investigate the Cauchy problem for a semilinear spatio--temporal fractional diffusion equation with a time-dependent forcing term: \[ \partial_t^\alpha u + (-\Delta)^{\mathsf{s}} u = |u|^p + t^{\sigma}\,\mathbf{w}(x), \quad (t,x) \in…

We describe partial differential operators for which we can construct generalised integral means satisfying Pizzetti-type formulas. Using these formulas we give a new characterisation of summability of formal power series solutions to some…

Vertex scaling of piecewise linear metrics on surfaces introduced by Luo is a straightforward discretization of smooth conformal structures on surfaces. Combinatorial $\alpha$-curvature for vertex scaling of piecewise linear metrics on…

We analyze a finite-difference approximation of a functional of Ambrosio-Tortorelli type in brittle fracture, in the discrete-to-continuum limit. In a suitable regime between the competing scales, namely if the discretization step $\delta$…

In this paper we prove convergence results for homogenization problem for solutions of partial differential system with rapidly oscillating Dirichlet data. Our method is based on analysis of oscillatory integrals. In the uniformly convex…

We prove a mixed joint discrete universality theorem for a Matsumoto zeta-function $\varphi(s)$ (belonging to the Steuding subclass) and a periodic Hurwitz zeta-function $\zeta(s,\alpha;{\mathfrak{B}})$. For this purpose, certain…

The divergent integral $\int_a^b f(x)(x-x_0)^{-n-1}\mathrm{d}x$, for $-\infty<a<x_0<b<\infty$ and $n=0, 1, 2, \dots$, is assigned, under certain conditions, the value equal to the simple average of the contour integrals $\int_{C^{\pm}}…

In this paper we give an explicit solution of Dzherbashyan-Caputo-fractional Cauchy problems related to equations with derivatives of order $\nu k$, for $k$ non-negative integer and $\nu>0$. The solution is obtained by connecting the…

Using a different approach, we derive integral representations for the Riemann zeta function and its generalizations (the Hurwitz zeta, $\zeta(-k,b)$, the polylogarithm, $\mathrm{Li}_{-k}(e^m)$, and the Lerch transcendent,…

The physical meaning, the properties and the consequences of a discrete scalar field are discussed; limits for the validity of a mathematical description of fundamental physics in terms of continuous fields are a natural outcome of discrete…

We establish an extension of Liouville's classical representation theorem for solutions of the partial differential equation $\Delta u=4 e^{2u}$ and combine this result with methods from nonlinear elliptic PDE to construct holomorphic maps…

The present paper is a continuation of our work [11], where we introduced a fractional operator calculus related to a fractional ${\psi}-$Fueter operator in the one-dimensional Riemann-Liouville derivative sense in each direction of the…

We introduce the sub-lattice approach, a procedure to generate, from a given integrable lattice, a sub-lattice which inherits its integrability features. We consider, as illustrative example of this approach, the discrete Moutard 4-point…

High-order derivatives of analytic functions are expressible as Cauchy integrals over circular contours, which can very effectively be approximated, e.g., by trapezoidal sums. Whereas analytically each radius r up to the radius of…

The main goal of this article is to bring together the theories of holomorphic iteration in the unit disc and semigroups of holomorphic functions. We develop a technique that allows us to partially embed the orbit of a holomorphic self-map…

In this work, we show a convergence result for the discrete formulation of the generalised KPZ equation $\partial_t u = (\Delta u) + g(u)(\nabla u)^2 + k(\nabla u) + h(u) + f(u)\xi_t(x)$, where the $\xi$ is a real-valued random field,…