Related papers: Tangential limits for harmonic functions with resp…

The effective potential for the local composite operator $\phi^{2}(x)$ in $\lambda \phi^{4}$-theory is investigated at finite temperature in an approach based on path-integral linearisation of the $\phi^4 $-interaction. At zero temperature,…

The existence and nonexistence of $\lambda$-harmonic functions in unbounded domains of $\mathbb{H}^n$ are investigated. We prove that if the $(n-1)/2$ Hausdorff measure of the asymptotic boundary of a domain $\Omega$ is zero, then there is…

We determine the scaling dimension $\Delta_n$ for the class of composite operators $\phi^n$ in the $\lambda \phi^4$ theory in $d=4-\epsilon$ taking the double scaling limit $n\rightarrow \infty$ and $\lambda \rightarrow 0$ with fixed…

In this note we investigate the behavior of harmonic functions at singular points of $\mathsf{RCD}(K,N)$ spaces. In particular we show that their gradient vanishes at all points where the tangent cone is isometric to a cone over a metric…

Parabolic Cylinder functions (PCFs) are classical special functions with applications in many different fields. However, there is little information available regarding simple uniform approximations and bounds for these functions. We obtain…

Exact results are derived, specifically the perihelion shift and the Kepler orbit, for a bound test particle in the Schwarzschild metric with cosmological constant $\Lambda=0$. A series expansion, of $\Delta\phi = 2(2(1-2M/p(3-e))^{-1/2}…

Let $G$ be a nonempty bounded domain in a finite-dimensional Euclidean space. The main results are general estimates from below at points from $G$ for an arbitrary subharmonic function $u\not\equiv -\infty$ on the closure of the domain $G$…

Let $\beta>1$, $I$ be the unite interval $[0,1)$ and $\phi$ be an integer function defined on $\mathbb{N}\setminus\{0\}$ satisfying $1\leq\phi(n)\leq n$. Denote by $A_\phi(x,\beta)$ the Erd\"{o}s-R\'{e}nyi average of $x\in I$ associated…

Assume $\alpha\geq p>1$. Consider the following $p$-th Yamabe equation on a connected finite graph $G$: $$\Delta_p\varphi+h\varphi^{p-1}=\lambda f\varphi^{\alpha-1},$$ where $\Delta_p$ is the discrete $p$-Laplacian, $h$ and $f>0$ are fixed…

We are concerned with solvability of nonlinear systems involving a discrete singular $\phi$-Laplacian operator of type \begin{equation*} u \mapsto \Delta\left[\phi(\Delta u(n-1))\right] \qquad (n\in \{1, \dots, T\}), \end{equation*}…

We study the singularity (multifractal) spectrum of the convex hull of the typical/generic continuous functions defined on $[0,1]^{d}$. We denote by ${\mathbf E}_ { { \varphi } }^{h} $ the set of points at which $ \varphi : [0,1]^d\to…

For any \alpha in (0, 2), a truncated symmetric \alpha-stable process is a symmetric Levy process with no diffusion part and with a Levy density given by c|x|^{-d-\alpha} 1_{|x|< 1} for some constant c. In previous paper we have studied the…

We present an $L_q(L_{p})$-theory for the equation $$ \partial_{t}^{\alpha}u=\phi(\Delta) u +f, \quad t>0,\, x\in \mathbb{R}^d \quad\, ;\, u(0,\cdot)=u_0. $$ Here $p,q>1$, $\alpha\in (0,1)$, $\partial_{t}^{\alpha}$ is the Caputo fractional…

By methods of stochastic analysis on Riemannian manifolds, we develop two approaches to determine an explicit constant $c(D)$ for an $n$-dimensional compact manifold $D$ with boundary such that $\frac{\lambda}{n}\,\|\phi\|_{\infty} \leq…

The rational Dunkl operators are commuting differential-reflection operators on the Euclidean space $\RR^d$ associated with a root system. The aim of the paper is to study local boundary behaviour of generalized harmonic functions…

We show how Lasry-Lions's result on regularization of functions defined on $\mathbb{R}^n$ or on Hilbert spaces by sup-inf convolutions with squares of distances can be extended to (finite or infinite dimensional) Riemannian manifolds $M$ of…

Let $\phi$ be an $L^2$-normalized Hecke--Maa{\ss} cusp form for $\mathrm{PGL}_n(\mathbb{Z}[i])$ on the locally symmetric space $X:=\mathrm{PGL}_n(\mathbb{Z}[i])\backslash \mathrm{PGL}_n(\mathbb{C}) / \mathrm{PU}_n$. If $\Omega$ is a compact…

We study the bound states of a quantum mechanical system consisting of a simple harmonic oscillator with an inverse square interaction, whose interaction strength is governed by a constant $\alpha$. The singular form of this potential has…

We derive general bounds on operator dimensions, central charges, and OPE coefficients in 4D conformal and N=1 superconformal field theories. In any CFT containing a scalar primary phi of dimension d we show that crossing symmetry of <phi…

In this paper, we analyze the solvability of the discrete nonlinear Schr\"odinger equation \begin{equation*} i\beta(\Delta_t+\nabla_t)\phi(t,k) +\gamma |\phi(t,k)|^2\phi(t,k) +\varepsilon \Delta_k^2\phi(t,k-1) = g(t,\phi(t,k)),…