We consider the stochastic Cahn-Hilliard equation with additive noise term $\varepsilon^\gamma g\, \dot{W}$ ($\gamma >0$) that scales interfacial width parameter $\varepsilon$. verify strong error estimates for a gradient flow structure-inheriting time-implicit discretization, where $\varepsilon^{-1}$ only enters polynomially; proof is based on higher-moment iterates, and (discrete) spectral es...

In the domain $\Omega =\left\{\left(t,\varepsilon _{1}, \varepsilon _{2} \right): t\in {\mathbb R},\varepsilon _{1}>0, >0\right\}$, we consider a linear singularly perturbed system with two small parameters \[ \left\{ \begin{array}{l} {\dot{x}_{0} =A_{00} x_{0} +A_{01} x_{1} +A_{02} x_{2},} \\ {\varepsilon _{1} \dot{x}_{1} =A_{10} +A_{11} +A_{12} \dot{x}_{2} =A_{20} +A_{21} +A_{22} \end{a...

Collisional energy loss of a fast particle in medium is mostly due to the polarization by electromagnetic fields particle. A small fraction carried away Cherenkov radiation. In chiral there an additional contribution induction anomalous current proportional magnetic field. It causes lose form We employ classical electrodynamics, adequate wide range energies, compute collisional homogenous plasm...

Abstract In the k -Connectivity Augmentation Problem we are given a -edge-connected graph and set of additional edges called links . Our goal is to find minimum size whose addition makes it ( + 1)-edge-connected. There an approximation preserving reduction from mentioned problem case = 1 (a.k.a. Tree or TAP) 2 Cactus CacAP). While several better-than-2 algorithms known for TAP, CacAP only recen...

In 2013, Masson and Siljander determined a method to prove that the minimal p-weak upper gradient $$g_{f_\varepsilon }$$ for time mollification $$f_\varepsilon $$ , $$\varepsilon >0$$ of parabolic Newton–Sobolev function $$f\in L^p_\mathrm {loc}(0,\tau ;N^{1,p}_\mathrm {loc}(\Omega ))$$ with $$\tau $$\Omega open domain in doubling metric measure space $$(\mathbb {X},d,\mu )$$ supporting weak (1...

In this paper, we consider the following logarithmic Schrödinger equation$ \begin{equation*} -\varepsilon^2\Delta u + V(x)u = u\log u^2\ \ \text{in}\ {\mathbb R}^N, \end{equation*} $where $ \varepsilon>0 $, N\ge 1 V(x)\in C({\mathbb R}) is a continuous potential which can be unbounded below. By variational methods and penalized idea, show that problem has family of solutions u_{\varepsilon} ...