This work deals with the asymptotic behaviour of electric field in transverse magnetic (TM) mode, propagating a bidimensional heterogeneous medium, composed by homogeneous linear dielectric isotropic material surrounded thin layer thickness $\varepsilon$ (destined to tend $0$) and embedded an ambient medium. Using tools multiscale analysis, expansion solution $u^{\varepsilon}$ Helmholtz problem...

Abstract We study infinite systems of globally coupled Anosov diffeomorphisms with weak coupling strength. Using transfer operators acting on anisotropic Banach spaces, we prove that the system admits a unique physical invariant state, $$h_\varepsilon $$ h ε . Moreover,...

This article is concerned with the approximation of unbounded convex sets by polyhedra. While there an abundance literature investigating this task for compact sets, results on case are scarce. We first point out connections between existing before introducing a new notion polyhedral called ($\varepsilon,\delta$)-approximation that integrates in meaningful way. Some basic about ($\varepsilon,\d...

We consider a class of equations in divergence form with singular/degenerate weight $$ -\mathrm{div}(|y|^a A(x,y)\nabla u)=|y|^a f(x,y)+\textrm{div}(|y|^aF(x,y))\;. Under suitable regularity assumptions for the matrix $A$, forcing term $f$ and field $F$, we prove H\"older continuity solutions which are odd $y\in\mathbb{R}$, possibly their derivatives. In addition, show stability $C^{0,\alpha}$ ...

We consider the 3D Gross-Pitaevskii equation \begin{equation}\nonumber i\partial_t \psi +\Delta \psi+(1-|\psi|^2)\psi=0 \text{ for } \psi:\mathbb{R}\times \mathbb{R}^3 \rightarrow \mathbb{C} \end{equation} and construct traveling waves solutions to this equation. These are of form $\psi(t,x)=u(x_1,x_2,x_3-Ct)$ with a velocity $C$ order $\varepsilon|\log\varepsilon|$ small parameter $\varepsilon...

Abstract Quantum walks are powerful tools for building quantum search algorithms. However, these success probabilities far below 1. Amplitude amplification is usually used to amplify probability, but the souffl'e problem follows. Only stop at right step can we achieve a maximum probability. Otherwise, as number of steps increases, probability may decrease, which will cause troubles in practical...

We investigate time complexities of finite difference methods for solving the high-dimensional linear heat equation, hyperbolic equation and multiscale system with quantum algorithms (hence referred to as "quantum methods"). For equations we study impact explicit implicit discretizations on advantages over classical method. problem, find complexity both treatment scheme scales $\mathcal{O}(1/\v...

A graph $G$ is \textit{$k$-critical} if $\chi(G) = k$ and every proper subgraph of $(k - 1)$-colorable, $L$ a list-assignment for $G$, then \textit{$L$-critical} not $L$-colorable but induced is. In 2014, Kostochka Yancey proved lower bound on the average degree an $n$-vertex $k$-critical tending to $k \frac{2}{k 1}$ large $n$ that tight infinitely many values $n$, they asked how their may be i...

In this paper we provide a proof of the Carleson $\varepsilon^2$-conjecture. This result yields characterization (up to exceptional sets zero length) tangent points Jordan curve in terms finiteness associated $\varepsilon^2$-square function.