In this paper, we study the line bundle mean curvature flow defined by Jacob and Yau. The is a kind of parabolic flows to obtain deformed Hermitian Yang-Mills metrics on given K\"ahler manifold. goal paper give an $\varepsilon$-regularity theorem for flow. To establish theorem, provide scale invariant monotone quantity. As critical point quantity, define self-shrinker solution Liouville type se...

We establish the existence of chimera states, simultaneously supporting synchronous and asynchronous dynamics, in a network consisting two symmetrically linked star subnetworks identical oscillators with shear Kuramoto--Sakaguchi coupling. show that states may be metastable or asymptotically stable. If intra-star coupling strength is order $\varepsilon$, persist on time scales at least $1/\vare...

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...