A temporal graph is a in which the edge set can change from one time step to next. The exploration problem TEXP of computing foremost schedule for graph, i.e., walk that starts at given start node, visits all nodes and has smallest arrival time. In first part paper, we consider only undirected graphs are connected each step. For such with $n$ nodes, show it \NP-hard approximate ratio $O(n^{1-\v...

Abstract Let X be a compact connected hyperbolic surface, that is, closed orientable smooth surface with Riemannian metric of constant curvature $$-1$$ - 1 . For each $$n\in {\mathbf {N}}$$ n ? N , let $$X_{n}$$ <mml:ms...

In this note, we consider the homogenization of compressible Navier-Stokes equations in a periodically perforated domain $\mathbb{R}^3$. Assuming that particle size scales like $\varepsilon^3$, where $\varepsilon>0$ is their mutual distance, and Mach number decreases fast enough, show limit $\varepsilon\to 0$, velocity density converge to solution incompressible with Brinkman term. We strongly ...

Let $u^\varepsilon$ and $u$ be viscosity solutions of the oscillatory Hamilton-Jacobi equation its corresponding effective equation. Given bounded, Lipschitz initial data, we present a simple proof to obtain optimal rate convergence $\mathcal{O}(\varepsilon)$ $u^\varepsilon \rightarrow u$ as $\varepsilon 0^+$ for large class convex Hamiltonians $H(x,y,p)$ in one dimension. This includes from cl...

We consider the following Schr\"odinger-Bopp-Podolsky system in $\mathbb R^{3}$ $$\left\{ \begin{array}{c} -\varepsilon^{2} \Delta u + V(x)u \phi = f(u)\\ \varepsilon^{4} \Delta^{2}\phi 4\pi\varepsilon u^{2}\\ \end{array} \right.$$ where $\varepsilon &gt; 0$ with $ V:\mathbb{R}^{3} \rightarrow \mathbb{R}, f:\mathbb{R} \mathbb{R}$ satisfy suitable assumptions. By using variational methods, we pr...

In this paper we consider the unfolding of saddle-node \[ X= \frac{1}{xU_a(x,y)}\Big(x(x^{\mu}-\varepsilon)\partial_x-V_a(x)y\partial_y\Big), \] parametrized by $(\varepsilon,\,a)$ with $\varepsilon \approx 0$ and $a$ in an open subset $A$ $ {\mathbb {R}}^{\alpha },$ study Dulac time $\mathcal {T}(s;\varepsilon,\,a)$ one its hyperbolic sectors. We prove (theorem 1.1) that derivative $\partial _...

For each given $n\ge 2$, we construct a family of entire solutions $u\_\varepsilon (z,t)$, $\varepsilon > 0$, with helical symmetry to the three-dimensional complex-valued Ginzburg–Landau equation $$ \Delta u+(1-|{u}|^2)u=0, \quad (z,t) \in \mathbb{R}^2\times \mathbb{R} \simeq \mathbb{R}^3. These are $2\pi/\varepsilon$-periodic in $t$ and have $n$ helix-vortex curves, asymptotic behavior, as $\...

In many cases, groundwater flow in an unconfined aquifer can be simplified to a one-dimensional Sturm-Liouville model of the form: \begin{equation*} x''(t)+\lambda x(t)=h(t)+\varepsilon f(x(t)),\hspace{.1in}t\in(0,\pi) \end{equation*} subject non-local boundary conditions x(0)=h_1+\varepsilon\eta_1(x)\text{ and } x(\pi)=h_2+\varepsilon\eta_2(x). this paper, we study existence solutions above pr...

We investigate the convergence rate of optimal entropic cost $$v_\varepsilon $$ to transport as noise parameter $$\varepsilon \downarrow 0$$ . show that for a large class functions c on $${\mathbb {R}}^d\times {\mathbb {R}}^d$$ (for which plans are not necessarily unique or induced by map) and compactly supported $$L^{\infty }$$ marginals, one has -v_0= \frac{d}{2} \varepsilon \log (1/\varepsil...

In this paper, we study the concentration and multiplicity of solutions to following fractional Schr\"{o}dinger-Poisson system \begin{equation*} \left\{ \begin{array}{ll} \varepsilon^{2s}(-\Delta)^su+V(x)u+\phi u=f(u)+u^{2_s^{\ast}-1} & \hbox{in $\mathbb{R}^3$,} \varepsilon^{2t}(-\Delta)^t\phi=u^2, u>0& \end{array} \right. \end{equation*} where $s>\frac{3}{4}$, $s,t\in(0,1)$, $\varepsilon>0$ is...