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 study the asymptotic behavior, when $$\varepsilon \rightarrow 0$$ , of minimizers $$\{u_\varepsilon \}_{\varepsilon >0}$$ for energy $$\begin{aligned} E_\varepsilon (u)=\int _{\Omega }\Big (|\nabla u|^2+\big (\frac{1}{\varepsilon ^2}-1\big )|\nabla |u||^2\Big ), \end{aligned}$$ over class maps $$u\in H^1(\Omega ,{{\mathbb {R}}}^2)$$ satisfying boundary condition $$u=g$$ on $$\partial \Omega ...

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...

for fr$acute{mathbf{text{e}}}$chet algebras $(a, (p_n))$ and $(b, (q_n))$, a linear map $t:arightarrow b$ is textit{almost multiplicative} with respect to $(p_n)$ and $(q_n)$, if there exists $varepsilongeq 0$ such that $q_n(tab - ta tb)leq varepsilon p_n(a) p_n(b),$ for all $n in mathbb{n}$, $a, b in a$, and it is called textit{weakly almost multiplicative} with respect to $(p_n)$ and $(q_n)$,...

We relate two complexity notions of bipartite graphs: the minimal weight biclique covering number Cov(G) and the minimal rectifier network size Rect(G) of a bipartite graph G. We show that there exist graphs with Cov(G) ≥ Rect(G). As a corollary, we establish that there exist nondeterministic finite automata (NFAs) with εtransitions, having n transitions total such that the smallest equivalent ...

From the interpretation of Linear Logic multiplicative disjunction as the ε-product defined by Laurent Schwartz, we construct several models of Differential Linear Logic based on usual mathematical notions of smooth maps. This improves on previous results in [BET] based on convenient smoothness where only intuitionist models were built. We isolate a completeness condition, called k-quasi-comple...

We study a two-level transition probability for finite number of avoided crossings with small interaction. Landau–Zener formula, which gives the one crossing as $$e^{-\pi \frac{\varepsilon ^{2}}{h}}$$ , implies that parameter h and interaction $$\varepsilon $$ play an opposite role when both tend to 0. The exact WKB method produces generalization formula under optimal regime $$\scriptstyle {\fr...