We consider the stochastic heat equation $$\partial _{s}u =\frac{1}{2}\Delta u +(\beta V(s,y)-\lambda )u$$ ? s = 1 2 ? + ( ? V , y ) - ? with a smooth space-time-stationary Gaussian random field V(s, y), in dimensions $$d\ge 3$$ d ? 3 an initial condition $$u(0,x)=u_0(\varepsilon x)$$ 0 x ? and suitably chosen $$\lambda \in {\mathbb {R}}$$ ? R . It is known that, for $$\beta $$ small enough, di...

This work is devoted to studying non-variational, nonlinear singularly perturbed elliptic models enjoying a double degeneracy character with prescribed boundary value in domain. In its simplest form, for each $\varepsilon>0$ fixed, we seek non-negative function $u^{\varepsilon}$ satisfying $$ \begin{cases} \left\[|\nabla u^{\varepsilon}|^p + \mathfrak{a}(x)|\nabla u^{\varepsilon}|^q \right] \De...

A triangle $T'$ is $\varepsilon$-similar to another $T$ if their angles pairwise differ by at most $\varepsilon$. Given a $T$, $\varepsilon>0$ and $n\in\mathbb{N}$, B\'ar\'any F\"uredi asked determine the maximum number of triangles $h(n,T,\varepsilon)$ being in planar point set size $n$. We show that for almost all there exists $\varepsilon=\varepsilon(T)>0$ such $h(n,T,\varepsilon)=n^3/24 (1+...

in this paper, we prove some results on characterization of $varepsilon$-simultaneous approximations of downward sets in vector lattice banach spaces. also, we give some results about simultaneous approximations of normal sets.

in this paper, we prove some results on characterization of $varepsilon$-simultaneous approximations of downward sets in vector lattice banach spaces. also, we give some results about simultaneous approximations of normal sets.

We investigate a degenerating elliptic problem in multi-structure $\Omega\_\varepsilon$ of $\mathbb{R}^3$, the framework thermal stationary conduction with highly contrasting diffusivity. Precisely, consists fixed basis $\Omega^-$ surmounted by thin cylinder $\Omega\_\varepsilon^+$ height $1$ and cross-section small diameter order $\varepsilon$. Moreover, $\Omega^+\varepsilon$ contains cylindri...

We study the existence, uniqueness and minimality of critical points form $m_{\varepsilon,\eta}(x) = (f_{\varepsilon,\eta}(|x|)\frac{x}{|x|}, g_{\varepsilon,\eta}(|x|))$ functional \[ E_{\varepsilon,\eta}[m] \int_{B^N} \Big[\frac{1}{2} |\nabla m|^2 + \frac{1}{2\varepsilon^2} (1 - |m|^2)^2 \frac{1}{2\eta^2} m_{N+1}^2\Big]\,dx \] for $m=(m_1, \dots, m_N, m_{N+1}) \in H^1(B^N,\mathbb{R}^{N+1})$ wi...

In this paper, we establish the discreteness of transmission eigenvalues for Maxwell's equations. More precisely, show that spectrum eigenvalue problem is discrete if electromagnetic parameters $\varepsilon, \, \mu, \hat \varepsilon, \mu$ in equations characterizing inhomogeneity and background are smooth some neighborhood boundary isotropic on boundary, satisfy conditions $\varepsilon \neq \va...