We describe the emergence of topological singularities in periodic media within Ginzburg-Landau model and core-radius approach. The energy functionals both models are denoted by $E_{\varepsilon,\delta}$, where $\varepsilon$ represent coherence length (in model) or size approach) $\delta$ denotes periodicity scale. carry out $\Gamma$-convergence analysis $E_{\varepsilon,\delta}$ as $\varepsilon\...

A bounded subset $M$ of a Banach space $X$ is said to be $\varepsilon $-weakly precompact, for given \geq 0$, if every sequence $(x_n)_{n\in \mathbb N}$ in $M$ admits subsequence $(x_{n_k})_{k\in such that $$ \limsup

We establish uniform error bounds of an exponential wave integrator Fourier pseudospectral (EWI-FP) method for the long-time dynamics nonlinear Klein--Gordon equation (NKGE) with a cubic nonlinearity whose strength is characterized by $\varepsilon^2$ $\varepsilon \in (0, 1]$ dimensionless parameter. When $0 < \varepsilon \ll 1$, problem equivalent to NKGE small initial data (and $O(1)$ nonlinea...

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 $L_2(\mathbb{R}^d;\mathbb{C}^n)$, we consider a selfadjoint matrix strongly elliptic second order differential operator $\mathcal{A}_\varepsilon$, $\varepsilon >0$. The coefficients of the $\mathcal{A}_\varepsilon$ are periodic and depend on $\mathbf{x}/\varepsilon$. We study behavior $\mathcal{A}_\varepsilon ^{-1/2}\sin (\tau \mathcal{A}_\varepsilon ^{1/2})$, $\tau\in\mathbb{R}$, in small p...

In this paper, we consider the homogenization problem for generalized elliptic systems$ \mathcal{L}_{ \varepsilon} = -\operatorname{div}[A(x/ \varepsilon)\nabla+V(x/ \varepsilon)]+B(x/ \varepsilon)\nabla+c(x/ \varepsilon)+\lambda I $with dimension two. Precisely, will establish $ W^{1, p} estimates, Hölder Lipschitz estimates and L^p convergence results with The operator has been studied by Qia...

Fix $\varepsilon&gt;0$ and a nonnull graph $H$. A well-known theorem of Rödl from the 80s says that every $G$ with no induced copy $H$ contains linear-sized $\varepsilon$-restricted set $S\subseteq V(G)$, which means $S$ induces subgraph maximum degree at most $\varepsilon |S|$ in or its complement. There are two extensions this result:&#x0D; &#x0D; quantitatively, Nikiforov (and later Fox Suda...

Let $\Gamma$ be a group and $\mathscr{C}$ class of groups endowed with bi-invariant metrics. We say that is $\mathscr{C}$-stable if every $\varepsilon$-homomorphism $\Gamma \rightarrow G$, $(G,d) \in \mathscr{C}$, $\delta_\varepsilon$-close to homomorphism, $\delta_\varepsilon\to 0$ when $\varepsilon\to 0$. If $\delta_\varepsilon < C \varepsilon$ for some $C$ we $ \mathscr{C} $-stable linear ra...