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 $\varepsilon>0$ 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:
 
 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...

We establish the error bounds of fourth-order compact finite difference (4cFD) methods for Dirac equation in massless and nonrelativistic regime, which involves a small dimensionless parameter $0 < \varepsilon \le 1$ inversely proportional to speed light. In this solution propagates waves with wavelength $O(\varepsilon)$ time $O(1)$ space, as well wave $O(1/\varepsilon)$ rapid outgoing waves. a...

A recent result of Alon, Ben-Eliezer and Fischer establishes an induced removal lemma for ordered graphs. That is, if \(F\) is graph \(\varepsilon›0\), then there exists \(\delta_{F}(\varepsilon)›0\) such that every \(n\)-vertex \(G\) containing at most \(\delta_{F}(\varepsilon) n^{v(F)}\) copies can be made \(F\)-free by adding/deleting \(\varepsilon n^2\) edges. We prove \(\delta_{F}(\varepsi...

We study the early work scheduling problem on identical parallel machines in order to maximize total work, i.e., parts of non-preemptive jobs that are executed before a common due date. By preprocessing and constructing an auxiliary instance which has several good properties, for any desired accuracy $$\varepsilon $$ , we propose efficient polynomial time approximation scheme with running $$O\l...

We are interested in the spectral properties of magnetic Schrödinger operator $H_\varepsilon$ a domain $\Omega \subset \mathbb{R}^2$ with compact boundary and field intensity $\varepsilon^{-2}$. impose Dirichlet conditions on $\partial\Omega$. Our main focus is existence description so-called edge states, namely eigenfunctions for $H_{\varepsilon}$ whose mass localized at scale $\varepsilon$ al...

We investigate tensor products of random matrices, and show that independence entries leads asymptotically to $\varepsilon$-free independence, a mixture classical free studied by M{\l}otkowski Speicher Wysocza\'nski. The particular $\varepsilon$ arising is prescribed the product structure chosen, conversely, we with suitable choices an arbitrary may be realized in this way. As result obtain new...

Let $p=\frac{1+\varepsilon}{n}$. It is known that if $N=\varepsilon^3n\to\infty$, then with high probability (w.h.p.) $G_{n,p}$ has a unique giant largest component. We show in addition, $\varepsilon=\varepsilon(n)\to 0$, w.h.p. the cover time of asymptotic to $n\log^2N$; previously Barlow, Ding, Nachmias, and Peres had shown this up constant multiplicative factors.

This paper provides an extended case study of the cutoff phenomenon for a prototypical class nonlinear Langevin systems with single stable state perturbed by additive pure jump L\'evy noise small amplitude $\varepsilon>0$, where driving process is layered type. Under drift coercivity condition associated family processes $X^\varepsilon$ turns out to be exponentially ergodic equilibrium distribu...