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...

We establish improved uniform error bounds on a second-order Strang time-splitting method which is equivalent to an exponential wave integrator for the long-time dynamics of nonlinear Klein--Gordon equation (NKGE) with weak cubic nonlinearity, whose strength characterized by $\varepsilon^2$ $0 < \varepsilon \leq 1$ dimensionless parameter. Actually, when \ll 1$, NKGE $O(\varepsilon^2)$ nonlinea...

We study absorbing phase transitions in the one-dimensional branching annihilating random walk with long-range repulsion. The repulsion is implemented as hopping bias such a way that particle more likely to hop away from its closest particle. strength due interaction has form $\varepsilon x^{-\sigma}$, where $x$ distance particle, $0\le \sigma \le 1$, and sign of $\varepsilon$ determines whethe...

Let f n ( z</mml...

&lt;p style='text-indent:20px;'&gt;In this paper, we consider the multiplicity of nodal solutions for following Kirchhoff type equations:&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE1"&gt; \begin{document}$ \begin{equation*} \left\{ \begin{array}{ll} -\varepsilon^2M\left(\varepsilon^{2-N}||\nabla u||^2_{L^2}\right)\Delta u+u = f\left(x\righ...

The impact of buoyancy on axisymmetric radiative Powell-Exring fluid flow towards a radially stretched surface is examined. To analyze the heat and mass transfer characteristics, thermal radiation, joule heating chemical reactions are incorporated to flow. equations governing obtained with help boundary layer approximation. modeled coupled PDE system transfigured as ODE by exerting adequate sim...