In this paper we study the following class of fractional relativistic Schr\"odinger equations: \begin{equation*} \left\{ \begin{array}{ll} (-\Delta+m^{2})^{s}u + V(\varepsilon x) u= f(u) &\mbox{ in } \mathbb{R}^{N}, \\ u\in H^{s}(\mathbb{R}^{N}), \quad u>0 \end{array} \right. \end{equation*} where $\varepsilon>0$ is a small parameter, $s\in (0, 1)$, $m>0$, $N> 2s$, $(-\Delta+m^{2})^{s}$ operato...

This paper studies the Dirichlet problem for Laplace’s equation in a domain $$\varOmega _{\varepsilon , \eta }$$ perforated with small holes, where $$\varepsilon $$ represents scale of minimal distances between holes and $$\eta ratio sizes . We establish $$W^{1, p}$$ estimates solutions bounding constants depending explicitly on parameters also show that these are either optimal or near optimal.

The projection constant $\Pi(E):=\Pi(E, \ell_\infty)$ of a finite-dimensional Banach space $E\subset\ell_\infty$ is by definition the smallest norm linear $\ell_\infty$ onto $E$. Fix $n\geq 1$ and denote $\Pi_n$ maximal value $\Pi(\cdot)$ amongst $n$-dimensional real spaces. We prove for every $\varepsilon >0$ that there exist an integer $d\geq subspace $E\subset\ell_1^d$ such $\Pi_n \leq \Pi(E...

In this paper, we present a new degree-based estimator for the size of maximum matching in bounded arboricity graphs. When graph is by $$\alpha $$ , gives +2$$ factor approximation size. For planar graphs, show does better and returns 3.5 Using estimator, get results approximating graphs streaming distributed models computation. particular, vertex-arrival streams, randomized $$O\left( \frac{\sq...

Let $P=(p_1, p_2, \dots, p_n)$ be a polygonal chain in $\mathbb{R}^d$. The stretch factor of $P$ is the ratio between total length and distance its endpoints, $\sum_{i = 1}^{n-1} |p_i p_{i+1}|/|p_1 p_n|$. For parameter $c \geq 1$, we call $c$-chain if $|p_ip_j|+|p_jp_k| \leq c|p_ip_k|$, for every triple $(i,j,k)$, $1 i<j<k n$. global property: it measures how close to straight line, involves al...

We describe a systematic approach to cast the differential equation for $l$-loop equal mass banana integral into an $\varepsilon$-factorised form. With known boundary value at specific point we obtain systematically term of order $j$ in expansion dimensional regularisation parameter $\varepsilon$ any loop $l$. The is based on properties Calabi-Yau operators, and particular self-duality.

We prove that for every integer $k$, there exists $\varepsilon > 0$ such n-vertex graph $G$ with no pivot-minor isomorphic to $C_k$, exist disjoint sets $A,B \subseteq V(G)$ $|A|,|B| \geq \varepsilon n$, and $A$ is either complete or anticomplete $B$. This proves the analog of Erd\H{o}s-Hajnal conjecture class graphs $C_k$.

We extend a discrepancy bound of Lagarias and Pleasants for local weight distributions on linearly repetitive Delone sets show that similar holds also the more general case without finite complexity if linear repetitivity is replaced by $\varepsilon$-linear repetitivity. As result we establish are some sufficiently small $\varepsilon$ rectifiable, incommensurable multiscale substitution tilings...