The shape of a sessile drop on horizontal substrate depends upon the Bond number $Bo$ and contact angle $\alpha$ . Inspired by puddle approximations at large (Quéré, Rep. Prog. Phys. , vol. 68, 2005, p. 2495), we address here limit small angles fixed volume arbitrary It readily leads to pancake approximation, where height radius scale as $\alpha ^{-1/2}$ respectively, with capillary forces bein...

Abstract In this paper, we consider the following fractional Kirchhoff equation with discontinuous nonlinearity $$\begin{aligned} \left\{ \begin{array}{ll} \left( \varepsilon ^{2\alpha }a+\varepsilon ^{4\alpha -3}b\int _{{\mathbb {R}}^3}|(-\Delta )^{\frac{\alpha }{2}} u|^2{{\mathrm{d}}}x\right) (-\Delta )^\alpha {u}+V(x)u = H(u-\beta )f(u) &{} \quad \text{ in }\,\,{\mathbb {R}}^3, \\ u\in H...

In an effort to determine subunit compositions of in vivo GABAA receptors, the cellular localization of 13 subunit encoding mRNAs (alpha 1-alpha 6, beta 1-beta 3, gamma 2-gamma 3, delta) was determined in the rat olfactory bulb and cerebellum. Cerebellar granule cells expressed significant quantities of alpha 1, alpha 6, beta 2, beta 3, gamma 2, and delta mRNAs. They contained very much lower l...

Relational diversity can be characterized by heterogeneous distributions of tie strengths in social networks and this diversity is present not only among humans, but throughout the animal world. We account for this observation by analyzing two network datasets from Facebook. We measure the strength of a tie by calculating the extent of overlap of friends between the two individuals. Based on th...

In this note, we prove weighted resolvent estimates for the semiclassical Schrodinger operator $-h^2 \Delta + V(x) : L^2(\mathbb{R}^n) \to L^2(\mathbb{R}^n)$, $n \neq 2$. The potential $V$ is real-valued, and assumed to either decay at infinity or obey a radial $\alpha$-Holder continuity condition, $0\leq \alpha \leq 1$, with sufficient of local $C^\alpha$ norm toward infinity. Note, however, t...

The ruthenium halide $\alpha$-RuCl$_{3}$ is a promising candidate for Kitaev spin liquid. However, the microscopic model describing still debated partly because of lack analogue materials $\alpha$-RuCl$_{3}$, which prevents tracking electronic properties as functions controlled interaction parameters. Here, we report successful synthesis RuBr$_{3}$. material RuBr$_{3}$~possesses BiI$_3$-type st...

Let f $f$ be a noncommutative polynomial of degree m ⩾ 1 $m\geqslant 1$ over an algebraically closed field F $F$ characteristic 0. If n − $n\geqslant m-1$ and α , 2 3 $\alpha _1,\alpha _2,\alpha _3$ are nonzero elements from such that + = 0 _1+\alpha _2+\alpha _3=0$ then every trace zero × $n\times n$ matrix can written as A _1 A_1+\alpha _2A_2+\alpha _3A_3$ for some i $A_i$ in the image M ( ) ...