The radius problem explaining the geometric properties of normalized forms special functions has been interest among Geometric function theories. In this paper, we consider Ma-Minda classes analytic $${\mathcal {S}}^{*}(\phi ):= \{f\in {\mathcal {A}} : ({zf'(z)}/{f(z)}) \prec \phi (z) \}$$ and {C}}(\phi (1+{zf''(z)}/{f'(z)}) defined on unit disk $${\mathbb {D}}$$ show that {S}}^{*}(1+\alpha z)$...

The antiferromagnetic $J_1-J_2$ model is a spin-1/2 chain with isotropic exchange $J_1 > 0$ between first neighbors and $J_2 = \alpha J_1$ second neighbors. supports both gapless quantum phases nondegenerate ground states gapped $\Delta(\alpha) doubly degenerate states. Exact thermodynamics limited to $\alpha 0$, the linear Heisenberg antiferromagnet (HAF). diagonalization of small systems at f...

the submodules with the property of the title ( a submodule $n$ of an $r$-module $m$ is called strongly dense in $m$, denoted by $nleq_{sd}m$, if for any index set $i$, $prod _{i}nleq_{d}prod _{i}m$) are introduced and fully investigated. it is shown that for each submodule $n$ of $m$ there exists the smallest subset $d'subseteq m$ such that $n+d'$ is a strongly dense submodule of $m$...

In this note we continue our study of unidirectional solutions to hydrodynamic Euler alignment systems with strongly singular communication kernels $\phi(x):=|x|^{-(n+\alpha)}$ for $\alpha\in(0,2)$. The describe parallel motion agents governing multi-dimensional collective behavior flocks. Here, consider the range $1<\alpha<2$ and establish global regularity smooth solutions, together a full de...

The alpha 6/beta 4 complex is a member of the integrin family of adhesion receptors. It is found on a variety of epithelial cell types, but is most strongly expressed on stratified squamous epithelia. Fluorescent antibody staining of human epidermis suggests that the beta 4 subunit is strongly localized to the basal region showing a similar distribution to that of the 230-kD bullous pemphigoid ...

In this paper, we introduce the dual notion of strongly top modules and study some of the basic properties of this class of modules.

In this paper, using the fractional difference operator and a modulus function we introduce concepts of $({}^{}_{2}{\Delta_{\beta}^{\tilde{\alpha}}},f)-$ statistical convergence, $({}^{}_{2}{\Delta^{\tilde{\alpha}}},f)-$ Cauchy p-strongly Cesàro summability, $(0&lt;p&lt;\infty)$ for double sequences. We also give some inclusion relations between convergence summability $(0&lt;p&lt;\infty)$.

The formation of $\ensuremath{\alpha}$ particles on nuclear surfaces has remained a fundamental problem since the dawn physics. This $\ensuremath{\alpha}$-particle strongly affects $\ensuremath{\alpha}$-decay lifetimes heavy and superheavy elements, level scheme light nuclei, synthesis elements in stars. However, despite its importance, medium-mass nuclei is seldom investigated. Based $^{48}\ma...