We consider the perturbed Stark operator $$H_q\varphi = -\varphi '' + x\varphi q(x)\varphi $$ , $$\varphi (0)=0$$ in $$L^2({\mathbb R}_+)$$ where q is a real function that belongs to $${\mathfrak {A}}_r =\left\{ q\in \mathcal {A}_r\cap \text {AC}[0,\infty ): q'\in {A}_r\right\} $$\mathcal {A}_r L^2_{\mathbb R}({\mathbb R}_+,(1+x)^r dx)$$ and $$r>1$$ arbitrary but fixed. Let $$\left\{ \lambda _n...

The flat torus $${{\mathbb T}}=\mathbb S^1\left( \frac{1}{2} \right) \times \mathbb $$ admits a proper biharmonic isometric immersion into the unit 4-dimensional sphere $$\mathbb S^4$$ given by $$\Phi =i \circ \varphi , where $$\varphi :{{\mathbb T}}\rightarrow S^3(\frac{1}{\sqrt{2}})$$ is minimal Clifford and $$i:\mathbb S^3(\frac{1}{\sqrt{2}}) \rightarrow small hypersphere. first goal of this...

Abstract Let D D be a digraph with vertex set V V and arc E E . For u u , the out-degree in-degree of are denoted by d + {d}_{u}^{+} − {d}_{u}^{-} respectively. A vertex-degree-based (VDB) topologic...

As a low-energy effective model emerging in disparate fields throughout all of physics, the ubiquitous $\varphi^4$-theory is one central models modern theoretical physics. Its topological defects, or kinks, describe stable, particle-like excitations that play role processes ranging from cosmology to particle physics and condensed matter theory. In these lecture notes, we introduce description k...

Extracting isolated rays from a symplectic manifold result in symplectomorphic to the initial one. The same holds for higher dimensional parametrized under an additional condition. More precisely, let $(M,\omega)$ be manifold. Let $[0,\infty)\times Q\subset\mathbb{R}\times Q$ considered as $[0,\infty)$ and $\varphi:[-1,\infty)\times Q\to M$ injective, proper, continuous map immersive on $(-1,\i...

We study a single field inflationary model modified by non-minimal coupling term between the Ricci scalar $R$ and $\varphi$ in context of constant-roll inflation. The first-order formalism is used to analyse inflation instead standard methods literature. In principle, considers two functions field, $W=W(\varphi)$ $Z=Z(\varphi)$, which lead reduction equations motion differential equations. appr...