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...

We prove higher integrability of the spatial gradient weak solutions to parabolic systems with $\phi$-growth, where $\varphi=\varphi(t)$ is a general Orlicz function. The need be neither degenerate nor singular. Our result generalized version one J. Kinnunen and Lewis [Duke Math. 102 (2000), no. 2, 253--271] for $p$-Laplace systems.

The aim of this paper is to present some results about the space $L^{\varPhi }(\nu ),$ where $\nu$ a vector measure on compact (not necessarily abelian) group and $\varPhi$ Young function. We show that under natural conditions, )$ becomes an $L^{1}(G)$ -module with respect usual convolution functions. also define one more structure ).$

In this short paper, we study sums of the shape $$\sum _{n\leqslant x}{f([x/n])}/{[x/n]},$$ where f is Euler totient function $$\varphi $$ , Dedekind $$\Psi sum-of-divisors $$\sigma or alternating $$\beta .$$ We improve previous results when $$f=\varphi and derive new estimates $$f=\Psi f=\sigma $$f=\beta