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Avancées en Mathématiques Pures et Appliquées

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Dans ce travail, nous étudions la commutativité des quasi-anneaux 3-premiers satisfaisant certaines identités différentielles sur des idéaux de Jordan impliquant certaines applications additives. Certains résultats bien connus caractérisant la commutativité des quasi-anneaux 3-premiers par l’action des dérivations et des multiplicateurs à gauche ont été étendus aux multiplicateurs à droite et aux dérivations généralisées à gauche. De plus, nous avons enrichi cet article par un exemple qui montre la nécessité de la 3-primalité mentionnée dans les hypothèses de nos théorèmes.

This paper studies the asymptotic behavior of energy solutions to the focusing non-linear generalized Hartree equation
$$$i u_t+\Delta u=-｜x｜^{-\varrho}｜u｜^{p-2}(\mathcal J_\alpha *｜\cdot｜^{-\varrho}｜u｜^p)u,\quad \varrho>0,\quad p\geq2.$$$
Here, $$$u:=u(t,x)$$$, where the time variable is $$$t \in ℝ$$$ and the space variable is $$$x\inℝ^2$$$.
The source term is inhomogeneous because $$$\varrho > 0$$$. The convolution with the Riesz-potential $$$\mathcal J_\alpha:=C_\alpha｜\cdot｜^{\alpha-2}$$$ for certain $$$0 < \alpha < 2$$$ gives a non-local Hartree type non-linearity. Taking account of the standard scaling invariance, one considers the inter-critical regime $$$1 + \frac{2-2\varrho + \alpha}2 < p < \infty$$$. It is the purpose to prove the scattering under the ground state threshold. This naturally extends the previous work by the first author for space dimensions greater than three (Scattering Theory for a Class of Radial Focusing Inhomogeneous Hartree Equations, Potential Anal. (2021)). The main difference is due to the Sobolev embedding in two space dimensions $$$H^1(ℝ^2)\hookrightarrow L^r(ℝ^2)$$$, for all $$$2 \leq r < \infty$$$. This makes any exponent of the source term be energy subcritical, contrarily to the case of higher dimensions. The decay of the inhomogeneous term $$$｜x｜^{-\varrho}$$$ is used to avoid any radial assumption. The proof uses the method of Dodson-Murphy based on Tao’s scattering criteria and Morawetz estimates.

In the paper, we introduce the notion of compression of generalized slant Toeplitz operators to the Hardy space of $$$n$$$-dimensional torus $$$\mathbb{T}^n$$$. It deals with characterizations of introduced operator with specific as well as general symbols. Certain algebraic and structural properties of considered operators are also investigated. Finally, we discuss few results related to essentially $$$k^{th}$$$-order $$$\lambda$$$-slant Toeplitz operator.

The purpose of this paper is to give some properties for the so-called ε-pseudo weakly demicompact linear operators acting on Banach spaces. Some sufficient conditions on the entries of an unbounded 2 × 2 block operator matrix $$$\mathcal{L}_{0}$$$ ensuring its ε-pseudo weak demicompactness are provided. In addition, we develop, in the bounded case, the class of ε-pseudo Fredholm perturbation to investigate the essential pseudo-spectra of $$$\mathcal{L}_{0}$$$. The results are formulated in terms of some denseness conditions on the topological dual space.

In this paper, we conduct a mathematical analysis of a tumor growth model with treatments. The model consists of a system that describes the evolution of metastatic tumors and the number of cells present in the primary tumor. The former evolution is described by a transport equation, and the latter by an ordinary differential equation of Gompertzian type. The two dynamics are coupled through a nonlocal boundary condition that takes into account the tumor colonization rate. We prove an existence result where the main difficulty is to handle the coupling and to take into account the time discontinuities generated by treatment terms. The proof is based on a Banach fixed point theorem in a suitable functional space. We also develop a computational code based on the method of characteristics and present numerical tests that highlight the effects of different therapies.