Mathematics

We exploit the connection between quantum dot Dirac operators and $\overline\partial$-Robin Laplacians. First, we find a graphical relation between their smallest positive eigenvalues, which allows us to deduce a recipe for translating…

We study Lusin-measurable functions with values in locally convex spaces. In particular, the behavior of pointwise limits of sequences of Lusin-measurable functions and exhibit pathological phenomena arising in the nonmetrizable setting.…

The scope of this text is to study a process that induces another proof of the Spectral Embedding Theorem: that any densely defined symmetric operator can be extended by a multiplication operator through an embedding of the Hilbert space…

We provide the first example of a finitely presented, and the first example of a simple, group of non-uniform exponential growth. The example is given by Thompson's group V.

Holographic coherent X-ray imaging enables nanoscale imaging of biological cells and tissues, rendering both phase and absorption contrast, i.e. real and imaginary parts of the refractive index. Unlike the standard model, which assumes a…

Consider, on the space of marked groups, the map $\mathrm{Res}_{\mathcal{C}}$ which associates to a marked group its greatest residually-$\mathcal{C}$ quotient, for different sets $\mathcal{C}$ of groups. Except for trivial cases, this map…

The shallow-water system is a standard model for long waves in shallow water. The system is hyperbolic and, for a large class of initial data, solutions develop steep gradients leading to shock formation in finite time. Since such…

We analyze how the interaction between local and nonlocal dispersions, combined with different types of nonlinearities, influences the smoothing effects of solutions. To achieve this goal, we consider a model that generalizes the KdV and…

We study positive solutions of the Dirichlet problem $-\Delta u = u^p$ in a uniformly convex domain $\Omega \subset \mathbb S^2$, $u= 0$ on $\partial\Omega.$ For $p=1$, we assume that the right-hand side is replaced by $\lambda_1 u$, where…

We characterize all lattices $\Lambda \subset \mathbb{R}^2$ and all compactly supported functions $g \in L^2(\mathbb{R})$ for which the Gabor system $\left \{ e^{2\pi i s x} g(x-t) : (t,s) \in \Lambda \right \}$ forms an orthonormal basis…

In this paper, we consider the generalized Navier-Stokes equations with fritional dissipation $(-\Delta)^{\beta}$ with $\beta>\frac{1}{2}$. When $\beta\in(1,2)$, We prove that smooth solutions of the generalized Navier-Stokes equations are…

We establish the existence of solutions of the 2D incompressible non-homogeneous Euler equations with $C^{0}_{t}C^{1,\,\sqrt{\frac{4}{3}}-1-\varepsilon}_{x}\cap C^{0}_{t}L^{2}_{x}$ source terms that develop a singularity in finite time. In…

We prove that the spaces $\ell_p(C(\alpha))$ and $\ell_p(C[0,1])$ have the uniform primary factorisation property whenever $\alpha$ is an ordinal and $1<p\leq\infty$. For the case $p=1$, we establish a general criterion ensuring that…

We show that groups with a mild form of non-positive curvature (a navigable path system) satisfy the weak rank rigidity conjecture: they either have linear divergence or a Morse element. This class includes discrete groups of projective…

In this note, we provide a family of $2\times 2$ tetrablock contractions that have tetrablock isometric dilation, but the corresponding fundamental operators do not commute. This answers a question raised by Bhattacharyya [Indiana Univ.…

In this work, we analyze a diffuse-interface model for tumor growth, subject to multiplicative white noises, posed on a bounded domain $\mathcal{O} \subset \mathbb{R}^d$, $d=2,3$. The model couples a stochastic incompressible convective…

The classes of abelian groups that are (uniformly) strongly Hopfian abelian groups, and dually, (uniformly) strongly co-Hopfian abelian groups have been studied by several authors, including Abdelalim (2015) and Abdelalim-Chillali-Essanouni…

We consider the weighted $p$-Laplacian associated with a measure $\mu$ that is absolutely continuous with respect to the Lebesgue measure on an open connected subset $X\subset\mathbb{R}^N$. We prove that Talenti's weighted…

We investigate critical polyharmonic equations of the following type: $$ Lu = |u|^{2^\sharp-2} u \quad \text{ in } \Omega $$ with Dirichlet boundary conditions, in a smooth bounded domain $\Omega$ of $\mathbb{R}^n$. Here $L$ is an elliptic…

This paper studies frames in Hilbert spaces generated by the orbits of (in)-finitely many vectors under a single operator, presenting new results on multiplication operators and operators composed of Jordan blocks, which generalizes…