Let $\mathbf{k}$ be a number field and $\mathbf{k}_M$ the Minkowski space associated to $\mathbf{k}$. Dirichlet's theorem in Diophantine approximation is generalized to the case of systems of linear forms with coefficients in $\mathbf{k}_M$. We study the set of singular systems in this setting. We generalize the transference principle, Dani's correspondence and give an estimate of the Hausdorff dimension of the set of singular systems from below.

We give an upper bound of the slope for finite cyclic covering fibrations of an elliptic surface which includes bielliptic fibrations. We also give an upper bound of the slope for triple cyclic covering fibrations of a ruled surface and hyperelliptic fibrations, which provides a new proof of Xiao's upper bound.

We introduce the affine ensemble, a class of determinantal point processes (DPP) in the half-plane $\mathbb{C}^{+}$ associated with the $ax + b$ (affine) group, depending on an admissible Hardy function $\psi$. We obtain the asymptotic behavior of the variance, the exact value of the asymptotic constant, and non-asymptotic upper and lower bounds for the variance on a compact set $\Omega \subset \mathbb{C}^{+}$. As a special case one recovers the DPP related to the weighted Bergman kernel. When $\psi$ is chosen within a finite family whose Fourier transform are Laguerre functions, we obtain the DPP associated to hyperbolic Landau levels, the eigenspaces of the finite spectrum of the Maass Laplacian with a magnetic field.

We introduce a new algebraic framework based on the deformation of pre-Lie products. This allows us to provide a new construction of the algebraic objects at play in regularity structures in the works by Bruned, Hairer and Zambotti (2019) and by Bruned and Schratz (2022) for deriving a general scheme for dispersive PDEs at low regularity. This construction also explains how the algebraic structure by Bruned et al. (2019) cited above can be viewed as a deformation of the Butcher–Connes–Kreimer and the extraction-contraction Hopf algebras. We start by deforming various pre-Lie products via a Taylor deformation and then we apply the Guin–Oudom procedure which gives us an associative product whose adjoint can be compared with known coproducts. This work reveals that pre-Lie products and their deformation can be a central object in the study of (S)PDEs.

In this paper, when a given symmetric Markov process X satisfies the stability of global heat kernel two-sided (upper) estimates by Markov perturbations (see Definition 1.2), we give a necessary and sufficient condition on the stability of global two-sided (upper) estimates for fundamental solution of Feynman–Kac semigroup of X. As a corollary, under the same assumptions, a weak type of global two-sided (upper) estimates holds for the fundamental solution of Feynman–Kac semigroup with (extended) Kato class conditions for measures. This generalizes all known results on the stability of global integral kernel estimates by symmetric Feynman–Kac perturbations with Kato class conditions in the framework of symmetric Markov processes.

We establish a sharp bilinear estimate for the Klein–Gordon propagator in the spirit of recent work of Beltran–Vega. Our approach is inspired by work in the setting of the wave equation due to Bez, Jeavons and Ozawa. As a consequence of our main bilinear estimate, we deduce several sharp estimates of null-form type and recover some sharp Strichartz estimates found by Quilodrán and Jeavons.

The global existence for semilinear wave equations with space-dependent critical damping $\partial_{t}^{2} u - \Delta u + \frac{V_0}{|x|} \partial_{t} u = f(u)$ in an exterior domain is dealt with, where $f(u) = |u|^{p-1} u$ and $f(u) = |u|^{p}$ are in mind. Existence and non-existence of global-in-time solutions are discussed. To obtain global existence, a weighted energy estimate for the linear problem is crucial. The proof of such a weighted energy estimate contains an alternative proof of energy estimates established by Ikehata–Todorova–Yordanov [J. Math. Soc. Japan (2013), 183–236] but the argument in this paper clarifies the precise dependence of the location of the support of initial data. The blowup phenomena are verified by using a test function method with positive harmonic functions satisfying the Dirichlet boundary condition.

Line congruences are 2-dimensional families of lines in 3-space. The singularities that appear in generic line congruences are folds, cuspidal edges and swallowtails. In this paper we give a geometric description of these singularities. The main tool used is the existence of an equiaffine pair defining a generic line congruence.

We first present some sufficient conditions for the construction of a robust family of exponential attractors for infinite dimensional dynamical systems with small time delay perturbation. In particular, we prove that this family of exponential attractors is stable in the sense of the symmetric Hausdorff distance as the delay effects vanish. The abstract result is then applied to two-dimensional nonlocal diffusion lattice systems with small delay.

In this work we approach the problem of determining which (compact) semialgebraic subsets of $\mathbb{R}^{n}$ are images under polynomial maps $f:\mathbb{R}^{m} \to \mathbb{R}^{n}$ of the closed unit ball $\overline{\mathcal{B}}_m$ centered at the origin of some Euclidean space $\mathbb{R}^{m}$ and that of estimating (when possible) which is the smallest $m$ with this property. Contrary to what happens with the images of $\mathbb{R}^{m}$ under polynomial maps, it is quite straightforward to provide basic examples of semialgebraic sets that are polynomial images of the closed unit ball. For instance, simplices, cylinders, hypercubes, elliptic, parabolic or hyperbolic segments (of dimension $n$) are polynomial images of the closed unit ball in $\mathbb{R}^{n}$.

The previous examples (and other basic ones proposed in the article) provide a large family of ‘$n$-bricks’ and we find necessary and sufficient conditions to guarantee that a finite union of ‘$n$-bricks’ is again a polynomial image of the closed unit ball either of dimension $n$ or $n + 1$. In this direction, we prove: A finite union $\mathcal{S}$ of $n$-dimensional convex polyhedra is the image of the $n$-dimensional closed unit ball $\overline{\mathcal{B}}_{n}$ if and only if $\mathcal{S}$ is connected by analytic paths.

The previous result can be generalized using the ‘$n$-bricks’ mentioned before and we show: If $\mathcal{S}_{1}, \ldots, \mathcal{S}_{\ell} \subset \mathbb{R}^n$ are ‘$n$-bricks’, the union $\mathcal{S} := \bigcup_{i=1}^{\ell} \mathcal{S}_i$ is the image of the closed unit ball $\overline{\mathcal{B}}_{n+1}$ of $\mathbb{R}^{n+1}$ under a polynomial map $f:\mathbb{R}^{n+1} \to \mathbb{R}^{n}$ if and only if $\mathcal{S}$ is connected by analytic paths.

We show finiteness results on torsion points of commutative algebraic groups over a $p$-adic field $K$ with values in various algebraic extensions $L/K$ of infinite degree. We mainly study the following cases: (1) $L$ is an abelian extension which is a splitting field of a crystalline character (such as a Lubin–Tate extension). (2) $L$ is a certain iterate extension of $K$ associated with points on Lubin–Tate formal groups.