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Comptes Rendus. Mathématique
Scopus
ISSN 1778-3569 · Académie des sciences · 771 articles in the index
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Journal Article
Fubini–Study forms on punctured Riemann surfaces
In this paper we consider a punctured Riemann surface endowed with a Hermitian metric that equals the Poincaré metric near the punctures, and a holomorphic line bundle that polarizes the metric. We show that the quotient of the induced Fubini–Study forms by Kodaira maps of high tensor powers of the…
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Journal Article
A direct proof of the weighted Pólya–Knopp inequality following Carleson’s method
The aim of the paper is to provide a direct proof of the weighted Pólya–Knopp inequality. This inequality (which is a limiting case of the Ariño–Muckenhoupt inequalities), involving non-increasing functions, was initially established by Sbordone–Wik, who proved its validity under the necessary and…
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Journal Article
Normal points on Artin–Schreier curves over finite fields
In 2022, S. D. Cohen and the two authors introduced and studied the concept of $(r, n)$-freeness on finite cyclic groups $G$ for suitable integers $r$, $n$, which is an arithmetic way of capturing elements of special forms that lie in the subgroups of $G$. Combining this machinery with some…
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Journal Article
On gelation for the Smoluchowski coagulation equation
Motivated by the recent results of Andreis–Iyer–Magnanini [2], we provide a short proof, revisiting the one of Escobedo–Mischler–Perthame [7], that for a large class of coagulation kernels, any weak solution to the Smoluchowski equation loses mass in finite time. The class of kernels we consider is…
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Journal Article
The cuspidal cubic and line arrangements with only triple points
We describe a new infinite family of line arrangements in the projective plane with only triple point singularities, and we recover previously known examples.
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Journal Article
Iteration of exponentials with sign changes
In this paper we consider the iteration of infinitely many signed exponentials with the same base but the signs may vary. We show that for every base in an explicit interval this iteration converges for any sequence of signs and all the real numbers are possible limits. We give some more results…
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Journal Article
Liouville theorems for $p$-Laplace equations with critical Hardy–Hénon exponents
This paper is concerned with possibly sign-changing solutions to the critical $p$-Laplace equation $-\Delta _p u = \vert {x}\vert ^\alpha \vert {u}\vert ^{p^*_\alpha -2}u$ in $\Omega $, $u=0$ on $\partial \Omega $, where $1-p$, $p^*_\alpha =\frac{(N+\alpha )p}{N-p}$, and $\Omega $ is a bounded or…
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Journal Article
The isospectral problem for $p$-widths: an application of Zoll metrics
We pose the isospectral problem for the $p$-widths: is a Riemannian manifold $(M^{n+1}, g)$ uniquely determined by its $p$-widths, $\bigl \lbrace \omega _p(M,g)\bigr \rbrace _{p=1}^{\infty }$? We construct many counterexamples on $S^2$ using Zoll metrics and the fact that geodesic $p$-widths are…
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Journal Article
Hessian regularity in Hölder spaces for a semi-linear bi-Laplacian equation
We examine a semi-linear variant of the bi-Laplacian equation in the superlinear, subquadratic setting and obtain $C^{2,\sigma }$-regularity estimates, depending on the growth regime of the nonlinearity. Our strategy is to render this fourth-order problem as a system of two Poisson equations and…
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Journal Article
A non-vanishing conjecture for cotangent bundles on elliptic surfaces
In this paper, we prove the non-vanishing conjecture for cotangent bundles on isotrivial elliptic surfaces. Combined with the result by Höring and Peternell, it completely solves the question for surfaces with Kodaira dimension at most $1$.
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Journal Article
Markov partitions for non-transitive expansive flows
In this note we construct Markov partitions for non-transitive expansive flows in dimension 3.
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Journal Article
A note on contractive semi-groups on a 1:1 junction for scalar conservation laws and Hamilton–Jacobi equations
We characterize the continuous semi-groups on $L^1(\mathbb{R})$ which coincide with a scalar conservation law $\rho _t+(f(\rho ))_x=0$ in $\mathbb{R}_+\times (\mathbb{R}\setminus \lbrace 0\rbrace )$ and are $L^1$-contracting. In a symmetric way, we characterize the continuous semi-groups on…
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Journal Article
On quasiconformal extension of harmonic mappings with nonzero pole
Let $\Sigma _H^k(p)$ be the class of sense-preserving univalent harmonic mappings defined on the open unit disk $\mathbb{D}$ of the complex plane with a simple pole at $z=p \in (0,1)$ that have $k$-quasiconformal extensions ($0\le k
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Journal Article
Toric polar maps, normal crossings and mixed volumes
Given a hypersurface in a complex projective space, we prove that the Chern–Schwartz–MacPherson class of a certain open set, namely the complement of the union of the hypersurface and the coordinate hyperplanes, is given (up to sign) by the multidegrees of associated toric polar map, in two…
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Journal Article
Existence of an extremal function of critical Sobolev embedding with an $\alpha $-homogeneous weight
In [Calc. Var. Partial Differ. Equ. 60 (2021), no. 1, article no. 16 (27 pages)], we revisited a critical Sobolev-type embedding for weighted Sobolev spaces as introduced in [J. Differ. Equations 255 (2013), no. 11, pp. 4312–4336] by Cabré and Ros-Oton. In the same paper, we announced to explore…
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Journal Article
Trivialité des groupes de Whitehead réduits avec applications à l’approximation faible et l’approximation forte
We prove some triviality results for reduced Whitehead groups and reduced unitary Whitehead groups for division algebras over a Henselian discrete valuation field whose residue field has virtual cohomological dimension or separable dimension $\le 2$. These results are applied to show strong…
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Journal Article
Non-uniform Berry–Esseen bounds via Malliavin–Stein method
In this paper, we establish non-uniform Berry–Esseen bounds by means of the Malliavin–Stein method. Applications to the multiple Wiener–Itô integrals and the exponential functionals of Brownian motion are given to illustrate the theory.
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Journal Article
The affine Grassmannian as a presheaf quotient
For a reductive group $G$ over a ring $A$, its affine Grassmannian $\operatorname{Gr}_G$ plays important roles in a wide range of subjects and is typically defined as the étale sheafification of the presheaf quotient $LG/L^+G$ of the loop group $LG$ by its positive loop subgroup $L^+G$. We show…
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Journal Article
The (co)homology of the Deligne–Mumford moduli spaces of marked rational curves
This informal note collects key results and open problems on the (co)homology of the Deligne–Mumford moduli spaces of real marked rational curves. The open problems are both of topological nature, aiming to investigate the (co)homology of these spaces further, and of algebraic nature, aiming to…
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Journal Article
The number system in rational base $3/2$ and the $3x+1$ problem
The representation of numbers in rational base $p/q$ was introduced in 2008 by Akiyama, Frougny & Sakarovitch, with a special focus on the case $p/q=3/2$. Unnoticed since then, natural questions related to representations in that specific base turn out to intimately involve the Collatz $3x+1$…