OpenScience

[FORTHCOMING] Hardy-Sobolev critical equations with totally geodesic singularities: existence via the mountain pass theorem

El Hadji Abdoulaye Thiam — 2026-06-19T13:35:34Z

We consider a compact Riemannian manifold $$$(M, g)$$$ of dimension $$$N \geq 3$$$ and $$$\Sigma$$$ a closed totally geodesic submanifold of dimension $$$1 \leq k \leq N-2$$$, and $$$h: M \to ℝ$$$ is a continuous function such that the linear operator $$$-Δ_g+h$$$ is coercive. We study existence of positive solutions $$$u \in H^1\left(M\right)$$$ to the following nonlinear PDE with two Hardy-Sobolev critical exponents:
(0.1) $$$ -\Delta_g u+h u=\lambda \rho_{\Sigma}^{-s_1} u^{2^*_{s_1}-1}+\rho_{\Sigma}^{-s_2} u^{2^*_{s_2}-1} \qquad \textrm{ in } (M, g)$$$
where $$$\lambda$$$ is a positive parameter, $$$0 < s_2 < s_1 < 2$$$, the $$$2^*_{s_i}:=\frac{2(N-s_i)}{N-2}$$$ $$$(i=1, 2)$$$ are two critical Hardy-Sobolev exponents and $$$\rho_\Sigma: \mathcal{M} \to ℝ$$$ is the distance function to $$$\Sigma$$$. In this paper, we give sufficient condition depending on the local geometries of the submanifold $$$\Sigma$$$ and the manifold $$$M$$$, for the existence of mountain pass solution to (0.1).

- Forthcoming papers / Mountain Pass Solution, Two Hardy-Sobolev critical exponents, Scalar Curvature, Riemannian curvature tensor, Submanifold

[FORTHCOMING] Équations critiques de Hardy–Sobolev avec singularités totalement géodésiques : existence par le théorème du col de montagne.

El Hadji Abdoulaye Thiam — 2026-06-19T13:35:22Z

We consider a compact Riemannian manifold $$$(M, g)$$$ of dimension $$$N \geq 3$$$ and $$$\Sigma$$$ a closed totally geodesic submanifold of dimension $$$1 \leq k \leq N-2$$$, and $$$h: M \to ℝ$$$ is a continuous function such that the linear operator $$$-Δ_g+h$$$ is coercive. We study existence of positive solutions $$$u \in H^1\left(M\right)$$$ to the following nonlinear PDE with two Hardy-Sobolev critical exponents :
(0.1) $$$ -\Delta_g u+h u=\lambda \rho_{\Sigma}^{-s_1} u^{2^*_{s_1}-1}+\rho_{\Sigma}^{-s_2} u^{2^*_{s_2}-1} \qquad \textrm{ in } (M, g)$$$
where $$$\lambda$$$ is a positive parameter, $$$0 < s_2 < s_1 < 2$$$, the $$$2^*_{s_i}:=\frac{2(N-s_i)}{N-2}$$$ $$$(i=1, 2)$$$ are two critical Hardy-Sobolev exponents and $$$\rho_\Sigma: \mathcal{M} \to ℝ$$$ is the distance function to $$$\Sigma$$$. In this paper, we give sufficient condition depending on the local geometries of the submanifold $$$\Sigma$$$ and the manifold $$$M$$$, for the existence of mountain pass solution to (0.1).

- Articles à paraître / Mountain Pass Solution, Two Hardy-Sobolev critical exponents, Scalar Curvature, Riemannian curvature tensor, Submanifold

[FORTHCOMING] Apprentissage géométrique et métriques de Finsler dans les espaces projectifs pondérés

Tanush Shaska — 2026-06-19T08:55:07Z

We introduce a hierarchical clustering framework for weighted projective spaces $$$ℙ_{\parallel}$$$ built on Finsler geometry. From an optimization-based Finsler norm that quotients out the weighted scaling action, we construct a scaling-invariant distance $$$d_F([z], [w])$$$ and a rational analogue $$$d_{F,ℚ}([z], [w])$$$ for points of $$$ℙ_{\parallel}(ℚ)$$$. The norm carries a shape parameter $$$p:$$$ the case $$$p=2$$$ is Riemannian and admits a closed-form distance, while $$$p\neq 2$$$ is genuinely Finsler, and the metric and clustering guarantees below hold for every $$$p\in[1,\infty)$$$. Whereas earlier work measured proximity in these spaces through non-metric dissimilarities, we prove that $$$d_F$$$ satisfies the triangle inequality and is therefore a genuine metric ; this is what equips the induced clustering with its theoretical guarantees, including monotone dendrograms and Gromov–Hausdorff stability under perturbation of the data. The metric respects the intrinsic scaling symmetry and weighted topology of $$$ℙ_{\parallel}$$$, avoiding the distortions of a flat-space embedding. We develop the framework's arithmetic applications—clustering rational points in the moduli space of genus two curves and analyzing rational functions in arithmetic dynamics—and indicate prospective extensions to quantum state spaces, where the weights $$${\parallel}$$$ model anisotropic noise. More broadly, the construction offers a rigorous metric foundation for graded neural networks and related machine-learning techniques on graded algebraic varieties.

- Articles à paraître / Geometric clustering, Weighted projective spaces, Finsler metrics, Non-Euclidean manifolds

[FORTHCOMING] Geometric learning and Finsler metrics in weighted projective spaces

Tanush Shaska — 2026-06-19T08:54:52Z

We introduce a hierarchical clustering framework for weighted projective spaces $$$ℙ_{\parallel}$$$ built on Finsler geometry. From an optimization-based Finsler norm that quotients out the weighted scaling action, we construct a scaling-invariant distance $$$d_F([z], [w])$$$ and a rational analogue $$$d_{F,ℚ}([z], [w])$$$ for points of $$$ℙ_{\parallel}(ℚ)$$$. The norm carries a shape parameter $$$p:$$$ the case $$$p=2$$$ is Riemannian and admits a closed-form distance, while $$$p\neq 2$$$ is genuinely Finsler, and the metric and clustering guarantees below hold for every $$$p\in[1,\infty)$$$. Whereas earlier work measured proximity in these spaces through non-metric dissimilarities, we prove that $$$d_F$$$ satisfies the triangle inequality and is therefore a genuine metric; this is what equips the induced clustering with its theoretical guarantees, including monotone dendrograms and Gromov–Hausdorff stability under perturbation of the data. The metric respects the intrinsic scaling symmetry and weighted topology of $$$ℙ_{\parallel}$$$, avoiding the distortions of a flat-space embedding. We develop the framework's arithmetic applications—clustering rational points in the moduli space of genus two curves and analyzing rational functions in arithmetic dynamics—and indicate prospective extensions to quantum state spaces, where the weights $$${\parallel}$$$ model anisotropic noise. More broadly, the construction offers a rigorous metric foundation for graded neural networks and related machine-learning techniques on graded algebraic varieties.

- Forthcoming papers / Geometric clustering, Weighted projective spaces, Finsler metrics, Non-Euclidean manifolds

From the French Artist Bouguereau to Modern Science: Why Are We Captivated by Butterfly Eyespots?

Md Jahir Rayhan , Vazrick Nazari — 2026-06-08T10:02:35Z

Md Jahir Rayhan
, University of Florida,USA,, Vazrick Nazari
, Università Degli Studi di Padova, Italy

De l'artiste français Bouguereau à la science modern : pourquoi sommes-nous captivés par les ocelles des ailes de papillon ?

Md Jahir Rayhan , Vazrick Nazari — 2026-06-08T10:02:30Z

Md Jahir Rayhan
, University of Florida,USA,, Vazrick Nazari
, Università Degli Studi di Padova, Italy,

The unique taste characteristics of West Indian rum

Alain MAURIN, Génica LAWRENCE — 2026-05-29T08:00:00Z

Cette revue vise à mettre en relief les spécificités sensorielles des rhums fabriqués aux Antilles, en mettant en lumière les facteurs principaux qui façonnent leur identité sensorielle et réglementaire. Le rhum antillais se distingue par une forte identité liée à la canne à sucre locale, aux techniques traditionnelles versus industrielles (procédés, levures de fermentation, ingrédients) et à la variabilité environnementale (climat, sol, …), conférant à chaque production une unicité difficilement reproductible. Cette revue met également en exergue l'importance de la dégustation et des critères sensoriels dans l'appréciation de ces spiritueux. Enfin, elle propose une réflexion sur le rôle du rhum dans la construction identitaire et économique des régions ultrapériphériques françaises.

- Issue 1 / rum, sensory, typicity, terroir

Les singularités gustatives dans l'offre des rhums antillais

Alain MAURIN, Génica LAWRENCE — 2026-05-29T08:00:00Z

- Numéro 1 / rhums, sensoriel, typicité, terroir

[À PARAITRE] L'émergence d'un technotype : la téléportation et sa genèse science-fictionnelle

Thomas Michaud — 2026-05-27T08:40:12Z

Cet article analyse la téléportation dans la science-fiction à travers le concept de « mythe techno-gestationnel ». Bien qu'elle soit techniquement irréalisable à l'heure actuelle et nécessiterait une rupture paradigmatique majeure, cette technologie fictive joue un rôle crucial en façonnant l'innovation très en amont des processus de recherche et développement. À travers l'étude d'oeuvres variées (comme la saga La Mouche, Star Trek, Everywhen, Terminus des Étoiles, ou Hypérion), l'article démontre la profonde ambivalence de cet imaginaire. D'une part, la téléportation incarne une innovation disruptive et utopique, promettant de révolutionner les mobilités, d'abolir les distances et de faciliter l'expansion cosmique de l'humanité. D'autre part, elle cristallise les angoisses liées à l'hubris scientifique, illustrant les dérives d'une science faustienne par le biais de mutations monstrueuses tragiques ou d'aliénations sociales dystopiques. En conclusion, la science-fiction s'affirme comme un puissant outil prospectif. En nommant l'impensable, elle génère des archétypes technologiques, ou « technotypes ». Ces représentations structurent l'imaginaire collectif et opèrent une fécondation in imago capable, à long terme, d'orienter et d'inspirer les programmes de recherche scientifique.

- Articles à paraître / Téléportation, Science-fiction, Technotype, Innovation, Mythe techno-gestationnel

[FORTHCOMING] The emergence of a technotype: teleportation and its science-fiction genesis

Thomas Michaud — 2026-05-27T08:40:07Z

This article analyses teleportation in science fiction through the concept of the techno-gestational myth. Although technically impossible at present and requiring a major paradigm shift, this fictional technology plays a crucial role in shaping innovation very early in the research and development process. Through the study of diverse works (such as the saga The Fly, Star Trek, Everywhen and Hyperion), the article demonstrates the profound ambivalence of this imagined phenomenon. On the one hand, teleportation embodies a disruptive and utopian innovation, promising to revolutionize mobility, abolish distances and facilitate humanity's cosmic expansion. On the other hand, it crystallizes anxieties linked to scientific hubris, illustrating the excesses of a Faustian science through monstrous mutations, tragic disappearances, and dystopian social alienation. In conclusion, science fiction proves to be a powerful tool for foresight. By naming the unthinkable, it generates technological archetypes, or technotypes. These representations structure the collective imagination and effect a kind of in imago fertilization capable, in the long term, of guiding and inspiring scientific research programs.

- Forthcoming papers / Teleportation, Science fiction, Technotype, Innovation, Techno-gestational Myth