arXiv subjects

Hông Vân Lê

Publications and source records attributed to Hông Vân Lê.

At least 37 records · Page 2Linked to original sources

arXiv2020

Epidemic Dynamics via Wavelet Theory and Machine Learning, with Applications to Covid-19

We introduce the concept of epidemic-fitted wavelets which comprise, in particular, as special cases the number $I(t)$ of infectious individuals at time $t$ in classical SIR models and their derivatives. We present a novel method for modelling epidemic dynamics by a model selection method using wavelet theory and, for its applications, machine learning based curve fitting techniques. Our universal models are functions that are finite linear combinations of epidemic-fitted wavelets. We apply our method by modelling and forecasting, based on the John Hopkins University dataset, the spread of the current Covid-19 (SARS-CoV-2) epidemic in France, Germany, Italy and the Czech Republic, as well as in the US federal states New York and Florida.

math.OC↗

arXiv2020

Formally integrable complex structures on higher dimensional knot spaces

Let $S$ be a compact oriented finite dimensional manifold and $M$ a finite dimensional Riemannian manifold, let ${\rm Imm}_f(S,M)$ the space of all free immersions $φ:S \to M$ and let $B^+_{i,f}(S,M)$ the quotient space ${\rm Imm}_f(S,M)/{\rm Diff}^+(S)$, where ${\rm Diff}^+(S)$ denotes the group of orientation preserving diffeomorphisms of $S$. In this paper we prove that if $M$ admits a parallel $r$-fold vector cross product $φ\in Ω^r(M, TM)$ and $\dim S = r-1$ then $B^+_{i,f}(S,M)$ is a formally Kähler manifold. This generalizes Brylinski's, LeBrun's and Verbitsky's results for the case that $S$ is a codimension 2 submanifold in $M$, and $S = S^1$ or $M$ is a torsion-free $G_2$-manifold respectively.

math.DG↗

arXiv2020

Strongly homotopy Lie algebras and deformations of calibrated submanifolds

For an element $Ψ$ in the graded vector space $Ω^*(M, TM)$ of tangent bundle valued forms on a smooth manifold $M$, a $Ψ$-submanifold is defined as a submanifold $N$ of $M$ such that $Ψ_{|N} \in Ω^*(N, TN)$. The class of $Ψ$-submanifolds encompasses calibrated submanifolds, complex submanifolds and all Lie subgroups in compact Lie groups. The graded vector space $Ω^*(M, TM)$ carries a natural graded Lie algebra structure, given by the Frölicher-Nijenhuis bracket $[-,- ]^{FN}$. When $Ψ$ is an odd degree element with $[ Ψ, Ψ]^{FN} =0$, we associate to a $Ψ$-submanifold $N$ a strongly homotopy Lie algebra, which governs the formal and (under certain assumptions) smooth deformations of $N$ as a $Ψ$-submanifold, and we show that under certain assumptions these deformations form an analytic variety. As an application we revisit formal and smooth deformation theory of complex closed submanifolds and of $φ$-calibrated closed submanifolds, where $φ$ is a parallel form in a real analytic Riemannian manifold.

math.DG↗

arXiv2020

Differentiation of measures on complete Riemannian manifolds

In this note we give a new proof of a version of the Besicovitch covering theorem, given in \cite{EG1992}, \cite{Bogachev2007} and extended in \cite{Federer1969}, for locally finite Borel measures on finite dimensional complete Riemannian manifolds $(M,g)$. As a consequence, we prove a differentiation theorem for Borel measures on $(M,g)$, which gives a formula for the Radon-Nikodym density of two nonnegative locally finite Borel measures $ν_1, ν_2$ on $(M, g)$ such that $ν_1 \ll ν_2$, extending the known case when $(M, g)$ is a standard Euclidean space.

math.FA↗

arXiv2020

Diffeological statistical models, the Fisher metric and probabilistic mappings

In this note we introduce the notion of a $C^k$-diffeological statistical model, which allows us to apply the theory of diffeological spaces to (possibly singular) statistical models. In particular, we introduce a class of almost 2-integrable $C^k$-diffeological statistical models that encompasses all known statistical models for which the Fisher metric is defined. This class contains a statistical model which does not appear in the Ay-Jost-Lê-Schwachhöfer theory of parametrized measure models. Then we show that for any positive integer $k$ the class of almost 2-integrable $C^k$-diffeological statistical models is preserved under probabilistic mappings. Furthermore, the monotonicity theorem for the Fisher metric also holds for this class. As a consequence, the Fisher metric on an almost 2-integrable $C^k$-diffeological statistical model $P \subset {\cal P}({\cal X})$ is preserved under any probabilistic mapping $T: {\cal X}\leadsto {\cal Y}$ that is sufficient w.r.t. $P$. Finally we extend the Cramér-Rao inequality to the class of 2-integrable $C^k$-diffeological statistical models.

math.ST↗

arXiv2019

Classification of $k$-forms on ${\bf R}^n$ and the existence of associated geometry on manifolds

In this paper we survey methods and results of classification of $k$-forms (resp. $k$-vectors on ${\bf R}^n$), understood as description of the orbit space of the standard ${\bf GL}(n, {\bf R})$-action on $Λ^k {\bf R}^{n*}$ (resp. on $Λ^k {\bf R}^n$). We discuss the existence of related geometry defined by differential forms on smooth manifolds. This paper also contains an Appendix by Mikhail Borovoi on Galois cohomology methods for finding real forms of complex orbits.

math.RT↗

arXiv2019

Frölicher-Nijenhuis cohomology on $G_2$- and ${\rm Spin}(7)$-manifolds

In this paper we show that a parallel differential form $Ψ$ of even degree on a Riemannian manifold allows to define a natural differential both on $Ω^\ast(M)$ and $Ω^\ast(M, TM)$, defined via the Frölicher-Nijenhuis bracket. For instance, on a Kähler manifold, these operators are the complex differential and the Dolbeault differential, respectively. We investigate this construction when taking the differential w.r.t. the canonical parallel $4$-form on a $G_2$- and ${\rm Spin}(7)$-manifold, respectively. We calculate the cohomology groups of $Ω^\ast(M)$ and give a partial description of the cohomology of $Ω^\ast(M, TM)$.

math.DG↗

arXiv2019

Deformations of Coisotropic Submanifolds in Jacobi Manifolds

In this paper, we attach an $L_\infty$-algebra to any coisotropic submanifold in a Jacobi manifold. Our construction generalizes and unifies analogous constructions by Oh-Park (symplectic case), Cattaneo-Felder (Poisson case), Lê-Oh (locally conformal symplectic case). As a new special case, we attach an $L_\infty$-algebra to any coisotropic submanifold in a contact manifold. The $L_\infty$-algebra of a coisotropic submanifold $S$ governs the (formal) deformation problem of $S$.

math.DG↗

arXiv2018

Frölicher-Nijenhuis bracket on manifolds with special holonomy

In this article, we summarize our recent results on the study of manifolds with special holonomy via the Frölicher-Nijenhuis bracket. This bracket enables us to define the Frölicher-Nijenhuis cohomologies which are analogues of the $d^c$ and the Dolbeault cohomologies in Kähler geometry, and assigns an $L_\infty$-algebra to each associative submanifold. We provide several concrete computations of the Frölicher-Nijenhuis cohomology.

math.DG↗

arXiv2018

A Novikov fundamental group

Given a $1$-cohomology class $u$ on a closed manifold $M$, we define a Novikov fundamental group associated to $u$, generalizing the usual fundamental group in the same spirit as Novikov homology generalizes Morse homology to the case of non exact $1$-forms. As an application, lower bounds for the minimal number of index $1$ and $2$ critical points of Morse closed $1$-forms are obtained, that are different in nature from those derived from the Novikov homology.

math.GT↗

arXiv2017

Jacobi bundles and the BFV-complex

We extend the construction of the BFV-complex of a coisotropic submanifold from the Poisson setting to the Jacobi setting. In particular, our construction applies in the contact and l.c.s. settings. The BFV-complex of a coisotropic submanifold $S$ controls the coisotropic deformation problem of $S$ under both Hamiltonian and Jacobi equivalence.

math.DG↗

arXiv2017

Frölicher-Nijenhuis bracket and geometry of $G_2$-and ${\rm Spin}(7)$-manifolds

We extend the characterization of the integrability of an almost complex structure $J$ on differentiable manifolds via the vanishing of the Frölicher-Nijenhuis bracket $[J, J] ^{FN}$ to an analogous characterization of torsion-free $G_2$-structures and torsion-free Spin(7)-structures. We also explain the Fernández-Gray classification of $G_2$-structures and the Fernández classification of Spin(7)-structures in terms of the Frölicher-Nijenhuis bracket.

math.DG↗

arXiv2017

The Cramér-Rao inequality on singular statistical models I

We introduce the notion of the essential tangent bundle of a parametrized measure model and the notion of reduced Fisher metric on a (possibly singular) 2-integrable measure model. Using these notions and a new characterization of $k$-integrable parametrized measure models, we extend the Cramér-Rao inequality to $2$-integrable (possibly singular) statistical models for general $φ$-estimations, where $φ$ is a $V$-valued feature function and $V$ is a topological vector space. Thus we derive an intrinsic Cramér-Rao inequality in the most general terms of parametric statistics.

math.ST↗

arXiv2017

Parametrized measure models

We develope a new and general notion of parametric measure models and statistical models on an arbitrary sample space $Ω$ which does not assume that all measures of the model have the same null sets. This is given by a diffferentiable map from the parameter manifold $M$ into the set of finite measures or probability measures on $Ω$, respectively, which is differentiable when regarded as a map into the Banach space of all signed measures on $Ω$. Furthermore, we also give a rigorous definition of roots of measures and give a natural definition of the Fisher metric and the Amari-Chentsov tensor as the pullback of tensors defined on the space of roots of measures. We show that many features such as the preservation of this tensor under sufficient statistics and the monotonicity formula hold even in this very general set-up.

math.DG↗

arXiv2017

Congruent families and invariant tensors

Classical results of Chentsov and Campbell state that -- up to constant multiples -- the only $2$-tensor field of a statistical model which is invariant under congruent Markov morphisms is the Fisher metric and the only invariant $3$-tensor field is the Amari-Chentsov tensor. We generalize this result for arbitrary degree $n$, showing that any family of $n$-tensors which is invariant under congruent Markov morphisms is algebraically generated by the canonical tensor fields defined in an earlier paper.

math.ST↗

arXiv2016

McLean's second variation formula revisited

We revisit McLean's second variation formulas for calibrated submanifolds in exceptional geometries, and correct his formulas concerning associative submanifolds and Cayley submanifolds, using a unified treatment based on the (relative) calibration method and Harvey-Lawson's identities.

math.DG↗

arXiv2016

The uniqueness of the Fisher metric as information metric

We define a mixed topology on the fiber space $\cup_μ\oplus^n L^n(μ)$ over the space $\mathcal{M}(Ω)$ of all finite non-negative measures $μ$ on a separable metric space $Ω$ provided with Borel $σ$-algebra. We define a notion of strong continuity of a covariant $n$-tensor field on $\mathcal{M}(Ω)$. Under the assumption of strong continuity of an information metric we prove the uniqueness of the Fisher metric as information metric on statistical models associated with $Ω$. Our proof realizes a suggestion due to Amari and Nagaoka to derive the uniqueness of the Fisher metric from the special case proved by Chentsov by using a special kind of limiting procedure. The obtained result extends the monotonicity characterization of the Fisher metric on statistical models associated with finite sample spaces and complement the uniqueness theorem by Ay-Jost-Lê-Schwachhöfer that characterizes the Fisher metric by its invariance under sufficient statistics.

math.ST↗

arXiv2015

Lagrangian submanifolds in strictly nearly Kähler 6-manifolds

Lagrangian submanifolds in strict nearly Kähler 6-manifolds are related to special Lagrangian submanifolds in Calabi-Yau 6-manifolds and coassociative cones in $G_2$-manifolds. We prove that the mean curvature of a Lagrangian submanifold $L$ in a nearly Kähler manifold $(M^{2n}, J, g)$ is symplectically dual to the Maslov 1-form on $L$. Using relative calibrations, we derive a formula for the second variation of the volume of a Lagrangian submanifold $L^3$ in a strict nearly Kähler manifold $(M^6, J, g)$. This formula implies, in particular, that any formal infinitesimal Lagrangian deformation of $L^3$ is a Jacobi field on $L^3$. We describe a finite dimensional local model of the moduli space of compact Lagrangian submanifolds in a strict nearly Kähler 6-manifold. We show that there is a real analytic atlas on $(M^6, J, g)$ in which the strict nearly Kähler structure $(J, g)$ is real analytic. Furthermore, w.r.t. an analytic strict nearly Kähler structure the moduli space of Lagrangian submanifolds of $M^6$ is a real analytic variety, whence infinitesimal Lagrangian deformations are smoothly obstructed if and only if they are formally obstructed. As an application, we relate our results to the description of Lagrangian submanifolds in the sphere $S^6$ with the standard nearly Kähler structure described in \cite{Lotay2012}.

math.DG↗