Schrödinger dynamics and optimal transport of measures

In this paper, we recover a class of displacement interpolations of probability measures, in the sense of the Optimal Transport theory, by means of semiclassical measures associated with solutions of Schrödinger equation defined on the flat torus. Moreover, we prove the completing viewpoint by proving that a family of displacement interpolations can always be viewed as a path of time-dependent semiclassical measures.

Topological Quantum Codes from Lattices Partition on the n-Dimensional Flat Tori

In this work, we show that an n-dimensional sublattice Λ′=mΛ of an n-dimensional lattice Λ induces a G=Zmn tessellation in the flat torus Tβ′=Rn/Λ′, where the group G is isomorphic to the lattice partition Λ/Λ′. As a consequence, we obtain, via this technique, toric codes of parameters [[2m2,2,m]], [[3m3,3,m]] and [[6m4,6,m2]] from the lattices Z2, Z3 and Z4, respectively. In particular, for n=2, if Λ1 is either the lattice Z2 or a hexagonal lattice, through lattice partition, we obtain two equivalent ways to cover the fundamental cell P0′ of each hexagonal sublattice Λ′ of hexagonal lattices Λ, using either the fundamental cell P0 or the Voronoi cell V0. These partitions allow us to present new classes of toric codes with parameters [[3m2,2,m]] and color codes with parameters [[18m2,4,4m]] in the flat torus from families of hexagonal lattices in R2.

A note on moduli spaces of conformal classes for flat tori of higher dimension and on their conformal multiplication

Motivated by the theory of complex multiplication of abelian varieties, in this paper we study the conformality classes of flat tori in $${\mathbb {R}}^{n}$$ R n and investigate criteria to determine whether a n-dimensional flat torus has non trivial (i.e. bigger than $${\mathbb {Z}}^{*}={\mathbb {Z}}{\setminus }\{0\}$$ Z ∗ = Z \ { 0 } ) semigroup of conformal endomorphisms (the analogs of isogenies for abelian varieties). We then exhibit several geometric constructions of tori with this property and study the class of conformally equivalent lattices in order to describe the moduli space of the corresponding tori.

On the Existence of Leray-Hopf Weak Solutions to the Navier-Stokes Equations

We give a rather short and self-contained presentation of the global existence for Leray-Hopf weak solutions to the three dimensional incompressible Navier-Stokes equations, with constant density. We give a unified treatment in terms of the domains and the relative boundary conditions and in terms of the approximation methods. More precisely, we consider the case of the whole space, the flat torus, and the case of a general bounded domain with a smooth boundary (the latter supplemented with homogeneous Dirichlet conditions). We consider as approximation schemes the Leray approximation method, the Faedo-Galerkin method, the semi-discretization in time and the approximation by adding a Smagorinsky-Ladyžhenskaya term. We mainly focus on developing a unified treatment especially in the compactness argument needed to show that approximations converge to the weak solutions.

Vortex Condensation in General U(1)×U(1) Abelian Chern-Simons Model on a flat torus

<p style='text-indent:20px;'>In this paper, we study an elliptic system arising from the U(1)<inline-formula><tex-math id="M2">\begin{document}$ \times $\end{document}</tex-math></inline-formula>U(1) Abelian Chern-Simons Model[<xref ref-type="bibr" rid="b25">25</xref>,<xref ref-type="bibr" rid="b37">37</xref>] of the form</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE123"> \begin{document}$ \begin{equation} \left\{\begin{split} \Delta u = &amp;\lambda \left(a(b-a)e^{u}-b(b-a)e^{v}+a^2e^{2u} -abe^{2v}+b(b-a)e^{u+v}\right)\\ &amp; +4\pi \sum\limits_{j = 1}^{k_1}m_j\delta_{p_j}, \\ \Delta v = &amp;\lambda \left(-b(b-a)e^{u}+a(b-a)e^{v}-abe^{2u} +a^2e^{2v}+b(b-a)e^{u+v}\right)\\ &amp; +4\pi \sum\limits_{j = 1}^{k_2}n_j\delta_{q_j}, \end{split}\right. \quad\quad\quad\quad (1)\end{equation} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>which are defined on a parallelogram <inline-formula><tex-math id="M3">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{R}^2 $\end{document}</tex-math></inline-formula> with doubly periodic boundary conditions. Here, <inline-formula><tex-math id="M5">\begin{document}$ a $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ b $\end{document}</tex-math></inline-formula> are interaction constants, <inline-formula><tex-math id="M7">\begin{document}$ \lambda&gt;0 $\end{document}</tex-math></inline-formula> is related to coupling constant, <inline-formula><tex-math id="M8">\begin{document}$ m_j&gt;0(j = 1,\cdots,k_1) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M9">\begin{document}$ n_j&gt;0(j = 1,\cdots,k_2) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M10">\begin{document}$ \delta_{p} $\end{document}</tex-math></inline-formula> is the Dirac measure, <inline-formula><tex-math id="M11">\begin{document}$ p $\end{document}</tex-math></inline-formula> is called vortex point. Concerning the existence results of this system over <inline-formula><tex-math id="M12">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula>, only the cases <inline-formula><tex-math id="M13">\begin{document}$ (a,b) = (0,1) $\end{document}</tex-math></inline-formula>[<xref ref-type="bibr" rid="b28">28</xref>] and <inline-formula><tex-math id="M14">\begin{document}$ a&gt;b&gt;0 $\end{document}</tex-math></inline-formula>[<xref ref-type="bibr" rid="b14">14</xref>] were studied in the literature. The solvability of this system (1) is still an open problem as regards other parameters <inline-formula><tex-math id="M15">\begin{document}$ (a,b) $\end{document}</tex-math></inline-formula>. We show that the system (1) admits topological solutions provided <inline-formula><tex-math id="M16">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula> is large and <inline-formula><tex-math id="M17">\begin{document}$ b&gt;a&gt;0 $\end{document}</tex-math></inline-formula> Our arguments are based on a iteration scheme and variational formulation.</p>