The reconstruction conjecture for finite simple graphs and associated directed graphs

In this paper, we study the Reconstruction Conjecture for finite simple graphs. Let Γ and ${\mathrm{\Gamma }}^{\prime }$ be finite simple graphs with at least three vertices such that there exists a bijective map $f:V(\mathrm{\Gamma })\to V({\mathrm{\Gamma }}^{\prime })$ and for any $v\in V(\mathrm{\Gamma })$, there exists an isomorphism ${?}_v:\mathrm{\Gamma }?v\to {\mathrm{\Gamma }}^{\prime }?f(v)$. Then we define the associated directed graph $\stackrel{?}{\mathrm{\Gamma }}=\stackrel{?}{\mathrm{\Gamma }}(\mathrm{\Gamma },{\mathrm{\Gamma }}^{\prime },f,{\{{?}_v\}}_{v\in V(\mathrm{\Gamma })})$ with two kinds of arrows from the graphs Γ and ${\mathrm{\Gamma }}^{\prime }$, the bijective map f and the isomorphisms ${\{{?}_v\}}_{v\in V(\mathrm{\Gamma })}$. By investigating the associated directed graph $\stackrel{?}{\mathrm{\Gamma }}$, we study when are the two graphs Γ and ${\mathrm{\Gamma }}^{\prime }$ isomorphic.     

New global bifurcation diagrams for piecewise smooth systems: Transversality of homoclinic points does not imply chaos

In this paper we consider some piecewise smooth 2-dimensional systems having a possibly non-smooth homoclinic $\overrightarrow{\gamma }(t)$. We assume that the critical point $\overrightarrow{0}$ lies on the discontinuity surface ${\mathrm{\Omega }}^0$. We consider 4 scenarios which differ for the presence or not of sliding close to $\overrightarrow{0}$ and for the possible presence of a transversal crossing between $\overrightarrow{\gamma }(t)$ and ${\mathrm{\Omega }}^0$. We assume that the systems are subject to a small non-autonomous perturbation, and we obtain 4 new bifurcation diagrams. In particular we show that, in one of these scenarios, the existence of a transversal homoclinic point guarantees the persistence of the homoclinic trajectory but chaos cannot occur. Further we illustrate the presence of new phenomena involving an uncountable number of sliding homoclinics.     

On the critical points of the flight return time function of perturbed closed orbits

We deal here with planar analytic systems $\dot{x}=X(x,\epsilon )$ which are small perturbations of a period annulus. For each transversal section Σ to the unperturbed orbits we denote by $T^{\mathrm{\Sigma }}(q,\epsilon )$ the time needed by a perturbed orbit that starts from $q\in \mathrm{\Sigma }$ to return to Σ. We call this the flight return time function. We say that the closed orbit Γ of $\dot{x}=X(x,0)$ is a continuable critical orbit in a family of the form $\dot{x}=X(x,\epsilon )$ if, for any $q\in \mathrm{\Gamma }$ and any Σ that passes through q, there exists $q^{\epsilon }\in \mathrm{\Sigma }$ a critical point of $T^{\mathrm{\Sigma }}(?,\epsilon )$ such that $q^{\epsilon }\to q$ as $\epsilon \to 0$. In this work we study this new problem of continuability.In particular we prove that a simple critical periodic orbit of $\dot{x}=X(x,0)$ is a continuable critical orbit in any family of the form $\dot{x}=X(x,\epsilon )$. We also give sufficient conditions for the existence of a continuable critical orbit of an isochronous center $\dot{x}=X(x,0)$.     

A generalization of lifting non-proper tropical intersections

Let X and $X^{\prime }$ be closed subschemes of an algebraic torus T over a non-archimedean field. We prove the rational equivalence as tropical cycles in the sense of [11, §2] between the tropicalization of the intersection product $X?X^{\prime }$ and the stable intersection $\mathrm{trop}(X)?\mathrm{trop}(X^{\prime })$, when restricted to (the inverse image under the tropicalization map of) a connected component C of $\mathrm{trop}(X)\cap \mathrm{trop}(X^{\prime })$. This requires possibly passing to a (partial) compactification of T with respect to a suitable fan. We define the compactified stable intersection in a toric tropical variety, and check that this definition is compatible with the intersection product in [11, §2]. As a result we get a numerical equivalence between $\stackrel{￣}{X}?{\stackrel{￣}{X}}^{\prime }{|}_{\stackrel{￣}{C}}$ and $\stackrel{￣}{\mathrm{trop}(X)?\mathrm{trop}(X^{\prime }){|}_C}$ via the compactified stable intersection, where the closures are taken inside the compactifications of T and ${\mathbb{R}}^n$. In particular, when X and $X^{\prime }$ have complementary codimensions, this equivalence generalizes [15, Theorem 6.4], in the sense that $X\cap X^{\prime }$ is allowed to be of positive dimension. Moreover, if $\stackrel{￣}{X}\cap {\stackrel{￣}{X}}^{\prime }$ has finitely many points which tropicalize to $\stackrel{￣}{C}$, we prove a similar equation as in [15, Theorem 6.4] when the ambient space is a reduced subscheme of T (instead of T itself).     

Optimal lower eigenvalue estimates for Hodge-Laplacian and applications

We consider the eigenvalue problem for Hodge-Laplacian on a Riemannian manifold M isometrically immersed into another Riemannian manifold $\overline{M}$. We first assume the pull back Weitzenböck operator of $\overline{M}$ bounded from below, and obtain an extrinsic lower bound for the first eigenvalue of Hodge-Laplacian. As applications, we obtain some rigidity results. Second, when the pull back Weitzenböck operator of $\overline{M}$ bounded from both sides, we give a lower bound of the first eigenvalue by the Ricci curvature of M and some extrinsic geometry. As a consequence, we prove a weak Ejiri type theorem, that is, if the Ricci curvature bounded from below pointwisely by a function of the norm square of the mean curvature vector, then M is a homology sphere. In the end, we give an example to show that all the eigenvalue estimates are optimal when $\overline{M}$ is the space form.