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 $\dot{x}=X(x,0)$.     

On Yau's problem of evolving one curve to another: Convex case

A revised Yau's Curvature Difference Flow is considered to deform one convex curve $X_0$ to another one $\stackrel{?}{X}$. It is proved that this flow exists globally on time interval $[0,+\infty )$ and the evolving curve, preserving its convexity and bounded area A, converges to a fixed limiting curve $X_{\infty }$ (congruent to $\sqrt{A/\stackrel{?}{A}}\stackrel{?}{X}$) as time tends to infinity, where $\stackrel{?}{A}$ is the area bounded by the target curve $\stackrel{?}{X}$.     

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.     

Characterization of f-extremal disks

We show uniqueness for overdetermined elliptic problems defined on topological disks Ω with $C^2$ boundary, i.e., positive solutions u to $\mathrm{\Delta }u+f(u)=0$ in $\mathrm{\Omega }?(M^2,g)$ so that $u=0$ and $\frac{?u}{?\overrightarrow{\eta }}=cte$ along ?Ω, $\overrightarrow{\eta }$ the unit outward normal along ?Ω under the assumption of the existence of a candidate family. To do so, we adapt the Gálvez–Mira generalized Hopf-type Theorem [19] to the realm of overdetermined elliptic problem.When $(M^2,g)$ is the standard sphere ${\mathbb{S}}^2$ and f is a $C^1$ function so that $f(x)>0$ and $f(x)\ge x{f}^{\prime }(x)$ for any $x\in {\mathbb{R}}_{+}^{?}$, we construct such candidate family considering rotationally symmetric solutions. This proves the Berestycki–Caffarelli–Nirenberg conjecture in ${\mathbb{S}}^2$ for this choice of f. More precisely, this shows that if u is a positive solution to $\mathrm{\Delta }u+f(u)=0$ on a topological disk $\mathrm{\Omega }?{\mathbb{S}}^2$ with $C^2$ boundary so that $u=0$ and $\frac{?u}{?\overrightarrow{\eta }}=cte$ along ?Ω, then Ω must be a geodesic disk and u is rotationally symmetric. In particular, this gives a positive answer to the Schiffer conjecture D (cf. [33], [35]) for the first Dirichlet eigenvalue and classifies simply-connected harmonic domains (cf. [28], also called Serrin Problem) in ${\mathbb{S}}^2$.     

Group-invariant solutions for the Ricci curvature equation and the Einstein equation

We consider the pseudo-Euclidean space $({\mathbb{R}}^n,g)$, $n\ge 3$, with coordinates $x=(x_1,\dots ,x_n)$ and metric $g_{ij}={\delta }_{ij}{?}_i$, ${?}_i=\pm 1$, where at least one ${?}_i$ is positive, and also tensors of the form $A={\sum }_{i,j}{A}_{ij}d{x}_{i}d{x}_j$, such that $A_{ij}$ are differentiable functions of x. For such tensors, we use Lie point symmetries to find metrics $\stackrel{￣}{g}=\frac{1}{u^2}g$ that solve the Ricci curvature and the Einstein equations. We provide a large class of group-invariant solutions and examples of complete metrics $\stackrel{￣}{g}$ defined globally in ${\mathbb{R}}^n$. As consequences, for certain functions $\stackrel{￣}{K}$, we show complete metrics $\stackrel{￣}{g}$, conformal to the pseudo-Euclidean metric g, whose scalar curvature is $\stackrel{￣}{K}$.     

On the Brauer p-dimension of Henselian discrete valued fields of residual characteristic p > 0

Let $(K,v)$ be a Henselian discrete valued field with residue field $\stackrel{?}{K}$ of characteristic $p>0$, and ${\text{Brd}}_p(K)$ be the Brauer p-dimension of K. This paper shows that ${\text{Brd}}_p(K)\ge n$ if $[\stackrel{?}{K}:{\stackrel{?}{K}}^p]=p^n$, for some $n\in \mathbb{N}$. It proves that ${\text{Brd}}_p(K)=\infty$ if and only if $[\stackrel{?}{K}:{\stackrel{?}{K}}^p]=\infty$.     

Maximizing the expected number of components in an online search of a graph

The following optimal stopping problem is considered. The vertices of a graph G are revealed one by one, in a random order, to a selector. He aims to stop this process at a time t that maximizes the expected number of connected components in the graph ${\stackrel{?}{G}}_t$, induced by the currently revealed vertices. The selector knows G in advance, but different versions of the game are considered depending on the information that he gets about ${\stackrel{?}{G}}_t$. We show that when G has N vertices and maximum degree of order $o(\sqrt{N})$, then the number of components of ${\stackrel{?}{G}}_t$ is concentrated around its mean, which implies that playing the optimal strategy the selector does not benefit much by receiving more information about ${\stackrel{?}{G}}_t$. Results of similar nature were previously obtained by M. Lasoń for the case where G is a k-tree (for constant k). We also consider the particular cases where G is a square, triangular or hexagonal lattice, showing that an optimal selector gains cN components and we compute c with an error less than 0.005 in each case.     

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).