A criticality result for polycycles in a family of quadratic reversible centers

We consider the family of dehomogenized Loud's centers $X_{\mu }=y(x?1){?}_x+(x+D{x}^2+F{y}^2){?}_y$, where $\mu =(D,F)\in {\mathbb{R}}^2$, and we study the number of critical periodic orbits that emerge or disappear from the polycycle at the boundary of the period annulus. This number is defined exactly the same way as the well-known notion of cyclicity of a limit periodic set and we call it criticality. The previous results on the issue for the family $\{X_{\mu },\mu \in {\mathbb{R}}^2\}$ distinguish between parameters with criticality equal to zero (regular parameters) and those with criticality greater than zero (bifurcation parameters). A challenging problem not tackled so far is the computation of the criticality of the bifurcation parameters, which form a set ${\mathrm{\Gamma }}_B$ of codimension 1 in ${\mathbb{R}}^2$. In the present paper we succeed in proving that a subset of ${\mathrm{\Gamma }}_B$ has criticality equal to one.     

Quadratic obstructions to small-time local controllability for scalar-input systems

We consider nonlinear finite-dimensional scalar-input control systems in the vicinity of an equilibrium.When the linearized system is controllable, the nonlinear system is smoothly small-time locally controllable: whatever $m>0$ and $T>0$, the state can reach a whole neighborhood of the equilibrium at time T with controls arbitrary small in $C^m$-norm.When the linearized system is not controllable, we prove that: either the state is constrained to live within a smooth strict manifold, up to a cubic residual, or the quadratic order adds a signed drift with respect to it. This drift holds along a Lie bracket of length $(2k+1)$, is quantified in terms of an $H^{?k}$-norm of the control, holds for controls small in $W^{2k,\infty }$-norm and these spaces are optimal. Our proof requires only $C^3$ regularity of the vector field.This work underlines the importance of the norm used in the smallness assumption on the control, even in finite dimension.     

Classification and evolution of bifurcation curves for the one-dimensional Minkowski-curvature problem and its applications

In this paper, we study the classification and evolution of bifurcation curves of positive solutions for the one-dimensional Minkowski-curvature problem

$\{\begin{array}{c}?{(u^{\prime }/\sqrt{1?u^{\prime 2}})}^{\prime }=\lambda f(u),\text{ in }(?L,L),\hfill \\ u(?L)=u(L)=0,\hfill \end{array}$

where $\lambda ,L>0$, $f\in C[0,\infty )\cap C^2(0,\infty )$ and $f(u)>0$ for $u\ge 0$. Furthermore, we show that, for sufficiently large $L>0$, the bifurcation curve $S_L$ may have arbitrarily many turning points. Finally, we apply these results to obtain the global bifurcation diagrams for Ambrosetti–Brezis–Cerami problem, Liouville–Bratu–Gelfand problem and perturbed Gelfand problem with the Minkowski-curvature operator, respectively. Moreover, we will make two lists which show the different properties of bifurcation curves for Minkowski-curvature problems, corresponding semilinear problems and corresponding prescribed curvature problems.     

General convergence analysis for two-step projection methods and applications to variational problems

First a general model for two-step projection methods is introduced and second it has been applied to the approximation solvability of a system of nonlinear variational inequality problems in a Hilbert space setting. Let $H$ be a real Hilbert space and $K$ be a nonempty closed convex subset of $H$. For arbitrarily chosen initial points $x^0,y^0\in K$, compute sequences $\left\{x^k\right\}$ and $\left\{y^k\right\}$ such that $x^{k+1}=(1-a^k)x^k+a^k{P}_K[y^k-\rho T\left(y^k\right)]\text{for }\rho >0$$y^k=(1-b^k)x^k+b^k{P}_K[x^k-\eta T\left(x^k\right)]\text{for }\eta >0\text{,}$ where $T:K\to H$ is a nonlinear mapping on $K,P_K$ is the projection of $H$ onto $K$, and $0\le a^k,b^k\le 1$. The two-step model is applied to some variational inequality problems.     

A sharp Balian–Low uncertainty principle for shift-invariant spaces

A sharp version of the Balian–Low theorem is proven for the generators of finitely generated shift-invariant spaces. If generators ${\{f_k\}}_{k=1}^K?L^2({\mathbb{R}}^d)$ are translated along a lattice to form a frame or Riesz basis for a shift-invariant space V, and if V has extra invariance by a suitable finer lattice, then one of the generators $f_k$ must satisfy ${\int }_{{\mathbb{R}}^d}|x||f_k(x){|}^{2}dx=\infty$, namely, $\stackrel{?}{f_k}?H^{1/2}({\mathbb{R}}^d)$. Similar results are proven for frames of translates that are not Riesz bases without the assumption of extra lattice invariance. The best previously existing results in the literature give a notably weaker conclusion using the Sobolev space $H^{d/2+?}({\mathbb{R}}^d)$; our results provide an absolutely sharp improvement with $H^{1/2}({\mathbb{R}}^d)$. Our results are sharp in the sense that $H^{1/2}({\mathbb{R}}^d)$ cannot be replaced by $H^s({\mathbb{R}}^d)$ for any $s<1/2$.     

Normal family of meromorphic functions sharing holomorphic functions and the converse of the bloch principle

In 1996, C. C. Yang and P. C. Hu [8] showed that: Let f be a transcendental meromorphic function on the complex plane, and a = 0 be a complex number; then assume that n ≥ 2, n_1, ···, n_k are nonnegative integers such that n_1+ ··· + n_k ≥1; thus f~n(f′)~(n_1)···(f~(k))~(n_k)-a has infinitely zeros. The aim of this article is to study the value distribution of differential polynomial, which is an extension of the result of Yang and Hu for small function and all zeros of f having multiplicity at least k ≥ 2. Namely, we prove that f~n(f′)~(n_1)···(f~(k))~(n_k)-a(z)has infinitely zeros, where f is a transcendental meromorphic function on the complex plane whose all zeros have multiplicity at least k ≥ 2, and a(z) ≡ 0 is a small function of f and n ≥ 2, n_1, ···, n_k are nonnegative integers satisfying n1+ ··· + n k ≥1. Using it, we establish some normality criterias for a family of meromorphic functions under a condition where differential polynomials generated by the members of the family share a holomorphic function with zero points. The results of this article are supplement of some problems studied by J. Yunbo and G. Zongsheng [6], and extension of some problems studied X. Wu and Y.Xu [10]. The main result of this article also leads to a counterexample to the converse of Bloch's principle.     

Improved FPT algorithms for weighted independent set in bull-free graphs

Very recently, Thomassé et al. (2017) have given an FPT algorithm for Weighted Independent Set in bull-free graphs parameterized by the weight of the solution, running in time $2^{O\left(k^5\right)}?n^9$. In this article we improve this running time to $2^{O\left(k^2\right)}?n^7$. As a byproduct, we also improve the previous Turing-kernel for this problem from $O\left(k^5\right)$ to $O\left(k^2\right)$. Furthermore, for the subclass of bull-free graphs without holes of length at most $2p?1$ for $p\ge 3$, we speed up the running time to $2^{O(k?k^{\frac{1}{p?1}})}?n^7$. As $p$ grows, this running time is asymptotically tight in terms of $k$, since we prove that for each integer $p\ge 3$, Weighted Independent Set cannot be solved in time $2^{o\left(k\right)}?n^{O\left(1\right)}$ in the class of $\{\text{bull},C_4,\dots ,C_{2p?1}\}$-free graphs unless the ETH fails.     

On k-symmetries of hyperbolic spaces

We determine all the possible pointwise k-symmetric spaces of negative constant curvature. In general, such spaces are not k-symmetric.In fact we show that, for all $n?3$, $k\ne 2$, $H^n$ is not k-symmetric, i.e., for any set of selected k-symmetries, one for each point of $H^n$, the regularity condition does not hold.     

Constructing archimedean copulas from diagonal sections

We introduce a family $?$ of functions called diagonal generators. These are convex functions with the properties of diagonal sections of archimedean copulas. We show that to each diagonal generator $f$ there corresponds an archimedean copula $C^f$ with the asymptotic representation $C^f(u_1,u_2)={lim}_{k\to \infty }{f}^k[f^{?k}\left(u_1\right)+f^{?k}\left(u_2\right)?1]$. Moreover, the diagonal section of $C^f$ equals $f$.We characterize archimedean copulas in terms of their asymptotic form. We construct a family ${?}_F$ of diagonal generators, induced by a regular distribution function $F$. We study a differential equation (depending on a function parameter), whose solution is $F$. We give four applications of diagonal generators: to concordance, quadrant dependence, measures of dependence and convergence of copulas.     

Wild theories with o-minimal open core

Let T be a consistent o-minimal theory extending the theory of densely ordered groups and let $T^{\prime }$ be a consistent theory. Then there is a complete theory $T^{?}$ extending T such that T is an open core of $T^{?}$, but every model of $T^{?}$ interprets a model of $T^{\prime }$. If $T^{\prime }$ is NIP, $T^{?}$ can be chosen to be NIP as well. From this we deduce the existence of an NIP expansion of the real field that has no distal expansion.     

Convergence rates and interior estimates in homogenization of higher order elliptic systems

This paper is concerned with the quantitative homogenization of 2m-order elliptic systems with bounded measurable, rapidly oscillating periodic coefficients. We establish the sharp $O(\epsilon )$ convergence rate in $W^{m?1,p_0}$ with $p_0=\frac{2d}{d?1}$ in a bounded Lipschitz domain in ${\mathbb{R}}^d$ as well as the uniform large-scale interior $C^{m?1,1}$ estimate. With additional smoothness assumptions, the uniform interior $C^{m?1,1}$, $W^{m,p}$ and $C^{m?1,\alpha }$ estimates are also obtained. As applications of the regularity estimates, we establish asymptotic expansions for fundamental solutions.