Weak version of restriction estimates for spheres and paraboloids in finite fields

We study $L^p-L^r$ restriction estimates for algebraic varieties in d-dimensional vector spaces over finite fields. Unlike the Euclidean case, if the dimension d is even, then it is conjectured that the $L^{(2d+2)/(d+3)}-L^2$ Stein–Tomas restriction result can be improved to the $L^{(2d+4)/(d+4)}-L^2$ estimate for both spheres and paraboloids in finite fields. In this paper we show that the conjectured $L^p-L^2$ restriction estimate holds in the specific case when test functions under consideration are restricted to d-coordinate functions or homogeneous functions of degree zero. To deduce our result, we use the connection between the restriction phenomena for our varieties in d dimensions and those for homogeneous varieties in $(d+1)$ dimensions.     

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.     

3-Restricted arc connectivity of digraphs

The $k$-restricted arc connectivity of digraphs is a common generalization of the arc connectivity and the restricted arc connectivity. An arc subset $S$ of a strong digraph $D$ is a $k$-restricted arc cut if $D?S$ has a strong component $D^{\prime }$ with order at least $k$ such that $D?V\left(D^{\prime }\right)$ contains a connected subdigraph with order at least $k$. The $k$-restricted arc connectivity ${\lambda }^k\left(D\right)$ of a digraph $D$ is the minimum cardinality over all $k$-restricted arc cuts of $D$.Let $D$ be a strong digraph with order $n\ge 6$ and minimum degree $\delta \left(D\right)$. In this paper, we first show that ${\lambda }^3\left(D\right)$ exists if $\delta \left(D\right)\ge 3$ and, furthermore, ${\lambda }^3\left(D\right)\le {\xi }^3\left(D\right)$ if $\delta \left(D\right)\ge 4$, where ${\xi }^3\left(D\right)$ is the minimum 3-degree of $D$. Next, we prove that ${\lambda }^3\left(D\right)={\xi }^3\left(D\right)$ if $\delta \left(D\right)\ge \frac{n+3}{2}$. Finally, we give examples showing that these results are best possible in some sense.     

Powerful numbers in (1ℓ + qℓ)(2ℓ + qℓ)⋯(nℓ + qℓ)

Let q be a positive integer. Recently, Niu and Liu proved that, if $n\ge \max ?\{q,1198?q\}$, then the product $(1^3+q^3)(2^3+q^3)?(n^3+q^3)$ is not a powerful number. In this note, we prove (1) that, for any odd prime power ? and $n\ge \max ?\{q,11?q\}$, the product $(1^{?}+q^{?})(2^{?}+q^{?})?(n^{?}+q^{?})$ is not a powerful number, and (2) that, for any positive odd integer ?, there exists an integer $N_{q,?}$ such that, for any positive integer $n\ge N_{q,?}$, the product $(1^{?}+q^{?})(2^{?}+q^{?})?(n^{?}+q^{?})$ is not a powerful number.     

The minimal measurement number for low-rank matrix recovery

The paper presents several results that address a fundamental question in low-rank matrix recovery: how many measurements are needed to recover low-rank matrices? We begin by investigating the complex matrices case and show that $4nr?4r^2$ generic measurements are both necessary and sufficient for the recovery of rank-r matrices in ${\mathbb{C}}^{n\times n}$. Thus, we confirm a conjecture which is raised by Eldar, Needell and Plan for the complex case. We next consider the real case and prove that the bound $4nr?4r^2$ is tight provided $n=2^k+r,k\in {\mathbb{Z}}_{+}$. Motivated by Vinzant's work [19], we construct 11 matrices in ${\mathbb{R}}^{4\times 4}$ by computer random search and prove they define injective measurements on rank-1 matrices in ${\mathbb{R}}^{4\times 4}$. This disproves the conjecture raised by Eldar, Needell and Plan for the real case. Finally, we use the results in this paper to investigate the phase retrieval by projection and show fewer than $2n?1$ orthogonal projections are possible for the recovery of $x\in {\mathbb{R}}^n$ from the norm of them, which gives a negative answer for a question raised in [1].     

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.     

A boundary Schwarz lemma for pluriharmonic mappings from the unit polydisk to the unit ball

In this article, we present a Schwarz lemma at the boundary for pluriharmonic mappings from the unit polydisk to the unit ball, which generalizes classical Schwarz lemma for bounded harmonic functions to higher dimensions. It is proved that if the pluriharmonic mapping f ∈ P(D~n, B~N) is C~(1+α) at z0 ∈ E_rD~n with f(0) = 0 and f(z_0) = ω_0∈B~N for any n,N ≥ 1, then there exist a nonnegative vector λ_f =(λ_1,0,…,λ_r,0,…,0)~T∈R~(2 n)satisfying λ_i≥1/(2~(2 n-1)) for 1 ≤ i ≤ r such that where z'_0 and w'_0 are real versions of z_0 and w_0, respectively.     

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.