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.     

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.     

Local kinetic energy and singularities of the incompressible Navier–Stokes equations

We study the partial regularity problem of the incompressible Navier–Stokes equations. A reverse Hölder inequality of velocity gradient with increasing support is obtained under the condition that a scaled functional corresponding the local kinetic energy is uniformly bounded. As an application, we give a new bound for the Hausdorff dimension and the Minkowski dimension of singular set when weak solutions v belong to $L^{\infty }(0,T;L^{3,w}({\mathbb{R}}^3))$ where $L^{3,w}({\mathbb{R}}^3)$ denotes the standard weak Lebesgue space.     

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 $s<1/2$.     

Construction of solutions via local Pohozaev identities

This paper deals with the following nonlinear elliptic equation

$?\mathrm{\Delta }u+V(|y^{\prime }|,y^{″})u=u^{\frac{N+2}{N?2}},u>0,u\in H^1({\mathbb{R}}^N),$

where $(y^{\prime },y^{″})\in {\mathbb{R}}^2\times {\mathbb{R}}^{N?2}$, $V(|y^{\prime }|,y^{″})$ is a bounded non-negative function in ${\mathbb{R}}^{+}\times {\mathbb{R}}^{N?2}$. By combining a finite reduction argument and local Pohozaev type of identities, we prove that if $N\ge 5$ and $r^{2}V(r,y^{″})$ has a stable critical point $(r_0,y_0^{″})$ with $r_0>0$ and $V(r_0,y_0^{″})>0$, then the above problem has infinitely many solutions. This paper overcomes the difficulty appearing in using the standard reduction method to locate the concentrating points of the solutions.     

The visible perimeter of an arrangement of disks

Given a collection of n opaque unit disks in the plane, we want to find a stacking order for them that maximizes their visible perimeter, the total length of all pieces of their boundaries visible from above. We prove that if the centers of the disks form a dense point set, i.e., the ratio of their maximum to their minimum distance is $O(n^{1/2})$, then there is a stacking order for which the visible perimeter is $\Omega (n^{2/3})$. We also show that this bound cannot be improved in the case of a sufficiently small $n^{1/2}\times n^{1/2}$ uniform grid. On the other hand, if the set of centers is dense and the maximum distance between them is small, then the visible perimeter is $O(n^{3/4})$ with respect to any stacking order. This latter bound cannot be improved either.Finally, we address the case where no more than c disks can have a point in common.These results partially answer some questions of Cabello, Haverkort, van Kreveld, and Speckmann.     

Higher genus Kashiwara–Vergne problems and the Goldman–Turaev Lie bialgebra

We define a family ${\mathrm{KV}}^{(g,n+1)}$ of Kashiwara–Vergne problems associated with compact connected oriented 2-manifolds of genus g with $n+1$ boundary components. The problem ${\mathrm{KV}}^{(0,3)}$ is the classical Kashiwara–Vergne problem from Lie theory. We show the existence of solutions to ${\mathrm{KV}}^{(g,n+1)}$ for arbitrary g and n. The key point is the solution to ${\mathrm{KV}}^{(1,1)}$ based on the results by B. Enriquez on elliptic associators. Our construction is motivated by applications to the formality problem for the Goldman–Turaev Lie bialgebra ${\mathfrak{g}}^{(g,n+1)}$. In more detail, we show that every solution to ${\mathrm{KV}}^{(g,n+1)}$ induces a Lie bialgebra isomorphism between ${\mathfrak{g}}^{(g,n+1)}$ and its associated graded $\mathrm{gr}{\mathfrak{g}}^{(g,n+1)}$. For $g=0$, a similar result was obtained by G. Massuyeau using the Kontsevich integral. For $g\ge 1$, $n=0$, our results imply that the obstruction to surjectivity of the Johnson homomorphism provided by the Turaev cobracket is equivalent to the Enomoto–Satoh obstruction.     

A Schauder and Riesz basis criterion for non-self-adjoint Schrödinger operators with periodic and antiperiodic boundary conditions

Under the assumption that $V\in L^2([0,\pi ];dx)$, we derive necessary and sufficient conditions in terms of spectral data for (non-self-adjoint) Schrödinger operators $?d^2/d{x}^2+V$ in $L^2([0,\pi ];dx)$ with periodic and antiperiodic boundary conditions to possess a Riesz basis of root vectors (i.e., eigenvectors and generalized eigenvectors spanning the range of the Riesz projection associated with the corresponding periodic and antiperiodic eigenvalues).We also discuss the case of a Schauder basis for periodic and antiperiodic Schrödinger operators $?d^2/d{x}^2+V$ in $L^p([0,\pi ];dx)$, $p\in (1,\infty )$.     

Minimal homogeneous Steiner 2-(v,3) trades

A Steiner 2-$(v,3)$ trade is a pair $(T_1,T_2)$ of disjoint partial Steiner triple systems, each on the same set of $v$ points, such that each pair of points occurs in $T_1$ if and only if it occurs in $T_2$. A Steiner 2-$(v,3)$ trade is called d-homogeneous if each point occurs in exactly d blocks of $T_1$ (or $T_2$). In this paper we construct minimal d-homogeneous Steiner 2-$(v,3)$ trades of foundation $v$ and volume $\mathit{dv}/3$ for sufficiently large values of $v$. (Specifically, $v>3(1.75d^2+3)$ if $v$ is divisible by 3 and $v>d(4^{d/3+1}+1)$ otherwise.)