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 $s<1/2$.     

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 $L^p([0,\pi ];dx)$, $p\in (1,\infty )$.     

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.     

On the Yamabe constants of S2×R3 and S3×R2

We compare the isoperimetric profiles of $S^2\times {\mathbb{R}}^3$ and of $S^3\times {\mathbb{R}}^2$ with that of a round 5-sphere (of appropriate radius). Then we use this comparison to obtain lower bounds for the Yamabe constants of $S^2\times {\mathbb{R}}^3$ and $S^3\times {\mathbb{R}}^2$. Explicitly we show that $Y(S^3\times {\mathbb{R}}^2,[g_0^3+d{x}^2])>(3/4)Y(S^5)$ and $Y(S^2\times {\mathbb{R}}^3,[g_0^2+d{x}^2])>0.63Y(S^5)$. We also obtain explicit lower bounds in higher dimensions and for products of Euclidean space with a closed manifold of positive Ricci curvature. The techniques are a more general version of those used by the same authors in Petean and Ruiz (2011) [15] and the results are a complement to the work developed by B. Ammann, M. Dahl and E. Humbert to obtain explicit gap theorems for the Yamabe invariants in low dimensions.     

On the positive radial solutions of a class of singular semilinear elliptic equations

In this paper, we consider the following elliptic equation(0.1)$\mathit{div}\left(A\right(|x|\left)\mathrm{?}u\right)+B(|x|)u^p=0\text{in}{\mathbb{R}}^n,$ where $p>1$, $n?3$, $A(|x|)>0$ is differentiable in ${\mathbb{R}}^n?\{0\}$ and $B(|x|)$ is a given nonnegative Hölder continuous function in ${\mathbb{R}}^n?\{0\}$. The asymptotic behavior at infinity and structure of separation property of positive radial solutions with different initial data for (0.1) are discussed. Moreover, the existence and separation property of infinitely many positive solutions for Hardy equation and an equation related to Caffarelli–Kohn–Nirenberg inequality are obtained respectively, as special cases.     

Homogenization of a Dirichlet semilinear elliptic problem with a strong singularity at u = 0 in a domain with many small holes

In the present paper we perform the homogenization of the semilinear elliptic problem

$\{\begin{array}{cc}{u}^{\epsilon }\ge 0\hfill & \text{in}{\mathrm{\Omega }}^{\epsilon },\hfill \\ ?divA(x)D{u}^{\epsilon }=F(x,u^{\epsilon })\hfill & \text{in}{\mathrm{\Omega }}^{\epsilon },\hfill \\ u^{\epsilon }=0\hfill & \text{on}?{\mathrm{\Omega }}^{\epsilon }.\hfill \end{array}$

In this problem $F(x,s)$ is a Carathéodory function such that $0\le F(x,s)\le h(x)/\mathrm{\Gamma }(s)$ a.e. $x\in \mathrm{\Omega }$ for every $s>0$, with h in some $L^r(\mathrm{\Omega })$ and Γ a $C^1([0,+\infty [)$ function such that $\mathrm{\Gamma }(0)=0$ and ${\mathrm{\Gamma }}^{\prime }(s)>0$ for every $s>0$. On the other hand the open sets ${\mathrm{\Omega }}^{\epsilon }$ are obtained by removing many small holes from a fixed open set Ω in such a way that a “strange term” $\mu u^0$ appears in the limit equation in the case where the function $F(x,s)$ depends only on x.We already treated this problem in the case of a “mild singularity”, namely in the case where the function $F(x,s)$ satisfies $0\le F(x,s)\le h(x)(\frac{1}{s}+1)$. In this case the solution $u^{\epsilon }$ to the problem belongs to $H_0^1({\mathrm{\Omega }}^{\epsilon })$ and its definition is a “natural” and rather usual one.In the general case where $F(x,s)$ exhibits a “strong singularity” at $u=0$, which is the purpose of the present paper, the solution $u^{\epsilon }$ to the problem only belongs to $H_{\text{loc}}^1({\mathrm{\Omega }}^{\epsilon })$ but in general does not belong to $H_0^1({\mathrm{\Omega }}^{\epsilon })$ anymore, even if $u^{\epsilon }$ vanishes on $?{\mathrm{\Omega }}^{\epsilon }$ in some sense. Therefore we introduced a new notion of solution (in the spirit of the solutions defined by transposition) for problems with a strong singularity. This definition allowed us to obtain existence, stability and uniqueness results.In the present paper, using this definition, we perform the homogenization of the above semilinear problem and we prove that in the homogenized problem, the “strange term” $\mu u^0$ still appears in the left-hand side while the source term $F(x,u^0)$ is not modified in the right-hand side.     

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.     

Regularity of random attractors for fractional stochastic reaction–diffusion equations on Rn

We investigate the regularity of random attractors for the non-autonomous non-local fractional stochastic reaction–diffusion equations in $H^s({\mathbb{R}}^n)$ with $s\in (0,1)$. We prove the existence and uniqueness of the tempered random attractor that is compact in $H^s({\mathbb{R}}^n)$ and attracts all tempered random subsets of $L^2({\mathbb{R}}^n)$ with respect to the norm of $H^s({\mathbb{R}}^n)$. The main difficulty is to show the pullback asymptotic compactness of solutions in $H^s({\mathbb{R}}^n)$ due to the noncompactness of Sobolev embeddings on unbounded domains and the almost sure nondifferentiability of the sample paths of the Wiener process. We establish such compactness by the ideas of uniform tail-estimates and the spectral decomposition of solutions in bounded domains.     

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.     

Large deviations of kernel density estimator in L1(Rd) for uniformly ergodic Markov processes

In this paper, we consider a uniformly ergodic Markov process $(X_n{)}_{n⩾0}$ valued in a measurable subset E of ${\mathbb{R}}^d$ with the unique invariant measure $\mu (\mathrm{d}x)=f(x)\mathrm{d}x$, where the density f is unknown. We establish the large deviation estimations for the nonparametric kernel density estimator $f_n^{*}$ in $L^1({\mathbb{R}}^d,\mathrm{d}x)$ and for $6f_n^{*}-f{6}_{L^1({\mathbb{R}}^d,\mathrm{d}x)}$, and the asymptotic optimality $f_n^{*}$ in the Bahadur sense. These generalize the known results in the i.i.d. case.