Blow-up behavior of ground states for a nonlinear Schrödinger system with attractive and repulsive interactions

We consider a nonlinear Schrödinger system arising in a two-component Bose–Einstein condensate (BEC) with attractive intraspecies interactions and repulsive interspecies interactions in ${\mathbb{R}}^2$. We get ground states of this system by solving a constrained minimization problem. For some kinds of trapping potentials, we prove that the minimization problem has a minimizer if and only if the attractive interaction strength $a_i(i=1,2)$ of each component of the BEC system is strictly less than a threshold $a^{?}$. Furthermore, as $(a_1,a_2)\nearrow (a^{?},a^{?})$, the asymptotical behavior for the minimizers of the minimization problem is discussed. Our results show that each component of the BEC system concentrates at a global minimum of the associated trapping potential.     

Efficient exponential-time algorithms for edit distance between unordered trees

The edit distance problem for rooted unordered trees is known to be NP-hard. Based on this fact, this paper studies exponential-time algorithms for the problem. For a general case, an $O(\min ({1.26}^{n_1+n_2},2^{b_1+b_2}\cdot \text{<mi mathvariant="italic" is="true">poly</mi>}(n_1,n_2)))$ time algorithm is presented, where $n_1$ and $n_2$ are the numbers of nodes and $b_1$ and $b_2$ are the numbers of branching nodes in two input trees. This algorithm is obtained by a combination of dynamic programming, exhaustive search, and maximum weighted bipartite matching. For bounded degree trees over a fixed alphabet, it is shown that the problem can be solved in $O({(1+ϵ)}^{n_1+n_2})$ time for any fixed $ϵ>0$. This result is achieved by avoiding duplicate calculations for identical subsets of small subtrees.     

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 $s<1/2$.     

Pointwise asymptotic behavior of modulated periodic reaction–diffusion waves

By working with the periodic resolvent kernel and the Bloch-decomposition, we establish pointwise bounds for the Green function of the linearized equation associated with spatially periodic traveling waves of a system of reaction–diffusion equations. With our linearized estimates together with a nonlinear iteration scheme developed by Johnson–Zumbrun, we obtain $L^p$-behavior ($p?1$) of a nonlinear solution to a perturbation equation of a reaction–diffusion equation with respect to initial data in $L^1\cap H^2$ recovering and slightly sharpening results obtained by Schneider using weighted energy and renormalization techniques. We obtain also pointwise nonlinear estimates with respect to two different initial perturbations $|u_0|?E_0{e}^{?{|x|}^2/M}$, ${|u_0|}_{H^2}?E_0$ and $|u_0|?E_0{(1+|x|)}^{?r}$, $r>2$, ${|u_0|}_{H^2}?E_0$ respectively, $E_0>0$ sufficiently small and $M>1$ sufficiently large, showing that behavior is that of a heat kernel. These pointwise bounds have not been obtained elsewhere, and do not appear to be accessible by previous techniques.     

A ratio-dependent predator–prey model with diffusion

This paper deals with the existence and nonexistence of nonconstant positive steady-state solutions to a ratio-dependent predator–prey model with diffusion and with the homogeneous Neumann boundary condition. We demonstrate that there exists $a_0(b)$ satisfying $0<a_0(b)<m_1$ for $0<b<m_1$, such that if $0<b<m_1$ and $a_0(b)<a<m_1$, then the diffusion can create nonconstant positive steady-state solutions; whereas the diffusion cannot do provided $a>m_1$.     

On Birman's sequence of Hardy–Rellich-type inequalities

In 1961, Birman proved a sequence of inequalities $\{I_n\}$, for $n\in \mathbb{N}$, valid for functions in $C_0^n((0,\infty ))?L^2((0,\infty ))$. In particular, $I_1$ is the classical (integral) Hardy inequality and $I_2$ is the well-known Rellich inequality. In this paper, we give a proof of this sequence of inequalities valid on a certain Hilbert space $H_n([0,\infty ))$ of functions defined on $[0,\infty )$. Moreover, $f\in H_n([0,\infty ))$ implies $f^{\prime }\in H_{n?1}([0,\infty ))$; as a consequence of this inclusion, we see that the classical Hardy inequality implies each of the inequalities in Birman's sequence. We also show that for any finite $b>0$, these inequalities hold on the standard Sobolev space $H_0^n((0,b))$. Furthermore, in all cases, the Birman constants ${[(2n?1)!!]}^2/2^{2n}$ in these inequalities are sharp and the only function that gives equality in any of these inequalities is the trivial function in $L^2((0,\infty ))$ (resp., $L^2((0,b))$). We also show that these Birman constants are related to the norm of a generalized continuous Cesàro averaging operator whose spectral properties we determine in detail.     

Properties of Beurling-type submodules via Agler decompositions

In this paper, we study operator-theoretic properties of the compressed shift operators $S_{z_1}$ and $S_{z_2}$ on complements of submodules of the Hardy space over the bidisk $H^2({\mathbb{D}}^2)$. Specifically, we study Beurling-type submodules – namely submodules of the form $\theta H^2({\mathbb{D}}^2)$ for θ inner – using properties of Agler decompositions of θ to deduce properties of $S_{z_1}$ and $S_{z_2}$ on model spaces $H^2({\mathbb{D}}^2)?\theta H^2({\mathbb{D}}^2)$. Results include characterizations (in terms of θ) of when a commutator $[S_{z_j}^{?},S_{z_j}]$ has rank n and when subspaces associated to Agler decompositions are reducing for $S_{z_1}$ and $S_{z_2}$. We include several open questions.     

Global solvability and global hypoellipticity in Gevrey classes for vector fields on the torus

Let $L=?/?t+{\sum }_{j=1}^N(a_j+i{b}_j)(t)?/?x_j$ be a vector field defined on the torus ${\mathbb{T}}^{N+1}?{\mathbb{R}}^{N+1}/2\pi {\mathbb{Z}}^{N+1}$, where $a_j$, $b_j$ are real-valued functions and belonging to the Gevrey class $G^s({\mathbb{T}}^1)$, $s>1$, for $j=1,\dots ,N$. We present a complete characterization for the s-global solvability and s-global hypoellipticity of L. Our results are linked to Diophantine properties of the coefficients and, also, connectedness of certain sublevel sets.