SL(m,C)-equivariant and translation covariant continuous tensor valuations

The space of continuous, $\mathrm{SL}(m,\mathbb{C})$-equivariant, $m\ge 2$, and translation covariant valuations taking values in the space of real symmetric tensors on ${\mathbb{C}}^m?{\mathbb{R}}^{2m}$ of rank $r\ge 0$ is completely described. The classification involves the moment tensor valuation for $r\ge 1$ and is analogous to the known classification of the corresponding tensor valuations that are $\mathrm{SL}(2m,\mathbb{R})$-equivariant, although the method of proof cannot be adapted.     

Virtual resolutions of points in P1×P1

We explore explicit virtual resolutions, as introduced by Berkesch, Erman, and Smith, for ideals of finite sets of points in ${\mathbb{P}}^1\times {\mathbb{P}}^1$. Specifically, we describe a virtual resolution for a sufficiently general set of points X in ${\mathbb{P}}^1\times {\mathbb{P}}^1$ that only depends on $|X|$. We also improve an existence result of Berkesch, Erman, and Smith in the special case of points in ${\mathbb{P}}^1\times {\mathbb{P}}^1$; more precisely, we give an effective bound for their construction that gives a virtual resolution of length two for any set of points in ${\mathbb{P}}^1\times {\mathbb{P}}^1$.     

Non-integrated defect relation for meromorphic mappings on Kähler manifolds intersecting non-subdegenerate families of hyperplanes

Let M be a complete Kähler manifold, whose universal covering is biholomorphic to a ball ${\mathbb{B}}^m(R_0)$ in ${\mathbb{C}}^m$ ($0<R_0\le +\infty$). Our purpose of this article is to establish a non-integrated defect relation with truncated level for a meromorphic mapping on M intersecting a family of hyperplanes in ${\mathbb{P}}^n(\mathbb{C})$ which is non-subdegenerate with respect to the mapping.     

Time-frequency analysis on the adeles over the rationals

We show that the construction of Gabor frames in $L^2(\mathbb{R})$ with generators in ${\mathbf{S}}_0(\mathbb{R})$ and with respect to time-frequency shifts from a rectangular lattice $\alpha \mathbb{Z}\times \beta \mathbb{Z}$ is equivalent to the construction of certain Gabor frames for $L^2$ over the adeles over the rationals and the group $\mathbb{R}\times {\mathbb{Q}}_p$. Furthermore, we detail the connection between the construction of Gabor frames on the adeles and on $\mathbb{R}\times {\mathbb{Q}}_p$ with the construction of certain Heisenberg modules.     

Vector-valued nonstationary Gabor frames

As an extension of Gabor frames, nonstationary Gabor (NSG) frames were recently introduced in adaptive signal analysis. They allow for efficient reconstruction with flexible sampling and varying window functions. In this paper we generalize the notion of NSG frames from $L^2(\mathbb{R})$ to the vector-valued Hilbert space $L^2(\mathbb{R},{\mathbb{C}}^L)$, and investigate the resulting vector-valued NSG frames. We derive a Walnut's representation of the mixed frame operator, and provide some necessary/sufficient conditions for a vector-valued NSG system to be a frame for $L^2(\mathbb{R},{\mathbb{C}}^L)$. Furthermore, we show the existence of painless vector-valued NSG frames, and of vector-valued NSG frames with fast decaying window functions.     

The Orlicz inequality for multilinear forms

The Orlicz $({?}_2,{?}_1)$-mixed inequality states that${\left(\sum _{j_1=1}^n{\left(\sum _{j_2=1}^n|A(e_{j_1},e_{j_2})|\right)}^2\right)}^{\frac{1}{2}}\le \sqrt{2}6A6$ for all bilinear forms $A:{\mathbb{K}}^n\times {\mathbb{K}}^n\to \mathbb{K}$ and all positive integers n, where ${\mathbb{K}}^n$ denotes ${\mathbb{R}}^n$ or ${\mathbb{C}}^n$ endowed with the supremum norm. In this paper we extend this inequality to multilinear forms, with ${\mathbb{K}}^n$ endowed with ${?}_p$ norms for all $p\in [1,\infty ]$.     

Long-time behavior for 3D MHD equations with nonlinear damping

The existence of global attractors is proved for the MHD equations with damping terms $|u{|}^{\alpha ?1}u$ and $|B{|}^{\beta ?1}B$ $(\alpha ,\beta ?1)$ on a bounded domain $\mathrm{\Omega }?{\mathbb{R}}^3$. First we establish the well-posedness of strong solutions. Then, the continuity of the corresponding semigroup is verified under the assumption $\alpha ,\beta <5$, which is guided by Gagliardo-Nirenberg inequality. Finally, the system is shown to possess an $(\mathbb{V},\mathbb{V})$-global attractor and an $(\mathbb{V},{\mathbf{\text{H}}}^2)$-global attractor.     

Galois LCD codes over mixed alphabets

In this work we give a characterization of Galois Linear Complementary Dual codes and Galois-invariant codes over mixed alphabets of finite chain rings, which leads to the study of the Gray image of ${\mathbb{F}}_p{\mathbb{F}}_p[\theta ]$-linear codes, where $p\in \{2;3\}$ and $\theta \ne {\theta }^2=0$ that provides LCD codes over ${\mathbb{F}}_p$.     

Positive solutions for a class of singular quasilinear Schrödinger equations with critical Sobolev exponent

We prove the existence of positive solutions of the following singular quasilinear Schrödinger equations at critical growth

$?\mathrm{\Delta }u?\lambda c(x)u?\kappa \alpha (\mathrm{\Delta }(|u{|}^{2\alpha }))|u{|}^{2\alpha ?2}u=|u{|}^{q?2}u+|u{|}^{2^{?}?2}u,u\in D^{1,2}({\mathbb{R}}^N),$

via variational methods, where $\lambda \ge 0$, $c:{\mathbb{R}}^N\to {\mathbb{R}}^{+}$, $\kappa >0$, $0<\alpha <1/2$, $2<q<2^{?}$. It is interesting that we do not need to add a weight function to control $|u{|}^{q?2}u$.     

Classification of cyclic codes over a non-Galois chain ring Zp[u]/〈u3〉

We explicitly determine generators of cyclic codes over a non-Galois finite chain ring ${\mathbb{Z}}_p[u]/〈u^3〉$ of length $p^k$, where p is a prime number and k is a positive integer. We completely classify that there are three types of principal ideals of ${\mathbb{Z}}_p[u]/〈u^3〉$ and four types of non-principal ideals of ${\mathbb{Z}}_p[u]/〈u^3〉$, which are associated with cyclic codes over ${\mathbb{Z}}_p[u]/〈u^3〉$ of length $p^k$. We then obtain a mass formula for cyclic codes over ${\mathbb{Z}}_p[u]/〈u^3〉$ of length $p^k$.     

Simply connected open 3-manifolds with slow decay of positive scalar curvature

The goal of this paper is to investigate the topological structure of open simply connected 3-manifolds whose scalar curvature has a slow decay at infinity. In particular, we show that the Whitehead manifold does not admit a complete metric whose scalar curvature decays slowly, and in fact that any contractible complete 3-manifolds with such a metric is diffeomorphic to ${\mathbb{R}}^3$. Furthermore, using this result, we prove that any open simply connected 3-manifold M with ${\pi }_2(M)=\mathbb{Z}$ and a complete metric as above is diffeomorphic to ${\mathbb{S}}^2\times \mathbb{R}$.