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.     

On a characterization of path connected topological fields

The aim of this paper is to give a characterization of path connected topological fields, inspired by the classical Gelfand correspondence between a compact Hausdorff topological space X and the space of maximal ideals of the ring of real valued continuous functions $C(X,\mathbb{R})$. More explicitly, our motivation is the following question: What is the essential property of the topological field $F=\mathbb{R}$ that makes such a correspondence valid for all compact Hausdorff spaces? It turns out that such a perfect correspondence exists if and only if F is a path connected topological field.     

Type and conductor of simplicial affine semigroups

We provide a generalization of pseudo-Frobenius numbers of numerical semigroups to the context of the simplicial affine semigroups. In this way, we characterize the Cohen-Macaulay type of the simplicial affine semigroup ring $\mathbb{K}[S]$. We define the type of S, $\mathrm{type}(S)$, in terms of some Apéry sets of S and show that it coincides with the Cohen-Macaulay type of the semigroup ring, when $\mathbb{K}[S]$ is Cohen-Macaulay. If $\mathbb{K}[S]$ is a d-dimensional Cohen-Macaulay ring of embedding dimension at most $d+2$, then $\mathrm{type}(S)\le 2$. Otherwise, $\mathrm{type}(S)$ might be arbitrary large and it has no upper bound in terms of the embedding dimension. Finally, we present a generating set for the conductor of S as an ideal of its normalization.     

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.     

On the universal completion of pointfree function spaces

This paper approaches the construction of the universal completion of the Riesz space $\mathrm{C}(L)$ of continuous real functions on a completely regular frame L in two different ways. Firstly as the space of continuous real functions on the Booleanization of L. Secondly as the space of nearly finite Hausdorff continuous functions on L. The former has no counterpart in the classical theory, as the Booleanization of a spatial frame is not spatial in general, and it offers a lucid way of representing the universal completion as a space of continuous real functions. As a corollary we obtain that $\mathrm{C}(L)$ and $\mathrm{C}(M)$ have isomorphic universal completions if and only if the Booleanization of L and M are isomorphic and we characterize frames L such that $\mathrm{C}(L)$ is universally complete as almost Boolean frames. The application of this last result to the classical case $\mathrm{C}(X)$ of the space of continuous real functions on a topological space X characterizes those spaces X for which $\mathrm{C}(X)$ is universally complete. Finally, we present a pointfree version of the Maeda-Ogasawara-Vulikh representation theorem and use it to represent the universal completion of an Archimedean Riesz space with weak unit as a space of continuous real functions on a Boolean frame.     

Irreducible polynomials from a cubic transformation

Let $R(x)=g(x)/h(x)$ be a rational expression of degree three over the finite field ${\mathbb{F}}_q$. We count the irreducible polynomials in ${\mathbb{F}}_q[x]$, of a given degree, that have the form $h{(x)}^{\mathrm{deg}f}\cdot f(R(x))$ for some $f(x)\in {\mathbb{F}}_q[x]$. As an example of application of our results, we recover the number of irreducible transformation shift registers of order three, which were computed by Jiang and Yang in 2017.     

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.     

Infinitely many nodal solutions with a prescribed number of nodes for the Kirchhoff type equations

In this paper, we investigate the existence of multiple radial sign-changing solutions with the nodal characterization for a class of Kirchhoff type problems$\{\begin{array}{cc}\hfill & ?(a+b|\mathrm{?}u{|}_{L^2}^2)\mathrm{\Delta }u+V(|x|)u=K(|x|)f(u)\text{in}{\mathbb{R}}^N,\hfill \\ \hfill & u\in H^1({\mathbb{R}}^N),\hfill \end{array}$ where $N=1,2,3,a,b>0$, $V,K$ are radial and bounded away from below by positive numbers. Under some weak assumptions on $f\in C^0(\mathbb{R};\mathbb{R})$, by taking advantage of the Gersgorin disc's theorem and Miranda theorem, we develop some new analytic techniques and prove that this problem admits infinitely many nodal solutions $\{U_k^b\}$ having a prescribed number of nodes k, whose energy is strictly increasing in k. Moreover, the asymptotic behaviors of $U_k^b$ as $b\to 0_{+}$ are established. These results improve and generalize the previous results in the literature.     

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 $|u{|}^{q?2}u$.