Global well-posedness for the 3D magneto-micropolar equations with fractional dissipation

This paper focus on the Cauchy problem of the 3D incompressible magneto-micropolar equations with fractional dissipation in the Sobolev space. Liu, Sun and Xin obtained the global solutions to the 3D magneto-micropolar equations with $\alpha=\beta=\gamma=\frac{5}{4}$. Deng and Shang established the global well-posedness of the 3D magneto-micropolar equations in the case of $\alpha\geq\frac{5}{4}$, $\alpha+\beta\geq\frac{5}{2}$ and $\gamma\geq2-\alpha\geq\frac{3}{4}$. In this paper, we establish the global well-posedness of the 3D magneto-micropolar equations with $\alpha=\beta=\frac{5}{4}$ and $\gamma=\frac{1}{2}$, which improves the results of Liu-Sun-Xin and Deng-Shang by reducing the value of $\gamma$ to $\frac{1}{2}$.

Constructing uncountably many groups with the same profinite completion

Two constructions are described: one gives soluble groups of derived length 4, the other uses groups acting on a rooted tree.

Generating wreath products of symmetric and alternating groups

We show that the wreath product of two finite symmetric or alternating groups is 2-generated.

The norming set of a symmetric 3-linear form on the plane with the $l_1$-norm

An element $(x_1, \ldots, x_n)\in E^n$ is called a {\em norming point} of $T\in {\mathcal L}(^n E)$ if $\|x_1\|=\cdots=\|x_n\|=1$ and$|T(x_1, \ldots, x_n)|=\|T\|,$ where ${\mathcal L}(^n E)$ denotes the space of all continuous $n$-linear forms on $E.$For $T\in {\mathcal L}(^n E),$ we define $${Norm}(T)=\Big\{(x_1, \ldots, x_n)\in E^n: (x_1, \ldots, x_n)~\mbox{is a norming point of}~T\Big\}.$$${Norm}(T)$ is called the {\em norming set} of $T$. We classify ${Norm}(T)$ for every $T\in {\mathcal L}_s(^3 l_{1}^2)$.

Invariance and normality of projections in the dual of Banach algebras

We study the classes of invariant and natural projections in the dual of a Banach algebra $A$. These type of projections are relevant by their connections with the existence problem of bounded approximate identities in closed ideals of Banach algebras. It is known that any invariant projection is a natural projection. In this article we consider the issue of when a natural projection is an invariant projection.

A reciprocal relation for Hermite polynomials

For $x\in \mathbb{R}$, the ordinary Hermite polynomial $H_k(x)$ can be written\begin{eqnarray*}\displaystyleH_k(x)= \mathbb{E} \left[ (x + {\rm i} N)^k \right] =\sum_{j=0}^k {k\choose j} x^{k-j} {\rm i}^j \mathbb{E} \left[ N^j \right],\end{eqnarray*}where ${\rm i} = \sqrt{-1}$ and $N$ is a unit normal random variable. We prove the reciprocal relation\begin{eqnarray*}\displaystylex^k=\sum_{j=0}^k {k\choose j} H_{k-j}(x)\ \mathbb{E} \left[ N^j \right].\end{eqnarray*}A similar result is given for the multivariate Hermite polynomial.

Local X-ray Transform on Asymptotically Hyperbolic Manifolds via Projective Compactification

We prove local injectivity near a boundary point for the geodesic X-ray transform for an asymptotically hyperbolic metric even mod $O(\rho^5)$ in dimensions three and higher.

Embedding Heegaard Decompositions

A smooth embedding of a closed $3$-manifold $M$ in $\mathbb{R}^4$ may generically be composed with projection to the fourth coordinate to determine a Morse function on $M$ and hence a Heegaard splitting $M=X\cup_\Sigma Y$. However, starting with a Heegaard splitting, we find an obstruction coming from the geometry of the curve complex $C(\Sigma)$ to realizing a corresponding embedding $M\hookrightarrow \mathbb{R}^4$.

Global aspects of measure preserving equivalence relations and graphs

This paper is an introduction and survey of a “global” theory of measure preserving equivalence relations and graphs. In this theory one views a measure preserving equivalence relation or graph as a point in an appropriate topological space and then studies the properties of this space from a topological, descriptive set theoretic and dynamical point of view.

Neumann problem for Monge-Ampere type equations revisited.

This paper concerns a priori second order derivative estimates of solutions of the Neumann problem for the Monge-Amp\`ere type equations in bounded domains in n dimensional Euclidean space. We first establish a double normal second order derivative estimate on the boundary under an appropriate notion of domain convexity. Then, assuming a barrier condition for the linearized operator, we provide a complete proof of the global second derivative estimate for elliptic solutions, as previously studied in our earlier work. We also consider extensions to the degenerate elliptic case, in both the regular and strictly regular matrix cases.