On the Bures–Wasserstein distance between positive definite matrices

The metric $d(A,B)={\left[\text{tr}A+\text{tr}B?2\text{tr}{\left(A^{1∕2}B{A}^{1∕2}\right)}^{1∕2}\right]}^{1∕2}$ on the manifold of $n\times n$ positive definite matrices arises in various optimisation problems, in quantum information and in the theory of optimal transport. It is also related to Riemannian geometry. In the first part of this paper we study this metric from the perspective of matrix analysis, simplifying and unifying various proofs. Then we develop a theory of a mean of two, and a barycentre of several, positive definite matrices with respect to this metric. We explain some recent work on a fixed point iteration for computing this Wasserstein barycentre. Our emphasis is on ideas natural to matrix analysis.     

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 ${\mathbb{P}}^1\times {\mathbb{P}}^1$.     

Panchromatic 3-colorings of random hypergraphs

The paper deals with panchromatic 3-colorings of random hypergraphs. A vertex 3-coloring is said to be panchromatic for a hypergraph if every color can be found on every edge. Let $H(n,k,p)$ denote the binomial model of a random $k$-uniform hypergraph on $n$ vertices. For given fixed $c>0$, $k⩾3$ and $p=cn∕\left(\genfrac{}{}{0ex}{}{n}{k}\right)$, we prove that if $c<\frac{ln3}{3}\cdot {\left(\frac{3}{2}\right)}^k-\frac{ln3}{2}-O\left({\left(\frac{\sqrt{3}}{2}\right)}^k\right)$then $H(n,k,p)$ admits a panchromatic 3-coloring with probability tending to 1 as $n\to \infty$, but if $k$ is large enough and $c>\frac{ln3}{3}\cdot {\left(\frac{3}{2}\right)}^k-\frac{ln3}{2}+O\left({\left(\frac{3}{4}\right)}^k\right)$then $H(n,k,p)$ does not admit a panchromatic 3-coloring with probability tending to 1 as $n\to \infty$.     

The dual code of any (δ+αu2)-constacyclic code over F2m[u]∕〈u4〉 of oddly even length

Let ${\mathbb{F}}_{2^m}$ be a finite field of cardinality $2^m$, $R={\mathbb{F}}_{2^m}\left[u\right]∕〈u^4〉={\mathbb{F}}_{2^m}+u{\mathbb{F}}_{2^m}+u^2{\mathbb{F}}_{2^m}+u^3{\mathbb{F}}_{2^m}$ $(u^4=0)$ which is a finite chain ring, and $n$ is an odd positive integer. For any $\delta ,\alpha \in {\mathbb{F}}_{2^m}^{\times }$, an explicit representation for the dual code of any $(\delta +\alpha u^2)$-constacyclic code over $R$ of length $2n$ is given. And some dual codes of $(1+u^2)$-constacyclic codes over $R$ of length 14 are constructed. For the case of $\delta =1$, all distinct self-dual $(1+\alpha u^2)$-constacyclic codes over $R$ of length $2n$ are determined.     

Solution of the Navier–Stokes problem

A new a priori estimate for solutions to Navier–Stokes equations is derived. Uniqueness and existence of these solutions in ${\mathbb{R}}^3$ for all $t>0$ is proved in a class of solutions locally differentiable in time with values in $H^1\left({\mathbb{R}}^3\right)$, where $H^1\left({\mathbb{R}}^3\right)$ is the Sobolev space. By the solution a solution to an integral equation is understood. No smallness restrictions on the data are imposed.     

On a Cheeger type inequality in Cayley graphs of finite groups

Let $G$ be a finite group. It was remarked in Breuillard et al. (2015) that if the Cayley graph $C(G,S)$ is an expander graph and is non-bipartite then the spectrum of the adjacency operator $T$ is bounded away from $-1$. In this article we are interested in explicit bounds for the spectrum of these graphs. Specifically, we show that the non-trivial spectrum of the adjacency operator lies in the interval $\left[-1+\frac{h{\left(\mathbb{G}\right)}^4}{\gamma },1-\frac{h{\left(\mathbb{G}\right)}^2}{2d^2}\right]$, where $h\left(\mathbb{G}\right)$ denotes the (vertex) Cheeger constant of the $d$ regular graph $C(G,S)$ with respect to a symmetric set $S$ of generators and $\gamma =2^9{d}^6{(d+1)}^2$.     

Non parametric estimation of the diffusion coefficients of a diffusion with jumps

In this article, we consider a jump diffusion process ${\left(X_t\right)}_{t\ge 0}$, with drift function $b$, diffusion coefficient $\sigma$ and jump coefficient ${\xi }^2$. This process is observed at discrete times $t=0,\Delta ,\dots ,n\Delta$. The sampling interval $\Delta$ tends to 0 and the time interval $n\Delta$ tends to infinity. We assume that ${\left(X_t\right)}_{t\ge 0}$ is ergodic, strictly stationary and exponentially $\beta$-mixing. We use a penalized least-square approach to compute adaptive estimators of the functions ${\sigma }^2+{\xi }^2$ and ${\sigma }^2$. We provide bounds for the risks of the two estimators.     

Uniqueness in Harper’s vertex-isoperimetric theorem

For a set $A\subseteq Q_n={\left\{0,1\right\}}^n$ the $t$-neighbourhood of $A$ is $N^t\left(A\right)=\left\{x:d\left(x,A\right)\le t\right\}$, where $d$ denotes the usual graph distance on $Q_n$. Harper’s vertex-isoperimetric theorem states that among the subsets $A\subseteq Q_n$ of given size, the size of the $t$-neighbourhood is minimised when $A$ is taken to be an initial segment of the simplicial order. Aubrun and Szarek asked the following question: if $A\subseteq Q_n$ is a subset of given size for which the sizes of both $N^t\left(A\right)$ and $N^t\left(A^c\right)$ are minimal for all $t>0$, does it follow that $A$ is isomorphic to an initial segment of the simplicial order?Our aim is to give a counterexample. Surprisingly it turns out that there is no counterexample that is a Hamming ball, meaning a set that lies between two consecutive exact Hamming balls, i.e. a set $A$ with $B\left(x,r\right)\subseteq A\subseteq B\left(x,r+1\right)$ for some $x\in Q_n$. We go further to classify all the sets $A\subseteq Q_n$ for which the sizes of both $N^t\left(A\right)$ and $N^t\left(A^c\right)$ are minimal for all $t>0$ among the subsets of $Q_n$ of given size. We also prove that, perhaps surprisingly, if $A\subseteq Q_n$ for which the sizes of $N\left(A\right)$ and $N\left(A^c\right)$ are minimal among the subsets of $Q_n$ of given size, then the sizes of both $N^t\left(A\right)$ and $N^t\left(A^c\right)$ are also minimal for all $t>0$ among the subsets of $Q_n$ of given size. Hence the same classification also holds when we only require $N\left(A\right)$ and $N\left(A^c\right)$ to have minimal size among the subsets $A\subseteq Q_n$ of given size.