Compact simple Lie groups admitting left-invariant Einstein metrics that are not geodesic orbit

In this article, we prove that the compact simple Lie groups $\mathrm{S}\mathrm{U}(n)$ for $n\ge 6$, $\mathrm{S}\mathrm{O}(n)$ for $n\ge 7$, $\mathrm{S}\mathrm{p}(n)$ for $n\ge 3$, ${\mathrm{E}}_6,{\mathrm{E}}_7,{\mathrm{E}}_8$, and ${\mathrm{F}}_4$ admit left-invariant Einstein metrics that are not geodesic orbit. This gives a positive answer to an open problem recently posed by Nikonorov.     

Primeness property for graded central polynomials of verbally prime algebras

Let F be an infinite field. The primeness property for central polynomials of $M_n(F)$ was established by A. Regev, i.e., if the product of two polynomials in distinct variables is central then each factor is also central. In this paper we consider the analogous property for $M_n(F)$ and determine, within the elementary gradings with commutative neutral component, the ones that satisfy this property, namely the crossed product gradings. Next we consider $M_n(R)$, where R admits a regular grading, with a grading such that $M_n(F)$ is a homogeneous subalgebra and provide sufficient conditions – satisfied by $M_n(E)$ with the trivial grading – to prove that $M_n(R)$ has the primeness property if $M_n(F)$ does. We also prove that the algebras $M_{a,b}(E)$ satisfy this property for ordinary central polynomials. Hence we conclude that, over a field of characteristic zero, every verbally prime algebra has the primeness property.     

A note on panchromatic colorings

This paper studies the quantity $p(n,r)$, that is the minimal number of edges of an $n$-uniform hypergraph without panchromatic coloring (it means that every edge meets every color) in $r$ colors. If $r\le c\frac{n}{lnn}$ then all bounds have a type $A_1(n,lnn,r){\left(\frac{r}{r?1}\right)}^n\le p(n,r)\le A_2(n,r,lnr){\left(\frac{r}{r?1}\right)}^n$, where $A_1$, $A_2$ are some algebraic fractions. The main result is a new lower bound on $p(n,r)$ when $r$ is at least $c\sqrt{n}$; we improve an upper bound on $p(n,r)$ if $n=o\left(r^{3∕2}\right)$.Also we show that $p(n,r)$ has upper and lower bounds depending only on $n∕r$ when the ratio $n∕r$ is small, which cannot be reached by the previous probabilistic machinery.Finally we construct an explicit example of a hypergraph without panchromatic coloring and with ${(\frac{r}{r?1}+o\left(1\right))}^n$ edges for $r=o\left(\sqrt{\frac{n}{lnn}}\right)$.     

Superconcentration,and randomized Dvoretzky's theorem for spaces with 1-unconditional bases

Let n be a sufficiently large natural number and let B be an origin-symmetric convex body in ${\mathbb{R}}^n$in the ?-position, and such that the space $({\mathbb{R}}^n,{6?6}_B)$ admits a 1-unconditional basis. Then for any $\epsilon \in (0,1/2]$, and for random $c\epsilon \log ?n/\log ?\frac{1}{\epsilon }$-dimensional subspace E distributed according to the rotation-invariant (Haar) measure, the section $B\cap E$ is $(1+\epsilon )$-Euclidean with probability close to one. This shows that the “worst-case” dependence on ε in the randomized Dvoretzky theorem in the ?-position is significantly better than in John's position. It is a previously unexplored feature, which has strong connections with the concept of superconcentration introduced by S. Chatterjee. In fact, our main result follows from the next theorem: Let B be as before and assume additionally that B has a smooth boundary and ${\mathbb{E}}_{{\gamma }_n}{6?6}_B\le n^c{\mathbb{E}}_{{\gamma }_n}{6{\mathrm{grad}}_B(?)6}_2$ for a small universal constant $c>0$, where ${\mathrm{grad}}_B(?)$ is the gradient of ${6?6}_B$ and ${\gamma }_n$ is the standard Gaussian measure in ${\mathbb{R}}^n$. Then for any $p\in [1,c\log ?n]$ the p-th power of the norm ${6?6}_B^p$ is $\frac{C}{\log ?n}$-superconcentrated in the Gauss space.     

The edge spectrum of the saturation number for small paths

Let $H$ be a simple graph. A graph $G$ is called an $H$-saturated graph if $H$ is not a subgraph of $G$, but adding any missing edge to $G$ will produce a copy of $H$. Denote by $SAT(n,H)$ the set of all $H$-saturated graphs $G$ with order $n$. Then the saturation number $sat(n,H)$ is defined as ${min}_{G\in SAT(n,H)}\left|E\left(G\right)\right|$, and the extremal number $ex(n,H)$ is defined as ${max}_{G\in SAT(n,H)}\left|E\left(G\right)\right|$. A natural question is that of whether we can find an $H$-saturated graph with $m$ edges for any $sat(n,H)\le m\le ex(n,H)$. The set of all possible values $m$ is called the edge spectrum for $H$-saturated graphs. In this paper we investigate the edge spectrum for $P_i$-saturated graphs, where $2\le i\le 6$. It is trivial for the case of $P_2$ that the saturated graph must be an empty graph.     

The expansion of a chord diagram and the Tutte polynomial

A chord diagram is a set of chords of a circle such that no pair of chords has a common endvertex. A chord diagram $E$ is called nonintersecting if $E$ contains no crossing. For a chord diagram $E$ having a crossing $S=\{x_1{x}_3,x_2{x}_4\}$, the expansion of $E$ with respect to $S$ is to replace $E$ with $E_1=(E?S)\cup \{x_2{x}_3,x_4{x}_1\}$ or $E_2=(E?S)\cup \{x_1{x}_2,x_3{x}_4\}$. For a chord diagram $E$, let $f\left(E\right)$ be the chord expansion number of $E$, which is defined as the cardinality of the multiset of all nonintersecting chord diagrams generated from $E$ with a finite sequence of expansions.In this paper, it is shown that the chord expansion number $f\left(E\right)$ equals the value of the Tutte polynomial at the point $(2,?1)$ for the interlace graph $G_E$ corresponding to $E$. The chord expansion number of a complete multipartite chord diagram is also studied. An extended abstract of the paper was published (Nakamigawa and Sakuma, 2017) [13].