Group gradings on the superalgebras M(m,n), A(m,n) and P(n)

We classify gradings by arbitrary abelian groups on the classical simple Lie superalgebras $P(n)$, $n\ge 2$, and on the simple associative superalgebras $M(m,n)$, $m,n\ge 1$, over an algebraically closed field: fine gradings up to equivalence and G-gradings, for a fixed group G, up to isomorphism. As a corollary, we also classify up to isomorphism the G-gradings on the classical Lie superalgebra $A(m,n)$ that are induced from G-gradings on $M(m+1,n+1)$. In the case of Lie superalgebras, the characteristic is assumed to be 0.     

Clique number of Xor products of Kneser graphs

In this article we investigate a problem in graph theory, which has an equivalent reformulation in extremal set theory similar to the problems researched in “A general 2-part Erd?s-Ko-Rado theorem” by Gyula O.H. Katona, who proposed our problem as well. In the graph theoretic form we examine the clique number of the Xor product of two isomorphic $KG(N,k)$ Kneser graphs. Denote this number with $f(k,N)$. We give lower and upper bounds on $f(k,N)$, and we solve the problem up to a constant deviation depending only on k, and find the exact value for $f(2,N)$ if N is large enough. Also we compute that $f(k,k^2)$ is asymptotically equivalent to $k^2$.     

The structure of the minimal free resolution of semigroup rings obtained by gluing

We construct a minimal free resolution of the semigroup ring $k[C]$ in terms of minimal resolutions of $k[A]$ and $k[B]$ when $〈C〉$ is a numerical semigroup obtained by gluing two numerical semigroups $〈A〉$ and $〈B〉$. Using our explicit construction, we compute the Betti numbers, graded Betti numbers, regularity and Hilbert series of $k[C]$, and prove that the minimal free resolution of $k[C]$ has a differential graded algebra structure provided the resolutions of $k[A]$ and $k[B]$ possess them. We discuss the consequences of our results in small embedding dimensions. Finally, we give an extension of our main result to semigroups in ${\mathbb{N}}^n$.     

Graphs with nullity 2c(G)+p(G)−1

Let G be a connected graph with $n(G)$ vertices and $e(G)$ edges. The nullity of G, denoted by $\eta (G)$, is the multiplicity of eigenvalue zero of the adjacency matrix of G. Ma, Wong and Tian (2016) proved that $\eta (G)\le 2c(G)+p(G)?1$ unless G is a cycle of order a multiple of 4, where $c(G)=e(G)?n(G)+1$ is the elementary cyclic number of G and $p(G)$ is the number of leaves of G. Recently, Chang, Chang and Zheng (2020) characterized the leaf-free graphs with nullity $2c(G)?1$, thus leaving the problem to characterize connected graphs G with nullity $2c(G)+p(G)?1$ when $p(G)\ne 0$. In this paper, we solve this problem completely.     

Spanning trees and spanning closed walks with small degrees

Let G be a graph and let f be a positive integer-valued function on $V(G)$. In this paper, we show that if for all $S?V(G)$, $\omega (G?S)<{\sum }_{v\in S}(f(v)?2)+2+\omega (G[S])$, then G has a spanning tree T containing an arbitrary given matching such that for each vertex v, $d_T(v)\le f(v)$, where $\omega (G?S)$ denotes the number of components of $G?S$ and $\omega (G[S])$ denotes the number of components of the induced subgraph $G[S]$ with the vertex set S. This is an improvement of several results. Next, we prove that if for all $S?V(G)$, $\omega (G?S)\le {\sum }_{v\in S}(f(v)?1)+1$, then G admits a spanning closed walk passing through the edges of an arbitrary given matching meeting each vertex v at most $f(v)$ times. This result solves a long-standing conjecture due to Jackson and Wormald (1990).     

Annihilator condition does not pass to polynomials and power series

In this paper we construct a ring A which has annihilator condition (a.c.) and we show that neither $A[x]$ nor $A[[x]]$ has this property. This answers in negative a question asked by Hong, Kim, Lee and Nielsen. We also show that there is an algebra A which does not have annihilator condition while both $A[x]$ and $A[[x]]$ have this property.     

On the smoothness of some quotients of Banach-Lie groups

Quotients of Banach-Lie groups are regarded as topological groups with Lie algebra in the sense of Hofmann-Morris on the one hand, and as Q-groups in the sense of Barre-Plaisant on the other hand. For the groups of the type $G/N$ where $N\subseteq G$ is a pseudo-discrete normal subgroup, their Lie algebra in the sense of Q-groups turns out to be isomorphic to the Lie algebra of G, which is in general merely a dense subalgebra of the Lie algebra of $G/N$ when regarded as a topological group with Lie algebra. The submersion-like behavior of quotient maps of Banach-Lie groups is also investigated. The two aforementioned approaches to the Lie theory of the quotients of Banach-Lie groups thus lead to differing results and the Lie theoretic properties of quotient groups are more accurately described by the Q-group approach than by the approach via topological groups with Lie algebras.     

Maximal nonassociativity via nearfields

We say that $(x,y,z)\in Q^3$ is an associative triple in a quasigroup $Q(⁎)$ if $(x⁎y)⁎z=x⁎(y⁎z)$. It is easy to show that the number of associative triples in Q is at least $|Q|$, and it was conjectured that quasigroups with exactly $|Q|$ associative triples do not exist when $|Q|>1$. We refute this conjecture by proving the existence of quasigroups with exactly $|Q|$ associative triples for a wide range of values $|Q|$. Our main tools are quadratic Dickson nearfields and the Weil bound on quadratic character sums.     

Multiplicity results for the Yamabe equation by Lusternik–Schnirelmann theory

Let $(M,g)$ be any closed Riemannianan manifold and $(N,h)$ be a Riemannian manifold of constant positive scalar curvature. We prove that the Yamabe equation on the Riemannian product $(M\times N,g+\delta h)$ has at least $\mathit{Cat}(M)+1$ solutions for δ small enough, where $\mathit{Cat}(M)$ denotes the Lusternik–Schnirelmann-category of M. The solutions obtained are functions of M and $Cat(M)$ of them have energy arbitrarily close to the minimum.