Bulletin of Mathematical Sciences

A finite permutation group $G\le \mathrm{\text{Sym}}(\mathrm{\Omega })$ is called $2$-closed if $G$ is the largest subgroup of $\mathrm{\text{Sym}}(\mathrm{\Omega })$ which leaves invariant each of the $G$-orbits for the induced action on $\mathrm{\Omega }\times \mathrm{\Omega }$. Introduced by Wielandt in 1969, the concept of $2$-closure has developed as one of the most useful approaches for studying relations on a finite set invariant under a group of permutations of the set; in particular for studying automorphism groups of graphs and digraphs. The concept of total $2$-closure switches attention from a particular group action, and is a property intrinsic to the group: a finite group $G$ is said to be totally $2$-closed if $G$ is $2$-closed in each of its faithful permutation representations. There are infinitely many finite soluble totally $2$-closed groups, and these have been completely characterized, but up to now no insoluble examples were known. It turns out, somewhat surprisingly to us, that there are exactly $6$ totally $2$-closed finite nonabelian simple groups: the Janko groups ${\mathrm{\text{J}}}_1$, ${\mathrm{\text{J}}}_3$ and ${\mathrm{\text{J}}}_4$, together with $\mathrm{\text{Ly}}$, $\mathrm{\text{Th}}$ and the Monster $𝕄$. Moreover, if a finite totally $2$-closed group has no nontrivial abelian normal subgroup, then we show that it is a direct product of some (but not all) of these simple groups, and there are precisely $47$ examples.

In the course of obtaining this classification, we develop a general framework for studying $2$-closures of transitive permutation groups, which we hope will prove useful for investigating representations of finite groups as automorphism groups of graphs and digraphs, and in particular for attacking the long-standing polycirculant conjecture. In this direction, we apply our results, proving a dual to a 1939 theorem of Frucht from Algebraic Graph Theory. We also pose several open questions concerning closures of permutation groups.

Classical moment functionals (Hermite, Laguerre, Jacobi, Bessel) can be characterized as those linear functionals whose moments satisfy a second-order linear recurrence relation. In this work, we use this characterization to link the theory of classical orthogonal polynomials and the study of Hankel matrices whose entries satisfy a second-order linear recurrence relation. Using the recurrent character of the entries of such Hankel matrices, we give several characterizations of the triangular and diagonal matrices involved in their Cholesky factorization and connect them with a corresponding characterization of classical orthogonal polynomials.

Let $A$ and $B$ be bounded operators in a Banach space. For the equation $du(t)/dt=Au(t)+Bu(t-h)(h>0)$ we derive sharp explicit delay-dependent exponential stability conditions via the commutator $AB-BA$. The stability conditions in terms of the commutator enable us to construct a feedback with delay, which stabilizes a system. Gives us a possibility construct Stabilization of Systems with Retarded Feedback in a Banach Space. Our results are new even in the finite-dimensional case.

We obtain sharp Hardy inequalities on antisymmetric functions, where antisymmetry is understood for multi-dimensional particles. Partially it is an extension of the paper [Th. Hoffmann-Ostenhof and A. Laptev, Hardy inequalities with homogeneous weights, J. Funct. Anal. 268 (2015) 3278–3289], where Hardy’s inequalities were considered for the antisymmetric functions in the case of the 1D particles. As a byproduct we obtain some Sobolev and Gagliardo–Nirenberg type inequalities that are applied to the study of spectral properties of Schrödinger operators.

By using linking and $\nabla$-theorems in this paper we prove the existence of multiple solutions for the following nonlocal problem with mixed Dirichlet–Neumann boundary data,