相关论文: Completely invariant sets of normality for rationa…

Let G be a semigroup of rational functions of degree at least two, under composition of functions. Suppose that G contains two polynomials with non-equal Julia sets. We prove that the smallest closed subset of the Riemann sphere which…

Let $\mathbb{N}$ be a set of the natural numbers. Symmetric inverse semigroup $R_\infty$ is the semigroup of all infinite 0-1 matrices $[g_{ij}]$ with at most one 1 in each row and each column such that $g_{ii}=1$ on the complement of a…

Let G be a finite group. A subgroup M of G is said to be an NR-subgroup if, whenever K is normal in M, then K^G\cap M=K, where K^G is the normal closure of K in G. Using the Classification of Finite Simple Groups, we prove that if every…

Let T(X) be the semigroup of full transformations on a finite set X with n elements. We prove that every subsemilattice of T(X) has at most 2^{n-1} elements and that there are precisely n subsemilattices of size exactly 2^{n-1}, each…

It is proved that, given a (von Neumann) regular semigroup with finitely many left and right ideals, if every maximal subgroup is presentable by a finite complete rewriting system, then so is the semigroup. To achieve this, the following…

Let a torus $T$ act freely on a closed manifold $M$ of dimension at least two. We demonstrate that, for a generic $T$-invariant Riemannian metric $g$ on $M$, each real $\Delta_g$-eigenspace is an irreducible real representation of $T$ and,…

In this article, we first prove that the type of an affine semigroup ring is equal to the number of maximal elements of the Ap\'ery set with respect to the set of exponents of the monomials, which form a maximal regular sequence. Further,…

We characterize left and right amenable semigroups of polynomials of one complex variable with respect to the composition operation. We also prove a number of results about amenable semigroups of arbitrary rational functions. In particular,…

Consider the action of a subgroup $G$ of the permutation group on the polynomial ring $S := k[x_{1}, \ldots, x_{n}]$ via permutations. We show that if $k$ does not have characteristic two, then the following are independent of $k$: the…

For a positive integer $k$, a group $G$ is said to be totally $k$-closed if in each of its faithful permutation representations, say on a set $\Omega$, $G$ is the largest subgroup of $\operatorname{Sym}(\Omega)$ which leaves invariant each…

We consider the connection of functional decompositions of rational functions over the real and complex numbers, and a question about curves on a Riemann sphere which are invariant under a rational function.

Let $A$ and $G$ be finite groups such that $A$ acts coprimely on $G$ by automorphisms. We provide a complete classification of a finite group $G$ in which every maximal $A$-invariant subgroup containing the normalizer of some $A$-invariant…

For a finite group $G$, we study the probability $sp(G)$ that, given two elements $x,y \in G$, the cyclic subgroup $\langle x \rangle$ is subnormal in the subgroup $\langle x, y \rangle$. This can be seen as an intermediate invariant…

Over an algebraically closed base field $k$ of characteristic 2, the ring $R^G$ of invariants is studied, $G$ being the orthogonal group O(n) or the special orthogonal group SO(n) and acting naturally on the coordinate ring $R$ of the…

Let $M$ be a $G$-covering of a nilpotent orbit in $\g$ where $G$ is a complex semisimple Lie group and $\g=\text{Lie}(G)$. We prove that under Poisson bracket the space $R[2]$ of homogeneous functions on $M$ of degree 2 is the unique…

For a closed Riemannian manifold $M$ with a compact Lie group $G$ acting by isometries, we show that there are infinitely many $G$-invariant minimal hypersurfaces. Under the assumption that $M$ contains at most a finite number of minimal…

A classical (or quantum) superintegrable system on an n-dimensional Riemannian manifold is an integrable Hamiltonian system with potential that admits 2n-1 functionally independent constants of the motion that are polynomial in the momenta,…

Given a Baumslag-Solitar group, we study its space of subgroups from a topological and dynamical perspective. We first determine its perfect kernel (the largest closed subset without isolated points). We then bring to light a natural…

Let (G, *) be a semigroup, D subset of G, and n >= 2 be an integer. We say that (D, *) is an n-closed subset of G if a_1* ... *a_n in D for every a_1, ..., a_n in D. Hence every closed set is a 2-closed set. The concept of n-closed sets…

Let $g \in L^2(\mathbb{R})$ be a rational function of degree $M$, i.e. there exist polynomials $P, Q$ such that $g = {{P} \over {Q}}$ and $deg(P) < deg(Q) \leq M$. We prove that for any $\varepsilon>0$ and any $M \in \mathbb{N}$ there…