a conjecture is proposed, bounding the number of cycles with label $w^n$ in a labeled directed graph. some partial results towards this conjecture are established. as a consequence, it is proved that $\langle a_1, a_2, \ldots\,{\mid}\,w^n\rangle$ is coherent for $n\,{\geq}\,4$. furthermore, it is coherent for $n\,{\geq}\,2$, provided that the strengthened hanna neumann conjecture holds.

Consider a scalar differential equation $\dot{x}\,{=}\,f(t,x), (t,x)\,{\in}\,I\,{\times}\,\R$, where $I$ is an open interval containing $[0,\,T]$. Assume that $f(t,x)$ is continuous with a continuous derivative $f^{\prime}_{x}(t,x)$, and weakly concave (or weakly convex) in $x$ for all $t \in I$, though strictly concave (or strictly convex) for some $t\,{\in}\,[0,\,T$]. It is well known that in this case there can be either no, one or two closed solutions; that is, solutions $\varphi(t)$ for which $\varphi(0)=\varphi$$(T)$ If there are two closed solutions, then the greater has a negative characteristic exponent and the smaller has a positive one. It is easily seen that this is equivalent to a statement on localization of closed solutions. It is shown how this statement can be generalized to systems of differential equations $\dot{\underline x} = \underline{f}(t,\underline{x}), (t,\underline{x})\in I \times \R^{n}$. The requirements are that the coordinate functions ${f}_{j}(t,\underline{x})$ be continuous with continuous derivatives with respect to $x_{1},x_{2},\ldots,x_{n}$, that the $f_j$ are weakly concave (or weakly convex) in $\underline{\it x}$, and that a certain condition pertaining to strict concavity (or strict convexity) is fulfilled.

We prove that if H is a finite-dimensional semisimple Hopf algebra acting on a PI-algebra R of characteristic 0, and R is either affine or algebraic over k, then the Jacobson radical of R is H-stable. Under the same hypotheses, we show that the smash product algebra R#H is semiprimitive provided that R is H-semiprime. More generally we show that the ‘finite’ Jacobson radical is H-stable, and that R#H is semiprimitive provided that R is H-semiprimitive and all irreducible representations of R are finite-dimensional. We also consider R#H when R is an FCR-algebra. Finally, we prove a general relationship between stability of the radical and semiprimeness of R#H; in particular if for a given H, any action of H stabilizes the Jacobson radical, then also any action of H stabilizes the prime radical.

A proof is given that the quasivariety of groups generated by finite and torsion-free groups does not contain the class of periodic groups. This result is related to (and inspired by) the solvability of equations over groups. The proof uses the Feit–Thompson theorem on the solvability of finite groups of odd order as well as Kostrikin–Zelmanov results on the restricted Burnside problem, and applies technical details of a recent construction of weakly finitely presented periodic groups.

a tits alternative theorem is proved in this paper for groups acting on cat(0) cubical complexes. that is, a proof is given to show that if $g$ is assumed to be a group for which there is a bound on the orders of its finite subgroups, and if $g$ acts properly on a finite-dimensional cat(0) cubical complex, then either $g$ contains a free subgroup of rank 2, or $g$ is finitely generated and virtually abelian. in particular, the above conclusion holds for any group $g$ with a free action on a finite-dimensional cat(0) cubical complex.

For a semigroup $S$, the finitary power semigroup of $S$, denoted $P_f(S)$, consists of all finite subsets of $S$ under the usual multiplication. The main result of this paper asserts that $P_f(G)$ is not finitely generated for any infinite group $G$.

In this paper, the minimal positive entropy of pure braids is shown to be greater than $\log(2+\sqrt5)$, independent of the braid index. For pure 3-braids, the minimal entropy is shown to be equal to $\log(3+2\sqrt2)$.

The famous Burnside–Schur theorem states that every primitive finite permutation group containing a regular cyclic subgroup is either 2-transitive or isomorphic to a subgroup of a 1-dimensional affine group of prime degree. It is known that this theorem can be expressed as a statement on Schur rings over a finite cyclic group. Generalizing the latter, Schur rings are introduced over a finite commutative ring, and an analogue of this statement is proved for them. Also, the finite local commutative rings are characterized in permutation group terms.

This paper proves that a subgroup of finite index in a positively finitely generated profinite group has maximal subgroup growth at most $n^{\log (n\!)}$. In particular such a subgroup cannot be free, answering a question by L. Pyber.

The determination of the generalized Springer correspondence for disconnected reductive groups is completed here by being achieved in the case of disconnected groups of types $D_4$ in characteristic 3 and $E_6$ in characteristic 2. It is expected that this correspondence will provide an important ingredient for the computation of character values of finite groups of Lie type. Some evidence is provided in support of the latter statement.

the cohomology of a right-angled artin group with group ring coefficients is explicitly presented in terms of the cohomology of its defining flag complex.

A group $G$ is said to have periodic cohomology with period $q$ after $k$ steps, if the functors $H^{i}(G,-)$ and $H^{i+q}(G,-)$ are naturally equivalent for all $i>k$. Mislin and the author have conjectured that periodicity in cohomology after some steps is the algebraic characterization of those groups $G$ that admit a finite-dimensional, free $G$-CW-complex, homotopy equivalent to a sphere. This conjecture was proved by Adem and Smith under the extra hypothesis that the periodicity isomorphisms are given by the cup product with an element in $H^q (G,\mathbb{Z})$. It is expected that the periodicity isomorphisms will always be given by the cup product with an element in $H^q (G,\mathbb{Z})$; this paper shows that this is the case if and only if the group $G$ admits a complete resolution and its complete cohomology is calculated via complete resolutions. It is also shown that having the periodicity isomorphisms given by the cup product with an element in $H^q (G,\mathbb{Z})$ is equivalent to silp $G$ being finite, where silp $G$ is the supremum of the injective lengths of the projective $\mathbb{Z}G$-modules.

It is proved that two random elements of the Nottingham group generate a free group with probability 1. This answers a question of A. Shalev.

The special values of multiple polylogarithms, which include multiple zeta values, appear in several fields of mathematics and physics. Many kinds of their linear relations have been investigated, as well as their algebraic relations. In this paper, from the viewpoint of a connection matrix of Fuchsian equations, two kinds of duality of these values are derived.

The above-titled paper of mine appeared in the Bulletin of the London Mathematical Society, 32 (2000) 297–304. Regrettably, there is a careless error in the proofs of Theorems 6 and 8. In line 6 of the proof of Theorem 6, it is claimed that a certain subset must be a subgroup. For this to hold, the subset must contain the zero element. This need not be the case; the true deduction is that the subset is a coset, say $M+h$, of a subgroup $M$. Now $M$ and $M+h$ contain the same number of elements, and so the deduction that $M$ has $p$ elements is still correct.

A new inequality concerning generalized characters of $p$-groups is obtained and applications to bounding the number of irreducible characters in blocks of finite groups are given.

The Poincaré series of an irreducible plane curve singularity equals the $\zeta$-function of its monodromy, by a result of Campillo, Delgado and Gusein-Zade. This fact is derived in this paper from a formula of Ebeling and Gusein-Zade, relating the Poincaré series of a quasi-homogeneous complete intersection singularity to the Saito dual of a product of $\zeta$-functions.

The optimal orders are determined for the best constants in the non-commutative Burkholder–Gundy, Doob and Stein inequalities obtained recently in non-commutative martingale theory.

in two papers, littlewood studied seemingly unrelated constants: (i) the best $\alpha$ such that for any polynomial $f$, of degree $n$, the areal integral of its spherical derivative is at most $\const\cdot n^\alpha$, and (ii) the extremal growth rate $\beta$ of the length of green's equipotentials for simply connected domains. these two constants are shown to coincide, thus greatly improving known estimates on $\alpha$.

In this paper, a proof is given that there exists a universal Taylor series – in the sense of Nestoridis – in the complement of a disk with respect to every center. This answers a question asked by G. Costakis.