Bulletin of the London Mathematical Society最新文献

We define a Grothendieck ring of varieties for log schemes. It is generated by one additional class “$�P$” over the usual Grothendieck ring. We show the naïve definition of log Hodge numbers does not make sense for all log schemes. We offer an alternative that does.

We define the orbit morphism of partial dynamical systems and prove that an orbit morphism being an isomorphism in the category of partial dynamical systems and orbit morphisms is equivalent to the existence of a continuous orbit equivalence between the given partial dynamical systems that preserves the essential stabilisers. We show that this is equivalent to the existence of a diagonal-preserving isomorphism between the corresponding crossed products when the essential stabilisers of partial actions are torsion-free and abelian. We also characterise when an étale groupoid is isomorphic to the transformation groupoid of some partial action. Additionally, we explore the implications in the context of semi-saturated orthogonal partial dynamical systems over free groups, establishing connections with Deaconu–Renault systems and the concept of eventual conjugacy.

It is proved that if a finite group $�G$ has an automorphism of order $�n$ with $�m$ fixed points, then $�G$ has a soluble subgroup whose index and Fitting height are bounded in terms of $�m$ and $�n$. As a corollary, a problem of B. Hartley is solved in the affirmative: if a locally finite group $�G$ has an element with finite centralizer, then $�G$ has a subgroup of finite index which has a finite normal series with locally nilpotent factors.

This paper is concerned with random holomorphic functions over the infinite-dimensional polydisc. The celebrated Billard's theorem states, among other things, that for a random holomorphic function in a single variable—obtained by placing a random sign before each Taylor coefficient—the boundedness of the function can be automatically improved to continuity. We extend Billard's theorem in three aspects: first, we extend it to the case of infinitely many variables. Second, we establish an inverse result, that is, not merely a sufficient condition. Such results are rare in the literature. In particular, we define Billard's property (BP) for random variables and prove: (1) all symmetric random variables have BP; (2) a square-integrable random variable has BP if and only if it is centered. Third, another novelty of our results is that they encompass applications to random Dirichlet series.

Complementing our previous results, we give a classification of all isometries (not necessarily surjective) of the metric space consisting of ball-bodies, endowed with the Hausdorff metric. ‘Ball-bodies’ are convex bodies which are intersections of translates of the Euclidean unit ball. We show that any such isometry is either a rigid motion or a rigid motion composed of the $�c$-duality mapping. In particular, any isometry on this metric space has to be surjective.

Let $�G$ be a permutation group on the finite set $�\Omega$. We prove various results about partitions of $�\Omega$ whose stabilizers have good properties. In particular, in every solvable permutation group there is a set-stabilizer whose orbits have length at most 6, which is best possible and answers two questions of Babai. Every solvable maximal subgroup of any almost simple group has derived length at most 10, which is best possible. In every primitive group with solvable stabilizer, there are two points whose stabilizer has derived length bounded by an absolute constant.

We study standing waves of the Schrödinger equation with constant magnetic field and combined power nonlinearities, which describes a single non-relativistic quantum particle in the presence of an electromagnetic field. We develop the local minima geometry method to establish the existence, estimates and mass collapse results for this equation, which generalize exiting results in the literature.

We show that the complex of partial bases of the free group of rank $�n$, where vertices are seen up to conjugation, is Cohen–Macaulay of dimension $��n�-�1�$. This confirms a conjecture by Day and Putman. We prove our results in the more general context of freely decomposable groups.