Hybrid journal (It can contain Open Access articles) ISSN (Print) 0024-6115 - ISSN (Online) 1460-244X Published by Oxford University Press[396 journals]

Authors:Menne U. Pages: 725 - 774 Abstract: This paper introduces first-order Sobolev spaces on certain rectifiable varifolds. These complete locally convex spaces are contained in the generally non-linear class of generalised weakly differentiable functions and share key functional analytic properties with their Euclidean counterparts.Assuming the varifold to satisfy a uniform lower density bound and a dimensionally critical summability condition on its mean curvature, the following statements hold. Firstly, continuous and compact embeddings of Sobolev spaces into Lebesgue spaces and spaces of continuous functions are available. Secondly, the geodesic distance associated to the varifold is a continuous, not necessarily Hölder continuous Sobolev function with bounded derivative. Thirdly, if the varifold additionally has bounded mean curvature and finite measure, then the present Sobolev spaces are isomorphic to those previously available for finite Radon measures yielding many new results for those classes as well.Suitable versions of the embedding results obtained for Sobolev functions hold in the larger class of generalised weakly differentiable functions. PubDate: Fri, 11 Nov 2016 00:00:00 GMT DOI: 10.1112/plms/pdw023 Issue No:Vol. 113, No. 6 (2016)

Authors:Cellarosi F; Marklof J. Pages: 775 - 828 Abstract: Hardy and Littlewood's approximate functional equation for quadratic Weyl sums (theta sums) provides, by iterative application, a powerful tool for the asymptotic analysis of such sums. The classical Jacobi theta function, on the other hand, satisfies an exact functional equation, and extends to an automorphic function on the Jacobi group. In the present study we construct a related, almost everywhere non-differentiable automorphic function, which approximates quadratic Weyl sums up to an error of order one, uniformly in the summation range. This not only implies the approximate functional equation, but also allows us to replace Hardy and Littlewood's renormalization approach by the dynamics of a certain homogeneous flow. The great advantage of this construction is that the approximation is global, that is, there is no need to keep track of the error terms accumulating in an iterative procedure. Our main application is a new functional limit theorem, or invariance principle, for theta sums. The interesting observation here is that the paths of the limiting process share a number of key features with Brownian motion (scale invariance, invariance under time inversion and non-differentiability), although time increments are not independent and the value distribution at each fixed time is distinctly different from a normal distribution. PubDate: Mon, 24 Oct 2016 00:00:00 GMT DOI: 10.1112/plms/pdw038 Issue No:Vol. 113, No. 6 (2016)

Authors:Araújo J; Bentz W, Cameron PJ, et al. Pages: 829 - 867 Abstract: Let $\Omega $ be a set of cardinality $n$, $G$ be a permutation group on $\Omega $ and $f:\Omega \to \Omega $ be a map that is not a permutation. We say that $G$synchronizes$f$ if the transformation semigroup $\langle G,f\rangle $ contains a constant map, and that $G$ is a synchronizing group if $G$ synchronizes every non-permutation.A synchronizing group is necessarily primitive, but there are primitive groups that are not synchronizing. Every non-synchronizing primitive group fails to synchronize at least one uniform transformation (that is, transformation whose kernel has parts of equal size), and it had previously been conjectured that this was essentially the only way in which a primitive group could fail to be synchronizing, in other words, that a primitive group synchronizes every non-uniform transformation.The first goal of this paper is to prove that this conjecture is false, by exhibiting primitive groups that fail to synchronize specific non-uniform transformations of ranks 5 and 6. As it has previously been shown that primitive groups synchronize every non-uniform transformation of rank at most 4, these examples are of the lowest possible rank. In addition, we produce graphs with primitive automorphism groups that have approximately $\sqrt {n}$non-synchronizing ranks, thus refuting another conjecture on the number of non-synchronizing ranks of a primitive group.The second goal of this paper is to extend the spectrum of ranks for which it is known that primitive groups synchronize every non-uniform transformation of that rank. It has previously been shown that a primitive group of degree $n$ synchronizes every non-uniform transformation of rank $n-1$ and $n-2$, and here this is extended to $n-3$ and $n-4$.In the process, we will obtain a purely graph-theoretical result showing that, with limited exceptions, in a vertex-primitive graph the union of neighbourhoods of a set of vertices $A$ is bounded below by a function that is asymptotically $\sqrt { A }$.Determining the exact spectrum of ranks for which there exist non-uniform transformations not synchronized by some primitive group is just one of several natural, but possibly difficult, problems on automata, primitive groups, graphs and computational algebra arising from this work; these are outlined in the final section. PubDate: Mon, 03 Oct 2016 00:00:00 GMT DOI: 10.1112/plms/pdw040 Issue No:Vol. 113, No. 6 (2016)

Authors:Griffeth S; Norton E. Pages: 868 - 906 Abstract: We study blocks of category $\mathcal {O}$ for the Cherednik algebra having the property that every irreducible module in the block admits a BGG resolution, and as a consequence prove a character formula conjectured by Oblomkov–Yun. PubDate: Mon, 24 Oct 2016 00:00:00 GMT DOI: 10.1112/plms/pdw044 Issue No:Vol. 113, No. 6 (2016)