Journal of the London Mathematical Society-Second Series最新文献

One formulation of Marstrand's slicing theorem is the following. Assume that $��t�\in �(�1�,�2�]�$, and $��B�\subset �R^2�$ is a Borel set with $��H^t��\left(�B�\right)��<�\infty �$. Then, for almost all directions $��e�\in �S^1�$, $�H^t$ almost all of $�B$ is covered by lines $�\ell$ parallel to $�e$ with $��{dim}_H��(�B�$

A celebrated theorem of Pippenger, and Frankl and Rödl states that every almost-regular, uniform hypergraph $�H$ with small maximum codegree has an almost-perfect matching. We extend this result by obtaining a conflict-free matching, where conflicts are encoded via a collection $�C$ of subsets $��C�\subseteq �E�\left(�H�\right)�$. We say that a matching $��M�\subseteq �E�\left(�H�\right)�$ is conflict-free if $�M$ does not contain an element of $�C$ as a subset. Under natural assumptions on $�C$, we prove that $�H$ has a conflict-free, almost-perfect matching. This has many applications, one of which yields new asymptotic results for so-called ‘high-girth’ Steiner systems. Our main tool is a random greedy algorithm which we call the ‘conflict-free matching process’.

We extend the characterization of context-free groups of Muller and Schupp in two ways. We first show that for a quasi-transitive inverse graph $�\Gamma$, being quasi-isometric to a tree, or context-free in the sense of Muller–Schupp (finitely many end-cone up to end-isomorphism), or having the automorphism group $��Aut�\left(�\Gamma �\right)�$ that is virtually free, are all equivalent conditions. Furthermore, we add to the previous equivalences a group theoretic analog to the representation theorem of Chomsky–Schützenberger that is fundamental in solving a weaker version of a conjecture of Brough which also extends Muller and Schupp's result to the class of groups that are virtually finitely generated subgroups of the direct product of free groups. We show that such groups are precisely those whose word problem is the intersection of a finite number of languages accepted by quasi-transitive, tree-like inverse graphs.

Let $�K$ be a field with $��G_K��\left(�2�\right)��\simeq �G_{Q_2}��\left(�2�\right)��$, where $��G_F��\left(�2�\right)��$ denotes the maximal pro-2 quotient of the absolute Galois group of a field $�F$. We prove that then $�K$ admits a (non-trivial) valuation $�v$ which is 2-henselian and has residue field $�F_2$. Furthermore, $��v�\left(�2�\right)�$ is a minimal positive element in the value group $�{\Gamma }_v$

By juxtaposing ideas from fractal geometry and dynamical systems, Furstenberg proposed a series of conjectures in the late 1960's that explore the relationship between digit expansions with respect to multiplicatively independent bases. In this work, we introduce and study — in the discrete context of the integers — analogs of some of the notions and results surrounding Furstenberg's work. In particular, we define a new class of fractal sets of integers that parallels the notion of $��\times �r�$-invariant sets on the 1-torus and investigate the additive and geometric independence between two such fractal sets when they are structured with respect to multiplicatively independent bases. Our main results in this direction parallel the works of Furstenberg, Hochman–Shmerkin, Shmerkin, Wu, and Lindenstrauss–Meiri–Peres and include:

We study transport equations on the unit tangent bundle of a closed oriented Riemannian surface and their links to the transport twistor space of the surface (a complex surface naturally tailored to the geodesic vector field). We show that fibrewise holomorphic distributions invariant under the geodesic flow — which play an important role in tensor tomography on surfaces — form a unital algebra, that is, multiplication of such distributions is well defined and continuous. We also exhibit a natural bijective correspondence between fibrewise holomorphic invariant distributions and genuine holomorphic functions on twistor space with polynomial blowup on the boundary of the twistor space. Additionally, when the surface is Anosov, we classify holomorphic line bundles over twistor space which are smooth up to the boundary. As a byproduct of our analysis, we obtain a quantitative version of a result of Flaminio [C. R. Acad. Sci. Paris Sér. I Math. 315 (1992) no. 6, 735–738] asserting that invariant distributions of the geodesic flow of a positively curved metric on $�S^2$ are determined by their zeroth and first Fourier modes.

Evolutionary relationships between species are represented by phylogenetic trees, but these relationships are subject to uncertainty due to the random nature of evolution. A geometry for the space of phylogenetic trees is necessary in order to properly quantify this uncertainty during the statistical analysis of collections of possible evolutionary trees inferred from biological data. Recently, the wald space has been introduced: a length space for trees which is a certain subset of the manifold of symmetric positive definite matrices. In this work, the wald space is introduced formally and its topology and structure is studied in detail. In particular, we show that wald space has the topology of a disjoint union of open cubes, it is contractible, and by careful characterisation of cube boundaries, we demonstrate that wald space is a Whitney stratified space of type (A). Imposing the metric induced by the affine invariant metric on symmetric positive definite matrices, we prove that wald space is a geodesic Riemann stratified space. A new numerical method is proposed and investigated for construction of geodesics, computation of Fréchet means and calculation of curvature in wald space. This work is intended to serve as a mathematical foundation for further geometric and statistical research on this space.

We describe a relationship between the monopole Floer homology of three-manifolds and the geometry of Riemann surfaces. For an automorphism $�\phi$ of a compact Riemann surface $�\Sigma$ with quotient $�P^1$, there is a natural correspondence between theta characteristics $�L$ on $�\Sigma$ which are invariant under $�\phi$ and self-conjugate $�{\text{spin}}^c$ structures $�s_L$ on the mapping torus $�M_{\phi }$ of $�\phi$. We show that the monopole Floer homology groups of $��(�M_{\phi }�,�s_L�)�$ are explicitly determined by the eigenvalues of the (lift of the) action of $�\phi$ on $�$

In this paper, a new general approach is developed to construct and study Lebesgue-type decompositions of linear operators or relations $�T$ in the Hilbert space setting. The new approach allows to introduce an essentially wider class of Lebesgue-type decompositions than what has been studied in the literature so far. The key point is that it allows a nontrivial interaction between the closable and the singular components of $�T$. The motivation to study such decompositions comes from the fact that they naturally occur in the corresponding Lebesgue-type decomposition for pairs of quadratic forms. The approach built in this paper uses so-called complementation in Hilbert spaces, a notion going back to de Branges and Rovnyak.

We find a finite free resolution of the counit of the free unitary quantum groups of van Daele and Wang and, more generally, Bichon's universal cosovereign Hopf algebras with a generic parameter matrix. This allows us to compute Hochschild cohomology with one-dimensional coefficients for all these Hopf algebras. In fact, the resolutions can be endowed with a Yetter–Drinfeld structure. General results of Bichon then allow us to compute also the corresponding bialgebra cohomologies. Finding the resolution rests on two pillars. We take as a starting point the resolution for the free orthogonal quantum group presented by Collins, Härtel, and Thom or its algebraic generalization to quantum symmetry groups of bilinear forms due to Bichon. Then, we make use of the fact that the free unitary quantum groups and some of its non-Kac versions can be realized as a glued free product of a (non-Kac) free orthogonal quantum group with $�Z_2$, the finite group of order 2. To obtain the resolution also for more general universal cosovereign Hopf algebras, we extend Gromada's proof from compact quantum groups to the framework of matrix Hopf algebras. As a by-product of this approach, we also obtain a projective resolution for the freely modified bistochastic quantum groups. Only a special subclass of free unitary quantum groups and universal cosovereign Hopf algebras decompose as a glued free product in the described way. In order to verify that the sequence we found is a free resolution in general (as long as the parameter matrix is generic, two conditions which are automatically fulfilled in the free unitary quantum group case), we use the theory of Hopf bi-Galois objects and Bichon's results on monoidal equivalences between the categories of Yetter–Drinfeld modules over universal cosovereign Hopf algebras for different parameter matrices.