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

For a fixed prime power $�q$ and natural number $�d$, we consider a random polynomial

We show that some subgroups of the orientation-preserving circle homeomorphism group with invariant veering pairs of laminations are hyperbolic 3-orbifold groups. On the way, we show that from a veering pair of laminations, one can construct a loom space (in the sense of Schleimer–Segerman) as a quotient. Our approach does not assume the existence of any 3-manifold to begin with, so this is a geometrization-type result, and supersedes some of the results regarding the relation among veering triangulations, pseudo-Anosov flows, taut foliations in the literature.

Motivated by a conjecture concerning Igusa local zeta functions for intersection posets of hyperplane arrangements, we introduce and study the Poincaré-extended $��a�b�$-index, which generalizes both the $��a�b�$-index and the Poincaré polynomial. For posets admitting $�R$-labelings, we give a combinatorial description of the coefficients of the extended $��a�b�$-index, proving their nonnegativity. In the case of intersection posets of hyperplane arrangements, we prove the above conjecture of the second author and Voll as well as another conjecture of the second author and Kühne. We also define the pullback $��a�b�$-index, generalizing the $��c�d�$-index of face posets for oriented matroids. Our results recover, generalize, and unify results from Billera–Ehrenborg–Readdy, Bergeron–Mykytiuk–Sottile–van Willigenburg, Saliola–Thomas, and Ehrenborg. This connection allows us to translate our results into the language of quasisymmetric functions, and — in the special case of symmetric functions — pose a conjecture about Schur positivity. This conjecture was strengthened and proved by Ricky Liu, and the proof appears as an appendix.

We consider the Hausdorff dimension of random covering sets formed by balls with centres chosen independently at random according to an arbitrary Borel probability measure on $�R^d$ and radii given by a deterministic sequence tending to zero. We prove, for a certain parameter range, the conjecture by Ekström and Persson concerning the exact value of the dimension in the special case of radii $�{�\left(�n^{�-�\alpha �}�\right)�}_{�n�=�1�}^{\infty }$. For balls with an arbitrary sequence of radii, we find sharp bounds for the dimension and show that the natural extension of the Ekström–Persson conjecture is not true in this case. Finally, we construct examples demonstrating that there does not exist a dimension formula involving only the lower and upper local dimensions of the measure and a critical parameter determined by the sequence of radii.

We consider the determinantal point processes associated with the spectral projectors of a Schrödinger operator on $�R$, with a smooth confining potential. In the semiclassical limit, where the number of particles tends to infinity, we obtain a Szegő-type central limit theorem for the fluctuations of smooth linear statistics. More precisely, the Laplace transform of any statistic converges without renormalisation to a Gaussian limit with a $�H^{�1�/�2�}$-type variance, which depends on the potential. In the one-well (one-cut) case, using the quantum action-angle theorem and additional micro-local tools, we reduce the problem to the asymptotics of Fredholm determinants of certain approximately Toeplitz operators. In the multi-cut case, we show that for generic potentials, a similar result holds and the contributions of the different wells are independent in the limit.

Every saturated fusion system corresponds to a group-like structure called a regular locality. In this paper we study (suitably defined) normalizers and centralizers of partial subnormal subgroups of regular localities. This leads to a reasonable notion of normalizers and centralizers of subnormal subsystems of fusion systems.

Let $�Y_2$ be the Yangian associated to the general linear Lie algebra $�{\mathrm{gl}}_2$, defined over an algebraically closed field $�k$ of characteristic $��p�>�0�$. In this paper, we study the representation theory of the restricted Yangian $�Y_2^{�\left[�p�\right]�}$. This leads to a description of the representations of $�{\mathrm{gl}}_{�2�n�}$, whose $�p$-character is nilpotent with Jordan type given by a two-row partition $��(�n�,�n�)�$.

A classical problem, raised by Fuchs in 1960, asks to classify the abelian groups which are groups of units of some rings. In this paper, we consider the case of finitely generated abelian groups, solving Fuchs' problem for such groups with the additional assumption that the torsion subgroups are small, for a suitable notion of small related to the Prüfer rank. As a concrete instance, we classify for each $��n�⩾�2�$ the realisable groups of the form $��Z�/�n�Z�\times �Z^r�$. Our tools require an investigation of the adjoint group of suitable radical rings of odd prime power order appearing in the picture, giving conditions under which the additive and adjoint groups are isomorphic. In the last section, we also deal with some groups of order a power of 2, proving that the groups of the form $��Z�/�4�Z�\times �Z�/�2^u�Z�$ are realisable if and only if $��0�⩽�u�⩽�3�$ or $��2^u�+�1�$ is a Fermat prime.