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

I study several problems about the symmetric space associated with the Lie group $��S�L�(�n�,�R�)�$. These problems are connected to an algorithm based on Poincaré's Fundamental Polyhedron Theorem, designed to determine generalized geometric finiteness properties for subgroups of $��S�L�(�n�,�R�)�$. The algorithm is analogous to the original one in hyperbolic spaces, while the Riemannian distance is replaced by an $��S�L�(�n�,�R�)�$-invariant premetric. The main results of this paper are twofold. In the first part, I focus on questions that occurred in generalizing Poincaré's Algorithm to my symmetric space. I describe and implement an algorithm that computes the face-poset structure of finitely sided polyhedra, and construct an angle-like function between hyperplanes. In the second part, I study further questions related to hyperplanes and Dirichlet–Selberg domains in my symmetric space. I establish several criteria for the disjointness of hyperplanes and classify particular Abelian subgroups of $��S�L�(�3�,�R�)�$ based on whether their Dirichlet–Selberg domains are finitely sided or not.

We introduce a notion of tame ramification for general finite covers. When specialized to the separable case, it extends to higher dimensions the classical notion of tame ramification for Dedekind domains and curves and sits nicely in between other notions of tame ramification in arithmetic geometry. However, when applied to the Frobenius map, it naturally yields the notion of center of $�F$-purity (aka compatibly $�F$-split subvariety). As an application, we describe the behavior of centers of $�F$-purity under finite covers — it all comes down to a transitivity property for tame ramification in towers.

We investigate the distribution of values of cubic Dirichlet $�L$-functions at $��s�=�1�$. Following ideas of Granville and Soundararajan for quadratic $�L$-functions, we model the distribution of $��L�(�1�,�\chi �)�$ by the distribution of random Euler products $��L�(�1�,�X�)�$ for certain family of random variables $��X�\left(�p�\right)�$ attached to each prime. We obtain a description of the proportion of $��\left|�L�\right(�1�,�\chi �\left)�\right|�$ that is larger or that is smaller than a given bound, and yield more light into the Littlewood bounds. Unlike the quadratic case, there is an asymmetry between lower and upper bounds for the cubic case, and small values are less probable than large values.

Using a two-dimensional version of the delta method, we establish an asymptotic formula for the number of rational points of bounded height on non-singular complete intersections of cubic and quadric hypersurfaces of dimension at least 23 over $��F_q��\left(�t�\right)��$, provided $��char�\left(�F_q�\right)�>�3�$. Under the same hypotheses, we also verify weak approximation.

Let $�A$ be a Noetherian domain and $�R$ be a finitely generated $�A$-algebra. We study several features regarding the generic freeness over $�A$ of an $�R$-module. For an ideal $��I�\subset �R�$, we show that the local cohomology modules $��H_I^i��\left(�R�\right)��$ are generically free over $�A$ under certain settings where $�R$ is a smooth $�A$-algebra. By utilizing the theory of Gröbner bases over arbitrary Noetherian rings, we provide an effective method to b make explicit the generic freeness over $�A$ of a finitely generated $�R$-module.

In this work, we prove the existence of time-periodic solutions to a model describing a ferrofluid flow heated from below. Navier–Stokes equations satisfied by the fluid velocity are coupled to the temperature equation and the magnetostatic equation satisfied by the magnetic potential. The magnetization is assumed to be parallel to the magnetic field and is given by a nonlinear magnetization law generalizing the Langevin law. The proof is based on a semi-Galerkin approximation and regularization methods together with the fixed point method.

The $�q$-Araki-Woods factor associated to a group of orthogonal transformations on a real separable Hilbert space $�H_R$ is full as soon as $��dim�H_R�⩾�2�$.