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

Let $��(�K�,�\sigma �)�$ be a simple-artinian ring with involution. This means that $�K$ is isomorphic to a matrix ring over a ring $�k$ that is either a skew field or the direct sum of a skew field and its opposite, and $�\sigma$ is given in terms of an involution $�\tau$ of $�k$. We show that an arbitrary pseudo-quadratic module $�\Theta$ defined over $��(�K�,�\sigma �)�$ can be obtained by a tensor product construction from a pseudo-quadratic module defined over $��(�k�,�\tau �)�$ and we apply this result to give a uniform description of arbitrary pseudo-maximal parabolic subgroups of arbitrary classical groups in terms of pseudo-quadratic modules.

We study an optimal partition problem on the sphere, where the cost functional is associated with the fractional $�Q$-curvature in terms of the conformal fractional Laplacian on the sphere. By leveraging symmetries, we prove the existence of a symmetric minimal partition through a variational approach. A key ingredient in our analysis is a new Hölder regularity result for symmetric functions in a fractional Sobolev space on the sphere. As a byproduct, we establish the existence of infinitely many solutions to a nonlocal weakly coupled competitive system on the sphere that remain invariant under a group of conformal diffeomorphisms and we investigate the asymptotic behavior of least-energy solutions as the coupling parameters approach negative infinity.

In the Lebesgue decomposition of a lower semibounded sesquilinear form, the corresponding regular and singular parts are mutually singular. The more general Lebesgue-type decompositions studied here allow components that need not be mutually singular anymore. In the new situation, the earlier basic orthogonal space decomposition in the background is now replaced by a nonorthogonal decomposition in the sense of de Branges and Rovnyak. The relevant theory is based on Lebesgue-type decompositions for linear operators and relations via a so-called representing map. This map also makes it possible to formulate explicit analogs for representation theorems for lower semibounded forms that are not necessarily closed or closable. This new representation also appears naturally in the convergence of monotone sequences of lower semibounded forms.

For any compactly generated triangulated category, we introduce two topological spaces, the shift spectrum and the shift-homological spectrum. We use them to parametrise a family of thick subcategories of the compact objects, which we call radical. These spaces can be viewed as non-monoidal analogues of the Balmer and homological spectra arising in tensor-triangular geometry: we prove that for monogenic tensor-triangulated categories, the Balmer spectrum is a subspace of the shift spectrum. To construct these analogues, we utilise quotients of the module category, rather than the lattice theoretic methods which have been adopted in other approaches. We characterise radical thick subcategories and show in certain cases, such as the perfect derived categories of tame hereditary algebras or monogenic tensor-triangulated categories, that every thick subcategory is radical. We establish a close relationship between the shift-homological spectrum and the set of irreducible integral rank functions, and provide necessary and sufficient conditions for every radical thick subcategory to be given by an intersection of kernels of rank functions. In order to facilitate these results, we prove that both spaces we introduce may equivalently be described in terms of the Ziegler spectrum.

Studying Manin's program for a family of spherical log Fano threefolds, we determine the asymptotic number of integral points whose height associated with an arbitrary ample line bundle is bounded. This confirms a recent conjecture by Santens and sheds new light on the logarithmic analog of Iitaka fibrations, which have not yet been adequately formulated.

This article investigates the splitting problem for finitely generated projective modules $�P$ over affine algebras over algebraically closed fields and their polynomial extensions. We then address an open question due to M. Roitman on monic inversion principle for projective modules and prove it in the affirmative for finitely generated rings. For affine algebras over $�{\overline{F}}_p$, we prove a monic inversion principle for ideals. We also exhibit some applications.

Very recently in [Das et al., Expo. Math. 43(3):125653, 2025], statistically characterized subgroups have been investigated for certain types of nonarithmetic sequences. Building on this work, we investigate further and demonstrate that, for a particular class of nonarithmetic sequences, the statistically characterized subgroups coincide with the corresponding characterized subgroups. It had already been shown that statistically characterized subgroups corresponding to arithmetic sequences cannot be characterized by any sequence [Das et al., Bull. Sci. Math. 179(2):103157, 2022], and further, they are always of the size of the continuum. From the very beginning, it has been an open question as to whether statistically characterized subgroups can be small in size, that is, countably infinite. Our observation thus sheds new light on the crucial role of sequences generating subgroups of the circle group, and at the same time, one can subsequently identify a class of sequences for which statistically characterized subgroups are countably infinite. This result provides a negative solution to Problem 2.16 posed in [Das et al., Expo. Math. 43(3):125653, 2025] and Question 6.3 from [Dikranjan et al., Fund. Math. 249:185-209, 2020]. Additionally, our findings resolve several open problems from [Dikranjan et al., Topo. Appl., 2025].