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

We study the topological basis problem under ZF, the Zermelo-Fraenkel axiomatic set theory without the Axiom of Choice. We prove that under AD+$�{\mathrm{DC}}_R$, the class of regular topologies on $�R$ has a three element basis and the class of Hausdorff topologies on $�{\omega }_1$ has a single-element basis. In particular, AD+$��V�=�L�\left(�R�\right)�$ implies that the class of uncountable regular spaces has a four element basis.

We define floating bodies in the class of $�n$-dimensional ball-convex bodies. A right derivative of volume of these floating bodies leads to a surface area measure for ball-convex bodies which we call relative affine surface area. We show that this quantity is a rigid motion invariant, upper semicontinuous valuation.

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.