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

In the setting of length PI spaces satisfying a suitable deformation property, it is known that each isoperimetric set has an open representative. In this paper, we construct an example of a length PI space (without the deformation property) where an isoperimetric set does not have any representative whose topological interior is nonempty. Moreover, we provide a sufficient condition for the validity of the deformation property, consisting in an upper Laplacian bound for the squared distance functions from a point. Our result applies to essentially nonbranching $��\mathrm{MCP}�(�K�,�N�)�$ spaces, thus in particular to essentially nonbranching $��\mathrm{CD}�(�K�,�N�)�$ spaces and to many Carnot groups and sub-Riemannian manifolds. As a consequence, every isoperimetric set in an essentially nonbranching $��\mathrm{MCP}�(�K�,�N�)�$ space has an open representative, which is also bounded whenever a uniform lower bound on the volumes of unit balls is assumed.

We study representations of finite groups on Stanley–Reisner rings of simplicial complexes and on lattice points in lattice polytopes. The framework of translative group actions allows us to use the theory of proper colorings of simplicial complexes without requiring an explicit coloring to be given. We prove that the equivariant Hilbert series of a Cohen–Macaulay simplicial complex under a translative group action admits a rational expression whose numerator is a positive integer combination of irreducible characters. This implies an analogous rational expression for the equivariant Ehrhart series of a lattice polytope with a unimodular triangulation that is invariant under a translative group action. As an application, we study the equivariant Ehrhart series of alcoved polytopes in the sense of Lam and Postnikov and derive explicit results in the case of order polytopes and of Lipschitz poset polytopes.

In this work, we classify the thick subcategories of the bounded derived category of dg modules over a Koszul complex on any list of elements in a regular ring. This simultaneously recovers a theorem of Stevenson when the list of elements is a regular sequence and the classification of thick subcategories for an exterior algebra over a field (via the BGG correspondence). One of the major ingredients is a classification of thick tensor submodules of perfect curved dg modules over a graded commutative noetherian ring concentrated in even degrees, recovering a theorem of Hopkins and Neeman. We give several consequences of the classification result over a Koszul complex, one being that the lattice of thick subcategories of the bounded derived category is fixed by Grothendieck duality.

We give an upper bound for the norm of the determinant of additively indecomposable, totally positive definite quadratic forms defined over the ring of integers of totally real number fields. We apply these results to find lower and upper bounds for the minimal ranks of $�n$-universal quadratic forms. For $��Q��\left(�\sqrt{�2�}�\right)��,��Q��\left(�\sqrt{�3�}�\right)��,��Q��\left(�\sqrt{�5�}�\right)��,��Q��\left(�\sqrt{�6�}�\right)��$, and $��Q�\left(�\sqrt{�21�}�\right)�$, we classify, up to equivalence, all classical, additively indecomposable binary quadratic forms.

Let $�x_{�k�\times �p�}$ be a $��k�\times �p�$ matrix of variables and let $��F�\left[�x_{�k�\times �p�}�\right]�$ be the polynomial ring in these variables. Given two weak compositions $��\alpha �,�\beta �{\vDash }_0�n�$ of lengths $��\ell �\left(�\alpha �\right)�=�k�$ and $��\ell �\left(�\beta �\right)�=�p�$, we study the ideal $��I_{�\alpha �,�\beta �}�\subseteq �F��[�x_{�k�}$

We consider the relativistic Vlasov–Maxwell system in three dimensions and study the limiting asymptotic behavior as $��t�\to �\infty �$ of solutions launched by small, compactly supported initial data. In particular, we prove that such solutions scatter to a modification of the free-streaming asymptotic profile. More specifically, we show that the spatial average of the particle distribution function converges to a smooth, compactly supported limit and establish the precise, self-similar asymptotic behavior of the electric and magnetic fields, as well as, the macroscopic densities and their derivatives in terms of this limiting function. Upon constructing the limiting fields, a modified $�L^{\infty }$ scattering result for the particle distribution function along the associated trajectories of free transport corrected by the limiting Lorentz force is then obtained. When the limiting charge density does not vanish, our estimates are sharp up to a logarithmic correction. However, when this quantity is identically zero in the limit, the limiting current density and fields may also vanish, which gives rise to decay rates that are faster than those attributed to the dispersive mechanisms in the system.

Let $�V$ be an affine space over field $�k$, which is characteristic zero. Let $��G�\subseteq �\mathrm{SL}�\left(�V�\right)�$ be a finite abelian group, and denote by $�S$ the $�G$-invariant subring of the polynomial ring $��k�\left[�V�\right]�$. It is shown that the singularity category $��D_{�s�g�}��\left(�S�\right)��$ recovers the reduced singular locus of $��\mathrm{Spec}�\left(�S�\right)�$.

We commence by constructing the mirabolic quantum Schur algebra, utilizing the convolution algebra defined on the variety of triples of two $�n$-step partial flags and a vector. Subsequently, we employ a stabilization procedure to derive the mirabolic quantum $�{\mathrm{gl}}_n$. Then we present the geometric approach of the mirabolic Schur–Weyl duality of type $�A$.

We verify part of a conjecture of Campana predicting that rational points on the weakly special nonspecial simply connected smooth projective threefolds constructed by Bogomolov–Tschinkel are not dense. To prove our result, we establish fundamental properties of moduli spaces of orbifold maps, and prove a dimension bound for such moduli spaces by using the recent extension of Kobayashi–Ochiai's finiteness theorem for Campana's orbifold pairs.

We prove a theorem on the intersection theory over a Noetherian local ring $�R$, which gives a new proof of a classical theorem of Rees about degree functions. To obtain this, we define an intersection product on schemes that are proper and birational over such rings $�R$, using the theory of rational equivalence developed by Thorup, and the Snapper–Mumford–Kleiman intersection theory for proper schemes over an Artinian local ring. Our development of this product is essentially self-contained. As a central component of the proof of our main theorem, we extend to arbitrary Noetherian local rings a formula by Ramanujam that computes Hilbert–Samuel multiplicities. In the final section, we express mixed multiplicities in terms of intersection theory and conclude from this that they satisfy a certain multilinearity condition. Then we interpret some theorems of Rees and Sharp and of Teissier about mixed multiplicities over 2-dimensional excellent local rings in terms of our intersection product.