Mathematische Nachrichten最新文献

In this work, we investigate the Sobolev space $��H^1��\left(�\partial �\Omega �\right)��$ on a strongly Lipschitz boundary $��\partial �\Omega �$, that is, $�\Omega$ is a strongly Lipschitz domain (not necessarily bounded). In most of the literature, this space is defined via charts and Sobolev spaces on flat domains. We show that there is a different approach via differential operators on $�\Omega$ and a weak formulation directly on the boundary that leads to the same space. This second characterization of $��H^1��\left(�\partial �\Omega �\right)��$ is in particular of advantage, when it comes to traces of $��H�(�curl�,�\Omega �)�$ vector fields.

The Dirichlet form is a generalization of the Laplacian, heavily used in the study of many diffusion-like processes. In this paper, we present a nonstandard representation theorem for the Dirichlet form, showing that the usual Dirichlet form can be well-approximated by a hyperfinite sum. One of the main motivations for such a result is to provide a tool for directly translating results about Dirichlet forms on finite or countable state spaces to results on more general state spaces, without having to translate the details of the proofs. As an application, we compare the Dirichlet forms of two general Markov processes by applying the transfer of the well-known comparison theorem for finite Markov processes.

In a recent paper, we showed that a subspace of a real $�{\mathrm{JBW}}^{\ast }$-triple is an $�M$-summand if and only if it is a $�{\mathrm{weak}}^{\ast }$-closed triple ideal. As a consequence, $�M$-ideals of real $�{\mathrm{JB}}^{\ast }$-triples, including real $�C^{\ast }$-algebras, real $�{\mathrm{JB}}^{\ast }$-algebras and real TROs, correspond to norm-closed triple ideals. In this paper, we extend this result by identifying the $�M$-ideals in (possibly non-self-adjoint) real operator algebras and Jordan operator algebras. The argument for this is necessarily different. We also give simple characterizations of one-sided $�M$-ideals in real operator algebras, and give some applications to that theory.

In this work, we study the existence of various classes of standing waves for a nonlinear Schrödinger system with quadratic interaction, along with a harmonic or partially harmonic potential. We establish the existence of ground-state normalized solutions for this system, which serve as local minimizers of the associated functionals. To address the difficulties raised by the potential term, we employ profile decomposition and concentration-compactness principles. The absence of global energy minimizers in critical and supercritical cases leads us to focus on local energy minimizers. Positive results arise in scenarios of partial confinement, attributed to the spectral properties of the associated linear operators. Furthermore, we demonstrate the existence of a second normalized solution using the Mountain-pass geometry, effectively navigating the difficulties posed by the nonlinear terms. We also explore the asymptotic behavior of local minimizers, revealing connections with unique eigenvectors of the linear operators. Additionally, we identify global and blow-up solutions over time under specific conditions, contributing new insights into the dynamics of the system.

We establish two-pointed Prym–Brill–Noether loci with special vanishing at two points, and determine their K-theory classes when the dimensions are as expected. The classes are derived by the applications of a formula for the K-theory of certain vexillary degeneracy loci in type D. In particular, we show a two-pointed version of the Prym–Petri theorem on the expected dimension in the general case, with a coupled Prym–Petri map. Our approach is inspired by the work on pointed cases by Tarasca, and we generalize unpointed cases by De Concini-Pragacz and Welters.

Based on variational methods, we study the spectral problem for the subelliptic $�p$-Laplacian arising from smooth Hörmander vector fields. We derive the smallest eigenvalue, prove its simplicity and isolatedness, establish the positivity of the first eigenfunction, and show Hölder regularity of eigenfunctions with respect to the control distance. Moreover, we determine the best constant for the $�L^p$-Poincaré–Friedrichs inequality for Hörmander vector fields as a byproduct.

We construct a correspondence between Mori dream regions arising from small modifications of normal projective varieties and $�C^{\ast }$-actions on polarized pairs which are bordisms. Moreover, we show that the Mori dream regions constructed in this way admit a chamber decomposition on which the models are the geometric quotients of the $�C^{\ast }$-action. In addition, we construct, from a given $�C^{\ast }$-action on a polarized pair for which there exist at least two admissible geometric quotients, a $�C^{\ast }$-equivariantly birational $�C^{\ast }$-variety, whose induced action is a bordism, called the pruning of the variety.