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

We prove Pólya's conjecture for the eigenvalues of the Dirichlet Laplacian on annular domains. Our approach builds upon and extends the methods we previously developed for disks and balls. It combines variational bounds, estimates of Bessel phase functions, refined lattice point counting techniques and a rigorous computer-assisted analysis. As a by-product, we also derive a two-term upper bound for the Dirichlet eigenvalue counting function of the disk, improving upon Pólya's original estimate.

The main goal of this paper is to establish the higher dimensional Nevanlinna theory in tropical geometry. We first develop a theory of tropical meromorphic functions (tropical holomorphic maps) in several real variables, such as the proximity function, counting function and characteristic function, the first main theorem, higher dimensional tropical versions of the logarithmic derivative lemmas. Based on this, for algebraically non-degenerate tropical holomorphic maps $�f$ with subnormal growth from $�R^n$ into tropical projective space $�{\mathrm{TP}}^m$ intersecting tropical hypersurfaces $�{�\left\{�V_{P_j}�\right\}�}_{�j�=�1�}^q$ with degree $�d_j$, we then obtain the second main theorem

Replacing operators with continuous operator-valued functions, we prove time-dependent versions of well-known results on compressions and diagonals of bounded operators. The setting of smooth functions is also addressed. Our results have no analogues in the literature and rely on a new technique. The results are especially transparent for self-adjoint operators.

We revisit certain one-parameter families of affine covers arising naturally from Euler's integral representation of hypergeometric functions. We introduce a partial compactification of this family. We show that the zeta function of the fibers in the family can be written as an explicit product of $�L$-series attached to nondegenerate hypergeometric motives and zeta functions of tori, twisted by Hecke Grössencharacters. This permits a combinatorial algorithm for computing the Hodge numbers of the family.

We study the weighted constant scalar curvature Kähler equations on mildly singular Kähler varieties. Assuming the existence of a suitable resolution of singularities, we establish the existence of singular weighted cscK metrics when the weighted Mabuchi functional is coercive for an extremal weight. This extends the works of Chen–Cheng and He to the singular weighted setting. Moreover, we provide a method for constructing examples of singular cscK metrics inspired by the work of Arezzo–Pacard. In contrast to the usual gluing techniques, our approach does not require a precise understanding about of the metric behavior near the singular locus.

We prove that globally hyperbolic compact anti-de Sitter (2+1)-spacetimes with a strictly convex spacelike boundary that is either smooth or polyhedral and whose holonomy is close to Fuchsian are determined by the induced metric on the boundary.

In this paper, we study quadratic growth solutions $�u$ of fully nonlinear elliptic equations of the form $��F�\left(�D^2�u�\right)�=�f�$ in $�R^n$, where $�f$ is periodic and $�F$ may be not uniformly elliptic. The existence of solutions and Liouville-type results in the whole space and exterior domains are established, which generalize the classical results when $�f$ is constant. As applications, the corresponding results are given to $�k$-Hessian equations, which include the celebrated results for Monge–Ampère equations.

Multi-layered structures are ubiquitous in the construction of metamaterial devices to enable advanced waveguiding functionalities. This paper presents several contributions to the mathematical understanding of subwavelength resonances in multi-layered nested structures with arbitrarily shaped resonators. First, by employing the Dirichlet-to-Neumann map and a variational framework, we derive an original formula for the subwavelength resonance frequencies in terms of the eigenvalues of a new type of capacitance matrix, thereby extending the celebrated Minnaert resonance from single-layered to general nested configurations. We then introduce a finite-layered dimer-type metamaterial in which the geometric symmetry is deliberately broken to form a ring defect, and rigorously demonstrate that such a defect acts as an effective radial laminate waveguide, supporting wave localization and propagation along the defect. Furthermore, for structures with a sufficiently large number of layers, we prove the existence and uniqueness of an eigenvalue in the spectral gap of the defect structure, establishing the existence of a unique localized defect mode. Finally, numerical simulations are provided to support and illustrate the theoretical results.