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

We establish the scaling limit of the geodesics to the root for the first-passage percolation distance on random planar maps. We first describe the scaling limit of the number of faces along the geodesics. This result enables us to compare the metric balls for the first-passage percolation and the dual-graph distance. It also enables us to give an upper bound for the diameter of large random maps. Then, we describe the scaling limit of the tree of first-passage percolation geodesics to the root via a stochastic coalescing flow of pure jump diffusions. Using this stochastic flow, we also construct some random metric spaces which we conjecture to be the scaling limits of random planar maps with high degrees. The main tool in this work is a time reversal of the uniform peeling exploration.

In this work, we study the Artin–Mazur zeta function for piecewise monotone functions acting on a compact interval of real numbers. In the case of unimodal maps, Milnor and Thurston [On iterated maps of the interval, in Dynamical systems (College Park, MD, 1986–87), vol. 1342 of Lecture Notes in Math., pp. 465–563. Springer, Berlin, 1988] gave a characterization for the rationality of the Artin–Mazur zeta function in terms of the orbit of the unique turning point under certain smoothness assumptions. We give a characterization for unimodal maps that does not depend on the smoothness of the map, and implies the previous result. We also show that for multimodal maps, the previous characterization does not hold. In the space of real polynomials of a given degree which is bigger than two, with all critical points being real, and having fixed multiplicities (that is known to be a smooth real manifold), there are real-analytic subvariety of codimention 1 such that every map of this subvariety has the same Artin–Mazur zeta function, which is a rational function. Moreover, all but one critical points of this family undergo independent bifurcations.

Let $�M_{�g�,�m�+�n�}$ be the moduli space of algebraic curves of genus $�g$ with $��m�⩾�1�$ boundaries and $��n�⩾�0�$ marked points, and $��H_c^{•}��\left(�M_{�m�+�n�}�\right)��$ its compactly supported cohomology group. We prove that the collection of $��S_m^{�o�p�}�\times �S_n�$-modules

We perform a systematic variational method for functionals depending on eigenvalues of Riemannian manifolds. It is based on a new concept of Palais–Smale (PS) sequences that can be constructed thanks to a generalization of classical min-max methods on $�C^1$ functionals to locally Lipschitz functionals. We prove convergence results on these PS sequences emerging from combinations of Laplace eigenvalues or combinations of Steklov eigenvalues in dimension 2.

We show that there exists a quasi-isometric embedding of the product of $�n$ copies of $�H_R^2$ into any symmetric space of non-compact type of rank $�n$, and there exists a bi-Lipschitz embedding of the product of $�n$ copies of the 3-regular tree $�T_3$ into any thick Euclidean building of rank $�n$ with co-compact affine Weyl group. This extends a previous result of Fisher–Whyte. The proof is purely geometrical, and the result also applies to the non–Bruhat–Tits buildings.

In this article we present new proofs for the boundedness and the compactness on $�{\ell }^2$ of the Rhaly matrices, also known as terraced matrices. We completely characterize when such matrices belong to the Schatten class $��S^q��\left(�{\ell }^2�\right)��$, for . Finally, we apply our results to study the Hadamard multipliers in weighted Dirichlet spaces, answering a question left open by Mashreghi–Ransford.

In our previous paper [Fei Hou, Fei Tao, Huicheng Yin, Global existence and scattering of small data smooth solutions to a class of quasilinear wave systems on $��R^2�\times �T�$, Preprint (2024), arXiv:2405.03242], for the $�Q_0$-type quadratic nonlinearities, we have shown the global well-posedness and scattering properties of small data smooth solutions to the quasilinear wave systems on $��R^2�\times �T�$. In this paper, we start to solve the global existence problem for the remaining $�Q_{�\alpha �\beta �}$-type nonlinearities. By combining these results, we have established the global well-posedness of small solutions on $��R^2�\times �T�$ for the general 3-D quadratically quasilinear wave systems when the related 2-D null conditions are fulfilled.

It is a major problem in analysis on metric spaces to understand when a metric space is quasisymmetric to a space with strong analytic structure, a so-called Loewner space. A conjecture of Kleiner, recently disproven by Anttila and the second author, proposes a combinatorial sufficient condition. The counterexamples constructed are all topologically one-dimensional, and the sufficiency of Kleiner's condition remains open for most other examples. A separate question of Kleiner and Schioppa, apparently unrelated to the problem above, asks about the existence of ‘analytically one-dimensional planes’: metric measure spaces quasisymmetric to the Euclidean plane but supporting a one-dimensional analytic structure in the sense of Cheeger. In this paper, we construct an example for which the conclusion of Kleiner's conjecture is not known to hold. We show that either this conclusion fails in our example or there exists an ‘analytically one-dimensional plane’. Thus, our construction either yields a new counterexample to Kleiner's conjecture, different in kind from those of Anttila and the second author, or a resolution to the problem of Kleiner–Schioppa.