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

In this article, we prove that Ramsey's theorem for pairs and two colors is a $��\forall �{\Pi }_4^0�$ conservative extension of $��{\mathrm{RCA}}_0�+�B�{\Sigma }_2^0�$, where a $��\forall �{\Pi }_4^0�$ formula consists of a universal quantifier over sets followed by a $�{\Pi }_4^0$ formula. The proof is an improvement of a result by Patey and Yokoyama and a step toward the resolution of the longstanding question of the first-order part of Ramsey's theorem for pairs. For this, we introduce a new general technique for proving $�{\Pi }_4^0$-conservation theorems.

Let $��f�:�N�\to �(�M�,�g�)�$ be a two-sided, complete, stable, minimal, immersed hypersurface. In this paper, we establish various vanishing theorems for the space of $�L^2$-harmonic forms and spinors (when $�M$ is additionally spin) under suitable positive curvature assumptions on the ambient manifold. Our results in the setting of forms extend to higher dimensions and more general ambient Riemannian manifolds previous vanishing theorems due to Tanno [J. Math. Soc. Japan 48 (1996), no. 4, 761–768] and Zhu [Nonlinear Anal. 75 (2012), no. 13, 5039–5043]. In the setting of spin manifolds, our results allow to conclude, for instance, that any oriented, complete, stable, minimal, immersed hypersurface of $�R^m$ or $�S^m$ carries no non-trivial $�L^2$-harmonic spinors. Finally, analogous results are proved for strongly stable constant mean curvature hypersurfaces.

We develop a new approach to the study of spectral asymmetry. Working with the operator $��curl�:�=�\ast �d�$ on a connected oriented closed Riemannian 3-manifold, we construct, by means of microlocal analysis, the asymmetry operator — a scalar pseudodifferential operator of order $��-�3�$. The latter is completely determined by the Riemannian manifold and its orientation, and encodes information about spectral asymmetry. The asymmetry operator generalises and contains the classical eta invariant traditionally associated with the asymmetry of the spectrum, which can be recovered by computing its regularised operator trace. Remarkably, the whole construction is direct and explicit.

We develop a probabilistic approach to the celebrated Jacobian conjecture, which states that any Keller map (i.e. any polynomial mapping $��F�:�C^n�\to �C^n�$ whose Jacobian determinant is a non-zero constant) has a compositional inverse which is also a polynomial. The Jacobian conjecture may be formulated in terms of a problem involving labellings of rooted trees; we give a new probabilistic derivation of this formulation using multi-type branching processes. Thereafter, we develop a simple and novel approach to the Jacobian conjecture in terms of a problem involving shuffling subtrees of $�d$-Catalan trees, that is, planar $�d$-ary trees. We also show that, if one can construct a certain Markov chain on large $�d$-Catalan trees which updates its value by randomly shuffling certain nearby subtrees, and in such a way that the stationary distribution of this chain is uniform, then the Jacobian conjecture is true. Finally, we use the local limit theory of large random trees to show that the subtree shuffling conjecture is true in a certain asymptotic sense, and thereafter use our machinery to prove an approximate version of the Jacobian conjecture, stating that inverses of Keller maps have small power series coefficients for their high-degree terms.

We investigate the Hénon–Lane–Emden system defined by

We investigate projection constants within classes of multivariate polynomials over finite-dimensional real Hilbert spaces. Specifically, we consider the projection constant for spaces of spherical harmonics and spaces of homogeneous polynomials as well as for spaces of polynomials of finite degree on the unit sphere. We establish a connection between these quantities and certain weighted $�L_1$-norms of specific Jacobi polynomials. As a consequence, we present exact formulas, computable expressions, and asymptotically accurate estimates for them.

The seminal paper (Stafford, J. Lond. Math. Soc. (2) 18 (1978), no. 3, 429–442) was a major step forward in our understanding of Weyl algebras. Beginning with Serre's Theorem on free summands of projective modules and Bass' Stable Range Theorem in commutative algebra, we attempt to trace the origins of this work and explain how it led to Stafford's construction of non-holonomic simple modules over Weyl algebras. We also describe Bernstein–Lunts' geometric construction of infinite families of non-holonomic simple modules. We recall more recent developments related to Weyl algebras, especially that of parametrising right ideals in the first Weyl algebra and its relation to Calogero–Moser spaces. Finally, we revisit Stafford's results in the context of quantised symplectic singularities, where they lead naturally to open problems on the behaviour of simple modules.

Sunflowers, or $�\Delta$-systems, are a fundamental concept in combinatorics introduced by Erdős and Rado in their paper: [J. London Math. Soc. (1) 35 (1960), 85–90]. A sunflower is a collection of sets where all pairs have the same intersection. This paper explores the wide-ranging applications of sunflowers in computer science and combinatorics. We discuss recent progress toward the sunflower conjecture and present a short elementary proof of the best known bounds for the robust sunflower lemma.