Bulletin of the London Mathematical Society最新文献

In this paper, we analyze the Cauchy problem for the two-component Camassa–Holm (2CH) system in weighted Sobolev spaces, achieving two key results. First, we establish a new irregular Riemann–Hilbert (RH) problem, without the normalization conditions, for the associated Cauchy problem. Second, we identify six main asymptotic sectors in the space–time region and derive the leading-order asymptotics for these sectors using the $�\overline{\partial }$-nonlinear steepest descent method. This confirms the soliton resolution conjecture for the 2CH system. Notably, the 2CH system exhibits unique characteristics within the RH formalism, distinguishing it from the classical Camassa–Holm (CH) equation. We prove that the reconstruction process remains unique even when the RH problem lacks uniqueness. Additionally, numerical simulations are provided to validate these asymptotic results.

The purpose of this paper is to establish a connection between a one-dimensional random walk in a random sparse environment and the random pinning model. We show that the grand canonical partition function of the pinning model coincides with the mean number of returns to the origin for a random walk in a random sparse environment averaged over the randomness location. We obtain thereof some information on the integrability of the number of return times in the annealed and partially annealed setups.

A matrix $��A�\in �R^{�m�\times �n�}�$ is strictly sign regular (or sign regular) if for each $��1�⩽�k�⩽�\min �\{�m�,�n�\}�$, all (nonzero) $��k�\times �k�$ minors of $�A$ have the same sign. This class of matrices contains the totally positive matrices, and was first studied by Schoenberg in 1930 to characterize variation diminution, a fundamental property in total positivity theory. In this article, we classify all surjective linear mappings $��L�:�R^{�m�\times �n�}�\to �R^{�m�\times �n�}�$ that preserve: (i) sign regularity and (ii) sign regularity with a given sign pattern, as well as (iii) strict versions of these.

For any given set $��E�\subset �[�1�,�2�]�$, we discuss a fractal frequency-localized version of the $�L^p$ local smoothing estimates for the half-wave propagator with times in $�E$. A conjecture is formulated in terms of a quantity involving the Assouad spectrum of $�E$ and the Legendre transform. We validate the conjecture for radial functions. We also prove a similar result for fractal-time $��L^2�\to �L^q�$ and square function bounds, for arbitrary $�L^2$ functions and general time sets. We formulate a conjecture for $��L^p�\to �L^q�$ generalizations.

Stable envelopes, introduced by Maulik and Okounkov, provide a family of bases for the equivariant cohomology of symplectic resolutions. They are part of a fascinating interplay between geometry, combinatorics and integrable systems. In this expository article, we give a self-contained introduction to cohomological stable envelopes of type $�A$ bow varieties. Our main focus is on the existence and the orthogonality properties of stable envelopes for bow varieties. The restriction to this specific class of varieties allows us to illustrate the theory combinatorially and to provide simplified proofs, both laying a basis for explicit calculations.

We exhibit finitely generated torsion-free groups for which any action on any finite-dimensional CW-complex with finite Betti numbers has a global fixed point.

We give an unconditional proof of the Coba conjecture for wonderful compactifications of adjoint type for semisimple Lie groups of type $�A_n$. We also give a proof of a slightly weaker conjecture for wonderful compactifications of adjoint type for arbitrary Lie groups.

We prove the optimal $�C^{�1�,�1�}$ regularity of the volume function on the big cone of a projective manifold, and investigate its regularity when restricted to segments moving in ample directions.

We show that the intersection of the rational derived series of a one-relator group is rationally perfect and is normally generated by a single element. As a corollary, we characterise precisely when a one-relator group is residually rationally solvable.

Let $�A$ be a standard graded Artinian $�K$-algebra over a field of characteristic zero. We prove that the failure of strong Lefschetz property (SLP) for $�A$ is equivalent to the osculating defect of a certain rational variety. Our results extend previous works by eliminating the need for the defining ideal of $�A$ to be equigenerated and by extending the study of weak Lefschetz property/SLP in a fixed degree to the study of the SLP in any degree and any range. As a consequence, we provide a geometric interpretation of the vanishing of a higher Hessian, extending the classical Gordan–Noether criterion. Moreover, we reobtain some foundational results, including the presence of SLP for codimension 2 Artinian algebras, and the SLP for Artinian Gorenstein algebras with Hilbert function (1,3,6,6,3,1).