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

A nonlinear version of Roth's theorem states that dense sets of integers contain configurations of the form $�x$, $��x�+�d�$, $��x�+�d^2�$. We obtain a multidimensional version of this result, which can be regarded as a first step toward effectivising those cases of the multidimensional polynomial Szemerédi theorem involving polynomials with distinct degrees. In addition, we prove an effective “popular” version of this result, showing that every dense set has some non-zero $�d$ such that the number of configurations with difference parameter $�d$ is almost optimal. Perhaps surprisingly, the quantitative dependence in this result is exponential, compared to the tower-type bounds encountered in the popular linear Roth theorem.

The Vlasov–Maxwell–Landau (VML) system and the Vlasov–Maxwell–Boltzmann (VMB) system are fundamental models in dilute collisional plasmas. In this paper, we are concerned with the hydrodynamic limits of both the VML and the noncutoff VMB systems in the entire space. Our primary objective is to rigorously prove that, within the framework of Hilbert expansion, the unique classical solution of the VML or noncutoff VMB system converges globally over time to the smooth global solution of the Euler–Maxwell system as the Knudsen number approaches zero. The core of our analysis hinges on deriving novel interplay energy estimates for the solutions of these two systems, concerning both a local Maxwellian and a global Maxwellian, respectively. Our findings address a problem in the hydrodynamic limit for Landau-type equations and noncutoff Boltzmann-type equations with a magnetic field. Furthermore, the approach developed in this paper can be seamlessly extended to assess the validity of the Hilbert expansion for other types of kinetic equations.

Let $�X$ be a subset of a Hilbert space. We prove that if $��v�:�X�\to �R^m�$ is such that

Moreover, if either $��m�\in �\{�1�,�2�,�3�\}�$ or $�X$ is convex, we prove the converse: We show that a map $��v�:�X�\to �R^m�$ that allows for a 1-Lipschitz, uniform distance preserving extension of any 1-Lipschitz map on a subset of $�X$ also satisfies the above set of inequalities. We also prove a similar continuity result concerning extensions of monotone maps. Our results hold true also for maps taking values in infinite-dimensional spaces.

We introduce a notion of algorithmic randomness for algebraic fields. We prove the existence of a continuum of algebraic extensions of $�Q$ that are random according to our definition. We show that there are noncomputable algebraic fields which are not random. We also partially characterize the index set, relative to an oracle, of the set of random algebraic fields computable relative to that oracle.

In order to carry out this investigation of randomness for fields, we develop computability in the context of the infinite Galois theory (where the relevant Galois groups are uncountable), including definitions of computable and computably enumerable Galois groups and computability of Haar measure on the Galois groups.

We show that, given $��k�⩾�3�$, there exist $�k$-player quantum XOR games for which the entangled bias can be arbitrarily larger than the bias of the game when the players are restricted to separable strategies. In particular, quantum entanglement can be a much more powerful resource than local operations and classical communication to play these games. This result shows a strong contrast to the bipartite case, where it was recently proved that, as a consequence of a noncommutative version of Grothendieck theorem, the entangled bias is always upper bounded by a universal constant times the one-way classical communication bias. In this sense, our main result can be understood as a counterexample to an extension of such a noncommutative Grothendieck theorem to multilinear forms.

We consider a nematic liquid crystal flow with partially free boundary in a smooth bounded domain in $�R^2$. We prove regularity estimates and the global existence of weak solutions enjoying partial regularity properties, and a uniqueness result.

For a log Calabi Yau pair ($��X�,�D�$) with $��X�\setminus �D�$ smooth affine, satisfying either a maximal degeneracy assumption or contains a Zariski dense torus, we prove under the condition that D is the support of a nef divisor that the structure constants defining a trace form on the mirror algebra constructed by Gross–Siebert are given by the naive curve counts defined by Keel–Yu. As a corollary, we deduce that the equality of the mirror algebras constructed by Gross–Siebert and Keel–Yu in the case $��X�\setminus �D�$ contains a Zariski dense torus. In addition, we use this result to prove a mirror conjecture proposed by Mandel for Fano pairs satisfying the maximal degeneracy assumption.