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

We study the Hartree type Lane–Emden conjecture, which states the nonexistence of the positive classical solutions for the following Hartree type system

We investigate the integrability of the noncommutative leapfrog map in this paper. First, we derive the explicit formula for the noncommutative leapfrog map and corresponding discrete zero-curvature equation by employing the concept of noncommutative cross-ratio. Then we revisit this discrete map, as well as its continuous limit, from the perspective of noncommutative Laurent bi-orthogonal polynomials. Finally, the Poisson structure for this discrete noncommutative map is formulated with the help of a noncommutative network. Through these constructions, we aim to enhance our understanding of the integrability properties of the noncommutative leapfrog map and its related mathematical structures.

We describe a connection between the subjects of cluster algebras, polynomial identity algebras, and discriminants. For this, we define the notion of root of unity quantum cluster algebras and prove that they are polynomial identity algebras. Inside each such algebra we construct a (large) canonical central subalgebra, which can be viewed as a far reaching generalization of the central subalgebras of big quantum groups constructed by De Concini, Kac, and Procesi and used in representation theory. Each such central subalgebra is proved to be isomorphic to the underlying classical cluster algebra of geometric type. When the root of unity quantum cluster algebra is free over its central subalgebra, we prove that the discriminant of the pair is a product of powers of the frozen variables times an integer. An extension of this result is also proved for the discriminants of all subalgebras generated by the cluster variables of nerves in the exchange graph. These results can be used for the effective computation of discriminants. As an application we obtain an explicit formula for the discriminant of the integral form over $��Z�\left[�\epsilon �\right]�$ of each quantum unipotent cell of De Concini, Kac, and Procesi for arbitrary symmetrizable Kac–Moody algebras, where $�\epsilon$ is a root of unity.

For a fixed prime power $�q$ and natural number $�d$, we consider a random polynomial

We show that some subgroups of the orientation-preserving circle homeomorphism group with invariant veering pairs of laminations are hyperbolic 3-orbifold groups. On the way, we show that from a veering pair of laminations, one can construct a loom space (in the sense of Schleimer–Segerman) as a quotient. Our approach does not assume the existence of any 3-manifold to begin with, so this is a geometrization-type result, and supersedes some of the results regarding the relation among veering triangulations, pseudo-Anosov flows, taut foliations in the literature.

Motivated by a conjecture concerning Igusa local zeta functions for intersection posets of hyperplane arrangements, we introduce and study the Poincaré-extended $��a�b�$-index, which generalizes both the $��a�b�$-index and the Poincaré polynomial. For posets admitting $�R$-labelings, we give a combinatorial description of the coefficients of the extended $��a�b�$-index, proving their nonnegativity. In the case of intersection posets of hyperplane arrangements, we prove the above conjecture of the second author and Voll as well as another conjecture of the second author and Kühne. We also define the pullback $��a�b�$-index, generalizing the $��c�d�$-index of face posets for oriented matroids. Our results recover, generalize, and unify results from Billera–Ehrenborg–Readdy, Bergeron–Mykytiuk–Sottile–van Willigenburg, Saliola–Thomas, and Ehrenborg. This connection allows us to translate our results into the language of quasisymmetric functions, and — in the special case of symmetric functions — pose a conjecture about Schur positivity. This conjecture was strengthened and proved by Ricky Liu, and the proof appears as an appendix.

We consider the Hausdorff dimension of random covering sets formed by balls with centres chosen independently at random according to an arbitrary Borel probability measure on $�R^d$ and radii given by a deterministic sequence tending to zero. We prove, for a certain parameter range, the conjecture by Ekström and Persson concerning the exact value of the dimension in the special case of radii $�{�\left(�n^{�-�\alpha �}�\right)�}_{�n�=�1�}^{\infty }$. For balls with an arbitrary sequence of radii, we find sharp bounds for the dimension and show that the natural extension of the Ekström–Persson conjecture is not true in this case. Finally, we construct examples demonstrating that there does not exist a dimension formula involving only the lower and upper local dimensions of the measure and a critical parameter determined by the sequence of radii.

We consider the determinantal point processes associated with the spectral projectors of a Schrödinger operator on $�R$, with a smooth confining potential. In the semiclassical limit, where the number of particles tends to infinity, we obtain a Szegő-type central limit theorem for the fluctuations of smooth linear statistics. More precisely, the Laplace transform of any statistic converges without renormalisation to a Gaussian limit with a $�H^{�1�/�2�}$-type variance, which depends on the potential. In the one-well (one-cut) case, using the quantum action-angle theorem and additional micro-local tools, we reduce the problem to the asymptotics of Fredholm determinants of certain approximately Toeplitz operators. In the multi-cut case, we show that for generic potentials, a similar result holds and the contributions of the different wells are independent in the limit.

Every saturated fusion system corresponds to a group-like structure called a regular locality. In this paper we study (suitably defined) normalizers and centralizers of partial subnormal subgroups of regular localities. This leads to a reasonable notion of normalizers and centralizers of subnormal subsystems of fusion systems.