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

A realisation of a graph in the plane as a bar-joint framework is rigid if there are finitely many other realisations, up to isometries, with the same edge lengths. Each of these finitely many realisations can be seen as a solution to a system of quadratic equations prescribing the distances between pairs of points. For generic realisations, the size of the solution set depends only on the underlying graph so long as we allow for complex solutions. We provide a characterisation of the realisation number — that is the cardinality of this complex solution set — of a minimally rigid graph. Our characterisation uses tropical geometry to express the realisation number as an intersection of Bergman fans of the graphic matroid. As a consequence, we derive a combinatorial upper bound on the realisation number involving the Tutte polynomial. Moreover, we provide computational evidence that our upper bound is usually an improvement on the mixed volume bound.

In 2012, Matsuda introduced the class of weakly closed graphs and investigated when binomial edge ideals are F-pure. He proved that weakly closed binomial edge ideals are F-pure whenever the base field has positive characteristic. He conjectured that: (i) when the base field has characteristic 2, every F-pure binomial edge ideal comes from a weakly closed graph; and (ii) that every binomial edge ideal is F-pure provided that the characteristic of the residue field is sufficiently large. In this paper, we resolve both of Matsuda's conjectures. We confirm Matsuda's first conjecture, showing that the binomial edge ideal of a graph defines an F-pure quotient in characteristic 2 if and only if the graph is weakly closed. We also show that Matsuda's second conjecture is false in a very strong way by showing that graphs containing asteroidal triples, such as the net, define non-F-pure binomial edge ideals in any positive characteristic. Our results yield a complete classification of F-pure binomial edge ideals of chordal graphs as well as large families of standard graded algebras that are F-injective but neither F-pure nor F-rational in all characteristics.

In 2012, Matsuda introduced the class of weakly closed graphs and investigated when binomial edge ideals are F-pure. He proved that weakly closed binomial edge ideals are F-pure whenever the base field has positive characteristic. He conjectured that: (i) when the base field has characteristic 2, every F-pure binomial edge ideal comes from a weakly closed graph; and (ii) that every binomial edge ideal is F-pure provided that the characteristic of the residue field is sufficiently large. In this paper, we resolve both of Matsuda's conjectures. We confirm Matsuda's first conjecture, showing that the binomial edge ideal of a graph defines an F-pure quotient in characteristic 2 if and only if the graph is weakly closed. We also show that Matsuda's second conjecture is false in a very strong way by showing that graphs containing asteroidal triples, such as the net, define non-F-pure binomial edge ideals in any positive characteristic. Our results yield a complete classification of F-pure binomial edge ideals of chordal graphs as well as large families of standard graded algebras that are F-injective but neither F-pure nor F-rational in all characteristics.

In this paper, we focus our attention on the positive solutions to second-order nonlinear ordinary differential equations of the form $��u^{�\prime �\prime �}�+�q��\left(�t�\right)��g��\left(�u�\right)��=�0�$, where $�q$ is a sign-changing weight and $�g$ is a superlinear function. We exploit the classical shooting approach and the comparison theorem to present non-degeneracy and exact multiplicity results for positive solutions. This completes the multiplicity results obtained by Feltrin and Zanolin. Numerical examples and some related open problems are also discussed.

We prove the existence of a non-round Einstein metric $�g$ on $�S^{12}$ that is invariant under the usual cohomogeneity one action of $��O�\left(�3�\right)�\times �O�\left(�10�\right)�$ on $��S^{12}�\subset �R^{13}�=�R^3�\oplus �R^{10}�$. The proof involves using several rigorous numerical analysis techniques to produce a Riemannian metric $�\hat{g}$ which approximately satisfies the Einstein condition to known high precision, and then demonstrating that $�\hat{g}$ can be perturbed into a true Einstein metric $�g$.

Generalising the Heilmann–Lieb theorem from statistical physics, Chudnovsky and Seymour [J. Combin. Theory Ser. B, 97 (2007), no. 3, 350–357] showed that the univariate independence polynomial of any claw-free graph has all of its zeros on the negative real line. In this paper, we show that for any fixed subdivided claw $�H$ and any $�\Delta$, there is an open set $��F�\subseteq �C�$ containing $��[�0�,�\infty �)�$ such that the independence polynomial of any $�H$-free graph of maximum degree $�\Delta$ has all of its zeros outside of $�F$. We also show that no such result can hold when $�H$ is any graph other than a subdivided claw or if we drop the maximum degree condition. We also establish zero-free regions for the multivariate independence polynomial of $�H$-free graphs of bounded degree when $�H$ is a subdivided claw. The statements of these results are more subtle, but are again best possible in various senses.

This paper addresses the quantitative stability for a Yamabe-type functional on compact manifolds with boundary introduced by Escobar. Minimizers of the functional correspond to scalar-flat metrics with constant mean curvature on the boundary. We prove that the deficit controls the distance to the minimizing set to a suitable power by reducing the problem to the analogous question for an effective functional on the boundary.

Generalising the uniform companion for large fields with a single derivation, we construct a theory $��U�C_D�$ of fields of characteristic 0 with free operators—operators determined by a homomorphism from the field to its tensor product with $�D$, a finite-dimensional $�Q$-algebra—which is the model companion of any theory of a field with free operators whose associated difference field is difference large and model complete. Under the assumption that $�D$ is a local ring, we show that simplicity is transferred from the theory of the underlying field to the theory of the field with operators, and we use this to study the model theory of bounded, PAC fields with free operators.

We characterise quantitative semi-uniform stability for $�C_0$-semigroups arising from port-Hamiltonian systems, complementing recent works on exponential and strong stability. With the result, we present a simple universal example class of port-Hamiltonian $�C_0$-semigroups exhibiting arbitrary decay rates slower than $�t^{�-�1�/�2�}$. The latter is based on results from the theory of Diophantine approximation as the decay rates will be strongly related to approximation properties of irrational numbers by rationals given through cut-offs of continued fraction expansions.