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

We prove a Hurewicz-type theorem for the dynamic asymptotic dimension originally introduced by Guentner, Willett, and Yu. Calculations of (or simply upper bounds on) this dimension are known to have implications related to cohomology of group actions and the $�K$-theory of their transformation group $�C^{\ast }$-algebras. Moreover, these implications are relevant to the current classification program for $�C^{\ast }$-algebras. As a corollary of our main theorem, we show that the dynamic asymptotic dimension of actions by groups on profinite completions along sequential filtrations by normal subgroups is often subadditive over extensions of groups, which shows that many such actions by elementary amenable groups are finite dimensional. We combine this extension theorem with other novel results relating the dynamic asymptotic dimension of such actions to the asymptotic dimension of corresponding box spaces. This allows us to give upper bounds on the asymptotic dimension of many box spaces (including those of infinitely many groups with exponential growth). For some of these examples, we can also find lower bounds by utilizing the theory of ends of groups.

New sharp affine isoperimetric inequalities for volume decomposition functionals $�X_2$ and $�X_3$ in $�R^n$ are established. To fulfil this task, we prove recursion formulas for volume decomposition functionals and find out the connections between the domains of these functionals and matroid polytopes. Applications of matroid theory to convex geometry are presented.

In this paper, we consider a quasilinear elliptic and critical problem with Dirichlet boundary conditions in presence of the anisotropic $�p$-Laplacian. The critical exponent is the usual $�p^{★}$ such that the embedding $��W_0^{�1�,�p�}��\left(�\Omega �\right)��\subset �L^{p^{★}}��\left(�\Omega �\right)��$ is not compact. We prove the existence of a weak positive solution in presence of both a $�p$-linear and a $�p$-superlinear perturbation. In doing this, we have to perform several precise estimates of the anisotropic Aubin–Talenti functions which can be of interest for further problems. The results we prove are a natural generalization to the anisotropic setting of the classical ones by Brezis–Nirenberg (Comm. Pure Appl. Math. 36 (1983), 437–477).

We give a description of finitely generated prosoluble subgroups of the profinite completion of 3-manifold groups and toral relatively hyperbolic virtually compact special groups.

Recently, B. Yang and J. Yang derived a family of rational solutions to the Sasa–Satsuma equation, and showed that any of its members constitutes a partial-rogue wave provided that an associated generalised Okamoto polynomial has no real roots or no imaginary roots. In this paper, we derive exact formulas for the number of real and the number of imaginary roots of the generalised Okamoto polynomials. On the one hand, this yields a list of partial-rogue waves that satisfy the Sasa–Satsuma equation. On the other hand, it gives families of rational solutions of the fourth Painlevé equation that are pole-free on either the real line or the imaginary line. To obtain these formulas, we develop an algorithmic procedure to derive the qualitative distribution of singularities on the real line for real solutions of Painlevé equations, starting from the known distribution for a seed solution, through the action of Bäcklund transformations on the rational surfaces forming their spaces of initial conditions.

We prove the Kazhdan–Lusztig correspondence for a class of vertex operator superalgebras that, via the work of Costello–Gaiotto, arise as boundary vertex operator algebra (VOAs) of the topological B twist of 3d $��N�=�4�$ abelian gauge theories. This means that we show equivalences of braided tensor categories of modules of certain affine vertex superalgebras and corresponding quantum supergroups. We build on the work of Creutzig–Lentner–Rupert for this large class of VOAs and extend it since, in our case, the categories do not have projective objects, and objects can have arbitrary Jordan–Hölder length. Our correspondence significantly improves the understanding of the braided tensor category of line defects associated with this class of topological quantum field theory (TQFT) by realizing line defects as modules of a Hopf algebra. In the process, we prove a special case of the conjecture of Semikhatov–Tipunin, relating logarithmic conformal field theory (CFTs) to Nichols algebras of screening operators.

We construct a relative Hodge–Tate spectral sequence for any smooth proper morphism of rigid analytic spaces over a perfectoid field extension of $�Q_p$. To this end, we generalise Scholze's strategy in the absolute case by using smoothoid adic spaces. As our main additional ingredient, we prove a perfectoid version of Grothendieck's “cohomology and base-change”. We also use this to prove local constancy of Hodge numbers in the rigid analytic setting, and deduce that the relative Hodge–Tate spectral sequence degenerates.