We prove that the Heisenberg group admits infinitely many inequivalent equivariant compactifications into for all . This result provides a non-commutative analog of Hassett–Tschinkel's classical result, which shows that there exist infinitely many inequivalent equivariant compactifications of vector groups into projective spaces of dimension at least 6.
We prove a criterion for the possible locations of zeros of type I and type II multiple orthogonal polynomials in terms of normality of degree 1 Christoffel transforms. We provide another criterion in terms of degree 2 Christoffel transforms for establishing zero interlacing of the neighboring multiple orthogonal polynomials of type I and type II. We apply these criteria to establish zero location and interlacing of type I multiple orthogonal polynomials for Nikishin systems. Additionally, we recover the known results on zero location and interlacing for type I multiple orthogonal polynomials for Angelesco systems, as well as for type II multiple orthogonal polynomials for Angelesco and AT systems. Finally, we demonstrate that normality of the higher-order Christoffel transforms is naturally related to the zeros of the Wronskians of consecutive orthogonal polynomials.
In this paper, we establish some new Liouville-type theorems for nonnegative weak solutions to fractional elliptic systems with different orders. To prove our result, we will use the local realization of fractional Laplacian, which can be constructed as Dirichlet-to-Neumann operator of a degenerate elliptic equation by the extension technique. Our proof is based on Alexandrov–Serrin method of moving planes based on some maximum principles that obtained by establishing some key integral inequalities.
We show that the quasi-local algebra of a coarse disjoint union of expander graphs does not contain a Cartan subalgebra isomorphic to . Ozawa has recently shown that these algebras are distinct from the uniform Roe algebras of expander graphs, and our result describes a further difference.
In this paper, we establish global boundedness results for the jet determination order of CR maps between (not necessarily bounded) generic submanifolds in complex space. A key feature of our results is that they apply to Nash submanifolds, but do not extend to arbitrary real-analytic submanifolds.
We classify all contact structures on 3-manifolds that admit a nonfree toric action, up to contactomorphism, and present them through explicit topological descriptions. Our classification is based on Lerman's classification of toric contact 3-manifolds up to equivariant contactomorphism [Lerman, J. Symplectic Geom. 1 (2003), 785–828]. We also prove that every contact 3-manifold with a nonfree toric action arises as the concave boundary of a toric linear plumbing over spheres inspired by Marinković et al. As a corollary of both results, we classify which contact 3-manifolds arise as the concave boundary of a linear plumbing of spheres.
We give two short proofs of the abelian Livšic theorem of Gogolev and Rodriguez Hertz. We show that these proofs may be extended to give new abelian Livšic theorems for positive density sets of null-homologous orbits and for amenable covers.
In this paper, we investigate a special autonomous Schwarzian differential equation
We show that the ‘profinite direct sum’ is a good notion of infinite direct sums for profinite modules, having properties similar to those of direct sums of abstract modules. For example, the profinite direct sum of projective modules is projective, and there is a Mackey's formula for profinite modules described using these sums. As an application, we prove that the class of profinite groups of type is closed under subgroups.
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