The antiferromagnetic Heisenberg spin chain of odd spin S is in the Haldane phase with several defining physical properties, such as thermodynamical ground-state degeneracy, symmetry-protected edge states, and nonzero string order parameter. If nonzero hole concentration δ and hole hopping energy t are considered, the spin chain is replaced by a spin-S t-J chain. The motivation of this paper is to generalize the discussions of the Haldane phase to the doped spin chain. The first result of this paper is that, for the model considered here, the \mathbb{Z}_2 sign structure in the usual Ising basis can be totally removed by two consecutive unitary transformations consisting of a spatially local one and a nonlocal one. Direct from the sign structure, the second result of this paper is that the Marshall theorem and the Lieb-Mattis theorem for pure spin systems are generalized to the t-J chain for arbitrary S and δ. A corollary of the theorem provides us with the ground-state degeneracy in the thermodynamic limit. The third result of this paper is about the phase diagram. We show that the defining properties of the Haldane phase survive in the small t/J limit. The large t/J phase supports a gapped spin sector with similar properties (ground-state degeneracy, edge state, and string order parameter) of the Haldane chain, although the charge sector is gapless.

Arthur classified the discrete automorphic representations of symplectic and orthogonal groups over a number field by that of the general linear groups. In this classification, those that are not from endoscopic lifting correspond to pairs (ϕ,b), where ϕ is an irreducible unitary cuspidal automorphic representation of some general linear group and b is an integer. In this paper, we study the local components of these automorphic representations at a nonarchimedean place, and we give a complete description of them in terms of their Langlands parameters.

We prove that the number of critical points of a Li–Tam Green’s function on a complete open Riemannian surface of finite type admits a topological upper bound, given by the first Betti number of the surface. In higher dimensions, we show that there are no topological upper bounds on the number of critical points by constructing, for each nonnegative integer N, a Riemannian manifold diffeomorphic to Rn (n > 3) whose minimal Green’s function has at least N non-degenerate critical points. Variations on the method of proof of the latter result yield contractible n-manifolds whose minimal Green’s functions have level sets diffeomorphic to any fixed codimension 1 compact submanifold of R^n.

We introduce and analyze two new semi-discrete numerical methods for the multi-dimensional Vlasov-Poisson system. The schemes are constructed by combing a discontinuous Galerkin approximation to the Vlasov equation together with a mixed finite element method for the Poisson problem. We show optimal error estimates in the case of smooth compactly supported initial data. We propose a scheme that preserves the total energy of the system.

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