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.

Let E be a quadratic algebra over a number field F. Let E(g, s) be an Eisenstein series on GL2(E), and let F be a cuspidal automorphic form on GL2(F). We will consider in this paper the following automorphic integral: ∫ZAGL2(F)∖GL2(AF)F(g)E(g,s)dg. This is in some sense the complementary case to the well-known Rankin–Selberg integral and the triple product formula. We will approach this integral by Waldspurger’s formula, giving a criterion about when the integral is automatically zero, and otherwise the L-functions it represents. We will also calculate the local integrals at some ramified places, where the level of the ramification can be arbitrarily large.

In this paper, we present an experimental analysis of HFE Challenge 2 (144 bit) type systems. We generate scaled versions of the full challenge fixing and guessing some unknowns. We use the MXL_3 algorithm, an efficient algorithm for computing Gröbner basis, to solve these scaled versions. We review the MXL_3 strategy and introduce our experimental results.

We construct 2-surfaces of prescribed mean curvature in 3manifolds carrying asymptotically flat initial data for an isolated gravitating system with rather general decay conditions. The surfaces in question form a regular foliation of the asymptotic region of such a manifold. We recover physically relevant data, especially the ADM-momentum, from the geometry of the foliation. For a given set of data (M, g,K), with a three dimensional manifoldM, its Riemannian metric g, and the second fundamental form K in the surrounding four dimensional Lorentz space time manifold, the equation we solve is H+P = const or H−P = const. Here H is the mean curvature, and P = trK is the 2-trace of K along the solution surface. This is a degenerate elliptic equation for the position of the surface. It prescribes the mean curvature anisotropically, since P depends on the direction of the normal.

In this note the fractional analytic index, for a projective elliptic operator associated to an Azumaya bundle, of [5] is related to the equivariant index of [1, 6] for an associated transversally elliptic operator.