We consider the solution of the rational matrix equations, or generalized algebraic Riccati equations with rational terms, arising in stochastic optimal control in continuous- and discrete-time. Fixed-point iteration and (modified) Newton's methods will be considered. In particular, the convergence results of a new modified Newton's method, for both continuous- and discrete-time rational Riccati equations, will be presented.

We study a ground-state ansatz for the single-hole doped t-J model in two dimensions via a variational Monte Carlo (VMC) method. Such a single-hole wave function possesses finite angular momenta generated by hidden spin currents, which give rise to a novel ground state degeneracy in agreement with recent exact diagonalization (ED) and density matrix renormalization group (DMGR) results. We further show that the wave function can be decomposed into a quasiparticle component and an incoherent momentum distribution in excellent agreement with the DMRG results up to an 8×8 lattice. Such a two-component structure indicates the breakdown of Landau's one-to-one correspondence principle, and in particular, the quasiparticle spectral weight vanishes by a power law in the large sample-size limit. By contrast, turning off the phase string induced by the hole hopping in the so-called σ⋅t-J model, a conventional Bloch-wave wave function with a finite quasiparticle spectral weight can be recovered, also in agreement with the ED and DMRG results. The present study shows that a singular effect already takes place in the single-hole-doped Mott insulator, by which the bare hole is turned into a non-Landau quasiparticle with translational symmetry breaking. Generalizations to pairing and finite doping are briefly discussed.

THE MAIN PURPOSE of this paper is to give a simple and unified new proof of the Witten rigidity theorems, which were conjectured by Witten and first proved by Taubes [25], Bott-Taubes [S], Hirzebruch Cl23 and Krichever [15]. Our proof shows that the modular invariance, which is the intrinsic symmetry of elliptic genera, actually implies their rigidity. Some new properties of elliptic genera and their relationships with theta-functions are also discussed. We remark that our proof makes essential uses of the new feature of loop groups and loop spaces, the modular invariance. We note that, with the help of the modular group, we can catch the topological information on loop space by simply working on finite-dimensional manifold. By developing this idea further, in [21] we have proved the rigidity of the Dirac operator on loop space twisted by higher-level loop group representations, while the Witten ridigity theorems are the special cases of level 1. Many topological vanishing theorems are also derived in [21] by refining the argument in this paper, especially an &-vanishing theorem for loop space. In [19] modular invariance is used again to establish a general miraculous cancellation formula, relating the Hirzebruch L-form to certain twisted &-forms, which has many interesting topological results as consequences. These results were announced in [20].

Computational conformal geometry focuses on developing the computational methodologies on discrete surfaces to discover conformal geometric invariants. In this work, we briefly summarize the recent developments for methods and related applications in computational conformal geometry. There are two major approaches, holomorphic differentials and curvature flow. The holomorphic differential method is a linear method, which is more efficient and robust to triangulations with lower quality. The curvature flow method is nonlinear and requires higher quality triangulations, but more flexible. The conformal geometric methods have been broadly applied in many engineering fields, such as computer graphics, vision, geometric modeling and medical imaging. The algorithms are robust for surfaces scanned from real life, general for surfaces with different topologies. The efficiency and efficacy of the algorithms

We study a one-dimensional transport equation with nonlocal velocity which was recently considered in the work of Crdoba, Crdoba and Fontelos [A. Crdoba, D. Crdoba, MA Fontelos, Formation of singularities for a transport equation with nonlocal velocity, Ann. of Math.(2) 162 (3)(2005) 13771389]. We show that in the subcritical and critical cases the problem is globally well-posed with arbitrary initial data in H max {3/2 , 0}. While in the supercritical case, the problem is locally well-posed with initial data in H 3/2 , and is globally well-posed under a smallness assumption. Some polynomial-in-time decay estimates are also discussed. These results improve some previous results in [A. Crdoba, D. Crdoba, MA Fontelos, Formation of singularities for a transport equation with nonlocal velocity, Ann. of Math.(2) 162 (3)(2005) 13771389].