On length spectrum metrics and weak metrics on Teichmüller spaces of surfaces with boundary

We define and study metrics and weak metrics on the Teichmüller space of a surface of topologically finite type with boundary. These metrics and weak metrics are associated to the hyperbolic length spectrum of simple closed curves and of properly embedded arcs in the surface. We give a comparison between the defined metrics on regions of Teichmüller space which we call $\varepsilon_0$-relative $\epsilon$-thick parts} for $\epsilon >0$ and $\varepsilon_0\geq \epsilon>0$

Length spectra and the Teichmüller metric for surfaces with boundary

International audienceWe consider some metrics and weak metrics defined on the Teichmmüller space of a surface of finite type with nonempty boundary, that are defined using the hyperbolic length spectrum of simple closed curves and of properly embedded arcs, and we compare these metrics and weak metrics with the Teichmüller metric. The comparison is on subsets of Teichmüller space which we call ''$\varepsilon_0$-relative $\epsilon$-thick parts", and whose definition depends on the choice of some positive constants $\varepsilon_0$ and $\epsilon$. Meanwhile, we give a formula for the Teichmüller metric of a surface with boundary in terms of extremal lengths of families of arcs

Design of a helicopter autopilot by means of linearizing transformations

An automatic flight control systems design methods for aircraft that have complex characteristics and operational requirements, such as the powered lift STOL and V/STOL configurations are discussed. The method is effective for a large class of dynamic systems that require multiaxis control and that have highly coupled nonlinearities, redundant controls, and complex multidimensional operational envelopes. The method exploits the possibility of linearizing the system over its operational envelope by transforming the state and control. The linear canonical forms used in the design are described, and necessary and sufficient conditions for linearizability are stated. The control logic has the structure of an exact model follower with linear decoupled model dynamics and possibly nonlinear plant dynamics. The design method is illustrated with an application to a helicopter autopilot design

Nonlinear control of aircraft

Transformations of nonlinear systems were used to design automatic flight controllers for vertical and short takeoff aircraft. Under the assumption that a nonlinear system can be mapped to a controllable linear system, a method using partial differential equations was constructed to approximate transformations in cases where exact ones cannot be found. An application of the design theory to a rotorcraft, the UH-1H helicopter, was presented

Canonical forms for nonlinear systems

Necessary and sufficient conditions for transforming a nonlinear system to a controllable linear system have been established, and this theory has been applied to the automatic flight control of aircraft. These transformations show that the nonlinearities in a system are often not intrinsic, but are the result of unfortunate choices of coordinates in both state and control variables. Given a nonlinear system (that may not be transformable to a linear system), we construct a canonical form in which much of the nonlinearity is removed from the system. If a system is not transformable to a linear one, then the obstructions to the transformation are obvious in canonical form. If the system can be transformed (it is called a linear equivalent), then the canonical form is a usual one for a controllable linear system. Thus our theory of canonical forms generalizes the earlier transformation (to linear systems) results. Our canonical form is not unique, except up to solutions of certain partial differential equations we discuss. In fact, the important aspect of this paper is the constructive procedure we introduce to reach the canonical form. As is the case in many areas of mathematics, it is often easier to work with the canonical form than in arbitrary coordinate variables

Applications to aeronautics of the theory of transformations of nonlinear systems

The development of the transformation theory is discussed. Results and applications concerning the use of this design technique for automatic flight control of aircraft are presented. The theory examines the transformation of nonlinear systems to linear systems. The tracking of linear models by nonlinear plants is discussed. Results of manned simulation are also presented

Approximating linearizations for nonlinear systems

The following problem is examined: given a nonlinear control system dot-x(t) = f(x(t)) + the sum to m terms(i=1) u sub i (t)g sub i (x(t)) on R(n) and a point x(0) in R(n), approximate the system near x(0) by a linear system. One approach is to use the usual Taylor series linearization. However, the controllability properties of both the nonlinear and linear systems depend on certain Lie brackets of the vector field under consideration. This suggests that a linear approximation based on Lie bracket matching should be constructed at x(0). In general, the linearizations based on the Taylor method and the Lie bracket approach are different. However, under certain mild assumptions, it is shown that there is a coordinate system for R(n) near x(0) in which these two types of linearizations agree. The importance of this agreement is indicated by examining the time responses of the nonlinear system and its linear approximation and comparing the lower order kernels in Volterra expansions of each

On indecomposable modules over the Virasoro algebra

It is proved that an indecomposable Harish-Chandra module over the Virasoro algebra must be (i) a uniformly bounded module, or (ii) a module in Category $\cal O$, or (iii) a module in Category ${\cal O}^-$, or (iv) a module which contains the trivial module as one of its composition factors.Comment: 5 pages, Latex, to appear in Science in China

Transport Properties in the "Strange Metal Phase" of High Tc Cuprates: Spin-Charge Gauge Theory Versus Experiments

The SU(2)xU(1) Chern-Simons spin-charge gauge approach developed earlier to describe the transport properties of the cuprate superconductors in the ``pseudogap'' regime, in particular, the metal-insulator crossover of the in-plane resistivity, is generalized to the ``strange metal'' phase at higher temperature/doping. The short-range antiferromagnetic order and the gauge field fluctuations, which were the key ingredients in the theory for the pseudogap phase, also play an important role in the present case. The main difference between these two phases is caused by the existence of an underlying statistical $\pi$-flux lattice for charge carriers in the former case, whereas the background flux is absent in the latter case. The Fermi surface then changes from small ``arcs'' in the pseudogap to a rather large closed line in the strange metal phase. As a consequence the celebrated linear in T dependence of the in-plane and out-of-plane resistivity is shown explicitly to recover. The doping concentration and temperature dependence of theoretically calculated in-plane and out-of-plane resistivity, spin-relaxation rate and AC conductivity are compared with experimental data, showing good agreement.Comment: 14 pages, 5 .eps figures, submitted to Phys. Rev. B, revised version submitted on 24 Oc