Immersion Anomaly of Dirac Operator on Surface in R^3

In previous report (J. Phys. A (1997) 30 4019-4029), I showed that the Dirac field confined in a surface immersed in $R^3$ by means of a mass type potential is governed by the Konopelchenko-Kenmotsu-Weierstrass-Enneper equation. In this article, I quantized the Dirac field and calculated the gauge transformation which exhibits the gauge freedom of the parameterization of the surface. Then using the Ward-Takahashi identity, I showed that the expectation value of the action of the Dirac field is expressed by the Willmore functional and area of the surface.Comment: AMS-Tex Us

On Density of State of Quantized Willmore Surface-A Way to Quantized Extrinsic String in R^3

Recently I quantized an elastica with Bernoulli-Euler functional in two-dimensional space using the modified KdV hierarchy. In this article, I will quantize a Willmore surface, or equivalently a surface with the Polyakov extrinsic curvature action, using the modified Novikov-Veselov (MNV) equation. In other words, I show that the density of state of the partition function for the quantized Willmore surface is expressed by volume of a subspace of the moduli of the MNV equation.Comment: AMS-Tex Us

Statistical Mechanics of Elastica on Plane as a Model of Supercoiled DNA-Origin of the MKdV hierarchy-

In this article, I have investigated statistical mechanics of a non-stretched elastica in two dimensional space using path integral method. In the calculation, the MKdV hierarchy naturally appeared as the equations including the temperature fluctuation.I have classified the moduli of the closed elastica in heat bath and summed the Boltzmann weight with the thermalfluctuation over the moduli. Due to the bilinearity of the energy functional,I have obtained its exact partition function.By investigation of the system,I conjectured that an expectation value at a critical point of this system obeys the Painlev\'e equation of the first kind and its related equations extended by the KdV hierarchy.Furthermore I also commented onthe relation between the MKdV hierarchy and BRS transformationin this system.Comment: AMS-Tex Us

Generalized Weierstrass Relations and Frobenius reciprocity

This article investigates local properties of the further generalized Weierstrass relations for a spin manifold $S$ immersed in a higher dimensional spin manifold $M$ from viewpoint of study of submanifold quantum mechanics. We show that kernel of a certain Dirac operator defined over $S$, which we call submanifold Dirac operator, gives the data of the immersion. In the derivation, the simple Frobenius reciprocity of Clifford algebras $S$ and $M$ plays important roles.Comment: 17pages. to be published in Mathematical Physics, Analysis and Geometr

Hyperelliptic Solutions of KdV and KP equations: Reevaluation of Baker's Study on Hyperelliptic Sigma Functions

Explicit function forms of hyperelliptic solutions of Korteweg-de Vries (KdV) and \break Kadomtsev-Petviashvili (KP) equations were constructed for a given curve $y^2 = f(x)$ whose genus is three. This study was based upon the fact that about one hundred years ago (Acta Math. (1903) {\bf{27}}, 135-156), H. F. Baker essentially derived KdV hierarchy and KP equation by using bilinear differential operator ${\bold{D}}$, identities of Pfaffians, symmetric functions, hyperelliptic $\sigma$-function and $\wp$-functions; $\wp_{\mu \nu} = -\partial_\mu \partial_\nu \log \sigma$ $= - ({\bold{D}}_\mu {\bold{D}}_\nu \sigma \sigma)/2\sigma^2$. The connection between his theory and the modern soliton theory was also discussed.Comment: AMS-Tex, 12 page

Design of a Model Reference Adaptive Controller for an Unmanned Air Vehicle

This paper presents the "Adaptive Control Technology for Safe Flight (ACTS)" architecture, which consists of a non-adaptive controller that provides satisfactory performance under nominal flying conditions, and an adaptive controller that provides robustness under off nominal ones. The design and implementation procedures of both controllers are presented. The aim of these procedures, which encompass both theoretical and practical considerations, is to develop a controller suitable for flight. The ACTS architecture is applied to the Generic Transport Model developed by NASA-Langley Research Center. The GTM is a dynamically scaled test model of a transport aircraft for which a flight-test article and a high-fidelity simulation are available. The nominal controller at the core of the ACTS architecture has a multivariable LQR-PI structure while the adaptive one has a direct, model reference structure. The main control surfaces as well as the throttles are used as control inputs. The inclusion of the latter alleviates the pilot s workload by eliminating the need for cancelling the pitch coupling generated by changes in thrust. Furthermore, the independent usage of the throttles by the adaptive controller enables their use for attitude control. Advantages and potential drawbacks of adaptation are demonstrated by performing high fidelity simulations of a flight-validated controller and of its adaptive augmentation

Verification and Tuning of an Adaptive Controller for an Unmanned Air Vehicle

This paper focuses on the analysis and tuning of a controller based on the Adaptive Control Technology for Safe Flight (ACTS) architecture. The ACTS architecture consists of a nominal, non-adaptive controller that provides satisfactory performance under nominal flying conditions, and an adaptive controller that provides robustness under off-nominal ones. A framework unifying control verification and gain tuning is used to make the controller s ability to satisfy the closed-loop requirements more robust to uncertainty. In this paper we tune the gains of both controllers using this approach. Some advantages and drawbacks of adaptation are identified by performing a global robustness assessment of both the adaptive controller and its non-adaptive counterpart. The analyses used to determine these characteristics are based on evaluating the degradation in closed-loop performance resulting from uncertainties having increasing levels of severity. The specific adverse conditions considered can be grouped into three categories: aerodynamic uncertainties, structural damage, and actuator failures. These failures include partial and total loss of control effectiveness, locked-in-place control surface deflections, and engine out conditions. The requirements considered are the peak structural loading, the ability of the controller to track pilot commands, the ability of the controller to keep the aircraft s state within the reliable flight envelope, and the handling/riding qualities of the aircraft. The nominal controller resulting from these tuning strategies was successfully validated using the NASA GTM Flight Test Vehicle

Parametrically controlling solitary wave dynamics in modified Kortweg-de Vries equation

We demonstrate the control of solitary wave dynamics of modified Kortweg-de Vries (MKdV) equation through the temporal variations of the distributed coefficients. This is explicated through exact cnoidal wave and localized soliton solutions of the MKdV equation with variable coefficients. The solitons can be accelerated and their propagation can be manipulated by suitable variations of the above parameters. In sharp contrast with nonlinear Schr\"{o}dinger equation, the soliton amplitude and widths are time independent.Comment: 4 pages, 5 eps figure

Toda Equations and σ\sigma-Functions of Genera One and Two

We study the Toda equations in the continuous level, discrete level and ultradiscrete level in terms of elliptic and hyperelliptic $\sigma$ and $\psi$ functions of genera one and two. The ultradiscrete Toda equation appears as a discrete-valuation of recursion relations of $\psi$ functions.Comment: 16 page

On Some Classes of mKdV Periodic Solutions

We obtain exact periodic solutions of the positive and negative modified Kortweg-de Vries (mKdV) equations. We examine the dynamical stability of these solitary wave lattices through direct numerical simulations. While the positive mKdV breather lattice solutions are found to be unstable, the two-soliton lattice solution of the same equation is found to be stable. Similarly, a negative mKdV lattice solution is found to be stable. We also touch upon the implications of these results for the KdV equation.Comment: 8 pages, 3 figures, to appear in J. Phys.