470 research outputs found
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 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 immersed in a higher dimensional
spin manifold from viewpoint of study of submanifold quantum mechanics. We
show that kernel of a certain Dirac operator defined over , which we call
submanifold Dirac operator, gives the data of the immersion. In the derivation,
the simple Frobenius reciprocity of Clifford algebras and plays
important roles.Comment: 17pages. to be published in Mathematical Physics, Analysis and
Geometr
Abelian Functions for Trigonal Curves of Genus Three
We develop the theory of generalized Weierstrass sigma- and \wp-functions
defined on a trigonal curve of genus three. In particular we give a list of the
associated partial differential equations satisfied by the \wp-functions, a
proof that the coefficients of the power series expansion of the sigma-function
are polynomials of moduli parameters, and the derivation of two addition
formulae.Comment: 32 pages, no figures. Revised version has the a fuller description of
the general (3,4) trigonal curve results, the first version described only
the "Purely Trigonal" cas
On discrete constant principal curvature surfaces
Recently, it is discovered that a certain class of nanocarbon materials has
geometrical properties related to the discrete geometry, pre-constant discrete
principal curvature [9] based on the discrete surface theory proposed on
trivalent graphs by Kotani, Naito and Omori [10]. In this paper, with the aim
of an application to the nanocarbon materials, we will study discrete constant
principal curvature (CPC) surfaces. Firstly, we developed the discrete surface
theory on a full 3-ary oriented tree so that we define a discrete analogue of
principal directions on them to investigate it. We also construct some
interesting examples of discrete constant principal curvature surfaces,
including discrete CPC tori.Comment: 13 pages, 9 figure
Toda Equations and -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 and
functions of genera one and two. The ultradiscrete Toda equation appears as a
discrete-valuation of recursion relations of functions.Comment: 16 page
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 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 , identities of Pfaffians, symmetric
functions, hyperelliptic -function and -functions; . The connection between his theory and the modern
soliton theory was also discussed.Comment: AMS-Tex, 12 page
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
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