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

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

Breather lattice and its stabilization for the modified Korteweg-de Vries equation

We obtain an exact solution for the breather lattice solution of the modified Korteweg-de Vries (MKdV) equation. Numerical simulation of the breather lattice demonstrates its instability due to the breather-breather interaction. However, such multi-breather structures can be stabilized through the concurrent application of ac driving and viscous damping terms.Comment: 6 pages, 3 figures, Phys. Rev. E (in press

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.

Abelian functions associated with a cyclic tetragonal curve of genus six

We develop the theory of Abelian functions defined using a tetragonal curve of genus six, discussing in detail the cyclic curve y^4 = x^5 + λ[4]x^4 + λ[3]x^3 + λ[2]x^2 + λ[1]x + λ[0]. We construct Abelian functions using the multivariate sigma-function associated with the curve, generalizing the theory of theWeierstrass℘-function. We demonstrate that such functions can give a solution to the KP-equation, outlining how a general class of solutions could be generated using a wider class of curves. We also present the associated partial differential equations satisfied by the functions, the solution of the Jacobi inversion problem, a power series expansion for σ(u) and a new addition formula

Annual variation in the levels of transcripts of sex-specific genes in the mantle of the common mussel, Mytilus edulis

Mytilus species are used as sentinels for the assessment of environmental health but sex or stage in the reproduction cycle is rarely considered even though both parameters are likely to influence responses to pollution. We have validated the use of a qPCR assay for sex identification and related the levels of transcripts to the reproductive cycle. A temporal study of mantle of Mytilus edulis found transcripts of male-specific vitelline coat lysin (VCL) and female-specific vitelline envelope receptor for lysin (VERL) could identify sex over a complete year. The levels of VCL/VERL were proportional to the numbers of sperm/ova and are indicative of the stage of the reproductive cycle. Maximal levels of VCL and VERL were found in February 2009 declining to minima between July - August before increasing and re-attaining a peak in February 2010. Water temperature may influence these transitions since they coincide with minimal water temperature in February and maximal temperature in August. An identical pattern of variation was found for a cryptic female-specific transcript (H5) but a very different pattern was observed for oestrogen receptor 2 (ER2). ER2 varied in a sex-specific way with male > female for most of the cycle, with a female maxima in July and a male maxima in December. Using artificially spawned animals, the transcripts for VCL, VERL and H5 were shown to be present in gametes and thus their disappearance from mantle is indicative of spawning. VCL and VERL are present at equivalent levels in February and July-August but during gametogenesis (August to January) and spawning (March to June) VCL is present at lower relative amounts than VERL. This may indicate sex-specific control mechanisms for these processes and highlight a potential pressure point leading to reduced reproductive output if environmental factors cause asynchrony to gamete maturation or release