© Hindawi Publishing Corp. A NOTE ON ASYMPTOTIC HELIX AND QUANTUM MECHANICAL STRUCTURE

Using the formulation of a moving curve, we 
demonstrate that an asymptotic helix goes over to the linear 
time-dependent Schrodinger equation as shown by Dmitriyev 
(2002).


Introduction.
Here we consider unbounded, inviscid, and incompressible fluid flows.In the absence of external force, the motion of such fluid with unit density is described by the Euler equations where u(x,t) is the velocity, x = (x 1 ,x 2 ,x 3 ) is the position, p is the pressure, and ∇ is the differentiation vector.The curl of the velocity field is the vorticity.The velocity u(x) can be determined from the vorticity ω(x) through the Biot-Savart law: If the vorticity is concentrated on a single thin filament C of circulation Γ , equation (1.3) becomes If one computes the self-induced motion of the filament by evaluating the velocity from (1.4) on the filament itself, then the result is logarithmically infinite if the filament is curved and zero if it is straight.So the self-induced motion occurs only for curved filaments.We must focus our attention on a very slender vortex filament.
One can avoid singularity in (1.4) by simply ignoring the nonlocal contribution of the filament and replace the Biot-Savart law by a velocity expression that depends only on the local curvature of the vortex filament.If a vortex filament is described by r(s), where s is an arc length parameter measured along the filament, then r(s) is the position vector.Let t, n, and b denote the unit tangent, normal, and binormal vectors, respectively.
Hence this leads to the local induction approximation, this is, where κ is the curvature.Differentiating both sides with respect to arc length s yields the Heisenberg spin chain equation in terms of tangent vector: In this paper, we investigate the relation between filament equation and Schrödinger equation of quantum mechanics.We consider the (stationary) Schrödinger equation This equation can easily be transformed into where ι = / √ 2m and ι∂ X = ∂ x .Similarly, we can scale the time-dependent Schrödinger equation We scale hence (1.9) becomes Then for ιx = X and iιt = T , one has It is well known that under the connection between the Hasimoto vortex solitons [8] and the nonlinear Schrödinger equation (NLSE), the basis in the Frenet-Serret frame (t,n,b) is associated to certain quantities in NLS equations.Subsequently, Lamb [9] demonstrated that various integrable systems could also be associated with the motion of a curve.In recent years, there has been a tremendous growth of interest in the applications of the Frenet-Serret equations for a curve to various contexts of integrable systems [3,6,7,10,11,12].
Balakrishnan and Dandoloff [2] have shown that the time-dependent Schrödinger equation for a particle in a potential V (s,u), u denoting time, can be interpreted geometrically as a moving curve whose Fermi-Walker phase density is given by −(∂V /∂s).In an extremely interesting paper, Dmitriyev [4] showed that the small amplitude to thread ratio helical configuration of a vortex filament in the ideal fluid behaves exactly as de Broglie wave.In this paper, using the formulation of a moving curve, we give a direct proof of how an asymptotic helix goes over to the linear time-dependent Schrödinger equation as shown by Dmitriyev.

Frenet-Serret frame.
We consider a space curve γ(s), parametrized by arc length s; κ and τ are the curvature and the torsion of the space curve.We denote by t the unit tangent vector to this curve and by n and b its principle normal and binormal, respectively.Then the Frenet frame {t, n, b} forms a moving triad of the curve and they form a right-handed system of axes, that is, A space curve embedded in 3D is described using the usual Frenet-Serret equations (see, e.g., [5]): and these underlie central investigations in the theory of space curves.From the equation, κ and τ are given by (2.3) If we introduce the Darboux vector (s-angular velocity of Frenet frame) then the Frenet-Serret equations may be written as If the curve moves with time u, then all quantities in (2.2) become functions of both s and u, and the time evolution is given by This set of equations may be written as where Ω is another Darboux vector given by The compatibility condition of (2.2) and (2.6) leads to (2.9)

Lamb formalism of moving curves.
The Frenet-Serret equations can be combined to give one complex and one real equations [7] (n + ib) s + iτ(n + ib) = −κt, t s = κn. (2.10) Following Lamb, we introduce a complex quantity Definition 2.1.The Hasimoto transformation is a map which assigns to a space curve a complex curvature function via the formula 12) The image of the Hasimoto map is defined up to an arbitrary constant phase φ.
The Frenet-Serret equations in terms of {t, N, N * } are (2.13) Lemma 2.2.The new Frenet frame {t, N, N * } satisfies the following conditions: (2.14) Then this complex basis {t, N, N * } can be identified with SO(3), where t is an element of the unit sphere and we regard (N, N * ) as a positive orthonormal frame of the tangent space of S 2 at t.

Equations of curve.
The derivatives of N and t with respect to u may be written, in general, as (2.15) Using (2.13) and the compatibility conditions of t us = t su , these equations boil down to ) where C = C(s, u) is a real function.
Using the compatibility condition N us = N su , we obtain one complex and one real equations ) In Section 3, we will see that the integrodifferential equation connects the moving space curve equations and other evolutionary systems.Since t • t u = 0, we may assume that Then from describe the motion of moving curves.

Examples
Thus we can identify f = −κτ, g = κ s , and Thus the equation of motion becomes NLSE (2.28)

Connection to quantum mechanics.
We consider an approximation of (2.18).Suppose that g f , then It is easy to see that, for g f , Thus (2.17) becomes Proof.The proof is given by direct computation.
In the event, κ τ tends to an asymptotic helix.This is the humped helix approximated by the wave packet, as shown by Dmitriyev [4].
We consider the solutions of the Schrödinger equation and φ = ψ 1 (s, u) + iψ 2 (s, u), then this denotes a helix rotating counterclockwise around the s-axis with the constant angular velocity ω = τ 2 .

Schrödinger equation with potential and geometric phase.
We consider a generalized Hasimoto map where λ is some parameter.Then the zero-curvature equation becomes This results from the compatibility condition of So the Schrödinger equation becomes Thus we can introduce a potential V (s,u).In this prototypical case, V (s,u) = λq s .We can manipulate λ to get a slightly more general potential V (s,u).Thus, we obtain iq u + q ss + V (s,u) = 0.
Finally, following Balakrishnan and Dandoloff [2], we present an intimate connection with the potential function V (s,u) and the Fermi-Walker phase.It is clear from (2.5) and (2.7) with their Darboux vectors that if we work in a Fermi-Walker frame, t essentially gets parallel transported and the triad undergoes a rotation τ ds about t.Similarly, the triad undergoes a rotation hdu as one moves along the temporal curve by an interval du.If we preform a Hasimoto transformation to iq s + q ss + V (s,u) = 0, the real part yields and our result immediately follows from the above.
Thus, in this paper, we formulated a direct and more geometrical connection between an evolution of a space curve satisfying the Heisenberg spin chain and quantum mechanics.

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.
However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning

Example 2 . 4 .Example 2 . 5 .
We first consider the Belavin-Polyakov equation t u = −t × t s .In this case, f = 0 and g = −k.Hence γ = iq and C = − s ∞ |q| 2 ds .Thus the equation of motion is q u − iq s − iq s ∞ |q| 2 ds .(2.25)This equation was introduced in [1].The real part of this equation coincides with the elliptic Liouville equation.We consider the Heisenberg spin chain equation t u = t × t ss .It is easy to see that t × t ss = κ s b − κτn.

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Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation