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The energy of stochastic vibration system of a class of protein base on wavelet.

INTRODUCTION

The study of protein system are very important [1-4]. In this paper, we study the energy of a stochastic vibration system of a class of protein through wavelet alternation.

With the rapid development of computerization, computerized scientific instruments come a wide variety of interesting problems for data analysis and signal processing. In fields ranging from Extragalactic Astronomy to Molecular Spectroscopy to Medical Imaging to computer vision, one must recover a signal, curve, image, spectrum, or density from incomplete, indirect, and noisy data. Wavelets have contributed to this already intensely developed and rapidly advancing field.

Wavelet has its energy concentrated in time to give a tool for the analysis of transient, nonstationary, or time-varying phenomena. It still has the oscillating wavelike characteristic but also has the ability to allow simultaneous time and frequency analysis with a flexible mathematical foundation. We take wavelet and use them in a series expansion of signals or functions much the same way a Fourier series the wave or sinusoid to represent a signal or function.

Wavelet analysis consists of a versatile collection of tools for the analysis and manipulation of signals such as sound and images as well as more general digital data sets, such as speech, electrocardiograms, images. Wavelet analysis is a remarkable tool for analyzing function of one or several variables that appear in mathematics or in signal and image processing. With hindsight the wavelet transform can be viewed as diverse as mathematics, physics and electrical engineering. The basic idea is always to use a family of building blocks to represent the object at hand in an efficient and insightful way, the building blocks themselves come in different sizes, and are suitable for describing features with a resolution commensurate with their size.

There are two important aspects to wavelets, which we shall call "mathematical" and "algorithmical". Numerical algorithms using wavelet bases are similar to other transform methods in that vectors and operators are expanded into a basis and the computations take place in the new system of coordinates. As with all transform methods such as approach hopes to achieve that the computation is faster in the new system of coordinates than in the original domain, wavelet based algorithms exhibit a number of new and important properties. Recently some persons have studied wavelet problems of stochastic process or stochastic system [5-13].

We know [4] the equation of the protein vibration system

is:

dx(t)/dt = A - Bx(t) - cx (t - [tau]) (1)

We study its stochastic action as follow.

Let p(n, t) is the probability of system at time t, its reflect equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where, p (n, t; m, t - [tau]) express the joint probability of system have n molecular at time t and have m molecular at time t - [tau]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

p(n, t)=0, p (n, t; m, t - [tau]) = 0

Supose, p (n, t; m, t - [tau]) = p (n, t) p (m, t - [tau])

Then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

We introduction function:

G(s,t) = [[infinity].summation over (k=0)] [S.sup.k] p(k, t1n, 0

Use the main equation [4]:

[partial derivative](y,t)/[partial derivative]t = [integral]{w(y | y')p(y',t) - w(y' | y)p(y,t)dy

We can obtain:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

We can obtain solution of (4):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

Where [A.sub.1] and [A.sub.2] are Constant.

THE ENERGY OF THE PROTEIN SYSTEM BASE ON WAVELET

Let function

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We call it as Haar wavelet [9].

The wavelet alternation of G(t) is

Ws = [[integral].sup.1.sub.-1] [psi] (t)G(t)dt

Ws is the energy of system (4)

We have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Use [integral] [x.sup.n][e.sup.ax] dx = 1/a [x.sup.n][e.sup.ax] - n/a [integral] [x.sup.n-1][e.sup.ax]dx

we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then, we have

Ws = F ([I.sub.1]-[I.sub.2])

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

WAVELET EXPANSIONS OF SYSTEM

For processes that are continuous in mean square,

i.e. E[[absolute value of G(t)].sup.2] < [infinity] and E [[absolute value of G(t) - G(s)].sup.2] [right arrow] 0 as [right arrow] s. [5]

we consider wavelet expansions of stochastics processes and show that for certain wavelets, the coefficients of the expansion have negligible correlation for different scales. We can introduce a modification of the wavelets. Certain nonstationary processes the wavelets may be chosen to give uncorrelated coefficients.

We observe that the approximation of G(t) by [G.sub.m](t), where

[G.sub.m](t)=[summation over (n)][a.sub.mn][[phi].sub.mn](t), [a.sub.mn] = [integral] G(t)[[phi].sub.mn](t)dt

is mean square for any [phi] [member of] Sr, that is: E[G(t)- [G.sub.m](t)] [right arrow] 0, as m [right arrow] [infinity], t [member of] R, we express [G.sub.m](t) in the wavelet series

is, [G.sub.m](t)= [m-1.summation over (k=-[infinity])][[infinity].summation over (n=- [infinity])[[b.sub.kn][[phi].sub.kn](t),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

DOI: 10.3968/j.ans.1715787020120502.1055

REFERENCES

[1] Tanaka, D. (2007). General Chemotactic Model of Oscillators. Phys. Rev. Lett., 99, 134103.

[2] Bratsun, D., Volfson, D., Tismring, L. S., & Hasty, J. (2005). Delay-Induced Stochastic Oscillations in Gene Expression. Pnas, 102, 14593-14598.

[3] Tsimring, L. S., & Pikovisky, A. S. (2001). Noise-Induced Dynamics in Bistable System with Delay. Phys. Rev. Lett., 87, 250602.

[4] Warmflash, A., & Dinner, A. R. (2008). Signatures of Combinatorial Regulation in Intrinsic Biological Noise. Proc. Natl. Acad. Sci., 105, 17262-17267.

[5] Cambancs (1994). Wavelet Approximation of Deterministic and Random Signals. Ieee Tran. on Information Theory, 40(4), 1013-1029.

[6] Flandrin (1992). Wavelet Analysis and Synthesis of Fractional Brownian Motion. Ieee Tran. on Information Theory, 38(2), 910-916.

[7] Krim (1995). Multire Solution Analysis of a Class of Nonstationary Processes. Ieee Tran. on Information Theory, 41(4), 1010-1019.

[8] REN, Haobo (2002). Wavelet Estimation for Jumps on a Heterosedastic Regression Model. Acta Mathematica Scientia, 22(2), 269-277.

[9] XIA, Xuewen (2005). Wavelet Analysis of the Stochastic System With Coular Stationary Noise. Engineering Science, 3, 43-46.

[10] XIA, Xuewen (2008). Wavelet Density Degree of Continuous Parameter AR Model. International Journal Nonlinear Science, 7(4), 237-242.

[11] XIA, Xuewen (2007). Wavelet Analysis of Browain Motion. World Journal of Modelling and Simulation, (3), 106-112.

[12] XIA, Xuewen, & DAI, Ting (2009). Wavelet Density Degree of a Class of Wiener Processes. International Journal of Nonlinear Science, 7(3), 327-332.

[13] Hendi, A. (2009). New Exact Travelling Wave Solutions for Some Nonlinear Evolution Equations. International Journal of Nonlinear Science, 7(3), 259-267.

XIA Xuewen [a], *

[a] Hunan Institute of Engineering, Xiangtan 411104, China; Hunan Normal University, Changsha 410082, China.

* Corresponding author.

Received 12 December 2011; accepted 20 February 2012
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Author:Xia, Xuewen
Publication:Advances in Natural Science
Article Type:Report
Geographic Code:9CHIN
Date:Jun 30, 2012
Words:1178
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