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User:Egm4313.s12.team8/R4

=R4.1 - Set up a matrix for solving coefficients for general form of polynomial excitation=

Problem Statement
Set up the matrix A as in (1) p.7-21 for the general case, with the matrix coefficients for rows 1, 2, 3, (n-2), (n-1), and n filled in.

Solution
The general form of (2) p.7b-13 is shown on p.7c-20 and is as follows:
 * $$\sum_{j=0}^{n-2}[c_{j+2} (j+2) (j+1) + a c_{j+1} (j+1) + bc_j]x^j

+ac_nnx^{n-1} + b[c_{n-1}x^{n-1} + c_nx^n] = \sum_{j=0}^{n} d_jx^j $$

When j=0:
 * $$2c_{2}+ac_{1}+bc_{0}=d_{0}$$

When j=1:
 * $$6c_{3}+2ac_{2}+bc_{1}=d_{1}$$

When j=2:
 * $$12c_{4}+3ac_{3}+bc_{2}=d_{2}$$

When j=(n-2):
 * $$[c_{n}(n)(n-1)+ac_{n-1}(n-1)+bc_{n-2}]=d_{n-2}$$

When j=(n-1):
 * $$ac_{n}n+bc_{n-1}=d_{n-1}$$

When j=n:
 * $$bc_{n}=d_{n}$$

Where $$c=[c_0, c_1, ..., c_{n-1}, c_n]^{T}$$ and $$d=[d_0, d_1, ..., d_{n-1}, d_n]^{T}$$

and $$Ac=d$$

The general form of matrix A can be created from the coefficients shown. $$A= \begin{bmatrix} b &     a &  2     &      0 &       0 &         &       0 \\ 0 &     b & 2a     &      6 &       0 &         &       0 \\ 0 &     0 &  b     &     3a &      12 &         &       0 \\ &       &        & \ddots & \ddots  &  \ddots &       0 \\ 0 &     0 &      0 &      0 &       b & a (n-1) & n (n-1) \\ 0 &     0 &      0 &      0 &       0 &       b & a n     \\ 0 &     0 &      0 &      0 &       0 &       0 &   b \end{bmatrix} $$

=R4.2 - ODE with $$\sin x$$ as its excitation=

Problem Statement
Consider the L2-ODE-CC with sinx as its excitation.

$$\ y''-3y'+2y=r(x)=sinx$$

Initial Conditions:

$$\ y(0)=0$$, $$\ y'(0)=1$$

Part 1) Use the Taylor series for sinx to reproduce the figure on p. 7-24 in lecture notes.

Part 2) Find the overall solution to the L2-ODE-CC above by substituting the Taylor series approximations for sinx at n=3,5, and 9 for the excitation equation. Plot the solutions over the interval $$\ [0,4 \pi]$$.

Part 3) Find the exact overall solution to the L2-ODE-CC above by using the function sinx as the excitation equation. Plot the solution and compare it to the solutions from Part 2.

Part 1
The Taylor series expansion of sinx takes the form:

$$\ sinx=\sum_{j=0}^{k}\frac{{}(-1)^{j}x^{2j+1}}{(2j+1)!}$$

Expanding this summation for j=6 gives:

$$\ sinx=\frac{x}{1!}-\frac{x^{3}}{3!}+\frac{x^{5}}{5!}-\frac{x^{7}}{7!}+\frac{x^{9}}{9!}-\frac{x^{11}}{11!}+\frac{x^{13}}{13!}$$

$$\ sinx=x-\frac{x^{3}}{6}+\frac{x^{5}}{120}-\frac{x^{7}}{5040}+\frac{x^{9}}{362880}-\frac{x^{11}}{39916800}+\frac{x^{13}}{6227020800}$$

By seperating and graphing each of the terms, we can see how the sum of the polynomials equal the sin function.

 * The value for n in the plot is equal to 2j+1, not the number of terms in the series.

Part 2
To find the general solution to the ODE, we first must find the homogeneous solution:

$$\ y''-3y'+2y=0$$

$$\ \lambda^2-3\lambda+2=0$$

$$\ (\lambda-1)(\lambda-2)=0$$

$$\ \lambda_{1}-1=0  =>  \lambda_{1}=1$$

$$\ \lambda_{2}-2=0  =>  \lambda_{2}=2$$

$$\ y_{h}(x)=c_{1}e^x+c_{2}e^{2x}$$

n=3
Because n=3, the excitation equation approximating sinx becomes:

$$\ r(x)=x-\frac{x^{3}}{3!}$$

The general form of $$\ y_p$$ for a polynomial excitation equation is a polynomial with the same highest degree.

$$\ y_{p}(x)=Ax^3+Bx^2+Cx+D$$

Differentiating $$\ y_p$$ and its derivative gives:

$$\ y'_{p}(x)=3Ax^2+2Bx+C$$

$$\ y''_{p}(x)=6Ax+2B$$

By substituting these equations into the original ODE we get:

$$\ y''-3y'+2y=r(x)=x-\frac{x^{3}}{3!}$$

$$\ [6Ax+2B]-3[3Ax^2+2Bx+C]+2[Ax^3+Bx^2+Cx+D]=x-\frac{x^{3}}{3!}$$

Combining like terms gives:

$$\ 2Ax^3+(2B-9A)x^2+(6A-6B+2C)x+(2B-3C+2D)=x-\frac{x^{3}}{3!}$$

Equating like terms gives:

$$\ 2Ax^3=-\frac{x^{3}}{3!}   =>   A=-\frac{1}{12}$$

$$\ (2B-9A)x^2=0      =>     B=-\frac{3}{8}$$

$$\ (6A-6B+2C)x=x         =>      C=-\frac{3}{8}$$

$$\ (2B-3C+2D)=0         =>    D=-\frac{1}{16}$$

So the particular solution becomes:

$$\ y_{p}(x)=-\frac{1}{12}x^3-\frac{3}{8}x^2-\frac{3}{8}x-\frac{1}{16}$$

The general solution is the sum of the homogeneous and particular soluions:

$$\ y(x)=c_{1}e^x+c_{2}e^{2x}-\frac{1}{12}x^3-\frac{3}{8}x^2-\frac{3}{8}x-\frac{1}{16}$$

Differentiating gives:

$$\ y'(x)=c_{1}e^x+2c_{2}e^{2x}-\frac{1}{4}x^3-\frac{3}{4}x^2-\frac{3}{8}$$

Applying initial conditions gives:

$$\ y(0)=1=c_{1}+c_{2}-\frac{1}{16}$$

$$\ y'(0)=0=c_{1}+2c_{2}-\frac{3}{8}$$

Solving the system of equations gives:

$$\ c_{1}=2$$

$$\ c_{2}=-\frac{13}{16}$$

The final approximated solution for n=3 is:

$$\ y(x)=2e^x-\frac{13}{16}e^{2x}-\frac{1}{12}x^3-\frac{3}{8}x^2-\frac{3}{8}x-\frac{1}{16}$$

n=5
Because n=5, the excitation equation approximating sinx becomes:

$$\ r(x)=x-\frac{x^{3}}{3!}+\frac{x^{5}}{5!}$$

The general form of $$\ y_p$$ for a polynomial excitation equation is a polynomial with the same highest degree.

$$\ y_{p}(x)=Ax^5+Bx^4+Cx^3+Dx^2+Ex+F$$

Differentiating $$\ y_p$$ and its derivative gives:

$$\ y'_{p}(x)=5Ax^4+4Bx^3+3Cx^2+2Dx+E$$

$$\ y''_{p}(x)=20Ax^3+12Bx^2+6Cx+2D$$

By substituting these equations into the original ODE we get:

$$\ y''-3y'+2y=r(x)=x-\frac{x^{3}}{3!}+\frac{x^{5}}{5!}$$

$$\ [20Ax^3+12Bx^2+6Cx+2D]-3[5Ax^4+4Bx^3+3Cx^2+2Dx+E]+2[Ax^5+Bx^4+Cx^3+Dx^2+Ex+F]=x-\frac{x^{3}}{3!}+\frac{x^{5}}{5!}$$

Combining like terms gives:

$$\ 2Ax^5+(2B-15A)x^4+(2C-12B+20A)x^3+(2D-9C+12B)x^2+(2E-6D+6C)x+(2F-3E+2D)=x-\frac{x^{3}}{3!}+\frac{x^{5}}{5!}$$

Equating like terms gives:

$$\ 2Ax^5=\frac{x^{5}}{5!}   =>   A=\frac{1}{240}$$

$$\ (2B-15A)x^4=0      =>     B=\frac{1}{32}$$

$$\ (2C-12B+20A)x^3=-\frac{x^{3}}{3!}    =>      C=\frac{1}{16}$$

$$\ (2D-9C+12B)x^2=0         =>    D=\frac{3}{32}$$

$$\ (2E-6D+6C)x=x       =>      E=\frac{19}{32}$$

$$\ (2F-3E+2D)=0       =>       F=\frac{51}{64}$$

So the particular solution becomes:

$$\ y_{p}(x)=\frac{1}{240}x^5+\frac{1}{32}x^4+\frac{1}{16}x^3+\frac{3}{32}x^2+\frac{19}{32}x+\frac{51}{64}$$

The general solution is the sum of the homogeneous and particular soluions:

$$\ y(x)=c_{1}e^x+c_{2}e^{2x}+\frac{1}{240}x^5+\frac{1}{32}x^4+\frac{1}{16}x^3+\frac{3}{32}x^2+\frac{19}{32}x+\frac{51}{64}$$

Differentiating gives:

$$\ y'(x)=c_{1}e^x+2c_{2}e^{2x}+\frac{1}{48}x^4+\frac{1}{8}x^3+\frac{3}{16}x^2+\frac{3}{16}x+\frac{19}{32}$$

Applying initial conditions gives:

$$\ y(0)=1=c_{1}+c_{2}+\frac{51}{64}$$

$$\ y'(0)=0=c_{1}+2c_{2}+\frac{19}{32}$$

Solving the system of equations gives:

$$\ c_{1}=1$$

$$\ c_{2}=-\frac{51}{64}$$

The final approximated solution for n=5 is:

$$\ y(x)=2e^x-\frac{51}{64}e^{2x}+\frac{1}{240}x^5+\frac{1}{32}x^4+\frac{1}{16}x^3+\frac{3}{32}x^2+\frac{19}{32}x+\frac{51}{64}$$

n=9
Because n=9, the excitation equation approximating sinx becomes:

$$\ r(x)=x-\frac{x^{3}}{3!}+\frac{x^{5}}{5!}-\frac{x^{7}}{7!}+\frac{x^{9}}{9!}$$

The general form of $$\ y_p$$ for a polynomial excitation equation is a polynomial with the same highest degree.

$$\ y_{p}(x)=Ax^9+Bx^8+Cx^7+Dx^6+Ex^5+Fx^4+Gx^3+Hx^2+Jx+K$$

Differentiating $$\ y_p$$ and its derivative gives:

$$\ y'_{p}(x)=9Ax^8+8Bx^7+7Cx^6+6Dx^5+5Ex^4+4Fx^3+3Gx^2+2Hx+J$$

$$\ y''_{p}(x)=72Ax^7+63Bx^6+42Cx^5+30Dx^4+20Ex^3+Fx^2+6Gx+2H$$

By substituting these equations into the original ODE we get:

$$\ y''-3y'+2y=r(x)=x-\frac{x^{3}}{3!}+\frac{x^{5}}{5!}-\frac{x^{7}}{7!}+\frac{x^{9}}{9!}$$

$$\ [72Ax^7+63Bx^6+42Cx^5+30Dx^4+20Ex^3+Fx^2+6Gx+2H]-3[9Ax^8+8Bx^7+7Cx^6+6Dx^5+5Ex^4+4Fx^3+3Gx^2+2Hx+J]+2[Ax^9+Bx^8+Cx^7+Dx^6+Ex^5+Fx^4+Gx^3+Hx^2+Jx+K]=x-\frac{x^{3}}{3!}+\frac{x^{5}}{5!}-\frac{x^{7}}{7!}+\frac{x^{9}}{9!}$$

Combining like terms gives:

$$\ 2Ax^9+(2B-27A)x^8+(2C-24B+72A)x^7+(2D-21C+63B)x^6+(2E-18D+42C)x^5+(2F-15E+30D)x^4+(2G-12F+20E)x^3+(2H-9G+12F)x^2+(2J-6H+6G)x+(2K-3J+2H)=x-\frac{x^{3}}{3!}+\frac{x^{5}}{5!}-\frac{x^{7}}{7!}+\frac{x^{9}}{9!}$$

Equating like terms gives:

$$\ 2Ax^9=\frac{x^{9}}{9!}   =>   A=\frac{1}{725760}$$

$$\ (2B-27A)x^8=0      =>     B=\frac{1}{53760}$$

$$\ (2C-24B+72A)x^7=-\frac{x^{7}}{7!}    =>      C=\frac{1}{13440}$$

$$\ (2D-21C+63B)x^6=0         =>    D=\frac{1}{3840}$$

$$\ (2E-18D+42C)x^5=\frac{x^{5}}{5!}     =>    E=\frac{19}{3840}$$

$$\ (2F-15E+30D)x^4=0        =>       F=\frac{17}{512}$$

$$\ (2G-12F+20E)x^3=-\frac{x^{3}}{3!}     =>      G=\frac{17}{256}$$

$$\ (2H-9G+12F)x^2=0           =>         H=\frac{51}{512}$$

$$\ (2J-6H+6G)x=x          =>       J=\frac{307}{512}$$

$$\ (2K-3J+2H)=0             =>     K=\frac{819}{1024}$$

So the particular solution becomes:

$$\ y_{p}(x)=\frac{1}{725760}x^9+\frac{1}{53760}x^8+\frac{1}{13440}x^7+\frac{1}{3840}x^6+\frac{19}{3840}x^5+\frac{17}{512}x^4+\frac{17}{256}x^3+\frac{51}{512}x^2+\frac{307}{512}x+\frac{819}{1024}$$

The general solution is the sum of the homogeneous and particular soluions:

$$\ y(x)=c_{1}e^x+c_{2}e^{2x}+\frac{1}{725760}x^9+\frac{1}{53760}x^8+\frac{1}{13440}x^7+\frac{1}{3840}x^6+\frac{19}{3840}x^5+\frac{17}{512}x^4+\frac{17}{256}x^3+\frac{51}{512}x^2+\frac{307}{512}x+\frac{819}{1024}$$

Differentiating gives:

$$\ y'(x)=c_{1}e^x+2c_{2}e^{2x}+\frac{1}{80640}x^8+\frac{1}{6720}x^7+\frac{1}{1920}x^6+\frac{1}{640}x^5+\frac{19}{640}x^4+\frac{17}{128}x^3+\frac{51}{256}x^2+\frac{51}{256}x^2+\frac{307}{512}$$

Applying initial conditions gives:

$$\ y(0)=1=c_{1}+c_{2}+\frac{819}{1024}$$

$$\ y'(0)=0=c_{1}+2c_{2}+\frac{307}{512}$$

Solving the system of equations gives:

$$\ c_{1}=1$$

$$\ c_{2}=-\frac{819}{1024}$$

The final approximated solution for n=9 is:

$$\ y(x)=e^x-\frac{819}{1024}e^{2x}+\frac{1}{725760}x^9+\frac{1}{53760}x^8+\frac{1}{13440}x^7+\frac{1}{3840}x^6+\frac{19}{3840}x^5+\frac{17}{512}x^4+\frac{17}{256}x^3+\frac{51}{512}x^2+\frac{307}{512}x+\frac{819}{1024}$$

This image is an unzoomed graph of the three general solutions to the L2-ODE-CC above on the interval $$\ [0,4 \pi]$$.

This image is a zoomed picture of the left side of the graph above. Without zooming, it is impossible to tell the differences, whereas the differences are obvious here.

This image is a zoomed picture of the right side of the graph above. In this graph, the three solutions are labeled by their power. (n=3,5,9)

Part 3
In finding the exact overall solution, we no longer use Taylor series approximations. Instead we use sinx as the excitation and solve accordingly.

$$\ y''-3y'+2y=sinx$$

Our choice for $$\ y_{p}$$ is based on r(x) and can be found in K 2011 p.82 Table 2.1.

$$\ y_{p}(x)=Asinx+Bcosx$$

Differentiating $$\ y_{p}(x)$$ and its derviate yields:

$$\ y'_{p}(x)=-Bsinx+Acosx$$

$$\ y''_{p}(x)=-Asinx-Bcosx$$

Then we substitute the derivatives into the ODE:

$$\ [-Asinx-Bcosx]-3[-Bsinx+Acosx]+2[Asinx+Bcosx]=sinx$$

Combining like terms:

$$\ (3B+A)sinx+(B-3A)cosx=sinx$$

Equating like terms:

$$\ (3B+A)sinx=sinx  => 3B+A=1$$

$$\ (B-3A)cosx=0     => B-3A=0$$

Solving the system of equations gives:

A=0.1 B=0.3

$$\ y_{p}(x)=\frac{1}{10}sinx+\frac{3}{10}cosx$$

The general solution becomes:

$$\ y(x)=y_{h}(x)+y_{p}(x)=c_{1}e^x+c_{2}e^{2x}+\frac{1}{10}sinx+\frac{3}{10}cosx$$

Differentiating the general solution:

$$\ y'(x)=c_{1}e^x+2c_{2}e^{2x}+\frac{1}{10}cosx-\frac{3}{10}sinx$$

Apply initial conditions:

$$\ y(0)=c_{1}e^{0}+c_{2}e^{0}+\frac{1}{10}sin0+\frac{3}{10}cos0=c_{1}+c_{2}+\frac{3}{10}=1 =>  c_{1}+c_{2}=\frac{7}{10}$$

$$\ y'(0)=c_{1}e^{0}+2c_{2}e^{0}+\frac{1}{10}cos0-\frac{3}{10}sin0=0 => c_{1}+2c_{2}=-\frac{1}{10}$$

Solving the system of equatiions:

$$\ c_{1}=1.5$$

$$\ c_{2}=-\frac{8}{10}$$

The final, exact, general solution is:

$$\ y(x)=1.5e^x-\frac{8}{10}e^{2x}+\frac{1}{10}sinx+\frac{3}{10}cosx$$

This image shows the unzoomed graph of all three approximations and the exact solution to the L2-ODE-CC on the interval $$\ [0,4 \pi]$$. However, it is hard to distinguish the graphs without zooming in.

This image is a zoomed part of the graph above. The n=9 solution (green) is much more accurate than the n=5 solution (blue) and the n=3 solution (red) as it is closer to the exact solution (magenta).

=R4.3 - ODE with $$\log (1+x)$$ as its excitation=

Problem
Consider the L2-ODE-CC below with $$ \displaystyle log(1+x)$$ as excitation:

$$ \displaystyle y''-3y'+2y = r(x)$$ $$ \displaystyle r(x) = log(1+x)$$

and the initial conditions $$ \displaystyle y(-\frac3{4})=1, y'(-\frac3{4})=0 $$

1. Develop $$ \displaystyle log(1+x)$$ in Taylor series about $$ \displaystyle \widehat{x}=0$$ to reproduce the figure below

2. Let $$ \displaystyle r_n(x)$$ be truncated Taylor series, with n terms--which is also the highest degree of the Taylor (power) series--of $$ \displaystyle log(1+x)$$. Find $$ \displaystyle y_n(x),$$ for n = 4, 7, 11, such that: $$ \displaystyle y_{n}''+ay_{n}'+by_{n}=r_n(x)$$ with the same initial conditions mentioned earlier. Plot $$ \displaystyle y_{n}(x)$$ for n = 4, 7, 11 for $$ \displaystyle x$$ in $$ \displaystyle [-\frac3{4},3].$$

3. Use the matlab command ode45 to integrate numerically the given equation to obtain the numerical solution for $$ \displaystyle y(x).$$ Plot $$ \displaystyle y(x)$$ in the same figure with $$ \displaystyle y_n(x).$$

Solution
1. The Taylor series about a point $$ \displaystyle \hat{x} $$ is defined as $$ \displaystyle  f(x) = \sum_{n=0}^\infty \frac{f^{n} (\hat x)}{n!} (x - \hat x)^n $$

For $$ \displaystyle \hat{x}=0, $$ we can expand the series as $$ \displaystyle f(x) = \sum_{n=0}^\infty \frac{f^{n}(0)}{n!} x^n = \frac{f^{0}(0)}{0!} x^0 + \frac{f^{1}(0)}{1!} x^1 + \frac{f^{2}(0)}{2!} x^2 + \frac{f^{3}(0)}{3!} x^3 + \cdots + \frac{f^{n}(0)}{n!} x^n $$

Deriving $$ \displaystyle ln(1+x) $$ yields $$ \displaystyle \frac{1}{x+1} $$

Plugging the derivative into the expanded series yields $$ \displaystyle log(1+x) = \sum_{n=0}^\infty \frac{f^{0}(0)}{n!}x^0 = \frac{1}{0!}ln(1)x^0 + \frac{1}{1!}\frac{1}{0+1}x^1 - \frac{1}{2!}\frac{1}{(0+1)^2}x^2 + \frac{1}{3!}\frac{2}{(0+1)^3}x^3 \cdots + \frac{1}{n!}\frac{f^n(0)}{(1)^n}x^n $$

Simplified gives us $$ \displaystyle ln(1+x) = \sum_{n=0}^\infty \frac{f^{0}(0)}{n!}x^0 = 0+ \frac{x^1}{1} - \frac{x^2}{2} + \frac{x^3}{3} + \cdots + \frac{x^n}{n} $$

Or from the Mercator series, $$\log(1+x)=\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} x^n = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots \quad$$ so, T4 (n = 4) $$ \displaystyle ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4}$$

T7 (n = 7) $$ \displaystyle ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \frac{x^5}{5} - \frac{x^6}{6} + \frac{x^7}{7}$$

T11 (n = 11) $$ \displaystyle ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \frac{x^5}{5} - \frac{x^6}{6} + \frac{x^7}{7} - \frac{x^8}{8} + \frac{x^9}{9} - \frac{x^{10}}{10} + \frac{x^{11}}{11}$$

T16 (n = 16) $$ \displaystyle ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \frac{x^5}{5} - \frac{x^6}{6} + \frac{x^7}{7} - \frac{x^8}{8} + \frac{x^9}{9} - \frac{x^{10}}{10} + \frac{x^{11}}{11} - \frac{x^{12}}{12} + \frac{x^{13}}{13} - \frac{x^{14}}{14} + \frac{x^{15}}{15} - \frac{x^{16}}{16}$$

Creating a function in MATLAB, we can code and plot

2. To find $$ \displaystyle y_n(x), $$ we must first determine the homogeneous and particular solutions. We know the characteristic equation to be $$ \displaystyle \lambda^2 -3\lambda + 2 = 0, $$ so $$ \displaystyle (\lambda - 2)(\lambda - 1) = 0, $$

Knowing $$ \displaystyle \lambda = 1,2, $$ the homogeneous solution is defined as $$ \displaystyle y_h = c_1e^x + c_2e^{2x}$$

n = 4 The particular solution for n = 4 can be obtained from the following: $$ \displaystyle y_p = K_4x^4 + K_3x^3 + K_2x^2 + K_1x^1 + K_0. $$ Also, $$ \displaystyle y_p' = 4K_4x^3 + 3K_3x^2 + 2K_2x + K_1 $$ and $$ \displaystyle y_p'' = 12K_4x^2 + 6K_3x + 2K_2. $$

Plugging back into the given ODE, and equating to the excitation gives us $$ \displaystyle (12K_4x^2 + 6K_3x + 2K_2) - 3(4K_4x^3 + 3K_3x^2 + 2K_2x + K_1) + 2(K_4x^4 + K_3x^3 + K_2x^2 + K_1x^1 + K_0) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} $$

Simplified, $$ \displaystyle (12K_4x^2 + 6K_3x + 2K_2) - (12K_4x^3 + 9K_3x^2 + 6K_2x + 3K_1) + (2K_4x^4 + 2K_3x^3 + 2K_2x^2 + 2K_1x^1 + 2K_0) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} $$

Creating a matrix for Matlab to solve for the unknown values of K $$ \begin{bmatrix} 2 & -3 & 2 & 0 & 0 \\ 0 & 2 & -6 & 6 & 0 \\ 0 & 0 & 2 & -9 & 12 \\ 0 & 0 & 0 & 2 & -12 \\ 0 & 0 & 0 & 0 & 2 \end{bmatrix}

\begin{bmatrix} K_0 \\ K_1 \\ K_2 \\ K_3 \\ K_4 \end{bmatrix} =

\begin{bmatrix} 0 \\ 1 \\ -\frac{1}{2} \\ \frac{1}{3} \\ -\frac{1}{4} \end{bmatrix}$$

Now, we have $$ \displaystyle y_p = 0.0543x^4 + 0.3981x^3 + 1.5743x^2 + 3.7458x^1 + 4.0444 $$ and our final solution is $$ \displaystyle y_n(x) = c_1e^x + c_2e^{2x} + 0.0543x^4 + 0.3981x^3 + 1.5743x^2 + 3.7458x^1 + 4.0444 $$

Evaluating at our initial conditions gives $$ \displaystyle y_n(-\frac3{4}) = c_1e^{-\frac3{4}} + c_2e^{2(-\frac3{4})} + 0.0543(-\frac3{4})^4 + 0.3981(-\frac3{4})^3 + 1.5743(-\frac3{4})^2 + 3.7458(-\frac3{4})^1 + 4.0444 = 1 $$ and

$$ \displaystyle y'_n(-\frac3{4}) = c_1e^{-\frac3{4}} + 2c_2e^{2(-\frac3{4})} + 4(0.0543)(-\frac3{4})^3 + 3(0.3981)(-\frac3{4})^2 + 2(1.5743)(-\frac3{4}) + 3.7458 = 0 $$

Solving for the constants gives us $$ \displaystyle c_1 = -4.4579 $$ $$ \displaystyle c_2 = 0.0526 $$

Therefore, $$ \displaystyle y_n(x) = -4.4579e^x + 0.0526e^{2x} + 0.0543x^4 + 0.3981x^3 + 1.5743x^2 + 3.7458x^1 + 4.0444 $$ and

n = 7 The particular solution for n = 7 can be obtained from the following: $$ \displaystyle y_p = K_7x^7 + K_6x^6 + K_5x^5 + K_4x^4 + K_3x^3 + K_2x^2 + K_1x^1 + K_0. $$ Also, $$ \displaystyle y_p' = 7K_7x^6 + 6K_6x^5 + 5K_5x^4 + 4K_4x^3 + 3K_3x^2 + 2K_2x + K_1 $$ and $$ \displaystyle y_p'' = 42K_7x^5 + 30K_6x^4 + 20K_5x^3 + 12K_4x^2 + 6K_3x + 2K_2. $$

Plugging back into the given ODE, and equating to the excitation gives us $$ \displaystyle (42K_7x^5 + 30K_6x^4 + 20K_5x^3 + 12K_4x^2 + 6K_3x + 2K_2) - 3(7K_7x^6 + 6K_6x^5 + 5K_5x^4 + 4K_4x^3 + 3K_3x^2 + 2K_2x + K_1) + \cdots $$ $$ \displaystyle 2(K_7x^7 + K_6x^6 + K_5x^5 + K_4x^4 + K_3x^3 + K_2x^2 + K_1x^1 + K_0) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \frac{x^5}{5} - \frac{x^6}{6} + \frac{x^7}{7} $$

Simplified, $$ \displaystyle (42K_7x^5 + 30K_6x^4 + 20K_5x^3 + 12K_4x^2 + 6K_3x + 2K_2) - (21K_7x^6 + 18K_6x^5 + 15K_5x^4 + 12K_4x^3 + 9K_3x^2 + 6K_2x + 3K_1) + \cdots $$ $$ \displaystyle (2K_7x^7 + 2K_6x^6 + 2K_5x^5 + 2K_4x^4 + 2K_3x^3 + 2K_2x^2 + 2K_1x^1 + 2K_0) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \frac{x^5}{5} - \frac{x^6}{6} + \frac{x^7}{7} $$

Creating a matrix for Matlab to solve for the unknown values of K $$ \begin{bmatrix} 2 & -3 &  2 &   0 &   0 &  0 &  0 &  0 \\ 0 &   2  & -6 &  6 &   0 &  0 &  0 &  0 \\ 0 &   0  &   2 & -9 &  12 &  0 &  0 &  0 \\ 0 &   0  &   0 &   2 & -12 & 20 &  0 &  0 \\ 0 &   0  &   0 &   0 &   2 & -15 &  30 &  0 \\ 0 &   0  &   0 &   0 &   0 &  2 & -18 &  42 \\ 0 &   0  &   0 &   0 &   0 &  0 &  2 & -21 \\ 0 &   0  &   0 &   0 &   0 &  0 &  0 &  2 \end{bmatrix}

\begin{bmatrix} K_0 \\ K_1 \\ K_2 \\ K_3 \\ K_4 \\ K_5 \\ K_6 \\ K_7 \end{bmatrix} =

\begin{bmatrix} 0 \\ 1 \\ -\frac{1}{2} \\ \frac{1}{3} \\ -\frac{1}{4} \\ \frac{1}{5} \\ -\frac{1}{6} \\ \frac{1}{7} \end{bmatrix}$$

Now, we have $$ \displaystyle y_p = 0.0310x^7 + 0.3619x^6 + 2.6492x^5 + 14.4946x^4 + 60.5479x^3 + 185.6066x^2 + 375.3933x + 377.4833 $$

and our final solution is $$ \displaystyle y_n(x) = c_1e^x + c_2e^{2x} + 0.0310x^7 + 0.3619x^6 + 2.6492x^5 + 14.4946x^4 + 60.5479x^3 + 185.6066x^2 + 375.3933x + 377.4833 $$

Evaluating at our initial conditions gives $$ \displaystyle y_n(-\frac3{4}) = c_1e^{-\frac3{4}} + c_2e^{2(-\frac3{4})} + 0.0310(-\frac3{4})^7 + 0.3619(-\frac3{4})^6 + 2.6492(-\frac3{4})^5 + \cdots $$ $$ \displaystyle 14.4946(-\frac3{4})^4 + 60.5479(-\frac3{4})^3 + 185.6066(-\frac3{4})^2 + 375.3933(-\frac3{4}) + 377.4833 = 1$$  and

$$ \displaystyle y'_n(-\frac3{4}) = c_1e^{-\frac3{4}} + 2c_2e^{2(-\frac3{4})} + 7(0.0310)(-\frac3{4})^6 + 6(0.3619)(-\frac3{4})^5 + \cdots $$ $$ \displaystyle 5(2.6492)(-\frac3{4})^4 + 4(14.4946)(-\frac3{4})^3 + 3(60.5479)(-\frac3{4})^2 + 2(185.6066)(-\frac3{4}) + 375.3933 = 0 $$

Solving for the constants gives us $$ \displaystyle c_1 = -2.6757 $$ $$ \displaystyle c_2 = -375.1733 $$

Therefore, $$ \displaystyle y_n(x) = -2.6757e^x - 375.1733e^{2x} + 0.0310x^7 + 0.3619x^6 + 2.6492x^5 + 14.4946x^4 + 60.5479x^3 + 185.6066x^2 + 375.3933x + 377.4833 $$ and

n = 11 The particular solution for n = 11 can be obtained from the following: $$ \displaystyle y_p = K_11x^11 + K_10x^10 + K_9x^9 + K_8x^8 + K_7x^7 + K_6x^6 + K_5x^5 + K_4x^4 + K_3x^3 + K_2x^2 + K_1x^1 + K_0. $$ Also, $$ \displaystyle y_p' = 11K_11x^10 + 10K_10x^9 + 9K_9x^8 + 8K_8x^7 + 7K_7x^6 + 6K_6x^5 + 5K_5x^4 + 4K_4x^3 + 3K_3x^2 + 2K_2x + K_1 $$ and $$ \displaystyle y_p'' = 110K_11x^9 + 90K_10x^8 + 72K_9x^7 + 56K_8x^6 + 42K_7x^5 + 30K_6x^4 + 20K_5x^3 + 12K_4x^2 + 6K_3x + 2K_2. $$

Plugging back into the given ODE, and equating to the excitation gives us $$ \displaystyle (110K_11x^9 + 90K_10x^8 + 72K_9x^7 + 56K_8x^6 + 42K_7x^5 + 30K_6x^4 + 20K_5x^3 + 12K_4x^2 + 6K_3x + 2K_2) - 3(11K_11x^10 + \cdots $$ $$ \displaystyle 10K_10x^9 + 9K_9x^8 + 8K_8x^7 + 7K_7x^6 + 6K_6x^5 + 5K_5x^4 + 4K_4x^3 + 3K_3x^2 + 2K_2x + K_1) + 2(K_11x^11 + K_10x^10 + K_9x^9 + K_8x^8 + K_7x^7 + K_6x^6 + K_5x^5 + \cdots $$ $$ \displaystyle K_4x^4 + K_3x^3 + K_2x^2 + K_1x^1 + K_0) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \frac{x^5}{5} - \frac{x^6}{6} + \frac{x^7}{7} - \frac{x^7}{8} + \frac{x^7}{9} - \frac{x^7}{10} + \frac{x^7}{11} $$

Simplified, $$ \displaystyle (110K_11x^9 + 90K_10x^8 + 72K_9x^7 + 56K_8x^6 + 42K_7x^5 + 30K_6x^4 + 20K_5x^3 + 12K_4x^2 + 6K_3x + 2K_2) - \cdots $$ $$ \displaystyle (33K_11x^10 + 30K_10x^9 + 27K_9x^8 + 24K_8x^7 + 21K_7x^6 + 18K_6x^5 + 15K_5x^4 + 12K_4x^3 + 9K_3x^2 + 6K_2x + 3K_1) + \cdots $$ $$ \displaystyle (2K_11x^11 + 2K_10x^10 + 2K_9x^9 + 2K_8x^8 + 2K_7x^7 + 2K_6x^6 + 2K_5x^5 + 2K_4x^4 + 2K_3x^3 + 2K_2x^2 + 2K_1x^1 + 2K_0) = \cdots $$ $$ \displaystyle x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \frac{x^5}{5} - \frac{x^6}{6} + \frac{x^7}{7} - \frac{x^7}{8} + \frac{x^7}{9} - \frac{x^7}{10} + \frac{x^7}{11}$$

Creating a matrix for Matlab to solve for the unknown values of K

Now, we have $$ \displaystyle y_p = 0.0197x^11 + 0.3474x^10 + 4.1499x^9 + 40.4165x^8 + \cdots $$ $$ \displaystyle 335.6321x^7 + 2392.5102x^6 + 20348.2561x^5 + 140399.9629x^4 + 638917.2891x^3 + 2032728.1323x^2 + 4181432.7467x + 4239420.9877 $$

and our final solution is $$ \displaystyle y_n(x) = c_1e^x + c_2e^{2x} + 0.0197x^11 + 0.3474x^10 + 4.1499x^9 + \cdots $$ $$ \displaystyle 40.4165x^8 + 335.6321x^7 + 2392.5102x^6 + 20348.2561x^5 + 140399.9629x^4 + 638917.2891x^3 + 2032728.1323x^2 + 4181432.7467x + 4239420.9877 $$

Evaluating at our initial conditions gives $$ \displaystyle y_n(-\frac3{4}) = c_1e^{-\frac3{4}} + c_2e^{2(-\frac3{4})} + 0.0197(-\frac3{4})^11 + 0.3474(-\frac3{4})^10 + 4.1499(-\frac3{4})^9 + 40.4165(-\frac3{4})^8 + \cdots $$ $$ \displaystyle 335.6321(-\frac3{4})^7 + 2392.5102(-\frac3{4})^6 + 20348.2561(-\frac3{4})^5 + 140399.9629(-\frac3{4})^4 + 638917.2891(-\frac3{4})^3 +  \cdots $$ $$ \displaystyle 2032728.1323(-\frac3{4})^2 + 4181432.7467(-\frac3{4}) + 4239420.9877 = 1 $$   and

$$ \displaystyle y'_n(-\frac3{4}) = c_1e^{-\frac3{4}} + 2c_2e^{2(-\frac3{4})} + 11(0.0197)(-\frac3{4})^10 + 10(0.3474)(-\frac3{4})^9 + \cdots $$ $$ \displaystyle 9(4.1499)(-\frac3{4})^8 + 8(40.4165)(-\frac3{4})^7 + 7(335.6321)(-\frac3{4})^6 + 6(2392.5102)(-\frac3{4})^5 + 5(20348.2561)(-\frac3{4})^4 + \cdots $$ $$ \displaystyle 4(140399.9629)(-\frac3{4})^3 + 3(638917.2891)(-\frac3{4})^2 + 2(2032728.1323)(-\frac3{4}) + 4181432.7467 = 0 $$

Solving for the constants gives us $$ \displaystyle c_1 = -484 $$ $$ \displaystyle c_2 = -1753750 $$

Therefore, $$ \displaystyle y_n(x) = -484e^x - 1753750e^{2x} + 0.0197x^11 + 0.3474x^10 + 4.1499x^9 + 40.4165x^8 + \cdots $$ $$ \displaystyle 335.6321x^7 + 2392.5102x^6 + 20348.2561x^5 + 140399.9629x^4 + 638917.2891x^3 + 2032728.1323x^2 + \cdots $$ $$ \displaystyle 4181432.7467x + 4239420.9877 $$ and 3. The Following Matlab function uses ode45 to calculate and graph alongside the functions previously obtained.

=R4.4 - Extend the Accuracy of the Solution Beyond $$\hat x = 0$$=

Problem Statement
Back up away a little from the brink of non-convergence at $$x = 1$$ for the Taylor series of $$\log (1+x)$$ about $$\hat x = 0$$, and consider the point $$x_1 = 0.9$$.

Find the value of $$y_n(x_1)$$, $$y_n'(x_1)$$ that will serve as initial conditions for the next iteration to extend the domain of accuracy of the analytical solution. Find $$n$$ sufficiently high so that $$y_n(x_1)$$ and $$y_n'(x_1)$$ do not differ from the numerical solution by more than $$10^{-5}$$.

Solution
The following MATLAB code generates the numerical solution to the problem:

The following MATLAB function will generate the output, as well as the coefficients, of a series solution to the ODE:

Plotting the absolute error at $$x_1$$ as a function of $$n$$ gives:

The minimum error is thus found at $$n = 15$$ for both $$y_n(x_1)$$ and $$y_n'(x_1)$$ and is roughly:

This is a couple orders of magnitude greater than the desired $$10^{-5}$$. Error increases for $$n > 15$$ due to the truncation error inherent in computing numerical answers.

Therefore, the initial conditions of the next iteration so as to extend the domain of accuracy of the analytical solution are: $$\begin{align} y_{15}(x_1) &= -22.1530 \\ y_{15}'(x_1) &= -55.9732 \end{align}$$

Problem Statement
Develop $$ \displaystyle log(1+x)$$ in a Taylor Series about â=1.

Compute for n= 4, 7, 11.

Plot these three functions along with the real solution.

Find the domain of convergence.

Solution
Taylor Series expansions follow the form:
 * $$ \ \sum_{n = 0}^{\infty} \frac { f^n(a) (x-a)^n }{n!}\ $$

For our problem this translates to:
 * $$ \ \sum_{n = 1}^{\infty} (-1)^{n+1}\frac { (x-1)^n }{n}\ $$

We start the summation at 1 because log(0) is not defined.

For n=4 terms this expands to:
 * $$ (x-1)- \frac {(x-1)^2}{2} + \frac {(x-1)^3}{3} - \frac {(x-1)^4}{4} $$

For n=7 terms this expands to:
 * $$ (x-1)- \frac {(x-1)^2}{2} + \frac {(x-1)^3}{3} - \frac {(x-1)^4}{4} + \frac {(x-1)^5}{5} - \frac {(x-1)^6}{6} + \frac {(x-1)^7}{7}  $$

For n=11 terms this expands to:
 * $$ (x-1)- \frac {(x-1)^2}{2} + \frac {(x-1)^3}{3} - \frac {(x-1)^4}{4} + \frac {(x-1)^5}{5} - \frac {(x-1)^6}{6} + \frac {(x-1)^7}{7} - \frac {(x-1)^8}{8} + \frac {(x-1)^9}{9} - \frac {(x-1)^{10}}{10} + \frac {(x-1)^{11}}{11} $$

Graph
Now these 3 functions will be plotted against the actual solution:

It can then be found from observation of the graph that the domain of convergence is (0,2).

Problem Statement
Find $$\displaystyle y_{n}(x) $$ for n=4,7,11 such that: $$\displaystyle y''_{n}+ay'_{n}+by_{n}=r_{n}(x) $$       (1) for x in [.9,3] with the initial conditions found i.e, $$\displaystyle y_{n}(x_{1}), y'_{n}(x_1). $$ Plot $$\displaystyle y_{n}(x) $$  for n=4,7,11 for x in [.9,3].

Solution
Using a modified version of the Matlab code used in 4.4.1, shown here below, we can solve our coefficients for each y equation (at the specified n values): Our previously solved initial conditions are: $$\displaystyle y_{n}(x_{1})=-22.153,y'_{n}(x_{1})=-55.9732 $$ Using the above code, we can solve for the coefficients and final y equation to be as shown below: For n=4: $$\displaystyle y_n(x) = 8.9210e^x - 5.6353e^{2x} -0.125x^4 - 0.583x^3 -2.125x^2 - 4.125x^1 - 4.0625 $$     (4.4.3.1) For n=7: $$ \displaystyle y_n(x) = -615.0725e^x - 3.3884e^{2x} + 0.07143x^7 + 0.6666x^6 + 4.6x^5 + 24.375x^4 + 100.4166x^3 + 305.375x^2 + 615.375x + 617.6875 $$    (4.4.3.2) For n=11: $$ \displaystyle y_n(x) = -3301815e^x + 734.893e^{2x} + 0.04545x^{11} + 0.7x^{10} + 8.0556x^9 + \cdots $$ $$ \displaystyle 77.1875x^8 + 636.3214x^7 + 4520x^6 + 27318x^5 + 137082x^4 + 549316x^3 + 1649429x^2 + 3300389x + 3301079 $$    (4.4.3.3)

Problem Statement
Use the matlab command ode45 to integrate numerically using the initial conditions over x in [0.9, 3] and obtain the numerical solution $$y(x)$$. Plot $$y(x)$$ in the same figure as the above $$y_n(x)$$.

Solution
The following MATLAB code using the function coeffs defined above in R4.4.3 obtains us the following graph:

=Team Contributions=

title: User:Egm4313.s12.team8/R4


license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

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MakerBot/How to Download 3D programs to build STL files

How to download Auto CAD

 * 1. Open Google Chrome or Mozilla Fire Fox.
 * 2. Go to http://usa.autodesk.com
 * 3. In the upper right hand corner in the search bar type "download autocad 2011"
 * 4. Click on the first link
 * 5. Close all open applications
 * 5. Fill out the information and click download.
 * 6. Click Save File; when file is saved, click the .exe icon in the dialog box. and select Install

How to Download Blender

 * 1. Open Google Chrome or Mozilla Fire Fox.
 * 2. Go to blender http://www.blender.org/
 * 3. In the second tab click on download
 * 4. Download the one that fits your machine
 * 5. Click on save file, when the file is saved, click the .exe icon in the dialog box. and select Install

How to Download Google Sketch Up

 * 1. Open Google Chrome or Mozilla Fire Fox.
 * 2. Go to http://sketchup.google.com/
 * 3. Click on "Downloading Google SketchUp"
 * 4. Click on "Downloading Google SketchUp" again
 * 5. Select the options the matches with your computer
 * 6. Agree to the terms and conditions.]]
 * 7. Click on save file, when the file is saved, click the .exe icon in the dialog box. and select Install


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User:Fading chaos

The following is a basic overview of my beliefs, and should provide enough infomation for now :)

My beliefs are on all based on personal experience and observation of social dynamics. For me, there has been only one source of spiritual knoweldge, and that source is commonley called Life. As, for me at least, it is Life itself, that has and still shares how to gain an understanding of not only myself, but also directing an awakeaning to the spirtual knowledge and insights I have for myself. The realization that Life iteself, is not only guiding me though circumstances, but then utilizes the expiernece as maternails to which insights are realized and immideatlhy understoon. This is how I expiereince some of the lessons Life has shared with me, and am hope that it is the same for others. I believe that Life, is of itself, a personal journey. The goal being personal growth, understanding, spiritual development, and raising one's conciousness Only though learning can one only develop new ideas, thoughts, analyize their pervious moments in whatever why the believe to be right for them. All I do know is that, for me life has and is full of patterns and parrelles within my existnace. Thouogh analysis of my Life pattern identifciation and the knowledge gained has been possible. Only though self reflection of the events from my childhood to present was I able to perform the analiitical process Life used as a covert teqnique in shaing knowledge. This process provided me with insights I previously was unaware of. Such as the realization that Life's journey is individualized. And as such an evolution in regards to individual conciousness and a coinsiding awakeing of one's spirituality can (and in my case does) occur. This is one example of how Life shares inights knowledge with me. Additoinally, this also can show that one's beliefs are subject to change. Change is the only thing that is static, besides values to numericial notation. Furthermore, all of the insightful knoweldege Life has and still shares with me it only possible due to the human expierence. Only though the human expierence the common phiolosophical views of compassion and noon-violence havee devoloped. Karmic lessons, based on my expiericne and mine anlone, I have funlly understoond because of my documentatoin and sub-sequent analysis of moments....Streaming thoughts documented over years is an invaluable resource Life provided me, As it is Life itself that has conditinoed me to either think outloud or type my thoughts....and it is this conditiong that Life has allowed me to provide myself with the raw data for all of this. Only I can understand my mental processes and only I am able to understand and provide accurate analyzses of documetned menta states, but again, unless one can fully and COMPLETELY understnad how they function mentally, how can they begin to comprehend another?

to the knoweldge that has been carried over from spiritual deveolpment of my inner-being. Again, this is how I see it for myself, and can only provide objective insights based on my expierence, and mine alone. Also the following quote contains knowledge I am always mindful of, as with out it grouwth and development could become delayed by the human Ego :

"“Having the knowledge that personal beliefs can be flawed, is the key to personal growth.” --- Anonymous

Additionally.... ""My existance is only possible, via the allowance, provided to by the surrounding envrionemtnt.""---Anonymous

title: User:Fading chaos


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User:Prativendra.singh

Prativendra Singh

Address: 92/1 Mo. Jawahar Nagar Town Kursath, Hardoi-241126, Uttar Pradesh, India Phone: +91-9802762436 Email: prativendra.singh@gmail.com FB Acadamia linkedin

EDUCATION

'Master Of Technology' National Institute of technology, Kurukshetra (India) Nanotechnology by Department of Physics, August 2012 to ongoing

'Bachelor Of Engineering' The University Of Burdwan, Burdwan (India) Applied Electronics And Instrumentation Engineering, August 2008 to July 2012

PROJECT WORK EXPERIENCE

Simulation on Flow parameter for Nano-Particle transport in a Micro-Channel, July 2013 to ongoing: National Institute of technology, Kurukshetra (India) Developed the model and method for Targeted Drug Delivery system during the master’s dissertation work.

Skin Response Meter using Silver /Silver Chloride ECG Electrodes and display using Microcontroller, February 2012 to June 2012: University Institute OF Technology, Burdwan

Skin Response Meter using IC CA3140, August 2011 to January 2012: University Institute Of Technology, Burdwan

Vertigo: The Manually Controlled Robot, Dec 2011 to January 2012: Indian Institute Of Technology, Kharagpur

SEMINAR PRESENTATIONS

Seminar on “Renewable Energy Solution From Solar cell to Nano-leaves (Green Energy Model)”, November 2012 Seminar on “Study of Velocity Profile and Pressure Gradient for Nano-fluid in a micro-channel”, December 2013 Seminar on “Liquiglide: The Slippery Surface Solution”, May 2013 Seminar on “Genomic Data Mining using ANN”, June 2012 Seminar on “Electrical Motor Starters”, January 2012

POSTER PRESENTATION IN CONFERENCES

 National Conference of NCNIT-2013 on the topic "Advance Stress level measurement using Silver/silver chloride electrodes and Microcontroller", March 2013  National Conference of NCNIT-2013 on the topic " Review on ZnO coated Carbon Nanotubes as Piezoelectric material and analysis”, March 2013  Frontiers of Microscopy Virtual Conference 2013 organized by Materials Today – UK on the topic “Green Energy Model on Renewable Energy Solution From Solar cells to Nano-leaves”, April 2013  INTERNATIONAL CONFERENCE on Nanotechnology in the service of health, environment & society 2014 on the topic “Study of Velocity Profile and Pressure Gradient for Nano-fluid in a micro-channel”.  Nanomaterials and Instrumentation 2014 on the topic “Study of Particle trajectory for the fluid flow within A Channel” in NATIONAL CONFERENCE

APPROVED POSTER  International congress of Mathematicians SEOUL ICM 2014, Korea on the topic “Modelling of magnetic particle trajectory in a microvasculature for targeted drug delivery”.  International Conference on Mathematical Science ICMS 2014,India on the topic“Computational Simulation on particle transport in a microchannel”

TRAINING Short Term Course on “Emerging materials: characterization and applications” at NIT Kurukshetra, 13st June 2014 to 17rd June 2014 Short Term Course on “Nanomaterials: Synthesis and Characterization” at NIT Kurukshetra, 11st March 2014 to 15rd March 2014 Short Term Course on “Nanomaterials and Their Characterization tools” at NIT Kurukshetra, 1st June 2013 to 3rd June 2013 Nominated by “Logicon Automation Limited” Lucknow for the One month Industrial Summer training, 27th June 2011 to 27th July 2011. Nominated by “Diesel Shed Locomotive” Siliguri for Short-term Vocational Training, 20th Oct 2011 to 29th oct 2011 Selected for 2 Week Industrial Training in “Durgapur Chemicals Ltd”,Durgapur, 27th Feb 2012 to 10th march 2012.

LABORATORY EXPERIENCE

 Template Synthesis method for Nanowire Deposition  SEM  TGA  Sample preparation and their XRD analysis  Sample preparation and their XRF analysis  Thin film deposition by Thermal Vapor Deposition and also created the Pattern by thermal Vapor Deposition using mask.  Analysis of the Alfa particles over LR-115 film and etching the sample then counting the tracks using Optical Microscope  UV- Analysis of the sample using UV Spectrometer  To analyze the sample solution using Colorimeter  Analysis of the different ferroelectric materials by PE Loop setup etc  PCB Fabrication  On T-Spice designed all the logic gates in addition to few logic functions like adder, subtractor, waveform generator using CMOS circuits.  Using MATLAB implemented the design of FIR filters, IIR filters, algebraic, logarithmic functions, step response of first order and second order system with and without feedback.  Acquaintance with PLC (programmable logic controller) and DCS (distributed control system).  Worked with various types of transducers and signal conditioning elements used in industrial, analytical and biomedical instrumentation.  Well versed with the assembly language programming of 8085 and 8086 microprocessor and its interfacing.  Implemented various applications of 8051 microcontroller using suitable assembly language programming.  A good experience of many experiments related to General Engineering, Basic Electrical, Basic Electronics, Analog circuits, Digital circuits during the first two years of the undergraduate course.

SOFTWARE EXPERIENCE

Good working knowledge of some programming languages such as C, C++. I have also worked on Other softwares such as  MATLAB 77  Multisim2003  Tspice13  Express PCB  Protious8.0  Virtual nanolab 12.8.1  Keil uVision4  Mplab etc.  Comsol Multiphysics 4.4a.

RESEARCH INTERESTS

 Application of carbon nanotubes and grapheme  Sensors(MEMS/NEMS)  Nanoelectronic Devices  Nano-sensors in bio-medical applications  Piezo-Electric material application in Bio-medical Sensor  Nano-material applications in Solar cell  Nano-Fluidic devices

ADDITIONAL QUALIFICATION/ACHIEVEMENTS/CERTIFICATIONS

 Good athlete (participated in Regional School level, District level Cycle racing).  Active member in college programs like fest, fresher’s, NSS,  E-leader of NEN. towards SANKALP (UIT E-Cell)  Class and Placement cell Representative of the College  PAC(Placement Advisory committee ) member of NIT Kurukshetra  Student Co-ordinator of Nano-Shrasti at NIT KKR

title: User:Prativendra.singh


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Draft:Original research/Phosphate budgets

"The amount of phosphate needed or available for a purpose, including estimates of phosphate in and phosphate out, and the phosphate form, determine the phosphate budget for a cell or an entire organism." Bold added. For a "standard man" of 70 kg the available phosphate is ~1.52 x 1025 molecules of phosphate in some form. The available phosphate of an adult human female may differ from the "standard man".

Adult humans
"A well-fed adult in the industrialized world consumes and excretes about 1-3 g of phosphorus per day in the form of phosphate (2-6 x 1022 molecules). Per the elemental composition of the "standard man" of 70 kg, phosphorus is 780 g or 1.1% (as 1.52 x 1025 molecules of phosphate). Of this 1.4 g/kg (98 g, 1.9 x 1024 molecules of phosphate) are present in soft tissue with the remainder (1.33 x 1025 molecules of phosphate) in mineralized tissue such as bone and teeth. Only about 0.1% of body phosphate (about 2 x 1022 molecules) circulates in the blood, but this amount reflects the amount of phosphate available to soft tissue cells. Blood plasma contains orthophosphate (as HPO42-) and H2PO4- in the ratio of about 4:1. "

"The total quantity of ATP in the human body is about 0.1 mole (about 6 x 1022 molecules). This ATP is constantly being broken down into ADP, and then converted back into ATP. At any given time, the total amount of ATP + ADP remains fairly constant. The energy used by human cells requires the hydrolysis of 100 to 150 moles (6 to 9 x 1025 molecules) of ATP daily which is around 50 to 75 kg. Typically, a human will use up their body weight of ATP over the course of the day. This means that each ATP molecule is recycled 1000 to 1500 times daily, or about once every minute."

Orthophosphoric acids
Def. a "colourless liquid, H3PO4" is called phosphoric acid.

Def. "[o]rdinary phosphoric acid, H3PO4" is called orthophosphoric acid.

Orthophosphoric acid is a non-toxic, inorganic, rather weak triprotic acid, that is a very polar molecule soluble in water. Triprotic means that an orthophosphoric acid molecule can dissociate up to three times, giving up an H+ each time, which typically combines with a water molecule, H2O, as shown in these reactions:

 * H3PO4(s)  + H2O(l) <=> H3O+(aq) + H2PO4−(aq)       Ka1= 7.25×10−3

 * H2PO4−(aq)+ H2O(l) <=> H3O+(aq) + HPO42−(aq)      Ka2= 6.31×10−8

 * HPO42−(aq)+ H2O(l) <=> H3O+(aq) + PO43−(aq)        Ka3= 3.98×10−13

"The anion after the first dissociation, H2PO4−, is the dihydrogen phosphate anion. The anion after the second dissociation, HPO42−, is the hydrogen phosphate anion. The anion after the third dissociation, PO43−, is the phosphate or orthophosphate anion. For each of the dissociation reactions shown above, there is a separate acid dissociation constant, called Ka1, Ka2, and Ka3 given at 25 °C. Associated with these three dissociation constants are corresponding pKa1=2.12, pKa2=7.21 , and pKa3=12.67 values at 25 °C."

"For a given total acid concentration [A] = [H3PO4] + [H2PO4−] + [HPO42−] + [PO43−] ([A] is the total number of moles of pure H3PO4 which have been used to prepare 1 liter of solution), the composition of an aqueous solution of phosphoric acid can be calculated using the equilibrium equations associated with the three reactions described above together with the [H+][OH−] = 10−14 relation and the electrical neutrality equation. Possible concentrations of polyphosphoric molecules and ions is neglected. The system may be reduced to a fifth degree equation for [H+] which can be solved numerically, yielding:"

Pyrophosphoric acids
Def. the "[syrupy liquid] acid formed by the dehydration of [i.e., removing a molecule of water from] two molecules of phosphoric acid [to form one molecule of] H4P2O7" is called pyrophosphoric acid.

Oligophosphoric acids
Def. a series of phosphoric acids condensed into one molecule when the number of phosphoric acids is small, per the general formula Hn+2PnO3n+1, where n is usually greater than 5, is called an oligophosphoric acid.

Phosphates
Notation: let the symbols ATP, CTP, GTP, NTP, and UTP stand for adenosine triphosphate, cytidine triphosphate, guanosine triphosphate, nucleotide triphosphate, and uridine triphosphate, respectively.

Notation: let the symbols ADP, CDP, GDP, NDP, and UDP stand for adenosine diphosphate, cytidine diphosphate, guanosine diphosphate, nucleotide diphosphate, and uridine diphosphate, respectively.

Notation: let the symbols AMP, CMP, GMP, NMP, and UMP stand for adenosine monophosphate, cytidine monophosphate, guanosine monophosphate, nucleotide monophosphate, and uridine monophosphate, respectively.

Def. "any salt or ester or phosphoric acid" is called phosphate.

Def. "any salt or ester of pyrophosphoric acid" is called a pyrophosphate.

Def. any "compound or anion containing several phosphate groups" is called an oligophosphate.

Def. "any of a class of inorganic polymers containing linked phosphate groups" is called a polyphosphate.

Nucleotides
When nucleotide hydrolysis is coupled to the polymerization process, the two ends of the polymer behave differently, and the critical concentration required for assembly at one end is different from that required for the opposite end. However, NTP hydrolysis is usually associated but not directly coupled with polymerization. When there are NTP-bound subunits at the ends, the polymer is stable and continues to grow slowly. When at the polymer ends there are subunits containing NDP, the polymer is in a shrinking phase, depolymerizes rapidly and completely, and much faster at lower mer concentrations.

Def. a "monomer constituting DNA or RNA biopolymer molecules, [consisting] of a nitrogenous heterocyclic base (or nucleobase), which can be either a double-ringed purine or a single-ringed pyrimidine; a [five-carbon] pentose sugar (deoxyribose in DNA or ribose in RNA); and a phosphate [group]" is called a nucleotide.

Catalytic phosphates
The catalytic phosphate in a nucleotide is usually the phosphate farthest from the nucleoside.

"The γ phosphate of ATP is the catalytic phosphate."

"A phosphate which affects the catalytic activity of an enzyme is a catalytic phosphate. Occasionally, an enzyme contains a structural phosphate and a catalytic phosphate. "

"Catalytic phosphates are acid-labile and base-stable. Structural phosphates are acid-stable, base-labile. "

Structural phosphates
Structural phosphate may be inside a cell, "structural to a nucleic acid such as DNA and RNA or a phospholipid. Outside the cell, phosphate may be dissolved in extracellular fluid (ECF) or form structures such as bone and teeth. Bringing phosphate in any form into the cell from a phosphate containing structure or for such a structure and when needed transporting phosphate out of the cell perhaps to a structure is a necessary activity of phosphate homeostasis for that cell."

Structural phosphate is a part of each polymer whether it is a microfilament, microtubule, phospholipid, or nucleic acid. Inorganic phosphates such as Pi and PPi can be precipitated with divalent cations like Ca2+, Mg2+, or others, e.g. Mn2+, to form a variety of tissue and mineral composites: cartilage, teeth, andbone. These in turn eventually become localized to the natural environment and form phosphorites.

Bones
Bones are rigid organs that constitute part of the endoskeleton of vertebrates. They support and protect the various organs of the body, produce red and white blood cells and store minerals. Bones act as reserves of minerals important for the body, most notably calcium and phosphorus. Bone controls phosphate metabolism by releasing fibroblast growth factor – 23 (FGF-23), which acts on kidneys to reduce phosphate reabsorption]]. "The primary tissue of bone, osseous tissue, is mostly made up of a composite material incorporating the mineral calcium phosphate in the chemical arrangement termed calcium hydroxylapatite (this is the osseous tissue that gives bones their rigidity) and collagen, an elastic protein which improves fracture resistance."

"Osteoblasts are mononucleate bone-forming cells that descend from osteoprogenitor cells. They are located on the surface of osteoid seams and make a protein mixture known as osteoid, which mineralizes to become bone. ... Osteoblasts also manufacture hormones, such as prostaglandins, to act on the bone itself. They robustly produce alkaline phosphatase, an enzyme that has a role in the mineralisation of bone, as well as many matrix proteins."

"Mineralisation involves osteoblasts secreting vesicles containing alkaline phosphatase. This cleaves the phosphate groups and acts as the foci for calcium and phosphate deposition."

Teeth
"A tooth (plural teeth) is a small, calcified, whitish structure found in the jaws (or mouths) of many vertebrates and used to break down food. Some animals, particularly carnivores, also use teeth for hunting or for defensive purposes. The roots of teeth are covered by gums. Teeth are not made of bone, but rather of multiple tissues of varying density and hardness. The cellular tissues that ultimately become teeth originate from the embryonic germ layer, the ectoderm".

"The general structure of teeth is similar across the vertebrates, although there is considerable variation in their form and position. The teeth of mammals have deep roots, and this pattern is also found in some fish, and in crocodilians. In most teleost fish, however, the teeth are attached to the outer surface of the bone, while in lizards they are attached to the inner surface of the jaw by one side. In cartilaginous fish, such as sharks, the teeth are attached by tough ligaments to the hoops of cartilage that form the jaw. "

"Mammals are diphyodont, meaning that they develop two sets of teeth. In humans, the first set (the "baby," "milk," "primary" or "deciduous" set) normally starts to appear at about six months of age, although some babies are born with one or more visible teeth, known as neonatal teeth. Normal tooth eruption at about six months is known as teething and can be painful."

Cartilage
Def. a "type of dense, non-vascular connective tissue, usually found at the end of joints, the rib cage, the ear, the nose, in the throat and between intervertebral disks" is called cartilage.

"Ossification (or osteogenesis) is the process of laying down new bone material by cells called osteoblasts. It is synonymous with bone tissue formation. There are two processes resulting in the formation of normal, healthy bone tissue: Intramembranous ossification is the direct laying down of bone into the primitive connective tissue (mesenchyme), while endochondral ossification involves cartilage as a precursor. ... Calcification is synonymous with the formation of calcium-based salts and crystals within cells and tissue. It is a process that occurs during ossification, but not vice versa."

"[M]atrix vesicles (MV) are primary initiators of extracellular mineral deposition in endochondral calcification. ... direct cellular metabolism of Ca2+ and inorganic phosphate (Pi), and cellular interaction with the matrix, are involved in the formation of calcifiable MV. ... chondrocytes in growth plate (GP) cartilage ... induce the formation of calcificable MV. ... GP chondrocytes are depleted of ATP and have elevated cytosolic Pi, a condition prerequisite to formation of Ca2+-acidic phospholipid (APL)-Pi complex-primed MV. [The] interaction between the extracellular matrix and chondrocytes [may] facilitate Ca2+ loading of chondrocytes, formation of Ca2+ and Pi-primed MV and rapid induction of mineralization in GP cartilage."

The spotted gar larva at right is a "living fossil" that shows cartilage beginning to be transformed into bone by endochondral calcification followed by extracellular mineralization of bone composed of calcium phosphate.

Microtubules
"During polymerization, both the α- and β-subunits of the tubulin dimer are bound to a molecule of GTP. The GTP bound to α-tubulin is stable, but the GTP bound to β-tubulin may be hydrolized to GDP shortly after assembly. The kinetics of GDP-tubulin are different from those of GTP-tubulin; GDP-tubulin is prone to depolymerization. A GDP-bound tubulin subunit at the tip of a microtubule will fall off, though a GDP-bound tubulin in the middle of a microtubule cannot spontaneously pop out. Since tubulin adds onto the end of the microtubule only in the GTP-bound state, there is generally a cap of GTP-bound tubulin at the tip of the microtubule, protecting it from disassembly. When hydrolysis catches up to the tip of the microtubule, it begins a rapid depolymerization and shrinkage. This switch from growth to shrinking is called a catastrophe. GTP-bound tubulin can begin adding to the tip of the microtubule again, providing a new cap and protecting the microtubule from shrinking. This is referred to as rescue."

Microfilaments
"In vitro actin polymerization, nucleation, starts with the self-association of three G-actin monomers to form a trimer. ATP-actin then binds the plus (+) end, and the ATP is subsequently hydrolyzed with a half time of about 2 seconds and the inorganic phosphate released with a half-time of about 6 minutes, which reduces the binding strength between neighboring units and generally destabilizes the filament. In vivo actin polymerization is catalyzed by a new class of filament end-tracking molecular motors known as actoclampins (see next section). Recent evidence suggests that ATP hydrolysis can be prompt in such cases (i.e., the rate of monomer incorporation is matched by the rate of ATP hydrolysis)."

"ADP-actin dissociates slowly from the minus end, but this process is greatly accelerated by ADP-cofilin, which severs ADP-rich regions nearest the (–)-ends. Upon release, ADP-actin undergoes exchange of its bound ADP for solution-phase ATP, thereby forming the ATP-actin monomeric units needed for further (+)-end filament elongation. This rapid turnover is important for the cell's movement. End-capping proteins such as CapZ prevent the addition or loss of monomers at the filament end where actin turnover is unfavourable like in the muscle apparatus."

Phospholipids
"Phospholipids are a class of lipids that are a major component of all cell membranes as they can form lipid bilayers. Most phospholipids contain a diglyceride, a phosphate group, and a simple organic molecule such as choline; one exception to this rule is sphingomyelin, which is derived from sphingosine instead of glycerol. ... The structure of the phospholipid molecule generally consists of hydrophobic tails and a hydrophilic head. Biological membranes in eukaryotes also contain another class of lipid, sterol, interspersed among the phospholipids and together they provide membrane fluidity and mechanical strength. Purified phospholipids are produced commercially and have found applications in nanotechnology and materials science. "

Nucleic acids
Def. "[a]ny acidic, chainlike biological macromolecule consisting of multiply repeat units of phosphoric acid, sugar and purine and pyrimidine bases" occurring in cell nuclei is called a nucleic acid.

Def. a nucleic acid "in which the sugar component is threose" is called threose nucleic acid, or threonucleic acid.

Phosphate reactions
A phosphate reaction is a chemical reaction that leads to the transformation of one set of phosphates into another.

Each phosphate reaction that occurs in living cells is a part of phosphate metabolism. These reactions are the basis of life, allowing cells to grow and reproduce, maintain their structures, and respond to their environments. Metabolism is usually divided into two categories: catabolism which yields energy such as the breakdown of food in cellular respiration, and anabolism which uses this energy to construct components of cells such as proteins and nucleic acids.

The KEGG database contains biochemical phosphate reactions, some of which may include a nucleotide such as nucleoside monophosphate (NMP). Various possible phosphate reactions number up to 2371 as of October 20, 2009. Not all of these occur in humans. A list of biochemical phosphate reactions may contain these phosphate reactions. Those that have human genes or have products occurring in humans contribute to human life.

Catabolism
Def. "destructive metabolism, usually including the release of energy and breakdown of materials" is called catabolism.

An decrease in the number of phosphates per molecule is phosphate catabolism.

For example, in phosphate catabolism, ATP is reduced to ADP with the release of an orthophosphate (Pi) and energy.

Def. any "enzyme that hydrolyzes an organic phosphate group" is called a phosphohydrolase.

Def. any "of several enzymes that hydrolyze phosphate esters" is called a phosphatase.

Anabolism
Def. the "constructive metabolism of the body" is called anabolism.

An increase in the number of phosphates per molecule is phosphate anabolism.

For example, in phosphate anabolism, GDP is oxidized to GTP with the use of energy and the bonding of an orthophosphate (Pi) to GDP.

Def. any "enzyme that catalyzes the production of" a phosphate by the addition of an inorganic phosphate to a molecule is called a phosphorylase.

Phosphate utilization rates
Phosphate in the nucleosol is utilized at a rate λ which represents the uptake of nutrients. The timescale 1/λ gives a measure of the residence time for phosphate in the surface layers. The maximum uptake rate is modulated by a periodic cycle, such as the daily feedings, and decreases exponentially with interiority.

Remineralization
"In biogeochemistry, remineralisation (UK spelling; US remineralization) refers to the transformation of organic molecules to inorganic forms, typically mediated by biological activity. "

"Usually remineralisation relates to organic and inorganic molecules involving biologically important elements such as carbon, nitrogen and phosphorus. For example, the following simplified equation shows the complete remineralisation of organic material with a standard Redfield ratio to oxidised inorganic minerals such as carbon dioxide, nitrate (nitric acid) and phosphate (phosphoric acid). "

(C106H124O36) (NH3)16 (H3PO4) + 150 O2 $$\rightarrow$$ 106 CO2 + 16 HNO3 + H3PO4 + 78 H2O + energy

"In reality, such complete remineralisation is likely to involve several stages each involving different organisms and metabolic pathways. For example, in the case of nitrogen, its transformation from ammonia (NH3) in the equation above, to nitrate involves the process of nitrification, usually mediated by a series of bacteria. "

"The remineralization of teeth is a process in which minerals are returned to the molecular structure of the tooth itself, and can reverse bacterial infestation of the enamel of a tooth. This process cannot replace lost tooth material (that is, it won't fill a cavity that has developed into a hole).  The human body naturally remineralizes teeth through the use of carbonic acid ."

Remineralization rates
The remineralization rate (r) is given by r = w/z, where w is the sinking rate and z is the optimum depth reached before conversion from organic phosphate to inorganic phosphate.

A simplified phosphate-cycling model is used to examine how changes in circulation might influence phosphate export production via the supply of phosphate to the phosphate utilization zone and the preservation of organic phosphate which is controlled by the oxygen content. The model uses coupled equations: phosphate utilization, phosphate remineralization, and the oxygen concentration. The initial conditions for all the integrations of the model consist of uniform values for the concentration of each component: [PO4] = 2 x 10-4 mol m-3, [D] = 0, and [O2] = 0.1 mol m-3, where D refers to the organic phosphate.

Dynamic models
“The relations between the external nutrient loading of lakes, recycling through sediments and the resulting productivity are complicated by feed-back mechanisms, seasonal variations and trends.” In a dynamic phosphate budget model the variables “include inorganic and organic particulate phosphate and dissolved o-phosphate, in both sediments and overlying water. Sediments may be aerobic or anaerobic, depending on topography, temperature and composition. The major processes described are primary production, mineralization, sedimentation, adsorption and diffusion. ... The sediment dilution rate, the extent of anaerobic conditions and the number and character of adsorption sites are important controlling factors.” Units of loading are g P m-2 yr-1.

"A dynamic phosphate budget can be modeled by determining the relations between the external phosphate loading, recycling through the formation of structural phosphates, and the resulting productivity, which are complicated by feed-back mechanisms, periodic variations and trends. In a dynamic phosphate budget model the variables include inorganic and organic particulate phosphate and dissolved phosphate, in both structural phosphate and nucleosol. Structural phosphate may be aerobic or anaerobic, depending on localization, temperature (or pH) and composition. The major describeable processes are primary production, mineralization, precipitation, adsorption and diffusion. The precipitate dilution rate, the extent of anaerobic conditions and the number and character of adsorption sites are important controlling factors. "

Minerals
"Apatite is a group of phosphate minerals, usually referring to hydroxylapatite, fluorapatite and chlorapatite, named for high concentrations of OH−, F− and Cl− ions, respectively, in the crystal. The formula of the admixture of the four most common endmembers is written as Ca10(PO4)6(OH,F,Cl)2, and the crystal unit cell formulae of the individual minerals are written as Ca10(PO4)6(OH)2, Ca10(PO4)6(F)2 and Ca10(PO4)6(Cl)2."

"Apatite is one of a few minerals produced and used by biological micro-environmental systems."

"Hydroxyapatite, also known as hydroxylapatite, is the major component of tooth enamel and bone mineral. A relatively rare form of apatite in which most of the OH groups are absent and containing many carbonate and acid phosphate substitutions is a large component of bone material."

"Fluorapatite [a sample of which is shown at right] ... is a mineral with the formula Ca5(PO4)3F (calcium fluorophosphate). ... Fluorapatite as a mineral is the most common phosphate mineral. It occurs widely as an accessory mineral in igneous rocks and in calcium rich metamorphic rocks. It commonly occurs as a detrital or diagenic mineral in sedimentary rocks and is an essential component of phosphorite ore deposits. It occurs as a residual mineral in lateritic soils. "

At lower left is another fluorapatite example that is violet in color on quartz crystals.

"Torbernite ... is a radioactive, hydrated green copper uranyl phosphate mineral, found in granites and other uranium-bearing deposits as a secondary mineral. Torbernite is isostructural with the related uranium mineral, autunite. The chemical formula of torbenite is similar to that of autunite in which a Cu2+ cation replaces a Ca2+. The number of water hydration molecules can vary between 12 and 8, giving rise to the variety of metatorbernite when torbernite spontaneously dehydrates."

Phosphorites
"Phosphorite is a phosphate-rich sedimentary rock, that contains between 18% and 40% P2O5. The apatite in phosphorite is present as cryptocrystalline masses referred to as collophane."

Hypotheses

 * 1) The phosphate budget required to run a default human genome daily is the same as running any daily human genome.


text: User:Mwnuk/myWork/Week4 Photos

title: User:Mwnuk/myWork/Week4 Photos


text:
What Matters/regrets of the dying

Regrets of the Dying
When palliative care nurse Bronnie Ware asked her dying patients what regrets they had or anything they would do differently, common themes surfaced again and again. Here are the most common five:

1. I wish I'd had the courage to live a life true to myself, not the life others expected of me.

This was the most common regret of all. When people realize that their life is almost over and look back clearly on it, it is easy to see how many dreams have gone unfulfilled. Most people had not honored even a half of their dreams and had to die knowing that it was due to choices they had made, or not made.

It is very important to try and honor at least some of your dreams along the way. From the moment that you lose your health, it is too late. Health brings a freedom very few realize, until they no longer have it.

2. I wish I didn't work so hard.

This came from every male patient that I nursed. They missed their children's youth and their partner's companionship. Women also spoke of this regret. But as most were from an older generation, many of the female patients had not been breadwinners. All of the men I nursed deeply regretted spending so much of their lives on the treadmill of a work existence.

By simplifying your lifestyle and making conscious choices along the way, it is possible to not need the income that you think you do. And by creating more space in your life, you become happier and more open to new opportunities, ones more suited to your new lifestyle.

3. I wish I'd had the courage to express my feelings.

Many people suppressed their feelings in order to keep peace with others. As a result, they settled for a mediocre existence and never became who they were truly capable of becoming. Many developed illnesses relating to the bitterness and resentment they carried as a result.

We cannot control the reactions of others. However, although people may initially react when you change the way you are by speaking honestly, in the end it raises the relationship to a whole new and healthier level. Either that or it releases the unhealthy relationship from your life. Either way, you win.

4. I wish I had stayed in touch with my friends.

Often they would not truly realise the full benefits of old friends until their dying weeks and it was not always possible to track them down. Many had become so caught up in their own lives that they had let golden friendships slip by over the years. There were many deep regrets about not giving friendships the time and effort that they deserved. Everyone misses their friends when they are dying.

It is common for anyone in a busy lifestyle to let friendships slip. But when you are faced with your approaching death, the physical details of life fall away. People do want to get their financial affairs in order if possible. But it is not money or status that holds the true importance for them. They want to get things in order more for the benefit of those they love. Usually though, they are too ill and weary to ever manage this task. It is all comes down to love and relationships in the end. That is all that remains in the final weeks, love and relationships.

5. I wish that I had let myself be happier.

This is a surprisingly common one. Many did not realise until the end that happiness is a choice. They had stayed stuck in old patterns and habits. The so-called 'comfort' of familiarity overflowed into their emotions, as well as their physical lives. Fear of change had them pretending to others, and to their selves, that they were content. When deep within, they longed to laugh properly and have silliness in their life again.

When you are on your deathbed, what others think of you is a long way from your mind. How wonderful to be able to let go and smile again, long before you are dying.

Life is a choice. It is your life. Choose consciously, choose wisely, choose honestly. Choose happiness.

title: What Matters/regrets of the dying


text:
Survey research and design in psychology/Assessment/Lab reports/3/Item correlations

 * How to put a correlations table from a factor analysis into a lab report appendix

Presenting a large number of correlations in a single table can be challenging.

Here are some tips for using SPSS to create a table that is relatively easy to present in an Appendix:
 * 1) Generate a correlation table in SPSS via Analyze - Data Reduction - Factor - Descriptives - Correlation Matrix - Coefficients. Note to use the data for your final model (after some variables have been dropped).
 * 2) Select the Correlation table in the Output (by left clicking on the Table)
 * 3) Export the table via: File - Export - Objects to Export - Selected - Options - Change Options - Wide Pivot Tables - Wrap table to fit within page margins (should be the default setting) - Page set-up for export - Landscape. You may also wish to adjust the filename and location for the file to be saved.
 * 4) In your main lab report document, create a section break (this allows you to have Portrait layout for one section and Landscape layout for another section), then insert (or copy and paste) the exported correlation table from SPSS.

Alternatively, create a table in your lab report document from scratch and type the correlations in, rounding to two decimal places. It might be quicker, easier, and better presented.


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Progress and Prospects in Parkinson's Research/Causes/Infection

< Back to Causes

Background
The ‘Causes’ page on this wiki states categorically that Parkinson’s Disease is not contagious, and most neurologists would agree with this. However, perversely, the disease can sometimes appears as a set of secondary symptoms of another disease which is contagious. The most noteworthy example of this was an outbreak of a disease with extreme PD symptoms called Encephalitis Lethargica. This was first described by Von Economo in 1912 and, following a pandemic of cases, which lasted from 1916 to 1926, he produced a monograph. The victims found it impossible to achieve any cognitive movement and were condemned to exist as living statues for decades.

Dr. Oliver Sachs describes the outbreak in his book “Awakenings” (1973), which includes the moving account of his rescue of the victims from their plight by administering massive doses of the newly discovered drug levodopa.

The outbreak disappeared as mysteriously as it first appeared. There was an unproven hypothesis that it was somehow linked to the devastating pandemic of flu in 1919, which killed more people than the combined casualties of World War One.

Research
2003

Wharton reviewed all literary references to encephalitis lethargica and identified the diagnostic criteria. He then set about reviewing modern diagnoses to see whether any met the criteria and turned up 19 cases since 1982. His paper suggest that a further widespread outbreak is real possibility.

2006

Field et al produced evidence that prenatal depression in a mother induced a biochemical/physiological profile in the foetus that mimics their mothers' prenatal biochemical/physiological profile including elevated cortisol,  lower levels of dopamine  and serotonin, greater relative right frontal EEG activation and lower vagal tone.

2012

H.V. et al studied patients in Belgrade during the period 2001-2005, comprising 110 subjects diagnosed for the first time as PD cases, and 220 controls chosen among patients with degenerative joint disease and some diseases of digestive tract.

"According to logistic regression analysis PD was significantly related to mumps (Odds ratio adjusted on occupation and family history of PD - aOR = 7.86, 95% confidence interval - CI = 3.77-16.36), scarlet fever (aOR = 12.18, 95%CI = 1.97 -75.19), influenza (aOR = 8.01, 95% CI = 4.61 -13.92), whooping cough (aOR = 19.90, 95% CI = 2.07 -190.66) and herpes simplex infections (aOR = 11.52, 95% CI = 2.25 -58.89). Tuberculosis, measles and chicken pox were not associated with PD. Other infectious diseases we asked for were not reported (12 diseases), or were too rare (4 diseases) to be analysed."

De Chiara et al evaluated connections between infective conditions and neurodegenerative diseases.

"The activation of inflammatory processes and host immune responses cause chronic damage resulting in alterations of neuronal function and viability, but different pathogens can also directly trigger neurotoxic pathways. Indeed, viral and microbial agents have been reported to produce molecular hallmarks of neurodegeneration, such as the production and deposit of misfolded protein aggregates, oxidative stress, deficient autophagic processes, synaptopathies and neuronal death. These effects may act in synergy with other recognized risk factors, such as aging, concomitant metabolic diseases and the host’s specific genetic signature. "

2013

Tome et al have carried out a comprehensve review of the role of inflammation and alpha synuclein in the development of PD and their conclusions were:-

"Although α-syn aggregation appears to be central to the development and progression of PD pathology, inflammation also plays an important role in this disease. Immune activation in the GI tract or in the olfactory system via chemical or virus exposure might trigger α-syn misfolding, subsequent aggregation, and propagation to the brain. It is also possible that neuroinflammation promotes the prion-like transfer of α-syn by increasing its release, increasing its uptake, or both. However, further studies are needed to support this hypothesis. If correct, it will open up new avenues for early detection and therapeutic intervention. For example, one could speculate that it will be possible to develop therapies which slow down the progression of PD by reducing the underlying inflammation and mitigating its effects on cell-to-cell α-syn transfer. In addition, it is conceivable that it would be possible to adopt a preventive strategy and lower the risk of developing PD by treating the triggers of inflammation in the olfactory or gastrointestinal system."


text:
Template:PD Infection Pages

Causes > Infection

Sub Pages:
 * Melanoma  -   Whooping cough  -  Helicobacter pylorii - Lipopolysaccharide

title: Template:PD Infection Pages


text:
Progress and Prospects in Parkinson's Research/Causes/Infection/Melanoma

"There appear to be a growing consensus of epidemiological evidence for a link between the risk of melanoma and the risk of PD. There is as yet no working hypothesis to explain this although the common themes of UV processing of Vitamin D3 may be significant."

Background
Melanoma is the most dangerous form of skin cancer and is frequently fatal. The risk of getting it increases with age and once initiated it can spread very rapidly. It can have genetic associations but the principle causative factor is exposure of the skin to sunlight or tanning machines and in particular over-exposure to ultra-violet radiation.

Research
2005

Freedman, et al followed 1.9 million cancer survivors who were diagnosed between 1973 and 2000 and who survived at least 1 year, through the year 2000.

"In this exploratory study, we observed a modest, significant association between melanoma and both ALS and Parkinson's disease mortality. It would be useful to explore these findings in other large national databases."

2006

Olsen et al carried out a Danish study which investigated the prevalence of malignant melanoma, skin carcinoma, and other cancers before a first hospitalization or outpatient visit for Parkinson disease.

"We observed an increased prevalence of malignant melanoma and skin carcinoma prior to the first hospital contact for Parkinson disease, with overall odds ratios of 1.44 (95% confidence intervals = 1.03-2.01) and 1.26 (1.11-1.43), respectively."

2007

Baade et al conducted a study of all people diagnosed as having melanoma in Australia since 1982 (n = 127,037). They then compared their mortality risk of amyotrophic lateral sclerosis (ALS) and PD to that of the general population. There were a total of 53 ALS deaths and 129 deaths due to PD. Although the absolute risk is small, the melanoma cohort had a risk of death due to ALS 70% higher (standardised mortality ratio = 169.4, 95% CI = 127-221) than the general population, and nearly a 3-fold increased risk of dying from PD (standardised mortality ratio = 266.3, 95% CI = 222-317).

2009

Gao et al examined the associations between a family history of melanoma and PD among 157,036 men and women. Their conclusions state:-

“In this prospective analysis, we found an approximately twofold increase in risk of PD among individuals who reported a family history of melanoma in a first-degree relative, as compared with those without such a family history”

Gao et al examined whether hair colour, one of the most important phenotypes of pigmentation and a risk factor for melanoma, was associated with PD risk. They found that:- "PD risk increased with decreasing darkness of hair color. Pooled RRs for PD were 1 (reference), 1.40, 1.61, and 1.93 (95% confidence interval, 1.1-3.4) for black, brown, blond, and red hair, respectively, after adjusting for age, smoking, ethnicity, and other covariates. The associations between hair color and PD were particularly strong for relatively younger onset of PD (<70 yr) (adjusted RR for red vs black hair = 3.83; 95% confidence interval, 1.7-8.7). In the case-control study, participants with Cys/Cys genotype, which was associated with red hair, had a greater PD risk, relative to the Arg/Arg genotype (adjusted RR, 3.15; 95% confidence interval, 1.1-9.4)."

2011

These conclusions were supported by research by Pan et al whose findings stated that:-

''On the basis of published findings, we conclude that (i) changes in pigmentation including melanin synthesis and/or melanin synthesis enzymes, such as tyrosinase and tyrosine hydroxylase, play important roles in altered vulnerability for both PD and melanoma; (ii) changes of PD-related genes such as Parkin, LRRK2 and α-synuclein may increase the risk of melanoma; (iii) changes in some low-penetrance genes such as cytochrome p450 debrisoquine hydroxylase locus, glutathione S-transferase M1 and vitamin D receptor could increase the risk for both PD and melanoma and (iv) impaired autophagy in both PD and melanoma could also explain the association between PD and melanoma.... ... From a clinical point of view, early diagnosis of melanoma in PD patients is critical and can be enhanced by periodic dermatological surveillance, including skin biopsies.''

Rui et al assessed the epidemiological evidence for a relationship and melanoma. The outcome:-

''Collective epidemiologic evidence supports an association of PD with melanoma. Further research is needed to examine the nature and mechanisms of this relationship.''

2012

Pan et al evaluated the relationship of the alpha synuclein protein and melanin in dopaminergic cells. Their findings:-

" We found that α-Syn expression reduced UVB light-induced increase of melanin synthesis and that melanin content was lower when melanoma cells were expressed with α-Syn, indicating that α-Syn may have inhibitory effects on melanin synthesis in melanoma cells……We concluded that α-Syn could be one of the points responsible for the positive association between PD and melanoma via its differential roles in melanin synthesis in melanoma cells and in dopaminergic neuronal cells."

Related Pages
Vitamin D


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Progress and Prospects in Parkinson's Research/Causes/Infection/Whooping cough

General
Whooping cough (also known as pertussis) is a highly contagious bacterial infection. The bacterium responsible is Bordetella pertussis. It is transmitted by coughing and sneezing, It affects mainly  young children, although adult cases are not unknown. In the past it occurred in periodic epidemics but these have greatly diminished since effective vaccination was introduced in the 1960s. In the United Kingdom the annual count of reported cases has dropped from 120,000 p.a. to 600 p.a. The estimated annual number of cases worldwide is 48 million.

Research
De Pedro-Cuesta et al (1996) analysed statistics on medical prescriptions for levodopa in Denmark and found that the highest incidence in proportion to the population was in Iceland. Icelandic health records are well documented and a notable feature is the occurrence of several outbreaks of whooping cough. The graph of these outbreaks was plotted against the graph of onset of PD cases and it was demonstrated that the peaks were replicated. No hypothesis could be advanced to account for this phenomenon, which at present constitutes an isolated piece of research.


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Template:PD Deficiencies Pages

Causes > Deficiencies

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 * Blood  -   Melatonin - Norepinephrine   -   Vitamin_D - Zinc

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Progress and Prospects in Parkinson's Research/Causes/Deficiencies/Blood

Background
There appears to have been very little analysis of blood disorders in PD patients. However the study described below may have some significance.

Research
2009

Savica et al investigated the association of anaemia with the subsequent risk of PD using a case-control study involving 196 PD patients and a corresponding number of matching controls. They found that anaemia was more common in the history of PD cases than of controls. The association was not significantly different between men and women, or between PD patients with or without rest tremor. Generally anaemia started 20 to 29 years before the onset of PD.Hemoglobin levels were slightly but consistently lower in PD cases than in controls across all ages


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Progress and Prospects in Parkinson's Research/Causes/Deficiencies/Melatonin

The hypothesis that melatonin deficiency might be a causative factor for Parkinson’s Disease was stated by Sandyk (1990)  Definitive proof is currently lacking, but there is sufficient circumstantial evidence to make it worthy of  serious consideration.

The Pineal Gland
In humans the pineal gland reaches its maximum size at the age of 2 and is about the size of a grain of rice. It is reddish-gray in colour, shaped like a pine cone, and tucked away in the centre of the brain at the top of the spinal column between the two hemispheres. It is a singular object: the only part of the brain not to have bipolar symmetry. Also, unlike the rest of the brain, the pineal gland is not insulated from the rest of the body by the blood brain barrier. It is rich in trace elements (zinc, iron, manganese, magnesium, strontium and copper) and its function could be affected if any of these prove to be deficient. It also accounts for a vigorous blood flow.

The anatomical structure of the gland shows it to have many features in common with the eye, and it may in the course of its evolution have been involved in visualisation. This, its concealed location, and its uniqueness have led several religions and beliefs to equate it with the concepts of The Third Eye and Second Sight. These give indoctrinates mystical powers of observation, insight and clairvoyance. Others earmark it as the seat of the Soul. Eastern statues show this as an outline of an eye in the centre of the forehead and adherents to the Hindu religion denote its presence by a ‘tilaka’, a crimson dot placed on the brows. There are nine ‘gates’ or openings into the body and the third eye is conceptualized as the tenth gate.

Where the pineal gland is severely damaged in children, it results in accelerated development of the sexual organs and the skeleton. In animals, the pineal gland appears to play a major role in sexual development, hibernation, metabolism, and seasonal breeding.

The gland has been shown to be susceptible to calcification and the degree of calcification increases steadily with age.

Melatonin
One of the gland’s main functions is the production of a hormone, a derivative of serotonin, called N-acetyl-5-methoxy-tryptamine, better known as melatonin. This production is geared to light reception in the retinas such that it follows a 24 hour cycle known as the Circadian Rhythm. The gland is active primarily during the hours of darkness. Secretion of melatonin as well as its level in the blood, peaks in the middle of the night, and gradually falls during the second half of the night.

Arendt and Skene (2005) have assessed the daily output of melatonin in mammals as a nocturnal maximum of about 200 pg/ml and a daytime nadir of less than 10 pg/ml.

It is principally blue light, around 460 to 480nm. that suppresses melatonin production. Special photoreceptive retinal ganglion cells containing melanopsin as a photopigment are involved in the projection from retina. Fibres from the suprachiasmatic nucleus pass through a circuitous route involving the paraventricular nucleus of the hypothalamus, medial forebrain bundle, reticular formation, lateral horn cells of the spinal cord, superior cervical ganglion, and then proceed to innervate pineal gland as postganglionic sympathetic fibres. Wearing glasses that block blue light in the hours before bedtime may avoid melatonin loss. The use of blue-blocking goggles in the last hours before bedtime has also been advised for people who need to adjust to an earlier bedtime such as dealing with jet lag, as melatonin promotes sleepiness. 5mg. of melatonin taken half an hour before retiring is sometimes prescribed as a treatment for sleep disorders. Its use at other times of the day is not recommended because of its potential narcoleptic effects.

Melatonin was isolated and named by Aaron B. Lerner and colleagues at Yale University in (1958). It performs a number of functions around the body, not all of which are fully understood. It has been shown that melatonin may play a significant role in modulating the effects of drugs of abuse such as cocaine. It also determines the diurnal sleep pattern.

Some supplemental melatonin users report an increase in vivid dreaming. Extremely high doses of melatonin (50 mg) dramatically increased REM sleep time. Many psychoactive drugs such as cannabis and lysergic acid diethylamide (LSD), increase melatonin synthesis.

Individuals with autism have been found to have reduced levels of melatonin and have been successfully treated with melatonin supplements.

Melatonin has been shown to have anti-aging properties. Its supplemental use has been shown to increase the levels of gene expression in older people.

Incomplete clinical trials have shown that melatonin may interact with the immune system and induce cytokine production. Some studies also suggest that melatonin might be useful fighting infections including viral diseases, such as HIV, and bacterial infections, and potentially in the treatment of cancer.

Melatonin and neuroprotection
2002

Cuzzocrea and Reiter reviewed the evidence for the neuroprotective qualities of melatonin and concluded that:-

" A vast number of experimental and clinical studies implicates oxygen-derived free radicals (especially, superoxide and the hydroxyl radical) and high energy oxidants (such as peroxynitrite) as mediators of acute and chronic inflammation."

"Reactive oxygen species can modulate a wide range of toxic oxidative reactions. These include initiation of lipid peroxidation, direct inhibition of mitochondrial respiratory chain enzymes, inactivation of glyceraldehyde-3-phosphate dehydrogenase, inhibition of membrane sodium/potassium ATPase activity, inactivation of membrane sodium channels, and other oxidative modifications of proteins."

"Reactive oxygen species (e.g., superoxide, peroxynitrite, hydrogen peroxide and hydroxyl radical) are all potential reactants capable of initiating DNA single strand breakage, with subsequent activation of the nuclear enzyme poly (ADP ribose) synthetase (PARS), leading to eventual severe energy depletion of the cells, and necrotic-type cell death. These toxic reactions are likely to play a role in the pathophysiology of inflammation. Melatonin has been shown to possess both in vitro and in vivo important antioxidant activities as well as to inhibit the activation of poly (ADP ribose) synthetase. A large number of experimental studies have documented that melatonin exerts important anti-inflammatory actions."

2003

Mayo et al have evaluated the neuroprotective qualities of melatonin and concluded;-

"The experimental data collectively suggest melatonin use by PD subjects would reduce their disease burden. Additionally, epidemiological studies of individuals who use melatonin daily regularly should be carried out. The devastating nature of PD and the lack of currently available means of preventing the disease call for the use of imaginative treatments.    Melatonin, as noted, has very low toxicity and is much less expensive than prescription drugs and, thus, it should be tested against the development or progression of PD."

2010

Reiter et al describe the toxic residue left after the processing of oxygen by cells:-

"This high utilization of oxygen, however, comes at a heavy biological price. As important as oxygen is for the survival of neurons and glia, it also indirectly contributes to their destruction and death over time. The reason for this is that a small percentage (an estimated 1-4%) of the oxygen that enters cells is metabolized to derivatives that gradually erode and destroy essential molecules.. These destructive derivatives of oxygen are often referred to as free radicals (although some are not radicals per se) or reactive oxygen species (ROS)."

And goes on to add:-

"N-acetyl-5-methoxytryptamine (melatonin), was discovered as a potent antioxidant in 1993 . Since then, melatonin’s ability to protect all cells and organs from oxidative/nitrosative damage has been confirmed in more than a thousand publications."

2011

Srinivasan et al have published a good description of the Physiology of melatonin. There appears to be no consensus at present on levels of melatonin to be found in PWP compared to controls.

Srinivasan (Ibid) includes the following concluding remarks:-

"Nonmotor symptoms of PD such as RBD, occur in many PD patients, and predate the manifestation of motor symptoms. Their early diagnosis and treatment are essential for improving the quality of life in PD patients. Melatonin and melatonin agonists can be useful tools in treating sleep and associated disorders in PD. ...The available evidence supports the inference that melatonin activity in PD can be substantial, and, further, that clinical investigations into the nature of this activity are warranted."

Research
1984

Lewy et al  have suggested that although evening bright light exposure produces a momentary suppression of melatonin, it actually causes a rebound increase in melatonin secretion late in the night.

1999

Willis and Armstrong tested  the effects of  melatonin on experimental models of Parkinson's disease.

Sprague-Dawley rats were exposed to intracerebroventricular implants of slow release melatonin, pinealectomy (PX), or constant light (LL) and then injected with central 6-hydroxydopamine (6-OHDA) or i.p. 1-methyl-4-phenyl,1-1,2,3,6-tetrahydropyridine (MPTP). The resulting impairment of motor function and related behavioural impairment were exacerbated by melatonin implantation, while PX and exposure to LL significantly reduced the severity of experimental PD. Their observations included the statement:-

"These findings illustrate that melatonin is not the universal remedy that it is currently claimed to be, and may pose considerable problems in neurological diseases characterised by dopamine degeneration."

2005

Some 90% of PD patients report sleep disorders among their prominent non-motor symptoms. Dowling et al carried out a multi-site double-blind placebo-controlled cross-over trial with 40 subjects over a period of 10-week protocol. Doses of melatonin from 5mg. to 50mg. were administered and nocturnal sleep and daytime sleepiness and function were assessed. All subjects were taking stable doses of antiparkinsonian medications during the course of the study. Their results showed that:-

"There was a significant improvement in total nighttime sleep time during the 50 mg melatonin treatment compared to placebo. There was significant improvement in subjective sleep disturbance, sleep quantity, and daytime sleepiness during the 5 mg melatonin treatment compared to placebo. as assessed by the GSDS."

2007

Willis and Turner carried out an experiment with Light Therapy (LT). Twelve patients diagnosed with PD were exposed to white fluorescent light for 1-1.5 h at an intensity of 1000 to 1500 lux once daily commencing 1 h prior to the usual time of sleep onset, approximately 22:00 h in most patients. All patients were assessed before LT commenced and at two weeks, five weeks, and regular intervals thereafter. Their findings were:-

"Within two weeks after commencing LT, marked improvement in bradykinaesia and rigidity was observed in most patients. Tremor was not affected by LT treatment; however, agitation, dyskinaesia, and psychiatric side effects were reduced, as verified by decreased requirement for DA replacement therapy. Elevated mood, improved sleep, decreased seborrhea, reduced impotence, and increased appetite were observed after LT. LT permitted the reduction of the dose of L-dopa, bromocriptine, or deprenyl in some patients by up to 50% without loss of symptom control."

2010

Reiter (Ibid) evaluated the results of a series of pieces of research in which neurons were exposed to toxins and then treated with melatonin. The toxins were Methamphetamine, Aminolevulinic Acid, 6-Hydroxydopamine, Rotenone, Iron (Fe), chromium (Cr), aluminum (Al), copper (Cu), vanadium (V) and cobalt (Co) mercury (Hg), cadmium (Cd) and nickel (Ni), toluene, Polychlorinated BiPhenyls, Kainic acid, Domoic acid, Cyanide

In summarising the results they stated:-

"Almost uniformly, although there are exceptions, melatonin demonstrated its efficacy in safeguarding neurons and glia from the persistent molecular disfiguring that would otherwise occur. The problem with loss of neurons is that, again with few exceptions, once these cells undergo either apoptosis or necrosis, they are not replaced. Interestingly, neuron precursors in the brain in those select areas where cellular proliferation does occur, are also stimulated by melatonin."

Summary
The hypothesis that melatonin deficiency is a primary cause of Parkinson’s Disease may be summed up thus:-

Oxygen, the principal fuel that drives animal cells is a toxin. However evolution has produced a system whereby individual cells metabolise oxygen into Adenosine TriPhosphate (ATP), which is better tolerated. There is, however, a toxic residue in the form of Reactive Oxygen Species (ROS), which have the capability of inducing cell death. Melatonin plays a vital neuroprotective role in rendering ROS harmless. The substantia nigra is a major consumer of melatonin. The pineal gland, which is the prime source of melatonin and is not protected by the blood/brain barrier, calcifies through aging, which can reduce its production of melatonin to the point where neuroprotection breaks down and all of the other PD causative factors identified in other pages, which are waiting in the wings, are given free reign to attack neurons in a variety of ways.

The Holy Grail of PD research has been the search for an environmental toxin which initiates sporadic forms of the diseases. This hypothesis offers a candidate for such a toxin: it is air!

The hypothesis offers scope for further research in the pathological examination of the pineal glands of PD patients, the use of light therapy, and the clinical trials of melatonin-based drugs.

Related Pages
Melatonin,


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Progress and Prospects in Parkinson's Research/Causes/Deficiencies/Norepinephrine

Background
Norepinephrine is also known as noradrenline and more commonly as the 'fright and flight' compound.

Norepinephrine is synthesized from dopamine by dopamine β-hydroxylase. It is released from the adrenal medulla into the blood as a hormone, and is also a neurotransmitter in the central nervous system and sympathetic nervous system where it is released from noradrenergic neurons in the locus coeruleus. The actions of norepinephrine are carried out via the binding to adrenergic receptors.

A sudden shock produces an autonomic reaction in the sympathetic nervous system, generating a pulse of norepinephrine, which speeds up the action of the heart and charges the limbs with fresh, oxygenated blood, ready to run or take other reactive action.

Research
2002

The National Institute of Neurological Disorders and Stroke – NINDS gives an account of the research of Dr. David Goldstein Chief of the Clinical Neurocardiology Section at NINDS. Using positron emission tomography (PET) scans, researchers at the National Institute of Neurological Disorders and Stroke (NINDS) found that nearly all of the 29 Parkinson's disease patients they studied had decreased numbers of norepinephrine-producing nerve endings in the heart. This can lead to a sudden drop in blood pressure termed ‘ orthostatic hypotension’.