## How to train your children…

Rule 1: Don’t ever tell them what you want them to do, instead if possible state the opposite.

My teenage kid touches my feet several times a day (in India, it is a mark of respect) once I made it clear to him that I dislike the practice.

Rule 2: Turn work into play

This is not me really, it is Mark Twain. My 10 year old regularly helped me cut veggies until he realized that it was work.

While we all love to gift our kids, there is a difference between a brand new mobile phone and vacation on the beach.

Especially working with little kids advertisement is everything. Believe it or not, milk tastes way… different through a straight straw and a zig-zag one.

Good luck and feel free to share your secrets too.

## The Story of a Witch

She was considered to be a witch and was prosecuted for practicing witch-craft. In reality she probably knew about chemicals andtheir medicinal values. She used her knowledge to cure people. Her son Johannes Kepler worked out the basic laws governing planetary orbits and provided a strong support for the Sun-centric planetary system.

At times public perception and reality may be miles away. It takes a discerning mind to see the difference.

## Understanding Fourier Series

Comparing Functions and vectors.

$\vec{v}$ function f(x)

Finite dimensional Infinite dimensional

A vector can be written in the following different ways,
$\vec{V} = V_x \hat{x} + V_y \hat{y} + V_z \hat{z}$
$\hskip .5cm = (V \cdot \hat{x}) \hat{x} + (V \cdot \hat{y}) \hat{y} + (V \cdot \hat{z}) \hat{z}$
If the decomposition is along an orthogonal frame along the vectors $\vec{a},\, \vec{b}$ and $\vec{c}$ then the expression would be,
$\vec{V} = (\vec{V} \cdot \hat{a}) \hat{a} + (\vec{V} \cdot \hat{b}) \hat{b} + (\vec{V} \cdot \hat{c}) \hat{c}$
$\hskip .5cm = \frac{\vec{V} \cdot \vec{a}}{\vec{a}\cdot\vec{a}} \vec{a} + \frac{\vec{V} \cdot \vec{b}}{\vec{b}\cdot\vec{b}} \vec{b} + \frac{\vec{V} \cdot \vec{c}}{\vec{c}\cdot\vec{c}} \vec{c}$

In general the dot product of two $n-$dimensional vectors $\vec{V} = (V_1, V_2,...,V_n)$ and $\vec{W} = (W_1,W_2,...,W_n)$, can be written as,
$\vec{V} \cdot \vec{W} = \sum_{i=1}^n V_i W_i.$

It is useful to think of a real function $f(x)$ over an interval
$[a,b]$ as a vector with infinite components. Here the argument serves
as an index and the function value as the vector component. Analogous to vector dot product, the dot product between two functions $f$ and $g$ defined over the same interval can be written as,
$(f,g) = \int_a^b f(x) g(x) dx.$

Using this definition of the dot product, one can show that the following functions
are orthogonal to each-other (mutual dot products are zero) on the interval
$[0, 2\pi]$.
$f_1(x) = 1, \sin{x}, \sin{2x}, \sin{3x}, ...,\cos{x}, \cos{2x}, \cos{3x},...$

Thus in parallel with writing a vector in terms of it’s components, one can write any (finite, smooth and continuous on $[0, 2\pi]$ (I am not trying to be mathematically precise, the aim is to give an intuitive feel)) function in terms of
the above basis functions in the same manner,
$f(x) = \frac{(f(x),1)}{(1,1)} 1 + \frac{(f(x),\sin(x)}{(\sin(x), \sin(x))} \sin(x) + \frac{(f(x),\sin(2x)}{(\sin(2x), \sin(2x))} \sin(2x) + ...$
$\hskip.2cm + \frac{(f(x),\cos(x)}{(\cos(x), \cos(x))} \cos(x) + \frac{(f(x),\cos(2x)}{(\cos(2x), \cos(2x))} \cos(2x) +....$
Notice the similarity of the expression of a function in terms of it’s components and a vector in terms of it’s components. Hence decomposition of a function in its Fourier components is quite akin to decomposition of a vector in its Cartesian components.