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>Hi all,
>I have done extensive research on meta physics... and couldn't find
>any single reason to support the existence of extra dimension... Its
>totally an absured concept given by some stupids...
>Its an advice not to think abt dimensions.... be happy with the
>dimensions given to u by GOD( the only 3 dimension and not even the
>fourth time dimension)
>i dont know why some ppl are trying to make the nature complex...

First of all, science is not constructing nature, it's about
'''explaining and predicting''' nature. And when ordinary explanations
fail, u need to invent better ones that give more accurate
predictions.
----------------------

multi dimensional analysis seems non-intuitive, if u seperate the math
from what a dimension is or means.

A concept of dimension is related to the concept of orthogonality or
independence.

Orthogonality means that the dot product of one dimension with another
dimensions yeilds 0

that is

x''hat''i dot x''har''j = Delta_ij

where Delta_ij is 1 if and only if i=j and 0, if otherwise.

The concept of multi dimensional analysis, also has influence from
notion of independent variables.

For example,

Let's take a store like sears and TV sales.

The sales,on a single day, may be related to

the average number of people entering the store [[ N]]
the average salary of money they each get paid [[ S ]]
the average number of TVs on displays[[ T ]]
the month of purchase [[ M]]
the average price of TV [[ pTV ]]

We can assume that these events are independent. For example,
independence in this context means, that number of people entering the
store does not depend on the month they enter. Ofcourse, you may
realize that people do a lot more shopping during holiday seasons that
they do other times of the year. For now, we will ignore this effect.

Statiscally, independence means 1 thing. No correlation

E[[ ( N- <N>) ( M - <M> ) ]] = 0

I shall come back to it expression later and how it is useful in
orthogonality analysis or multi dimension analysis.

Now,the plot of the sales function SALES( N, S, T, M , pTV) requires
a multi dimensonal graph (6-dimensonal) . However, it does not mean
that the SALES function is non-intuitive concept.

Now, in mathematical analysis,

If you studied dot products in your vector algebra class

a dot b = ||a|||b|| cos ( heta)

heta is angle between the 2 vectors. Now if theta = 90 degrees,

a dot b = 0

In a way, 2 vectors are orthogonal. Also note that dot of a vector
purely along x-axis with a vector purely along y-axis is 0.

Now, we can extend this concept to functions. Two real functions a(x),
b(x) are orthogonal if

integrate''[[x<code>-infty, infty]] a(x) b(x) dx </code> Delta''ab

That means that, it is just simlar to saying that the functions are at
90 degrees to each other.

Moreover, we also require

integrate_[[x<code>infty, infty]] a(x) dx </code> 0

and

integrate_[[x<code>infty, infty]] b(x) dx </code> 0

So, There is no net energy inside them.

However,

integrate_[[x<code>-infty, infty]] a(x)^2 dx </code> 1

This follow from

integrate''[[x<code>-infty, infty]] a(x) b(x) dx </code> Delta''ab

We can treat the series of functions a(x), b(x), c(x), which statisfy
the above conditions as dimensions.

Now let's look at our expression

E[[ ( N- <N>) ( M - <M> ) ]] = E[[ N * M ]] - E[[N]]E[[M]]

You must notice something interesting there. If N and M are
othorgonal,

that means that

integrate_[[x<code>-infty, infty]] N(x) M(x) dx </code> 0

E[[N]] <code> 0, E[[M]] </code>0 since

integrate_[[x<code>infty, infty]] N(x) dx </code> 0

Here, x is supposedly a variable that influences both N and M. It
could even be N or M.

So, that means, that if N is orthogonal to M, N is not correlated to M
and so, N and M are independent, and can change without influencing
each other.

The concept of a mutli dimensonal the sales function SALES( N, S, T, M
, pTV) is not counter-intuitive. It is definitely not possible, in
our 3d world to plot this functions. That does not make it unreal or
ridiculous.

Moreover, you need to realize that if the SALES function is only of 3
variables
for example, of

the average number of people entering the store [[ N]]
the average salary of money they each get paid [[ S ]]
the average number of TVs on displays[[ T ]]

The SALES functions may be compromizing some details and parts of it
may appear to be random because the other variables are not accounted
for as observable effects correctly.

Similarly, when physicists talk about more than 3 dimensional space,
they
are talking about more than 3 othogonal dimensions that each satisfy
the dot product requirment

x''i dot x''j = Delta_ij

There is no restriction on the number of x_i, as along as they satisfy
the above condition. From a mathematical stand point, multi
dimensional analysis, is very real, and is in fact, already has many
practical applications. For example, consider the problem of optimal
packaging. My math prof told me that there are hunderds of factors
that can influence the arrangement of spaces taken by boxes.

-suresh

Why do you need more dimensions (q) - Revision history

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