Math

Z transform of and exponentially decaying sequence

The series:   x^n      doubleleftright 1+ x + x^2 + x^3 + ....  = 1/{1-x}    n=0,1,2...   this converges for x < 1 : Both of these expressions are the Z transform of the  x^n   exponential decay sequence.  The first expression is easier to deal with because it is smaller and easier to work with.  The following diagram uses a decay sequence with  x = 0.75 

 

The filter takes what ever input it is fed and every interval multiplies it by 0.75.   To see the filters time response you can thus feed in a single ping.  This results in the filter output tracing out a special response.  This  is called the impulse response. It is the same as the filter plot above.

The exponential decay is maximum entropy.  That is to say this is how concentrated things soak out into the rest of the world as they become more dilute. 

By Fudgy McFarlen, ago

Convolution of time signals using polynomials-The Super Easy Z transform

1+x+x^2 *     1+x+x^2 =   1+2x+3x^2+2x^3+x^4

  • The filter is the sum of the last 3 signal samples. 
  • The signal is a pulse set of three ones.  The signal arrives 1 sample at a time.

Polynomial convolution diagram showing how coefficient of multiplied polynomials correspond to signal amplitudes

Notice the coefficients of the multiplied polynomials are equal to the signal output values at times 0 through 4.

If instead of using X as our variable we could use Z and we would see that all this is the Z transform.   The following principle is true for the above signal and filter.  It is true in general.

    Signal convolved with Filter    doubleleftright Transform of signal *  Transform of filter

Notice that the powers of X perform the function of place holding for location in time.  They keep track of what we can tally and these tallies correspond in power of X to time.  Time=4 corresponds to powers of 4 of X.

By Fudgy McFarlen, ago

Two 2 dimensional determinant of a matrix animation showing it is equal to the area of the parallelogram

The 2 dimensional determinant of a matrix can be interpreted as the area of a parallelogram as shown in the following diagram.

Numeric Example

  det (matrix{2}{2}{2 1 1 2}) = 2 * 2 - 1 * 1 = 3

Compare that with the old fashioned area of two triangles that make up the parallelogram:

  Area Paralelogram = 2 * {1/2} * Base * Height

Using Pythagoras:

Height = {3/sqrt{2}} 

Base  = sqrt{2} 

 Area Parallelogram = sqrt{2} * {3/sqrt{2}} = 3

 

 

This carries on through higher dimensions.  Below depicts a 3 variable system.

The rows r1, r2, r3 are vectors each. The various summations taken 1, 2 and 3 at a time define a parallelepiped. 

 

The following excerpt is from X and may yield some insight when maximum entropy principle is applied. ( still working on this )

By Fudgy McFarlen, ago

WordPress WP-Latex Plugin for rendering Mathematics

I found another math rendering plugin refered to here: http://www.illigal.uiuc.edu/web/kumara/2007/04/10/latex-math-plugin-for-wordpress/

It is based on LaTeX.

Note To Self:  Kumara Sastry  appears to be an interesting an talented person.  He studies in the area of genetic algorithms.  An idea I have is to study talented people and make a blog and the subsequently a company based around these people. 

Easy LaTeX – author interesting blog to punch around also

By Fudgy McFarlen, ago

Cube Sequences

13=1

13+23=9  = 32                         

13+23+33=36 = 62

13+23+33+43=100 = 102

13+23+33+43+53=225 = 152

…. 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16….

These are triangular numbers are in bold: 

  • 1+2 =3
  • 1+2+3=6
  • 1+2+3+4=10

          ( n(n+1)/2 )2= n2(n+1)2/4

A question I have in my mind is that Fermat's last theorem states: If an integer n is greater than 2, then the equation an + bn = cn has no solutions in non-zero integers a, b, and c. 

But how about  a3 + b3 + c3 = d3        … Are there any integer solutions to this?   I ask this because geometrically speaking volume is 1 degree of freedom more than area.

      33+43+53=63             …. = 152 – 32

 

By Fudgy McFarlen, ago

Derivation of the Normal Gaussian distribution from physical principles – Maximum Entropy

Research Links

 

In many physical systems the question arises what is the probability distribution that describes a system with a given expected energy E  over the interval from -infinity to + infinity?     Again you will use the maximum entropy principle to determine this.

The constraints are as follows:

 

By Fudgy McFarlen, ago

What is Entropy?

There are many mathematical definitions of entropy.  The mental picture I find most useful is to imagine the following:  

  • you are put in a room and your job is to label everything in the room with a sharpy indelible marker and masking tape.  
  • You are asked to label everything in the room using the binary numbering system.  This binary number will be that particular objects I.D.

As you go about this you may want to number the objects you most commonly refer to with the lower digits that have less length.  That way since you mention "FORK" much more often than "NUMBER 6 SCREW" you will end up having to say less digits.

The measure of entropy in this room is the number of binary digits required to number all the objects.  This is entropy.  The formula for this sentence that I just said is:

             Entropy ~= log2N    where N is the number of different types of objects in the room

Now in a probabalistic situation with outcomes  x1 , x2 ….  xn    with P(xi) = probability of xi

…..more

By Fudgy McFarlen, ago