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Do a long hand division of  

x greater than or equal to 1 does not result in convergence of this sum.  However this algorithm can still be used to do some interesting things.  Let us use a complex value of   

Each power of x yields a result one step around this unit circle. Thus this series is the Z transform of the associated sequence.  [1,0] , [0.707,0.707] , [0,1] ……. This sequence is  

Thus the z transform of this sequence is:    

If you want to get express in terms of n instead of n-1 you can multiply by 1/x.  Since x is the place holder it is easy to see if you want to slide a series one unit to the left by dividing by x. 

   : note this series starts at 45 degrees phase!

More information:

• Generating Functions
• Z Transform

The series:          this converges for x < 1 : Both of these expressions are the Z transform of the  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

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. 

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

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 )

Likely you have heard this old adage about security and liberty.  The security they talk about is economic.  Things are likely to get tougher in the near future.   We are shipping too much money out of the country for oil.   We are too dainty to allow it to be drilled on our coasts and in ANWR.   Thus things will deteriorate more until the pain threshold is found. 

Many people will want to vote for the politician that says it will be ok if we only regulate this or tax that.  These are the unDemocrats who will say anything to get elected.  Say anything to get the power.  I would like to think you as an electorate will have the sense not to listen.  But I’m doubtful about this.  You listened in the past and you likely will in the future.   The problem is this sort of thing never helps economically and in fact damages something much more precious than your temporary economic circumstances and that would be your liberty.  I know this is true.  My favorite home away from home country is Brazil.   As currently constituted Brazil has a fully implemented form of government that the unDemocrats want to have here.  The people know their government and changing it is hopeless.  Nothing will ever change in that area and the Brazilians know it.   With liberty you can improve your economic circumstances.   With economic wellbeing you can not necessarily do diddly squat about your liberty.  I know its tough to be a good soldier.  You have to be brave.  And clearly about 50% of the population lacks any impulse toward liberty. Quite the opposite.  They lick their chops at the thought of another round of incremental carving up of liberty for the sake of financial gain. 

Remember losing a job is temporary.  Living in a land of curtailed liberty is pretty much for your entire lifetime.    So don’t be dour Gus Halls who secretly long for a beer, a recliner and a black and white television.   For an inspired life you have to dream of liberty.

In a previous article entropy was defined as the expected number of bits in a binary number required to enumerate all the outcomes.  This was expressed as follows:

entropy= H(x)=  

In physics ( nature ) it is found that the probability distribution that represents a physical process is the one that has the maximum entropy given the constraints on the physical system.   What are constraints?  An example of a probabalistic system is a die with 6 sides.  For now pretend you do not know that it is equally likely to show any 1 of the 6 faces when you roll it.  Assume only that it is balanced.

In the case of a die the above summation is equivalent to the following sort of computation:

• Initial assumption set of 6 probabilities that  sum up = 1  … this is a given as it has to be at least one of the 6 faces unless it stands on edge Twilight Zone style.  Lets assume P(x_i) = 0.05, 0.05, 0.05, 0.05,0.05, 0.75  …. you know instinctively this is not correct but demonstrates the maximum entropy principle

The total entropy given these probabilities = (.05) * (4.322) * 5 + 0.75 * (.415)= 1.0805 + .311= 1.39 bits

Let us use our common sense now.  We know there are 6 equally probable states that can roll up.  So its easy to calculate the number of bits required.  

• Bits required = log₂6 = 2.585 bits 

Thus we can see our initial assumption of probabilities yields an entropy number less than we would expect from common sense.   How do we find the maximum entropy possible? 

• Use the Langrangian maximization method. 
• Maximize the entropy phrase with the constraint that  

                 …. sum over all probabities must = 1

The langrangian is formed as follows:

Now differentiating the langrangian and setting the derivative = 0 we can find the maximal entropic probability

         solving for the P_i  yields

        All the P_i= the same constant with the probabilities summing to 1….Thus P_i=1/6 since N=6

While this is alot of work to derive the obvious it there is a purpose. In the case of more complicated situations where the probability distribution is not obvious this method works.  For example in the case of the Black Body emission curve of Planck.  Given just the quantization of energy levels you can derive the black body curve!!  This principle is woven all through nature.  Learn it because it will serve you well. 

Some interesting Notes to myself — myself? I meant me.

• Maximum Entropy Modeling - including some open source software

The base state of an electron in an infinite potential well has the most "space" for the electron state.  Thus it has the maximum entropy. Take that same state and imagine pinching the electrons existence to nil in the middle of the trough.  Now you have state-2.  The electron now exists in a smaller entropic state and guess what?  It contains exploitable energy now. This is like a spring compressed.  The electron can decompress and exert force / expend energy.  For example in an interaction with another atom possibly a recoil could occur.   In a crystal lattice an electron can transfer its energy to the atom next door and in effect yield conduction.  All these are preliminary suppositions subject to more scrutiny. As mentioned before since the electron exists in this potential well in the form of free fall it can not have any acceleration.  Thus its distribution must thoroughly avoid the edges of the well were it would indeed experience accelerations by bouncing and recoiling off of the walls.