A MarketPlace of Ideas                        

I always wondered why a freeway with no accidents or other driver diversions such as a policeman pulling over a speeder can come to an almost complete halt.  I think if have figured out this mystery.  The following graph is velocity versus distance on a freeway.  Here in Phoenix there is a location on the freeway where you can be driving at a normal highway rate and then encounter a traffic jam.  Once through the 4 miles of the traffic jam the freeway opens up again to acceptable conditions.

Note the oscillatory behavior in the jammed area where cars alternate between a maximum low speed and complete stop.

If we make the simplifying assumption that

Then we can say

Some example numbers are

                           Using the 2 second driving rule

Which is close enough to claim agreement.  

My strategy in a traffic jam is to not follow the car in front very closely.   I leave alot of space in an effort to recreate the conditions that result in 60 mph driving!

An interesting observation.   The two second rule would give a spacing of 32 feet at 10 mph in a traffic jam.  My observation is that the cars are packed tighter than 30 feet and thus causing the conditions that set up the braking wave.   People are driving too close to one another and it is bound to result in the driver being forced to throw on the brakes.  In Normal 60 mph driving following another car with a 2 second lead its very rare that you need to throw on the brakes.  Going the other direction if you drive 60mph and only leave a 1 second lead you're much more likely to end up needing to throw on the breaks. 

                                1      4      9      16      25     36      49  ….

                     1       3      5      7       9      11     13               

               2      2      2      2       2       2  

For example:

And it is interesting to note that 9 is an integer square all by itself so

         or

Which is an integer solution of Pythagorean relation.   Its easy to see that that right hand side must be an odd number in this case of a single term of the summation being a perfect square.  Of course this changes when you ask that 2 terms sum up to a perfect square.

         or         which is  

It appears from this point of view that the numbers in both sets must  be from a contiguous series or the sum of the set will go off the perfect square grid.   If you remove the sequence of the smaller square from a bigger square all that is left is the higher terms and no way of swapping any terms to get the same answer.

Pythagorean Triples in Wikipedia

Fermats last theorem states that there are no integer solutions for

              where  

The point of view above can be used to illustrate a good view of why.   I will use the perfect cubes to illustrate

                                1      8      27       64      125    216      343       512    729    1000

        1        7       19     37      61       91      127       169             

                                6      12      18     24       30      36        42 

                     6        6      6        6         6          6          

In the first situation the summed term simply incremented by 2 each time and was bound to hit on perfect integer squares.  In this example the term really starts moving along and is much less likely to encounter a perfect cube.  In fact we know it to never encounter one.  It only gets worse at higher integer exponents.  

Links

• Fermats last theorem blog – lots of number theory
  
• Latex Syntax
  

Robert McNamara has died.  Unfortunately his brand of social engineering is alive and well.   You may know a couple of Roberts "innovations".  They include the F-111.

 And the ever jamming M-16 rifle.

His life demonstrates the abject failure of when the unwashed hands of government apparatiks touch the inner workings of our society. 

Read George Wills Article to understand better how McNamera introduced his own particular version of errors into the system. 

The apogee of McNamara’s professional life, in the first half of the 1960s, coincided, not coincidentally, with the apogee of the belief that behavioralism had finally made possible a science of politics. Behavioralism held — holds; it is a hardy perennial — that the social and natural sciences are not so different, both being devoted to the discovery of law-like regularities that govern the behavior of atoms, hamsters, humans, whatever.

This sort of social meddling has not ceased.  Unfortunately it has found a new home in the left.  Unable to fathom the mathematics of chaos they straddle the bucking bronco wild horse proclaiming over and over it will end differently this time.  This time we can break this wild horse called human reality.  This from the people who claim there is no god on one hand but on the other purport to have god like qualities of to-the-core understanding.  I think that is unlikely.

I guess all we can do is wait for Obama to be tossed off this pitch black mare with the bewitching eyes.  It will come sure as the sun rises.

Show that the curve:

contains only one set of three distinct points, A, B, and C, which are vertices of an equilateral triangle, and find its area.

The “curve” is actually reducible, because the left side factors as
Moreover, the second factor is

so it only vanishes at . Thus the curve in question consists of the single point together with the line

To form a triangle with three points on this curve, one of its vertices must be . The other two vertices lie on the line so the length of the altitude from is the distance from to   or  The area of an equilateral triangle of height h is    , so the desired area is .

Remark:

The factorization used above is a special case of the fact that

 .

Where omega denotes a primitive cube root of unity. That fact in turn follows from the evaluation of the determinant of the circulant matrix

by reading off the eigenvalues of the eigenvectors for  

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 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.

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    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.

Graph Theory and Its Engineering Applications  By Wai-Kai Chen

What is most interesting about the following geometric proof is that it uses a shearing transform.  This transform exploits the fact that a parallelogram of equal height and length has equal area.  This gives it a fluid pouring aspect.

Alot of different proofs of Pythagorean theorem.

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 )