Rotations

 

Twist and Shout

Well, shake it up, baby, now
Twist and shout
Come on, come on, come, come on, baby, now
Come on and work it on out

Is life twisted?

The surface, with a coefficient of the imaginary axis=0 and a rotation of that imaginary axis of π, passes through the origin of a two-sheeted hyperboloid where those two sheets intersect at that origin. This surface in complex Cartesian coordinates can be described as -r +0*i.  where r is the real radius and i is the imaginary number, √-1.  But in doing so there is the need for a hyperbolic trigonometric function, the hyperbolic cosine, on that surface to be rotated, twisted, by π.  This is because of the nature of that function, which is undefined, and undefinable, for negative numbers.   In order to not allow negative  numbers, the hyperbolic cosine must be twisted, rotated, by π, in order to pass from one sheet of that two-sheeted hyperboloid to the other sheet through that origin.  It must also be translated at a point which is equal to the variance of that sheet, when the value of that hyperbolic cosine is equal to the variance of that sheet.

This sounds complicated but it is not.   A fluid in a pipe or channel passes from a laminar, calm, domain to a turbulent, white water, domain, when the flow in that pipe, or channel approaches its capacity.  Flow in traffic passes between a congested and an uncongested domain when the volume of traffic on a road is equal to the capacity.  When passing between these domains the fluid flow and traffic volume appear to follow a hyperbolic cosine.  

The variance of the surface is always a multiple of .5²*π²/3, in other words n*.25*π²/3.  This has a minimum value of 0.822467...., when n=1. 

This means that the hyperbolic cosine on the surface which includes our universe has both a mean/median locational position and a non-negative variance.  Since it has both parameters, that makes it a random function.  While that mean/median location can take on many values, if the variance was zero, it would be flat-lined, i.e. dead, not alive. By having a non-zero vaviance and twisting, it is alive and able to shout.

 

Harvard President Resigns

 

Branded

Branded, scorned as the one who ran.
What do you do when you're branded, and you know you're a man.

Wherever you go, for the rest of your life
You must prove, you're a man.

My sympathy for former Harvard President Claudine Gay

It is hard for a Brown alum who had to endure chats from the Harvard attendees at sporting events of “If it’s Brown, flush it down” and “You may have beat us now, but you’ll work for us someday” to defend a Harvard President, but the “Christian” Nationalists who have demanded her resignation and appear to have won, have not been very Christian.  I believe that Christ’s own words were “Judge not, lest ye be judged” Matthew 7:1-3 and “He that is without sin among you, let him cast the first stone" John 8:6-7.  But the criticism was coming mostly from the Two Corinthians crowd, so clearly the Bible means nothing to them, despite what they claim.

Real

 

Compared to What?

I love the lie and lie the love
A-Hangin' on, with push and shove
Possession is the motivation
That is hangin' up the God-damn nation
Looks like we always end up in a rut (everybody now!)
Tryin' to make it real, compared to what? C'mon baby!

Real compared to Imaginary?

Euler’s formula, e^ix=cos(x)+sin(x)*i, can be multiplied by a variable r on each side of the equal sign.  It then becomes r*e^ix=r*cos(x)+r*sin(x)*i.  This is also the statement of a complex number in polar coordinates where r is the radius in polar coordinates and x is the angle of rotation of the imaginary axis. In Cartesian, rectangular, coordinates, this is a+b*i, where a = r*cos(x) and b=r*sin(x).  The angle x, is tan^-1(r*sin(x)/(r*cos(x))) which is simply x  The radius in polar coordinates is r²=r²*cos²(x)+r²*sin²(x).  This is traditionally solved as r=√(a²+b²)=r*√(cos²(x)+sin²(x)) where a = r*cos(x) and b=r*sin(x).  Because of the identity for a circle, cos²(x) + sin²(x)=1 for all x, this means r=r, and thus the square root is the solution on a flat surface for all values of x and is the formula on a spherical surface when the radius of the spsherical surface is very large compared to the polar radius.

However this is not the solution on a hyperbolic surface.  This has the solution r=cosh^‑1(cosh(a)*cosh(b)).  When sin(0)=0 and cos(0)=1, this also has the solution r*eⁱ⁰=r +0*i and since eⁱ⁰=1, this also means r²=r²  and thus r=r.  But sin(π) is also zero, but cos(π)=-1 this means that as a complex number r*e^iπ=-r+0*i.    This can only be solved by the hyperbolic cosine and requires that r=ln(cosh(-r) ± sinh(-r)).  If -r is further replaced by –(x²+y²), this becomes ln(cosh(x²+y²) ± sinh(x²+y²)).  The single solution on a flat or spherical surface, when the coefficient of the imaginary axis is 0 and the angle of rotation of the imaginary axis is π, becomes a logarithm of cosh(x²+y²), the constant term, plus or minus an uncertainty, which is sinh(x²+y²), on a hyperbolic surface. However if the uncertainty is greater than the constant term, the logarithm is NOT imaginary.  It is undefined.

If a light cone is constructed which has three dimensions: space, time, and imagination, and space is measured on an absolute scale, i.e. there are no negative numbers, then the surface that light follows defines the edge of that volume.  The prohibition against negative numbers is merely an indication that the position in space is a vector, not a scalar.  The surface may not be flat, but may be hyperbolic. If that surface is hyperbolic, then the volume is not a cone connected to an inverted cone at the origin, but a two-sheeted hyperboloid where the two sheets connect at a single point at the origin.

The position in space as a vector is the velocity as a vector multiplied by time, v*t.  As a scalar this is often expressed as v*t, with the provision that v>0.  This effectively combines the dimensions of space, x,  with that of  time,  t.  Reality is defined as the position on the imaginary axis where i=0.  Thus the expression v*t+0*i is a statement of a position in real spacetime.  With v*t = a and with b=0, this is a complex number.  Using the conversion of a complex number in Cartesian coordinates, to one in polar coordinates this is re^ix=a+b*i where x is tan^-1(b/a) and r²=a²+b².  According to Euler’s Formula, there are two solutions when the coefient of the imaginary axis is 0: reⁱ⁰=v*t+0*i, where r²=(v*t)²+0² and tan^‑1(0/(v*t))=0; and re^iπ=-v*t+0*i where r²=(‑v*t)²+0² and  tan^-1(0/(-v*t))=π.  If r is on a flat, Euclidean, surface, r could be solved by Pythagoras’ Theorem.  However while t can be positive or negative, sin(x), the coefficient of the imaginary axis is cyclical, which means that while x can be positive or negative, it is properly x=n*π where n can be positive or negative and has a maximum and minimum value at integer multiples of π.

Einstein’s Theory of Realitivity can be derived from Einstein’s Triangle,  (mc²)²=(mvc)²+(m₀c²)², where v is the velocity of a particle, m is the relativistic mass of a particle, m₀ is the rest mass of that particle, and c is the speed of light.  This is conventionally solved on a flat, Euclidean, surface, but if Einstein’s Triangle is solved for a hyperbolic surface, the correct formulation would be

m=m_0* ln(cosh(1-v²/c²)±sinh(1-v²/c²))

or

 v=c*ln(cosh(1-m/m₀)±sinh(1-m/m₀))

If the Cartesian coordinates are converted to polar coordinates, and x is an integer multiple of π,  then the polar radius can be solved as

r=ln(cosh(-v*t))*cosh(n*π) ± sinh(-v*t) * sinh(n*π)).

Further if cosh(n*π)=0 and sinh(n*π)=-1, since sinh(-x)=-sinh(x), then this can be stated as 

r=ln(0±sinh(v*t)))

Since sinh(x)=-cos(i*x), while the radius can be defined as the natural logarithm of a regular cyclical trigonometric function, that function only repeats, is cyclical, in imaginary, not real, planes.  It is a single value in the real plane where the coefficient of the imaginary axis is zero.

Complex

 

Complicated.

Why'd you have to go and make things so complicated?

I see the way you're acting like you're somebody else

Gets me frustrated.

Life's like this you,

And you fall, and you crawl, and you break

And you take what you get, and you turn it into

Honesty and promise me I'm never gonna find you faking

No, no, no

What if things are complex, complicated, and you have to consider imaginary numbers.

The formula for the hypotenuse, c, of a right triangle with sides a and b,  on a hyperbolic surface is

cosh(c)=cosh(a) * cosh(b)

Using the formula for the inverse of the hyperbolic cosine this becomes

c=ln(cosh(a) * cosh(b) ± √((cosh(a) * cosh(b))²-1))

Using the identity for  hyperbola, cosh²-sinh²=1 which also means that cosh²-1=sinh², this becomes

c=ln(cosh(a) * cosh(b) ± sinh(a) * sinh(b)).

Using Euler’s Formula which involves imaginary numbers, e^iz=cos(z)+sin(z)*i; the definitions of cosh(x) = ½(e^x+e^-x) and sinh(x) = ½(e^x‑e^-x); and letting z = ix, this becomes

cosh(-x) = sin(ix) and sinh(-x) = cos(ix).

Using the fact that cosh is symmetrical, reflects, with respect to 0, while sinh is not symmetrical with respect to 0, this becomes

cosh(x) = sin(ix) and sinh(x) = -cos(ix)

As a result,  the hypotenuse of a right triangle on a hyperbolic surface can be stated as the natural logarithm of regular circular trigonometric functions as

c=ln(sin(ia)*sin(ib) ± cos(ia)*cos(ib)).

For the case when a, or b, = π,  since cos(iπ) = -1 and sin(iπ)=0, this then means that

c=ln(0 ± cos(ai)) if b= π, or ln(0 ± cos(bi)) if a=π.

Equation 1

For the case when a, or b, =0,  since cos(i0)=1 and sin(i0)=0, this then means that

c=ln(sin(ai) ± 0) if b= 0, or ln(sin(bi) ±0)) if a=0.

Equation 2

This shows that for a hypotenuse as a complex number, if its real portion has a coefficient of zero, then the imaginary coefficient can take on any value, and there is zero standard deviation, i.e. the second eqaution.  If instead the real portion has a non-zero coefficient, then the imaginary coefficient must be zero, and there must be a non-zero standard deviation, i.e. the first equaion.
 
 
 
 
 

Friends

 

Friendship

If you're ever in a jam, here I am.
If you're ever in a mess, S.O.S.
If you're so happy, you land in jail. I'm your bail.
It's friendship, friendship, just a perfect blendship.
When other friendships are soon forgot, ours will still be hot.

Is the enemy of your enemy your friend?

As in the Cole Porter song, a friend values your User Optimal almost as much as you do in ALL things.  An enemy might share the same User Optimal as you with respect to your enemy, but that is NOT all things. He still values his User Optimal more than yours.  It is purely a transactional “friendship”.  If his User Optimal with respect to your enemy changes and is no longer shared by you, he will act in his own interests, not yours.

In this sad definition of “friendship”, today’s "friend" may be tomorrow’s enemy.  Thus when the Mujahidin in Afghanistan were fighting the Soviet Union, the United States acted liked they were our friend.  When they became the Taliban and sheltered Osama Bin Laden, they became the enemy of the United States.

When Saddam Hussein was fighting the enemy of the United States, the Islamic Republic of Iran, he was treated as our friend.  When he invaded Kuwait, he became the enemy of the United States.

The saying is that you should keep your friends close, but your enemies closer.  A distinction should be made between real friends, and purely transactional friends.  You would be wise to keep transactional friends, who are only the enemy of your enemy, almost as close as enemies. The difference is that you can trust real friends.  You can’t trust transactional friends.

Solutions

 

                                                     You Can’t Always Get What You Want

I saw her today at the reception
A glass of wine in her hand
I knew she would meet her connection
At her feet was her footloose man
No, you can't always get what you want
You can't always get what you want
You can't always get what you want
But if you try sometime you'll find
You get what you need 

What you want is your User Optimal.  What you get is a Nash Equilibrium.

By profession and training I am a traffic engineer.  When there is merging traffic ahead, “Road narrows by one lane ahead”, what are traffic engineers trained to believe will happen? 

Traffic engineers are trained to believe that everyone follows the best strategy, and there is a rolling merge in which traffic continues in the merging lane until there is a safe gap in the continuing lane (and there should always be a safe gap).  This is an System Optimal, SO, solution and has the highest capacity for the merging section.  (i.e. utilize the capacity in the ending lane as long as possible).   

What do people really do?  Each individual car will try to follow their own User Optimal, UO, strategy.  For some cars, that UO strategy will be to stay in the moving lane until the last minute and then force their way into the continuing lane, even when there is no safe gap.  To prevent this, all other cars can move into the continuing lane as soon as they see the merge notice, and ensure there is no safe gap and punish the last minute switchers, i.e. block their UO solution, by NOT letting them in.  What all cars adopt is called a Nash Equilibrium in that most cars will not follow yheir owb UO soluion, will block some from their UO solution, and will adopt a Nash Equilibrium instead. 

Traffic engineers are trained to believe that everyone will adopt the most efficient, SO, solution.  But individuals will see their own UO solution as the most efficient.  Unfortunately this leads to Nash Equilibriums, where individuals can block each other from that most efficient solution. They are called Nash Equilibriums because they were descrbed by mathematician John Nash, the subject of the movie, A Beautiful Mind. No one gets what they want. (The Blonde, if everyone goes after the Blonde, because they will block each other.)  Instead people will get what they need. (Nash Equilibriums).  https://www.youtube.com/watch?v=vCyZvfRHkC4

Fools

What A Fool Believes

But what a fool believes, he sees
No wise man has the power to reason away
What seems to be
Is always better than nothing
Than nothing at all

Are you a fool to always believe a GPS?

A GPS is marvelous device.  But while it might be marvelous, it can be wrong.  In 2006, when GPS in cars were very new, I was riding with my brother-in-law to his son’s wedding rehearsal dinner.  We were both from out of town, so he was showing off the GPS in his new car to get directions to the restaurant where the dinner was being held.  He was following the GPS’s directions like a good soldier.  But when the GPS said to make a left turn into the restaurant, I stopped him because that that was the lobby of the restaurant, and the parking lot was across the street.  The GPS knew where the restaurant was, but it didn’t account for the fact that a car had to park in the restaurant’s parking lot.

A few years later on a family vacation on the Pacific Coast, we had rented a car but taken our own portable GPS device with us.  It flawlessly predicted that we were stuck in traffic on I-5 way south of Portland, Oregon because of an accident  on the Columbia River Bridge.  It re-routed us to get off at the next exit, onto Wheatland Ferry road and onto an agricultural ferry, along with farm equipment,  to cross the Willamette River.  Because of the GPS we were able to reach our destination on-time, despite a 5 hour back up on I-5. So the GPS was very smart, correct?

The very next day, we needed some supplies and asked the GPS for directions to a store.  We followed the GPS’s directions into a forest and onto what became a logging road.  The GPS had assumed that our rental car could use this logging road to cross a mountain.  To avoid a claim with the rental car company, I demured and forgot about that store.

A GPS has lots more knowledge. But it is a fool.  It believed a restaurant’s lobby was the same as its parking lot, and a rental car was the same as a logging  truck.  Be wise and listen to a fool, but don’t always believe a fool.