Complexity

 

Just My Imagination (Running Away with Me)

But it was just my imagination
Runnin' away with me
It was just my imagination
Runnin' away with me

Reality is just one component of a complex number that requires an imagination.

A complex number, x, has a real component and an imaginary component, a+bi, where a is the coeffcint of the real component and b is the coeffcient of the imaginary component, i.  This can also be expressed as re^iθ, where r is the real component and θ is the angle to the imaginary axis.  It is thus proposed that x, the relationship to the absolute, is complex, and has both a real and an imaginary component.

For example, the Probability Density Function, PDF, of an exponential distribution, e.g. the formula for radioactive decay, is λe^-λx, where λ is the decay rate and x >0.  This has a Cumulative Distribution Function, CDF, which is the integral of the PDF, of 1-e^-λx.  This can be translated along the x-axis by an amount µ as PDF = λe^-λ(x-µ) which makes the CDF =1-e^-λ(x-µ).  The median is when the CDF=50%.  Thus 0.5=1-e^-λ(x-µ) requires that the median occurs when x-µ=-ln(0.5)/λ.  If µ=0, which means that x is a relationship to absolute zero, the median is -ln(0.5)/λ. But for an exponential distribution with a µ of zero, the median has previously been defined as 1/λ.  To resolve this apparent paradox, where the median must be both -ln(0.5)/λ and 1/λ, it is proposed that the median is a complex number and that 1/λ is only the real component of that complex number, -ln(0.5)/λ.  This means that -ln(0.5)/λ =(1/λ)e^iθ,  and that the angle θ, in radians, of the imaginary axis, is thus ln(-ln(0.5))=-0.36651i. Also, according to Euler’s Formula, iθ=ln(cos(θ)+isin(θ)), which means that (1/λ)e^-0.36651i= a*cos(θ)+i*b*sin(θ). This means that a decay rate, λ, defines the complex number, the angle of the imaginary axis, and the translation along the x-axis of the complex number x.

Coincidence

 

It’s All Over Now, Baby Blue

The highway is for gamblers, better use your sense
Take what you have gathered from coincidence
The empty-handed painter from your streets
Is drawing crazy patterns on your sheets
This sky, too, is folding under you
And it's all over now, baby blue

Coincidence is God’s sense of humor

I have been musing on Pythagoras’ Theorem.  The Transportation Modeling Improvement Program included a post from someone who wanted to know whether he should get a Professional Engineering, PE, license.  In responding I was going to quote from the movie the Wizard of Oz https://www.youtube.com/watch?v=uCOxU2rKLas. When given a diploma, e.g. a PE, the Scarecrow  quotes Pythagoras’ Theorem, and to me that is the coincidence.  

The Scarecrow had to already know that Theorem.  The Wizard's diploma only said that he could be trusted to know it.  Getting a PE does not add to your wisdom.  It does serve as a shorthand that others have tested your wisdom and it can be trusted.    The reality is that you had to already know Pythagoras’ Theorem BEFORE you get your “Doctor of Thinkology, Th.D.”,…. or a PE.  A PE says that your judgement has been judged by others.  It may be that you will someday be in a position where you are not known, and your opinion may have to be quickly judged.  In that case a PE may serve as that proof. Getting a PE to me is like giving Chicken Soup to a dead man.  Will it help?  It couldn’t hurt!

Climate Change

 

Sinnerman

Well, I run to the river
It was boilin', I run to the sea
It was boilin', I run to the sea
It was boilin', all on that day
So I ran to the Lord
I said Lord, hide me
Please hide me
Please help me, all on that day
He said, hide?
Where were you?
When you oughta have been prayin'
I said Lord, Lord
Hear me prayin', Lord, Lord
Hear me prayin', Lord, Lord
Hear me prayin', all on that day
Sinnerman, you oughta be prayin'
Oughta be prayin', sinnerman
Oughta be prayin', all on that day

When the hurricane and earthquake hits SoCal
and the wildfire hits Maui and Canada, where you gonna run to?

Are you looking for someplace to hide from Climate Change?  Good luck with that!

Cordinate Transformation

 

Reflections

Through the mirror of my mind
Time after time
I see reflections of you and me
Reflections of
The way life used to be
Reflections of
The love you took from me

But can you reflect an absolute?

Mathematically you can translate a function with respect to an x-axis as g(x)=f(x-µ), where µ is the new location.  You can rotate a function with respect to an x-axis as g(x)=sin(θ)*f(x) where θ is the angle between the rotated and the original x-axis.  You can reflect a function with respect to an x-axis as g(x)=-f(x). But if x is the relationship to an absolute then that can restrict the ability to translate, rotate, or reflect a function of x, f(x). Translation, rotation, and reflection together can be called cordinate transformation

The simplest one to explain is the translation.  There is an absolute zero temperature.  It is measured in degrees Kelvin.  There is a 0° Kelvin but by definition you can not have a temperature below 0 degrees Kelvin.  It is more convenient for us to use a translated temperature scale where the origin, zero, has been translated by 273.15 degrees Kelvin, to the freezing point of water.  You can observe a temperature of -273° Celsius but you can’t observe a temperature of -274° Celsius because that would be below absolute zero.

Rotating a function in one direction is equivalent to rotating a function in the opposite direction by 2*π because the period of the sine function is 2π.  This has an impact on Euler’s Formula e^ix=cos(x)+isin(x) which is a rotation of the imaginary axis by x degrees.  If x is the relationship to the absolute, then x can not be less than zero because you can’t rotate the imaginary axis by something less than absolute zero.  That is why e^ix cannot be defined as a real number, or else it would create the paradox which would require both that cos(-x)= cos (x) and cos(-x) = -cos(x). 

This same restriction applies to the reflection.  If the origin of zero has been translated to µ, then you can reflect that function between µ and 0, but you can not reflect that function below 0 or else it would be a reflection below absolute zero.

Since a random number has the parameters µ and σ, and the variance, σ², defines the range of that random number, a random number must be µ>3σ or else the random number could be below absolute zero.  

Strategy v. Tactics

 

Climb Every Mountain

Climb every mountain
Ford every stream
Follow every mountain (every mountain)
Don't you ever give up, no ohh
Climb every mountain (every mountain)
There's a brighter day on the other side
Follow every rainbow
'Till you find your dream

It is NOT winning every battle!

Winning every battle might seem like a winning strategy, but it is not.  Bad strategies include Pyrrhic victories and "Winning the battle but losing the war". Winning a battle, being a good field general, a great tactician, does not mean that you would be a good strategist.  Frank Robinson might have been a great player and great manager, but the more likely examples are Ted Williams who was a great player/tactician, but a lousy manager/strategist and Joe Torre who was a lousy player/tactician but a great coach/strategist.  Dynasties are more likely when a great player is teamed with a great coach,  e.g. Tom Brady and Bill Belichick, than one without the other.

Winning every battle is a good strategy only if you prevent the other side in the battle from ever contending again.  But then you have to go all scorched earth on this, a la Rome v. Carthage. For example, while Dunkirk may have been a tactical loss for the British, destroying the British Army might have only emboldened the small ship captains who particpated in the miracle at Dunkirk and other civilians who would have ultimately defeated Hitler.  I think that this is a lesson that Vladmir Putin is learning in his ill-fated invasion of Ukraine.  The fact that the leader of Ukraine is a ordinary former TV comedian, and not a bully, only reinforces the fact that bullying/dominating alone does not always work.

Sometimes sacrificing a tactical victory is the better strategy.  In the 2004 American League Baseball Championship Series, Tim Wakefield was scheduled to be the Game 4 starting pitcher. But when Bronson Arroyo was knocked out of Game 3 after only two innings and the relievers were taking a beating in the game that the Sox would go on to lose 19-8, Wakefield sacrificed his chance to start Game 4 and volunteered to "take one for the team." He ended up pitching 3 1/3 innings in Game 3, allowing manager Terry Francona to keep the rest of the Red Sox bullpen rested. Relief pictchers Keith Foulke and Mike Timlin didn't have to pitch at all that day, and Alan Embree faced only four batters. When the next two games of the series went extra innings, it was crucial to have rested relievers, and Francona and his Red Sox teammates were quick to point out that without Wake's selfless act, the comeback would not have been possible.  In other words, Wakefield knew that a strategic series victory for the team was more important than his own tactical win in Game 4. And in doing so, he climbed the mountain and found every Red Sox fan’s dream.

Absolutely Nothing

 

I’ve Got Plenty of Nuttin’

Oh, I got plenty o' nuttin'
An' nuttin's plenty fo' me
I got my gal, I got my song
Got Hebben de whole day long
No use complainin'
Got my gal,
Got my Lawd,
Got my song!

In Other Words, NOT absolutely nuttin’

Porgy in this song from Porgy and Bess, states that he has nothing on the scale of plenty.  But plenty excludes Heaven, his Lord, his gal, his song, and many other things. Plenty thus must be measured on a relative scale, not an absolute scale.  It is like saying that the temperature is below zero outside. On the absolute Kelvin scale it can by definition never be below zero.  Saying that a temperature is below zero on the Centigrade scale only means that zero on that scale has been relatively set as the freezing point of water. 

Saying that there is a temperature of absolute zero must therefore mean that at the opposite end there is an absolute temperature.  The fact that we have never observed an absolute temperature does not mean that there isn’t one.  There are absolutes that we can observe such as light.  Einstein’s Theory of Relativity says that the speed of light is an absolute that can be approached, but not exceeded.  An exponential equation, including radioactive decay, measures the relationship to an absolute that can be approached, but not exceeded.  The question becomes how many absolutes are there?  An answer is that there is only one absolute, which can be perceived differently, but each perception is only one aspect of a single absolute. If there is only one absolute then if x is measured on an absolute scale then the integral of any function of x, f(x), should not be which ∫_-∞^∞ f(x) dx which implies the existent of two absolutes, infinity and its negative, but instead should be ∫₀^∞ f(x)dx, which implies the existence of only one absolute, infinity and its absence, zero.

This has an implication on the computation of the Cumulative Distribution Function, CDF, for all distributions.  The Probability Density Function, PDF, is defined without consideration of positive or negative x, but the CDF, the integral of that PDF, is then also only defined for x>0, i.e. x less than zero is undefined.  If the PDF is a normal random logistic function, its PDF is 1/(4s)*sech²(0/2s), and its CDF is ½* tanh(0/2s)+1/2.  A physicist might say that if s=0 and 0/0 is defined by its limit as 1, then the PDF is 0 but the CDF is ½, but a mathematician might say that if s=0 and 0/0 is undefined, then both the PDF and CDF are undefined.

Free Speech?

 

I Swear

And I swear by the moon and the stars in the sky
I'll be there
I swear like the shadow that's by your side
I'll be there

I swear to tell the truth, the whole truth, and nothing but the truth.

Not A truth. Not what you BELIEVE to be the truth. Not YOUR truth, but THE truth. And that is because a trial is about establishing CERTAINTY. You are not asked for your opinion on the evidence. You are not asked for your opinion of the witnesses. You are not asked for your opinion on the officers of the court who are charged with protecting the witness and seeing that the evidence is being fairly presented. You are not asked for your opinion on the jury who will be ruling on that evidence. Your opinions may be free speech, but they could also be viewed as witness tampering, interfering with a fair trial, or jury intimidation. If you wish, then you are free to express your opinions under oath, on the stand, in front of that jury.

Shouting “Fire” in a crowded theater when there is no fire, using the phase “Your money or your life” during a robbery, or as King Henry II famously ranted leading to the murder of Thomas Becket, “Who will rid me of this troublesome priest” are all speech. But they led to actions that are crimes.

Free speech is protected because the Constitution has rightfully judged that the state can not determine what is the truth. However whether it is true or not, …and there is cetainly the possiblilty that that the speech was a lie, that does not absolve you of the consequences of that speech.

King Henry II might have only uttered words, not committed the murder, and might not even intended the murder, but his contemporaries and history have judged him to have been wrong. King Henry publicly did penance for his words. Just as the state can not prevent your speech because you might be telling the truth, you also can not state with certainty that what you said was THE truth. That certainty is why there is a trial.