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.

Don't Dwell in the Past

 

Dog Days Are Over

Run fast for your mother, run fast for your father
Run for your children, for your sisters and brothers
Leave all your love and your longing behind
You can't carry it with you if you want to survive
The dog days are over
The dog days are done
Can you hear the horses?
'Cause here they come

The worst is behind you.

The dog days or dog days of summer are the hot, sultry days of summer. They were historically the period following the rising of the star system Sirius (known as the "Dog Star"), which was connected with heat, drought, sudden thunderstorms, lethargy, fever, mad dogs, and bad luck. They are now taken to be the hottest, most uncomfortable part of summer. If you have survived those days, then cooler, better days are ahead. The dog days will come again next year, but the important thing to remember is that you have survived. Do things in the days to come that ensure that you will survive the dog days next year. You have a whole year to get ready.

Competition

 

All I Really Want To Do

Now I ain't lookin' to compete with you
Beat on, cheat on, mistreat you
Simplify you, classify you
Deny, defy, mystify you
All I really want to do
Is baby, be friends with you
Baby, be friends with you

Reality is NOT a competition.  No wagering please!

Reality is random.  Fair betting requires random events.  Therefore there should be betting on reality, correct?  Not exactly.  There is betting by individuals, but collectively there should be no betting.  Betting requires a zero sum game: i.e. winners AND losers.  Reality is NOT a zero sum game, there is expected to be at least a change, and hopefully growth, from the current reality. So betting individually may make sense, but betting collectively does not make sense.  If there is change, and again hopefully growth, it should be shared by both winners AND losers, because in a fair game, the winners in the past could be losers in the current contest.

If the total value of a system increases, then the median should increase.  However it is possible that the increase could go only to the current winners, or a subset of the current winners. In this case, the mean might increase while the median decreases, or stays the same.  The problem is that popular wisdom is that the average is the mean, because that is easiest to compute, i.e. system total divided by individuals in the system. Actually, average is the centrality of the normal, which is by definition the median, where half of the individuals have more than the median and half have less than the median. If the mean is the median, as it is in normal distributions, that popular wisdom isn’t a problem.  But if there is a gap between the mean and the median, it indicates that the situation is NOT normal.  If the gap between the median and the mean is increasing, this also indicates that the situation is becoming more ABnormal.

I wish I could say that the gap has been decreasing, but based on US Census reports, the gap has been increasing. 

This is not only because people have confused the mean and the median  https://dbeagan.blogspot.com/2018/08/wonderful-world-dont-know-much-about.html , inflation has masked the impact, and the tax code has exacerbated the effect, https://dbeagan.blogspot.com/2020/06/taxman.html.

Competition should be between groups, not between individuals within groups. Let’s not compete. Let’s be friends.

 

Art v. Science

 

It’s So Easy

People tell me love's for fools.
So, here I go, breaking all the rules.
It seems so easy (Seems so easy, seems so easy)
Oh so doggone easy (Doggone easy, doggone easy)

Breaking rules isn’t easy, but it is necessary.

“Art is in the eye of the beholder”.  “There are rules”.  And this describes the apparent conflict between the arts and the sciences.  Artists are individuals, breakers of rules.  Scientists are also individuals, discoverers of rules.  Those of you who are old enough to remember the classic TV show The Prisoner with Patrick McGoohan, playing Number 6, might remember the phrase.  “I am not a number.  I am a free man.”  Those of you who also remember the finale will remember that the elusive Number One was shown also to be Patrick McGoohan.  The finale seemed to imply that I am a free man because I am a number.  The problem is that man is an individual animal, but man is also a group animal, a member of a group.

The phrase is “All for One, and One for All”.  In an ideal world, All, the group, supports the individual, One and the individual, One, supports the group, All.  Or as I learned in Cub Scouts, “The Cub Scout helps the Pack go. The Pack helps the Cub Scout grow”.  A group, such as the United States in its Constitution, “Promote(s) the Progress of Science and useful Arts.”  It does so by promoting those who discover new rules and those who break old rules. To do so you don’t force old rules on individuals because those rules might be wrong or incomplete.  You also don’t prevent individuals from breaking old rules, again because those rules might be wrong or incomplete.  There should be no conflict between arts and sciences. They both have the same end goal, the making of new rules, that improve on, and may replace, old rules.