Variance II

 

Ain’t We Got Fun

There's nothing surer
The rich get rich and the poor get poorer
In the meantime, in between time
Ain't we got fun?

Do the rich have to get richer, and the poor get poorer?

Given my recent blog post that variance ( e.g. the difference between rich and poor) should be a constant, https://dbeagan.blogspot.com/2025/02/variance.html. it is appropriate to review an earlier blog post on the difference in wealth among nations.  https://dbeagan.blogspot.com/2021/11/distribution-of-wealth-iii.html .  As well as a blog  on the consequences of keeping the location parameter constant and allowing the variance to increase to accommodate growth.  https://dbeagan.blogspot.com/2023/09/distribution-of-income-ii.html

The fact that variance is a constant and location is not, can lead to bad policy decisions if it is wrongly assumed that location is constant and variance is not.  It is totally correct that things were better from 1946 to 1964, and that things were much worse from 1996 to 2013, just as it is true that Christmas is better than the Dog Days of Summer.  But making every day Christmas is a stupid and childish response, just as MAGA is.  The grown‑up response is to work hard during the Dog Days of Summer so that the following Christmas can be as good, or even better than, the past Christmases that you do remember.  The fact that humans don’t live through more than portion of two cycles of historical periods of 100 years does not mean that the cycle is not there, just that a single individual can not experience multiple cycles. 

Mark Twain said that “History doesn't repeat itself, but it often rhymes.”  William Faulkner said that the “The past is never dead. It's not even past”.  Just because you can’t see other cycles, does not mean that other cycles do not exist, only that you can’t personally experience them. Learn from a study of the historical past.  Grow up! It can’t be Christmas every day!  Don’t be a such a child!

Surfaces

 

Blue Horizon

Beyond the blue horizon
Waits a beautiful day
Goodbye to things that bore me
Joy is waiting for me
I see a new horizon
My life has only begun
Beyond the blue horizon lies a rising sun

The horizon is why Pythagoras Theorem applies Locally by not Globally

There are three kinds of surfaces. 1) Spherical, 2) Flat (Euclidean) and 3) Hyperbolic. While it seems like we live on a flat Earth, the existence of the horizon is actually one of the proofs that the surface of the Earth is actually a sphere, not flat. An observer can perceive 50% of any object that is between the observer and the horizon. An observer can not perceive the object at all if it does not extend above the horizon. However the observer can only perceive 50% of the portion of any object that extends above the horizon. That is why the mast of a sailing ship on the horizon is visible before the rest of the ship, because at that point  the mast extends above the horizon, but the rest of the ship does not. The horizon is defined by the radius of the spherical surface. If the sphere is large enough, then most objects which are perceived will be between the observer and the horizon. But that does not mean that the horizon does not exist, just that it is not always perceived. A Flat surface is thus only the limit of the spherical domain, not a domain itself. The surface will have a curvature defined by the major and minor axes, a and b, of the equations describing the eccentricity of the surface. If a and b are both less than infinity, then a spherical surface is described, and the eccentricity is less than 1. If a and b are both infinite, then a hyperbolic surface is described, and the eccentricity is greater than 1. A flat surface is thus only the boundary between these two conditions where the eccentricity is exactly equal to 1.

For any hyperbolic surface, regardless of the curvature, only 5/6 or 1/6 of an object will be perceived, depending on which side of the hyperbolic surface is being perceived by an observer. That is because a hyperbola has two solutions, one which is the opposite sign of the other. But the solutions are similar to the spherical solution until the numbers involved are very large: e.g. 2/3 the size of the universe or 5/6 the speed of light. Thus could it  also be said that the universe is flat locally, but hyperbolic universally.

Variance I

 

Too Much of Nothing

Too much of nothin' can turn a man into a liar
It can cause some man to sleep on nails
Another man to eat fire
Everybody's doin' somethin', I heard it in a dream
But when it's too much of nothin', it just makes a fella mean
Say hello to Valerie, say hello to Marion
Send them all my salary on the waters of oblivion 

Say hello to Variance too! 

A normal random distribution is defined by its location, often called the Mean/Median/Mode, μ,  and its Variance, σ². The mean/median/mode are not equal, except at infinity. The mean, often called the average, is computed from the total of the observations divided by the number of observations. The variance is the range of the observations but is not as easy to compute. An observation consists of where you are observing a thing,  its x value, and what is the attribute of the thing that you are observing, its f(x). Just because the x-axis is infinite it does NOT mean that the function on that x-axis is also infinite. For example, a wave is a repeating function, f(x)= cos(x). In this case x can be any number, e.g., infinity, ∞, but the value of f(x) can only be between -1 and 1. Mathematically this would be stated that the domain is infinite, but the range is finite and between -1 and 1.  The variance, the range, can be finite even if the input variable, the domain. is infinite.

It is fairly easy to compute the mean of a variable. It is the total of all of the observations divided by the number of the observations. It is a little harder to compute the variance of the observations, but not impossible. If an infinite number of variables is computed then the mean by definition is half of infinity. But it is important to differentiate between infinite variables and infinite domains. Just because the domain is infinite that does not imply that the mean/median/mode is infinite or that the variance is infinite. These are characteristics are of the range, not the domain. The range can be finite even if the domain is infinite, for example, the cosine (x). The mean/median/mode of the cos(x) is sine (x). The variance of the cosine, and the variance of the sine for that matter which is also the mean/median/mode of the cosine, is ½.

The variance is thus a constant. The variance of the range is always a constant, even  if the domain is infinite. Thus it is not inconsistent to say that the variance of infinity is a finite number. The mean of cos(x) is sin(x) whose domain is also infinite, but the variance of that cosine function, and its mean the sine, is, ½,  which is finite.

Cycles

 

It’s Still Rock and Roll to Me

Don't waste your money on a new set of speakers
You get more mileage from a cheap pair of sneakers
Next Phase, New Wave, Dance craze, anyways
It's still Rock and Roll to me

And it’s still Mathematics to me!

I believe that there are historical cycles of  approximately 100 years. I say approximately, because the length of the historical cycle is a random phenomena whose average is 90 ±10 years. Half of a full period of a cycle is thus 45±5 years and 1/6 of a half cycle would thus be on average 8 1/3 years long. If the last Turning (end of a cycle) was in October of 1929, then you would expect an upward slope from approximately 1929 to 1979. And  a downward slope from approximately 1979 to 2029. In the first 1/6 of the first half cycle, e.g. ~1929 to 1938,  there would be slow to virtually no growth and that growth may be virtually indistinguishable from the decline at end of the last cycle. To an observer in the midst of a cycle, it may appear as if the cycle ended sometime in the middle of that period, e.g. 1932 ( i.e. it appears like a lag variable). 

There should be moderate growth for the next 1/6 of the half cycle (~1938 to 1947). There would be twice the rate of growth for the next 1/6 of a half cycle ( ~1947 to 1956) and that rate of growth would be accelerating. There would be almost the same rate of growth during the next 1/6 of the half cycle (~1956 to 1964), but that growth would be decelerating. During the first 1/6 of the next half cycle the growth is declining but the change is virtually indistinguishable from the last 1/6 of the previous half cycle. The rate of decline accelerates during the next 1/6 of a cycle, (~ 1979-1986). The rate of decline virtually doubles during the next 1/6 of the cycle, ( ~1986-2004). The highest rate of decline happens during the next 1/3 of a half cycle, (~2004 to 2020). The rate of decline decelerates during the next 1/6 of the half cycle, (~ 2020-2029). The last phase of the cycle has virtually no growth, just like the first 1/6 of the next half cycle where growth returns.

Yes, the length of each phase can vary. Wars, economic downturns, terrorist attacks, extreme weather  and other random events can occur. What about on average is so hard to understand?

There is a danger zone about 80% through a declining half cycle where things may have gotten so inequitable that revolution against the sovereign can happen. That puts the danger zone at somewhere between 2020 and 2029 where there is a strong possibility that the subjects of the sovereign will say off with their heads.

Standard Deviation

 

A.D.H.D.

Fuck that, eight doobies to the face
Fuck that, twelve bottles in the case nigga. Fuck that
Two pills and a half, wait nigga, fuck that.
Got a high tolerance when your age don't exist

And tolerance is where it’s at!

I would be lying if I said I understood anything that Kendrik Lamar said during the Super Bowl LIX halftime show. I don’t speak Rap. But Kendrick’s lyrics as quoted above are wise beyond words, The language is very rough, but….maybe I am merely showing my age. Tolerance is what engineers call Standard Deviation, the square root of variance. A problem is acting like mean and the midpoint of tolerance squared, variance, are the same. They are NOT, except for the absolute. There are three outcomes to any contest: win, lose, and tie. There is an average, mean of that contest. The average plus the tolerance should include the entirety. But because the mean, average, is defined as half of the absolute, it is confused that this requires that mean be constant, when it is NOT or the tolerance to be NOT constant, when it is. One is subject to growth, the average, and one is NOT subject to growth, the tolerance.

According to L’Hôpital’s rule, the limit of the average is the mean AND the median. But for anything less than the absolute, the mean and median can, and will, be different. The mean of an even number is half of that even number. The mean of an odd number is NOT an odd number, it is half of the original number, which makes it an even number. Thus saying the variance is one third of the absolute, while the location, is half of the absolute seems like it is a contradiction but the mean is subject to growth, and the variance is a constant and is NOT subject to growth.

Mathematically x>μ AND σ=μ/3 is true for the absolute but that does not mean that variance increases as the mean, location, increases. The variance is a constant, but the location can change with growth. The problem is that the limit of N/2, the mean, as N approaches the absolute is the absolute , but the limit of the Standard Error, what engineers call tolerance, the square root of the variance is SE=σ/√N, zero. This is true if the absolute is zero, but it is also true if the absolute is NOT zero, x>μ and σ=μ/3 is true not only for N=0, but it also is true for any value of the number N. There is no contradiction,  The mean is a function and changes. The tolerance is a constant and can NOT change. And apparently by not growing up in Compton like Lamar, I missed that.

Atheists

 

Universal Soldier

He's a Catholic, a Hindu, an Atheist, a Jain,
A Buddhist and a Baptist and a Jew.
And he knows he shouldn't kill,
And he knows he always will,
Kill you for me my friend and me for you.

Not only are Scientists NOT Atheists, but Evangelicals must be Pro-Choice!

Just because the words are different doesn’t mean that it is not the same concept. Scientists say that they are Atheists but they probably also believe in infinity, an absolute. Those are the same thing, just using different words. Not only do scientists believe in an absolute, but they also believe in only one absolute.

Scientists believe in space-time. Space, x,  has an absolute: that is x>0. Time, t,  is related to space by the constant speed of light: c=∆x/∆t .  Speed is always defined as the change in space divided by the change in time. If the speed of light is a constant, then an absolute in space is the same absolute in time. But there is an infinite amount of time, before the time which is Now, and an infinite amount of time after Now. Thus Now is a relative zero but the absolute, infinity, is the same based on the constant speed of light. If space is absolute, and time is based on the same absolute, then there is only one absolute: a “mono” absolute. If you say absolute or theist, then you are effectively saying the same thing.

If time is a relative zero and space is an absolute zero, then scientists and all theists must also believe in choice. The future is merely all choices. The past is only the choices which have been made, but the amount of those choices in the past are the same as in the future. Thus saying that you believe in a future means that you believe in choices. So not only are scientists NOT Atheists but Evangelicals must be Pro-choice if they are not Atheists.

Rules

One, Two, Three

One, two, three
Oh, that's how elementary
It's gonna be
Come on let's fall in love
It's easy
Like taking candy
From a baby.

THREE is not only elementary, but it's also fundamental.

The ancient East Asian painting of the Three Vinegar tasters is supposed to be a representation of 1) Confucius, (Kung Fu Tse); 2) Lord Buddha; and 3) Lao Tse (the founder of Taoism) tasting vinegar from the same jar.

1. Confucius says that the vinegar tastes bitter which demonstrates to him that rules are necessary to impose order over chaos in reality and that it is up to you to impose that order.
2. Lord Buddha says that the vinegar tastes sour which demonstrates to him that all reality has to be endured so that you can pass from reality to the unobserved.
3. Lao Tse says that the vinegar tastes just like it is supposed to taste in reality. There is both reality, the observed, and the unobserved, but they are different to him.

Of the three, I’m on Team Tao. The fact that reality, the observed, is different than the unobserved does NOT mean that one or the other is better, only that they are different. Vinegar tastes how it is supposed to taste in reality, but that doesn’t mean it won’t taste differently in some place other than reality. What is “order” in reality,  might be considered “chaos” in the unobserved.