Planning

 

I Want To Be Happy

I'm a very ordinary man
Trying to work out life's happy plan
Doing unto others as I'd like to have them doing unto me

In life's happy  plan, are things random or deterministic?

By definition, a random number has two parameters: its location, μ, and its scale, σ. To define a deterministic number, the scale must be equal to zero, which is equivalent to saying that there is only one parameter, its location.

If the scale is non-zero, the system is random. If the scale is zero, the system is deterministic. Is the universe in which we exist random, or deterministic? By this it is meant the universe as a whole, not an object within that universe. The properties of an object within the universe can be deterministic even when the shape of the universe is random. The scientific standard for the properties of an object is 3 Sigma, 3σ, that is setting σ as close to zero as possible. This means that you are 99.7% certain that the location is the only value. In particle physics, an even more strict standard applies, 5 Sigma, 5σ. This means that you are 99.99994% certain that the location is the only value.

These percentages are due to the 68/95/99 rule of normal distributions. That is

·        68% of the observations are within 1 Standard Deviation, σ, of the mean, μ; that is μ ± σ,

·        95% of the observations are within 2 Standard Deviations, 2σ, of the mean, μ, that is μ ± 2σ,

·        99.7% of the observations are within 3 Standard Deviations, 3σ, of the mean, μ, that is μ ± 3σ.

There are distributions which have only one parameter. The most common of these is the Exponential Distribution. Its Probability Density Function, PDF, can be stated using a single parameter, λ, as

 λ*e^-λ*x for x>0 and x∈ R.

Its Cumulative Distribution Function, CDF, which is the integral of its PDF, is

1- e^-λ*x for x>0 and x∈ R.

Its mean is 1/λ. Its median is 1/λ.   Its variance, which is the square of the Standard Deviation, is 1/λ². Its skewness is 2. It does not meet the 68/95/99 requirement and is therefore not normal.

This can be translated on the x- axis from an origin of (0,0) to an origin of (μ,0). Then it has two parameters, λ and μ. Its PDF can be stated as y= λ*e^-λ*(x-μ) for x>μ and x∈ R. With these two parameters, the mean is μ+1/λ. Its median is μ+1/λ. But its variance remains 1/λ² and its skewness remains 2. And it still is not normal.

The Gaussian Normal Distribution also has two parameters, μ and σ. Its PDF is conventionally stated as

1/(σ√(2π)) e^{-½*((x-μ)/σ)} for x∈ R.

With these two parameters, the Gaussian Normal Distribution’s mean is μ. Its median is μ. Its variance, whose square is the Standard Deviation, is σ. Its skewness is 0.

The Exponential Distribution is often used in analyses where the location, µ, is known to be zero, for example, the average travel distance from the home in any direction. In this case the home is defined as at location = 0. Fitting the observations of the trip length with respect to the home to an Exponential Distribution gives the mean and median trip lengths.

However when the location is not zero, then an Exponential Distribution should not be used, for example, observations of the size of widgets that are being manufactured. In this case, you would like the observations to tell you the location and standard deviation of the observations and see if those meet the location and variance requirements for the size.

To combine the features of the exponential and random distributions, it has been proposed that there be an Exponentially Modified Gaussian distribution, EMG, which includes all three parameters, but the Probability Density Function and the Cumulative Distribution Function become very complex. The PDF is

λ/2 e^(λ/2 (2*μ+λ*σ^2-2x) ) erfc((μ+λ*σ^2-x)/(√2 *σ))

where erfc(x), the complementary error function, is 1-erf(x), or

(2/√π)* ∫_x^∞ e^(-(t^2) )

The Cumulative Distribution Function of the EMG, the EMG’s CDF, the integral of its PDF, is

Φ(x, μ, σ)-1/2 e^(λ/2 (2μ+λ*σ^2-2x) ) erfc((μ+λ* σ^2-x)/(√2* σ))

where Φ(x, μ, σ) is the is the CDF of a Gaussian distribution.

The mean of an EMG is μ+1/λ. Its median is the value of x at which the EMG’s CDF is 50%. The variance is σ²+1/λ². Its skewness is (2/(σ³λ³)) (1+1/(σ²λ²))^-3/2.  By inspection, the skewness takes on a value between 0.00 and 0.31.

The Gaussian Distribution is not the only Normal Distribution. The Logistics Distribution, also known as the Sech Squared, the square of the hyperbolic secant, Distribution, is a normal distribution which has a PDF of

(1/(4*s))* sech² ((x-μ)/(2*s))

where s is a scale parameter which is equal to √3/π *σ. Its CDF is

(1/2)+(1/2)*tanh((x-μ)/(2*s))

Its mean is μ. Its median is μ. Its variance is s²π²/3. Its skewness is zero.

The CDF of the Logistics Distribution is identical to a logit choice of 0 or 1 at a location of μ where the odds of making a choice of 1 are 50%. In this case the value of s is 0.5, 50%.

A doubling of the logit choice CDF is almost identical to the CDF of a translated exponential distribution. A Standard Normal Logit Choice Distribution is proposed to always have a
σ = √3/0.5π= .91 and have a PDF of sech²(√(√3/π) (x-μ)) for x>μ and x∈ R. This means that it has a CDF of tanh(√((√3)/0.5π) (x-μ)) for x>μ and x∈ R. 

The plot of an exponential association, which is the CDF a conventional exponential distribution, and, the CDF of a Standard Normal Logit Choice Distribution, where μ=0, is shown in Figure 1. The correlation between the two equations between x=0 and x=3 is 0.999. 

Figure 1 Exponential Association and  CDF of Standard Normal Logit Distribution

This suggests that the universe is random, and not deterministic, because the universe has a σ  = 0.91.

New Ideas

 

You’re So Vain

Well, you're where you should be all the time
And when you're not, you're with some underworld spy
Or the wife of a close friend, wife of a close friend, and
You're so vain
You probably think this song is about you
You're so vain (so vain)
I bet you think this song is about you
Don't you, don't you, don't you?

I am so vain.

An article by Ethan Siegel asked why scientist are so hostile to new ideas. https://medium.com/starts-with-a-bang/the-good-reasons-scientists-are-so-hostile-to-new-ideas-27aa237c0375. I have the bad feeling that I triggered this article, I am so vain, because I sent an email to Dr. Siegel about whether the universe could be hyperbolic instead of flat, admittedly a new idea.

Dr Siegel asked some questions in his article.

“What is the problem you’re considering that motivated this idea?

How does this idea compare to the prevailing theory when applied to this specific phenomenon?

How does this idea compare to the prevailing theory when applied to the other major successes of the prevailing theory?

And what are some critical tests that you can legitimately perform (with current or near-future technology) to further discern your idea versus the prevailing theory?

As Richard Feynman once put it so eloquently, “The first principle is that you must not fool yourself — and you are the easiest person to fool.””

Challenge accepted. I will try to answer those questions.

What is the problem you’re considering that motivated this idea?

To start at the very beginning, I was born in Providence, Rhode Island. Oh, not that beginning. How about this one? In my senior year at Brown University, I took a class that made me embark on a career in Transportation Engineering. But that was over 50 years ago, so what was I looking at lately? I was trying to address a problem with the relationship between speed and volume, commonly known as congestion. There was an equation that was simple, but wrong, proposed in the 1930s. There was another equation that was much more complex, at least the relationship was not understandable to me. It was, to use Dr. Siegel’s words, “like Johannes Kepler, who threw away his “beautiful” theory of nested spheres and perfect solids before settling on his “ugly” theory of elliptical orbits that fit the data better,” and I was looking for an ugly theory that might fit the data better. The problem was that the data fit in the normal domain very well where volume on a road is less than the capacity on that road but did not fit the observations in over capacity conditions. While most of traffic engineering is concerned with making things better before capacity is reached, i.e. operations/tactics; I work in the branch of traffic engineering that deals, when, as the great philosopher Jimmy Buffet puts it “Shit hits the fan”, traffic volumes exceed capacity, i.e. planning/ strategy.

In trying to improve, or at least understand, the relationship between speed and volume, I wanted to examine the idea that the relationship, which appeared to be a rotated asymmetrical parabola, might instead be a rotated asymmetrical hyperbola.

This was not the first time that I encountered hyperbolas. The firm at which I have worked for almost 25 years, became known for its ability to explain choices.  In fact one of the early employees of that firm, Daniel McFadden, won the Nobel Prize in Economics for his advances on the theory of choices. A normal distribution based on the percentage making a choice is the logistics distribution. One of alternative names for the logistic distribution is the hyperbolic secant squared distribution. The Cumulative Distribution Function, CDF, of this hyperbolic secant squared distribution is also a scaled version of the hyperbolic tangent.

The hyperbolic tangent looks almost like an exponential association, which is itself the CDF of an exponential distribution, but the exponential association is not normal. I had previously proposed that the relationship between reliability, the ability to achieve an on-time performance, and the mean time on a road was an exponential association.

I was also tasked with looking into improving the way that traffic makes a choice among competing routes. This is when I am forced to employ the old chestnut that coincidences are an example of God’s sense of humor. Remember that course that influenced my career choice. It was taught by the late Dr. Stella Dafermos, who proposed the current method used in those route choices. Here is where I get into a second coincidence. Dr. Dafermos was a member of the Applied Math (popularly known by the students as Apple Math) Department at Brown University. My late mother retired from a position as a clerk in what was effectively the mail room of that department (I was not admitted due to nepotism. She started working there after I graduated). I thus felt a special affinity for that Department. In the mid 1980s I was working at a public agency that had an exceptionally good library. I went looking for articles published by those in that Department at Brown that were also in my field. I came across an article written by a post-doctoral student of Dr. Dafermos, Anna Nagurney. In that article, she explained that the impedance function used in route choice, which worked best with Dr. Dafermos’ method, was a fourth power function of volume, i.e. a parabola, but could not explain why. Shortly before this, the Bureau of Public Roads, the predecessor of the Federal Highway Administration before the formation of the US Department of Transportation, USDOT, had proposed an empirical volume diversion equation, known in the profession as the BPR curve, which was a fourth power function of volume. Despite the warnings of Dr. Alan Horowitz, that this equation had only been observed in conditions less than capacity and should not be used in over capacity conditions, the USDOT’s Travel Model Improvement Program, TMIP, recommended the use of a variation of this curve in the traffic models based on Dr. Dafermos’ method in all conditions.

I participated in a study that related the choices with observations of mean travel time and the planning time (the mean time plus reliability, expressed in units of time). We found that route choice was better correlated with planning time, than with mean time alone. So I was faced with choice (which was based on a hyperbola), mean time (and that is simply the inverse of speed, which appeared to be a hyperbola) and reliability, which also seemed to follow a hyperbola. Given that, I did a Google search on hyperbola to find where else it was used and found a link a paper by Dr. Mabkhout which suggests that the shape of the universe was hyperbolic.

So I did not propose that the universe is hyperbolic. I came across the ideas because I was studying relationships in traffic engineering.

How does this idea compare to the prevailing theory when applied to this specific phenomenon?

The idea that the shape of the universe is hyperbolic means that it is open, i.e. at infinity it will also be infinite, but also that it is not flat, i. e. has a curve. It means that it also should begin at a single point, e.g. a Big Bang. This hyperbolic shape explains cosmic inflation in the early universe, as well as the continued expansion of the universe. It is an “ugly” theory because it explains these without resorting to the “beautiful” additions of Dark Energy and Dark Matter. 

How does this idea compare to the prevailing theory when applied to the other major successes of the prevailing theory?

The hyperbolic universe is consistent with the age and size of the observable universe.  It is also consistent with small scales, i.e. the Planck length is consistent with the Planck energy.

A hyperbolic shape also explains the observations of galaxy rotation while a flat universe does not.

And what are some critical tests that you can legitimately perform (with current or near-future technology) to further discern your idea versus the prevailing theory?

I have been involved in using GPS data as traffic observations. The GPS records the latitude and longitude of a point. The distance between observations is not a straight line, because the earth is not flat, it is the Great Circle Distance, a non-Euclidean solution, because the earth only appears locally to be flat but is in fact a sphere. This same phenomena also have a bearing the shape of the universe. If the universe appears to be flat locally, but is in fact hyperbolic, then Pythagoras’ Theorem, which only applies in flat space, should not be used. Instead, the shortest distance between two points in a hyperbolic universe should use Pythagoras’ hyperbolic theorem. Any rotation by n/2π to n/π where n is odd in a flat universe using conventional trigonometry may change a real number to an imaginary number. However a rotation in a hyperbolic universe will never result in changing a real solution to an imaginary solution because the hyperbolic trigonometry used does not change the set of the solutions. Thus the Lorentz Transform, which in a flat Euclidean universe is √(1-(v/c)²⁾, in a hyperbolic universe would be 1+ln(cosh(v/c)±sinh(v/c)) The results are not appreciably different until the ratio of velocity to the speed of light, v/c, exceeds 85%. ( Acrtually 82.2%, but close enough for government work.)

If the universe is hyperbolic, then it would be random, not deterministic. Determinism requires a flat universe where at a Standard Deviation from the mean of zero, there is only one solution. In a hyperbolic universe, the square root of the variance, which is usually thought to be the Standard Deviation from the mean should always be apprimtely 0.91. You can approach zero but can never reach zero, and thus there is not always a single solution.

If the universe is hyperbolic, curved, then gravity might be an apparent force, where two masses are just following a geodesic to a point of lower entropy. It only appears to be a force because in a flat frame of reference, the masses would not appear to move unless they were acted on by a force.

Speaking of entropy, many of the solutions that appear deterministic in my field are in fact maximum entropy solutions.  That is they are the mesostate that has the greatest number of microstates.   This sounds very complex but is actually very simple.  In a game of craps (i.e. a macrostate), the most probable roll (i.e. a mesostate) is a seven because there are more combinations (i.e. microstates) that total seven than any other roll.  The fact that there is entropy means that there is at least one microstate.  Only when there are no microstates will the be no entropy. The fact that we exist (are a microstate) AND there is entropy, must mean that the universe is random.  

Christian Nationalists?

 

Onward Christian Soldiers

Onward Christian soldiers!
Marching as to war,
With the cross of Jesus
Going on before.
Christ, the royal Master,
Leads against the foe;
Forward into battle,
See, His banners go!

Isn’t “Christian Soldiers” an oxymoron?

The Christian Nationalists are at it again. Fact challenged Sarah Huckabee, the current Governor of Arkansas, gave the rebuttal to the State of the Union address. Her father, the former Governor, is an unapologetic Christian Nationalist.  Representatives Marjorie Taylor Greene and Lauren Bobert have claimed that the US is a Christian Nation. Governor Ron DeSantis is taking over the New College in Florida and trying to turn it into what has been characterized as a right-wing Christian school.

Notable in all of their rantings is quoting from the Old Testament, and the Epistles from the New Testament, but noticeably absent are the words of Jesus Christ himself. IMHO, this is not an accident. Christian Nationalists feel that the end times are imminent, and they will all be received into Heaven in the Rapture. I think that Jesus said something about “Knowing not the hour.”

The Christian Nationalist are also intolerant, and worthy successors to the Pharisees. Jesus spoke often against the Pharisees in the Gospels (Matthew 23:1-36; Mark 12:36-40; Luke 11:37-52; Luke  20:45‑47). Jesus spoke against domination.  Soldiers and War are all about domination. Boy are Christian Nationalists going to be disappointed when Jesus finds that they have been taking his name in vain. Maybe they should start calling themselves “Biblicists” or something other than Christians.

Immigrants

 

How Have You Been 

How have you been my darling children,
While I have been away in the west?
Though you are strangers,
I feel that I know you.
By the way that you treat me
 and offer to feed me
 and eagerly ask if I'll stay for a rest. 

How should you treat strangers? 

The United States has citizens and wards.   (Monarchies and other forms of government may have subjects and wards). Before the 14^th Amendment, slaves were wards of the United States and the property of a slave owner.  The apportionment value of a slave was only 3/5 of a person (citizens plus wards).  Before the 19^th Amendment, women did not have the right to vote ( it took 42 years to certify this Amendment). Even after this Amendment, married women were considered to be property of their husbands, a situation agreed to by the state, so arguably women were still wards of the state.  The Equal Rights Amendment, ERA, would have changed this status (but opponents of the ERA had learned from the 19^th Amendment, so while the ERA was  originally passed by Congress in 1972, it had a deadline for ratification by March 1979.)  Children are wards of the state and while there is no formal age in which they are granted full rights as citizens, various rights are granted at various ages 16, 18, 21, etc.  There is considerable disagreement from parents who consider children to be their property whose  interest overrides any authority by the state.  The state considers children to be wards of the parents and the state has a secondary claim which is not excised unless the rights of the parent are terminated.   In no cases are children ever considered to be property of the state, nor does the state acknowledge that they are property of their parents. 

I would be remiss if I did not mention that non viable fetuses  are NOT considered to be wards of the state by the Constitution.  They might be viewed by many as persons, but that is an ethical opinion, not a legal opinion. In the same fashion, Corporations are neither citizens nor wards of the state. 

Prisoners who are incarcerated, and those with mental or physical disabilities can be wards of the state, even after they have passed the normal age of full citizenship.  People who are not born in the United States can be granted citizenship if they have lived in the United States a certain number of years, but  meanwhile they are wards of the state ( e.g. Resident Aliens). 

If individuals are citizens, subjects, or wards of other states, they can be visitors to the United States.  Only in rare cases, such as diplomatic immunity, would a visitor not be treated as a ward, subject of, the United States.  If they have evil intentions, visitors are considered to be intruders to the United States.  Those who are strangers may be treated as visitors or intruders.  Most major religions and cultures have an admonition to treat strangers as visitors.  Those who cross the border into the United States are strangers, but are they visitors or intruders? Gov. DeSantis flying asylum seekers to Martha’s Vineyard was treating them as intruders, but the residents of Martha’s Vineyard treated them as visitors. If we treat strangers only as intruders, then not only are we failing to act as major religions and cultures want us to act, but we are exercising a poor strategy. 

The opening move in any game with three or more players should to first be nice, e.g. assume that the stranger is a visitor not an intruder .  Only in a two-player game would always treating the stranger as an intruder be a winning strategy. The last I looked, there were a lot more than two of us.

Board of Directors II

 

Take The “A” Train

Hurry, get on board, it's comin'
Listen to those rails a-thrumming
All aboard, get on the "A" train
Soon you will be on Sugar Hill in Harlem

Who is getting on a Corporation’s Board?

The Board of Directors of a Corporation should represent ALL of the Corporation, not merely the shareholders of the Corporation. The shareholders own the ASSETS of the Corporation, but they are NOT the Corporation. The Board of Directors should therefore include the representatives of the shareholders, but it should also include others.

A Corporation is a special legal entity chartered by a state (including our federal government which is a group of United States). That Corporation protects the assets of the shareholders from being commingled with the assets of the Corporation, i.e. Limited Liability. If the state provides this protection, then it has an interest in the governance of the Corporation and should arguably have seats on its Board of Directors.

A Corporation is a producer. The factors of production are land, labor, capital, and entrepreneurship but this is an antiquated definition based on agricultural production. I am proposing treating Capital as a single factor (combining land and capital), and am proposing calling entrepreneurship, Intellectual Property, for example trademarks, goodwill, brand names, trade secrets, etc. Intellectual Property is an asset of the Corporation whose asset owners should be, and are already, represented on the Board.

Not represented on the Board, but arguably it should be, is Labor, the employees of the corporation. If a Corporation has gone public (i.e. shares of the assets of the corporation are liquid and can be sold to others of the public) then the employees of the corporation probably do not have a seat on the Board governing the corporation, but arguably Labor should be involved in governing the Corporation. A special class of corporation, Employee Stock Ownership Plan, ESOP, often has employees on its Board, or at least employees always elect members of the Board because those employees are shareholders.

The economic transactions of the Corporation should produce profits. The profits of a Corporation are taxed. Thus the government already should be receiving a share of those profits. The remaining profits are either retained to increase the assets of the corporation (shareholder equity), are distributed to the shareholders in the form of dividends, or distributed to employees in the form of bonuses, at the direction of the Board. When a Corporation is private, then it is more likely to award bonuses, share the profits with all of the employees of the Corporation. When a Corporation is public, then it is less likely to do so.

Having seats on the Board does not compel those directors in those seats to always be in attendance and vote on the actions of the Corporation. The Labor seats and the Government seats can vote present or be absent and, in that case, can give their vote as proxies to other Board members. However having a seat means that those members can, and should, participate in debates on Corporate actions.

Outcomes II

 

Paint it Black

I see a red door
And I want it painted black
No colors anymore
I want them to turn black.

If you only see things in Black and White, you can be missing quite a lot.

Black and White might seem like two  dimensions.  It is not. It is only one dimension,  choice, and black is not making that choice and white is making that choice.  But Black is only the choice of one outcome.  If you look at multiple choices, then the most probable choice is 50% Black and 50% White, in other words gray.  If various shades of gray were allowed, for example 50 as in the popular book series, then only one of those 50 is 100% black and one of those 50 is 100% white.

Microsoft Word allows for three colors, dimensions: Red, Green, and Blue. It allows for 256 graduations within these colors.  This allows for 256³ gradations within these three dimensions, or 16,777,216 colors.  And the most common color is still grey. It does not take many dimensions and many gradations within those dimensions to give rise to many, many possible outcomes.

Outcomes

 

Impossible

But the world is full of zanies and fools
Who don't believe in sensible rules
And won't believe what sensible people say
And because these daft and dewy eyed dopes keep building up impossible hopes
Impossible things are happening every day\

There is a difference between impossible and improbable.

Improbable means that there is only one outcome, or a few outcomes, out of the myriad of outcomes that could happen.  Impossible means that there is NO  outcome.  The problem is when the number of outcomes gets very large, or the popular wisdom rules out certain outcomes as impossible, people confuse impossible with improbable.

Automatic sliding doors that detect the approach of a person who wishes to use that door, did not exist at the time of the production of the original Star Trek TV series.  To appear futuristic, the doors on the StarShip Enterprise appeared to open automatically.  In reality, this was a practical visual special effect where an off-screen stagehand pulled the door open when an actor approached.  Automatic door opening was an improbable outcome in 1966, not an impossible outcome.  Now most retail stores have automatic doors, sliding or folding, and what seemed improbable in 1966, has become so possible that it is common place.  Before 1969, a man walking on the moon might have seemed an impossible outcome, but now we know that it was merely an improbable outcome.

Impossible things can never happen.  But many improbable things are often incorrectly characterized as impossible.