Classifications

 

Do I Love You Because You’re Beautiful

Do I want you because you're wonderful,
Or are you wonderful because I want you?
Are you the sweet invention of a lover's dream
Or are you really as beautiful as you seem?

Why can’t it be both? Wonderful AND Beautiful.

There is a basic tendency to classify things. If you are to classify anything there have to be at least two classifications: odd and even; day and night; yin and yang; side one and side two; etc. There can be more than two classifications, but not fewer.  But how should an absolute be classifed? There was an advertising campaign for Miller Lite Beer in the 1980s: Less Filling, Tastes Great. The advertisements typically took place in the form of an argument by each side as to which side the beer belonged. The punch line is that it belonged to both. It was Less Filling AND Tastes Great.

The concept of being both things is hardly unique to Miller Lite Beer. Certs Mints had a similar advertising campaign. Is it a candy mint? Is it a breath mint? It’s two, two, two mints in one. 

In mathematics, infinity is an absolute. An integer might be classified as odd or even. So is infinity odd or even? This is not a trivial question. If an integer is even, then at half of that integer there is an equal amount above and below that integer. And half of an even integer is still an integer. By contrast, if an integer is odd, then at half of that integer there is also an equal amount above and below that value.  But half of an odd integer is no longer an integer. If odd and even behave differently, how is infinity to be classified? The answer is that it is classified as BOTH. For any odd or even integer, the median and the mean need not be the same. But at infinity they are defined as being the same, and thus infinity can be defined as both odd and even.

This insight is not unique to advertising and mathematics. St. Paul’s letter to the Galatians, 3:18, according to the King James version is translated as

There is neither Jew nor Greek, there is neither bond nor free, there is neither male nor female: for ye are all one in Christ Jesus.

That does not mean that there are no differences. People can have a religion or an ethnicity. People can be free or slave. People can be male or female. But as far as the absolute is concerned, these classifications do not matter and they certainly don’t apply to the absolute. The absolute is both.

Power And Glory

 

Sympathy for The Devil

Please allow me to introduce myself
I'm a man of wealth and taste
I've been around for a long, long year
Stole many a man's soul and faith

Do you really need that warning about wealth and taste?

The absolute’s Power and Glory is an output of the Truth and Justice of the absolute. It is proper to worship an absolute. But not all of the Power and Glory of an individual is because of their Truth and Justice. An individual can lie, cheat, and steal their way to Power and Glory. Don't forget that one of the titles of the Devil is the Prince Of Lies.

Some basic wisdoms are “All that glitters is not gold” and “Don’t judge a book by its cover.” IOW an individual’s wealth and taste is NOT any indication that their Power and Glory is because they are an absolute.

Not all heroes are famous, and not all famous are heroes. Just because someone has wealth and taste, e.g. is famous, it doesn’t mean that they are a hero. Remember that before you worship, or follow, anyone.

Truth and Justice

 

The Power

Like the crack of the whip, I Snap attack
Front to back, in this thing called rap
Dinging like a cymbal, rhyme devil on the heavenly level
Bang the bass, turn up the treble
Radical mind, day and night all the time
7:14, wise divine
Maniac brainiac, winning the game
I'm the lyrical Jesse James

I’ve got the Power!
I’ve got the Power!

Power is an output, NOT an input.

Power and glory are the outputs. Truth and justice are the inputs. Absolute power is correctly associated with the absolute. So if we have absolute power, then we can be absolute too? Uh no, it doesn’t work that way.

If there is no truth and justice, then there is no power and glory. Lord Acton’s adage is that “Power tends to corrupt,  and absolute power corrupts absolutely”.  I don’t think that there is a similar saying that truth tends to corrupt, and absolute truth corrupts absolutely. So the goal should be to create truth and justice, and then hope to get power and glory. Don’t ever accept power and glory without truth and justice.

The Long Term

 

A Hazy Shade of Winter

Time, time, time,
See what's become of me
While I looked around
for my possibilities
I was so hard to please
Don't look around
The leaves are brown
And the sky is a hazy shade of winter 

What seems like a good decision at the time, might not be so good in the long term.

The list of movies that did NOT get an Academy Award at the time of their release is most impressive.  https://www.usatoday.com/story/entertainment/movies/2024/03/07/best-movies-that-never-won-an-oscar/72836637007/.  But in the long run they have become some of the most important and beloved of films. So it often goes. 

What seems like a good decision in the short term, is not always such a good decision in the long term.  Remember that on election day.  You are not just voting for the short term.  You are voting for the long term.  There are no do-overs or mulligans.  The old saying is  "Marry in haste, and repent at leisure".

Pi

Can't Help Myself

Sugar pie, honey bunch
You know that I love you
I can't help myself
I love you and nobody else.

But what about Sugar Pi?

I love pie, but I love Pi, π,  even more. A wave has a repeating form that can be described by its amplitude, its wavelength, and its phase and that typically involves Pi.

Sinusoidal wave functions, such as the sine and cosine, have several distinct characteristics: 

• They are periodic functions with a period of 2 π.
•  The domain of each function is (-∞, ∞) and the range is [-1,1]
•  The graph of y=sin(x)=sin(-x) is symmetric about the origin because it is an odd function.
• The graph of y=cos(x)=-cos(-x) is symmetric about the y-axis because it is an even function.
• A cosine is a sine that has been phase shifted by π/2, one quarter of its period.

It is thus not surprising that hyperbolic sinusoidal wave functions, such as the hyperbolic sine, sinh, and the hyperbolic cosine, cosh, have similar characteristics.

• They are periodic functions with a period of 2 π i.
•  The domain of the hyperbolic sine, sinh, is (-∞, ∞) and the range is [-1,1] and while the domain of the hyperbolic cosine, cosh, is also (-∞, ∞) its range is [1, ∞].
• The graph of y=cosh(x)=cosh(-x) is symmetric about the origin because it is an odd function.
• The graph of y=sinh(x)=-sinh(-x) is symmetric about the y-axis because it is an even function.
• A hyperboic cosine is a hyperbolic sine that has been phase shifted by π/2 i, one quarter of its period.

A hyperbola is a function that does not change signs because of the sign of its input. A negative or a  positive input always yields a positive output OR a negative or a positive input always yields a negative output. An ellipse can change signs. A negative input can yield a negative or a positive output AND a positive input can yield a negative or a positive output. A hyperbola can change signs, if it is rotated by π radians. (which is equivalent to a rotation of π/2 radians AND a reflection). Euler's Formula is true in an elliptical domain in all cases and in a hyperbolic domain if the phase is less than π/2. It is not true for a hyperbolic domain with a phase greater than π/2. For example, for a complex number that is x+0i, if the phase, rotation of the imaginary axis is π, then Euler’s formula is e^ix which should be positive but is negative, cos(π) + sin(π)*i= -1. It should be restated in a hyperbolic domain as e^ix=cosh(x)‑sinh(x)i and then with a rotation of π when traversing domains, it would be e^ix=1 in both domains.

It is proposed that the entire universe consists of a two sheeted hyperboloid, while in one sheet it is true that e^ix=cos(x)+sin(x)i due to the elliptical identity cos²+sin²=1 while in the other sheet the hyperbolic identity cosh²- sinh²=1 applies. The observable universe is one sheet of this hyperboloid, the sheet in which the hyperbolic identity applies. The two sheets of the hyperboloid connect at the origin. Thus a 2-D Minkowski space becomes a 3-D two-sheeted hyperboloid when an imaginary axis is added to the two axes of space and time, and a hyperbolic surface is rotated by 2π on this imaginary axis. Then the two cones, one of which is inverted, become two connecting sheets of a two sheeted hyperboloid. A hyperbolic surface which passes through the imaginary axis at π and the origin would satisfy e^ix=cos(x)+sin(x)i in one sheet and e^ix=cosh(x)-sinh(x)i in the other sheet. For any equation to be valid in both sheets, that equation would require a rotation by π when it passes between the two sheets at the origin. And that rotation is important and why I love Pi.

 

 

 

 

 

 

 

$$

 

(−∞,∞)

·        The graph of y=cosh(x

$$

·        The graph of y=sinh (x

·        A hyperbolic cosine is a hyperbolic sine that has been phase shifted by πi/2, one quarter of its period.

 

A hyperbola is a function that does not change signs. A negative or positive input always yields a positive outputs OR a negative or positive input  always yields a negative output. An ellipse  can change signs. A negative input can yield negative or positive output and a positive input can yields a negative or positive output. A hyperbola can change signs, if it is rotated by π radians. (which is equivalent to a rotation of π/2 radians AND a reflection). Eulers’s formula is true in an elliptical domain in all cases and in a hyperbolic dominance if the phase is less than π/2. It is not true for a hyperbolic domain with a phase greater than π/2. For example, for a complex number that is x+0i, if the phase, rotation of the imaginary axis is π, then Euler’s formula is e^ix which should be positive but is cos(π) + sin(π)*i= -1. It should be restated in a hyperbolic domain as e^xit=cosh(x)‑sinh(x)i=and with a rotation of π it would be e^ix=1 in either domain.

 

It is proposed that the unobservable universe consists of a tow sheeted hyperboloid, when in one sheet it is true that e^ix=cos(x)+sin(x)i due to the elliptical identity cos²+sin²=1 while in the other sheet the hyperbolic identity cosh²-sinh²=1 applies. Th observable universe is one sheet of this hyperboloid,  the sheet where the hyperbolic identity applies. The two sheets of the hyperboloid connect at the origin. Thus a 2-D Minkowski space become a 3-D hyperboloid when an imaginary axis is added to the two axis of space and time, and  hyperbolic surface is rotated by 2π on this imaginary axis, and the two cones, one of which is inverted, becomes two connecting sheets of a two sheeted hyperboloid. Then  a hyperbolic surface which passes through the imaginary axis at π, and the origin would satisfy e^ix=cos(x)+sin(x)i in one sheet and e^ix=cosh(x)-sinh(x)i in the other sheet. For any equation to be valid in both sheets, that equation would require a rotation by π when it passes between the two sheets at the origin. And that rotation is important and why I love Pi.

 

 

 

 

 

 

 

$$

 

Nash Equilbriums

 

I Am the Very Model of a Modern Major-General

I know our mythic history, King Arthur's and Sir Caradoc's,
I answer hard acrostics, I've a pretty taste for paradox,
I quote in elegiacs all the crimes of Heliogabalus,
In conics I can floor peculiarities parabolous.
I can tell undoubted Raphaels from Gerard Dows and Zoffanies,
I know the croaking chorus from the Frogs of Aristophanes,
Then I can hum a fugue of which I've heard the music's din afore,
And whistle all the airs from that infernal nonsense Pinafore.

Sometime a Paradox is hiding a deeper truth.

Governments in the US are individuals acting as if they were a system. Individuals CAN function as a system. The “wave” at many sports events is an example of individuals acting together as if they were a system.

Individuals might function as if they are a system, but they are NOT a system. If they were truly individuals, they should adopt  User Optimal solutions. If they were a system, they should adopt a System Optimal solution. Since they are not a system, it  is not surprising that the sum of their User Optimal solutions for a system is more than the System Optimal solution. But to act as if they are a system, the users can adopt a Nash Equilibrium, often called a User Equilibrium, because in that solution no individual can chose a solution that is better for themselves. It is a Nash Equilibrium that is observed. It can be summed, and it will be found to be greater than the System Optimal solution. However the Nash Equilibrium is unique to each system. If you change the system, then you change the Nash/User Equilibrium.

This is the basis for the Braess Paradox. https://en.wikipedia.org/wiki/Braess’s_paradox. When examining a traffic network, “Dietrich Braess, a mathematician at Ruhr University, Germany, noticed the flow in a road network could be impeded by adding a new road, when he was working on traffic modelling. His idea was that if each driver is making the optimal self-interested decision as to which route is quickest, a shortcut could be chosen too often for drivers to have the shortest travel times possible. More formally, the idea behind Braes' discovery is that the Nash equilibrium may not equate with the best overall flow through a network.”  Thus it can be argued that Braess’s Paradox is because people were confusing a Nash/User Equilibrium with a System Optimal.

If you’re unfamiliar with a Nash Equilibrium, Adam Smith was correct that everyone should do what is best for themselves, and while this is a User Optimal, it is incomplete.  Karl Marx was correct that everyone should do what is best for the common good and while that is a System Optimal, it is also incomplete. John Nash appears to be correct AND complete in that everyone should do what is best for them AND the common good, which is a Nash Equilibrium.    https://www.youtube.com/watch?v=vCyZvfRHkC4

Dafermos and Saprrow (Dafermos & Sparrow, 1969) developed what came to be the basis for what is known as an User (Nash) Equilibrium in Travel Demand Modeling. It solved the problem that the impedance on a link of a network depends on how it reponds to the volume on that link, but how the link responds to volume is not known, and thus an iterative algorithm is required to solve for the volume. Nagurney, (Nagurney, 1984), while a post doctoral student of  Dafermos, showed that while the exact response may not be known, the most efficient response could be found, and that a fourth power function, such as the Bureau of Public Roads, BPR, curve, is an efficient solution.  Azizi and Beagan, (Azizi & Beagan, 2022) showed that a discontinuous function, where the discontinuity happens at the link capacity, is an even more efficient solution than the BPR curve. Arguably the most efficient response is the correct response.

It is hardly surprising that the Nash/User Equilibrium is specific to each system. And therefore  if you change the system, you change the Nash Equilibrium. However the Nash Equilibrium is also only possible if some members of the System forego their own User Optimal and all Users block others from pursing their own User Optimal. The farther each User Optimal is from that Nash Equilibrium, the more likely it is for some users to leave that system and seek their own User Equilibrium.

Beagan (Beagan, 2016) appeared to shown that the equation which is used in User Equilibrium is itself a function of the Standard Deviation, σ aka SD, of the system, which is the reliability time that is used is a function of the 95^th percentile time, the mean time of a normal system plus two Standard Deviations.  Reducing the error of a system depends on increasing the number of individuals, n, in the system, i.e. Standard Error = SD/√n. Losing individuals in a system by straying too far from their UO solution can led to increasing error, or even to competing Systems, despite what Braaess's Paradox seems to suggest.

It is suggested that the Nash/User Equilibrium should not be reduced so far from the UO solution that individuals are tempted to leave the system. Thus making the sum of NE, UE, closer to a SO solution may not be desirable if it leads to fewer users in the system. What is instead desired is not NO government, and only UO solutions, but a government which is for ALL individuals. To parrot Lincoln "government of the people, by the people, and for the people". But fooling people, by convincing them to leave the system, is an example Lincoln's “You can fool all of the people some of the time, and some of the people all of the time, but you can’t fool all of the people all of the time.” Vote for the People, not just as a Republican or a Democrat, when you are choosing a government.

Works Cited

Azizi, L., & Beagan, D. (2022, January). Inclusion of Reliability in the Volume Delay Function. Poster Presented at Annual TRB Meeting.

Beagan, D. (2016). Including Reliability in VDF Curves. Prestentation to the 6th TRB Conference on Innovations in Transportation Modeling. Denver, Colorado.

Dafermos, S., & Sparrow, F. (1969). The Traffic Assignment Problem for a General Network. Journal of Research of the National Bureau of Standards, 73B, 91-118.

Nagurney, A. B. (1984). Comparative Tests of Muitimodal Equilibium. Transportation Research Part B: Methodological, 18B(No 6), 469-486.

 

 

 

Waves

 

Anchors Aweigh

Anchors aweigh, my boys, anchors aweigh
Farewell to foreign shores, we sail at break of day, of day
Through our last night ashore, drink to the foam
Until we meet again, here's wishing you a happy voyage home

Let’s hear it for those who sail the waves. Not only are those waves alive, but those on those waves also keep us alive!

The dictionary definition of flatline is

“to die or be so near death that the display of one's vital signs on medical monitoring equipment shows a flat line rather than peaks and troughs. to remain at a continuous low level.”

It does not sound like being flat is such a good thing. Thus it is surprising that much of mathematics is based on the principle that a surface is flat, Euclidean. On the contrary it sounds like life should be a wave, not a flat line.

A single wave has peaks and throughs. Except on the wave itself, it is empty in the time between those peaks and troughs. An absolute should be a wave to be alive, but it should have no peaks or throughs. Rather than a single value, it should be any value within its amplitude, and thus it needs to fill in those empty spaces. An absolute should have an amplitude, but it should be the sum of all waves such that there are no empty spaces.

So why does this matter? Because every wave has a variance, which is twice the square of its amplitude. If an absolute is a wave, then by having an amplitude it by definition has a variance, which is the square of Standard Deviation. It is not only possible, but by its very definition, that an absolute, infinity, has no error. Standard Error, SE, is the Standard Deviation, SD, divided by the square root of its sample size. The limit of error as the sample size approaches infinity is zero. This is because the limit of SD/√n as n approaches ∞ is zero. It is NOT because its SD is zero. If that were true, then the SE for all values of n would be zero. Saying that the absolute is a wave and has zero Error does NOT mean that the absolute has zero Deviation.