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.

 

 

 

 

 

 

 

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(−∞,∞)

·        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.

 

 

 

 

 

 

 

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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.

Serenity

 

Mblem

Lord give mе serenity to accept the things I can't change
And the courage to change the things I can, yeah
And if in life on my journey if I should stumble and fall
Make me wiser than the man I already am, yeah, yeah

This I pray.

Yet another threesome. This time it is serenity, courage, and wisdom.  I guess it really is true that good things come in threes. 

Among the things I can’t change are uncertainty, randomness.  I might not like that things can be random and uncertain, but I need the serenity, courage, and wisdom to accept that uncertainty. 

Stories

 

Camelot (reprise)

Ask every person if he's heard the story,
And tell it strong and clear if he has not,
That once there was a fleeting wisp of glory
Called Camelot.
Camelot! Camelot!
Now say it out with pride and joy!

What is important is the story!

My heroes are Shakespeare, Shaw, Bob Dylan, Cole Porter, Joni Micthell, Picasso, Frank Capra, among many, many others. I wish that I could tell stories like they do with words, songs, or pictures. But that is not how I roll. My stories use numbers and variables, mathematics, but they are stories nonetheless. And I would like to believe that those stories are important and add to the glory too!

Stories are important because they require imagination. And while we live in a real world where the coefficient of imagination is zero, that does NOT mean that imagination does not exist. If it does exist and has to be considered, then the implications are tremendous.

Imagination might be why randomness, entropy, gravity, etc. exist in the first place. We ignore imagination at our peril. It is convenient to pretend that the square of any number, x², should be solved by simply taking its square root, but the square root is only real if imagination is NOT considered. Take the variance, σ², for example. It is often assumed that the Standard Deviation, SD, is the square root of the variance. But this is strictly only true for flat, Euclidean, surfaces. In the surface is flat, then the variance is σ²= (SD)² +0²*i does indeed have the solution, SD=√σ², but is only because on a flat surface cos(σ)=cos(SD)*cos(0). On a hyperbolic surface it should be cosh(σ)=cosh(SD)*cosh(0). This does NOT become a single value but two values, σ=ln(cosh(SD)±sinh(SD)). When sinh(SD) is very small, that term  can be ignored and then the mid point of this range approximates SD = √σ². It is appropriate to think of the term cosh(SD) as the location, µ, parameter in a random equation where the range parameter, the standard devation , σ, sinh(SD) is the other parameter.

The problem is that no number is exact. There is always a Standard Error term, SE. The definition of SE is SD/√n, where n is the size of the sample population. If there is growth, a value that is outside those values range, then a Growth Factor, GF, has to be applied to the values to include that growth, GF*(x-SE)<y<GF*(x+SE). X is the average, mean, value of a series of numbers, corresponding to the sample population, (∑x_i)/n. If n, the sample size of the population has not changed, then the Growth Factor should only be applied to every value of x_i.  If the Growth Factor is also applied to SE, and if the sample population has not changed, then it has to be applied  to the SD. This new SD may now have become so large that the uncertainty of the range can no longer be ignored. Otherwise, to keep the Standard Error the same, the size of the population has to be decreased.

Sounds like the “rINO, republicans IN Name Only” response to growth, decreasing the size of the population? The problem is that if the size of the population is decreased, then the location term, µ, has to decreased by even more because it is divided by n not √n. So you get into a mathematical death spiral. In order to accommodate growth, without increasing error, you have to constantly decrease the size of the population.

Mathematics is a harsh mistress, and imagination is ignored at all of our peril. That is my story, and I am sticking to it!

 

 

The Middle II

 

You Do Something to Me

You do something to me
Something that simply mystifies me
Tell me, why should it be
You have the power to hypnotize me?

We are in the middle of a battle between “Do Something” versus “Destroy Everything”

Progressive Democrats such as Alexandria Ocasio-Cortez, Bernie Sanders, Elizabeth Warren  seem to be believers in “Do Something”.  The Freedom Caucus and their allies such as Matt Gaetz, Majorie Taylor Greene and Ted Cruz seem to be believers in “Destroy Everything”.  It is up to the moderate and conservative Democrats, and the remaining liberal and moderate Republicans (who are believers in our constitutional republic and are not “republicans In Name Only” ) to ensure that some things that have been done will be destroyed, but not everything that has been done will be destroyed. 

Neither of the two opposing sides should win.  It is that middle that should win because the middle is most of us.

Certainty IV

 

Michael Row the Boat Ashore

Jordan's river is chilly and cold, hallelujah
Chills the body but not the soul, hallelujah

Michael row the boat ashore, hallelujah
Michael row the boat ashore, hallelujah
Sister help to trim the sail, hallelujah
Sister help to trim the sail, hallelujah

What position in Heaven is the Archangel Michael?

In a previous blog post I claimed that the Top is one of the Top Three. IWAS WRONG!!!!. While it is easy to compute dominance, it is the winner, first place, it is harder to calculate certainty, the best. The problem is that in any contest it could be determined by pure luck and the best could finish second. The Trinitarian in me made the mistake of thinking that the Top Three in Heaven were obviously the Father, Son, and Holy Ghost Spirit, which makes the Archangel Michael, the Head of the Heavenly Host, number four. The problem is that while it is true that they are the Top Thee, a contest can not include the absolute. So what is the certainty of the best being in the Top X finishers, excluding the absolute.

This is a statistics problem. If points are awarded to finishers based on the inverse of the number of places receiving points e.g., if 10 places receive points, then 10 points for a first-place finish, 9 points for second place, etc. What I forgot to  consider previously is that any finisher can only finish, get points, once, e.g., you can’t get points for first AND second place ....unless you are the absolute and the Pauli Exclusion principle doesn’t apply. Thus you know the total points which can be awarded (the sum of the number of places getting points) divided into the number of points awarded for a mean-finish. The certainty is 100% minus this number.

While it is true that if only the first place is awarded points, that first space is 100% dominant, but the certainty of its being the best is only 50%, no better than luck. The certainty increases if points are awarded to the first two places, such that the certainty that the best is in that Top 2 has increased to 67%. It is 75% certain that the best is in the Top 3. The number of places that receive points increases the certainty that the best is among those receiving points, but that increase is at a decreasing rate as shown below for the top 20 places.

There is of course a limit, and that limit is 100% certainty that the best is among the places receiving points, as the number of places being awarded points approaches infinity. However there always has to be more contestants in the contest than those receiving points so absolute certainty can never be achieved. So while the Top Three in Heaven may have certainty because they are an absolute, and Michael may be the best of the angels, but the certainty that he finishes in the money, gets a medal, etc. in any contest of the angels has no more than 75% certainty as long as 4 or more angels compete.