Monday, July 29, 2024

Discontinuity

 

Turn the World Around

We are of the spirit Truly of the spirit Only can the spirit Turn the world around

There is no discontinuity in spirit!

A rebound, bounce, occurs at a physical discontinuity. The motion approaching the discontinuity is rotated and reflected at the discontinuity. When that motion is linear, then the rotated reflection will also be linear because a linear motion is symmetric with respect to the discontinuity. If the motion approaching the discontinuity is parabolic, which is also symmetric with respect to the discontinuity, then its rotated reflection will be parabolic. But a hyperbolic motion approaching a physical discontinuity is NOT reflected as a symmetrical hyperbolic motion in a rotated direction. It is NOT symmetric in real physical dimensions, but it may be  symmetric in the imaginary dimension.

A discontinuity need not be only physical. It can be an apparent, unseen, discontinuity. When approaching this unseen discontinuity, the motion can appear to be linear, but the rotated and reflected motion can appear to be parabolic. Rotation by π/2 is identical to the inverse of a function, and the inverse of a parabolic function, rotation by π/2, is undefined with a change of signs. A change of signs is required at a discontinuity from being observed to unobservable. This most probably indicates that the approaching motion is probably hyperbolic, and the rotated and reflected motion is also hyperbolic. Hyperbolic motion is NOT symmetrical in a real plane with respect to a discontinuity.

If the discontinuity is one of measurement, for example traffic departure volume approaching the capacity of a road, the departure volume may be hyperbolic with a discontinuity, but that is only because you are observing departure volume. Departure volume is equal to arrival volume before the discontinuity. If you could observe arrival volume, which can never be expected to be negative,  its motion might be expected to continue past the discontinuity.

That discontinuity can be expected to be the  separation between domains. For departure volumes it can be volume as uncongested traffic in the laminar domain before the discontinuity, which occurs at the transition speed and as congested traffic in the turbulent domain which occurs below the transition speed. Again linear and parabolic motion are symmetrical with respect to this transition speed, but hyperbolic motion is NOT symmetrical with respect to this real transition speed. The observations of traffic may appear linear before the discontinuity and may appear parabolic after the discontinuity. This has been interpreted by Van Aerde (Van Aerde, 1995) and others that the speed-volume curve is an asymmetrical parabola, which appears almost linear above the transition speed and more parabolic below the transition speed. It is instead proposed that the motion is hyperbolic which only appears linear in the uncongested traffic above the transition speed and appears parabolic in the congested traffic below the transition speed.

It is convenient to place the origin at the discontinuity which creates four domains, quadrants, with respect to the axis of observability ( the y-axis) and the  axis of transition ( the x- axis). These quadrants will be then:

1)     unobservable and laminar;

2)     observable and laminar;

3)     observable and turbulent; and

4)     unobservable and turbulent.

If an object approaches the discontinuity in quadrant 2), then it will rebound, leave the discontinuity, in quadrant 3) remaining in an observable quadrant. If an object approaches the discontinuity in quadrant 2) and passes THROUGH the discontinuity ( travelling across the axis of observability) it will still transition but it will leave in the unobservable quadrant 4).

Why would traffic, or any particle, have hyperbolic motion? If the geodesic is on a flat surface, then the object will not leave that geodesic unless it is acted upon by a force. If that particle is on a hyperbolic surface, the geodesic will be hyperbolic and no force is required for an object to change that hyperbolic, exponential, motion. Thus vehicles in traffic or any particle, in the absence of force, might be expected to have hyperbolic motion if the shape of the surface on which it is traveling is hyperbolic.

This has implications for traffic engineering. A speed-volume curve can be expected to have a discontinuity at its capacity. In travel demand modeling, the Volume Delay Function, VDF, appears to include both time and reliability (expressed as a time). (Azizi & Beagan, 2022) . The VDF arguably has three periods governed by the discontinuity of time and when the reliability budget has been exhausted. This means that from a volume to capacity ratio, v/c, from  0 to 1, the volume will be less than the discontinuity at the capacity and the reliability budget has not been exhausted. From a v/c of 1 to a v/c of 1.216,  the volume has passed through the  discontinuity, but  the reliability time budget has not been exhausted. For volumes above a v/c ratio of 1.216, the volume discontinuity has been passed and the reliability time budget has been exhausted.

This also has implications for the overflow delay at an intersection because delay, which is measurement of time, and it is the inverse of speed. This has traditionally been solved as a rotation of a domain with a random departure queue and a constant flat peak arrival volumes to a deterministic queuing departure. (Hurdle, 1984).  Instead, if a transition at a discontinuity is assumed, there need be no restriction on arrival volumes while preserving random departures and deterministic departures.

This discontinuity is suggested to also appear in other situations where an observable non‑physical  discontinuity exists, such as in fluid dynamics and its separation of regular laminar fluid flow from white water, turbulent flow. If particles travel on geodesics on a hyperbolic surface rather than geodesics on a flat surface, what is perceived as the force of gravity on a flat surface might instead be motion without a force, and there thus need be only three fundamental (the electrical, the strong nuclear, and the weak nuclear) forces with gravity as only an apparent force, similar to centrifugal force resulting from motion on the hyperbolic surface.

The solution to the Pythagoras’ Theorem for a triangle would be different on a curved hyperbolic surface than on a flat Euclidean surface. This new solution would be c=ln(0±2*cosh(√(a2+b2))). This would be virtually indistinguishable from the traditional solution for Pythagoras‘ Theorem when the distances involved are far less than 5/6 times the size of the observable universe. In other words, the universe would be locally flat, but universally hyperbolic. The rth moment of a mean would not be 0 for odd powers and highly negative for even powers, but would be  0 for all powers of r. The Standard Deviation would not require Bessel’s Adjustment of n/(n-1). The relativistic dilation factor, the Lorentz Transform, would be ln(0±2*cosh(√(1-v2/c2))). The observable universe would not be flat but might be only one sheet of a two‑sheeted hyperboloid where its two sheets intersect at the origin, the Big Bang. That the observable universe is hyperbolic was proposed by Mabkhout (Mabkhout, 2012), in which he also proposes that if Einstein’s tensors are solved for a hyperbolic surface, Dark Energy and Dark Matter are not required.

It is suggested that Euler’s Formula is because a complex number is being transformed from cylindrical coordinates to Cartesian coordinates, and that reality is a surface intersecting the origin of that cylindrical volume formed by the three dimensions of space, time and imagination where the coefficient of imagination is zero, r*eix=r*cos(x)+r*i*sin (x). This can be restated narratively as reality is the negative sheet of a two-sheeted hyperboloid, having a negative coefficient, with the other sheet being positive, and has a zero coefficient of the imaginary axis, i.e., r*e=-r +02i, because sin(π)=0 and cos(π)=-1.

It is proposed that the apparent discontinuity is only an illusion which comes from the intersection of tanh(x-μ) and ‑tanh(x-x0-μ), both where x>x0, and x0 is -CAP, capacity, for traffic.  The discontinuity appears at zero, but that is only because the x coordinate axis has been translated.  The actual equations, if the translation is removed, are  tanh(x), x>0 and ‑tanh(x-x0), x>0. If laminar real behavior continues THROUGH the discontinuity, then it follows ‑tanh(x-x0) where x is always positive.

In each case, the amplitude, A, is 2. Given that σ2=½A2,this means the variance, σ2, is 1. This  means that the Standard Deviation, σ, the square root of the variance, is √(2/2), 1, if the imaginary dimension is NOT considered.  But since tanh repeats only in imaginary planes, the standard deviation considering the imaginary axis should be ln(2*cosh(√(2/2)), 1.058

Assuming that the mean is only zero leads to making the mistake that luck must be destiny.  Assuming that the absolute has no error and thus its Standard Deviation, SD, and variance must be zero is confusing the limit of error, Standard Error, SE=SD/√n where n→∞, with deviation. The discontinuity shows that variance is not equal to zero AND the applicable mean is not equal to zero.

References

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

Hurdle, V. F. (1984). Signalized intersection delay models–a primer for the uninitiated. Transportation Research Record, 971(112), 89.

Mabkhout, S. (2012). The infinite distance horizon and the hyperbolic inflation in the hyperbolic universe. Phys. Essays, 25(1), p.112.

Van Aerde, M. (1995). A single regime speed-flow-density relationship for freeways and arterials. Washington D. C.,: Presented at the 74th TRB Annual Meeting,.

 

 

Saturday, July 27, 2024

Pardon Me!

 

Goin’ Home

I'm goin' home, my baby
I'm goin' home, to see my baby
Goin' home, my baby
Gonna see my baby, see my baby fine
Take my baby, take my baby mine
Gonna tell your mama how good that love of ours 

https://www.youtube.com/watch?v=ee20LF-bY6U

And let’s get it correct this time.

In 1968, the Democratic party had a presidential nominating convention in Chicago, the incumbent President, Lyndon Johnson, chose not to run. His Vice President, Hubert Humphrey,  was forced to endure a brutal primary season that divided the party, the delegates, and the country. That incumbent Vice President selected as his running mate Senator Muskie and lost the election to the Republican nominee, Richard Nixon. In 1972, the year of Watergate, the Democratic nominee for president was George McGovern and Hubert Humphrey declined to serve as his running mate. Richard Nixon won a virtually unprecedented percentage of the electoral vote. (Prompting the “Don’t Blame Me I’m from Massachusetts” bumper stickers when Richard “I am not a crook” was proven to be a liar). While Richard Nixon was pardoned by Gerald Ford, his second Vice President, (let’s not forget that Spiro Agnew, Nixon’s first vice President, had to resign because he was caught accepting bribes). You need not accept a pardon if you are innocent. He was saved from being the worst President in modern times by Donald Trump, a convicted felon, an adjudicated rapist, a liar, and a fraudster.

Thus it is fitting that the current incumbent present has chosen not to seek reelection. And that the incumbent Vice President will also be selected as the Democratic nominee of President at a return to a national convention in Chicago. Given that Chicago in 1968 was vital to the election of Richard Nixon, it is fitting that Chicago in 2024 will also be pivotal to the defeat of Donald Trump. Dorothy in the Wizard of Oz was correct. There is NO place like home.

Thursday, July 25, 2024

Nash Equilibriums II

 

Auld Lang Syne

For auld lang syne, my dear
For auld lang syne
We'll take a cup of kindness yet
For auld lang syne.

https://www.youtube.com/watch?v=lxNXtjGY_Us

To Joe Biden, the Richest Man in town!

During yesterday’s address by President Biden from the Oval Office, I could not get the ending to It’s A Wonderful Life out of my head.  I was watching it unfold in real time.  Mathematician John Nash defined a User Optimal solution, of doing the best for yourself as described by Adam Smith, as incomplete.  He did not say, but the System Optimal solution, of doing the best for the common good as described by Karl Marx, is also incomplete.  In order for individual users to act as a system they should follow neither a User Optimal nor a System Optimal solution.  Instead they should follow a solution which he described as the best for the user AND the best for the common good. This is nothing new.  The author T.H. White would describe King Arthur’s  Round Table as “Might For Right, not Might Makes Right”.  Knute  Rockne might describe it as “Taking one for the Team.”  Charles Dicken would describe it as “A far, far better thing.”  The mathematical solution has been given the name as a Nash Equilibrium.  I would like to propose that after yesterday’s speech it be henceforth be described as a Biden Equilibrium.  Let’s raise a toast to Joe Biden, a far greater President than we might have ever realized. Thank you Joe, for Making America Great Again, by looking ahead.  And with Kamala, we are NOT going back!

Monday, July 22, 2024

Pretending

 

California Dreamin’

Stopped into a church
I passed along the way
Well, I got down on my knees
And I pretend to pray
You know the preacher like the cold
He knows I'm gonna stay
California dreamin'
On such a winter's day 

When is pretending passing? 

There is a nasty phrase “passing for white”’.  It refers to someone who is not white, but is intentionally passing for white because those people have privileges.  It generally is done by someone whose features are mostly viewed as “white”.  Passing is done intentionally, and there are consequences upon being discovered.  But at least passing was done intentionally.  Accepting privileges because people have made a mistake as to your identity is something of which I am guilty.  Unintentionally obtaining privileges is probably even worse than intentionally obtaining those same privileges. 

My last name is Beagan, produced BEE- Gan.  During the time in which I was a member of a Massachusetts Republican administration, I let it be pronounced  as RAY-Gan, after the Republican President, even though I secretly despise Ronald Reagan.  During that same time I accepted my colleagues mocking undocumented immigrants, although my maternal grandparents were immigrants from Poland and my paternal great-grandparents were immigrants from Ireland.  My paternal grandfather was even an undocumented illegal alien from Canada pretending to have been born in the United States.  And while my Irish ancestors might have spoken English, my maternal grandparents never learned to speak English.  But my last name is taken to be as American as apple pie. 

I am a straight, cis, male. But my brother, the best man at my wedding and the godfather of my eldest son, is gay. Some of my favorite co-workers are transgender.  The persons I most admire are female.  I am accepted by Christian Nationalists because I am Catholic, but I happily celebrate Rosh Hashanah, and Ramadan.  I was a manager and have been sued by union workers, but my father was a member of the Steel Workers Union and my sympathies are mostly with laborers.  I have lacked the courage to correct those who grant me privileges that I know that they do not extend to others.  By accepting those privileges I am accepting that they can be denied to others.  I have found that pretending by omission is worse than intentionally passing, it certainly is more cowardly.

 

Kamala Harris

 

Into The West

And all will turn To silver glass A light on the water Grey ships pass Into the West

JD Vance is a LOTR fan?

One of the most disturbing images that came out of the Republican National convention was JD Vance in Gandalf’s’ hat blowing smoke rings like a hobbit.  This was an illustration in a story supposedly because JD Vance is a Lord of The Rings, LOTR, fan.  Challenge accepted, fan boy to fan boy.  Sir, I have read LOTR many times and you, sir, are no Gandalf or Frodo.  If you appear to be anyone it is Denethor the fallen Steward of Gondor; Saruman of Many Colors, Grima Wormtongue, or any of the anti-heroes who were deceived into thinking that they alone could fix things.  Time that the voters do a little Scouring of the Shire ourselves.

My favorite characters in the LOTR embody Truth (Treebeard), Justice (Samwise Gamgee) and the American Way of inclusion (the shieldmaiden Eowyn).  In fact if there is an embodiment of a LOTR character in the 2024 presidential campaign my vote is for Kamala Harris.  When the old King of Rohan,  Theoden, is struck down by the Lord of the Nazguls in battle, she heroically defends him in her guise as a simple soldier.  When the Lord of the Nazgul says that “No man can kill him” and is about to choke her, she takes off her helmet and reveals that “she is no man” as she slays him.  https://youtu.be/8Q_HM5EMX6E?si=I4IehTWOs8Xd1Tk8

LFG Kamala.

Joe Biden

 

Hey, Joe

Hey, Joe Where you gonna run to now? Where you gonna run to? Hey Joe, I said Where you gonna run to now? Where you, where you gonna go?

Wherever you go, or whatever you do, thank you Joe!

President Joseph Biden shocked the political world when he walked away from his campaign for re-election. He thus deprived the voters of saying a resounding thank you for all that he has done. Does he deserve to be Grandpa Joe? Absolutely. He has more than earned all that and much, much more. Contrast this behavior with that of January 6th , 2021. Joe Biden willingly gave up re-election to the office of President for the good of the country he loves. God bless you, Joe. To Sir with Love.

And Donald Trump has the nerve to say that the Republican Party should be reimbursed for fraud for all of its campaigning against President Biden. And here I thought given all of Donald Trump’s criminal and civil court cases that  understood all about fraud by now. Maybe DJT should listen instead of sleeping in court?

Monday, July 15, 2024

Vice President

 

Simply The Best

You're simply the best
Better than all the rest
Better than anyone
Anyone I've ever met

How about simply the worst?

Donald Trump has done it again. Of all of the possible candidates of those who might have been selected as his running mate, he has picked the worst one. It is perhaps not surprising. Marco Rubio and Doug Burnham had the possibility of pulling a Mike Pence on Donald Trump and putting the constitution above Trump’s interest. No such worries with J.D. Vance.  He will put personal ambition over constitutional duty every single time. But if I were Donald Trump, I would use taste testers before accepting any food or drink from Senator Vance. He has to be very careful that J.D. Vance’s personal ambition does not overcome any loyalty to his President. The Vice President is only a heartbeat away from the Presidency, which does not speak very well for Donald Trump’s heart, if he has one.