Tuesday, December 5, 2023

Parabolas

 

She’s Like A Rainbow

She's like a rainbow Coming, colours in the air Oh, everywhere She comes in colours

Rainbows are pretty, but they are illusions.

To paraphrase Lady MacBeth, out damned parabola, out I say.

There are two forms of growth: geometric, also known as parabolic, and exponential. In geometric growth the assumption is that the rate of increase is continuous AND constant. This means that in the presence of a limit to growth, that eventually the growth will exceed that limit and the growth will become imaginary. By contrast exponential growth is continuous but NOT constant. It will approach the limit but will not exceed it and will never become imaginary.

A parabola is the midpoint between an ellipse (a circle is a perfect ellipse) that has an eccentricity less than 1, and a hyperbola that has an eccentricity greater than 1. A parabola is a formula/curve with an eccentricity exactly equal to one. However I would suggest that it is metastable to be exact, and the universe is thus proabaly stably hyperbolic as proposed by Mabkhout (Mabkhout, 2012). This may mean that parabolas, Gravity’s Rainbow, and gravity itself, are illusions. They are useful illusions, but illusions, nonetheless.

Just as there are those individuals that believe in a Flat Earth, because Pythagoras’ Theorem which only applies on a flat surface, gives good results, compared to the correct spherical formula. (Actually those individuals probably do not even know that there is a Pythagoras’ Theorem!). The reason it gives good results is that when the spherical surface is very large, compared to the values in Pythagoras’ formula, then there is an imperceptible difference between a flat and spherical surface. However commercial airplane pilots will tell you that you that the Great Circle Distance is much more useful than Pythagoras’ formula over distances between continents.

Similarly Newton proposed a Law of Gravity. (An aside. It is only called a Law because of the History of Science. Today it would be called a Theory). Newton was proven incomplete by Einstein. Newton’s Law assumes a constant mass, and Einstein’s Theory shows that the relative mass of an object increases as that object approaches the speed of light.  But when an object is moving at very low speeds compared to the speed of light, there is virtually no difference between Newton’s Law and Einstein’s Theory. But Einstein’s Theory was still applied in a flat universe and therein lies the rub. In a non-flat universe, the increase in relative mass might not follow Lorentz’s adjustment, but the difference might be imperceptible at the speeds commonly encountered.

However if the universe has a hyperbolic shape, then gravity itself might be an illusion where two or more objects approach a common center on that curved, hyperbolic, surface. In this case, just as Einstein’s Theory is more correct than Newton’s Law, Newton’s Law continues to be used because it is simpler to apply, even if Gravity is an illusion, Newton’s Law and Einstein’s Theory might be “wrong” but in the words of George Box, they might be useful.

If the universe has an absolute, then the distribution of objects in that universe can be expected to follow a distribution within that absolute. For an arbitrary absolute, for example π, objects within that absolute should be expected to follow a random distribution. One such normal random distribution is the logistics, also known as the hyperbolic secant squared, distribution, whose Probability Density Function, PDF, is (1/(4*s))*sech2((x‑µ)/(2*s)), where s and µ are parameters of that distribution. Its Cumulative Distribution Function, CDF, is also a hyperbolic trigonometric function, ½*tanh((x‑µ)/(2*s))+½. If objects are uniformly distributed, then the mean, and the median, of that normal logistics distribution, µ, is at half of the absolute, or in this example, π/2. When x is at that mean, median, there should be 50% of all objects, a PDF of 0.5, which requires that in this example s=0.5. Its variance, σ2, is s2π2/3 or 0.822.

Another normal distribution is the Gaussian distribution. Its Probability Density Function, PDF, is 1/(σ*√(2*π))*e(-0.5*((x-µ)/σ)^2), where its parameters are σ and µ. Its Cumulative Distribution Function, CDF, is 0.5*(1+ERF((x-µ)/(σ*√(2)))), where ERF is the standard error function.  As shown in the figure below when the PDF of the normal logistics function at the mean, median is 0.5, the PDF of the Gaussian is 0.44.  If the PDF of the Gaussian is 0.5 at the mean, median, then the s parameter of the logistics distribution must instead be 0.44 and its PDF is then 0.57 instead of 0.5.  Neither the logistics nor the Gaussian distribution are zero at value of x of zero or the absolute. The Gaussian distribution on a flat surface has almost the same values as the Gaussian distribution on a hyperbolic surface. 


Also shown in that figure is a parabola with a coefficient of 1 which has a value of 0 at an x of 0.  This does NOT have a value of 50% at the median, mean, µ.  In fact it becomes very large near the absolute.  A parabola can be made to take on a value of 50% at the median.  The reflection of the adjusted parabola, which creates a discontinuity, can be used at the mean, median. This will result in a value of zero at the absolute.  However a parabola and its reflection is not as simple as a logistics distribution. A logistics distribution is also smooth and does not create a discontinuity at the mean, median. However using a simple parabola can highlight how the integral of the PDF, the CDF can be viewed.


The CDF, integral or area under the curve, of the PDF of a simple parabola is identical to the formula for the area of a triangle.   However, as shown, it has a height of 1.57, π/2,  rather than a height of 1 at the absolute, if the absolute is assumed in this example to be π, which should be the CDF and is approximately the value of the CDF of the normal logistics distribution on all surfaces and the normal Gaussian distribution on a flat or hyperbolic surface. However if the CDF of a parabola, the formula for the area of a triangle, is translatedon the y axis, reduced, by a value of .29, it becomes almost identical to the CDF of the normal distributions near the mean, median.  It continues to approximate the CDF of the normal distributions up to a distance of π/6 from the mean, median, of π/2.  At this distance the slope changes to become approximately half of the previous slope and this continues for a distance of π/6 from the last change.  At this point, the slope again changes to become approximately half of the previous slope. This continues to an x of the absolute and an x of zero, the absence of the absolute. 

The fact that the formula for the hypotenuse of a triangle on a flat surface must be adjusted suggests that the correct formula should not be on a flat, surface.  The logistic distribution is consistent with, and uses, hyperbolic trigonometric functions.  It has parameters of s=0.5 and µ=the absolute divided by 2.  It is observed that an s of 0.5 is also consistent with the mean of a single choice of that absolute.  It is also suggests that the Gaussian distribution was an attempt to derive a normal distribution on a flat surface, when it should have been derived for a hyperbolic surface.  It is also observed that the 68/95/99 rule of a Gaussian distribution on a flat surface corresponds to a 52/85/100 rule for a logistics distribution on the  hyperbolic surface of the universe ( 52% percent of the values fall within ±1/3 of the mean, median; 85% of the values fall within ± 2/3 of the mean,median; and 100% of the values fall within ± 3/3 of the mean, median.  Rather than an arbitrary variance, when the choice parameter, s, is equal to 0.5, the variance, σ2, has a fixed value of 0.822467 and the square root of the variance, also known as the Standard Deviation, has a fixed value of 0.9069.  

The 68/95/99 rule is for the multiples of the standard deviation of a Guassian distribution on a flat surface.  For a logistics distribution on a hyperbolic surface, when 100% of the values fit within the range of the absolute, 

  • 70.6% are within the mean, median, ±  σ; 
  • 94.3% are within the mean, median, ±2σ; and 
  • 99.0% are within the mean, median, ±3σ.

It is suggested that the surface of the universe is hyperbolic.  It is suggested that the distribution of objects follows a logistics distribution.  It is suggested the parameters s, variance, and standard deviation must all take on nonzero values in reality.  

At the mean, median, the dominance is 100% and it remains this value for any outcome.  However 

  • at the mean, median, ± 0/3 of the mean, median, there is  25% certainty; 
  • at the mean, median, ± 1/3 of the mean, median, there is  52% certainty; 
  • at the mean, median, ± 2/3 of the mean, median, there is  85% certainty; and 
  • at the mean, median, ± 3/3 of the mean, median, there is 100% certainty.  

If the distances commonly encountered are less than 1/3 of the range of the absolute, then there is no appreciable difference between the results for a flat or hyperbolic surface.

Parabolas may be an illusion.  We may aspire to live on circles.  But we appear to live on a hyperbola.

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

 

 


 



Sunday, December 3, 2023

Dimensions

Medley: Aquarius/Let the Sunshine In (The Flesh Failures)

Now let me tell you one thing (let the sunshine) I want you to sing along with The 5th Dimension (let the sunshine in) Hey, open up your heart (the sunshine in) Come on! (let the sunshine) And let it shine every day (let the sunshine in)

So how many dimensions are there?

The song above was a Grammy Record of the Year for the Fifth Dimension in 1970 .  Are there five dimensions? There are the three dimensions of space, x, y, z also known as length, width and depth. There is the dimension of time.  If we add the dimension of the imagination then there are five dimensions. If reality is defined as space and time and zero imagination, that zero is only the coefficient of the axis, dimension, of  imagination and there are thus five dimensions of reality.  So the name of the singing group is more than appropriate.  Let the Sunshine In!


Saturday, December 2, 2023

Show Your Work

 

Show Me

Sing me no song! Read me no rhyme!
Don't waste my time, Show me!
Don't talk of June, Don't talk of fall!
Don't talk at all! Show me!

Show your work!

The formulae shown in red in this post have been corrected.

When I was in college, I was famous for submitting tests with minus the correct answer or with some other simple mistake.  On numerous tests, I was saved by the fact that my instructors took pity on me and gave me credit for showing my work.  Ironically, I now find myself with what appears to be the correct answer for a hypotenuse on a hyperbolic surface, but because I did not save my work, I can not prove it.

On a flat surface the hypotenuse/radius of a triangle is r=√(a2+b2).  I would like to say that on a hyperbolic surface this is r=ln(0±2*cosh(√(a2+b2))), which I arrived as a solution and it seems to work, but I did not save my work.  Thus because I did not save my work, I am unable to prove that on a hyperbolic surface it is r=ln(0± 2*cosh(√(a2+b2))), instead of r=√(a2+b2). Being unable to show my work means that I am unable to prove this equation and unable to show that:

The Lorentz transform, which is typically given as√(1-(v/c)2), where in this case v is the velocity of an object and c is the speed of light, should instead be r=ln(0±2*cosh(√(1-(v/c)2))).

In a curved universe, gravity should be an apparent force and should NOT be combined with the three intrinsic (electric, weak nuclear, and strong nuclear) forces in a Unified Field Theory.

In a hyperbolic universe, there is a discontinuity at the Big Bang and our universe may be only one sheet of a asymmetrical two-sheeted  hyperboloid.

In a hyperbolic universe, there is only one absolute, and that absolute is both random AND deterministic.

If there is one absolute, then there is also only one choice: choosing that absolute, and not choosing that absolute, aka absolute zero. 

In a hyperbolic universe, regressions and statistics using least squares should be redone; the formula for what is called the standard deviation is in fact the formula for error; and the Bessel adjustment, n/(n-1), is not necessary.

In a hyperbolic universe, when there is no error, every moment about the mean should be 0, not just odd movements where the moment is expressed as powers of i; and even powers which are multiples of i2 , are expressed as a real number, - 1.

The universe has a variance of .822,  and thus its standard deviation can never be 0.

So being unable to show my work, I can only assume that the above are true, but I can not prove that the above are true.

The Future

 

It’s About Time

It's about time we begin to turn the world around
It's about time we start to make it the dream we've always known
It's about time we start to live the family of man
It's about time, it's about changes and it's about time
It's about peace and it's about plenty and it's about time
It's about you and me together and it's about time.

That time is NOT only the present, it is the present AND the future.

Republicans are supposedly conservatives and Democrats are supposedly liberals.  Conservatives are supposed to seek the lowest total cost associated with an action.  However the lowest total cost at the moment, may not be the lowest total cost over the life of that action. Case in point, a story when I was an appointed government bureaucrat serving in a conservative Republican administration.  The manger, to whom I reported, had been upset by one of the mangers who reported to me.  She thought that the manager, who reported to me, had been insubordinate to her and as punishment wanted him fired.  That manager, who reported to me, had Civil Service status.

I pointed out to my manager that if I fired the manager who reported to me, that manager would revert to his Civil Service status and be transferred to another office in our department.  I also pointed out that his salary as a Civil Service employee was higher than the salary he was receiving as a manager who reported to me. Thus if I fired him, the long term result was that he would receive a pay raise.  Upon reflection my manager thought about this long term impact, reconsidered, and directed me to tell him that as punishment, she was directing that he NOT be fired.  Since I did not agree that the manager who reported to me had been insubordinate to my manager, I was more than happy to follow this direction.  When I explained the conversation to the manger who reported to me, he could not stop laughing. As punishment, he was not to be fired?

There is a reason that the popular sayings are "Look before you leap", "Marry in haste and repent at leisure",  "A stitch in time saves nine", "For want of  a nail..", etc. It might be conservative to seek the action that has the lowest cost.  But it is the TOTAL cost associated with that action, now and in the future, not merely the cost at the present.

Love and War

 

War

It ain't nothing but a heart-breaker
(War) It's got one friend that's The Undertaker
Oh, war, has shattered many a young man's dreams
Made him disabled, bitter and mean
Life is much too short and precious
To spend fighting wars each day
War can't give life
It can only take it away, oh

Is it true that “All’s fair in love and war”?

War seems like a simple matter of two parties: the American Colonists v. the British Empire, the North v. the South, the Allies v. the Axis, etc. But it is not just two parties if there are rules and a party to enforce the rules. A third party has stated the rules, e.g. the Geneva Convention, and a third party also ensures that war crimes have not been committed by either of the two combatants, e.g. the International Criminal Court in the Hague.

The same is true of the opposite of War: “Make Love, not War.”  This saying shows that while War and Love are not the same, they are the same in that they are both contests with three players. Again, one of the players is not often recognized but it is always there …society. If love is “You and Me against the World” then there are admittedly three parties.  If society will either bless or condemn your love, then there is always a third party.

Rather than the popular wisdom, there is no diffence between love, war, and any other contest.  It should be “Even in love and war, you should always play fair.

Wednesday, November 22, 2023

Prosperity Gospel

 

Money (That’s What I Want)

Money don't get everything, it's true What it don't get, I can't use Now give me money, (That's what I want) That's what I want

Is the love of money good for you?

“My kingdom is not of this world”. John 18:36

It is easier for a camel to pass through the eye of a needle, than for a rich man to enter into the kingdom of heaven.  Matthew 19:24

So how can these words be reconciled with the Prosperity Gospel, which is the fast-growing theologically conservative movement frequently associated with evangelicalism and charismatic Christianity. It emphasizes believers' abilities to become rich in this world through devotion and positive confession.  Does Jesus wish to punish those believers in the Prosperity Gospel, by making them rich in this world, or did those believers have a “Come to Satan” moment?

Saturday, November 18, 2023

Continuing Resolutions

 

I Think It’s Gonna Rain Today

Lonely, lonely
Tin can at my feet
Think I'll kick it down the street
That's the way to treat a friend
Bright before me the signs implore me
To help the needy and show them the way
Human kindness is overflowing
And I think it's going to rain today

Continuing Resolutions are Congress’s way of kicking the can down the street.

MAGA Mike Johnson’s innovation in Continuing Resolutions is a laddered approach, which was making two Continuing Resolutions, which each fund approximately half of the Government and expire respectively on January 19, 2024 and February 2, 2024.  Uh, as Steve Colbert observed in his Late Night monologue, isn’t this merely dividing the can in half and kicking both halves down the street.

The old joke goes

“A man had offended the king and was sentenced to death. He fell to his knees before the king and implored, "Oh your majesty! Spare me but for one year, and I will teach your horse to talk!" The king was amazed and granted his wish.

The man's close friend and brother upbraided him, saying, "Why did you make such an absurd promise?"

The man shrugged and replied, "In a year, the king may die. In a year, I may die. In a year, the horse may talk!"

I've got some facts for MAGA Mike.  This horse is NEVER going to talk. Tomorrow may be another day and you might not want to think of unpleasant facts until tomorrow, but “Frankly, Scarlett, I don’t give a damn.”  Kicking the can down the street only moves the can, it does not make the can go away.  Running out the clock, hoping that you or the king may die, is the coward's way.  Deal with the can.  THAT is the way to treat a friend.