Sides

 

Both Sides Now 

I've looked at life from both sides now
From win and lose and still somehow
It's life's illusions I recall
I really don't know life at all 

You don’t need to pick a side, or repeat both sides.  Life IS both sides. 

If a surface is hyperbolic then there is a hyperbolic, non-Euclidean solution for the hypotenuse of a triangle on that surface. If life is a complex number, x + y*i, and we are living in reality, y=0, then that solution is cosh(life)=cosh(x)*cosh(y), which is life=ln(cosh(x) ± sinh(x)) because cosh(0)=1. By subtracting cosh(x) from each solution, and because cosh(x) is symmetrical, that is cosh(-x) is equal to -cosh(x), this can also be stated as life=ln(0 ± (sinh(x)+cosh(x))). If an absolute can be approached but not obtained, then the behavior approaching that absolute is said to be exponential. Since both sinh and cosh can be stated as exponentials, the two solutions are x and -x. This, and the surface being hyperbolic, is consistent with the two sheets of a hyperboloid. If our universe is one of those sheets, then it has the opposite sign of the other sheet. If our universe is observable, then there must exist an unobservable universe that has the opposite sign of any value in our universe. 

Or to put it in New Testament terms, “The first shall be last, and the last shall be first.”  And life always has both a certain term, cosh(x) and an uncertainty term, sinh(x), which means that uncertainty is a consequence of being on a hyperbolic surface. Any value from an absolute will have two solutions, the value and its opposite. Or from an absolute, again in New Testament terms Galatians 3:28, “There is neither Jew nor Gentile, neither slave nor free, nor is there male and female, for you are all one in Christ Jesus”. In other words, remember on Election Day, there is neither liberal nor conservative, neither Democrat nor Republican, for you are all one in America.

Affirmative Action III

 

I’m Sorry

I'm sorry, so sorry
Please accept my apology
But love was blind
And I was too blind to see

Oops I made a mistake, but IMHO so did the Supreme Court!

I took the position that Affirmative Action is always a good thing in a previous blog post,  https://dbeagan.blogspot.com/2022/08/affirmative-action.html, while the US Supreme Court overturned Affirmative Action in its 2024 opinion on college admissions.  IMHO, we were both guilty of being false. I was being a false positive. I was saying that Affirmative Action was always being used for good, and therefore the decision was wrong. The Supreme Court was saving that Affirmative Action is always bad, and IMHO that is a false negative. As it turns out MIT and other elite schools are finding that the racial constitution of the incoming freshman classes is less diverse racially, but is more diverse economically.  Huh?

There is a wealth inequality in the United States that has a racial component. On average Whites are wealthier than Blacks and Hispanics.  But that doesn’t mean that some Blacks and Hispanics aren’t wealthier than some Whites. What the elite colleges were apparently doing is admitting the wealthiest minorities. In that fashion they could be more racially diverse, but NOT more economically diverse. If they had used NO bias, the incoming class should be just like the overall society. By only admitting wealthy minorities, those colleges could be more diverse racially, while being not being economically diverse. The lack of Affirmative Action, as it was being used in admissions, just  might have highlighted the fact that colleges were being less diverse than the general population when it comes to wealth, even if they appeared to be closer to the general position for minority status. In the words of Martin Luther King Jr., the goal is to judge on “the content of one’s character, not the color of one’s skin.”  

So is the goal of Affirmative Action wrong? IMHO, not in the least. 

Was the way in which it was being administered producing a false positive? Apparently.

Division?

 

Born Free

Stay free
Where no walls divide you
You're free as a roaring tide
 So there's no need to hide
Born free
And life is worth living
But only worth living
'Cause you're born free

Walls ARE meant to divide you.

The opposite of freedom is to have walls.  The phrase after all is “Divide and Conquer.”  If you accept walls, then you are accepting being conquered.  If life is a Zero-Sum game with walls, then when you win, you take something from others.  If life is a game of growth, not Zero-Sum,  then everyone could have more at the end than at the beginning.

A motto of the United States is “E pluribus unum.  Out of many, one” .  It is the United States, NOT the Divided States. Don’t build or accept walls.

Confused

 

Dazed and Confused

Been dazed and confused for so long it's not true
Wanted a woman, never bargained for you
Lots of people talk and few of them know
Soul of a woman was created below, yeah

Are you confused?

Many of the problems of the world are that some very basic concepts are being confused.

Mean and median.  Mixing up these terms, which sound similar, can lead to unintended things. https://dbeagan.blogspot.com/2018/08/wonderful-world-dont-know-much-about.html

Effective and Marginal.  The US Tax Code changed in the 1980s because people did not understand the difference between effective (e.g. first derivative) and marginal ( e.g. second derivative) and that has led to more than 40 years of extra income for high income taxpayers.  https://dbeagan.blogspot.com/2024/08/laffer-curve.html

Dominance and Certainty.  The ruling of a panel of judges should be certain.  It should not merely be dominant. A unanimous jury approaches certainty.  A 5-4 panel of judges is merely dominant.  https://dbeagan.blogspot.com/2024/06/dominance.html

If we have dazed and confused ourselves, shame on us.  If others have taken advantage of us because they are dazing and confusing us, shame on them. And shame on us.  Don’t be confused.  The first thing a stage magician learns is to distract the audience so they can't tell they are being tricked.  If you don’t know, take the time to find out before you act, you are effectively letting others act for, and possibly trick, you.

Nash Equilibriums III

 

All about the Bass

Because you know I'm all about that bass
'Bout that bass, no treble
I'm all about that bass, 'bout that bass, no treble
I'm all about that bass, 'bout that bass, no treble
I'm all about that bass, 'bout that bass, hey

I beg to differ. It’s all about the Math!

I know that Math is Hard, but bear with me. Math may be the explanation, and the solution, to our current problems. This might sound strange but what mathematicians would call infinity, a scientist might call an absolute, an ethicist might call Truth, and an Evangelical would call God, But all can be analyzed/approached the same way.

Let’s suppose the existence of infinity/absolute/Truth/God. The absence of these things would be death, flat-lined. A flat-line has no Amplitude, no variance. The presence of these things must therefore have an Amplitude according to the formula ½A²=σ², where A is the amplitude and σ² is the variance. Infinity/absolute/truth/God as a wave, the opposite of a straight line, must be ...doh...infinite. A wave is an infinitely repeating function of π, which is merely saying that while a wave is infinite, there might be no difference between its behavior at π, 2π, 3π, …nπ, …∞π. If there is an absolute that repeats as a function of π, it must have a repeating median/mean, μ, of π /2.

Individuals can act like a wave. (Don’t believe me! Have you never seen a crowd at a stadium doing the wave!). So how do you get many individuals to perform like a wave. A normal (The mathematical name for it. Not a moral judgement!) distribution of individuals is the logistics, hyperbolic secant squared,  distribution, 1/4s*sech²((x-μ)/2s). S is a function of the variance, according to the formula σ²=s²π²/3. That function is also known as the Probability Distribution Function, PDF.

While ordinary wave functions repeat with a period related to π,  hyperbolic wave functions, such as sech, repeat with a period related to πi, where i is the imaginary number, i=√-1. ( I know math is hard, but the kids in Algebra know this!). Also for functions of x, f(x), there is a derivative of that function, f’(x), which is the slope of that function, and an integral of that function F(x)=∫f(x)δx, which is the area under that function. The derivative of the logistics distribution is
-1/8s²*sech²((x-μ)/(2s))*tanh((x-μ)/(2s)). The integral of the logistics function, which also goes by the name Cumulative Distribution Function, CDF, is 1/2*tanh((x-μ)/(2s))+1/2. ( again Math is Hard. Feel free to check with the kids taking AP Math!)

All of this to say that a wave function, its derivative, and its integral can be defined for any value of x given only two parameters: the median, μ,  which is akin to the phase of the wave, and the range, s, which defines the amplitude of the wave. There are three behaviors that are of interest. The first is User Optimal, UO,  defining only the median, as μ=0 and allowing any value for s. This can be defined as “Only I matter,” “Second place is first loser”, “Winning is the only thing”, “All’s fair” , etc. The second behavior is System Optimal, SO where you accept infinity/absolute/Truth/God/π and thus the median must be π/2, but you say that  you always are 100% certain. The problem is that since you are only an individual, according to the logistics distribution you must also be 40% certain at either 1/3π or 2/3π, and 6% certain at either 1/6π or 5/6π, etc. This means that you then have to be much more than 100% certain if you add all of these together. While being more than 100% certain sounds great, it is mathematically impossible. In fact the integral, CDF, of the logistic function when μ=π/2 and s=.25, which is consistent with being 100% certain at μ, starts at 50% certain at x=0, while the certainty should not be 50% until  x=π/2.

This is like the scene at the door to the back room in Casablanca. UO behavior is asking ”Do you know who I am?”  SO behavior is throwing all UOs out of the Club. The third behavior, the Nash Equilibrium (for reasons that would make your head hurt) is saying that μ=π/2 and s= 0.51451 and this is like saying “I know who you are. You’re lucky your cash is good at the bar.”     https://www.youtube.com/watch?v=aALbiGJpw7c.  An ideal single wave would be s=.55139 which is consistent with a value of s=.5 (twice the SO value) on a hyperbolic non-Euclidean surface because of the use of the hyperbolic secant, from a perfect multiple wave. The derivative of this function and its integral under all of these behaviors are shown below. 

Figure 1 Derivative, dPDF, of logistic function

Figure 2 Certainty, PDF, of logistic function 

Figure 3 Integral, CDF, of logistics function

Those following a User Optimal, UO,  strategy will  try to get those following a System Optimal,  SO, strategy to join them by saying that "I will act like you are correct ( the quiet part NOT out loud being especially since as a UO, I can accept any s) if you act like my median of μ=0 is correct". Instead if the SO wants to be like perfection, then it should adopt a Nash Equilibrium, NE, strategy. The best strategy is not to win, UO, and not the common good, SO, but to quote mathematician John Nash from Ron Howard’s Oscar winning A  Beautiful Mind, is  “to win for the common good,” a Nash Equilibrium. If you are a SO plan to join with NEs, NOT with UOs, if you want to be perfect.

Rebound

 

Red Rubber Ball

And I think it's gonna be all right
Yeah, the worst is over now
The mornin' sun is shinin' like a red rubber ball

But how does that red rubber ball bounce?

A bounce, rebound, occurs when an object, such as a particle, encounters a discontinuity. That discontinuity can be a physical surface, or it can be merely observational, that is the ability to observe, and measure, may be the actual reason that there appears to be a  discontinuity..

If a particle is moving, and is not acted upon by a force, that particle moves in a straight line. That is Newton’s Law of Inertia. However this is only true if space is flat. It is more proper to say that a particle moves along the geodesic in its space. If the space is flat, then the geodesic is a straight line. But if that space is not flat, for instance is spherical or hyperbolic, then it is non-Euclidean, and only flat space is Euclidean.

We say that the Earth is a sphere, and we live on the Earth’s spherical surface. That is why the shortest distance between two points on earth is more properly a Great Circle Distance. While this is true, it might be only of interest to airplane pilots and others who measure vast distances. When the distances involved are far less than the radius of the Earth, then the solution for the hypotenuse of a triangle on that spherical surface is cos(c/R)=cos(a/R)*cos(b/R), where R is the radius of the Earth/spherical surface, and it is virtually identical to the solution on a classical flat Euclidean surface, cos(c)=cos(a)*cos(b), as can be verified by using the series for the trigonometric functions. Both of these are equal to Pythagoras’ Theorem, c=√(a²+b²). It is therefore customary to say that the distances on Earth  are spherical globally but are flat locally. Might this also be true for space?

The solution for the hypotenuse of a triangle on a hyperbolic surface uses hyperbolic trigonometric functions, cosh(c)=cosh(a)*cosh(b). This has a different solution than the classical solution. The classical solution relies on the circular identity, cos²+sin²=1.  In hyperbolic space the identity cosh²-sinh²=1 applies. Additionally space may not be merely what can be observed, it might be that which can not be observed, i. In this case reality having a coefficient of zero for that which can not be observed can be expressed as a complex number which is reality plus zero imagination, r+0*i. If reality is the solution of a triangle r²=(a²+b²)+0^2*i, then its solution in hyperbolic space is
ln(cosh(√(a²+b²)) ± sinh(√(a²+b²))) because cosh(0²) is 1,  where the ± indicates that there are two solutions. Because cosh is symmetrical while sin is symmetrical, and for small values of a²+b² compared to the size of the universe, sinh(√(a²+b²)), the uncertainty, is also small. This can be also expressed as a single solution, ln(cosh(√(a²+b²))-sinh(√(a²+b²))), if reality is one solution. This is no different than electrical engineering where some solutions have real and imaginary components, and the imaginary component is ignored. The single solution merely says that the real solution has the opposite sign of the imaginary solution, and that the imaginary solution is being ignored.

A rebound in flat space is symmetrical because a straight line is symmetrical about that discontinuity. Hyperbolic motions are NOT symmetrical as real numbers. They are almost linear on one side of the discontinuity and almost parabolic on the other side of the discontinuity. If there is a linear motion on one side of a discontinuity and that discontinuity is NOT a surface and the motion is parabolic on the other side of the discontinuity, this probably is an indication that the motion is hyperbolic, NOT a highly skewed parabola. A parabola requires an imaginary solution if that motion passes through, is rotated by 180º or π. It is suggested that is more reasonable to assume that this discontinuity is because the observable behavior continues as unobservable behavior than it is to assume the behavior has become imaginary.

 

 

Engineers III

 

This is Me

When the sharpest words wanna cut me down
I'm gonna send a flood, gonna drown 'em out
I am brave, I am bruised
I am who I'm meant to be, this is me
Look out 'cause here I come
And I'm marching on to the beat I drum
I'm not scared to be seen
I make no apologies, this is me

And me (sic) IS an engineer!

An engineer has been defined as  someone who is  good at math and socially awkward.  I admit to being socially awkward and I am a Professional Engineer.  As to the good at math, here goes my feeble attempt.

The logistic distribution, also known as the hyperbolic secant squared distribution, is a normal distribution. Its Probability Density Functions, PDF, is f(x)=

1/(4s)*sech²((x-μ)/(2s))

and its Cumulative Distribution Function, CDF, which is the integral of f(x),  ∫f(x), is

½ tanh((x-μ)/(2s)) +½.

The derivative, f’(x), of the PDF is

-1/(8*s²)*sech²((x-μ)/(2s))*tanh((x-μ)/(2s)) = (-1/s)*PDF*(CDF-½).

Each of these are wave functions in hyperbolic space. They each have the same period of πi.  The hyperbolic tangent, tanh, also has a period of πi.  Each wave function has the same phase, μ/2s. For the Amplitude of each of these three waves to be the same, s must be equal to ½, in which case the phase for each wave would be just μ.

The PDF can be considered to be equivalent to momentum in classical Newtonian systems, a spring in a mechanical systems, a capacitor in electrical systems, etc. The derivative of the PDF can be considered to be distance in Newtonian and mechanical systems,  a resistor in electrical systems, etc. The CDF can be considered to energy in Newtonian systems, a dashpot in mechanical systems,  an inductor in electrical systems, etc. Since energy and mass are convertible according to Einstein’s Equation, E=mc², this also has implications for mass via relativity.

If s=½ is taken to be one volume divided into two sheets, then it could be on a two-sheeted hyperboloid. If space is then hyperbolic, not flat, then two Minkowski light cones intersecting  at an origin, could instead be considered not to be light travelling on a flat Euclidean surface where the geodesic is a straight line, but light traveling on a hyperbolic surface, where the geodesic is hyperbolic and therefore non-Euclidean. If, as suggested by Mabkhout , the universe is hyperbolic, it may be just one (observable) sheet of that hyperboloid. For a function to span both the observable and unobservable sheets there must be a transition/discontinuity between the two sheets.

Euler’s Formula is  e^ix=cos(x)+sin(x)*i. This can be viewed as a special case of a transformation of a complex number from cylindrical polar coordinates to Cartesian coordinates, r*e^ix=r*cos(x)+r*sin(x)*i, where there are three dimensions, the  dimension of space and dimension of time, reality r, where r²=(r*cos(x))²+(r*sin(x))²,  and an imaginary dimension, i , when r=1, and x is the angle of rotation of the imaginary axis. If reality has a coefficient of 0 for the imaginary dimension/axis, then both sin(0)=0 and sin(π)=0, but cos(0)=1 while cos(π)=-1. This means that if reality has a coefficient of the imaginary axis of zero, then there are two sheets forming that surface/plane; one sheet which has the opposite sign of the other.

A transition/discontinuity is observed in many applications. At a discontinuity, a particle can rebound from that discontinuity and still remain in the same space/sheet. However if a particle passes through that discontinuity, then it must be transformed, and unobservable from the original space/sheet. It is suggested that for many applications, such as fluid in a channel or pipe, or traffic on a road, a transition occurs at a discontinuity from laminar, uncongested to turbulent, congested conditions. This is most probably the consequence of remaining in the same space and infers the existence of an unobservable sheet to which a transition will occur.

If the discontinuity is physical, then the path after the discontinuity is a rotation by π/2, 90º.  This means that that a path passing through a discontinuity should then be two rotations by π/2, in other words,  a rotation by π or 180º. If a path appears to behave like it is encountering a discontinuity in the absence of a physical discontinuity, it is proposed that this is an observational discontinuity. What is not being observed could in fact pass through the observational discontinuity, as opposed to a physical discontinuity which will prevent passage.