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.