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 r^th 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*e^ix=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^iπ=-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,.