The opening gate

When I realised it was the opening gate, about three weeks ago, I knew it was full of dynamical information. It was about ten in the night, the night was dark and I was watching outside the windows with all the lights inside the house turned off. At some point, the light from outside started to alternate between bright light and deep darkness, it took me a while to figure it out what was the cause of this strange phenomena. The reason was a car with its powerful headlights on that was waiting a huge gate made from solid metal bars (of approximate 5cm x 5cm located at distance of about 10 cm of each other) to entire open. I was watching from an oblique direction.

I remember to have thought that this was something wonderful, and that somehow it must be related to a mathematical phenomena, but which one? I was entirely sure… This alternating between luminous night and dark night produced by the fact that the gate was at some times producing an overlapping between the metal bars so that the area of the surface not allowing  the light from the car headlights to pass was minimal and therefore the light that I was seeing maximal, and at other times, maximising this area of the surface, so that the light that I was seeing was minimal, and even more, I was seeing no light at all at some instants, because of the oblique direction I was located from the gate.

Short time ago, I was working with iterated functions schemes on the interval with overlapping, and I could learn about a similar phenomena, the non-continuity of the Hausdorff  dimension of the limit set as a function of the contraction rates of the function scheme. So this overlapping must be related to the overlap of the metal bars of the gate, and the continuity must be related to the intensity of the light, the change between intensity at some very specific points was entirely different, it was the difference between not light at all and light, this can be seen as the non-continuity of the function of light intensity. Then Pisot numbers, it happens that they are related with the distances between the metal bars of the gate… This was still to be just a bunch of formally related facts but it must be done more precisely, on there hand it was surely possible to write this phenomena more elegantly, in a simple way an general enough…

Then it happens, that I had a board game a cousin had giving me as a gift recently for my birthday. In order to play for the first time, a set of circular tokens had to be removed from two apparently equal rectangular cardboards.  So that, once removed the tokens, both cardboards remained full of circular holes of the same size. So this two cardboards with circular holes could produce a similar phenomena that the car waiting the gate to open…Indeed, turning off the light of the room, pointing a flashlight at the two cardboards plates with holes, and moving them one over the other, a similar phenomenon occurred, a non continuous map of the light intensity. This time, I could realise the size of the circular holes were essential, as well as their distribution on the cardboard plate, so this seems a kind of generalisation to more complicated geometries, surely the same phenomena could be observe in more dimensions, considering projections… It seemed at this point it was something too general not to have been studied before, many unrelated mathematical results should had exhibit a similar phenomena, but which ones? How close they were to this phenomena? Or maybe, in the most optimistic case (or pessimistic if you prefer to say)… It could be written as a something new? At least in some sense new?

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Headlight Guards

jeep

Picture from https://www.autoanything.com

Why headlight guards are so simple?

If they are supposed to protect the headlights then why they don´t have a more efficient geometry?

I was some long time ago astonishing about how simple headlight guards were, I now think I should have probably asked some jeep owner but I did not know anyone who was using any kind of headlight guard.

My short answer would be the following: The resistance of the glass (or plastic or whatever) of the headlights is known, therefore, it is possible to estimate the probability that an object of known: material, volume, mass, geometry and velocity breaks the headlight. I guess an accurate model may be difficult to build, so starting from experimentation with different objects, I guess the conclusion was that the volume of the object was fundamental, so small volumes cannot break the headlight so it was not necessary to have complex geometries for the guards. Anyway, I wonder how far this answer if from the reality? Can anybody provide a better answer?

-H- Problem

-H- Problem is a problem motivated by a road configurations that constitutes a rather typical pattern of streets intersection.

The -H- configurations consists of the following intersection of streets:

.                                                 N

.

.               | Street 1|                                    |Street 2|

.              |                |                                    |              |

————                     ——————————                  ——————–

W  Street 3                     Street 4                                 Street 5         E

———–                     ——————————                 ——————-

.              |              |                                     |              |

.              |Street 6|                                     |Street 7|

.

.                                                    S

The problems consists in coordinating car’s traffic simultaneously along the 7 streets of the picture in order to minimise the congestion. You must allow cars to travel from any parts of the city (North, South, West, East) to travel to any other part of the city (North, South, West, East).

This problem is far from being solved from a practical point of view.

Some long time ago I was in a bus along Street 7 going North from the south. In the configurations I was:

  • Street 3, 4 and 5: W<->E
  • Street 1 and 6:  N->S
  • Street 2 and 7:  S->N

And additionally, we could turn:

  • From Street 7 to Street 4 to the left.
  • From Street 7 to Street 5 to the right
  • From Street 5 to Street 2 to the right
  • From Street 4 to Street 6 to the left
  • From Street 1 to Street 3 to the right
  • From Street 1 to Street 4 to the left

It was a nightmare … I waited a long time to go from Street 7 to Street 2, because:

Cars going from Street 1 to Street 6 blocked cars traveling W<->E between Street 3 and 4. As a consequence, cars traveling W<->E between Street 4 and 5 were blocking cars traveling S->N from street 7 to street 2.

In practice, the results of the current design was a stationary solutions with cars not allowed to move in any direction. Cars going S<->N were blocking and blocked by cars traveling W<->E, and cars traveling W<->E were at the same time blocking and blocked the first, traveling S<->N.

I saw at the time that was completely impossible for any driver to do anything useful as an individual, in order to travel along the wanted direction without taking any decision that would had a completely disastrous consequence for the cars going along other directions.

I guess it is sad to admit that the same problem is probably repeated every single day at exactly the same time, so an efficient solution to this apparently simple but common street configuration could have valuable impact for solving transport problems in crowded cities.

El problema de la funda de auto

Ayer me vi enfrentado al siguiente problema que me pareció interesante.

Se tiene un pequeño auto cubierto enteramente con una gran funda, claramente diseñada para un auto más grande. Corre mucho viento, tanto así, que cada cierto rato la funda deja completamente al descubierto el auto.

Supongamos se asume que el auto, cuya geometría se conoce, se encuentra sobre una superficie horizontal de la tierra, se conoce además la geometría  y el material de la funda, y la velocidad del viento en cada punto sobre una superficie semi esférica de radio suficiente grande para incluir en su interior al auto y la funda, mientras ésta aún no deje al auto completamente descubierto.

Es posible, bajo todos los supuestos mencionados, determinar el tiempo necesario para que el auto quede al descubierto?

La solución de este problema, parte por considerar el caso más elemental imaginable. Consideremos la geometría del auto es semi esférica con radio r_1, y lo es también la funda con radio r_2>r_1, supongamos se conoce la velocidad del viento sobre la superficie semi-esférica de radio 2r_2. Suponemos el coeficiente de roce entre el auto y la funda es cero, y que la funda se comporta como un fluido con la misma densidad del aire.

Bajo estos supuestos, el problema puede describirse parcialmente! [KURT N. JOHNSON, CIRCULARLY SYMMETRIC DEFORMATION OF SHALLOW ELASTIC MEMBRANE CAPS, VOLUME LV, NUMBER 3, SEPTEMBER 1997, PAGES 537-550].

Metro Linea 5

This is the Santiago Metro Line 5:

Línea_5_-_Metro_de_Santiago

There are three kind of stations, R red, G green and M red/green. Between 6 and 9 hours and between 18 and 21 hours a train goes either along R and M stations, or along G and M stations. The rest of the time, trains stop at every station. This systems is part a “route express” program.

Is the route express system efficient ?

This turns out to be a highly non-trivial maths problem. First of all, we require to design an accurate model of the Santiago Metro Line 5.

Mathematical model:  

  1. We need to identify the passenger entrance rate to each station. It is certainly not a Poisson distribution, as there is not constant rate (it highly depends on the time) and moreover, the arrivals are not independent, because usually people enters in group. Actually, I currently do not know which discrete distribution would be the most appropriate. An obvious simplification is to consider it is Poisson with different rates depending on the time (example from 6 to 7, 7 to 8, etc.). In this case we could use some statistical test to adjust the rates at each station and at each hour. 
  2. We need to identify the passenger exit rate to each station. An ideal supposition is it that the exit rate at each station at the time n+12 \pmod{24} is equal to its entry rate at the time n. This would allows us to consider only entrance rates.
  3. Once we have established the entrance and exit rate we can evaluate the efficiency of the system. We consider a random passenger X, its expected travel time is \mathbb{E}(X)= \sum \mathbb{P}(going from the station a to the station b)\times Time(going from the station a to the station b), where the sum is over all possible combinations of a and b. The function Time does depend on the strategy of the model (i.e. which stations are R, G or M).

Choosing the optimal transportation plan:

A similar problem was addressed by the french mathematician Gaspard Monge in 1781.

  1. Each strategy of colouring stations by R, G, M will give a (optimal) travel time from the station a to the station b. We are no force in principle either to assign the same colouring for routes in opposite directions.
  2. In order to solve the problem we would need to check 2 \times 3^{30} different possibilities and choose the one that gives the smaller value of \mathbb{E}(X). As this is clearly impossible with modern computers, we need to design an efficient algorithm to solve this problem in a reasonable amount of time.

There is still the harder part of the work to be done in order to answer our original question. I wonder to know how strong are the assumptions by Metro de Santiago in order to determine its route express system.

Contraction Mapping Principle

The Contraction Mapping Principle is the most basic fixed point theorem in Analysis. It appeared in Banach Ph.D thesis published in 1922. We see this principle applied every time that we close an umbrella.

Theorem: Let (X, d) be a complete metric space and f:X\to X be a map such that  d(f(x),f(y) )< d(x,y)  for every x,y\in X. Then f has a unique fixed point in X. Moreover, for any x_0\in X the sequence of iterates x_0,f(x_0),f(f(x_0)),\ldots converges to the fixed point of f.

How is this related with closing an umbrella?

The contraction mapping principle tells us that we can close an umbrella, moreover, there is a unique direction along which it will (always) close.