# Is human behaviour mathematically predictable?

This is such a broad question that nobody can even pretend to answer it. Instead, a strategy is to try to solve partial problems, like, human: spending/investment behaviour, mobility, social public/private behaviour, etc…

For example, Albert-László Barabási et al. concluded that with 93% of efficiency, the future whereabouts of individuals can be determined for an horizon of time of one hour, based only in the previous trajectory.  Randomness and unpredictability seems therefore not to be related to short-time mobility patterns, and therefore, it should be possible to predict related human behaviour, like, traveling times, fuel consumption, contagious spreading diseases, place’s popularity, building usages, etc…

Paradoxically, I wonder, if is it true that the more options a person has the most predictable its behaviour becomes, and the opposite direction, the less options a person has the most unpredictable its behaviour is.

###### 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

• 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.

# Mathematics is becoming old but preserving beauty

Nowadays mathematical modelling that has allows us to make a huge progress during the last century is letting us helpless to solve quickly new challenges in accurate ways.

The over-simplifications assumptions that engineering were used to work with in the early 1900 are not longer the best solutions that we can give, considering the improvements in velocity of direct computing greedy algorithms that in many realistic scenarios can behave better than over-simplificated mathematical approaches.

This requires a new way of approaching problems that considers a huge amount of available previous research on similar lines. This phenomena is present in mathematical education among many others.

It seems that the level of abstraction at which extends modern mathematics works is not longer at the level of practical problems that arise in our daily life, but that approaches toward more abstract scenarios. On the other side, mathematics seems to be too rigid to adapt techniques to difficult problems for which technology can provide huge amount of variables and sufficiently accurate approximate solutions. The most elemental example is the plane industry. Mathematicians are still not been able to prove that a plane can flight, but engineering have been able to make plane flighting better. It is happening a phenomena that, not being an expert, I read about philosophy at the beginning of the 20th century. A big deception of the way philosophy was made. It attempted to solve such difficult problems that finished finding non-useful answers. Mathematics won over philosophy, by fixing the basis, and having more concrete goals. However, it seems the need of precision in maths is given us a huge price to pay in modern days,  as this translates into spending a huge amount of time. It seems to me, that the time required to find a solution of a real problem mathematically can be in many in other scale of magnitude of time than finding a good approximate solution using available technology. The combination of different sciences, is giving a huge time shortness in finding solutions to real problems. Mathematics is evolving slower, as an isolated group, because it seems that time has assumed a greater value compared to intellect.

Maybe, in the future, the unique reasson to study mathematics will be to admire its honest beauty. Efficiency was a borrow term believed to belong to optimisation, a mathematical object which beautifulness was beyond what computer scientists found interesting and were able to use it to defeat intellect, by using fast machines and clever algorithms that were built on a cooperative business.

# 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].

# Explaining maths without using them

At the time I was finishing my thesis some non mathematician friends asked me to explain them what I was actually doing.  In this post I write the explanation I elaborated for them. In one sentence, the first problem one need to know how to solve is the following.

How do you explain a mathematical result without using maths at all?

The answer is well known to everybody teaching maths,  however, I will elaborate a bit on this. The first step is to understand the historical roots of the problem you solved, together with the main motivations that people had to need to develop maths in such a determined direction that your problems appeared naturally. Once, you have understood completely this part, you need to associate each problem you solved to something that the people you want to explain the problem to can understand well. This means to associate each result to something completely different but that behaves exactly the same. Finally, you need to somehow transport all this information and deliver it to your audience in a comprehensive way. The last part of the process is well performed by an ”artistic drawing” (whatever that means) together with an explanation of each element. The hardest part of the communication is the explanation of the proof of each result, it happened indeed, that this becomes an issue of time.

In what follows I will answer a more specific question.

Problem:

Explain your thesis to your friends with time constrains conditions.

My thesis can be coded by the following drawing.

The drawing has 5 main elements: a black disk at the top left, a black disconnected curve at the top and bottom, a red apple shape at the left, a blue swirl at the bottom and a green tree shape in the centre and right of the drawing. I will explain each of the five element in what follows.

1. Black disk: If one goes back in time from the questions I faced in my thesis, we arrive to the three body problem. This is represented by the three white bodies inside the black disk. I can suggest you some nice reference to read about, like [Wikipedia, Celestial Encounters by F. Diacu and P. Holmes]. In a few words:  The n-body problem consists in determining the position of $n$ planet at any time $t$, given the initial conditions of known position and velocity at time $t=0.$ For $n=2$ the problem can be easily solved, one can prove indeed, that the trajectory of one planet with respect to the other always lies along a conic section. The problem for $n=3$ was addressed by Henri Poincaré and literally he found the chaos in it. He was not able to find a solution, but instead, introduces qualitative methods to understand the solution, like, periodic solutions, recurrence theorem, non existence of uniform integrals, asymptotic solutions, dependence of the solutions with respect to a parameter, homoclinic orbits, first return map, invariant curves and homoclinic tangles. These qualitative methods are still our tools to understand dynamic systems that are too complicated to be understand in a deterministic way, i.e. those dynamical systems which equations can be  solved.
2. Black disconnected curve:  This curve represents the continuous trajectory of a particle that moves because of some physical laws acting on it. We suppose that there is a finite measure on the drawn system, we assume further that it is invariant under the physical laws. Under these assumptions,  by the Poincare recurrence theorem, almost every particle will enter in a finite amount of time into a set $A,$ providing the measure of this set is strictly positive. The set $A$ in the picture corresponds to the black disk, and the black disconnected curve eventually enters $A$ after some (unknown) finite amount of time. The question that we face is how can we estimate the time that takes the particle to enter for first time $A.$ We solved this problem under very strong hypothesis on the physical laws. Estimations for this problem under our hypothesis are known, however, we refined the existing bound.
3. Blue swirl: This represents chaos in the classic sense of the diagram of phase of ODE´s (Ordinary differential equation). Recall that when we have an ODE $x'=Ax$ in $\mathbb{R}^2,$ the diagram of phase corresponds to the plot of the vectors $(y_1,y_2)$ in the $(x_1,x_2)$-plane, where we draw at the point $x= (x_1,x_2)$ the vector $(y_1,y_2)=Ax.$ This represent the velocity field of the solutions of the ODE.  We assume $A$ is not singular, so that the equilibrium is $x=0.$  The diagram of phase will be determined by the eigenvalues and eigenvectors $\lambda_1,\lambda_2$ of the matrix $A.$ There are 7 cases for the equilibrium state. The equilibrium state is:
1. Stable if $\lambda_1<\lambda_2<0.$
2. Unstable if $0<\lambda_1<\lambda_2.$
3. Saddle if $\lambda_1<0<\lambda_2.$
4. Degenerate if $\lambda_1=\lambda_2\in \mathbb{R}.$
5. Center if $\lambda_1=i \beta, \beta\in\mathbb{R}\setminus \{0\}.$
6. Stable spiral if $\lambda_1=\alpha+i \beta, \alpha<0, \beta\in\mathbb{R}\setminus \{0\}.$
7. Unstable spiral if $\lambda_1=\alpha+i \beta, \alpha>0, \beta\in\mathbb{R}\setminus \{0\}.$ In the drawing we represented this case.
4. Red apple shape: This represents a seed that gives birth to a tree after a process that allows to ”build” a complex structure from iterations of a simpler one. This is analogous to the system build by a simple iterated function scheme that consists of two disjoint contractions on the unit interval, these contractions have associated a unique non-empty closed invariant set. This sets comes from a structure similar to the one of a tree, where one branch divided into to smaller, and each smaller one into two smaller and so on.
5. Green tree shape: The green structure represent a tree that we consider analogous to the invariant set of an iterated function system. The problem that we face in the last chapter is: What happens if we perturb a little bit the seed of the tree, how different is going to be our tree? The analogy with this is: perturbing a little bit both maps of our iterative function and study how smoothly that affect the invariant set. A mathematical way to do this is by considering the for example the ”measure” (called Hausdorff dimension) of the invariant set.

# Metro Linea 5

This is the Santiago Metro Line 5:

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.