Swimming is not boring or maybe, it is…

Swimming allows you to solve the Navier-Stokes equation with different border conditions thousands of times, in particular, you can choose at any stroke a different border condition and check how the solution of the equation changed.

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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?

Intersection of Cantor Sets

I will write about an interesting talk given by Pablo Shmerkin.

Main goal: computing dim(A\cap B) for A,B\subset \mathbb{R} with dynamics properties.

Theorem [Marstrand, 1954] E\subset\mathbb{R}^2 be a Borelian set. Given \theta\in (0,\pi), we define L_{\theta} to be the family of all the lines on the plane with slope \theta. Then for almost all l\in L_{\theta} we have that dim(E\cap l)\leq \max (dim(E)-1,0).

Example
The graph G of a Brownian motion on the plane has dim(G)=\frac{3}{2}, however, vertical lines have intersection of HDIM equal to zero.

Theorem [Marstrand, 1954] E\subset\mathbb{R}^2 Borelian set. ess \mbox{ } \sup dim(E\cap l)=\max (dim(E)-1,0).

We can obtain the following corollaries for the HDIM of the intersection of subsets A,B\subset \mathbb{R}.

Corollary Let A,B\subset\mathbb{R} be Borelian set. Then for Leb almost all t we have that dim(A\cap (B+t))\leq \max (dim(A\times B)-1,0).

Remember that dim(A\times B)\geq dim(A)+dim(B), and that there is equality if dim(A)=\overline{dim_{BOX}(A)}. Therefore, we have the following corollaries.

Corollary If A\subset\mathbb{R} is a Borelian set such that dim(A)=\overline{dim_{BOX}(A)}, then for Leb almost all t we have that dim(A\cap (B+t))\leq \max (dim(A)+dim(B)-1,0).

And from the second theorem of Marstrand, we have the following corollary.

Corollary ess \mbox{ } \sup dim(A\cap g(B))=\max (dim(A\times B)-1,0), where g:\mathbb{R}\to\mathbb{R} is lineal.

Some examples.

Example
There exists compact A\subset \mathbb{R}, dim(A)=1 such that the cardinality of A\cap(A+t)\leq 1 such that dim(A\cap(A+t))=0 and \max (dim(A)+dim(B)-1,0)=1.

For all s\in[0,1], \exists A\subset \mathbb{R}, dim(A)=s and dim(A\cap (A+t))=s for all t\in\mathbb{R}. [Falconer example]

Furstenberg asked what happened if A,B were invariant for the map T_p(x)=px \mod1. Motivated by this questions, let us consider first the classic Cantor set C\subset \mathbb{R}, whose dim(C)=\overline{dim_{BOX}(C)}=\frac{\log 2}{\log 3}. Applying directly Marstrand’s theorem we obtain the bound dim(C\cap (C+t))\leq 2\frac{\log 2}{\log 3}-1\approx 0.262 for almost all t. There is however a better bound by Hawkes.

Theorem [Hawkes, 1975] dim(C\cap (C+t))= \frac{1}{3}\frac{\log 2}{\log 3}\approx 0.21 for almost all t.

Kenyon and Peres generalized further in 1991 the result to the case of p\geq 2, D_1,D_2\subset \{0,1,\ldots,p-1\}, A_i=\{\sum x_j p^{-j}: x_j\in D_i\}. Notice that C=\{\sum x_j 3^{-j}: x_j\in \{0,2\}\}.

Theorem [R. Kenyon and Y. Peres, 1991] For almost all t\in [0,1] dim(A_1\cap (A_2+t))= \frac{\lambda}{\log p}. Where \lambda= greater Lyapunov exponent for the product of iid matrices M_r, where M_r(i,j)=\# (D_1+i+r)\cap (D_2+j+p), r=0,\ldots, p-1, 0\leq i,j\leq 1.

Barany and Rams in 2014 considered the case of self-similar sets E\subset [0,1]^2 that are obtained by a generalization of the set C\times C (ex. consider the unit square [0,1]?2, divide it in 9 squares of side 1/3, numerate the squares by \{1,2,\ldots,9\}=\{0,1,3-1\}^2. Choose 4 squares, say \Omega=\{1,2,5,9\}, delete the rest of the squares and repeat the procedure on each of the 4 squares left, etc.).

Theorem [B. Barany and M. Rams, 2014] Let E\subset \mathbb{R}^2 a set constructed doing the described process where \Omega\subset \{0,1,\ldots,p-1\}^2, p\nmid \#\Omega and \#\Omega>p. If \tan \theta \in \mathbb{Q}, then \exists \alpha(\theta),\beta(\theta) with \alpha(\theta)>dim(E)-1>\beta(\theta), dim(E\cap l)=\alpha(\theta) for \nu-almost all l\in L_{\theta}, where \nu is a natural measure with support on E and dim(E\cap l)=\beta(\theta) for Leb-almost all l\in L_{\theta}.

The non rational case was recently studied by Pablo.

Theorem [P. Schmerkin, 2016] Let E\subset \mathbb{R}^2 a closed set and invariant for T_p(x,y)=(T_p(x),T_p(y)). If \tan \theta \notin \mathbb{Q} then dim(E\cap l)\leq \max(dim(E)-1,0) for all l\in L_{\theta}.

The proof of the theorems follows from an application of a tool now known as the BSG (Balog-Szemer\’edi-Gowers) theorem, which has found many further applications. In the words I can remember… is a theorem in combinatorics proved with graphs that allows to have a structure on the integeres, it allows to find lower bounds by convoluting indicatrix functions and obtained a l^2 norm that can by bound with an l^1 norm apart from a small set. Remember that the BSG theorem is the main ingredient of the 1998 proof of the first effective bounds for the Szemer\’edi theorem, showing that any subset A\subset\{1,2,\ldots,N\} free of k-term arithmetic progression has cardinality O(N(\log \log N)^{-c_k}) for an appropriate c_k>0.

Notice that the theorem works in particular for E=C\times C. Pablo was not convinced of the conjecture that indeed dim(E\cap l)\in \{dim(E)-1,0\}.

As a corollary we conclude that if \lambda\notin\mathbb{Q}, then for all t\in \mathbb{R}, dim(C\cap (\lambda C +t))\leq 2\frac{\log 2}{\log 3}-1.

The following theorem was obtained independently in 2016 by P. Schmerkin and M. Wu:

Theorem [P. Schmerkin, 2016] If p and q are not powers of the same integer, A,B are closed sets, T_p,T_q-invariants, then
dim(A\cap (\lambda B+t))\leq \max(dim(A)+dim(B)-1,0) for all \lambda\neq 0, t\in\mathbb{R}.

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.

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.

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 the only thing that will remain in the future to look at in mathematic will be 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.