Esoteric Mathematics and Life

One of the most useless mathematical exercises that engineering students do is that of calculating limit of a function using the epsilon-delta definition. The definition is complex, its motivation non-obvious and applications nearly non-existent. While it is true that in many important theoretical calculations this definition is useful, one is not quite sure, where in the day-to-day use would an engineer need to apply it.

However there is a closely related area of mathematics, pertaining to continuity and differentiability of functions which has come to the fore due to new engineering applications in 3D printing and bio engineering. Organ transplants save life, however there are not enough number of human donors to meet the demands.

3D printing of organ’s is an emerging technology that has immense potential. Human organs, however rarely fit into the standard geometric shapes. To design and analyze organs, the mathematics required is that of fractal geometry. Fractal geometry deals with self similar structures, where a big part of a structure looks like a scaled up version of its smaller part. Such structures naturally occur in our bodies, for example lungs, kidneys and the blood circulatory system. Thus to 3D print such organs, the code required will be more efficient if it uses the foundational ideas of fractal geometry, rather than those of the Euclidean geometry.

Ventilator Design

Many innovators around the world are busy working on affordable ventilators that can be mass produced. It does make one wonder as to why are these systems so expensive and difficult to make, given the advances in tech. tools and software over the last decade. Think of the following possibilities.

1. The ventilator has to be in-step with the breathing pattern of the patient, which keeps changing in time. Which requires continuous monitoring and fast adjustments.

2. The ventilator capacity should be adjustable to the lung capacity of the patient.

3. The on board arduino clock may require regular resetting which has to be done as unobtrusively as possible.

4. The optimun pressure range has to be maintained. More pressure may damage the lungs. Less pressure may cause difficulty in breathing.

5. Sufficient back-up mechanism should be there in case of a sensor failure.

6. Design should manage to avoid system hang-up.

These and similar other concerns require a careful design and testing and quality control before the devices can be used in field.

Math of covid-19

The rate at which microbes multiply is proportional to the microbes present at a given time, provided other factors like nutrients and temperature remain favorable.

$\frac{dN}{dt} = k N$

This equation has the exponential solution $N(t) = N_0 e^{kt}$. Spread of infections can also be modeled along similar lines, since the rate at which people become infected is expected to be proportional to people already infected.

Interestingly Nicolas Vandevalle has made a model to fit the covid-19 data for Belgium to an exponential curve.

As can be seen, the initial cases of infection follow an exponential curve to a very good approximation, however as the effect of social distancing comes into play, it is possible that the curve may become linear. The data was analyzed for Belgium. Right now, the data is insufficient to say if the curve is turning linear, but the implications are far-reaching.

A Mathematical Scam

Monetary scams by their nature are mathematical. Here is a story of an interesting one. Suppose you receive a mail from a company saying that share prices of X is going to go up the next day and offers services to help you invest, you being a wise person ignore the tip and find to your mild amazement that the prices actually did go up the next day. A week later you the whole process repeats and now you are a little more than mildly amazed that the prediction came true. Suppose this process goes on for five weeks, if you are not well trained in thinking rationally, you now trust the predictions to a large degree and contact the company for a decent sized investment.

Here is what is quite likely to be going on. The scammers send similar mails to say 100,000 people with randomized predictions of say 10 different types of shares going up. Out of which let us say 50,000 predictions came out to be true. The next mail goes only to those 50,000 and so on. At the end of the fifth round about 6250 of those brave-hearts who have not given-in to the desire of making big money are quite likely to do so now.

So while managing your money, if something seems too good to be true, your hunch may be correct.