Mathematics and HIV

World AIDS Day

There is a link between mathematics and HIV that goes beyond the geometric structure of the virus, based on the icosahedron. Denise Kirschner describes this relationship very well in Using Mathematics to Understand HIV Immune Dynamics:

Since the early 1980s there has been a tremendous effort made in the mathematical modeling of the human immunodeficiency virus (HIV), the virus which causes AIDS (Acquired Immune Deficiency Syndrome). The approaches in this endeavor have been twofold; they can be separated into the epidemiology of AIDS as a disease and the immunology of HIV as a pathogen (a foreign substance detrimental to the body).⁽¹⁾

The paper focuses on HIV immunology:

Our goal then is to better understand the interaction of HIV and the human immune system for the purpose of testing treatment strategies.⁽¹⁾

The behavior of the immune system is schematized in this way:

The entropy and the halting probability problem

The third law of thermodynamics states:

It is impossible for any procedure to lead to the isotherm \(T = 0\) in a finite number of steps.

The theorem, discovered by Walther Nernst, is equal to say:

It is impossible for any process, no matter how idealized, to reduce the entropy of a system to its zero point value in a finite number of operations.

In classical thermodynamics we can define entropy, or the variation of entropy \(\Delta S\), with the following equation: \[\Delta S = \frac{\Delta Q}{T}\] where \(\Delta Q\) is the heat's variation and \(T\) is the temperature.

The Berry's phase and the black hole

In quantum mechanics a geometric phase, also called Berry phase, is a phase difference that a given physical system acquires during a cycle in which the system itself is under the action of an adiabatic process. This phase is linked to the geometric properties of the system itself (which is a simplification, but for our purposes there is no need to go into too much detail).
It was discovered independently by Shivaramakrishnan Pancharatnam in 1956⁽¹⁾, Hugh Christopher Longuet-Higgins⁽²⁾ in 1958 and subsequently generalized by Michael Berry⁽³⁾ in 1984. This phase, although geometric, has measurable physical effects, for example in an interference experiment. An example of a geometric phase is Foucault's pendulum.
The most famous version of this experiment, designed by Léon Foucault, dates back to 1851 when the French physicist, with the aim of showing the rotation of the Earth around its axis, suspended a ball of 28 kilograms of lead coated with brass over a surface of sand using a 67 meter cable hooked to the top of the dome of the Panthéon in Paris. The plane of the pendulum was observed to rotate clockwise at approximately 11.3 degrees per hour, completing a full circle in 31.8 hours. A more refined examination shows that after 24 hours there is a difference between the initial and final orientation of the trace left on Earth which is equal to

Spider-man's magical snake

Ernő Rubik is one of the best known puzzle creators of the last 45 years: his best known puzzle, the Rubik's cube, was invented in 1974 and then marketed first as Hungarian Magic Cube in 1977 and then as Rubik's Cube in 1980. Rubik designed a second puzzle, dates to 1981, also based on the same principle of the Cube. The puzzle also had an exceptional testimonial, Spider-Man, in a one-page story: The mystery of the museum snakes. During the story, Spider-Man used the puzzle as the best trap to catch a gang of thieves.
But what is this new puzzle? Let's read it in the words of its creator:

Earth's albedo and global warming

It's actually quite concerning. For some time, many scientists had hoped that a warmer Earth might lead to more clouds and higher albedo, which would then help to moderate warming and balance the climate system. But this shows the opposite is true.

In this way Edward Schwieterman⁽¹⁾ commented the result of a new paper about the Earth's climate. But first of all we must say what is albedo:

(...) is the measure of the diffuse reflection of solar radiation out of the total solar radiation and measured on a scale from 0, corresponding to a black body that absorbs all incident radiation, to 1, corresponding to a body that reflects all incident radiation.

Now, a black body, an idealized opaque, non-reflective body, emits a thermal electromagnetic radiation that we could estimate also for the Earth. If we modelled it as a perfect black body, we find a temperature about 254.356 K, or -18.8 °C. But if we consider also, for example, the albedo, we can find a temperature of 245 K for albedo equals to 0.4, and a temperature of 255 K for albedo equals to 0.3. So, if the albedo decreases, Earth's temperature increases. And this is exactly what the researchers found.

Goode, P. R., Pallé, E., Shoumko, A., Shoumko, S., Montañes‐Rodriguez, P., & Koonin, S. E. (2021). Earth's Albedo 1998–2017 as Measured From Earthshine. Geophysical Research Letters, 48(17), e2021GL094888. doi:10.1029/2021GL094888

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1. Earth is dimming due to climate change ↩︎

Our flat, fractal universe

In order to evaluate the curvature of a space, we drawn a triangle and measure its internal angles. If the value is approximately 180°, the space is flat; if it is greater than 180 degrees, the space is like a sphere; if less than 180°, the space is a kind of saddle. To evaluate the curvature of a space, however, we need to find sufficiently large triangles: if we try to draw a triangle on the ground, it will most likely be a flat triangle, but if we try to draw a triangle, from space, with the extremes of the Sicily, we will have a spherical triangle. Similarly, for the universe, we must determine a triangle as large as possible. At this point we could take three stars and draw a triangle: the only complication is finding three stars that are at the same time from the moment the cosmic expansion began, and this thing is not exactly easy to determine. This forces us to examine a widespread signal that we are certain is from the same period in the universe timeline: the cosmic microwave background.

Leonardo, a comics genius

Léonard by Turk & De Groot is a particularly long-lived humorous series: after having made its debut in 1975 on the pages of Achille Talon magazine, it was subsequently serialized starting from March 1977 in a series of volumes, now in its 51st edition. June 2020. Now the first two volumes are also available in english thanks to the digital edition of Europe Comics (volume 1 and volume 2).
Originally Bob De Groot, the screenwriter, had imagined an inventor named Methuselah as the long-lived biblical character, but later opted to focus on Leonardo da Vinci. On the other hand, this initial idea leaves traces in the drawings of Philippe Liégeois, known as Turk: Leonardo, in fact, is outlined with a white bum constantly in motion.
The two authors focus above all on Leonardo the inventor, a choice that allows them to show the scientist's variety of interests and his brilliant and multifaceted mind. With an irreverent spirit, the two belgian cartoonists create a series of gags, some of a purely visual page, others developed over a dozen pages, in which one laughs not only with, but also about Leonardo.
A heartfelt tribute to one of the greatest geniuses in the history of Italy and the world.