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.

A chaotic balance

Our mathematical history begins in a discipline that, apparently, has very little to do with mathematics: biology. In 1975 on the journal Nature Robert May, an australian ecologist, publishes a review article with a rather indicative title: Simple mathematical models with very complicated dynamics⁽¹⁾. The heart of the paper is the following equation: \[x_{t+1} = a x_t (1 - x_t)\] The equation, or logistic map, this is its name, describes the rate of change of a population in function of the parameter \(t\) (the time), that varies in a discrete rather than continuous way, while \(a\) is a constant that identifies the growth rate of a population. Insteed \(x_t\) is the ratio between the existing population and the maximum possible population at time \(t\).
The model thus described is deterministic, i.e. the population at instant 0 determines the population at subsequent instants. The equation predicts the existence of a stationary state, i.e. a situation in which the population at time \(t + 1\) is equal to the population at time \(t\). This state is stable, that is, it is maintained for a sufficiently long time, but only for \(a\) lower than or equal to 3. However when the growth rate exceeds this value, the size of the population begins to oscillate between 0 and 1, apparently in a random way. But if we observe carefully, we notice small more or less periodic recurrences, which show how the behavior of the equation is actually chaotic.

Travelling on Sputnik 2

The Mysterious Traveler was a multimedia project as it could only be before the advent of the world wide web: it was a radio program, which started on the 5th december, 1943 and went on, with mixed fortunes, until the 16th september, 1952; an anthological magazine (on which Ray Bradbury among others wrote short stories), published between 1951 and 1952; and a comic book, also anthological, of which 13 issues were released every two months between august 1956 and june 1959 (not counting the two volumes of 1985). Published by Charlton, it had Steve Ditko as its leading artist (he was not the only one, anyway) and, like the other two products that preceded it, contained fantasy and science fiction stories with a hint of crime.
The protagonist of Tales of the Mysterious Traveler (this is the name of the comic book) is a... mysterious traveler in a raincoat and with a wide-brimmed hat pulled over his eyes. The mysterious traveler moved from the most disparate corners of the universe and there was no barrier capable stopping him, neither the boiling heart of a planet, nor the cold and dark desolation of outer space.
On #12, the mysterious traveler is sent (perhaps) by Joe Gill and Bill Molno aboard Sputnik 2. This was the second object launched by the Soviets into space, and the first to bring a living being on board, the dog Laika.

Fluorine in the early universe

Using ALMA (Atacama Large Millimeter/submillimeter Array), a team of astronomers has detected fluorine in a galaxy far, far away: its light reached us after a journey of over 12 billion years.
What we see of the NGP-190387 galaxy is a large cloud of gas crystallized at the time when the universe was only 1.4 billion years old. And since stars shed chemical elements into their surroundings only when they reach the end of their life, which generally ends explosively, the detection of fluorine in the gases of NGP-190387 implies that the stars in the galaxy must have lived relatively short lives. In particular these characteristics are possessed by the Wolf–Rayet stars.
Furthermore, the levels of fluorine in NGP-190387 (the first galaxy after the Milky Way where this chemical element was observed) are comparable to those of our galaxy, with the difference that the latter is older than a dozen and more than billions of years compared to NGP-190387.

We have shown that Wolf–Rayet stars, which are among the most massive stars known and can explode violently as they reach the end of their lives, help us, in a way, to maintain good dental health!

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- Maximilien Franco from the University of Hertfordshire in the UK

(ESO's press release)

The space eye of Sauron

When the first image of the black hole, M87*, was released, several memes circulated online that repositioned the photo in different contexts. One of the best known was the one that placed M87* in the center of Sauron's eye as it was displayed in Peter Jackson's The Lord of the Rings trilogy.
The photo I present above, however, taken in 2008 by the Hubble Space Telescope is much more reminiscent of the evil eye of Sauron. It represents the debris disk around the star Fomalhaut, a white star in the constellation of Piscis Austrinus approximately 25 light years away. In 2008, an exoplanet was also discovered, Fomalhaut b (also known as Dagon, a perfect name for Halloween parties!), although there are still doubts about its existence (probably it does not exist, at least not yet).
The curiosity about this star is that the protagonist of Stanislaw Lem's Return from the Universe returns to Earth after a space exploration trip right around Fomalhaut: the book is dated 1961, almost fifty years before astronomers discovered clues about the possible existence of Dagon.

Total's responses to global warming

Building upon recent work on other major fossil fuel companies, we report new archival research and primary source interviews describing how Total responded to evolving climate science and policy in the last 50 years. We show that Total personnel received warnings of the potential for catastrophic global warming from its products by 1971, became more fully informed of the issue in the 1980s, began promoting doubt regarding the scientific basis for global warming by the late 1980s, and ultimately settled on a position in the late 1990s of publicly accepting climate science while promoting policy delay or policies peripheral to fossil fuel control. Additionally, we find that Exxon, through the International Petroleum Industry Environmental Conservation Association (IPIECA), coordinated an international campaign to dispute climate science and weaken international climate policy, beginning in the 1980s. This represents one of the first longitudinal studies of a major fossil fuel company's responses to global warming to the present, describing historical stages of awareness, preparation, denial, and delay.

Bonneuil, C., Choquet, P. L., & Franta, B. (2021). Early warnings and emerging accountability: Total’s responses to global warming, 1971–2021. Global Environmental Change, 102386. doi:10.1016/j.gloenvcha.2021.102386

Rewind a black hole story

Glass bubbles from black hole

In terms of the length of human life, we can conclude that this is a bit impossible to reconstruct the story of a particular star. Observing all the star in the universe we can create a model about their evolution, but we observe with a great details cosmo only since a century or so. Now, thanks to a particular device, the LoFar, Low Frequency Array, a team of astronomers collected data about the last 100000 years of the black hole at the center of Nest200047.
LoFar is a radiotelescope that collects radiation produced by the oldest electrons that are in the neighbour of a cosmic object. In this way researchers can go literally back in time along the story of Nest200047*.
During its phases of activity, the black hole devours the surrounding material and in this process releases a large amount of energy, sometimes even in the form of jets of particles that move at the speed of light and emit radio waves. These jets generate bubbles of particles and magnetic fields which by expanding are able to heat and move the intergalactic medium that surrounds them, enormously influencing its evolution and therefore the rate at which stars are formed.

Butterflies, hurricanes and... pools!

Chaos is nothing more than order seen from the opposite side.

This defintion by Fethry Duck in the italian story Il mobile caotico (The chaotic furniture) can be considered very centered on the heart of chaos. And the mathematical tool that we used to study it is the theory of chaos.

Flapping the wings

What best identifies chaos theory is the butterfly effect, which identifies in a simple and effective way the strong dependence of chaotic systems on initial conditions. The name was first used by Edward Lorentz, who published the first article on this effect in 1963⁽¹⁾.
The popular version of the butterfly effect goes something like this: The flapping of a butterfly's wings in Brazil causes a hurricane in New York and the use of the butterfly was probably suggested to Lorentz from Ray Bradbury's 1952 short story A sound of thunder in which an unwary time traveler, stepping out of the path set by the travel agency and thus stepping on a butterfly, even manages to change the result of the last US presidential elections, allowing a fascist to become the most powerful man on the planet!
From a scientific point of view, one of the most typically chaotic problems is that of weather forecasts, because of the large amount of variables that are present. The appearance of chaotic behaviors, however, would not be so scientifically interesting if it were not for one of their particular characteristics: the fundamental laws that govern, for example, time are deterministic and individually easily solved, but by combining together a large number of such equations, not only the resolution of the system is more complicated, so much so that it is necessary to use electronic calculators, but also the solution shows a chaotic behavior graphically well identified by the Lorentz attractors:

42: A family portrait

No, this is not the towel day, but what ESO released yesterday is undoubtedly something quite useful for any space tourist: a series of 42 detailed images of the largest asteroids in the solar system.
The main asteroid belt, located just beyond the orbit of Mars, is made up of rocky objects of various sizes, reaching up to 200 km in diameter, without forgetting the largest of all, Ceres and Vesta, respectively 940 and 520 kilometers in diameter. The family portrait of the 42 asteroids was made using the Very Large Telescope:

Nobel Prize in Chemistry 2021: A scent of Feynman

One of the most famous speech by Richard Feynman is There's plenty of room at the bottom:

Now comes the interesting question: How do we make such a tiny mechanism? I leave that to you. However, let me suggest one weird possibility. You know, in the atomic energy plants they have materials and machines that they can’t handle directly because they have become radioactive. To unscrew nuts and put on bolts and so on, they have a set of master and slave hands, so that by operating a set of levers here, you control the “hands” there, and can turn them this way and that so you can handle things quite nicely.

The idea is to manipulate molecules to build, for example, an electric engine, or a book, or something else. The most curious fact about the Nobel Prize in Chemistry 2021 is that Johan Jarnestad has illustrated the work of Benjamin List and David MacMillan using a couple of workers, an image that, in a particular way, is very similar to Feynman's idea.

Building molecules is a difficult art. Benjamin List and David MacMillan are awarded the Nobel Prize in Chemistry 2021 for their development of a precise new tool for molecular construction: organocatalysis. This has had a great impact on pharmaceutical research, and has made chemistry greener.

I hope to write soon an article about Feynman and miniaturization obviously from the physics point of view.
Stay tuned!

Giorgio Parisi: A Nobel for complex systems

The last time an italian was awarded the Nobel Prize in physics was in 2002: Roberto Giacconi for his pioneering research in the field of X-ray radiation from the universe. Another italian research that probably could win the Prize was Adalberto Giazotto, who designed the VIRGO interferometer, that with LIGOs shared the first observation of gravitational waves. The Swedish Academy decided to assign the Prize to three of the LIGO's founders, Rainer Weiss, Barry Barish and Kip Thorne. But this is not a great problem: after all, the Nobel Prize serves to emphasize personal contributions, but also to establish key points in the knowledge, and in this sense, the role of Italy had already been indicated as fundamental.
Today, however, a long-awaited award arrives: Giorgio Parisi, theoretical physicist, whose works have provided important contributions to field theory and statistical physics, won the Nobel Prize in physics

for the discovery of the interplay of disorder and fluctuations in physical systems from atomic to planetary scales

Dark scent

The hand of creation by John Byrne from Green Lantern: Ganthet's Tale

The XENON1T experiment at Gran Sasso's laboratories in Italy is a liquid xenon detector designed for serach of the mysterious dark matter. About one year ago researchers observed an excess of events: 53 more than the 232 predicted⁽¹⁾.
There are several explanations: an unexpected noise' source; the passage of some axions, an hypothetical particle predicted by Roberto Peccei and Helen Quinn in 1977; some neutrinos that interact with matter in a new way.
Or dark matter's traces⁽²⁾. There are two new articles suggesting this; in particular the second suggests that the detected excess in XENON1T is dued by chameleons⁽³⁾.
The chameleon is an(other) hypothetical particle proposed in 2003 by Justin Khoury and Amanda Weltman, that couples to matter more weakly than gravity, and so a candidate for dark matter. It has a variable mass, so the hypothetical fifth force mediated by the chameleons can evade the constraints deduced by experiments on the equivalence principle. In this way chameleons could drive the observed acceleration of the universe's expansion, but it's very difficult to verify experimentally.

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1. XENON collaboration. (2020). Excess electronic recoil events in XENON1T. Physical Review D, 102(7). doi:10.1103/PhysRevD.102.072004 ↩︎
2. Aboubrahim, A., Klasen, M., & Nath, P. (2021). Xenon-1T excess as a possible signal of a sub-GeV hidden sector dark matter. Journal of High Energy Physics, 2021(2), 1-21. doi:10.1007/JHEP02(2021)229 ↩︎
3. Vagnozzi, S., Visinelli, L., Brax, P., Davis, A. C., & Sakstein, J. (2021). Direct detection of dark energy: the XENON1T excess and future prospects. Physical Review D, 104(6). doi:10.1103/PhysRevD.104.063023 ↩︎

The Case for Dwarf K Stars

61 Cygni, a binary K-type star system - via commons

The stellar main-sequence is a strip that cuts diagonally the Hertzsprung-Russell diagram, a star's plot of stellar color versus brightness. The main feature of main-sequence stars is that they burn hydrogen. In this group the K dwarfs, or orange dwarfs, are intermediate stars in size between red M stars (red dwarfs) and yellow G stars. Their mass is between 0.5 and 0.8 times the Sun's mass and surface temperature is bewteen 3900 and 5200 K.
In the last few years these type of stars have become particularly interesting to astronomers, as they appear to have the characteristics to host life-as-we-know:

Uchuu: Universes' creator

If you are a superheroes' comics readers, you probably know All-Star Superman by Grant Morrison and Frank Quitely (if you want, I could publish a review of this comic). At some point in the story, Superman designs a small cubic universe to see what would happen on a planet like Earth without his presence. The development of intelligent life was also included in the Superman's simulation, but in essence even those of astronomers are structured in the same way: a cube of space of finite dimensions whose evolution is driven by a network of dark matter and dark energy.
At the end of the july 2021 it was realased Uchuu, presented as a suite of large high-resolution cosmological N-body simulations, in practice, a simulation that shows the evolution of dark matter structures in a cube of 9.63 billion light years on each side and made up of 2.1 trillion particles.

Uchuu's main goal is to shed light on the dark matter halos surrounding galaxies, but the researchers think that another field of use for their simulation is the study of gravitational lenses.
In any case, it is a tool that could be very useful for improving the algorithms generally used in astronomy to process the data collected by instruments such as satellites and telescopes.

Ishiyama, T., Prada, F., Klypin, A. A., Sinha, M., Metcalf, R. B., Jullo, E., ... & Vega-Martínez, C. A. (2021). The Uchuu simulations: Data Release 1 and dark matter halo concentrations. Monthly Notices of the Royal Astronomical Society, 506(3), 4210-4231. doi:10.1093/mnras/stab1755 (arXiv)

Read also:
Skies & Universes
Uchuu project on Git-Hub

The dog-bone asteroid

Using the European Southern Observatory's Very Large Telescope (ESO's VLT), a team of astronomers have obtained the sharpest and most detailed images yet of the asteroid Kleopatra. The observations have allowed the team to constrain the 3D shape and mass of this peculiar asteroid, which resembles a dog bone, to a higher accuracy than ever before. Their research provides clues as to how this asteroid and the two moons that orbit it formed.

Kleopatra also possesses another characteristic: a two-moon system discovered in 2008 by Franck Marchis' team at the Keck Observatory.
It is interesting to observe that the dynamics of Kleopatra's three-body system and its moons turn out to be chaotic. I hope to soon publish an article on the problem of the three bodies to clarify this aspect.

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Read ESO's press release

Marchis, F., Jorda, L., Vernazza, P., Brož, M., Hanuš, J., Ferrais, M., ... & Yang, B. (2021). (216) Kleopatra, a low density critically rotating M-type asteroid. Astronomy&Astrophysics, 653. doi:10.1051/0004-6361/202140874
Broz, M., Marchis, F., Jorda, L., Hanuš, J., Vernazza, P., Ferrais, M., ... & Yang, B. (2021). An advanced multipole model for (216) Kleopatra triple system. Astronomy&Astrophysics, 653. doi:10.1051/0004-6361/202140901

Descamps, P., Marchis, F., Berthier, J., Emery, J. P., Duchêne, G., De Pater, I., ... & Macomber, B. (2011). Triplicity and physical characteristics of Asteroid (216) Kleopatra. Icarus, 211(2), 1022-1033. doi:10.1016/j.icarus.2010.11.016

Professor Politzer and the Rho Mesons: Simple Harmonic Oscillator

I want to talk today about things that shake and I hope my words aren't too opaque. One degree of freedom moving to and fro just how it moves we'd like to know we can represent all kinds of things by a single mass between ideal springs. Each spring's connected to a wall so the outer ends don't move at all
Let the mass be $m$ spring constant $k$ but don't let friction get in the way use Newton's laws and what have we got $F$ equals $m$ psi double dot that is also minus escape psi times 2 so now we have a diff eq and we can write down the general solution for the simple harmonic time evolution
Let omega be root 2 $k$ over $m$ here's the answer won't repeat again size a cosine omega t plus a phase call it $b$ so it's all very simple and you can see for any initial psi and velocity we can find the constants $a$ and $b$ and the equations exact for all time $t$
Now look again at the diff eq. It's homogeneous and linear too so if you add two solutions together there sums a solution that's even better. We call it the principle of superposition. You can use it to fit the boundary condition in fact there is no contradiction if we use it in a system that does have friction
In a real system nothing's perfect of course we have to include the frictional force suppose it goes as the velocity right minus $m$ gamma d psi dt now if the damping is not too strong our old solution is close but wrong see it starts out with some amplitude $a$ but after a while it just dies away
The amplitude decays exponentially as you can see experimentally as $e$ to the minus half gamma $t$. Now it's almost right but you see the frequency is lower as we can compute omega is now given by the square root of the quantity $k$ over $m$ times two minus quarter gamma squared now we're through
So now we have the complete solution for an oscillator's time evolution and when there's damping as everyone knows the amplitude decays and the frequency slows if we have two solutions no matter how chose you know we can always superpose and since you all find physics such fun to problems 12 18 and 21.

Class dismissed

In the video I use the notation used on en.wiki

Read also:
Explenation of the system with some animation | A simulation on Go-Lab | A simulation using GeoGebra

(audio source)