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)