Noether's theorem

The Noether's theorem, discovered by German mathematician Emmy Noether, is one of the most sophisticated theorems in physics, a way to see how group theory, a branch of mathematics believed by many to be abstract, can provide the basis for an important physical concept. The premises of the theory of groups, coupled with the calculus of variations, lead to the conclusions of the theorem: the existence, under certain conditions, of conserved quantities within physical systems.
First of all we start with symmetry, one of the most important concepts for physics, and also the subject og group theory studies. To realize, therefore, this close link, it is enough to have in mind the statement of the theorem:

If a physical system exhibits some continuous symmetry, then there are corresponding observables whose values are constant over time.

A more sophisticated formulation of the theorem, on the other hand, goes something like this:

To every differentiable symmetry generated by local actions there corresponds a conserved current.

This more technical statement links the theorem and the symmetries with some of the most important groups for physics, the Lie groups. In the abstract of the Noether's paper, Invariant Variationsprobleme, in fact, we can read:

The problems in variation here concerned are such as to admit a continuous group (in Lie's sense); the conclusions that emerge for the corresponding differential equations find their more general expression in the theorems formulated in Section I and proved in the following sections.

The Noether's theorem, therefore, ensures that, when a physical system is invariant under the action of the transformations belonging to a Lie group, that is a group in which we are able to differentiate functions, then it certainly exists at least one conserved quantity, and this quantity and its invariance are expressed in the following equation: \[\frac{\text{d}}{\text{d} t} \left ( \frac{\partial L}{\partial \dot x_k} \right ) = \frac{\text{d} p_k}{\text{d} t}\]

Emmy Noether (1918). Invariante Variationsprobleme. de.wikisource.
English translation by Mort Tavel on arXiv

Pi stories: Viète and the infinite series

Spiked Math

Ludolph van Ceulen in 1596 using the polygon method, first came to calculate 20 decimal digits, then 35. Van Ceulen wasn't the last to use the method: for example Willebrord Snellius in 1621 calculated 34 digits, while the Austrian astronomer Christoph Grienberger in 1630 reached a record 38 digits using a 1040-sided polygon: this result is the most accurate ever achieved using the polygon method.
The infinite series supplanted this method: the first to use them in Europe was the French mathematician François Viète in 1593 \[\frac2\pi = \frac{\sqrt2}2 \cdot \frac{\sqrt{2+\sqrt2}}2 \cdot \frac{\sqrt{2+\sqrt{2+\sqrt2}}}2 \cdots\] And in 1655 John Wallis \[\frac{\pi}{2} = \frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot \frac{6}{7} \cdot \frac{8}{7} \cdot \frac{8}{9} \cdots\] European mathematics, however, had come to this method only after Indian mathematics, albeit independently. In India, in fact, there is evidence of first approaches of this kind between 1400 and 1500. The first infinite series used to calculate \(\pi\) is found, in fact, on the pages of the Tantrasamgraha (literally "compilation of systems") of the Indian astronomer Nilakantha Somayaji, circa 1500-1501. The series, presented without any proof (later published in the Yuktibhāṣā, circa 1530), was attributed by Nilakantha to the mathematician Madhava of Sangamagrama, who lived between 1350 and 1425 circa. Apparently Madhava discovered several infinite series, including many that contain the sine, cosine, and tangent. The Indian mathematician used these series to reach up to 11 digits around 1400, a value that was improved around 1430 by the Persian mathematician Jamshīd al-Kāshī using the polygon method.

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 ↩︎