Pi and the Basel's problem

In 1644 the Italian mathematician Pietro Mengoli proposed the so-called Basel's problem, which asked for the exact solution to the square of the sum of the reciprocals of all the natural numbers: \[\sum_{n=1}^\infty \frac{1}{n^2} = \frac{1}{1^2} + \frac{1}{2^2} + \cdots\] The solution to the problem came in 1735 thanks to Leonard Euler, at the time at the beginning of his brilliant career as a problem solver. The Swiss mathematician proved that the exact sum of the series is $\pi^2 / 6$.
The Euler's demonstration, published in its final form in 1741, is particularly interesting: Euler supposed that it's possible to apply the rules of the finite polynomials even those endless.
We start with the development in Taylor series for the sine function in 0: \[\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots\] Dividing by $x$ both terms, we obtain: \[\frac{\sin(x)}{x} = 1 - \frac{x^2}{3!} + \frac{x^4}{5!} - \frac{x^6}{7!} + \cdots\] whose roots are $\pi$, $-\pi$, $2\pi$, $-2\pi$, $3\pi$, $-3\pi$, $\ldots$ By changing the variable as $z = x^2$, the polynomial above becomes: \[\frac{\sin(\sqrt{z})}{\sqrt{z}} = 1 - \frac{z}{3!} + \frac{z^2}{5!} - \frac{z^3}{7!} + \cdots\] whose roots are $\pi^2$, $4\pi^2$, $9\pi^2$, $\ldots$
Now, given a polynomial $a_n x^n + \cdots + a_3 x^3 + a_2 x^2 + bx + 1$, for the formulas of Viète, we have that the sum of the reciprocals of its roots has as result $-b$. Applying this result for finished polynomials to infinite polynomial in $z$ above, we get: \[\frac{1}{3!} = \frac{1}{6} = \frac{1}{\pi^2} + \frac{1}{4\pi^2} + \frac{1}{9\pi^2} + \frac{1}{16\pi^2} + \cdots\] and so: \[\frac{\pi^2}{6} = 1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \cdots = \sum_{n=1}^\infty \frac{1}{n^2}\] It's simple to observe the connection between Mengoli's series and Riemann's zeta \[\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}\] with $s=2$.
Last observation: in 1982 on the magazine Eureka, it appeared a rigorous proof of Euler's result signed by John Scholes, although it seems that such a demonstration circulated already to late sixties between Cambridge corridors.

JMP 58, 1: magnetic monopoles, spacetime and gravity

Just another selection of papers from the Journal of Mathematical Physics. I would start with the folowing paper:
Fine, D., & Sawin, S. (2017). Path integrals, supersymmetric quantum mechanics, and the Atiyah-Singer index theorem for twisted Dirac Journal of Mathematical Physics, 58 (1) DOI: 10.1063/1.4973368
Feynman’s time-slicing construction approximates the path integral by a product, determined by a partition of a finite time interval, of approximate propagators. This paper formulates general conditions to impose on a short-time approximation to the propagator in a general class of imaginary-time quantum mechanics on a Riemannian manifold which ensure that these products converge. The limit defines a path integral which agrees pointwise with the heat kernel for a generalized Laplacian. The result is a rigorous construction of the propagator for supersymmetric quantum mechanics, with potential, as a path integral. Further, the class of Laplacians includes the square of the twisted Dirac operator, which corresponds to an extension of $N = 1/2$ supersymmetric quantum mechanics. General results on the rate of convergence of the approximate path integrals suffice in this case to derive the local version of the Atiyah-Singer index theorem.
Kováčik, S., & Prešnajder, P. (2017). Magnetic monopoles in noncommutative quantum mechanics Journal of Mathematical Physics, 58 (1) DOI: 10.1063/1.4973503
We discuss a certain generalization of the Hilbert space of states in noncommutative quantum mechanics that, as we show, introduces magnetic monopoles into the theory. Such generalization arises very naturally in the considered model, but can be easily reproduced in ordinary quantum mechanics as well. This approach offers a different viewpoint on the Dirac quantization condition and other important relations for magnetic monopoles. We focus mostly on the kinematic structure of the theory, but investigate also a dynamical problem (with the Coulomb potential).

SphereX: the road to BB-8

BB-8 is the last Star Wars' droid introduced in The Force Awakens. It's a spherical robot with a free-moving head. Now, looking on arXiv, I found the proposal for SphereX, a new spherical robot for planetary explorations:
Wheeled planetary rovers such as the Mars Exploration Rovers (MERs) and Mars Science Laboratory (MSL) have provided unprecedented, detailed images of the Mars surface. However, these rovers are large and are of high-cost as they need to carry sophisticated instruments and science laboratories. We propose the development of low-cost planetary rovers that are the size and shape of cantaloupes and that can be deployed from a larger rover. The rover named SphereX is 2 kg in mass, is spherical, holonomic and contains a hopping mechanism to jump over rugged terrain. A small low-cost rover complements a larger rover, particularly to traverse rugged terrain or roll down a canyon, cliff or crater to obtain images and science data. While it may be a one-way journey for these small robots, they could be used tactically to obtain high-reward science data. The robot is equipped with a pair of stereo cameras to perform visual navigation and has room for a science payload. In this paper, we analyze the design and development of a laboratory prototype. The results show a promising pathway towards development of a field system.
The litle robot was tested under simulated lunar and martian gravity conditions, and the results are encouraging:
It was observed that as angle of separation between grouser decreases there is increase in average speed of robot and the power consumption remains almost constant. A hopping mechanism was developed for the robot that enables the robot to in theory perform unlimited hops. Currently the system is able to perform a hop of 8-10 cm under simulated Martian gravity. Extrapolating this, we would be able to achieve 16-20 cm hop in lunar conditions. The performance of hopping mechanism has to be improved to achieve the stated mission requirements. Based on power consumption for each hop and maximum power available, it was calculated that the robot would be able to produce maximum 208 hops in a single charge and robot would operate for 35 minutes of continuous hopping. The proposed SphereX design shows a promising pathway towards further maturation and testing of the technology in the field.

Life on Mars

This Voyager spacecraft was constructed by the United States of America. We are a community of 240 million human beings among the more than 4 billion who inhabit the planet Earth. We human beings are still divided into nation states, but these states are rapidly becoming a single global civilization.
We cast this message into the cosmos. It is likely to survive a billion years into our future, when our civilization is profoundly altered and the surface of the Earth may be vastly changed. Of the 200 billion stars in the Milky Way galaxy, some--perhaps many--may have inhabited planets and spacefaring civilizations. If one such civilization intercepts Voyager and can understand these recorded contents, here is our message:
This is a present from a small distant world, a token of our sounds, our science, our images, our music, our thoughts, and our feelings. We are attempting to survive our time so we may live into yours. We hope someday, having solved the problems we face, to join a community of galactic civilizations. This record represents our hope and our determination, and our good will in a vast and awesome universe.
Jimmy Carter

The IAU firmly opposes any discrimination based on factors such as ethnic origin, religion, citizenship, language, and political or other opinion and therefore expects U.S. officials to not discriminate on the basis of religion.
(read more)

JMP 57, 12: quantum mechanics for the universe

I will begin to publish posts in a more continuous way, also recovering some drafts that I'm not able to conclude in 2016. For now I propose you a selection of articles from the Journal of Mathematical Physics, vol.57, issue 12:
Andersson, A. (2016). Electromagnetism in terms of quantum measurements Journal of Mathematical Physics, 57 (12) DOI: 10.1063/1.4972287
We consider the question whether electromagnetism can be derived from the theory of quantum measurements. It turns out that this is possible, both for quantum and classical electromagnetism, if we use more recent innovations such as smearing of observables and simultaneous measurability. In this way, we justify the use of von Neumann-type measurement models for physical processes. We apply the operational quantum measurement theory to gain insight into fundamental aspects of quantum physics. Interactions of von Neumann type make the Heisenberg evolution of observables describable using explicit operator deformations. In this way, one can obtain quantized electromagnetism as a measurement of a system by another. The relevant deformations (Rieffel deformations) have a mathematically well-defined "classical" limit which is indeed classical electromagnetism for our choice of interaction.
Aerts, D., & Sassoli de Bianchi, M. (2016). The extended Bloch representation of quantum mechanics: Explaining superposition, interference, and entanglement Journal of Mathematical Physics, 57 (12) DOI: 10.1063/1.4973356
An extended Bloch representation of quantum mechanics was recently derived to offer a possible (hidden-measurements) solution to the measurement problem. In this article we use this representation to investigate the geometry of superposition and entangled states, explaining interference effects and entanglement correlations in terms of the different orientations a state-vector can take within the generalized Bloch sphere. We also introduce a tensorial determination of the generators of $SU(N)$, which we show to be particularly suitable for the description of multipartite systems, from the viewpoint of the sub-entities. We then use it to show that non-product states admit a general description where sub-entities can remain in well-defined states, even when entangled. This means that the completed version of quantum mechanics provided by the extended Bloch representation, where density operators are also considered to be representative of genuine states (providing a complete description), not only offers a plausible solution to the measurement problem but also to the lesser-known entanglement problem. This is because we no longer need to give up the general physical principle saying that a composite entity exists and therefore is in a well-defined state, if and only if its components also exist and therefore are also in well-defined states.

Super Science Friends: how to save Newton

There is a new animated series in the city(1): Super Science Friends. Created by Brett Jubinville of Tinman Creative Studios, it was funded the first short through Kickstarter, while the rest of the episodes can be supported on Patreon.
The series is evidently inspired by Super Friends, produced by Hanna-Barbera for DC Comics, with the place of the heroes of JLA taken by some iconic scientists: Marie Curie, with a radioactive ring similar to Green Lantern’s one; Charles Darwin, able to transform into any animal, like Beast Boy from the Teen Titans; Nikola Tesla, with electromagnetic powers, like Magneto from Marvel universe, historical X-Men enemy; Sigmund Freud, father of psychoanalysis with oversight of lust, to be seen as an alternative to Aquaman with his oversight of aquatic creatures.