Doc Madhattan: July 2014

A trigonometric proof of the pythagorean theorem

\[\alpha + \beta = \frac{\pi}{2}\] \[\sin (\alpha + \beta) = \sin \frac{\pi}{2}\] \[\sin \alpha \cdot \cos \beta + \sin \beta \cdot \cos \alpha = 1\] \[\frac{a}{c} \cdot \frac{a}{c} + \frac{b}{c} \cdot \frac{b}{c} = 1\] \[\frac{a^2}{c^2} + \frac{b^2}{c^2} = 1\]

\[a^2 + b^2 = c^2\]

via @MathUpdate

Fifty years of CP violation

The CP violation is a violation of the CP-symmetry, a combination between the charge conjugation symmetry (C) and the parity symmetry (P).

CP-symmetry states that the laws of physics should be the same if a particle is interchanged with its antiparticle, and then its spatial coordinates are inverted.

The CP violation is discovered in 1964 by Christenson, Cronin, Fitch, and Turlay (Cronin and Fitch awarded the Nobel Prize in 1980) studying the kaons' decays and it could have a key-role in the matter-antimatter imbalance.
Now the CERN Courier dadicated a special issue about the fifty years of the discovery (download here).

Christenson, J., Cronin, J., Fitch, V., & Turlay, R. (1964). Evidence for the 2π Decay of the K_{2}^{0} Meson Physical Review Letters, 13 (4), 138-140 DOI: 10.1103/PhysRevLett.13.138

Gods, phylosophy and computers

The ontological arguments for the existence of God was introduced for the first time by St. Anselm in 1078:

God, by definition, is that for which no greater can be conceived. God exists in the understanding. If God exists in the understanding, we could imagine Him to be greater by existing in reality. Therefore, God must exist.

There are a lot of phylosophies, mathematics and logicians that proposed their ontological argument, for example Descartes, Leibniz, Frege, and also Kurt Gödel, that proposed the most formal ontological proof:

The proof was published in 1987 (Godel died in 1978), and a lot of logicians discussed around it. One of the last papers published about the argument is an arXiv that suggested to Anna Limind to write that European Mathematicians ‘Prove’ the Existence of God, but the aim of the paper is to control the consistence of the proof and not the reality of the theorem (I think that the theorem is, simply, undecidable), and also to start a new discipline: the computer-phylosophy.
Indeed Benzmüller and Paleo developed an algorothm in order to use a computer to control the ontological proof. So the work:

(...) opens new perspectives for a computerassisted theoretical philosophy. The critical discussion of the underlying concepts, definitions and axioms remains a human responsibility, but the computer can assist in building and checking rigorously correct logical arguments. In case of logico-philosophical disputes, the computer can check the disputing arguments and partially fulfill Leibniz' dictum: Calculemus

Read also: Spiegel Online International

Christoph Benzmüller & Bruno Woltzenlogel Paleo (2013). Formalization, Mechanization and Automation of Gödel's Proof of God's Existence, arXiv: 1308.4526v4

How quantum mechanics explains global warming

The physician Mark Schleupner, a Ronaoke native, writes about global warming:

So, according to NASA scientists, if all the ice in 14 million sq km Antarctica melts, sea levels will rise more than 200 feet. Greenland alone has another huge chunk of the Earth’s water tied up in ice; some scientists say that its ice sheet has passed a tipping point and will be gone in the next centuries, raising ocean levels by 24 feet. These are scary amounts of sea level rise that put huge areas of population centers (New York, Boston, Miami, San Francisco, etc.) under water.
In the end, one can deny climate change (although I’d not recommend it), but one cannot deny math.

Well, it's really interesting, about the climate change, to see the following Ted-Ed lesson:

You've probably heard that carbon dioxide is warming the Earth. But how exactly is it doing it? Lieven Scheire uses a rainbow, a light bulb and a bit of quantum physics to describe the science behind global warming.

Mathematicians discuss the Snowden revelations

In the last period I cannot read the Notices of AMS, so I lost the most recent discussion on this journal about the revelations made by Edward Snowden about NSA. Thanks to the n-category Café I recover the letters about this topic:

In the first part of 2013, Edward Snowden, a former contractor for the National Security Agency (NSA), handed over to journalists a trove of secret NSA documents. First described in the media in June 2013, these documents revealed extensive spying programs of the NSA and other governmental organizations, such as the United Kingdom's GCHQ (Government Communications Headquarters). The disclosures reverberated around the world, influencing the bottom lines of big businesses, the upper echelons of international relations, and the everyday activities of ordinary people whose lives are increasingly mirrored in the Internet and on cell phone networks.
The revelations also hit home in the mathematical sciences community. The NSA is often said to be the world's largest employer of mathematicians; it's where many academic mathematicians in the US see their students get jobs. The same is true for GCHQ in the UK. Many academic mathematicians in the US and the UK have done work for these organizations, sometimes during summers or sabbaticals. Some US mathematicians decided to take on NSA work after the 9/11 attacks as a contribution to national defense.

The discussion on Notices: part 1, part 2

Beach sand for long cycle life batteries

This is the holy grail – a low cost, non-toxic, environmentally friendly way to produce high performance lithium ion battery anodes
Zachary Favors

Schematic of the heat scavenger-assisted Mg reduction process.

Herein, porous nano-silicon has been synthesized via a highly scalable heat scavenger-assisted magnesiothermic reduction of beach sand. This environmentally benign, highly abundant, and low cost SiO₂ source allows for production of nano-silicon at the industry level with excellent electrochemical performance as an anode material for Li-ion batteries. The addition of NaCl, as an effective heat scavenger for the highly exothermic magnesium reduction process, promotes the formation of an interconnected 3D network of nano-silicon with a thickness of 8-10 nm. Carbon coated nano-silicon electrodes achieve remarkable electrochemical performance with a capacity of 1024 mAhg⁻¹ at 2 Ag⁻¹ after 1000 cycles.

Favors, Z., Wang, W., Bay, H., Mutlu, Z., Ahmed, K., Liu, C., Ozkan, M., & Ozkan, C. (2014). Scalable Synthesis of Nano-Silicon from Beach Sand for Long Cycle Life Li-ion Batteries Scientific Reports, 4 DOI: 10.1038/srep05623

(via Popular Science)

Mesons produced in a bubble chamber

A bubble chamber is a pool filled with a liquid (typically hydrogen) such that its molecules are ionized to the passage of a charged particle, thus producing bubbles. In this way the trajectories of the particles are visible and it is possible to study the various decays⁽²⁾.
The bubble chamber was invented by Donald Glaser⁽¹⁾ in 1952, who win the Nobel Prize in 1960.

(1) Glaser, D. (1952). Some Effects of Ionizing Radiation on the Formation of Bubbles in Liquids Physical Review, 87 (4), 665-665 DOI: 10.1103/PhysRev.87.665
(2) Image from the italian version of Weisskopf, V. (1968). The Three Spectroscopies Scientific American, 218 (5), 15-29 DOI: 10.1038/scientificamerican0568-15

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