Rock, paper, scissors, lizard, Spock Goodbye #LeonardNimoy, #Spock from #StarTrek
One popular five-weapon expansion is "rock-paper-scissors-lizard-Spock", invented by Sam Kass and Karen Bryla, which adds "Spock" and "lizard" to the standard three choices. "Spock" is signified with the Star Trek Vulcan salute, while "lizard" is shown by forming the hand into a sock-puppet-like mouth. Spock smashes scissors and vaporizes rock; he is poisoned by lizard and disproven by paper. Lizard poisons Spock and eats paper; it is crushed by rock and decapitated by scissors. This variant was mentioned in a 2005 article of The Times and was later the subject of an episode of the American sitcom The Big Bang Theory in 2008.
The majority of such proposed generalizations are isomorphic to a simple game of modulo arithmetic, where half the differences are wins for player one. For instance, rock-paper-scissors-Spock-lizard (note the different order of the last two moves) may be modeled as a game in which each player picks a number from one to five. Subtract the number chosen by player two from the number chosen by player one, and then take the remainder modulo 5 of the result. Player one is the victor if the difference is one or three, and player two is the victor if the difference is two or four. If the difference is zero, the game is a tie.
Alternatively, the rankings in rock-paper-scissors-Spock-lizard may be modeled by a comparison of the parity of the two choices. If it is the same (two odd-numbered moves or two even-numbered ones) then the lower number wins, while if they are different (one odd and one even) the higher wins. Using this algorithm, additional moves can easily be added two at a time while keeping the game balanced:
  • Declare a move N+1 (where N is the original total of moves) that beats all existing odd-numbered moves and loses to the others (for example, the rock (#1), scissors (#3), and lizard (#5) could fall into the German well (#6), while the paper (#2) covers it and Spock (#4) manipulates it).
  • Declare another move N+2 with the reverse property (such as a plant (#7) that grows through the paper (#2), poisons Spock (#4), and grows through the well (#6), while being damaged by the rock (#1), scissor (#3), and lizard(#5)).

The dark side of the moon

#moon #astronomy #NASA #video #PinkFloyd

The first photo of the lunar far side taken by the Soviet spacecraft Luna 3 on Oct. 7, 1959 - via Universe Today

What Einstein thought about Galilei

about #AlbertEinsten #GalileoGalilei
Galileo's Dialogue Concerning the Two Chief World Systems is a mine of information for anyone interested in the cultural history of the Western world and its influence upon economic and political development.
(...) To begin with, the Dialogue gives an extremely lively and persuasive exposition of the then prevailing views on the structure of the cosmos in the large. The naïve picture of the earth as a flat disc, combined with obscure ideas about star-filled space and the motions of the celestial bodies, prevalent in the early Middle Ages, represented a deterioration of the much earlier conceptions of the Greeks, and in particular of Aristotle’s ideas and Ptolemy’s consistent spatial concept of the celestial bodies and their motions.
(...) In advocating and fighting for the Copernican theory Galileo was not only motivated by a striving to simplify the representation of the celestial motions. His aim was to substitute for a petrified and barren system of ideas the unbiased and strenuous quest for a deeper and more consistent comprehension of the physical and astronomical facts.
The form of dialogue used in his work may be partly due to Plato’s shining example; it enabled Galileo to apply his extraordinary literary talent to the sharp and vivid confrontation of opinion. To be sure, he wanted to avoid an open commitment in these controversial questions that would have delivered him to destruction by the Inquisition. Galileo had, in fact, been expressly forbidden to advocate the Copernican theory. Apart from its revolutionary factual content the Dialogue represents a down-right roguish attempt to comply with this order in appearance and yet in fact to disregard it. Unfortunately, it turned out that the Holy Inquisition was unable to appreciate adequately such subtle humor.
(...) It is difficult to us today to appreciate the imaginative power made manifest in the precise formulation of the concept of acceleration and in the recognition of its physical significance.
Once the conception of the center of the universe had, with good reason, been rejected, the idea of the immovable earth, and, generally, of an exceptional role of the earth, was deprived of its justification (...)
(...) Galileo takes great pains to demonstrate that the hypothesis of the rotation and revolution of the earth is not refuted by the fact that we do not observe any mechanical effects of these motions. Strictly speaking, such a demonstration was impossible because a complete theory of mechanics was lacking. I think it is just in the struggle with this problem that Galileo’s originality is demonstrated with particular force. Galileo is, of course, also concerned to show that the fixed stars are too remote for parallaxes produced by the yearly motion of the earth to be detectable with the measuring instruments of his time. This investigation also is ingenious, notwithstanding its primitiveness.
It was Galileo’s longing for a mechanical proof of the motion of the earth which misled him into formulating a wrong theory of the tides. The fascinating arguments in the last conversation would hardly have been accepted as proofs by Galileo, had his temperament not got the better of him. It is hard for me to resist the temptation to deal with this subject more fully.
The leitmotif which I recognize in Galileo’s work is the passionate fight against any kind of dogma based on authority. Only experience and careful reflection are accepted by him as criteria of truth. Nowadays it is hard for us to grasp how sinister and revolutionary such an attitude appeared at Galileo’s time, when merely to doubt the truth of opinions which had no basis but authority was considered a capital crime and punished accordingly. Actually we are by no means so far removed from such a situation even today as many of us would like to flatter ourselves; but in theory, at least, the principle of unbiased thought has won out, and most people are willing to pay lip service to this principle.
It has often been maintained that Galileo became the father of modern science by replacing the speculative, deductive method with the empirical, experimental method. I believe, however, that this interpretation would not stand close scrutiny. There is no empirical method without speculative concepts and systems; and there is no speculative thinking whose concepts do not reveal, on closer investigation, the empirical material from which they stem. To put into sharp contrast the empirical and the deductive attitude is misleading, and was entirely foreign to Galileo. Actually it was not until the nineteenth century that logical (mathematical) systems whose structures were completely independent of any empirical content had been cleanly extracted. Moreover, the experimental methods at Galileo’s disposal were so imperfect that only the boldest speculation could possibly bridge the gaps between the empirical data. (For example, there existed no means to measure times shorter than a second). The antithesis Empiricism vs. Rationalism does not appear as a controversial point in Galileo’s work. Galileo opposes the deductive methods of Aristotle and his adherents only when he considers their premises arbitrary or untenable, and he does not rebuke his opponents for the mere fact of using deductive methods. In the first dialogue, he emphasizes in several passages that according to Aristotle, too, even the most plausible deduction must be put aside if it is incompatible with empirical findings. And on the other hand, Galileo himself makes considerable use of logical deduction. His endeavors are not so much directed at "factual knowledge" as at "comprehension". But to comprehend is essentially to draw conclusions from an already accepted logical system.
(from the foreword to Dialogue Concerning the Two Chief World Systems: Ptolemaic and Copernican (1953), Einstein Archives 1-174 - via Open Parachute)
About the italian physicist, Galileo Galilei and the impossible biomechanics of giants is an interesting reading.

The mathematics of love

#ValentinesDay #mathematics
\[\left (x^2 + \frac{9}{4} y^2 + z^2 - 1 \right )^3 - x^2 z^3 - \frac{9}{200} y^2 z^3 = 0\]

Black holes and revelations: their large interiors

about #blackhole #cosmology #arXiv #abstract #CarloRovelli
The 3d volume inside a spherical black hole can be defined by extending an intrinsic flat-spacetime characterization of the volume inside a 2-sphere. For a collapsed object, the volume grows with time since the collapse, reaching a simple asymptotic form, which has a compelling geometrical interpretation. Perhaps surprising, it is large. The result may have relevance for the discussion on the information paradox.
Marios Christodoulou & Carlo Rovelli (2014). How big is a black hole?, arXiv:
A sphere $S$ on the event horizon bounds a spacelike hypersurface, a large portion of which coincides with an $r$ = constant hypersurface. We show this hypersurface with one dimension suppressed, and cut in the middle, omitting the long cylindrical part which gives the main contribution to its volume. We also illustrate the argument showing that most of the volume is contained in a region out of causal contact with matter that has advanced far into the black hole.
Ingemar Bengtsson & Emma Jakobsson (2015). Black holes: Their large interiors, arXiv:

Black holes and revelations: the seeds of the galaxies

about #blackhole #astronomy #arXiv #abstract

The centre of the Milky Way - via Nasa
In this paper we present a new scenario where massive Primordial Black Holes (PBH) are produced from the collapse of large curvature perturbations generated during a mild waterfall phase of hybrid inflation. We determine the values of the inflaton potential parameters leading to a PBH mass spectrum peaking on planetary-like masses at matter-radiation equality and producing abundances comparable to those of Dark Matter today, while the matter power spectrum on scales probed by CMB anisotropies agrees with Planck data. These PBH could have acquired large stellar masses today, via merging, and the model passes both the constraints from CMB distortions and micro-lensing. This scenario is supported by Chandra observations of numerous BH candidates in the central region of Andromeda. Moreover, the tail of the PBH mass distribution could be responsible for the seeds of supermassive black holes at the center of galaxies, as well as for ultra-luminous X-rays sources. We find that our effective hybrid potential can originate e.g. from D-term inflation with a Fayet-Iliopoulos term of the order of the Planck scale but sub-planckian values of the inflaton field. Finally, we discuss the implications of quantum diffusion at the instability point of the potential, able to generate a swiss-cheese like structure of the Universe, eventually leading to apparent accelerated cosmic expansion.

Sébastien Clesse & Juan García-Bellido (2015). Massive Primordial Black Holes from Hybrid Inflation as Dark Matter and the seeds of Galaxies, arXiv:

A probabilistic approach to the prime numbers distribution

by @ulaulaman about #prime_numbers #arXiv #mathematics
The prime numbers theorem states the asymptotic approximation for the prime-counting function. The first statement for the theorem is given by Euler in 1737 (pdf):
The sum of the series of the reciprocals of the prime numbers, \[\frac{1}{2} + \frac{1}{3} + \frac{1}{5} + \frac{1}{7} + \frac{1}{11} + \frac{1}{13} + \cdots\] is infinitely large, but it is infinitely many times less than the sum of the harmonic series, \[1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \cdots\] Furthemore, the sum of the former series is like the logarithm of the sum of the latter series.
After Euler, the most known attempt to evaluate the prime numbers distribution is given by Bernard Riemann, and the most recent is posted on arXiv some days ago: