Roger Penrose invented ingenious mathematical
methods to explore Albert Einstein’s general theory
of relativity. He showed that the theory leads to the
formation of black holes, those monsters in time
and space that capture everything that enters them.
Nothing, not even light, can escape.
Reinhard Genzel and Andrea Ghez each lead a
group of astronomers who have focused on a
region at the centre of the Milky Way since the
early 1990s. With increasing precision, they
have mapped the orbits of the brightest stars
that are closest to the centre. Both groups found
something that is both invisible and heavy,
forcing this jumble of stars to swirl around.
This invisible mass has about four million solar
masses squeezed together in a region no larger than our solar system. What is it that makes the stars
at the heart of the Milky Way swing around at such astonishing speeds? According to the current
theory of gravity, there is only one candidate – a supermassive black hole.
If you are aspiring for Nobel prize you should take this class seriously! Estimation of the mass of the supermassive blackhole from observed positions of stars circling around it is a nice example of data modelling, regression. :-)
A nice summary video (from 2009!) describing the discovery of our Milky Way’s supermassive black hole and the technologies that made it possible.
Maximum Likelihood Estimation vs. Least Square Fit summary slides and notebook
Can we improve simple algorithms like multiplication?
Nemeth Robert’s question from last class: Where can I find a derivation for the fourth order Runge-Kutta method which follows a similar logic as the one for rk2 in Landau’s book (in 9.5.2.)?
Answer: I thought that it is presented in Numerical Recipes, but it seems that derivation of the 4th order Runge-Kutta is more tedious than I thought. It is not derived in the NR book, but I have found this “simplified” derivation: https://www.researchgate.net/publication/49587610_A_Simplified_Derivation_and_Analysis_of_Fourth_Order_Runge_Kutta_Method
Machine learning in physics review: Carleo, G., Cirac, I., Cranmer, K., Daudet, L., Schuld, M., Tishby, N., Vogt-Maranto, L. and Zdeborová, L., 2019. Machine learning and the physical sciences. Reviews of Modern Physics, 91(4), p.045002. arxiv
2020.11.03
Fourier analysis
Viusal interactive explanation of Fourier analysis
In 1975, Dr. Feigenbaum, using the small HP-65 calculator discovered that the ratio of the difference between the values at which such successive period-doubling bifurcations occur tends to a constant of around 4.6692.
Sympletic integrators for nonlinear Hamiltonian systems: https://en.wikipedia.org/wiki/Symplectic_integrator
Corless, R.M., 1994. What good are numerical simulations of chaotic dynamical systems?. Computers & Mathematics with Applications, 28(10-12), pp.107-121. link
Li, X. and Liao, S., 2018. Clean numerical simulation: a new strategy to obtain reliable solutions of chaotic dynamic systems. Applied Mathematics and Mechanics, 39(11), pp.1529-1546. link
Boekholt, T.C.N., Portegies Zwart, S.F. and Valtonen, M., 2020. Gargantuan chaotic gravitational three-body systems and their irreversibility to the Planck length. Monthly Notices of the Royal Astronomical Society, 493(3), pp.3932-3937. link : “… using the accurate and precise N-body code Brutus, which goes beyond standard double-precision arithmetic … three massive black holes with zero total angular momentum, we conclude that up to five percent of such triples would require an accuracy of smaller than the Planck length in order to produce a time-reversible solution”
HW: Read Ch. 15
2020.11.24
2020.12.01
NEWS: The “holy grail” of molecular dynamics, protein folding is solved:
