Modeling Probability Distributions and Solving Differential Equations
See Spread Polynomials:
27:48 On factorization of spread-polynomials, see Lewis Carroll - Crocodile Story
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JPL Orbits & Ephemerides: https://ssd.jpl.nasa.gov/orbits.html
See also this very impressive student project, which is a digital orrery that uses the JPL Ephemerides: https://www.robots.ox.ac.uk/~mjc/Papers/3YP_Report.pdf This seems to be the basis for a way to make a space-flight simulator for planets.
See Trying to Make a Sci-Fi Movie for the context. See also Laplace Transform:
Algebraic construction
The Laplace transform can be alternatively defined in a purely algebraic manner by applying a field of fractions construction to the convolution ring of functions on the positive half-line. The resulting space of abstract operators is exactly equivalent to Laplace space, but in this construction the forward and reverse transforms never need to be explicitly defined (avoiding the related difficulties with proving convergence).
Mikusiński, Jan. Operational Calculus (14 July 2014).
Pure and Applied Mathematics, Volume 109: Operational Calculus, Second Edition. Volume I presents the foundations of operational calculus and its applications to physics and engineering. This book introduces the operators algebraically as a kind of fractions. Organized into three parts, this volume begins with an overview of the concept as well as the characteristics of a convolution of continuous functions. This text then examines the transitivity, associativity, and distributivity of convolution with regard to addition. Other parts consider the methods of solving other difference equations, particularly in the field of electrical engineering, in which the variable runs over integer values only. This book discusses as well the solution of differential equations under given initial conditions. The final part deals with the characteristic properties of a derivative and provides the definition of algebraic derivative to any operators. This book is a valuable resource for physicists, electrical engineers, mathematicians, and research workers.
See also p-norm, Field of Fractions, Convolution quotient and Pontryagin duality:
In mathematics, Pontryagin duality is a duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group (the multiplicative group of complex numbers of modulus one), the finite abelian groups (with the discrete topology), and the additive group of the integers (also with the discrete topology), the real numbers, and every finite-dimensional vector space over the reals or a p-adic field.
It sounds to me like I might have hit the nail on the head when I said that Lev Pontryagin's success in control theory was [partly!] a result of his earlier work on number theory. It's the only way I could explain how the Soviets produced such fantastically accurate control systems engineering for their solar-system exploration without having digital computers. They were doing all the calculations by hand, so they must have had some really efficient methods of managing them. See also Chris Mack on Career Planning and the Principles of Metrology. Why is his company called Fractilia?
I say partly because obviously he had a lot of students working with him, and he was blind, so he had to do a lot of stuff in his head.
Jade on The Gettier Problem
8:25 I think she means "A true belief counts as knowledge only if it not just because of some accident or luck that it is true." Otherwise she makes no sense at all, to me.
Her latest video, so you can think about what it means to know some situation is "fair":
Maybe you can use Pi Picos to simulate this stuff (in your car, as it drives down the road taking your kid to school): CAN 2.0 driver using PIO coprocessors on the RP2040.
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Normal Wildberger on Analytic versus Synthetic approaches to geometry:
See Pascal's theorem. Here he is talking about combinatorial approaches to functional relations as formal power series.
See Freya Holmér on Continuity of Splines for some more cute polynomials.
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Back to the Sci-fi movie: Jack Wisdom on two problems in the Solar System: chaotic planetary motions and chaotic climate models:
6:48 On transporting meteorites to earth orbit and removing material from the Kirkwood gaps: he seems to be suggesting that some of that material ended up on Earth:
Most of the Kirkwood gaps are depleted, unlike the mean-motion resonances (MMR) of Neptune or Jupiter's 3:2 resonance, that retain objects captured during the giant planet migration of the Nice model. The loss of objects from the Kirkwood gaps is due to the overlapping of the ν5 and ν6 secular resonances within the mean-motion resonances. The orbital elements of the asteroids vary chaotically as a result and evolve onto planet-crossing orbits within a few million years.
14:47 As to the two origins of Chaos Theory in conservative and dissipative systems in the early sixties: see Hénon–Heiles system (Hénon, M.; Heiles, C. (1964). "The applicability of the third integral of motion: Some numerical experiments". The Astronomical Journal. 69: 73–79.) and Lorenz system (Lorenz, Edward Norton (1963). "Deterministic nonperiodic flow". Journal of the Atmospheric Sciences. 20 (2): 130–141.). Lorenz' work on this started in the late fifties. See The statistical prediction of solutions of dynamic equations (1962)
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John F. Kennedy at Rice University, 12 September 1962:
In the last 24 hours we have seen facilities now being created for the greatest and most complex exploration in man's history. We have felt the ground shake and the air shattered by the testing of a Saturn C-1 booster rocket, many times as powerful as the Atlas which launched John Glenn, generating power equivalent to 10,000 automobiles with their accelerators on the floor. We have seen the site where five F-1 rocket engines, each one as powerful as all eight engines of the Saturn combined, will be clustered together to make the advanced Saturn missile, assembled in a new building to be built at Cape Canaveral as tall as a 48 story structure, as wide as a city block, and as long as two lengths of this field.
Within these last 19 months at least 45 satellites have circled the earth. Some 40 of them were "made in the United States of America" and they were far more sophisticated and supplied far more knowledge to the people of the world than those of the Soviet Union.
The Mariner spacecraft now on its way to Venus is the most intricate instrument in the history of space science. The accuracy of that shot is comparable to firing a missile from Cape Canaveral and dropping it in this stadium between the 40-yard lines.
Transit satellites are helping our ships at sea to steer a safer course. Tiros satellites have given us unprecedented warnings of hurricanes and storms, and will do the same for forest fires and icebergs.
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