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Hyperbolic angle - Wikipedia
A unit circle {\displaystyle x^{2}+y^{2}=1} x^2 + y^2 = 1 has a circular sector with an area half of the circular angle in radians. Analogously, a unit hyperbola {\displaystyle x^{2}-y^{2}=1} {\displaystyle x^{2}-y^{2}=1} has a hyperbolic sector with an area half of the hyperbolic angle.
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8 weeks ago by nhaliday
Stability of the Solar System - Wikipedia
The stability of the Solar System is a subject of much inquiry in astronomy. Though the planets have been stable when historically observed, and will be in the short term, their weak gravitational effects on one another can add up in unpredictable ways. For this reason (among others) the Solar System is chaotic,[1] and even the most precise long-term models for the orbital motion of the Solar System are not valid over more than a few tens of millions of years.[2]

The Solar System is stable in human terms, and far beyond, given that it is unlikely any of the planets will collide with each other or be ejected from the system in the next few billion years,[3] and the Earth's orbit will be relatively stable.[4]

Since Newton's law of gravitation (1687), mathematicians and astronomers (such as Laplace, Lagrange, Gauss, Poincaré, Kolmogorov, Vladimir Arnold and Jürgen Moser) have searched for evidence for the stability of the planetary motions, and this quest led to many mathematical developments, and several successive 'proofs' of stability of the Solar System.[5]

...

The planets' orbits are chaotic over longer timescales, such that the whole Solar System possesses a Lyapunov time in the range of 2–230 million years.[3] In all cases this means that the position of a planet along its orbit ultimately becomes impossible to predict with any certainty (so, for example, the timing of winter and summer become uncertain), but in some cases the orbits themselves may change dramatically. Such chaos manifests most strongly as changes in eccentricity, with some planets' orbits becoming significantly more—or less—elliptical.[7]

Is the Solar System Stable?: https://www.ias.edu/ideas/2011/tremaine-solar-system

Is the Solar System Stable?: https://arxiv.org/abs/1209.5996
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9 weeks ago by nhaliday
gn.general topology - Pair of curves joining opposite corners of a square must intersect---proof? - MathOverflow
In his 'Ordinary Differential Equations' (sec. 1.2) V.I. Arnold says "... every pair of curves in the square joining different pairs of opposite corners must intersect".

This is obvious geometrically but I was wondering how one could go about proving this rigorously. I have thought of a proof using Brouwer's Fixed Point Theorem which I describe below. I would greatly appreciate the group's comments on whether this proof is right and if a simpler proof is possible.

...

Since the full Jordan curve theorem is quite subtle, it might be worth pointing out that theorem in question reduces to the Jordan curve theorem for polygons, which is easier.

Suppose on the contrary that the curves A,BA,B joining opposite corners do not meet. Since A,BA,B are closed sets, their minimum distance apart is some ε>0ε>0. By compactness, each of A,BA,B can be partitioned into finitely many arcs, each of which lies in a disk of diameter <ε/3<ε/3. Then, by a homotopy inside each disk we can replace A,BA,B by polygonal paths A′,B′A′,B′ that join the opposite corners of the square and are still disjoint.

Also, we can replace A′,B′A′,B′ by simple polygonal paths A″,B″A″,B″ by omitting loops. Now we can close A″A″ to a polygon, and B″B″ goes from its "inside" to "outside" without meeting it, contrary to the Jordan curve theorem for polygons.

- John Stillwell
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11 weeks ago by nhaliday
multivariate analysis - Is it possible to have a pair of Gaussian random variables for which the joint distribution is not Gaussian? - Cross Validated
The bivariate normal distribution is the exception, not the rule!

It is important to recognize that "almost all" joint distributions with normal marginals are not the bivariate normal distribution. That is, the common viewpoint that joint distributions with normal marginals that are not the bivariate normal are somehow "pathological", is a bit misguided.

Certainly, the multivariate normal is extremely important due to its stability under linear transformations, and so receives the bulk of attention in applications.

note: there is a multivariate central limit theorem, so those such applications have no problem
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12 weeks ago by nhaliday
Best Topology Olympiad ***EVER*** - Affine Mess - Quora
Most people take courses in topology, algebraic topology, knot theory, differential topology and what have you without once doing anything with a finite topological space. There may have been some quirky questions about such spaces early on in a point-set topology course, but most of us come out of these courses thinking that finite topological spaces are either discrete or only useful as an exotic counterexample to some standard separation property. The mere idea of calculating the fundamental group for a 4-point space seems ludicrous.

Only it’s not. This is a genuine question, not a joke, and I find it both hilarious and super educational. DO IT!!
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october 2017 by nhaliday
Variance of product of multiple random variables - Cross Validated
prod_i (var[X_i] + (E[X_i])^2) - prod_i (E[X_i])^2

two variable case: var[X] var[Y] + var[X] (E[Y])^2 + (E[X])^2 var[Y]
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october 2017 by nhaliday
Power of a point - Wikipedia
The power of point P (see in Figure 1) can be defined equivalently as the product of distances from the point P to the two intersection points of any ray emanating from P.
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september 2017 by nhaliday
Isaac Newton: the first physicist.
[...] More fundamentally, Newton's mathematical approach has become so basic to all of physics that he is generally regarded as _the father of the clockwork universe_: the first, and perhaps the greatest, physicist.

The Alchemist

In fact, Newton was deeply opposed to the mechanistic conception of the world. A secretive alchemist [...]. His written work on the subject ran to more than a million words, far more than he ever produced on calculus or mechanics [21]. Obsessively religious, he spent years correlating biblical prophecy with historical events [319ff]. He became deeply convinced that Christian doctrine had been deliberately corrupted by _the false notion of the trinity_, and developed a vicious contempt for conventional (trinitarian) Christianity and for Roman Catholicism in particular [324]. [...] He believed that God mediated the gravitational force [511](353), and opposed any attempt to give a mechanistic explanation of chemistry or gravity, since that would diminish the role of God [646]. He consequently conceived such _a hatred of Descartes_, on whose foundations so many of his achievements were built, that at times _he refused even to write his name_ [399,401].

The Man

Newton was rigorously puritanical: when one of his few friends told him "a loose story about a nun", he ended their friendship (267). [...] He thought of himself as the sole inventor of the calculus, and hence the greatest mathematician since the ancients, and left behind a huge corpus of unpublished work, mostly alchemy and biblical exegesis, that he believed future generations would appreciate more than his own (199,511).

[...] Even though these unattractive qualities caused him to waste huge amounts of time and energy in ruthless vendettas against colleagues who in many cases had helped him (see below), they also drove him to the extraordinary achievements for which he is still remembered. And for all his arrogance, Newton's own summary of his life (574) was beautifully humble:

"I do not know how I may appear to the world, but to myself I seem to have been only like a boy, playing on the sea-shore, and diverting myself, in now and then finding a smoother pebble or prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me."

Before Newton

...

1. Calculus. Descartes, in 1637, pioneered the use of coordinates to turn geometric problems into algebraic ones, a method that Newton was never to accept [399]. Descartes, Fermat, and others investigated methods of calculating the tangents to arbitrary curves [28-30]. Kepler, Cavalieri, and others used infinitesimal slices to calculate volumes and areas enclosed by curves [30], but no unified treatment of these problems had yet been found.
2. Mechanics & Planetary motion. The elliptical orbits of the planets having been established by Kepler, Descartes proposed the idea of a purely mechanical heliocentric universe, following deterministic laws, and with no need of any divine agency [15], another anathema to Newton. _No one imagined, however, that a single law might explain both falling bodies and planetary motion_. Galileo invented the concept of inertia, anticipating Newton's first and second laws of motion (293), and Huygens used it to analyze collisions and circular motion [11]. Again, these pieces of progress had not been synthesized into a general method for analyzing forces and motion.
3. Light. Descartes claimed that light was a pressure wave, Gassendi that it was a stream of particles (corpuscles) [13]. As might be guessed, Newton vigorously supported the corpuscular theory. _White light was universally believed to be the pure form_, and colors were some added property bequeathed to it upon reflection from matter (150). Descartes had discovered the sine law of refraction (94), but it was not known that some colors were refracted more than others. The pattern was the familiar one: many pieces of the puzzle were in place, but the overall picture was still unclear.

The Natural Philosopher

Between 1671 and 1690, Newton was to supply definitive treatments of most of these problems. By assiduous experimentation with prisms he established that colored light was actually fundamental, and that it could be recombined to create white light. He did not publish the result for 6 years, by which time it seemed so obvious to him that he found great difficulty in responding patiently to the many misunderstandings and objections with which it met [239ff].

He invented differential and integral calculus in 1665-6, but failed to publish it. Leibniz invented it independently 10 years later, and published it first [718]. This resulted in a priority dispute which degenerated into a feud characterized by extraordinary dishonesty and venom on both sides (542).

In discovering gravitation, Newton was also _barely ahead of the rest of the pack_. Hooke was the first to realize that orbital motion was produced by a centripetal force (268), and in 1679 _he suggested an inverse square law to Newton_ [387]. Halley and Wren came to the same conclusion, and turned to Newton for a proof, which he duly supplied [402]. Newton did not stop there, however. From 1684 to 1687 he worked continuously on a grand synthesis of the whole of mechanics, the "Philosophiae Naturalis Principia Mathematica," in which he developed his three laws of motion and showed in detail that the universal force of gravitation could explain the fall of an apple as well as the precise motions of planets and comets.

The "Principia" crystallized the new conceptions of force and inertia that had gradually been emerging, and marks the beginning of theoretical physics as the mathematical field that we know today. It is not an easy read: Newton had developed the idea that geometry and equations should never be combined [399], and therefore _refused to use simple analytical techniques in his proofs_, requiring classical geometric constructions instead [428]. He even made his Principia _deliberately abstruse in order to discourage amateurs from feeling qualified to criticize it_ [459].

[...] most of the rest of his life was spent in administrative work as Master of the Mint and as President of the Royal Society, _a position he ruthlessly exploited in the pursuit of vendettas_ against Hooke (300ff,500), Leibniz (510ff), and Flamsteed (490,500), among others. He kept secret his disbelief in Christ's divinity right up until his dying moment, at which point he refused the last rites, at last openly defying the church (576). [...]
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august 2017 by nhaliday
Inscribed angle - Wikipedia
pf:
- for triangle w/ one side = a diameter, draw isosceles triangle and use supplementary angle identities
- otherwise draw second triangle w/ side = a diameter, and use above result twice
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august 2017 by nhaliday
Diophantine approximation - Wikipedia
- rationals perfectly approximated by themselves, badly approximated (eps~1/q) by other rationals
- irrationals well-approximated (eps~1/q^2) by rationals: https://en.wikipedia.org/wiki/Dirichlet%27s_approximation_theorem
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august 2017 by nhaliday
Subgradients - S. Boyd and L. Vandenberghe
If f is convex and x ∈ int dom f, then ∂f(x) is nonempty and bounded. To establish that ∂f(x) ≠ ∅, we apply the supporting hyperplane theorem to the convex set epi f at the boundary point (x, f(x)), ...
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august 2017 by nhaliday
co.combinatorics - Classification of Platonic solids - MathOverflow
My question is very basic: where can I find a complete (and hopefully self-contained) proof of the classification of Platonic solids? In all the references that I found, they use Euler's formula v−e+f=2v−e+f=2 to show that there are exactly five possible triples (v,e,f)(v,e,f). But of course this is not a complete proof because it does not rule out the possibility of different configurations or deformations. Has anyone ever written up a complete proof of this statement?!

...

This is a classical question. Here is my reading of it: Why is there a unique polytope with given combinatorics of faces, which are all regular polygons? Of course, for simple polytopes (tetrahedron, cube, dodecahedron) this is clear, but for the octahedron and icosahedron this is less clear.

The answer lies in the Cauchy's theorem. It was Legendre, while writing his Elements of Geometry and Trigonometry, noticed that Euclid's proof is incomplete in the Elements. Curiously, Euclid finds both radii of inscribed and circumscribed spheres (correctly) without ever explaining why they exist. Cauchy worked out a proof while still a student in 1813, more or less specifically for this purpose. The proof also had a technical gap which was found and patched up by Steinitz in 1920s.

The complete (corrected) proof can be found in the celebrated Proofs from the Book, or in Marcel Berger's Geometry. My book gives a bit more of historical context and some soft arguments (ch. 19). It's worth comparing this proof with (an erroneous) pre-Steinitz exposition, say in Hadamard's Leçons de Géométrie Elémentaire II, or with an early post-Steinitz correct but tedious proof given in (otherwise, excellent) Alexandrov's monograph (see also ch.26 in my book which compares all the approaches).

P.S. Note that Coxeter in Regular Polytopes can completely avoid this issue but taking a different (modern) definition of the regular polytopes (which are symmetric under group actions). For a modern exposition and the state of art of this approach, see McMullen and Schulte's Abstract Regular Polytopes.

https://en.wikipedia.org/wiki/Platonic_solid#Classification
https://mathoverflow.net/questions/46502/on-the-number-of-archimedean-solids
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july 2017 by nhaliday
Harmonic mean - Wikipedia
The harmonic mean is a Schur-concave function, and dominated by the minimum of its arguments, in the sense that for any positive set of arguments, {\displaystyle \min(x_{1}\ldots x_{n})\leq H(x_{1}\ldots x_{n})\leq n\min(x_{1}\ldots x_{n})} . Thus, the harmonic mean cannot be made arbitrarily large by changing some values to bigger ones (while having at least one value unchanged).

more generally, for the weighted mean w/ Pr(x_i)=t_i, H(x1,...,xn) <= x_i/t_i
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july 2017 by nhaliday
If there are 3 space dimensions and one time dimension, is it theoretically possible to have multiple time demensions and if so how would it work? : askscience
Yes, we can consider spacetimes with any number of temporal or spatial dimensions. The theory is set up essentially the same. Spacetime is modeled as a smooth n-dimensional manifold with a pseudo-Riemannian metric, and the metric satisfies the Einstein field equations (Einstein tensor = stress tensor).
A pseudo-Riemannian tensor is characterized by its signature, i.e., the number of negative quadratic forms in its metric and the number of positive quadratic forms. The coordinates with negative forms correspond to temporal dimensions. (This is a convention that is fixed from the start.) In general relativity, spacetime is 4-dimensional, and the signature is (1,3), so there is 1 temporal dimension and 3 spatial dimensions.
Okay, so that's a lot of math, but it all basically means that, yes, it makes sense to ask questions like "what does a universe with 2 time dimensions and 3 spatial dimensions look like?" It turns out that spacetimes with more than 1 temporal dimension are very pathological. For one, initial value problems do not generally have unique solutions. There is also generally no canonical way to pick out 1 of the infinitely many solutions to the equations of physics. This means that predictability is impossible (e.g., how do you know which solution is the correct one?). Essentially, there is no meaningful physics in a spacetime with more than 1 temporal dimension.
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june 2017 by nhaliday
Lecture 6: Nash Equilibrum Existence
pf:
- For mixed strategy profile p ∈ Δ(A), let g_ij(p) = gain for player i to switch to pure strategy j.
- Define y: Δ(A) -> Δ(A) by y_ij(p) ∝ p_ij + g_ij(p) (normalizing constant = 1 + ∑_k g_ik(p)).
- Look at fixed point of y.
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june 2017 by nhaliday
Archimedes Palimpsest - Wikipedia
Using this method, Archimedes was able to solve several problems now treated by integral calculus, which was given its modern form in the seventeenth century by Isaac Newton and Gottfried Leibniz. Among those problems were that of calculating the center of gravity of a solid hemisphere, the center of gravity of a frustum of a circular paraboloid, and the area of a region bounded by a parabola and one of its secant lines. (For explicit details, see Archimedes' use of infinitesimals.)

When rigorously proving theorems, Archimedes often used what are now called Riemann sums. In "On the Sphere and Cylinder," he gives upper and lower bounds for the surface area of a sphere by cutting the sphere into sections of equal width. He then bounds the area of each section by the area of an inscribed and circumscribed cone, which he proves have a larger and smaller area correspondingly. He adds the areas of the cones, which is a type of Riemann sum for the area of the sphere considered as a surface of revolution.

But there are two essential differences between Archimedes' method and 19th-century methods:

1. Archimedes did not know about differentiation, so he could not calculate any integrals other than those that came from center-of-mass considerations, by symmetry. While he had a notion of linearity, to find the volume of a sphere he had to balance two figures at the same time; he never figured out how to change variables or integrate by parts.

2. When calculating approximating sums, he imposed the further constraint that the sums provide rigorous upper and lower bounds. This was required because the Greeks lacked algebraic methods that could establish that error terms in an approximation are small.
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may 2017 by nhaliday
Lucio Russo - Wikipedia
In The Forgotten Revolution: How Science Was Born in 300 BC and Why It Had to Be Reborn (Italian: La rivoluzione dimenticata), Russo promotes the belief that Hellenistic science in the period 320-144 BC reached heights not achieved by Classical age science, and proposes that it went further than ordinarily thought, in multiple fields not normally associated with ancient science.

La Rivoluzione Dimenticata (The Forgotten Revolution), Reviewed by Sandro Graffi: http://www.ams.org/notices/199805/review-graffi.pdf

Before turning to the question of the decline of Hellenistic science, I come back to the new light shed by the book on Euclid’s Elements and on pre-Ptolemaic astronomy. Euclid’s definitions of the elementary geometric entities—point, straight line, plane—at the beginning of the Elements have long presented a problem.7 Their nature is in sharp contrast with the approach taken in the rest of the book, and continued by mathematicians ever since, of refraining from defining the fundamental entities explicitly but limiting themselves to postulating the properties which they enjoy. Why should Euclid be so hopelessly obscure right at the beginning and so smooth just after? The answer is: the definitions are not Euclid’s. Toward the beginning of the second century A.D. Heron of Alexandria found it convenient to introduce definitions of the elementary objects (a sign of decadence!) in his commentary on Euclid’s Elements, which had been written at least 400 years before. All manuscripts of the Elements copied ever since included Heron’s definitions without mention, whence their attribution to Euclid himself. The philological evidence leading to this conclusion is quite convincing.8

...

What about the general and steady (on the average) impoverishment of Hellenistic science under the Roman empire? This is a major historical problem, strongly tied to the even bigger one of the decline and fall of the antique civilization itself. I would summarize the author’s argument by saying that it basically represents an application to science of a widely accepted general theory on decadence of antique civilization going back to Max Weber. Roman society, mainly based on slave labor, underwent an ultimately unrecoverable crisis as the traditional sources of that labor force, essentially wars, progressively dried up. To save basic farming, the remaining slaves were promoted to be serfs, and poor free peasants reduced to serfdom, but this made trade disappear. A society in which production is almost entirely based on serfdom and with no trade clearly has very little need of culture, including science and technology. As Max Weber pointed out, when trade vanished, so did the marble splendor of the ancient towns, as well as the spiritual assets that went with it: art, literature, science, and sophisticated commercial laws. The recovery of Hellenistic science then had to wait until the disappearance of serfdom at the end of the Middle Ages. To quote Max Weber: “Only then with renewed vigor did the old giant rise up again.”

...

The epilogue contains the (rather pessimistic) views of the author on the future of science, threatened by the apparent triumph of today’s vogue of irrationality even in leading institutions (e.g., an astrology professorship at the Sorbonne). He looks at today’s ever-increasing tendency to teach science more on a fideistic than on a deductive or experimental basis as the first sign of a decline which could be analogous to the post-Hellenistic one.

Praising Alexandrians to excess: http://sci-hub.cc/10.1088/2058-7058/17/4/35
The Economic Record review: http://sci-hub.cc/10.1111/j.1475-4932.2004.00203.x

listed here: https://pinboard.in/u:nhaliday/b:c5c09f2687c1

Was Roman Science in Decline? (Excerpt from My New Book): https://www.richardcarrier.info/archives/13477
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may 2017 by nhaliday
Interview: Mostly Sealing Wax | West Hunter
https://soundcloud.com/user-519115521/greg-cochran-part-2
https://medium.com/@houstoneuler/annotating-part-2-of-the-greg-cochran-interview-with-james-miller-678ba33f74fc

- conformity and Google, defense and spying (China knows prob almost all our "secrets")
- in the past you could just find new things faster than people could reverse-engineer. part of the problem is that innovation is slowing down today (part of the reason for convergence by China/developing world).
- introgression from archaics of various kinds
- mutational load and IQ, wrath of khan neanderthal
- trade and antiquity (not that useful besides ideas tbh), Roman empire, disease, smallpox
- spices needed to be grown elsewhere, but besides that...
- analogy: caste system in India (why no Brahmin car repairmen?), slavery in Greco-Roman times, more water mills in medieval times (rivers better in north, but still could have done it), new elite not liking getting hands dirty, low status of engineers, rise of finance
- crookery in finance, hedge fund edge might be substantially insider trading
- long-term wisdom of moving all manufacturing to China...?
- economic myopia: British financialization before WW1 vis-a-vis Germany. North vs. South and cotton/industry, camels in Middle East vs. wagons in Europe
- Western medicine easier to convert to science than Eastern, pseudoscience and wrong theories better than bag of recipes
- Greeks definitely knew some things that were lost (eg, line in Pliny makes reference to combinatorics calculation rediscovered by German dude much later. think he's referring to Catalan numbers?), Lucio Russo book
- Indo-Europeans, Western Europe, Amerindians, India, British Isles, gender, disease, and conquest
- no farming (Dark Age), then why were people still farming on Shetland Islands north of Scotland?
- "symbolic" walls, bodies with arrows
- family stuff, children learning, talking dog, memory and aging
- Chinese/Japanese writing difficulty and children learning to read
- Hatfield-McCoy feud: the McCoy family was actually a case study in a neurological journal. they had anger management issues because of cancers of their adrenal gland (!!).
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may 2017 by nhaliday
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