fairy tale: bajka2
Learn from “Mathematical Omnibus: Thirty Lectures on Classic Mathematics”, Dmitry Fuchs, Serge Tabachnikov
- “The evolute of a curve is the locus of centers of curvature. A vertex of the curve corresponds to a singularity of the evolute, generically, a cusp”
- “The osculating circles of an arc with monotonic positive curvature are nested.”
- “If a differentiable function in the annulus is constant on each circle then this is a constant function.”
- “For every oval C, the outer billiard map is area preserving”
- “If two closed smooth curves have the same winding numbers then one can be continuously deformed to the other.”
- “Two generic smooth spherical curves can be continuously deformed to each other if and only if their numbers of double points are either both even or both odd.”
- “In space, typical curves have no points of zero curvature.”
- “The osculating plane is the plane through three infinitesimally close points of the curve.”
- “There are no non-planar triply ruled surfaces.”
- “A surface of degree 3 contains 27 or infinite number of straight lines.”
- Crofton formula (used for solving Hilbert’s fourth problem)
- Fary-Milnor: “If a closed spacial curve is knotted then its total curvature is greater than 4pi.”
- “There exist no simple closed geodesics on closed convex polyhedron”
- Gauss-Bonnet theorem, Dehn’s theorem
- “If a rectangle is tiled by squares then the ratio of its side lengths is a rational number.”
- “A rectangle R is tiled by rectangles each of which has an integer side. Then R has an integer side.”
- “If the corresponding faces of two convex polyhedra are congruent and adjacent in the same way then the polyhedra are congruent as well.”
- Cauchy: “Any convex polyhedron is rigid (cannot be bent).”
- Connelly: “There exists a (non-convex) flexible polyhedron.”
- “simple connectedness: every closed curve can be continuously deformed to one point.”
- “The equivipotential surfaces of the free distribution of charge on an ellipsoid are the confocal ellipsoids.”
- “We believe that most of the matter in the universe is dark, i.e. cannot be detected from the light which it emits (or fails to emit)”
- “Current indications from the cosmic microwave background are that the universe is spatially flat.”
- “Evidence of dark matter has been confirmed through the study of rotation curves. To make a rotation curve one calculates the rotational velocity of e.g. stars along the length of a galaxy by measuring their Doppler shifts, and then plots this quantity versus their respective distance away from the center.”
- “When studying other galaxies it is invariably found that the stellar rotational velocity remains constant, or “flat”, with increasing distance away from the galactic center. This result is highly counterintuitive since, based on Newton’s law of gravity, the rotational velocity would steadily decrease for stars further away from the galactic center”
- “In this way a plot of recession velocity (or redshift) vs. distance (a Hubble plot), which is a straight line at small distances, can tell us about the amount of matter in the universe and provide crucial information about dark matter.” – “gravity acts non-linearly — a large amount of mass will attract other mass around it, increasing the amount of mass at that position. This enhances the attraction to other nearby mass and so on.”
- “At early times the universe was both denser and smoother than it is today..Regions which were slightly overdense (compared to their surroundings) had a slightly larger than average gravitational potential and so accreted matter from the surroundings, in the process becoming even more overdense.”
- “The two most obvious means of studying clusters of galaxies are by observing the light emitted from the constituent galaxies or the X-ray emission from the hot intracluster gas.”
- “Obviously clusters trace out the large-scale structure of the universe just as do galaxies. However there are several cluster properties that are interesting in and of themselves. If clusters provide a “fair sample” of the universe, then the fraction of their mass in baryons should equal the universal baryon fraction, known as OmegaB. The present number density of clusters is a measure of the amplitude of fluctuations in the universe on scales of around 8Mpc (see also this paper). The evolution of this number density (vs mass or temperature) with redshift can determine the mass density parameter Omega and possibly determine the equation of state (and nature) of the dark energy believed to be causing the expansion of the universe to accelerate. “
- “Weak gravitational lensing denotes that regime where the gravitational deflection of light is so small that one does not see multiple images. Instead the images of background objects (galaxies) are sheared and magnified. “
- “The universe is filled with radiation at a temperature of 2.728K, whose spectrum peaks at about 300GHz. This radiation was first detected several decades ago and is known as the Cosmic Microwave Background (CMB).”
- “If we observe the microwave sky we find that the temperature of the CMB is not exactly the same in all directions: it is anisotropic. There are small fluctuations in the temperature across the sky at the level of about 1 part in 100,000: the microwave background temperature anisotropies.”
- “In the Hot Big Bang Model the universe was much hotter and denser in the past. It has been adiabatically cooling as it expands. At early times the universe was almost entirely ionized. Photons and baryons (protons and electrons) were tightly coupled by Compton scattering and electromagnetic interactions.At a redshift of about 1000 the universe cooled enough to form Hydrogen. With the rapid drop in the free electron density the photon-matter interactions effectively ceased… Cosmic Microwave Background (CMB) is a snapshot of the universe at redshift 1000. The fluctuations in temperature across the sky are the precursors of the large-scale structure we see around us today. These small fluctuations grow through gravitational instability from 1 part in 100,000 at redshift 1000 to highly concentrated structures today.”
- “The Universe’s light-element abundance is another important criterion by which the Big Bang hypothesis is verified. It is now known that the elements observed in the Universe were created in either of two ways. Light elements (namely deuterium, helium, and lithium) were produced in the first few minutes of the Big Bang, while elements heavier than helium are thought to have their origins in the interiors of stars which formed much later in the history of the Universe. Both theory and observation lead astronomers to believe this to be the case.”
- “..it is observed that upwards of 25% the Universe’s total matter consists of helium—much greater than predicted by theory! A similar enigma exists for the deuterium. According to stellar theory, deuterium cannot be produced in stellar interiors; actually, deuterium is destroyed inside of stars. Hence, the BBFH hypothesis could not by itself adequately explain the observed abundances of helium and deuterium in the Universe.”
- “Roughly three minutes after the Big Bang itself, the temperature of the Universe rapidly cooled from its phenomenal 10^32 Kelvin to approximately 10^9 Kelvin. At this temperature, nucleosynthesis, or the production of light elements, could take place”
- “The Big Bang Nucleosynthesis theory predicts that roughly 25% the mass of the Universe consists of Helium. It also predicts about 0.01% deuterium, and even smaller quantities of lithium. … the prediction depends critically on the density of baryons (ie neutrons and protons) at the time of nucleosynthesis. .. the required density of baryons is a few percent (the exact value depends on the assumed value of the Hubble constant). This relatively low value means that not all of the dark matter can be baryonic, ie we are forced to consider more exotic particle candidates.”
- ” The preponderance of observational evidence suggests that (when smoothed over large scales) the universe is homogeneous and isotropic. “
- “In general relativity, the presence of matter (energy density) can curve spacetime, and the path of a light ray will be deflected as a result. This process is called gravitational lensing and in many cases can be described in analogy to the deflection of light by (e.g. glass) lenses in optics”
- “The Lyman series is the series of energies required to excite an electron in hydrogen from its lowest energy state to a higher energy state”
- “Because the universe is expanding, one can learn more than just the number of neutral hydrogen atoms between us and the quasar. As these photons travel to us, the universe expands, stretching out all the light waves. This increases the wavelengths lambda and lowers the energies of the photons (`redshifting’).”
- “It is common to see a series of absorption lines, called the Lyman alpha forest. Systems which are slightly more dense, Lyman limit systems, are thick enough that radiation doesn’t get into their interior.”
- “One can use the ionization of neutral hydrogen to find galaxies as well”
- “we can limit how much neutral hydrogen is out there between us and the quasar and what its distribution is. It used to be thought that there was a smooth intergalactic medium (IGM) with regions embedded in it, and the smooth background would provide an absorption at all positions between us and the quasar (Gunn-Peterson effect). But observers only see evidence of lumpy regions. There isn’t evidence for a spatially smooth component of neutral hydrogen between us and the quasar sources. “
- “All viable models of strucute formation today are dominated by cold dark matter (whose velocity dispersion or pressure is negligible).”
- “While neutrinos could be massive, and thus give a component of hot dark matter, the upper limit on their contribution to the energy density of the universe is a few percent, and they are therefore not relevant to structure formation”
- “In this model a period of accelerated expansion in the early universe amplified quantum vacuum fluctuations into the seeds for the large-scale structure. Other models based on topological defects from a phase transition in the early universe fail to account for the clustering observed today or the anisotropies in the microwave background.”
- “The most successful theory of cosmological structure formation is inflationary cold dark matter (a.k.a CDM) in which nearly scale-invariant adiabatic initial fluctuations are set up by a period of inflation in the early universe. Thus the initial conditions are: (a) the fluctuations in the gravitational potential (which can be related to density fluctuation through Poisson’s equation) are almost independent of scale and (b) that the fluctuations in the pressure are proportional to those in the density (which keeps the entropy constant, hence the term adiabatic). A direct consequence of the adiabatic assumption is that a cosmic overdense region contains overdensities of all particle species. The alternative mode, where overdensities of one species counterbalance underdensities in another, is known as isocurvature because the spatial curvature is unchanged. Models incorporating isocurvature initial conditions fare very badly when compared to the observations.”
- weak clustering of absorbers, continuous density fields used for redshift analysis, gravitational clustering predicts a non-gaussian pattern
- effects of radiative transfer, reionization, “metal lines”, galaxy-idm connection – Wikipedia: “Einstein ring is the deformation of the light from a source (such as a galaxy or star) into a ring through gravitational lensing of the source’s light by an object with an extremely large mass (such as another galaxy, or a black hole)”
- John Wheeler: “Matter tells space how to curve, and space tells matter how to move.”
1. Why does our physical body die?
Tim Radford, The Guardian, “Organisms grow old because nature doesn’t need them any more. If the purpose of life is to procreate and replicate successfully – this is the logic of the so-called selfish gene theory – then it helps to stay healthy long enough to generate children and provide them with food. Immortality arrives with your offspring, and is only guaranteed when all your children also have children.”. Were that the case, we should never optimize for values that last only for the perceived life-time, but for what we perceive to be important. How to find those things?
2. How was the universe born?
Wikipedia: “The Big Bang theory is the prevailing cosmological model that describes the early development of the Universe. According to the theory, the Big Bang occurred approximately 13.798 ± 0.037 billion years ago, which is thus considered the age of the universe. At this time, the Universe was in an extremely hot and dense state and began expanding rapidly. After the initial expansion, the Universe cooled sufficiently to allow energy to be converted into various subatomic particles, including protons, neutrons, and electrons. Though simple atomic nuclei formed within the first three minutes after the Big Bang, thousands of years passed before the first electrically neutral atoms formed. The majority of atoms that were produced by the Big Bang are hydrogen, along with helium and traces of lithium. Giant clouds of these primordial elements later coalesced through gravity to form stars and galaxies, and the heavier elements were synthesized either within stars or during supernovae.”. But how did the “ extremely hot and dense state” appear there in the first place?
3. What is the best way to learn?
I cannot give a direct answer to this question. I cannot find the best way right now. One thing to note: we should be able to exploit the diversity of talents given (different beings, design thinking etc.) (1) to punish getting stuck at local minimas, evaluate axioms (2) (iterative axiom modelling combined with contradiction detection), exploit an efficient ways for modelling different problems (3) (see (2)).
For now, I will say that the ensemble of approaches exploiting a properly configured graph structure (built on top of some grammar rules). The ensemble itself can exploit, for instance, e.g. multi-armed bandits (or a better approach) (among humans: there are different humans and we can exploit it better, see: design thinking approach; certain humans get stuck with problems (local minima)). The graph structure may exploit Szemeredi theorem (we want to limit the amount of data via learning, which can be treated as a compression problem (mostly a loss-prone one)). The rules should be built by identifying the most informative features. Computation could be done in parallel.
If we want to exploit the pameter space more effectively, we should allow (1), which also means we should aim at creating a society of non-self bidders (optimizers) but people optimizing for the society. We have learnt that selfish bidders degrade system effectiveness – is it also case of further development?
4. What is the place for mathematics?
Some problems are undecidable. We face incompleteness. Axioms are likely leaky and we will more often approach mathematics from experimental perspective. We shall develop better axioms – but how?
Automated reasoning may help us to disclose vulnerabilities within our axiom systems. Machine learning will exploit the state-of-art of algorithms and will help us explore parameter spaces of problems more effectively. More structuring of knowledge is important. More innovative approaches to axiom systems are needed.
Tesla’s point about mathematics being (sometimes) a non-existent creature itself is valid, but I cannot agree that this is not science (still, I would also not assume that Tesla did not appreciate- as he put it- “metaphysics”). We explore problem spaces in different ways. In some cases we need more (in some less) model re-building. Mathematics requires much iterative work on axioms due to its cognitive structure. It will also profit from exploiting its secrets with an experimental approach.
5. Why should I care what happens after I die?
If you optimize for a sequence of values that can last only for your lifetime, then please name them. Comment this post and I will address each of the parameters that you optimize. Exemplary parameters: “health”, “family”, “amount of money”, etc.
Does it makes sense to optimize for values that will last longer? What are those values?
If you think that you were born in a place about which you know virtually nothing, the very first naive idea would be “to learn about this place”. Given the problem size, we should first exploit ways to learn more effectively. Given the efforts from many, we could now exploit the computational resources to extract knowledge faster.