====================================================================== This is a diary of facts and ideas I came across and wrote down for my own benefit; some may not make sense to you without further explanation. Nothing here is original but I left out all sources and references. Axel Boldt axelboldt@yahoo.com http://math-www.uni-paderborn.de/~axel/ ====================================================================== Nov 20, 2004 Humans are slowed down apes; most of babyhood should really happen in the mom's belly; some later stages such as ape body hair never happen in humans. Lengthening of child stage allows for growth of brain. Feb 20, 2005 The dual price p_i of a linear programming problem is the marginal increase in the optimal objective function value per increase of i-th constraint's right-hand side. They are also the solution of the dual problem: p_i is the "fair" price per unit of the resource described in the i-th constraint. Primal and Dual have the same optimal solution, the "fair price" of the whole operation. Mar 10, 2005 Appolonius circle: start with two points, consider the locus of all points whose product of the distances to those two points is constant. Mar 20, 2005 Fermat's principle in optics (light takes shortest-time path) explained by Feynman: light "tries out" all other paths to, but if a path's light-time is not stationary (variation = 0) then light travelling along nearby paths will be out of phase and will cancel out. He claims the same effect is at work in QED. Mar 25, 2005 Inverted repeats in DNA: the only point seems to be to be able to form cruciform structures, in order to excise or expose the parts of the DNA between the repeats. Mar 30, 2005 Huber-Dyson's conjecture in group theory: if a first-order statement is true about all groups, then this can be proven in ZFC. Not obvious: Diophantine incompleteness theorem yields a concrete polynomial with integer coefficients which doesn't have a zero yet there is no proof for this fact in ZFC. Apr 6, 2005 Little white dots before eyes when looking in blue light: "blue field entoptic phenomenon", due to white blood cells creating gaps in blood columns in capillaries before retina. Larger, slower moving "floaters" due to debris floating inside the eye's vitreous humor. Apr 10, 2005 Horoscopes and psychics use "cold reading" (vague general statements to elicit responses) and Forer effect (vague general personality statements are given high accuracy ratings by the subject). HIV-2 most common in West Africa, especially in Guinea-Bissau, former Portuguese colony. Thus Portugal has highest HIV-2 rate in Europe. Countries with ties to Portugal also have significant HIV-2: Angola, Mozambique, Brazil, Southwest India. http://www.socgenmicrobiol.org.uk/JGVDirect/18253/18253ft.htm Apr 16, 2005 Protein folding problem: Neuronal nets are used to classify the amino acids of a protein according to local characteristics (alpha helix, beta sheet, outside, etc.). Groups of related proteins can be described by Hidden Markov Models. Folding prediction with molecular mechanics way too slow; other methods not very good. Crystallographer takes 1-3 years for a single protein. Apr 20, 2005 In US, women without college degree have about 50% divorce rate, women with college degree about 25%. (NYT) Singular value decomposition: every linear map R^n -> R^m is described by a diagonal matrix with non-negative entries w.r.t. suitable orthonormal bases. This is the technique behind principal factor analysis. Apr 22, 2005 "How many children do you have?" - "Two". "Any daughters?" - "Yes, in fact I have to pick up my daughter from school in half an hour." Probability 1/3 that they have two daughters. "How many children do you have?" - "Two, in fact I have to pick up my daughter from school in half an hour." Probability 1/2 that they have two daughters. What if in front of a girl's school? Apr 24, 2005 Mozilla Bayesian spam filter: start with training set of classified spam/ham; tokenize messages; calculate for each token (=word) a subjective Pr(spam|token) (based on assumption that prior Pr(spam) = 50%). Then for incoming message combine the 15 most extreme token scores using naive Bayes theorem (ignoring dependence of words) to get a subjective Pr(spam|message). Apr 27, 2005 Left brain: details (e.g. logic, math, language syntax); right brain: big picture (spatial relations, emotional and metaphorical aspects of speech, context, meaning). Experimental confirmation: picture of big letter made of little letters. Later contradicted by big symbols made of little symbols! Left brain controls right side of body (and left side of the face), receives visual images from the right first (being connected to the left halves of both retinas). Added 3 Jan 2009: in heterosexual males and homosexual females, the right half is on average about 1% larger than the left; in the others the two halves are on average about the same size (std deviations are roughly 1-2 percentage points). In male monkeys, right side of brain has on average more receptors for male sex hormones. Jill Bolte Taylor relates how a left-side stroke created holistic feelings. Bolzmann's entropy theorem: if X is a random variable and you know E(f_j(X)) for functions f_j, then the density with largest entropy (if it exists) is proportional to exp(\sum \lambda_j f_j(x)). This is the default choice if no other information is available. Applications: * normal distribution if only mean and standard deviation are known * uniform distribution if supported on closed interval or finite set * exponential distribution if supported on [0,\infty) with given expected value * geometric distribution if supported on natural numbers with given expected value http://www.eecis.udel.edu/~jing/11.pdf May 2, 2005 Utility theory: to decide, don't maximize expected Dollar amount E(X) but expected utility E(U(X)). The function U(X) (typically concave for large X; often convex for small X except for risk-averse individuals) can be experimentally determined: U(0)=0 and U(1,000,000)=1 arbitrary, then ask: "Give Dollar amount x such that a lottery with .5 chance of winning 1,000,000 and .5 chance of winning 0 is worth x$ to you." Then U(x)=.5. Repeat to get more points on the graph of U. St. Petersburg paradox: keep flipping a coin; if first head occurs on k-th trial, you get 2^k dollars. How much do you pay to play? (Expected value infinite, but likely values very small; need to use expected utility not expected money). May 8, 2005 Leibniz' theory of the best possible world: most variety of phenomena with the simplest, most elegant laws. (-> simulated universe?) 1+2+3+4+... = -1/12 because zeta(-1)=-1/12. Probability distributions: Variable | discrete | continuous ====================================== number of | Binomial | Poisson events | | -------------------------------------- time until| Geometric | Exponential next event| | -------------------------------------- time until| Negative | Gamma (or k-th event| binomial | Erlang) Also: * Beta(alpha,beta) distribution useful for prior on [0,1] if you have seen alpha successes, beta failures. * Lifetimes: Weibull (exponential hazard function), lognormal (accumulating small multiplicative effects) or Gamma (sum of exponentials = sequence of constant-hazard processes) May 14, 2005 Catabolism: liberate energy by taking high energy electron from A and giving it to B, i.e. oxidize A and reduce B. Fermentation: A,B are organic, generate ATP by substrate-level phosphorylation (not very effective). Respiration: B inorganic, generate ATP by chemiosmosis: electron chain along a membrane pumps H+ out, let back in through ATP synthase. Table of reduction reactions, most difficult first H+ -> H2 CO2 -> CH4 (other organics here) sulfate/sulfur -> H2S nitrate -> nitrite O2 -> H20 For a respiration reaction, combine the inverse of one line with another line further down. This creates free energy. In photosynthesis, free energy is input from light, and we combine the inverse of one line with another line further UP. (June 17) E. coli also have citric acid cycle, electron transfer chain, and ATP synthase. Flagellum uses proton gradient to produce rotation (Aug 19); operation of motor not known, but system is hollow and probably evolved from a protein ejection system, coupled with a proton pump. (Aug 28) Retroviral Long Terminal Repeat (LTR): RNA: RB-XXX-AR DNA: ARB-XXX-ARB needed because A is promoter and B is end part which won't be copied into RNA by host's polymerase. during DNA synthesis, new RB piece jumps from left to right; later forms cycle. Gallo found a growth factor for CD4-cells (IL-2), then isolated the first two human retroviruses (growing on CD4 cells) with reverse transcriptase assay, then was able to productively infect an immortal (=cancerous) CD4-cell line with HIV. Greasemonkey: firefox extension that can apply javascript by URL to modify/extend given site Berkman Annotation Engine: proxy that allows users to leave and read annotations for arbitrary websites Wikalong: Firefox extension, accesses central wiki, gives every website a wiki border (added June 5) May 25, 2005 Taylor's theorem for analytic functions: exp(tD)f=f(.+t) (D=d/dx). More generally, exp describes the flow V(t,x) generated by an analytic vector field v: exp(tv)f = f(V(t,.)) for every analytic function f(x) on the manifold. Campbell-Hausdorff formula exp(v)exp(w)=exp(v+w+1/2[v,w]+...) is important here. May 28, 2005 Solar wind: hot charged particles that escape from the upper atmosphere of the Sun because of their high thermal energy. Water -> Oxygen photosynthesis was invented by cyanobacteria; needs two reaction centers, one from green and one from purple bacteria. Chloroplasts of green algae (leading to higher plants), red algae and glaucophytes were created by (just one event of?) primary endosymbiosis with eukaryotes (mitochondria, from bacteria that use oxygen to produce energy, were already in place). All other types of algae (protists) got their chloroplasts by secondary endosymbiosis. June 4, 2005 Descartes to overcome scepticism: cogito ergo sum plus God is not deceptive, so we can trust our senses. God exists: I can conceive of a perfect being, and only perfect causes can have perfect effects, so God caused my conception of him. Hume's criticism of induction: you need a principle that says "what happened in the past will likely continue in the future", but our only justification of that principle is inductive => circular reasoning. June 6, 2005 Cerebral cortex: Wrinkled 2mm thin outermost layer of the brain, about size of a pizza, responsible for higher processing. The cortex consists of neurons (grey matter) and covers white matter (myelinated axons). Almost all of it (neocortex) is only present in mammals and has six layers. ATP synthase: two coupled motors/generators: F0 and F1. F0 sits in membrane and converts H+ (sometimes Na+) gradient into rotation; F1 sits on top and converts rotation into ADP->ATP transformation. Present in mitochondria, chloroplasts, and bacteria. Can also operate in reverse: using ATP to pump H+, affecting pH (in human lysosomes, some archae). F0: 10-14 cyclically arranged identical units form a rotating wheel. Two fixed half-channels, open to opposite sides. Each unit is normally neutral and can gain/lose a protein only in the half channels; more likely to lose a proton in one and gain a proton in the other (because of proton concentration difference). In the middle a permanently positively charged residue. => Thermal movements of the wheel push it forward in one direction. F1: three fixed outside slots, each can be in three states: accepting ADP+Pi, converting into ATP, releasing ATP. Rotationary axis in the middle cyclically changes the states. DNA topoisomerases: let DNA strands move through each other; essential for replication of DNA rings, so that they don't stay interlocked. Also removes supercoiling. June 11, 2005 Fungi often have two names: anamorph (asexually reproducing) and telemorph (sexually reproducing). The two corresponding forms are not easily recognized. homolog: similar because of shared ancestry paralog: homolog after gene duplication ortholog: homolog after speciation June 20, 2005 Bacterial evolution: * fermentation of abundant organic molecules (produces organic acids) * ATP synthase, operating as proton pump to reduce intracellular acidity * electron chain to exploit redox reactions, better than fermentation; ATP synthase now operates the other way * H2S-based photosynthesis, using same electron chain * H2O-based photosynthesis (cyanobacteria), needs two chlorophylls -> chloroplasts * Oxygen first oxidizes Fe (iron ores), then emerges into the atmosphere * Modified electron chain used to oxidize organics (cyt c oxidase) with oxygen; Krebs cycle -> mitochondria July 6, 2005 Legendre trafo: convex f(x) <-> convex g(p) via g(p) = max_x (p.x - f(x)) This transforms Laplacian L:T(C)->R into Hamiltonian H:T*(C)->R. The maximizing x-value satisfies df_x = p; this gives a diffeomorphism between T(C) (q1,...,qn,q1dot,...,qndot) and T*(C) (q1,...,qn,p1,...,pn) in coordinates: pi=del L/del qidot. Earth magnetic field: hot solid inner core causes rising convection in outer liquid core; Coreolis force turns these into helices. Helical movement of fluid conductors can turn a pre-existing magnetic field by 90 degrees. Also: magnetic field lines in a moving liquid conductor are dragged along, creating a circular magnetic field along the equator in the core. Helices turn this to create Earth's magnetic field. (added 12-Nov-2005) August 2, 2005 Szostak has 14-base self-replicating RNA. PACE: project to construct life from scratch. Venter minimizes 517-gene Mycoplasma genitalium. Original mammal genome is being computed from extant mammals. Neanderthal genome is being reconstructed using human scaffold. August 3, 2005 Humans have 25000 genes (=DNA sequences that are transcribed into RNA). Introns are spliced out of pre-mRNA in nucleus. In complex organisms, Splicing Regulatory (SR) proteins bind to Exonic Splicing Enhancer (ESE) [or Exonic Splicing Suppressor (ESS)] sites within an exon, and thereby direct the spliceosome to the adjacent splice sites [or prevent it from attaching]. Humans have highest number of introns per gene. Avg gene is 28000 bp long, with 8.8 exons, each usually about 120 bp long. In simple organisms, introns are much shorter and introns, not exons, are recognized for splicing out (evolutionary origin). SRs can be controlled to yield alternative splicing, which may affect 75% of human genes. Introns (in primates often with the 300 bp long retrotransposon Alu) yield a playground for evolution because they can become alternatively spliced exons. August 18, 2005 Every square real matrix is product of two real symmetric matrices; every square complex matrix is product of two complex symmetric matrices. (Quick proof of the latter with Jordan Normal Form.) August 27, 2005 Jet streams: polar air meets tropic air, +Coriolis force => strong wind west to east, 12km up. Two in each hemisphere. Meandering shape. August 30, 2005 Kademlia: architecture for a distributed (key,value) database, used by (some) edonkey and bittorrent clients. Nodes have IDs which are of the same length as the keys; a (key, value) pair is stored at those nodes with closest IDs (metric: XOR, converted into int.) Sept 04, 2005 Oxytocin: hormone involved in birth and breastfeeding and love/trust. Also inhibits development of drug tolerance, suggesting function: no need for tolerance of love/sex endorphins which act on opiate receptors (Sept 11). Sept 21, 2005 Precession of the equinoxes: axis of Earth's rotation slowly precesses, completing one round every 26,000 years. Reason: Earth is not spherical but bulged at the equator, Sun's pull gives torque causing change in angular momentum vector. Sept 23, 2005 All (non-trivial) conics are the same in two-dimensional real projective space. In fact, if points there are identified with lines through the origin in R^2, they are nothing but a cone! Oct 1, 2005 p53 directs DNA repair in mammalian cells. It serves as a transcription factor for many genes; one of them is mdm2; mdm2 then degrades p53, resulting in a negative feedback loop. DNA damage leads to a number of discrete p53 pulses (same length and amplitude; strength of damage encoded in number of pulses). Oct 10, 2005 Heavy things fall faster than lighter things, because they attract the Earth more. There's a solution to Newtonian n-body which goes to infinity in finite time: two pairs of horizontally rotating pairs, one particle shuttling back and forth. (Works only for *point* particles!) Can't extend past the singularity. But physically it should be extendible. Since Newtonian mechanics is time-reversible, this means it's not quite deterministic: the end stage, reversed, would have a new body appear from infinity! Oct 11, 2005 Spiral arms of galaxies are moving regions of higher star density resulting from flight paths of the individual stars, which are *rotating* ellipses (galaxy cannot be simplified as a simple mass point like the solar system). Spiral galaxies slowly turn into barred spiral galaxies and back. Colliding spirals form elliptical galaxies. Oct 25, 2005 E=mc^2 is always involved, when chemical energy is converted into kinetic energy: we "melt" a bit of mass to get kinetic energy. (Added Nov 17:) More radically: the only thing that exists is energy; electrons and quarks are bundles of energy (and therefore have mass by Einstein's formula, but no extension). There is no hard "matter"; everything is energy and Einstein's formula just describes how much that energy weighs. In General Relativity, *spacetime* is curved, so the trajectories of particles are geodesics; parametrization is unimportant, speed can be read off from geodesic. Given a point and a direction in *spacetime*, there's just one geodesic that fits, so the particle's movement is determined; given a point and a direction in *space*, there are lots of particle paths that fit, with different speeds. Oct 30, 2005 Cloning is mainly useful to produce transgenic animals: transformation of somatic cells can be performed in vitro en masse, much easier than microinjection of one-cell embryos with low yield. In pineapples, pinecones, sunflowers etc., you have two sets of spirals; the numbers of them are always two consecutive Fibonacci numbers. Reason(?): the units emerge from the center, push others away, and grow as large as they can. Nov 22, 2005 Dopamine is the pleasure and reward neurotransmitter. low dopamine = Parkinson's, high dopamine = creativity/psychosis/schizophrenia. Ritalin/amphetamine/cocaine increase dopamine (block reuptake); Opiates+THC increase dopamine output (block inhibitory neurotransmitters); Antipsychotics like Haldol and friends block dopamine receptors. Ventral tegmental area and substantia nigra in the midbrain release dopamine. Low levels of the neurotransmitter Serotonin are believed to be involved in depression (no firm evidence though, added 17 Jan 2008), and also in explosive aggression (added 14 Sept 2006). Prozac and friends increase serotonin by blocking reuptake. LSD blocks some and excites other serotonin receptors. (added Apr 2, 2006) GABA is the primary inhibitory neurotransmitter. Benzodiazepines and other GABA analogues are GABA receptor agonists and thus have relaxing, sedating, anti-anxiety and anti-convulsive effects. Glutamate is standard excitorary neurotransmitter. Alcohol increases activity of GABA receptors and blocks glutamate receptors. Nov 23, 2005 We can take a photograph that records information about the direction light rays are coming from: take a grid of microlenses as your image plane, behind it many more actual light sensors. => Allows us later to focus on anything in the picture. Can sharpen ordinary picture with Fourier trafo and inverse convolution (because moving or unfocused camera can both be modeled as a convolution of the original picture). To remove noise in a photo use anisotropic diffusion PDE: first figure out direction of sharp lines, then diffuse in the direction (but not orthogonal to) these lines. (Added March 5, 2007.) Dec 17, 2005 Teilhard de Chardin: atomism is correct, the smallest indivisible unit is the universe; the radius of any atom is equal to the radius of the universe; cannot separate parts without making mistakes at the boundaries. (Wave function of atom is non-zero everywhere.) Dec 26, 2005 Franks were German tribe at lower Rhine, conquered France ~500-600. Became Roman Catholics (trinitarians, not arians like the other Germanics). Charlemagne, crowned emperor in 800, spoke Frankish/German and conquered and violently christianized Saxons, incorporated Bavaria. Afterwards empire breaks apart: Kingdom of France, Holy Roman Empire of German Nation. Dec 28, 2005 Senescence: death probability increases with age. Happens in all animals, possibly not in Hydra. Reason: cellular and molecular damages accumulate. In the wild, most deaths are caused by external factors, so repair (which is always possible) is waste of energy beyond a certain point. Better invest energy in reproduction (except during famine/caloric restriction). Jan 15, 2006 Metasploit: modular framework to hack into a machine; can choose from a large variety of exploits and a large number of payloads. RNAi: RNA interference, a mechanism in all(?) eukaryotes to fight off double-stranded RNA-viruses: dsRNA is recognized and cut into 20-25 bp pieces (called "small interfering" siRNA) by Dicer; those pieces attach to RISC, and the double strand is split into a single strand. Other copies of the matching single stranded RNA are then degraded by this activated RISC. Used by scientists to silence a gene: simply introduced a dsRNA of a part of the gene. (To fight disease or study function of a gene.) Mechanism is also used in embryonal development to shut down genes: a DNA region codes for a specific siRNA, and the RNAi machinery does the rest. Jan 20, 2006 Calabi-Yau manifold: compact, complex, with Riemannian and symplectic structure, and "flat" in some sense. 1-dim Calabi-Yau manifolds are just the elliptic curves. 3-dim Calabi-Yau manifold used in string theory for the tiny additional 6 dimensions in which the strings vibrate. Jan 25, 2006 Real photons are the quanta of electromagnetic radiation and constitute light; virtual photons transmit the electromagnetic force. Real gravitons are the quanta of gravitational radiation and constitute gravitational waves; virtual gravitons transmit gravitational force. Gravitons (and gravitational waves) are still hypothetical, but everyone believes in them. Virtual particles can also move faster than the speed of light. Uranus orbit measurements lead to predict Neptune, which then was found. Mercury orbit measurements (perihelion precession) lead to predict "Vulcan" which was not found; General Relativity was the answer. Feb 26, 2006 Amino acids in living things are all right-handed. Two theories: 1) They came from space, where circularly polarized light destroyed one type in our cloud. 2) Weak nuclear force is not symmetrical w.r.t. handedness. Could affect magnetic water molecules and dissolved amino acids. No need for space intervention. Feb 27, 2006 Buffer overflow exploit: send too long string to a function, which tries to store it on the stack, thereby overwriting the function's return address. Rest of string contains the exploit's "payload". In Unix: set return address directly to start of payload. In Windows: location of stack is not known, so find a register that points into the payload, then set return address to a known jump command in some DLL that uses said register. Mar 4, 2006 Lie bracket: if v and w are C^k vectorfields, then [v,w] is a C^{k-1} vector field. [v,w](p)(f)=v(p)(w(f))-w(p)(v(f)) at every point p; the vector [v,w](p) depends not just on v(p) and w(p) but on v and w in a whole neighborhood of p (except on Lie groups). Turns the set of C^\infty vector fields into a Lie algebra. Geometrically: 1) e^(tv)* w = w + t[v,w] + o(t) [pullback of w along v's integral curve for time t]. Alternatively: [v,w] = lim(t->0) (e^(tv)* w - w)/t 2) e^(tv)e^(tw) = e^(t(v + w) + 1/2 t^2 [v,w] + O(t^3)) [first following w, then v, ist the same as following v+w to first order; second order term is [v,w] 3) e^(-tv)e^(-tw)e^(tv)e^(tw) = e^(t^2 [v,w] +O(t^3)) [move around in a small square, alternatingly following v's and w's integral curves] Added Mar 21, 2006: Frobenius theorem. Suppose you're given a k-dimensional "distribution" in a C^\infty n-manifold M (i.e.: k-dimensional subspaces of M's tangent spaces). Then we can foliate M with k-dimensional submanifolds that have the given distribution as tangent spaces if and only if any two vector fields that live in the distribution have their Lie bracket also in the distribution. Proof: => clear. <= coordinatize M so that the last n-k coordinate vectors are transversal to the distribution. Project the first k coordinate vectors into the distribution, along the last n-k. Then these k will have zero Lie bracket amongst each other (follows from assumption and the fact that coordinate vectors have zero Lie brackets). Then we can use the k vectors to span our foliating submanifolds. Added Mar 29, 2006: Frobenius theorem generalizes fundamental theorem for vector fields: given a vector field V:M->TM, define a 1-d distribution on MxR: at the point (x,t), the space is spanned by (V(x)/|V(x)|,|V(x)|). Submanifold given by Frobenius is just the flow given by fundamental theorem. Mar 28, 2006 Hyperinflation in Germany: 1921-1923, ending with Rentenmark. Alcohol prohibition in U.S. 1920-1933 (amendment passed 1917, ratified 1919) Women's suffrage in U.S.: 1920 (amendment passed 1919, ratified 1920) Great depression: October 1929 (50% stock market crash) - late 1930s. Dust Bowl: 1930-1936 (drought + unsustainable agricultural practices; worst drought: Summer 1936) 1930-1931: bank runs, deflation. 1928-1932: Republican President Hoover, 1928-1930 large Republican majorities in both houses, 1930-1932 tiny majorities in both houses. 1930: Hoover increases tariffs with the Republican Smoot-Hawley bill, resulting in retaliation and contributing to world trade depression. Hoover also tried volunteer and government programs; didn't help. 1933: Democrat F. D. Roosevelt comes into office, with 25% unemployment rate. He has Democratic majorities in both houses until 1945. Roosevelt's New Deal: 1933 reform financial system, abandon gold standard to allow monetary policy, regulate prices and wages; 1935 Social Security. Federal expenditures tripled from 1933-39, financed by debt. 1937: "Second depression" after tax increase and rise in interest rates. Government expansion and spending during WWII ends depression. Hitler and Roosevelt both came to office January 33 and died in office April 45. Stalin came to power in 1922; big famine in 1932/33 caused by collectivization. Great Terror against party members starts 1937. Dies 1952. Plato: three forces propel man: reason, eros (passion), thymos (need to be recognized, to be valued, to be part of something important). Apr 7, 2006 A meme is like a virus: sometimes harmful to the host. Consider the religious meme "don't question received beliefs". Gromov space: the isometry classes of compact metric spaces form the points of a metric space; distance is given by the minimal Hausdorff distance achievable by isometrically embedding the two spaces in some other space. (Can be extended to locally compact pointed spaces and be used in group theory, on Cayley trees.) Apr 12, 2006 At a divergent boundary between two tectonic plates, molten rock rises from mantle and becomes crust, pulling the plates apart (e.g. Mid-Atlantic Ridge, growing in width several centimeters each year). At a subduction zone, one plate (usually the denser oceanic) slides underneath another, recycling the material back into the mantle (e.g. Japan, where Philipine and Pacific Plates subduct unterneath Eurasian and North-American Plates). These two processes power the movement of the plates. Oceanic crust: largely basalt. Continental crust: largely granite. Granite has more silicate and is not as dense as basalt. May 7, 2006 Degree of a map f between two connected oriented compact manifolds of the same dimension n: * globally defined as the integer describing the action on H_n = Z, * locally defined: pick any x such that f^-1(x) has no critical points, then for each element u of f^-1(x) add 1 if f preserves orientation and -1 otherwise. Index of a vector field at x: surround x with a small square (or triangle), put +1 at the center and the corners, -1 at the edge midpoints, let the vector field act and count the total "charge" inside the figure afterwards. Alternatively: surround x with a small sphere, and consider the normed vectorfield on that sphere as a map to the unit sphere; its degree is the index of the vector field at x. Poincare Theorem: If a Riemannian manifold is closed and the vector field has only isolated zeros, then the Euler characteristic = sum of all the indices. Gauss-Bonnet: closed 2-dim. Riemann manifold. 2 pi * Euler characteristic = integral of Gaussian curvature. Adopt children whose mothers agree and no father comes forward: they automatically become German citizens. You are responsible for child support, so proceed only if pension cannot be taken away. May 17, 2006 Compact connected 2-manifold without boundary: * orientable genus g: sphere with g handles, i.e. connected sum of g toruses; Euler characteristic 2-2g (if embedded: = 2* degree of Gauss map, which assigns to each point the normal vector) * non-orientable genus g: connected sum of g projective planes = sphere with g disks cut out, opposite points on each disk's boundary identified = star of g tubes, opposite points on each tube's boundary circle identified. Euler characteristic 2-g 2 projective planes = Klein bottle; 3 projective planes = torus + projective plane. August 14, 2006 Jimmy Gettys from OLPC project: cache misses are extremely slow, and often it is faster to recompute values than to retrieve them from memory. Programmers often have wrong assumptions. August 27, 2006 Entropy in thermodynamics: amount of energy not available for useful work. Entropy of a gas is proportional to the number of bits stored in the molecules positions and velocities. I.e. thermodynamic entropy = informational entropy. Pax-6 is the gene that switches on the construction of an eye. The versions of mice and fruit flies are interchangeable, even though the eyes are completely differently constructed! So they must have the same ancient origin. Don't they also use the same light-detector cells? Same is true for genes that construct appendages. September 29, 2006 Principal Component Analysis: given a set of points in R^n, find linear functions f1,f2,... such that f1 has highest variance, f2 next highest (subject to being uncorrelated to f1), and so on. These are the principal components. (added Nov 7:) The f's will be orthogonal, and projection via f1,f2 gives the 2-dim'l projection of the points which minimizes the sum of the squares of the distances between point and projected point. The f's are the eigenvectors of X^T*X, where the rows of the matrix X are the given points; moved so that all column averages are 0. PCA is different from *Factor analysis* where you try to find a small set of uncorrelated 'hidden' variables X1,...,Xp so that the observed variables are linear combinations of the hidden ones. Support Vector Machines: given training data (x_i, y_i) [where the x_i are typically from R^n, y_i \in {0,1} or \in R], we want to "generalize" and "learn" to find a good simple function that approximates the unknown underlying "true" function. Modeling with linear functions, and optimizing so that the separating line is furthest away from the data (or coefficients are smallest), is a quadratic optimization problem which is subject to duality. If expressed dually, it depends only on the inner products of the data points. To generalize to non-linear situations, we implicitly map to a high dimensional Hilbert space by using as "scalar product" a kernel K(x,y). In applications, choosing the kernel (often with a parameter) is the only important choice to be made, unlike in neural nets and other approaches, where many parameters need to be tuned. October 5, 2006 Supernova type Ia: a white dwarf (cooling, small, very dense, dead star, resisting gravity by Pauli pressure [electrons are fermions and can't sit on top of each other], consisting of carbon and oxygen leftover core of a small star after the red giant phase) pulls in matter from a companion star until it reaches the Chandrashekar limit of 1.4 solar masses (gravity overcomes electron pressure) and a fusion reaction of carbon etc. ignites, exploding the star. Simulations show that turbulence is important or else the shock wave will peter out; therefore 2-D simulations don't work. Supernova type II: as fuel of a massive (10 solar masses) star gets exhausted, the core collapses to a neutron star (merging protons and electrons into neutrons, releasing neutrinos). As the collapse stops, the infalling matter bounces off the neutron star and, together with the neutrinos, creates an explosion, again with turbulence and asymmetry being important. In larger stars (~100 solar masses) the core can also collapse all the way to a black hole, creating gamma bursts. [Added Jan 23, 2012] Stars are displayed in an HR-diagram: temperature=color on horizontal axis vs absolute magnitude=brightness on vertical axis. Temperature is related to energy output per surface area, while brightness is related to total energy output. Stars that fuse hydrogen into helium (which are most of them for most of their lifetime) are located on the Main Sequence, a diagonal strip in the HR-diagram. As stars are born they move on the Main Sequence, then they don't move much, until hydrogen fusion stops and they move off the Main Sequence. Upper left of Main Sequence = high mass = bright = hot = short life on Main Sequences = not dense. After leaving the main sequence, fusion in the core stops, material falls in, heats up, and hydrogen fusion in the outer layers starts up again, creating red giants (upper right of HR-diagram). October 16, 2006 Solomonoff's universal induction: imagine a bit sequence produced by a machine (representing the "environment") that, based on the previous bits, computes a probability for the next bit being a 1, then flips an appropriate coin. It's your job to predict the next bit, based on observation of the past string of bits w; you don't know the internal workings of the machine. Solomonov's approach, in several senses optimal, feeds a random input string to a pascal interpreter (note that end of a Pascal program is well-defined), considers only those whose output starts with w, and then asks whether the next output of these machines is 1 or 0. The probability of 1 determined this way approaches the true probability of 1 computed by the environment machine. So Solonomoff's prediction is the more likely of the two bits. Epicurus: "retain all theories that agree with the data" Occam: "retain only the simplest theory that agrees with the data" Solomonoff: "retain all theories, but put more trust into the simpler theories" November 6 Maximizing the area of a rectangle with given circumference gives the same result as minimizing the circumference of a rectangle with given area. That's a special case of a general result, consequence of Lagrange's multiplier method: the gradient of the objective function and the gradients of the constraints are linearly dependent at the optimum; it doesn't matter which you call objective function and which you call constraints. Or: imagine the level curves of f expanding until they touch a fixed level curve of g; alternatively imagine the level curves of g shrinking until they touch the fixed level curve of f: same result. TiddlyWiki: a personal wiki, all in one Javascript-filled HTML page. Several nodes can be displayed at once; all is searchable and taggable. Can be published on the web, but other people can't permanently edit it. November 7 Closures in javascript: at the moment an inner function is *defined*, all the free variables in its body are bound to the variables active in the then current execution context. These variables persist and can be read/changed by the inner function, even if the outer function has stopped execution (e.g. if the inner function has been passed to the caller via return, or has been assigned to some global variable/event handler). Several inner functions created this way can share access to the same variables, and communicate. var f,g; function Outer() { var x=12; //inner functions assigned to global variables; //form closure and keep access to variable x. f = function() { x++; print(x); } g = function() { x--; print(x); } x = 100; } Outer(); f(); => 101 //and not 13! f(); =>102 f(); =>103 g(); =>102 function Test(x) { print(x); //The x in the body of f does not refer to the variable //current at *execution* time, but to the one current at //*definition* time. f(); } Test(20); => 20 => 102 Another example: ---------------- function Outer(x){ var f=function() { x++; return function(){x++; print(x)} } return f; } var h=Outer(4); var g; function Outer2(x){ g=h(); } Outer2(400); g(); =>6 //and not 401, because all the x's in the body of f were bound to Outer's x at the time Outer was executed and f(=h) was defined. Closures work the same in Lisp/Scheme/Python/Smalltalk (called "blocks" in Smalltalk). November 9 Real-time PCR: after every round of PCR, measure the amount of DNA with some dye. So you can *quantify*, not just *detect*, specific sequences. You determine the Ct value: the number of cycles needed to cross a predetermined detection threshold, and you compare the Ct values of different samples. November 19 Key bumping: Take a key that fits into a lock but can't turn it, then file every position as low as possible (not too steep before and after the positions or else key won't go in and out the lock anymore). File a bit off the tip of the key and off the part where key meets outer lock. Insert in lock, hit slightly from the outside, and shortly afterwards turn: lock opens. Hitting the key pushes it further in (allowed by filing off the tip and the outer part), forcing the pins in the lock up the slope. This upward impulse is transmitted to the unconnected pins, which jump up, while the original pins stay down (kicking a billard ball that touches another will only move the second one, not the first). The lock can be turned in the short moment the unconnected rods are flying up. Lots of illustration videos on youtube, search for "bump key". November 25 If r is the outcome of a measurement process designed to estimate the true value R, then E[(r-R)^2] = Var(r) + Bias^2(r), the latter being (E[r] - R)^2. "Reliability vs. Validity": often increasing one decreases the other. To estimate probability densities, better than histograms: compute convolution of data distribution with some smooth kernel like Gaussian; increase variance of kernel to increase smoothness of estimated density. Can use density estimate to plot probability level curves in bivariate case. If X,Y,Z are random variables, need to distinguish between "X & Y are independent" and "X & Y are independent given Z". Neither statement implies the other! EM algorithm: Given observations x1,...,xn, you want to estimate the parameters pk, sk of a probability distribution of the form f(x)=sum(pk*fk(x,sk),k=1..K) (a mixture of K distributions fk whose shapes are assumed to be known; sum(pk,k=1..K)=1). 1. Pick all the pk, sk randomly. 2. Assuming pickings from previous step were correct, estimate the probability that data xi came from distribution k, Pr(k|xi) for i=1..n, k=1..K using Bayes. (This is the E step.) 3. Compute new updates of pk, sk based on result of 2. (This is the M step.) Then go back to 2. Once it converges, repeat with a different random starting point, to find the solution that maximizes likelihood of the observations. The computed Pr(k|x) can be used for clustering and classification of new data x. Other approaches to clustering: K-means (pick K random cluster centers, assign each point to the closest cluster center, then use averages of each cluster as new cluster centers and iterate); agglomerative (initially each point in its own cluster, then successively merge two clusters that are "close" together). Cross-validation: To check how good a classification/prediction algorithm performs on unseen data, divide off a chunk of the original data, construct the model based on the other data, and then evaluate its performance on the chunk. Particularly popular: leaving-one-out method where the chunk has size 1, and cross-validation is performed for all possible chunks, to find an average error rate of the algorithm, so that it can be compared with competing algorithms. Bootstrap method: to estimate how an algorithm's performance on the observed sample compares to its performance on the whole population, take a random subsample of the sample and compare the algorithm's performance on the subsample to that on the sample. To predict classification based on a training set: Logistic regression (for two classes); look at k closest neighbors of x in training set and base classification of x on those; support vector machines (see above; inefficient for large training sets); tree models (keep dividing a component of input space along one variable, so as to minimize entropy; each component is assigned to the class of the majority of training points in it); naive Bayesian (assume that variables are independent given the classes; use Bayes to find probability of a given class given an observation x; works well even if assumptions are wrong, can be improved with Bayesian networks where some variables are taken as dependent). Naive Bayesian and and closest neighbors can easily deal with missing values. December 18 In MySQL, use MyISAM tables for simple read-mostly applications, and InnoDB tables for read-write applications. The former has a locking mechanisms that slows down writing in many situations. InnoDB also allows transactions and foreign key constraints. In general, don't forget to index important rows and run EXPLAIN ANALYZE on your queries. "S LEFT (OUTER) JOIN T ON cond" returns all the rows from SxT that match the condition, plus all the rows from Sx{null} to make sure that every row of S shows up at least once in the result set. SELECT returns a table, so can be nested. MP3 creation has several quality settings (that take different amounts of time), so the bits/sec rate of an MP3 stream is not enough to tell the stream's quality. December 27 Plato: The creator god ("demiurge") is transcendental, sets moral law, is interested in human affairs, wants the good. Aristotle: World has existed forever, no need for creator. God is the ultimate mover, the reason behind the eternal order. God thinks theoretical thoughts, leading to the human ideal of understanding the world. Epicureans: World is infinite and made of moving atoms, everything is to be explained scientifically. God is tranquil, intelligent, made of fine atoms, not interested in us. Stoics: World is finite and made of continuous matter whose movement is caused by divinity present everywhere. Divinity set the rules and obeys them; no miracles. January 7 MHC are the genes that define the body's self-tag and determine selectivity against pathogens. The two parents' MHC versions operate co-dominantly, meaning parents with differing MHCs produce children able to withstand more diseases. Female mice can sniff the MHC of males based on urine and prefer MHC-different partners. (Similar in fish.) Human females also prefer the body oder of MHC-different partners. MCH-like human couples have more often difficulty conceiving and produce weaker babies. However: when women take contraceptives, they prefer the body odor of MHC-like men. Number and intensity of wars has declined significantly since the end of the Cold War, and Pinker argues that much longer, similar trends are at play (much fewer people die violent deaths). Rate of reported rapes in U.S. has decreased by 85% between 1980 and 2005. January 15 DSL: last-mile phone lines aren't built to cleanly transmit the high frequencies needed for DSL speeds. So the phone company measures the distortion produced by any single last-mile phone line, and puts in a box that precisely complensates for that distortion. 1/24/2007 1:34 PM God for the atheist (Seligman): our institutions get better (in the moral sense), we get more powerful (in the technological sense) and we learn more (scientifically). The logical end and limit point is a good, omnipotent, omniscient entity: we are creating God. 1/24/2007 10:13 PM Defeat any copy protection on video/audio: play the content on a certified software player in a virtual machine, copy material from the virtual screen/loudspeaker. 1/28/2007 9:24 PM Of the important human features (big brain, language, upright walk) the latter came first, in Lucy (3.2 mio years ago). Large animals have slower metabolism than smaller ones (ways are longer) but they all live about the same number of heart beats (humans live a bit longer than "allowed" by this equation, they develop extremely slowly.) 1/31/2007 10:51 PM "Shoplifting raises everyone's prices" is false. Shoplifter who would otherwise have bought the item but steals instead increases fixed costs and lowers the price-demand curve and therefore lowers price (and profits). Same for a thief who sells the stolen item afterwards. Advertising raises prices for the same reason, and college teachers selling free textbooks lower prices. Thieves who would never have bought the item do not affect prices as long as the cost function (cost per item depending on number of items ordered) is constant. If non-constant need to analyze more. Raising a constant cost function always increases prices. 2/24/2007 4:23 PM Free market does not always give Pareto optimal solution. Customers equally distributed on a linear road [0,1], two competing stores, every customer shops at the closest store and incurs travel cost proportional to distance to store. Both stores will settle in the middle (Nash equilibrium), but that's not Pareto optimal; better would be to locate the stores on 1/4 and 3/4. Generally, competing firms will tend to cluster at the center of their markets (not just geographically, but also in terms of product characteristics). Rational self-interest predicts that perfect competition (in an efficient market with full information) will drive the market prices down until they reach the no-profit situation (allowing for reasonable return on capital investment built into the cost function). 3/2/2007 3:42 PM NP complete problems are "on average" not that hard; good approximate solvers exist. The human brain is also pretty good at these. 3/14/2007 10:18 PM The buoyant force of an object is equal to the weight of the displaced water (Archimedes). Proof: imagine a parcel of water were in the place of the object; the water wouldn't move, so a force equal to its weight and directed upwards must be acting on it. 3/24/2007 11:29 PM Creativity: frontal lobe generates ideas, temporal lobe edits and evalutates ideas, dopamine from the limbic system provides drive, reward, low latent inhibition. 4/9/2007 11:42 PM Numeral systems: ~2000BC Sumer/Babylon: base 60 positional system. Digits are written in a base 10 additive system. They didn't use a zero place holder symbol until ~700BC. ~300BC Brought to Greece by Alexander but not adopted. Greeks used additive system with sequence of letters for digits and multiples of ten; later adopted and improved by Romans. ~500AD our base 10 number system with 0 in India, later adopted by Arabs ~1200AD Fibonacci writes book to popularize system in Europe, meets with resistance, some merchants and bankers adopt it. 4/12/2007 9:50 PM male: sex drive high and constant usually exclusively homo or hetero some images will arouse and some won't; knows the difference female: sex drive lower except around ovulation all sexual images will engorge genitals and lubricate vagina, even the onles they don't like (possibly defense against rape?) more often open to both sexes Males have higher IQ std dev than females. Reason: many brain genes are on X chromosome. 9/23/2007 12:53 AM Vitamin D (from food or sun light) may prevent LOTS of cancers (and bone loss in elderly women). Vitamin B12 is beneficial in the elderly, and folic acid (B9) for women. All other vitamin supplements don't help and some even hurt. (added 23 Nov 2008) 9/23/2007 12:54 AM Smog or second hand smoke: immune system reacts in the lung, leading to higher incidence of blood clots => heart attacks. Supported by tests in animals, people; statistics after smoking bans. 9/23/2007 12:58 AM Poincare-Hopf Theorem: Given compact C-inf manifold with boundary and C-inf vector field that at the boundary is normal to the boundary, and has only isolated singularities (i.e. zeros), then the sum of the indices at the zeros equals the Euler characteristic. (Index = degree of induced map from sphere to sphere near zero.) By applying it to a gradient, get "Mountaineer's Theorem": a landscape that smoothly levels off towards infinity will have (# of local maxima) - (# of saddle points) + (# of local minima) = 1 Fermat's principle in optics: light follows a path whose length is stationary when pertubed slightly. Reason: if it isn't stationary, the phase of nearby paths is different and light cancels. This principle also underlies gravitational lensing! 9/30/2007 2:32 PM Hydrolysis of water: * at the cathode, where electrons are pumped into the water, you have the reduction reaction 2 H2O + 2 e- -> H2 + 2 OH-, the gas H2 escapes and the environment around the cathode turns basic. * at the anode, where electrons are pumped out of the water, you have the oxidation reaction 2 H2O -> O2 + 4 H+ + 4 e-, oxygen escapes and the environment turns acidic. * Without any salt in the water, the buildup of H+ at the anode and OH- at the cathode will soon resist the further flow of current, until these ions diffuse away from the electrodes. This diffusion is slow, so pure water is a poor conductor. * With extra ions in the water (e.g. from salt or acid), the buildup of charges in the water around the electrodes is quickly neutralized by nearby ions rushing in or out. (The H+ and OH- ions naturally occurring in pure water could in principle do the same job, but they are far too rare.) * The extra ions could in principle also partake in reduction at the cathode or oxidation at the anode. However, these ions are far fewer than water molecules, and these reactions generally require more energy to carry out than the bread down of water. 10/20/2007 2:23 PM Spiral form of galaxies: each star has elliptic path, different stars have different ellipses, inside stars move faster; under some assumptions you get slowly rotating log-spiral density pattern. 11/1/2007 11:58 AM Phyllotaxis: the patterns of plant structures (leaves, seeds etc.). The plant grows at the tip of a cone; new structures arise there, want to stay away from previous structures as much as possible, then "move down the cone" as the cone grows further at its tip. This requires that they are placed at a turn of the golden ratio (tau) of a full circle relative to the previous one. (Tau is the "most irrational" number as it has the slowest continued fraction expansion.) This golden ratio pattern will create spirals whose numbers are two Fibonacci numbers (as their fractions best approximate tau.) Growth rate determines particular Fibonacci numbers; sometimes that rate changes, and e.g. sunflowers often show different numbers of spirals as you move out. Leaves along a stem and petals of a rose are also often turned tau*2pi with respect to each other, for the same reason. 11/1/2007 7:20 PM Lions don't usually hunt cooperatively; typically only 2 are active. A pride has one or two males, often brothers; must constantly defend harem against young outside challengers. Ovulation is stimulated: after the female is ready, she flirts and then they have sex for 200-300 times over 2-3 days. 11/16/2007 9:36 PM Stoicism: started by Greek Zeno 3rd century BC, later writers: Romans Seneca (tutor and advisor of Nero), Marc Aurel (emperor). Pantheistic materialism; everything is ordered for the best; can't be affected by loss or love; suffering is good for you; committing suicide is honorable. Follow nature, i.e. develop reason, do the best work at your position. People are fundamentally equal, strife for a global society (i.e.(?) Roman empire). 11/24/2007 1:26 PM MPTP (byproduct of some synthetic illicit drugs) specifically kills dopaminergic neurons in substantia nigra, causing Parkinson's in humans. It's manufactured as a herbicide. A pesticide (rotenone ) causes Parkinson's in rats. Maybe Parkinson's is caused by some environmental chemical? 12/19/2007 and 6/8/2008 and 8/18/2012 Magnetism is the relativistic correction for electrostatics. A test charge moving parallel to a current-carrying line will see different distances between the positive and negative charges, because of length contraction. This creates a net charge on the line and an electrostatic force in the rest frame of the test charge. In the rest frame of the line this is the magnetic force exerted by the current on the moving charge. A moving charge or changing electric field creates a magnetic field. A moving magnet or changing magnetic field creates an electric field. Details in Maxwell's equations. Ferromagnetism: Iron and some other metals are attracted to a magnet because the unpaired electrons' spin align to form magnetic domains, and an externally applied field aligns these domains, increasing the internal field (= huge relative permeability) and producing an attractive force. Above Curie temperature this won't happen. Paramagnetism: Magnetization is proportional to applied magnetic field (and inverse proportional to temperature) if magnetic moments of electrons don't cancel completely. Smaller relative permeability. Diamagnetism: Change in applied magnetic field will cause electrons to form current loops whose magnetic field opposes the applied change. So material is repelled by any magnet. Every material has this, but it's a much smaller effect than paramagnetism or ferromagnetism. Extreme in super conductors. B = "Magnetic field" = "magnetic induction" = "magnetic flux density"; occurs in Lorenz law describing force on moving charges H = "Magnetic field strength" = what is created by a current or magnet. In vacuum, H and B are basically the same (except for different units). If the field passes through a material though, the material may be magnetized, changing the field within it, and H has to be multiplied by the material's relative permeability mu to get the effective B. Non-zero nuclear spin can also turn a substance magnetic, but is much weaker (factor 1000). Protons pair up, and so do neutrons; if nucleus has an even number of protons and of neutrons, then nuclear spin will be 0. 1/7/2007 Everybody has about twice as many female ancestors as male ancestors, because in every generation a few men father many children. Smoking protects against Parkinson's. 1/20/2008 4:44 PM Cancer cells use lactose fermentation (in cytosol) rather than citric acid cycle (in mitochondria), even if oxygen is plentiful. This is the Warburg effect. Old theory: mitochondria of cancer cells are damaged and can't carry out citric acid cycle. New theory: mitochondria are responsible for apoptosis program, and inside a tumor there's not much oxygen, so it's in the interest of a cancer cell to shut off mitochondria. While fermentation is not as efficient as the citric acid cycle, it is much faster, and if the "egoistical" cells use a lot of sugar they will actually get more energy, and they will still have residual carbon skeletons left over to use for further growth. The lactic acid also harms surrounding healthy cells and weakens the extracellular matrix, allowing metastasis. Pyruvate hydrogenase kinase switches between lactose fermentation and citric acid cycle. Dichloroacetic acid (DCA) inhibits this enzyme, thereby increasing citric acid cycle and decreasing lactose fermentation. By restarting the mitochondria, DCA restores apoptosis, and cancer cells die. 3/21/2008 2:04 PM Nanoradio: one end of a nanotube is connected to a battery's electrode, the other end is free to vibrate near the other electrode. It will vibrate in tune of external electromagnetic radiation; resonance frequency can be adjusted by changing battery's voltage. Electrons tunnel from tube's tip to the electrode, giving a current which depends on distance of tip to electrode. This dependency is quadratic, allowing the device to demodulate and recover the original signal. [Use cos^2(x) = 1/2 ( 1 + cos (2x) ). ] U.S. life insurance policy will pay if bought 2 years before suicide. 3/27/2008 9:44 PM Bootstrap resampling: if you've got 100 data points, repeatedly draw 100 points from this set (with replacement), and compute their means. Order these means, then take appropriate percentiles to compute population mean confidence interval. Similar for hypothesis testing. No model assumption needed. 4/22/2008 12:17 PM Darwin's arguments about God/Christianity from his autobiography: 1) old testament is comparable to Hindu writings; despicable in that it presents a revengeful tyran god that condems non-believers to eternal torture; obviously factually wrong in manners of science. 2) If God created human suffering for moral uplifting of humans, why do sentient animals suffer so much? 3) Inward strong convictions about the existance of a higher being, found in all cultures, are of no evidentiary value as the various concepts of god differ widely. 4) Strongest argument for existence of God: the immense and wonderful universe is hard to conceive without an intelligent mind as first cause. But: we can't trust our mind in these matters, since our mind has evolved from lower animals whose convictions on metaphysics we certainly wouldn't trust. 6/8/2008 Thermodynamics. Assume that every isolated system (no exchange of heat/work/matter with environment) is an ergodic system with symmetric transition probabilities (motivated by QM, but also classically). This means that system approaches equilibrium where all states are equally likely. [For an ergodic system in equilibrium, time averages and ensemble averages are the same.] For an ensemble of such isolated systems, entropy is S = - k sum p_r ln(p_r) where p_r is probability of state r. S will increase over time ("H theorem"), until equilibrium is reached, where simply S = k ln(N). S is a function of the ensemble/macrostate. If N(E) is the number of states of the system, depending on its total energy E, we define (absolute) temperature T = 1/(k d ln(N)/dE). High temperature systems lose only few states (lose little entropy) if they give off a bit of energy dE. That's why systems in contact will exchange heat until the temperatures meet somewhere in the middle. Quantified: dS = dQ/T where dQ is the heat taken in. If a system A is in contact with a large heat bath A' which is kept at constant temperature T (or if only the ensemble average energy of A is known), then the states of A in equilibrium won't all be equally likely; instead state r with energy E_r will have probability p_r ~ exp(-E_r/kT) (Boltzmann/Maxwell). Lower energies are more likely, so systems "try to minimize their energy". In this case, consider partition function Z=sum exp(-beta E_r), which depends on beta=kT and on the external parameters x (which affect the E_r). Then avg E = - d ln(Z)/d beta; driving force d 1/beta ln(Z)/dx; entropy S = k ln Z + (avg E) / T. Simple heat [Carnot] engine: given two heat reservoirs R1, R2 at temperatures T1 Use of polarization filters in cameras and sun shades.) 10/31/2008 8:16 PM Migraine = temporal artery (just outside of skull) expands Normal headache = muscles attached to outside of skull contract 12/29/2008 1:00 PM Air at room temperature and atmospheric pressure: density: 1.3 kg / m^3 speed of a typical N2 molecule: 450 m/s = 1620 km/h speed of a typical H2 molecule: 1700 m/s = 6120 km/h mean free path: 70 nanometre ~ few hundred molecular radii 1/3/2009 3:37 PM Birds have a one-way lung with air entering air sacs and then passing almost constantly through the lung, allowing for a countercurrent system between blood and air: much more efficient than mammals' two-way lung. Lungs arose from fish lungs connected to the gut; in ray-finned fish these developed into swim bladders. In human embryos they're still connected to the gut. Aging is a tradeoff: an organism can either invest energy into quick procreation or into repair of accumulating damage to stave off aging. Animals that have lots of predators (early mammals) are better off doing the former. By contrast, many reptiles live long. 3/8/2009 9:13 PM odds(A|E) = P(A|E)/P(~A|E) = P(A and E)/P(~A and E) Bayes theorem for odds: odds(A|E) = P(E|A)/P(E|~A) * odds(A) 3/11/2009 7:30 PM Evo-Devo (Carroll): The genes involved in animal development (e.g. Hox) are largely conserved. Each one has numerous independent functions in development, acts as transcription factor for numerous other genes, and is affected by numerous other transcription factors. (By contrast, genes coding for ordinary structural proteins are regulated only by one or a few transcription factors.) The development genes thus form large Gene-Regulatory-Networks (GRNs). These GRNs evolve by mutations in the transcription-factor binding sites (CREs, Cis-regulatory elements). New CREs can arise from transposable elements. 3/19/2009 6:42 PM Archimedes' way to find the volume of a sphere: semisphere is equal to a cylinder minus an upside-down cone, since the horizontal slices of the two solids have equal areas. 3/23/2009 2:09 PM Moebius grp (connected complex Lie grp with complex dimension 3) = grp of Moebius trafos (by def.) = PGL(2,C) = SL(2,C)/{+-} (with universal cover being SL(2,C)) = grp of conformal orientation-preserving maps S^2 -> S^2 = Aut (P^1 C) (holomorphic automorphisms of the Riemann sphere) = grp of trafos in Minkowski spacetime preserving Lorentz form, the origin, the direction of time and orientation of space = SO+(1,3) (proper orthochronous or restricted Lorentz group, by def.) = component of identity in Lorentz group = grp of orientation-preserving isometries of hyperbolic 3-space (which we can think of as the interior of a 3-ball akin to the Poincare disk, with a S^2 as "boundary") * Poincare group = grp of all isometries in Minkowski spacetime * Lorentz group = O(1,3) = subgroup of Poincare grp that fixes the origin (observer) * Some authors define a (larger) Moebius group as the real Lie grp of conformal maps S^2->S^2 = O+(1,3) = grp of isometries of hyperbolic 3-space * the above (smaller) Moebius group is the identity component of this (larger) one. 5/6/2009 5:42 PM Possible explanation for 1) long period of thrusting during intercourse 2) refractory period after male orgasm 3) peculiar shape of glans Thrusting removes previous male's sperm from vagina. 8/29/2009 3:38 PM Mermin's thought experiment to show quantum theory paradox: Machine has one emitter and two distant detectors. Each detectors can be set to '1', '2', or '3' and has a red and a green light. If a button at the emitter is pressed, one of the two lights at each of the emitters goes off. No information from the detectors can flow back to the emitter (enforced by distance/speed of light). Observations: if both detectors are set to the same number, they will always show the same color light; if they are set to different numbers, they will show the same color light only 25% of the time. Classically, if you assume 3 bits of information travel from emitter to the detectors, you get that the latter probability should be at least 33% QM predicts that such a machine can actually be built: the emitter sends out two identical photons; the detectors, depending on number setting, measure spin in 0, 120 or 240 degree direction. Probability that two different settings will give same sping is cos^2(theta/2), where theta=120; this is 1/4. This is similar to EPR paradox and Bell's inequality: shows that local hidden variable theories can't exist; either some signal travels faster than light between the detectors, or the system doesn't have a definite state until it is measured. 8/30/2009 4:29 PM Limestone (CaCO3, calcium carbonate) is sedimentary rock from marine animal shells or microorganisms; chalk is a special form of soft limestone and marble is metamorphic (compressed) limestone. Heating limestone gives burnt lime (CaO), releaseing CO2. Treating burnt lime with water gives slaked or hydrated lime (calcium hydroxide, Ca(OH)2, very alkaline). Slaked lime is used in lime mortar and plaster: mixed with water and exposed to air, it will bind to CO2, harden and recreate calcium carbonate. Cement consists mainly of calcium silicates CaO.SiO2 made by heating ground limestone with clay. It hardens when water is added; the water serves as crystal water; no air is needed for this process. Gypsum (calcium sulphate hydrate, CaSO4.2H2O) is a mineral created when hot springs, vulcanos, seas etc. provide sulfur. For gypsum plaster, it is first heated to remove some of the crystal water; when water is later added, it hardens and gypsum is recreated. 9/7/2009 6:29 PM Ideas from Musser's "Idiot's Guide to String Theory" "Ontogeny recapitulates phylogeny" also in physics: the various forces (and particles) are ultimately seen as aspects of a single entity (phylogeny), and in the early development of the cosmos they were indeed unified, then separated as the energy dropped (ontogeny). In special relativity, space and time are combined to form spacetime. Different observers slice it differently into space and time. In General Relativity, where gravity is taken to be equivalent to acceleration, the accelerated observer keeps slicing spacetime with different time axes, leading to a curved coordinate system: mass causes gravity (which is equivalent to acceleration) causes curvature of spacetime. Equivalence of mass and energy: in a fast moving train, two equally massive balls are thrown, one from the front and one from the back. They meet in the middle, stick together and fall to the ground. For the static observer, it looks as if one ball is slow and the other is fast, and when they meet they move together with the train's speed. This observer must conclude that the faster one is heavier: kinetic energy has mass. When two bosons are interchanged, their wave stays the same. When two fermions are interchanged, their wave changes sign. Two fermions can't sit on top of each other: the wave would have to be its own negative, i.e. would have to be 0. All fermions come in two versions: left-handed and right-handed. They're further divided in matter and antimatter. Weak force only acts on left-handed particles; its bosons (W&Z) have mass, unlike gluons and photons. Electron charge appears smaller than it really is, because the naked electron is surrounded by virtual electrons and positrons, attracting the positrons and repelling the electrons. At high energies, we can muscle through to the real electron, causing the electromagnetic force to become somewhat stronger. Similar arguments show that the strong and weak forces get weaker at higher temperatures, and eventually the three of them roughly meet. Gravitation varies much more dramatically with scale/energy. At the Planck scale (10^-35m) all 4 forces become roughly equal. Probing even smaller scales would require so much energy that a black hole would be created. Cosmological constant problem: accelerating universe requires a small non-zero constant; quantum theory predicts a huge one (vacuum energy from virtual particles). Main argument for string theory: gravity was not built in from the beginning, but a spin-2 particle naturally turned up; the graviton is known to be spin-2. Apparently, consistency of string theory requires General Relativity to hold!? The various string theories (9 space + 1 time dimension) unify in M theory: 10 space + 1 time. Extra dimensions are required to make Special Relativity work; using 10 dimensions unifies the particles of the Standard Model. The extra dimensions could be tiny, in which case gravity acting within them could look like electromagnetism to us; or they could be relatively large, in which case the end points of our open strings could be "glued" to a 3-D brane floating in the higher-dimensional space. Closed strings, like gravitons, could move between branes: large dimensions could explain the weakness of gravity, and they could be confirmed if gravity increases on small scales. Multiverse could consist of numerous universes: pockets where inflation has halted, the laws froze into different spots on the string theory landscape [each spot corresponds to a different way the extra dimensions curl up] and a local "Big Bang" occurred; in between those pockets the empty space is still inflating. [Added 10/22/2012, 10/4/2015:] Inflation can occur if you have a region where an "inflation field" creates negative pressure (like a rubber band: you need to put energy in to expand it) and if the energy required to expand it to twice its volume is exactly enough to double the energy contained in the field. General Relativity says that negative pressure causes a negative gravitational force that pushes things apart and enlarges the region, creating a self-sustaining exponential growth called inflation. Inflation eventually halts in certain spots (which turn into pocket universes when the energy stored in the inflation field is turned into ordinary particles and radiation) because the negative-pressure field eventually decays away. But this decay is slower than the expansion of space, so you have "eternal inflation" surrounding the pocket universes. Energy conservation is not violated by inflation: the density of energy (which is stored as potential energy of the field) in the inflating patch stays constant, so the total energy in the inflating patch increases, but this is balanced by the growing negative energy of the gravitational field (every gravitational field stores a negative amount of potential energy). It's quite possible that the total energy of our pocket universe is 0. Evidence for inflation: microwave background radiation is very uniform; an ordinary Big Bang wouldn't be able to create that uniformity because particles would have to move faster than c; in inflation there is enough time to create homogeneity. But quantum fluctuations create slight inhomogeneities which inflation would enlarge; the microwave background shows exactly the right kind of fluctuations. The universe is flat (Omega close to 1) which in an ordinary Big Bang model is an unstable situation. Inflation creates Omega=1 automatically. Standard model claims protons are stable; string theory says they last on average 10^35 years before disintegrating. Supersymmetry: to every fermion there's a boson, to every boson there's a fermion. These partners are named with leading "s". None of them have been observed. Predicted by string theory and some other theories. They could be the cause of dark matter. The force related to supersymmetry might be gravity. The information content of a black hole (and of everything else) goes up with surface area and not with volume [massenergy ~ r and temperature ~ 1/r and information ~ massenergy / temperature]; this is an argument for the holographic principle: string theory in an n-dimensional spacetime with gravity is faithfully reflected by a string theory in a "surrounding" (n-1)-dimensional spacetime without gravity. We are sitting inside, but all information must also be stored in the outside. 9/8/2009 12:30 PM Proof of existence of dark matter: a galaxy cluster consists of dark matter, hot gas, and a few galaxies. The gas is the major part of the ordinary matter. Two clusters collided. The gas stayed in the middle (x-ray picture); the galaxies and dark matter moved through each other (confirmed by gravitational lensing). 11/7/2009 3:59 PM Steam power: 1) Newcomen engine (~1710). Driving the piston out of the cylinder and filling it with steam is *not* done by steam pressure but by action of a counter weight. The cylinder is then cooled with water that's sprayed in; the steam contracts and condenses and the resulting vacuum pulls the piston into the cylinder, pulling the counter weight up and driving the machine (which is used as a pump). Problem: energy is lost by repeatedly heating/cooling the cylinder. 2) Watt engine (~1775). The cooling/condensing of the steam is done in a separate cylinder; the first cylinder is always hot while the condensing cylinder is always cool. Two further Watt improvements: a) use bored instead of cast cylinder, to produce better vacuum. b) double acting engine: both halves of the main cylinder are alternatingly filled with steam => same time for up- and down-stroke, necessary for industrial applications other than pumps. 3) Cavish engine (~1800). Use high pressure steam for the first time. 4) Steam turbine. (~1880). Better steel allowed a turbine to be driven by high-pressure steam, giving many more revolutions per minute, suitable for driving an electrical dynamo. Several such turbines are placed in series, followed by a condenser, bringing the water back to the boiler. These are the highest efficiency generators available today (>40%). 1/1/2010 3:09 PM Plants, when attacked by herbivores such as caterpillars, produce chemicals that directly interfere with the attackers and also emit volatile chemicals that attract the natural enemies of the attackers. These signals may be picked up by nearby plants, causing them to start their own defense cascade. 2/13/2010 7:28 PM Common sedimentary rocks: * Limestone from shells and corals of reefs * Shale from mud in lagoons (between reef and beach) * Sandstone from sand at the beach * Coal from inland swamps So if the sea retreats and comes back, we often get a sequence limestone-shale-sandstone-coal-sandstone-shale-limestone-shale-... 2/14/2010 2:35 PM Buddhism * Theravada (Sri Lanka, Thailand, Burma, Laos, Cambodia) "Lesser Vehicle" - mostly focused on original teachings of the Buddha to attain individual enlightenment. Texts in Pali. * Mahayana (China, Korea, Japan, Tibet, Nepal, Vietnam) "Greater Vehicle" - mostly focused on attaining true buddhahood which aids in universal liberation and enlightenment. Accepts the later Lotus Sutra (1st century BC); defines Bodhisattvas as a being close to enlightenment who holds back to help others on the path to liberation. Many sects: Tibetan, Zen, etc. Texts in Sanskrit. * Vajrayana, or Tantric Buddhism. (Tibet) "Diamond vehicle". Sometimes considered part of Mahayana. Focused on rituals to achieve enlightenment. 3/14/2010 3:58 PM Airtraffic Control (ATC) and Airplane navigation * Every plane is assigned to an ATC region at all times. Communicate by radio. As the plane leaves one region, it is handed off to another, often communicating on a different frequency. VHF/UHF is used on land (line of sight); longer range HF over the ocean. * Each ATC group uses ordinary radar ("primary surveillance") to track the planes it is responsible for. It also uses secondary surveillance radar (SSR): a vertically rotating transmitter sends out a signal, and every (large) plane has a transponder that automatically responds to the signal with a message that includes identification of the plane and possibly the height above ground. Based on the time when this response arrives, ATC can figure out the plane's position. * Planes do not have radar and don't know what's happening around them. Except for the modern ADS-B technology that's being introduced on passenger aircraft: Each plane broadcasts its precise location to other planes and to ATC. The plane knows its location through GPS improved by a network of geostationary satellites (WAAS). * Navigation is most often through VORs: a station sends out a signal that varies in each direction; when receiving the signal you know at what direction you are from the VOR. With more expensive equipment you can also tell the distance to the VOR. * To guide a plane during landing, airports have instrument landing systems (ILS). Antennas at each side of the runway broadcast a signal that decreases in strength depending on direction; directly above the runway the two signals are equal in strength, and the plane (even small ones) can detect this. A similar system is used vertically; two signals are equal in strength iff the plane is on the right flight path down to the grown. 4/14/2010 3:46 PM Central limit theorem proof: probability distribution of the sum of independent variables is given by convolution of the distributions; Fourier transform turns this into multiplication. Now use the fact that if f(0)=1, f'(0)=0 and f''(0)=-1, then f^n(x/sqrt(n)) -> exp(-x^2/2). Inverse Fourier transform of a Gaussian is a Gaussian. 4/25/2010 3:05 PM From Landsburg's "The Armchair Economist": Indifference principle: For people with average tastes and abilities, all activities are equally desirable, taking all factors into account. (Else nobody would engage in the least desirable ones.) Only owners of resourses in fixed supply can avoid it. Economists call a policy inefficient if it creates "deadweight losses", i.e. losses (expressed in dollars) that are not offset by other's gains. When having to choose between several alternatives, ask each affected person how much they would be willing to pay to effect a particular alternative (benefit) or to prevent that alternative (cost). Add up all benefits and costs and pick the alternative that maximizes benefits-costs. Argument for this method: if another alternative is chosen, then one can theoretically construct a new alternative that everyone would prefer over the chosen one. (So in that sense the chosen alternative is inefficient.) First Fundamental Theorem of Welfare Economics: Competitive markets allocate resources efficiently. (Reason: if a product has a single market price, then all producers will operate at the same marginal cost, eliminating inefficiencies.) Rationality or biological evolution do not always lead to efficiency, see e.g. sexual selection leading to expensive signalling processes. Second Fundamental Theorem of Welfare Economics: every efficient resource allcation can be achieved by first redistributing income, and then letting the competitive market take over. Winner's curse: the winner of an auction valued the item higher than all the other (expert) bidders, so it's likely that he overvalued the item and will be disappointed by it. Same is true for every purchase you make: your quality estimate of an item is not always correct, sometimes too high and sometimes too low; you purchase only items whose quality you estimate to be very high, and those quality estimates are likely to be higher than the true quality. 5/8/2010 7:06 PM Aumann's "Agreeing to Disagree" theorem: two honest truthseekers should over time agree when discussing the truth of a statement, since each one will treat the other's opinion as new evidence, to be added to the private evidence. The "Delphi method" uses this idea to find a rough consensus among a group of experts: they all answer a questionnaire and give written reasons for their answers; these answers and reasons are then anonymized and distributed to each member of the group; then another iteration starts, the questionnaire is filled out again, etc. 9/19/2010 4:09 PM The only living dinoraurs (Dinosauria) are the birds. Crocodiles belong to the larger group Archosauria and are the closest living relatives of birds. The other reptiles are more distant relatives of birds/crocodiles. 10/3/2010, updated 3/15/2017 Homo erectus lived in Africa, Europe and Asia, 1.9-0.1 million years ago. They controlled fire and used sophisticated stone tools. Homo ergaster is probably identical to or an ancestor of H. erectus. H. ergaster/erectus gives rise to H. heidelbergensis in Africa (about 600,000 years ago). H. heidelbergensis moves to Europe. Neanderthals appear in Europe and the Near East about 400,000 years ago, having evolved there from H. heidelbergensis. They eat almost exclusively meat of animals that they hunt collaboratively with spears and other weapons. Evidence of cannibalism exists at numerous Neanderthal sites. They can speak but they don't have art and few ornamental objects. H. sapiens arises from H. heidelbergensis in Africa (about 200,000 years ago); about 40,000 years ago, during the last ice age (which lasted from 110,000 - 12,000 years ago), they start to move to Europe and Asia, displacing the Neanderthals. The last Neanderthals die out 30,000 years ago. As H. sapiens moved out of Africa, they interbred with Neanderthals, probably in the Middle East. Evidence: contemporary European, Asian and Australian H. sapiens share about 2-4% of their genome with Neanderthals (sequence completed in 2010), but sub-Saharan H. sapiens don't. 4/6/2011 9:58 PM Uran consists mostly of U-238 and a little U-235. Only the latter is useful for reactors or bombs. Have to use physical methods (centrifuges etc.) to enrich U-235 or separate the two. In a reactor, the U-235 molecules split when hit with a neutron, creating heat, fission products and two more neutrons. Some of the U-238 turns into Plutonium-239, which can also be used for reactors or bombs (nuclear reprocessing). Cesium rods and/or water with boron is used to slow down or swallow neutrons, to prevent an exponential chain reaction and explosion. The fission products need to be cooled for a long time since they keep decaying further. [Added 1/12/2012:] Critical mass is the smallest amount of material that starts a self-sustaining chain reaction. (17cm-diameter-sphere of U-235, 10cm-diameter-sphere of Pb-239). In an atomic bomb, the material is first kept apart in a subcritical configuration, than imploded (with some other conventional explosive) to reach critical mass and nuclear explosion. Modern nuclear weapons use the Teller-Ulam design: a conventional atomic bomb explodes and compresses a column of fusion fuel; the column contains a sub-critical core of fissile material which becomes critical after compression, explodes, and causes the fusion reaction to start. 5/28/2011 11:53 AM According to PPP (Purchasing Power Parity), 1 Euro in Germany in 2010 buys as much as $1.24 in the US. 6/26/2011 8:29 PM Basic personality traits (big five model): extraversion, emotional stability (neuroticism), agreeableness, conscientiousness, openness to experience Backfire effect: sometimes beliefs are strengthened when *contrary* evidence is presented. 6/28/2011 12:47 PM Germany loses battle of Stalingrad (turning point of war): 2/1942 Devastating US+UK bombings of Hamburg: 7/1943 Normandy landing: 6/1944 German officer's assassination attempt on Hitler: 7/1944 US+UK bombing of Dresden: 2/1945 War ends: 5/1945 8/1/2011 2:30 PM First well-defined alphabet: Phoenician, 14th century BC (right-to-left, consonants only). Possibly preceeded by a script used by Semitic workers in Egypt. Phoenician gives rise to Greek (8th century BC, left-to-right, with vowels) and that to Latin, Cyrillic etc. Phoenician also gives rise to Aramaic (8th century BC, right-to-left, without vowels) which developed into Hebrew and Arabic (same). Aramic also developed into Brahmi script in India (6th century BC, usually left-to-right, obligatory vowel notation) which gave rise to many Asian scripts: Devanagari, Bengali, Thai etc. 12/29/2011 2:49 PM Tower of Babel story from the Bible: humanity was united, speaking one language, building a tall tower. God said: "These people can accomplish anything!" So he decided to separate them and confuse them by giving them different languages. Rotational parabaloid, inscribed in a cylinder, will cut the cylinder's volume in half. Also works for base areas other than circles. 1/1/2012 7:38 PM Hawkins' "On Intelligence" describes cortex as a memory-prediction machine: Every region of cortex receives a sequence of input patterns from two or more lower regions and predictions from a higher region. It classifies each input patterns and processes the sequence of classifications (basically by memory lookup of learned correlations) into a more abstract and stable pattern ("invariant representations") which is sent on to the higher region. The current pattern classification and the prediction received from the higher region also serve to predict the next pattern classification, which is sent down to the lower regions. In the motor cortex, these "predictions" flowing down are actually commands. Consider the primary visual cortex V1 not as a single region, but as a collection of many small independent regions, connecting up to V2 subregions. Cortex has six vertical layers, from outside to inside: 1) Horizontal axons connecting different areas 2&3) Receive signals from layer 1 (predictions from higher regions and delayed input from own region) as well as from layer 4; send signals to higher regions, while also synapsing in layers 5&6. 4) Inputs from lower regions arrive at the neurons here; they send signals to layers 2&3. 5&6) send signals down the hierarchy, to layer 1 where the signal is spread out over a large area of the lower region. Receive signals from 2&3 (signals moving up) and from incoming axons that connect to layer 4. Some cells in 5 send axons to the thalamus, which sends delayed signals back to layer 1 of the same region. (Proposed mechanism for autoassociative learning.) A region is organized as a set of vertical columns; every column represents one classification; layer 4 cells in a column fire if that classification is received from below; layer 6 cells fire if that classification is received from below or predicted. Layer 2&3 produce the invariant representation. 12/28/2012 11:58 PM Turing's model for pattern generation in biology: two substances ("morphogens"), activator and inhibitor, interact by diffusion-reaction equation. Inhibitor reduces production of activator and diffuses faster than activator. Then you can get patterns of regions with high/low activator concentration. These patterns can be stable in time, or can be oscillating (chemical models exist for both situations). Minimal size of patterns is governed by diffusion coefficients (=> small animals tend to have fewer patterns, tails don't have dots but stripes). Not all biological patterns are generated like this, but some examples of Turing pattern generation have been found. 12/28/2012 The Milky Way (and most galaxies) started out as a homogeneous disk of stars rotating about the central black hole. A collision with small neighboring galaxy Sagittarius c. 2 billion years ago and possibly some other collisions disturbed the elliptical paths of the stars. Staggered ellipses resulted in spiral arms, regions of high star density that themselves rotate about the center. The circular rotation of the arms is independent of the elliptical rotation of the stars that move in and out of the arms. There is a bar in the center, two major arms attached to the bar's ends, and two minor arms between the major arms. The galaxy has mopped up smaller galaxies in the neighborhood for fuel. In 4 billion years it will collide with the larger Andromeda galaxy and turn into an elliptical galaxy (an ellipsoid of stars without spiral structure; each star on an elliptical path about the center). 2/28/2013 1:02 PM Snowball Earth: The last one occurred c. 650MYA, continents were close together at the equator, leading to lots of precipitation and high temperatures, which washes out lots of CO2 (arising from volcanos) from the atmosphere and binds it to minerals as limestone CaCO3 ("silicate weathering"). CO2 atmosphere levels drop, resulting in reduced greenhouse effect, lowering temperatures. Reflection from snow and ice cause positive feedback effect, resulting in snowball earth. Lasted 6-12MY. Ended when CO2 levels built up again. Bilateral animals appeared some 80MY later. 4/29/2013 2:41 PM Temperature T of a system with entropy S and energy E is given by dS = (dE)/T This is compatible with the fact that hotter systems lose energy to cooler systems, and that in any such interaction the total entropy increases: if system 1 is hotter than system 2, T1>T2, and system 1 gives energy dE to system 2, then system 1 loses entropy (dE)/T1 and system 2 gains entropy (dE)/T2 > (dE)/T1. Systems can have negative temperature: this happens if the number of states a particle can be in is bounded, which typically implies that the particles can't have much kinetic energy. Increasing the total energy of such a system pushes more and more particles in the high-energy states, which eventually leads to decreasing entropy. (Min entropy= all particles in lowest state or all in highest state; max entropy = particles equally distributed among the states.) Such a system will transfer energy to any positive-temperature system it is brought in contact with, and is therefore "hotter" than any such system. "Hottest" possible system: temperature just below 0. Intuitively, temperature is the average kinetic translational energy per particle (ignoring rotational and internal vibrational energy, if any.) This is not a definition but a consequence of the equipartition theorem, which doesn't apply to the situation described above. It says that the total energy per particle is equally distributed among the particle's degrees of freedom, and that temperature is proportional to average energy per degree of freedom. A molecule with more degrees of freedom (b/c of rotational or vibrational modes) therefore needs more energy (heat) input to rairse its temperature, since some of the energy will be given to degrees of freedom, such as rotation or internal vibration, that don't contribute to temperature. 5/5/2013 9:22 PM Laser cooling: use atoms whose excited energy level exceeds the ground level by deltaE, and shine laser photons on them whose energy (in rest frame) is a bit below deltaE. Then only atoms moving against the laser beam will absorb a photon (Doppler effect increases photon's energy in atom's frame; wavelength in atom's frame = wavelength in rest frame*(1-v/c)). Momentum of photon (p=deltaE/c) is transferred to atom, slowing it down; atom transitions to excited energy level. Later it emits a photon (wavelength hc/deltaE in its frame) in a random direction; in rest frame, Doppler effect causes emitted photons moving in direction of atom to have higher energy than those moving against the atom, so that the emitted photon's momentum again slows the atom down, on average. 3/29/2014 2:49 PM Monte Carlo method to play Go: Dynamically build a tree of positions and associated values by simulating many complete games that start at the current positions, which is the root of the tree. For each position p in the tree and each move m in position p record the values N(p,m) = # of times past simulations have chosen move m in positions p W(p,m) = # of Black wins resulting from past simulations that chose m in p The early phase of every simulation ("descent phase") uses the N,W values to chose promising moves, but employing a rule that ensures that rarely or never-before chosen moves are preferred. Once the simulation leaves the tree, it proceeds with a fixed randomized strategy for both sides ("roll-out phase"). (Better than complete randomization are roll-out strategies that take some pattern matching on master games into account.) Once the end of the simulated game is reached, one new leaf (the first position visited in roll-out) is added to the tree and the N,W values of all visited positions in the tree are updated ("update phase"). (One may also initialize N,W values of the new leaf with "virtual" values coming from some evaluation function learned from master games.) We continue simulating games until we run out of time, then choose our actual move based on the computed N,W values of the current position. 3/29/2014 3:35 PM Inflation at beginning of universe is needed to account for flatness or metric and uniformity of microwave background radiation. Compatible with General Relativity: energy and pressure (and all the other components of the stress-energy tensor) cause gravity; pressure can be negative but energy cannot. A region has negative pressure if expanding it will increase its energy (as in a rubber band). If you have a region with a bit of energy and a lot of negative pressure, GR will give a gravitational repulsion, so the region expands. This causes its energy to increase (because of the negative pressure). So if the initial combination of energy and negative pressure was just right, you end up with a larger region that has the same energy density and negative pressure as the original region. The process can continue, giving inflation. During inflation, the universe is very empty and cold. At end of inflation, the energy is converted into radiation and particles: very hot and dense, start of big bang. 380,000 years after inflation, atoms formed and the universe became transparent to electromagnetic radiation, which we now observe as the microwave background. 8/14/2014 5:05 PM Cells at the kidneys sense blood pressure; if it is too low, they cause the release of hormone renin. Renin increases blood pressure by a) contracting blood vessels, and b) causing the kidneys to pull less sodium out of the blood, resulting in more water in the blood (osmosis) and higher blood pressure. About 20% of the population have a genetic condition where increased salt intake is not excreted by the kidneys and therefore causes high blood pressure. Baroreceptors in carotid artery and aorta sense elevated blood pressure and relay the information to the brainstem which then downregulates blood pressure by activation of the parasympathetic and deactivation of the sympathetic nervous system. 11/2/2014 3:25 PM Energy comes only in two forms: motion and mass. "Motion" is momentum, and "mass" is rest mass, which includes potential energy, chemical energy, thermal energy etc. The formula is Einstein's E^2 = m^2 c^4 + p^2 c^2 (where E=total energy, m=(rest) mass, p=momentum, c=speed of light). In this equation, E and p depend on the observer; together they form the energy-momentum vector (E/c, p) whose length in the Minkowski metric is mc; this length is not observer dependent. Photons have zero mass but positive energy and non-zero momentum. A system of two photons, moving in opposite directions, has zero total momentum but positive energy, therefore it must have positive mass. 11/27/2014 6:11 PM Multi-level Price equation explains how altruism can evolve and group selection occurs: an altruistic gene lowers the fitness compared to other members of one's group, but may increase the overall fitness of the group. If the other groups shrink as a result and the own group grows, then the relative frequency of the altruistic gene in the overall population can increase. (added 12/25/2019) Trivers has a theory how "reciprocal altruism" can evolve: if organisms can recognize each other and repeatedly interact with each other, it can pay to perform an act that harms you a little while helping someone else a lot, in the expectation that they will pay back. (Similar to the successful tit-for-tat strategy in Axelrod's repeated prisoner's dilemma tournaments.) In this scenario, it's crucial to detect and punish cheaters and recognize friends, both of which are ubiquitous in human societies. 9/20/2015 4:53 PM Troposphere is the lowest layer of the atmosphere, up to ca. 7-20km. It's where all the weather occurs. Above it: stratosphere and mesosphere. Troposphere, stratosphere and mesosphere don't conduct electricity well; above them (at ~100km) is the ionosphere which is a good conductor, as is the Earth's surface. So we have a capacitor consisting of Earth's surface and the ionosphere, separated by tropo/strato/mesosphere. This capacitor is being charged by thunderstorms, leading to negatively charged surface and positively charged ionosphere. Friction in the clouds collects negative charges at the bottom of clouds and positive charges at the top; lightning then transfers the negative charges to Earth's surface; the positive charges travel to the ionosphere. There is also a "fair weather current" which constantly and slowly transports negative charges from the Earth back to the ionosphere. 11/29/2015 6:50 PM Betz law says that a wind turbine cannot extract more than 59% of wind power (since it isn't possible to bring the wind to full stop, so some kinetic energy must be left). This limit comes from energy considerations alone and doesn't take into account that some energy is also necessarily lost in turbulence at the blades. In realistic wind turbines, increasing the number of blades increases the efficiency slightly, decreases the optimal speed (thereby decreasing noise), increases torque, and increases cost significantly. Early wind turbines were two-bladers. Three-bladers are almost as efficient as two-bladers, easier to turn when wind direction changes, and not as noisy, which is why they are almost always used today. Recently, off-shore windparks have started to investigate two-bladers since they are easier to transport and install and noise is not an issue. 9/25/2016 2:47 PM When transmitting electrical power through a line, the current (not the voltage) causes the losses; that's why high voltage/low current schemes are used. Over long distances, DC lines are used, even though AC-DC conversion causes some additional losses. The reason is that DC doesn't have the problem of reactive currents ("Blindströme"). With AC, if an inductive or capacitive load is present, current and voltage will be out of phase, which means that the effective transmitted power is smaller than the product of effective current and effective voltage. A portion of the current, the "reactive current", does not contributes to power transmission but passes through the lines and causes losses. 11/6/2016 4:33 PM Method of least squares: we want to estimate a1,...,aq by measuring f1(a1,...,aq),...,fn(a1,...,aq). The functions fi are known, but measuring them gives values Mi which involve an error ei that we assume is normally distributed with mean 0 and standard deviation si. n is (often much) larger than q. The joint probability P(e1,...,en) has a density function that is proportional to exp(-e1^2/(2s1^2))*...*exp(-en^2/(2sn^2))= exp(-1/2*(e1^2/s1^2+...+en^2/sn^2)) and to maximize this (maximal likelihood principle) we have to minimize the sum of squares e1^2/s1^2+...+en^2/sn^2. So we pick as our best estimate those values a1,...,aq which minimize (f1(a1,...,aq)-M1)^2/s1^2+...+(fn(a1,...,aq)-Mn)^2/sn^2 This can be done efficiently if the functions fi happen to be linear, but the approach works in all generality. 1/6/2017 5:10 PM Structure of ice and snowflakes: normal water ice (hexagonal ice, ice Ih) consists of identical layers, with each layer consisting of hexagons. Each water molecule can be thought off as a tetrahedron with the oxygen at the center, two corners positively charged (hydrogens) and two negatively charged (lone electron pairs). Oppositely charged corners of different molecules form "hydrogen bonds". Six can form a (staggered, not flat) hexagon, and many such hexagons attach to from a layer. Half of the molecules in a layer form a hydrogen bond to the layer above, half to the layer below. Irregular or round sides of a growing ice crystal grow fastests since they offer many possibilities for new molecules to attach; these sides therefore tend to straighten out, and the basic form of an ice crystal is a hexagonal column. Snow crystals form in clouds when supercooled water vapor sublimes into ice. Temperature, humidity and pressure determine whether the crystal tends to grow orthogonal to the layers, forming long columns or needles, or parallel to the layers, forming thin hexagonal plates. In the latter case, it often happens that availability of vapor molecules is highest at the hexagon's corners, resulting in spikes growing out of these corners. Since the temp/humid/pressure conditions tend to be the same at the six corners of the small (ca. 1 mm) crystals, symmetrical growth results. 1/6/2017 5:58 PM Renormalization in Quantum Field Theory: From Feynman diagrams we need to compute certain expressions M(λ,p), where λ is a field-strength parameter, and p is a momentum vector. Unfortunately, M(0,p)=0 and M(λ,p)= ∞ if λ>0, which is physically meaningless. The expression M(λ,p) involves an integral over 4-dimensional spacetime, which can be solved analytically, even for dimensions other than d=4. In fact, the expression is analytic in the dimension d, which we now treat as a complex parameter. M_d(λ,p) is finite if d<>4. The idea now is to treat λ as a function of d, with lim_(d->4) λ = 0 und such that lim_(d->4) M_d(λ,p) gives a finite physically meaningful answer. For a particularly simple p0 we can measure m0=lim_(d->4) M_d(λ,p0), which can be solved for λ to determine the speed with which λ should approach 0 as d->4. This then gives physically meaningful values for all p, which can be checked against experiment. 3/11/18 Unruh effect: in an empty spacetime, an accelerated observer will observe particles (a very small effect). Intuitively: the virtual particles popping in and out of existence appear to him as real. This is closely related to Hawking radiation of a Black Hole: the static observer outside the Black Hole is "accelerated" and therefore observes particles. The standard explanation of "one partner of a virtual particle pair created near the horizon falls in, the other becomes real" is too simplistic: the radiation does not originate only near the horizon. 9/12/2019 In classical and also in intuitionist/constructivist logic, you prove ~p by showing that the assumption of p leads to a contradiction. In classical logical, you can use proof by contradiction: to prove p, you show that the assumption of ~p leads to a contradiction. In intuitionist logic, the rule of the excluded middle is not valid; in particular, the rule ~(~p) -> p is missing, and proof by contradiction is not accepted. This makes sense if (for example) one interprets p as "p can be proven" and ~p as "p leads to a contradiction": after all, there might be statements p that cannot be proven but also don't lead to a contradiction [in fact Gödel showed us such statements]. 9/19/2019 The current expansion rate of the universe ("Hubble parameter") is about 70 km/s/Mpc, which amounts to a distance dilation of 7% per billion years. This rate will slowly decrease over time and settle down at 6% per billion years eventually. This slowdown is caused by ordinary matter and energy. (Before we knew about dark energy, which pushes things apart and has constant density, we believed that the presence of ordinary matter and energy would eventually halt the expansion and cause the Hubble parameter to approach 0%/Gyr.) 9/19/2019 Haskell is a functional programming language: (almost) everything is done by composing functions without side effects; needed side effects (input/output) have to be specially marked. Syntactically, indentation is used to indicate parts of an expression that belong together. Application of the function f to the arguments x and y is written as f x y (left associative, using Currying). It is lazily evaluated: arguments to a function are only evaluated when and to the extend that the called function needs the values; this allows for arguments that are infinite sets or sequences. Functions can take functions as arguments and produce functions as values; e.g. map :: (a -> b) -> [a] -> [b] takes a function from type a to type b as input and produces a function from lists of type a to lists of type b as output; "->" is right associative. (This function declaration uses "type variables'', placeholders for actual types. Types start in upper case, type variables in lower case.) Functions can be defined using a list of pattern matching rules; the first rule in the list matching the current arguments is used. The user can define new types, even ones that include type variables, called parameterized types. Type definitions, like function definitions, may be recursive. A "type class" is a collection of types (including parameterized types) that all implement a certain interface; e.g. all the types in the type class Ord are ordered and implement a function GreaterThan. The parameterized types in a type class all must have the same "kind", i.e. the same number and order of parameters. When defining a function whose domain/codomain contains type variables, we can require that the types replacing those variables must belong to certain type classes, thereby ensuring that they implement certain required functions. E.g. iselem :: (Eq a) => a -> [a] -> Bool is a function that decides whether an item of type a is in a list of type [a]; the type a must belong to the Eq type class of types that implement =. Functor is a type class containing parameterized types f that for every type a produces a new type f a, and such that "mapping over it" is defined, i.e. there is defined a function fmap :: (Functor f) => (a -> b) -> f a -> f b that satisfies the axioms of a category-theoretical functor (where a Haskell functor is a category-theoretical functor from the category of Haskell types to the category of Haskell types, with Haskell functions as morphisms). Example functors are: lists of type a; trees of type a; Maybe a (the latter is just like a but with a new special value 'Nothing' thrown in). In one view, f a represents values of a "wrapped in" a new computational context provided by f. A function that reads user input or a random number or writes text output formally produces an "IO action" as output; such functions are considered "tainted" since they have side effects; these IO actions can be performed only in the main loop of the program. IO is an example of a monad. Monads form a type class of functors that are useful to chain actions together (as in IO, Maybe) or keep state or logging information when a sequence of actions is performed (as in a parser) or to keep track of several possible results of a computation (as in lists). The important operation of a monad is (>>=) :: (Monad m) => m a -> (a -> m b) -> m b [Think of the values of m a as "actions" that result in value(s) of type a when performed; the function a -> m b represents another action that modifies these a-values to b-values; the result of >>= is the composition of these actions: a value of type m b.] To make chaining of monadic actions easier, a special do/return/<- syntactic sugar is provided which allows for an imperative programming style. 9/6/2019 Origin of eukaryotes and the cell nucleus (following Thomas Cavalier-Smith): Eukaryotes and archaebacteria are sister groups with a common prokaryotic ancestor. The eukaryotic branch evolved phagocytocis (which is the key innovation enabling all the others), endomembranes, a nucleus and then captured a bacterium to serve as mitochondrion. In prokaryotes, the DNA is attached to the inside of the cell membrane. The ancestor of eukaryotes (probably an actinomycetes) had a single cell membrane, unlike most other bacteria which have a double cell membrane. This allowed phagocytocis to evolve. During phagocytocis, a part of the cell membrane with attached DNA can turn inward and form a vesicle inside the cell, still with DNA attached. Several such vesicles, when merging, can create the endoplasmic reticulum connected to a double-membraned nucleus with DNA inside, and nuclear pores at the locations where they merged. These pore complexes are crucially important to let RNA out of the nucleus. 12/25/2019 Unit quaternions form a 3-dimensional real Lie group, topologically S^3, algebraically SU(2). This group is the universal cover of SO(3) via a surjective 2-1 homomorphism SU(2)->SO(3), so that SO(3)=SU(2)/{+1,-1} can be thought off as the three-sphere with antipodal points identified. Every rotation in R^3 can be described by a unit vector (axis) u and an angle a, where (a,u) and (-a,-u) describe the same rotation in SO(3), but these two are distinguished in SU(2). 3/14/2020 Social network graphs in humans and primates show "degree assortativity": nodes tend to be connected to nodes of a similar degree, a result of status seeking. In most other naturally occurring graphs (airport connections, web links etc.) we see the opposite: degree disassortativity. 6/3/20 In Quantum Electro Dynamics (QED) the "field" view of classical electromagnetism and the "particle" view of mechanics are unified: both theories become quantum fields, which are operator-valued distributions on spacetime; the underlying Hilbert space is a Fock space. Electrons are the quanta of the electron (quantum) field and photons are the quanta of the electromagnetic (quantum) field; in both cases, the fields are the fundamental entities; in many states of the quantum fields, the number and types of these quanta are not defined (they are defined in the eigenstates of the Hamiltonian, where the energy is well-defined). While electrons have location, photons do not. Think of a photon as a "wave packet" that can have any of a large number of shapes allowed by the classical field equations. To arrive at the quantum field for electromagnetism, one starts with the classical Maxwell field equations; to arrive at the quantum field for electrons, one starts with the Dirac equation (which itself is a quantum wave equation for the electron and does not describe a classical field, therefore this process is called "second quantization"). The procedure to turn a field into a quantum field is similar in the two cases. The field is expanded as a linear combination of eigenfunctions, and the coefficients in that expansion are replaced by suitable operators. These operators turn out to be the raising and lowering operators on the Fock space constructed on the Hilbert space whose orthonormal basis is given by said eigenfunctions. For electromagnetism, the symmetric Fock space is used. For electrons, the antisymmetric Fock space is used. An element of the canonical orthonormal basis of this Fock space is a list |n_1,n_2,...> of non-negative integers, almost all zero, where n_i is interpreted as the number of field quanta belonging to the i-th eigenfunction. ====================================================================== This is a diary of facts and ideas I came across and wrote down for my own benefit; some may not make sense to you without further explanation. Nothing here is original but I left out all sources and references. Axel Boldt axelboldt@yahoo.com http://math-www.uni-paderborn.de/~axel/ ======================================================================