I am an Israeli algorithm designer, working on image analysis. I've done my PhD in physics, in string theory (or is it mathematics?), and did post-doc in biology. Also have studied math and philosophy.
So please don't argue with me about physics unless you're a physicist yourself.
- (Sorry for the arrogance but you may understand the need after reading this)
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Work to do
- Merge Luxon and Bradyon into Mass in special relativity if no objections are raised.
- String theory: Rewrite overview, replace first image, add AdS-CFT images.
- Write String theory: a technical review.
- Add general w=p/u into Thermodynamics of the universe, which covers also Cosmic inflation and Big rip cases.
- Supersymmetry breaking.
- Gravitational instanton, instanton, Gravitational singularity (general - add massless fields, conical - blow up of orbifold)
- Angular momentum: commutation relations of J with any vector operator are like those of J with itself.
- Commutator, Commutativity: add a part about significance and use in Quantum mechanics.
- Expand Particle physics in cosmology.
- Expand Lattice QCD some day.
- Lagrange multipliers - add intuition (show 2d case).
- Prove Snell's law using Fermat's principle and Lagrange multipliers (+ add link in the latter to this as an example), and using Maxwell's equations.
- Expand Biblical criticism.
- Add Socrates's debate over whether evil is ignorance in Unsolved problems in philosophy and Problem of evil?
- Mass–energy equivalence: add a section about the difference between E=MC2 and E=(1/2)Mv2.
- Expand the Meat article and add pictures of the production process.
- Mention this in Global warming.
(pages I've started or was significantly involved in their writing):
- String theory
- Nuclear magnetic moment
- Shell model
- Woods Saxon potential
- Measure (physics)
- Particle physics in cosmology
- E6 (mathematics)
- E7 (mathematics)
- E8 (mathematics)
- Thermodynamics of the universe
- Proof sketch for Gödel's first incompleteness theorem
- Lattice QCD
- Unitarity (physics)
- Gravitational singularity
- Entropy (arrow of time)
- Arrow of time
- Relationship between string theory and quantum field theory
- Particle statistics
- Picture (string theory)
- Ramond sector
- Neveu-Schwarz sector
- Conformal symmetry
- Friedmann-Lemaître-Robertson-Walker metric
- Euler number (topology)
- Deuterium#Nuclear properties
- Quantum number#Quantum numbers with spin-orbit interaction
- SO(10)_(physics)#Important subgroups
- Split supersymmetry#Advantages and drawbacks
- Physical cosmology#History of the Universe
- Tachyon#Modern interpretation: Quantum field theory
- Evolution as theory and fact#Predictive power
- Lindemann–Weierstrass theorem#A proof sketch
- Baryogenesis#Baryogenesis within the Standard Model
- Liénard–Wiechert potential#Derivation#Lorenz Gauge, Electric and Magnetic Fields
- Net (mathematics)#Properties (first proof: equivalence of continuity definitions)
- Von_Neumann_bicommutant_theorem#Proof (major revision)
- Riemann_integral#Integrability (rewrote proof which was wrong)
very short stubs which I may want to expand
- Dimensional reduction
- Secrets of the Baby Whisperer
A proof sketch that is irrational
We first consider the following matrix:
where i runs over all (2S-1)2 pairs (si, ti) with -S < si, ti < S, and j runs over all (L0+1)(L1+1) pairs (uj, vj) with 0 ≤ uj ≤ L0, 0 ≤ vj ≤ L1.
S is an arbitrarily large integer, L0 = S [Log(S)], L1 = [S /Log(S)]. (here [x] denotes the integer value closest to x from below).
Mij is a (2S-1)2 by L matrix, and since (2S-1)2 > L for large enough S, we may look at any determinant of an LXL sub-matrix of M, which we denote Δ.
|Δ| can be shown to satisfy an upper bound, |Δ|≤x-L2/2 for some x>1, as we later discuss.
Now let us assume that is rational. Then for some integers a, b with b>1, . Therefore:
, for some integers A, B, p.
For every x,b>1 there exists an S large enough such that: bSL1L< x-L2/2. We thus arrive at a contradiction, and therefore must be irrational.
We turn now to the upper bound of Δ. As S (and hence L) grow larger, the determinant Δ contains multiplication of more factors (which tend to increase its size), but it also contains more lines and columns which become closer and closer to being linearly dependent (which tend to reduce its size), and the second effect is shown to be more prominent .
To show this, we see that every entry in the matrix M satisfies: , with , and some . Note that for all i, j. Therefore:
where is the determinant of the LXL sub-matrice fj(zi) (instead of Mij).
An example for a Gödel number in Peano arithmetic
Let's try to write a concrete example for the Godel number required here, with Peano arithmetics and the decimal system. We will have to use abbrevations for lengthy expressions.
A k-ary relation
A list of words similar in Hebrew and Greek
α- alpha privativum, expressing want or absence. αγείρω (root ΑΓΕΡ-) to bring together, gather, assemble δωμα a house, a chief room, a hall, αδρός strong, great. Of persons: large, fine, well-grown. αγαπάω to treat with affection, love, be fond of έχω to have or to hold (Ionic imperative: εχεσκον) Iliad: to grip αυριον dawn, aurora, ουρανός the sky [perhaps source of light?], ελλος a young deer, fawn ελλοψ Hesiod = mute (for fish) ύλη a wood, forest, woodland. Homerus and Herodotus: forest trees. άνθρωπος man, or woman. (latin: homo, not vir) a human being οπτάω to cook by means of fire or dry heat. όφις a serpent, snake αράχνη a spider, a spider’s web αρά I. a prayer. Especially a prayer for evil, a curse, imprecation. II. The effect of the curse, bane. δόλιχος long έρα ground ερώ say, speak αίθω (1) to light up, kindle. (2) intransitive: to burn or blaze. αιθάλη soot. Αιθίοψ = Ethiopian = פני אש. αίθος = fire αστράπτω hurl lightnings (of Zeus) αστραπή = lightning, αστήρ =star σοφός skilledin any handicraft or art, cunning in his craft. Clever, wise. βαίνω (ΒΑ-) to walk, to step, to go, to come, to arrive, to depart, etc. φρέαρ Iliad etc,: a well βάραθρον a pit, (at Athens a left behing the Acropoliw, into which criminals were thrown
φαείνω Skt. *bha- to shine, give light בהר, בהק βύσσος a fine flax and the linen made from it (a later borrowing?) βωμός any raised platform, or stand, araised place for sacrificing, altar. πυρός wheat flee, flight Skt. palay- ברח (Gothic: pliuhan, etc.) πέσσω I : of the sun: to ripen. II. By the action of fire: to cook, bake. όχθη גדה γίγας mostly plural: Γίγαντες = Giants, a savage race destroyed by the Gods κυλίω to roll, גל (see, γογγύλος – עגול) γλάφω to scrape up (the ground). γλαφυρος = polished, finished. γλυπ-, γλυφ- (akin to γλάφω) to carve, cut out with a knife γοάω, γόος to wail, groan, weep Αίγυπτος Egypt (αια-γυπ) ארץ גושן γεφυρόω to bridge over, make passable by a bridge, γέφυρα = a bridge τιτθός a teat, nipple γύρος a ring, circle δύη a woe, misery, anguish, pain. θολός mud, dirt, the dark juice of the cuttle-fish. θολόω to make turbid τρέχω to run, move quickly έρχομαι (fut. ελθω) to come, to go ολολύζω, αλαλάζω (= crying aloud to the gods in prayer or thanksgiving) όρος a mountain, a hill (Cf. C.D. Buck, A Dictionary of Selected Synonyms in the Prinicipal Indo- European Languages: “Gr. όρος ..… from IE *er- ‘move, raise, rise’ seen in Skt. ŗ- , G r. όρνυμι , Lat. oriri.”. Heb. רם= high, also connected? αράω (old verb: to harm) σφάζω to slay, slaughter, properly by cutting the throat. σφαγή slaughter, σεύω (έσσευσα) to put in quick motion. σείω to move to and fro. σεισμός = earthquake σκώληξ a worm. μέλος (for the σ see μικρ-, σμικρ) a song, tune [ζέω ( = to boil) זעם (= anger)] ρέω (ροή) to flow, stream, gush σπείρω ( Lat. sero) to sow seed Akk. zāru זרע κόπτω to strike, smite, knock down, cut off chop down χωμα earth thrown up, a bank, a mound, a dike Lat. forare , Akk. harāru חרר ( hole, Old Eng.: hol, חור ) άλσος a glade or grove (see ύλη, אלה, etc.) γάλα (root: ΓΛΑΚ-, ΓΛΑΓ-) milk חלב connected to γλαύκος, γλυκύς, glow, etc. ? and to λευκός, -- לבן ? χαλάω Iliad: to slacken, loosen γάμος, γαμέω) a wedding, wedding-feast, marriage, wedlock. ξένος In Homer: a guest friend, i.e. any citizen of a foreign state, with whom one has a treaty of hospitality for self and heirs, a refugee. σκεπάω to cover, shelter. σκάπ-τω to dig σκότος darkness, gloom χήρα (-ος) a widow, bereaved Akk. Һarābu חרב and κάρφω to dry, destroy ορρωδέω to fear, dread, shrink from. (formed so as to express the shuddering of fear) χαράττω (ΧΑΡΑΚ-) to cut into, to sharpen, whet, scratch. (χαράσσω) αρόω to plow δείδω (epic aor. έδδεισα) to fear, dread ω εδνόω to betroth, to marry σφαγ- σφακ- slaughter (see above: טבח (זבח) δρόσος = dew, υδωρ = water, θάλασσα = sea, τέλμα = standing water, marsh, pool, pond. τρέφω (έθρεψα) to rear, feed. τροφή food, nourishment μογέω, μόγος to toil, to suffer *δάω (anc. root =learn, perf.: to know) Skt. vid- = to know Akk. idu ידע οίδα (perf. of to see= know) οίνος wine οιωνός = a bird of omen, and also omen. Aesch. οιωνός γόος, but the only bird that makes a wailing sound – γόος -- is the dove! Ιάπετος one of the Titans (sons of Uranus and Gaea), father of Atlas, Prometheus, etc. According to a tradition – rather late—he formed the body of the first man with earth and water (perhaps his own tears) into which Athena breathed soul and life. לא במקרה,הסופר בחר שם זה לאבא של פינוקיו! :Japetto ύπνος sleep, slumber ιασπις Akk. (j)ašpu ישפה (but not a basic, primitive word…) ιθύς straight, direct ιθύνω: make straight, guide in a straight line καίω to light kindle, Passive: to get burned, καύμα : burning heat. κικυς (in Homer) strength, vigour. (h)όλος whole, entire, Akk. kullatu כל κλείω to shut, close, bar, to confine. σκύλαξ (OE hwelp) a young dog, puppy, a whelp, generally: a dog -- Κέρβερος κλωβός a bird cage (El Amarna: kilubi) κάμνω (root: KAM-) Iliad: to work, of smith’s work. κύπελλον a big-bellied drinking vessel, a beaker, goblet, cup κυφός (r.: ΚΥΠ-) bent forward, bent, stooping κριός a ram χράω (1) to be eager to do (2) to furnish what is needed. χρη = must, need κείρω (I) to cut the hair (II) to cut, to hew out (of wood), destroy γρύψ , Eng. Griffin κρόκος, IE crocus Akk. kurkanu כרכם χιτών (κιθών) the garment worn next the skin, in early times: a man’s frock. Later worn by women as well. νάσσω, νάττω to press, squeeze, ναστός: a well kneaded cake λανθάνω (root: ΛΗΘ- ΛΑΘ) to escape notice, be unseen, etc. λέων a lion Akk. nesu ליש μανθάνω (r.: ΜΑΘ-) Iliad: to learn, to know, to understand. λέγω to say, to speak larynx = לע ( whence Arab. λάρυγξ = לע + ρύζω(=to growl) ? λωβάομαι to treat despitefully, to outrage, λώβη an outrage, dishonour λείχω to lick up γλω̃σσα the tongue, the organ of speech. Homerus: language μω̃̃̃μος a blame, riducule, disgrace. μωμεύω (Hom.) to find fault. μίσγω to mix one thing with another, join together, to mingle. μυελός = marrow, μύχιος = inward, inmost, μυχός = the innermost place. Connected to μυέω, μυστήρια? סתר – מוסתר?) Lat. amarus French: amer, = bitter μετρέω to measure πίμπλημι (r.:ΠΛΕ- or ΠΛΑ-) to fill full of a thing . Passive: to be filled. μέγαρον Homer: a large room or chamber, a hall. βασιλεύς a king, chief μάσσω to handle, touch εγγύς = near, near at hand ναίω, ναιετάω = dwell νέω to go, νέομαι to go or come έχις, έχιδνα an adder, viper (Skt. ahi) πίπτω to fall δάκνω (root: ΔΑΚ-) to bite δίδωμι (root: ΔΟ-) to give, to grant σηκός, σακός a pen for lambs) σής a moth (which eats woolen stuff) αγαπάω to treat with affection, to caress, love, be fond of.. γογγύλος round γερανός a crane κόραξ a raven, crow αιξ a goat αετός an eagle αχλύς a mist, a mist over the eyes of one dying άναξ a lord, master, being applied to the gods, esp. Apollo and Zeus. A title given to all men of rank and note.
άγχω to strangle, throttle, αγχόνη a throttling, strangling, hanging ενίπ-τω to reprove, upbraid ( connected to όνειδος ) ασκέω to work, to form by art, to fashion, to practice and art. ίδη in plural: a timber tree, in singular: a wood ακίς עקצ, עֺקץ σκορπίος Akk. aqrabu עקרב Έρεβος Akk. erebu ערב εγείρω Akk. êru ער (root: ,עור Aram. עיר ) IE *ger- Skt. jagr-, Av. gar- είς Akk. iśten עשתי- (found only in composites) [Καφτωρ Akk. Ištar עשתרת Was the island named in honour of the goddess (as Athens for Athena) and later renamed Crete under the influence of Ugarit (as St. Petersburg –Leningrad)? πεδίον Akk. šadu פדן [a different form of שדה ?] πω̃; όποι, etc. Lat. qu- פה πέρδομαι, Skt. pŗd- Akk. balātu פלט (fart, peter, etc.) πέλας, πελάζω פלש πατέω פסע (and in the Bible: [פשע βοάω, βοή, פעה πόρτις Akk. parru פר, פרה flight, Skt. palay- ברח ,פרח οπώρα, fruit Skt. phala- פרי (connected to root פרה = grow) Lat.flos, IE *bhle-, bhel- Akk.pirhu פרח πείθω פתה, פֽתה (later borrowing?) Πύθων Akk. bašmu פתן σχελίς Akk. şelu צלע σκοπέω Akk. šafu צפה (and ; שקפ (Latιn: aspicere) κάπρος = wild boar Lat. caper Akk. şapir צפיר ] Lat. strictus צרר [ = bound together, whence narrow: Skt : avi-stīrņa- and avistŗa צר (It. stretto, Fr : etroit ; etc ) καλέω to call, to summon, to call by name, to name, κλαίω to weep, lament, wail, cry. κράζω, κραυγή, cry out, καίνω ( = kill, in Iliad) קין (the name) ( Hitt.: kwen-) εύ-κολος easy δύσ-κολος difficult κέλυφος a sheath, case, pod, shell καλύπτω (r. KΑΛΥΒ-) to cover with a thing κάννα a reed κρύος icy cold, chill, frost κυρέω, κύρω meet with, light upon, fall in with, κέρας the horn of an animal, horn as a material, anything made of horn
οράω (and λάω) to see, look at, behold οργή Ι. natural impulse, temperament, ΙΙ. passion, anger, wrath οργίζω to provoke anger ορμάω to set in motion, spur on, push on, ορμή a violent movement onwards, onset, an attack έρπω to creep, crawl, to move slowly ρύπος dirt, filth, dirtiness ορχέομαι to dance in a row, and generally: to dance εθέλω or θέλω (Doric δήλομαι) to wish, will, purpose θήρ, φήρ awild beast, a beast of prey θηράω = to hunt or chase wild beasts θήραμα = the hunted animal ταυ̃ρος a bull σκηνή a covered place, a tent, a tented cover στέλλω I. to set in order, equip, furnish with II. to dispatch, send. In Medium: to to send for one. συλάω to strip off, esp. to strip off the arms of a slain enemy, to pillage, plunder, rob . λεηλατω̃ to drive away cattle as booty, to make booty, plunder, despoil λεία booty, plunder. ληστής robber, plundered. πλήρης full of a thing, complete. οδούς (gen. oδόντος) a tooth σινέομαι, σίνομαι to do one harm or mischief, πνίγω to choke, throttle, strangle θεάομαι to look on, gaze, behold. θύω to rush on or along, of a rushing wind. To storm, rage. θύελλα a furious storm, hurricane. θύρα a door. In Homer mostly of double doors πύλη one wing of a pair of double gates. In plural (in Iliad): the gates of a city έθειρα hair. In Homer in plural: either a horse;s mane, or horsehair crest on helmets. Εθειράζω = to have long hair. πηγάζω gush forth with water. πηγή = mostly in plural: running waters, streams σακίον, σάκκος a sack, bag. Akk. šaqqu שׂק έρπω to creep, crawl ρίζα a root (Lesb.: βρίσδα ) πορφύρα the purple-fish, purple dye, purple ίστημι (r. ΣΤΑ-) to stand. To make to stand. To set up. To raise, rouse, stir up. τίθημι (r. ΘΕ-) to set up, put, place πίνω (ρ. ΠΟ-) to drink. πόσις = a drinking, a drink, beverage. θαυμάζω to wonder, to marvel, be astonished. θαυ̃μα whatever one regards with wonder, a wonder, a marvel τρέπω to turn or direct towards a thing. θώς (Iliad) the jackal