n-body, near O(n)

After Newton discovered the equations behind Kepler's 2-Body solution, he ran afoul of the n-Body Problem. A limited solution presented itself in the equilibrium point; the full five solutions were offered by Lagrange. Further refinements handled special cases of the 4-body like when Laplace sussed out three of the heaviest (and largest) Jovian moons. So here we are in the computer age, when computers struggle to handle particles numbering more than maybe five. It's O(n*n).

Kepler and Newton assumed that the bodies are rigid points and that we ignore Einstein; so Io's tides and Mercury's depth in the well don't figure. Also computers have to approximate floating point numbers well above Zeno's quantum level (Planck?). That means we have already sold the horse for approximations. And that means we may as well approximate some more. Keyframe Codes has a little primer on the available algos.

Midlevel nerds have been using Barnes-Hut. This method runs on multipoles: it buckets the particles in squares-of-particles, finds the center of gravity, and runs the calc for the centers. The buckets are identified with quad-trees; Barnes-Hut ends up O(n * ln(n)). Leave aside for now if we want to run this 2D or 3D; we'll pretend perfect ecliptic. What if the cluster doesn't behave like a square (or cube). Real astrodynamicists need the particle swarm where most particles would be clustering. They will be looking first at Lagrange's L4 and L5 "Trojan" haloes especially for Jovians. These haloes are elongates.

In 1987 Leslie Greengard came up with something else, the "Fast Multipole Method" - usually FMM. Not only does it claim O(n) but, depending on how often you want to run each n, the error can arbitrarily diminish. This was Greengard's thesis and, it seems, got a refinement under one Vladimir Rokhlin. Discontents exist: Srinivas Aluru in "Greengard’s N-body Algorithm is not order N" (pdf) showed that the choice of precision must depend on n (at least ln n), therefore cf the title - to the tune of O(n ln²n). That extra ln(n) is, then, the price for accuracy Barnes-Hut must pay in the Trojan haloes. Dr Greengard's peers seem to have agreed; he got to keep his PhD and the IEEE considers FMM in the top ten of twentieth-century algos.

This starts by keeping each interaction as a field vector Φ and treating that vector as a number on the complex plane. The vector acting upon particle "i" is the gravitational parameter of counterpart "j", times the natural logarithm of the i-j distance (plus iθ); that's the force. The iθ exists, here, to make it a vector as to place the particle on the coördinate plane; a computer may ignore it now to figure the coörds when it's display-time.

Some might then use "numerical methods", which is mathematician for iteration. The FMM, in the complex plane, may use the Cauchy-Riemann equations. At this point I'm leaving the rest to the nerds.

The scouring of the land

I am continuing my scan of Elitzur's dataset. Yesterday I added some words I recognise in Arabic: BKY, ḌḤK; this morn have I removed his first example elón. I got intrigued with example #5 GLY so have held off to poast that, today, separately.

The Quran will use GLY for exile. The Arabs didn't, at least not in the Safa. There it meant cut off, erased, exposed. So when I look at Elitzur's data, he turns up Judges 18:30 for which authenticity he argues: which I think vocalised some gelút há-aretz in CBH construct. Clearly you can't exile the land. Meanwhile GLY as exile comes nowhere in the Hexateuch (to Joshua), despite exile being a central theme.

By parallel with Safaitic, gelút há-aretz implies a scouring. Elitzur applies this term to the land around Judges 18:30-1, whose candidate is Shiloh's temple which, by then, was dismantled. Whenever Judges thought that was: Gath's invasion? Hazael?

At this point I want to bring example #1, elón; I further suspect the há- as late in Hebrew's evolution (it was probably *han- prebiblically). Elitzur finds elón in Genesis and Deuteronomy, then through to 1 Samuel. Once David takes control we no longer read the word. For Elitzur it should be something like the Archaic Greek chora (district/ viable-flatland)... like aretz. Why does it end with Samuel, if it was Deuteronomic, he asks. I counter that everywhere, elon had been fossilised - into placenames (construct, without há-!) and/or semantically become "oak grove". These elón texts existed to place scenes into a landscape of memory. For Judges 18, either Elitzur must abandon its status as an ancient account or else he must admit that elón was already a fossil.

For my part I accept that Judges 18 be early. As for why the Pent-to-Hexateuch didn't use GLY, Elitzur would put that down to not knowing Judges 18. For Elitzur the Hexateuch is simply earlier. The Deuteronomic History, having read Judges 18, shifted GLY's semantics to exile. Elitzur fn. 17 is aware of GLY for exile in Akkadian (which might have followed a parallel shift) then Aramaic.

I won't argue that point for Deuteronomy alone. For the rest of Torah (and Joshua) I propose, rather: refusing this use of GLY. The authors and editors saw the term as belonging to a later time.

REPOST: I don't know WTF happened to the URL, but let's try to fix it

The end of the bombardment

We haven't discussed the "Late Heavy Bombardment" on this blog. As of 2024 we've rather been skeptical. We may, however, have some timeline for the end of some kind of bombardment: Universe Today reports from Tim[othy] Johnson. And from Hamed Gamaleldien and some others.

The issue here is that the Hadean has, in fact, ended - but the crusts came later than that ending, agreed 4.03 Gya. Crust first formed 3.9 Gya. There should have been, what, 400?500? My after the great Theia crash and all its debris; Earth should have cooled enough by then. Johnson argues that the crust had been heated exogenously. That is, Theia wasn't the last crash.

The "L" part of the LHB implies a surge at the 4.03-3.9 Gya ... I guess we can call it a "period", 130 My as it would be. We've been marking this period as the Archaean beginning. Johnson doesn't need a surge. A continual pounding from our uncaring universe would do the job just as well.

The constant heat over this extent of time continued the process of differentiation, at least at the crustal layers. We have a mostly "felsic" land surface up here, dominated by silicon and aluminium and all those lithophiles... just like the Lunar regolith. Granite looks compositionally felsic. The "mafic" rock, of iron and its siderophiles, clusters in some rare upwellings and on the ocean floor. That's básalt or should we say défault.

Our Hadean is defined by committee, basically: Ten oldest zircon U-Pb ages in the Acasta Gneiss up in Canada's NW Territory. Gneiss is metamorph granite so it was a crust once, exposed by glacier. These zircons are however preserving siderophilic (heavy) metals. So I dunno. They might be mafic and precede our crust - and the crust of Acasta.

Johnson and Gamaleldien earlier had evidence for water in those transCanadian zircons, which they haven't disavowed. So I guess they can continue to have a waterless Hadean, just like in the Gospel of Luke. Archaean is, then, defined by still-no-crust but now with uneasy pools of water. Maybe a steam period.

The implosive fusion pulse rocket

I woke up today with ToughSF poasting Gradient Field Imploding Liner Fusion Propulsion System. This report is fairly recent, like 2018. I wasn't quite blogging Atomic Rockets content then but, soon.

The "field" here means magnets. The liner is of some metal that can crumple when subjected to the Tesla. This implode-the-liner-with-dekamagnets tech is not new. Here is something from 1976. Also given the heat of the fusion around there, they cannot use superconductors - not 1976 niobium, not 1986 ceramics, nothin' under 125 K. Which puts MR Lapointe at some engineering conundra. For instance he only has 30 Tesla to play with, which he's wrapping externally. 1 T, internal.

Luckily for him and his, "magneto-inertial" designs had been an active, uh, field of research up to 2018. And he doesn' t have to keep the plasma: we're going to spaaace. The drive works by pulse like an Orion; unlike Orion they have to start in space.

This is about where nerds start talking z-pinch (or ζ-). This report is talking θ-pinch.

To load up these interplanetary missions, the 2018 paper suggests a couple of SLS boosts to eject 260 "mt" off our well. "mt" looks like "metric tonnes" so I'd say "T" or "Mg". As of last Friday Elon can sell you a Starship with throwaway booster (supposedly 188 T); he might soon be able to rent you one as well (for 115 T). They'd allow assemblage in orbit, some of those T being sufficient fuel and orbiting Starship to push your fusion rocket out of Earth's hair.

Where we speak of specific-impulse (I_sp) as usual the time-savings ramp up beyond Mars, like this takes 50-100 "mt" to Saturn in 200 days. In this context, 20 mt is a crewed mission. The report requires for the Earth-to-Saturn mission no less than 20000 s of I_sp, which I take as a function of deltaV more than of distance.

I_sp depends on what to use for the crumpling liner. Table 5 lists 32,200 s for Lithium (7.8 kN thrust), 17,145 s for Beryllium (14.45 kN); compare Sunbird's 15ks. So lithium is how we, Earthlings and Venereans, get the direct lines to Saturn and beyond. Definitely going to have to construct the former in some anhydrous and anoxic environment like, uh, in space. Table 5 further calculates the energy required, for Mars: lithium to expend 165 MJ.

Asimov-approved beryllium would appear to work best where we don't care about longterm I_sp. A three-month trip to Mars ain't that much better than a six-month. Here we're optimising for thrust and, it seems, for energy (153 MJ). Energy aside that looks better for anything as far as Jupiter; inner-planetary travel or Oberth toward the Sun will give us lots of that energy, as well. With the extra thrust comes more payload and, we must add, more payload affords more options for crew comfort. Like spinning the hab, protection from radiation, all of it.

Also, pipe dream here, if we had stations on Vesta or on Deimos, beryllium could probably hit up Saturn thence (and it might even be mineable on that differentiated silicon). Launch windows will be a bear out there however.

Age of Mangione

Sunnyright, who got linked by Stephen Green, points to a 60 Minutes ep where a company invested $3B to develop a drug, which company now has to recoup its expenses. Commenters, who Deserve treatment, are calling for Luigi. They mean, for an army of DavidsLuigi Mangiones.

Green assumes all these commenters be Left. They are not.

A few days ago Ace of Spades posted this juxtaposition:

I'm just going to say it: I am now ready, willing, and eager to begin inflicting punishments on communists for what I consider wrongthink. If they want to criminalize my thoughts, I am now ready to use state power to criminalize theirs.

Mike Lee wants to defund schools that insanely continue to force students to get vaccinated in contradiction of their religious beliefs.

Ace has been so "ready" for some time longer than just this past week, I suspect. This whole side of X has been sliding into this, yes, fascist direction.

Meanwhile Richard Hanania wonders how come Bill Gates Jr hasn't st[oo]d up for himself and for public health, generally. I might so wonder, too.

The evolution of Classical Biblical Hebrew

In 2022 Yoel Elitzur, or "Joel" in German, laid down a challenge. Google Scholar reveals basically nobody as take this seriously; I consider Joosten's cite as an opportunism (and Joosten knows better). So I'll try :-

Elitzur pinpoints 17 Hebrew words and expressions as appear over the early Biblical corpus, and where they appear. Many cluster in specific sections. Some are quoted later but otherwise don't appear (e.g. the Chronicler quoting Reigns), some change their triconsonantal root so probably pronunciation, some change semantics. From that, Elitzur argues for no need to budge the Masoretic (and, honestly, Septuagintal) order of the historical books from the Pentateuch through Joshua-Judges through Reigns. Nor to budge the historically-understood dates of the Prophets - with the dramatic exception of Isaiah 40+ (I've been arguing 34+). The Psalms don't seem to figure.

The paper admits that some cases be stronger than others. The infamous he who pisseth against the wall is used in the Saul-David cycle and not earlier nor since. You'd think there'd be occasion for an author to bring it up in Judges or in the Ahab-Jezebel sequence, or when the Assyrians arrived, or for one of the more visceral Prophets like Ezekiel. Elitzur concedes the possibility of one author later subsumed into a longer history. But that might even help the paper's case: he's arguing that the later editor, or perhaps Editor, assigned earlier content to their correct place.

The largest tub o' skub develops from the implication of no later major editorial intrusions. He might allow, oh, the tweaks to Joshua against 4QJoshᵃ or the airdropping of Judges 6:7-10. But he doesn't allow for the Documentary Hypothesis in the five books (claimed) before Joshua. They're historical books too and they end with the Testament of Moses which is Deuteronomy.

[We'll deal with elon and with GLY later.]

Elitzur brings also ḌḤK as a basic Semitism. This is ṢḤQ for Genesis, which needs to explain the name יצחק; the root is also in Exodus (thus Ezekiel) and once in Judges. Alternative Hebrew is ŚḤQ from... Judges again, and Jeremiah and 1-2 Reigns/Samuel, and the Psalter and Proverbs. By the way Psalm 105, Jeremiah, and Amos(!) agree that Isaac's name is with S not Ṣ. The majority principle leads to ŚḤQ for the actual Hebrew. 'Tis a little harder to figure in Arabic because I cannot find it in Safaitic. Once it gets into the postNabati script it could just as easily be ḌḤK - or ṢḤK.

BKY is a good Safaitic root for weeping, which these herders on the Nabati fringe did over cairns and graves. Elitzur sees this root in the earlier parts and, yeah, I suspect it was always Semitic. From Isaiah 1-39 on, Prophets like Jeremiah use DM῾ and it also appears once in Kings / 3-4 Reigns, then Ezekiel. If Torah knew DM῾ why didn't Torah use DM῾?

"Maidservant" amah and shaptah exist only in quoted dialog, like -na. In each case here, the woman is rendering an appeal. Elitzur here does not read these female words in Pentateuch, even in narrative sections where someone like Rebecca in Genesis 24:18-19, or especially Sarah [in Gen. 18], could have used it but did not. One might also ask after Rahab in Jericho; here, we can contrast that -na which does appear in Genesis. Elitzur here argues this means the usage (amah first) started in the Gibeah episode [of Judges 19] to continue throughout Reigns although maybe dropping off in final 2 Kings. As Joshua precedes all that; so does Torah.

asa for "fire-sacrifice" is of course Levitic. Elitzur however refuses it for P - he overall refuses all P - because it's also in Deuteronomy. Then it recurs in Joshua and even Samuel.

Elitzur brings a fair dataset. I think, however, his cases aren't as strong as he thinks. Just because a bundle with some word exists, that says nothing about the sequence before-and-after.

For "thy [maid]servant", we have outside data: Late Bronze Age texts, wherein Canaani or Ugariti supplicants abase themselves before some Pharaoh as "thy servant". The trope's persistence in the Iron Age would parallel the world of Judges 19-to-Reigns. For Elitzur the trope should be even stronger if Genesis truly was a text of early iron or even bronze.

Now take asa. Since Wellhausen, most understand that even without the "Holiness Code" P came from a central altar. D and D's history was arguing for that central altar. Just because a sacrifice was burned does not mean the law came from P. D would do just as well, in which case P simply developed the theme and drew up a whole ritual around it.

The story Nolan told

I still have no intention of watching Nolan as, I hear, Emily Wilson isn't bothering. I am still checking out the reactions; Benjamin Shapiro seems to have a good one. He argues Nolan has written a Christian story, inasmuch as the theme is Odysseus' sense of that Christian emotion guilt. Nolan might deny this himself.

Rebbe Shapiro has some interest in this era, as he claims descent from those who skipped out of the House of Ra around then. This blog has noted the earlier Libyan attack on the House of Ptaḥ. That pharaoh Merne Ptaḥ noted Israel, in his own time. Israel wasn't around for the great naval run against Ramses III "1177 BC", but the Tribes will have to face the Philistine allophyles. Back to Shapiro, I'll circle, below.

The Romans, who like the Jews are not Greeks, unlike the Jews did inherit the Greek pantheon or something close-enough. So they produced something suitably Roman: the Aeneid. Virgil set out to retell Sack of Troy but from the Trojan point-of-view. Aeneid was so good, at least books 2/4/6 were, that we no longer actually own Sack of Troy. Theodore Nash points out that Virgil had good Greek predecessors for his concern for Troy's victims.

Shapiro's people own their own Aeneid - on land: the Exagog(u)e, in Egyptian Greek. Shapiro himself prefers the Haggadah but I doubt he is ignorant at least of The Prince of Egypt.

You know what Virgil and Ezekiel didn't do? They didn't rewrite Odysseus' story for their own. These shenanigans did sometimes happen then but Virgil and Ezekiel were too good for that. They found or inherited, respectively, their own protagonists. Nolan, I submit, should have written his own fanfic, as Virgil had done. It may be that Nolan can't.

I wonder if Nolan has created here something like Cameron's Avatar. Big money now; big noise now. Low cultural impact.