In the most extreme case of this, you can use Counting Sort. Tangential to this, Spaghetti Sort makes me wonder about which parts of CS (especially the data structures, like arrays) are objective or just accidental.
If we're taking the pure math view, complexity analysis is agnostic to this. Strictly speaking you'd have to define how your abstract machine works: O(n log n) is (again, strictly speaking) literally just a set of functions. In practice we usually hand-wave this away (not without problems: arithmetic taking time O(log n) and hash table lookups taking time O(1) are not consistent with each other!)
The unstated implication is that the theory tracks with real world behavior. This is more or less the case for simple problems: your O(n^2) algorithm won't scale, your database query without an index will take forever and so on, it's definitely a useful and high-signal way of thinking about computational problems.
Obviously modern hardware isn't much like a 70s machine and things like "all memory accesses are O(1)" are so wrong it's not even funny.
It's a pure thought experiment with no possible counterfactual, but I think if you tried to develop basic CS theory from a purely mathematical angle (e.g. consider a machine defined so and so with basic operations costing 1, let's run with this ball without caring much about the real world) you'd naturally end up with some (but not all) of the concepts we're used to. For example, arrays and buffers are very natural. We need more space than our basic unit of memory can accomodate, let's use several in a row. Pointers also follow very nicely, and with them structures like linked lists. Other stuff like B-trees and to some extent hash-tables and friends very much less so, they're definitely "imported" from real-world usage.
Not sure where you got classism from, I've got a number of posh southern public schoolboy friends; after all, it's not really their fault that they mispronounce "butter" as "batter". What I object to is the term "received pronunciation" which implicitly places their "fwah fwah" as being in some way correct and standard.
First, let's try differential equations, which are also the point of calculus:
Idea 1: The general study of PDEs uses Newton(-Kantorovich)'s method, which leads to solving only the linear PDEs,
which can be held to have constant coefficients over small regions, which can be made into homogeneous PDEs,
which are often of order 2, which are either equivalent to Laplace's equation, the heat equation,
or the wave equation. Solutions to Laplace's equation in 2D are the same as holomorphic functions.
So complex numbers again.
Now algebraic closure, but better:
Idea 2: Infinitary algebraic closure. Algebraic closure can be interpeted as saying that any rational functions can be factorised into monomials.
We can think of the Mittag-Leffler Theorem and Weierstrass Factorisation Theorem as asserting that this is true also for meromorphic functions,
which behave like rational functions in some infinitary sense. So the algebraic closure property of C holds in an infinitary sense as well.
This makes sense since C has a natural metric and a nice topology.
Next, general theory of fields:
Idea 3: Fields of characteristic 0. Every algebraically closed field of characteristic 0 is isomorphic to R[√-1] for some real-closed field R.
The Tarski-Seidenberg Theorem says that every FOL statement featuring only the functions {+, -, ×, ÷} which is true over the reals is
also true over every real-closed field.
I think maybe differential geometry can provide some help here.
Idea 4: Conformal geometry in 2D. A conformal manifold in 2D is locally biholomorphic to the unit disk in the complex numbers.
Idea 5: This one I'm not 100% sure about. Take a smooth manifold M with a smoothly varying bilinear form B \in T\*M ⊗ T\*M.
When B is broken into its symmetric part and skew-symmetric part, if we assume that both parts are never zero, B can then be seen as an almost
complex structure, which in turn naturally identifies the manifold M as one over C.
Lack of informed consent is a serious possible issue here. Lack of accountability might be another one. Being cheaper, by contrast, shouldn't be seen as bad. But all of this needs to be proven.
Did you even read what you posted ? Talking about the 7 deaths:
> Five were evidently unrelated to the vaccine: One girl drowned in a quarry; another died from a snake bite; two committed suicide by ingesting pesticides; and one died from complications of malaria. The causes of death for the other two girls were less certain: one possibly from pyrexia, or high fever, and a second from a suspected cerebral hemorrhage. Government investigators concluded that pyrexia was "very unlikely" to be related to the vaccine, and likewise they considered a link between stroke and the vaccine as "unlikely."
Deaths of "many girls", when the parent comment said it was at most 1 out of 2300 participants (a suicide)? Those numbers might, however, be untrustworthy. I don't know India well enough to know how much to trust statistics compiled there.
He wasn't but in the decision between more taxes and war bonds, he chose bonds, against his advisors' wishes.
And the population proved that they are willing to make massive contributions to important causes, without having to be forced at gun-point with taxation.
I'm aware of that, so I don't understand what you're arguing about. The comment asked for an alternative to taxes for funding the army, and war bonds is the answer.
It's really amazing how successful war bonds have been in the 20th century. Before that, kings who wanted to wage war either taxed their population or borrowed money from people like Epstein for funding.
That's way higher than I thought. Is there any evidence? Dresden was 25,000, and the V2 and V1 campaigns had less numbers. So this is high even for an aerial bombing campaign.
[edit] I don't get why I'm getting downvoted. Are people making assumptions because I mentioned Dresden? Get a hold of yourself.
Aerial bombardments typically target areas with ~0.01 people/square meter, and those people are often in hardened shelters. A protest may have 1–4 people/square meter out in the open. Attacks targeting the latter cause orders of magnitude more casualties for the same amount of firepower.
And the crowd itself can be deadly if it gets too dense, due to panic or otherwise. For example, there have been at least two crowd collapse events with >1000 deaths in the Mecca pilgrimage.
I think it's just a stupid comparison. Aerial bombing campaign on a single city 80 years ago vs government coming down on protesters distributed in over 100 cities was the best reference you could find to doubt these numbers?
first: it's not in one day, it is over three weeks. Second, the 25000 is an extrapolation. Basically the Iran Islamic republic has a tendency to admit to 10-15% of the death toll, and they admit 3000 death.
Have you considered finding a conformal transformation† that maps a square to any other possible shape, as long as the shape doesn't have any holes? Such a transformation always exists by the Riemann Mapping Theorem, and is unique as long as you specify in addition (1.) which point the square's centre maps to, and (2.) the angle of rotation around that point. Not sure if anyone's ever tried that.
If you actually want more aesthetic freedom, you can compose with an arbitrary diffeomorphism of the square to itself. But I think that might usually look worse.
† - That is, preserving all angles, including right angles. The terminology stems from the output angles conforming (???) to the input angles.
i'm torn between my stupidity re:your first comment and my love of eliminating pinch points (now you're speaking my language). i need to read up a lot to better understand your suggestion!
(most pinch points are gone these days, it just required a lot of edge case hunting)
Always send "pragma foreign_keys=on" first thing after opening the db.
Some of the types sloppiness can be worked around by declaring tables to be STRICT. You can also add CHECK constraints that a column value is consistent with the underlying representation of the type -- for instance, if you're storing ip addresses in a column of type BLOB, you can add a CHECK that the blob is either 4 or 16 bytes.
The transdichotomous model looks interesting.
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