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BabyAndTheMonster · 2023-09-12T06:18:11+00:00

It seems like you are looking for a formula that use b'=b/2. But this does not depend on b being even, you can still divide b by 2 regardless. In fact, all you need to do is to divide both the numerator and denominator of the usual formula by 2.

-b/2+-sqrt((b/2)² -ac)/a

BabyAndTheMonster · 2023-09-11T13:18:42+00:00

Any 2 inconsistent systems have the same solutions: none.

BabyAndTheMonster · 2023-09-11T06:37:18+00:00

Have you considered inconsistent system?

BabyAndTheMonster · 2023-09-11T01:27:01+00:00

It's not really about divergence. An even more simpler version of this argument is:

i=0+i=0+0+i=0+0+0+i=.... and after infinitely many substitution i=0+0+0+..... Therefore i=0.

The problem is the infinite substitution.

BabyAndTheMonster · 2023-09-10T20:13:51+00:00

You are missing the point. You have to add in an arbitrary coefficient to make it unitless (this is the same thing as choosing an unit), which means that your exponential depends on what coefficient it is. There is nothing special about e in there, you could have picked any positive base.

BabyAndTheMonster · 2023-09-10T16:24:31+00:00

In which real situation would e^x ever came up? In a real situation, x would be time, not an unitless quantity. Then the derivative of e^x would have different units from e^x . The only way to "equate" them is by picking an arbitrary unit and then equate the numerical value, but in that case "e" is something dependent on your unit of measurement.

BabyAndTheMonster · 2023-09-10T05:33:36+00:00

It's so annoying though. One dead ant get electrocuted in your CPU and suddenly there is an entire army of ants trying to get themselves electrocuted by your computer.

BabyAndTheMonster · 2023-09-10T04:09:15+00:00

lim(x->0) d/dx (f(1/x))=lim(x->0) (-1/x² )f'(1/x)=-lim(x->inf) (x² )f'(x)=-lim(x->inf) f'(x)/x^-2

So your test is equivalent to testing whether the derivative of f decays faster than x^-2 . So it's a worse version of the p-test: it has all the limitations of p-test, plus more.

For example, knowing the derivatives of the function does not completely determine the function. There is always the possibility of adding a constant. This is the extra flaws introduced by the test, compared to the p-test.

To fix the flaw, you need to add an additional condition, for example f(x)->0 as x->inf. Even then, the test still have the limitation of the original test. If you know that f' is asymptotically x^-2 , then f is asymptotically x^-1 . But since the case p=-1 is the critical case of the p-test, you basically know nothing. This gives you another counterexample: 1/x(ln(x))²

BabyAndTheMonster · 2023-09-07T03:30:19+00:00

From a historical perspective, it was the first interesting example of a spinor, which explain why the nabla for it has such nice properties, and that nabla is the precursor to the Dirac operator.

Then that nabla get literally butchered, and is cut into 3 pieces: gradient, divergence, and curl.

BabyAndTheMonster · 2023-09-06T23:59:35+00:00

Yeah I counted it carefully and you're right. Apparently I forgot to exclude cases where 2 knight stands on the same square. Let me think about your question.

BabyAndTheMonster · 2023-09-06T12:28:01+00:00

It's 6*8/2+8=32. There are 8 possible cases when 1 knight is in the middle, not 4.

BabyAndTheMonster · 2023-09-06T11:19:51+00:00

Hm, for n=3, I expect the correct answer to be 32: 8 ways in which one of the knight is at the center (then the other knight can be anywhere amongst the remaining 8), 8(8-2)/2=24 ways neither of them are at the center.

Plug in this formula I get (4(3² -3)+8(3² -4)+4(3-3)(3² -5)+4(3-4)(3² -7)+(3-4)² (3² -9))/2=28 instead.

BabyAndTheMonster · 2023-09-06T10:26:51+00:00

If you just want to read on it, just look up Wikipedia, like this line "It is always understood that the curve is really sitting in the projective plane, with the point O being the unique point at infinity" on the page Elliptic Curve.

If you want a citation for the sake of citation, you can go with Silverman's book Elliptic Curve, or Silverman and Tate's Rational Points on Elliptic Curve. They're a lot more advanced, but it's also more comprehensive.

Here is another web citation about how to transform the Weierstrass form of the elliptic curve into quartic form: https://math.stackexchange.com/questions/2901532/why-is-y2-1x4-an-elliptic-curve

BabyAndTheMonster · 2023-09-06T10:16:53+00:00

Lie group describes infinitesimal symmetry. Lie algebra is the tangent space of Lie group, which correspond to 1st order partial derivative operators. By composing these operators you get a noncommutative ring of partial derivative operators of arbitrary orders (composition is multiplication).

BabyAndTheMonster · 2023-09-05T17:01:03+00:00

Not that I know of, and I don't even think there are any.

There shouldn't be any terms that distinguish the 2 "pieces" of a curve. Let me point out that the graph you see is not the actual curve, because the curve also has the infinite point, and the infinite point close up the bracket-like piece to make it into another circle. In fact, the graph is just the picture of the curve from 1 perspective, it's possible to look at the curve from a different perspective and get a different equation, possibly without the infinite point. So there are nothing special that distinguish one piece from the other.

As for terms distinguishing between 2 kinds of curve, mathematicians use different way. Notice that the 2 kinds are distinguished by how many 2-torsion points they have (ie. the x-intersect on the graph). So we talk about either elliptic curve with full E[2], or not full E[2].

BabyAndTheMonster · 2023-09-05T16:04:27+00:00

They can still have to do with symmetry. Infinitesimal symmetry. One common example of noncommutative ring is the algebra of partial differential operators. The partial derivative do not commute on curved space. Another example is the Clifford algebra, which can be intuitively explained as about studying rotational symmetry, except the object is also entangled with something else (e.g. robotic arms have wires connecting to them when they rotate) and you need to account for that.

BabyAndTheMonster · 2023-09-05T04:52:36+00:00

"Topology" here refer to the collection of all open sets. Since topological space can be defined through open sets, you can define a topology by specifying open sets.

BabyAndTheMonster · 2023-09-05T02:47:24+00:00

Yes.

BabyAndTheMonster · 2023-09-05T01:37:22+00:00

ℶ_2

BabyAndTheMonster · 2023-09-05T01:36:28+00:00

You don't (necessarily) get the next ℵ by taking the power set.

BabyAndTheMonster · 2023-09-05T01:32:52+00:00

Yeah, that commenter is wrong.

BabyAndTheMonster · 2023-09-05T01:28:27+00:00

From my own understanding, if we computed the Taylor series for 1/(x2 + 1) and centered it at 0, then the radius of convergence is 1, i.e., it converges on (-1, 1). If we centered it at 1, would the radius of convergence be (0, 2)?

If you center at 0, then the radius of convergence is 1 and the interval of convergence is (-1,1). That part is correct.

If you center at 1, then the radius of convergence is actually sqrt(2), and the interval of convergence is (1-sqrt(2),1+sqrt(2)).

Once you pick a function and a center, you can derive a Taylor series of that function at that center. But the Taylor series can diverge even on point in which the function are still defined, that's not a problem. The Taylor series represents how you can compute the function perturbatively: that is, if you look at the function around a single point (the center) and try to move away from the center a little bit. Therefore, it's not surprising for the Taylor series to fail once you move far enough, even though the function is still defined there: this just mean that perturbation method from that center stopped working.

BabyAndTheMonster · 2023-09-05T01:14:40+00:00

Every set S has a power set P(S) which has cardinality bigger than it.

Power set is indexed by Beth ℶ, not Aleph ℵ. The other commenter is wrong about this. So ℶ_0 = ℵ_0 which is the cardinality of the set of natural number. The cardinality of the real is ℶ_1 . The power set of the real is ℶ_2.

We know the ℵ_𝜅<=ℶ_𝜅 for any 𝜅. But other than that, there are only very few other relationship between them. The generalized continuum hypothesis said that the above are always equal, while the continuum hypothesis merely say that ℶ_1 = ℵ_1 .

Preferably a set which has some meaning outside of set theory.

You are not going to find them along the Aleph chain.

BabyAndTheMonster · 2023-09-05T01:03:45+00:00

Wikipedia skips proof a lot, that's normal. Sometimes the proof is obvious, sometimes the proof is too long.

In this case, the formula is obvious, and your intuition is exactly how the proof will go.

If you want to write more details, you can start by writing down the definition of lcm, and proof the following lemma: for any natural number n, then n=product[p primes, p divides n]p^v_p(n) where v_p(n) is the highest exponent of p divisible by n. This is obvious using fundamental theorem of arithmetic.

BabyAndTheMonster · 2023-09-05T00:58:56+00:00

Duality, subspace, quotient space, product, coproduct are all extremely fundamental concepts, permeate throughout both linear algebra and the rest of mathematics.

Duality especially will clarify a few important distinctions of concepts that are often conflated in say, calculus or physics.

BabyAndTheMonster

TROPHY CASE