eyeballfrog

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$Mathematics$ Discussion on question by user1293931…

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eyeballfrog

May 11, 2023 19:36

For $2\times 2$ matrices, if neither $A$ nor $B$ are multiples of the identity, it should be sufficient to check trace and determinant.

$Mathematics$ Discussion on question by cold_snow_2…

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eyeballfrog

Mar 19, 2023 00:40

@J.G. Indeed it doesn't always have brackets, but when it does have brackets I can't see how it's reasonable to ignore them and say it actually means something else.

eyeballfrog

Mar 19, 2023 00:40

@AnneBauval Wut. You're saying that rather than interpret the written parentheses as denoting the argument of $\sin$, I should assume they're meaningless and add in my own new parentheses, then say that's what the expression means.

eyeballfrog

Mar 19, 2023 00:40

@J.G. I mean, same applies. That's the square of the gamma function. I challenge you to find any reference in which that is used to mean $\Gamma((2x+1)^2)$.

eyeballfrog

Mar 19, 2023 00:40

@AnneBauval Of course it adds information. It says that $x$, and not $x^2$, is the argument of $\sin$. Would you say that $\Gamma(x)^2$ is equivalent to $\Gamma(x^2)$?

$Mathematics$ Comparing log(x) to log(x - 1)

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eyeballfrog

Mar 9, 2023 17:57

that series only works for $|x - 1| < 1$

eyeballfrog

Mar 9, 2023 17:28

Does the fact that \ln(1 - x) = - x - x^2/2 - x^3/3 -x^4/4 ... help in some way? seems like it might be relevant to your question

eyeballfrog

Mar 9, 2023 17:19

that is, what calculation are you trying to make?

eyeballfrog

Mar 9, 2023 17:18

but yes, what is the goal here?

eyeballfrog

Mar 9, 2023 17:18

I don't think it does because that's not true. You may be misinterpreting it

eyeballfrog

Mar 9, 2023 17:16

AKA the riemann sphere

eyeballfrog

Mar 9, 2023 17:16

complex infinity is part of the extended complex plane

eyeballfrog

Mar 9, 2023 17:16

+ and - infinity are part of the extended real line

eyeballfrog

Mar 9, 2023 17:16

no, complex infinity is different from + and - infinity

eyeballfrog

Mar 9, 2023 17:14

I'm willing to let complex infinity == complex infinity evaluate to true

eyeballfrog

Mar 9, 2023 17:13

not what?

eyeballfrog

Mar 9, 2023 17:11

hmm, the chat doesn't like TeX?

eyeballfrog

Mar 9, 2023 17:11

The range where $arcsin(sin\theta) = x$ is $-\pi/2<\theta<\pi/2$. On this range, the statement $\ln(2i\sin\theta) = \ln(e^{2i\theta}-1) - i\theta$ is indeed true.

$Mathematics$ Discussion on question by Amenhotep: …

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eyeballfrog

Mar 3, 2023 21:39

Any linear combination of the columns of $X$ should also be in the null space. Adding the first, fourth, and fifth columns gives a vector with all positive entries.

$Mathematics$ Discussion on question by jasmine: Is…

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eyeballfrog

Feb 7, 2023 22:17

In mathematics, $\log$ means natural log almost always. Why they don't use the perfectly good symbol $\ln$ is beyond me.

$Mathematics$ Discussion on answer by chrslg: What …

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eyeballfrog

Feb 4, 2023 17:55

@Mike Applying $A$ to that vector gives $(0,0,1,1)$.

eyeballfrog

Feb 4, 2023 17:55

That matrix also has $\lambda_3 = i$ and $\lambda_4 = -i$, so it's not a counterexample.

$Mathematics$ Discussion between eyeballfrog and Lu…

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eyeballfrog

Feb 3, 2023 21:17

this is the nyquist frequency

eyeballfrog

Feb 3, 2023 21:16

anyways, point is if you have N samples over an interval of [-5,5], your fourier transform is only good out to omega = N/20

eyeballfrog

Feb 3, 2023 21:12

your sampling rate is about 5, but [-5, 5] is width 10

eyeballfrog

Feb 3, 2023 21:12

so your samples only give valid frequency data over a window of width equal to the sampling rate

eyeballfrog

Feb 3, 2023 21:10

so your sampling rate is roughly 1/5, right?

eyeballfrog

Feb 3, 2023 21:10

wait, you're integrating from -5 to 5?

eyeballfrog

Feb 3, 2023 21:09

and thus is probably noise

eyeballfrog

Feb 3, 2023 21:09

but that's just the sampling rate

eyeballfrog

Feb 3, 2023 21:09

the Fourier transform has clear peaks at +/- 5

eyeballfrog

Feb 3, 2023 21:07

it's kind of noisy

eyeballfrog

Feb 3, 2023 21:06

with a source term that expresses perturbations to that mode

eyeballfrog

Feb 3, 2023 21:06

so, it sounds like you want to find a "dominant" wave mode

eyeballfrog

Feb 3, 2023 21:05

because you might be able to get better looking fs that way

eyeballfrog

Feb 3, 2023 21:05

which is why I was wondering if w was free

eyeballfrog

Feb 3, 2023 21:05

well, in that case you can pick any w and you'll get some f

eyeballfrog

Feb 3, 2023 21:05

just to be sure, u is draupnerfun

eyeballfrog

Feb 3, 2023 21:03

I was thinking of w as a constant

eyeballfrog

Feb 3, 2023 21:02

not saying you have to do one or the other, just wondering which model you're using

eyeballfrog

Feb 3, 2023 21:02

or the model is u'' + w^2 u = f, and we're trying to fit both w and f

eyeballfrog

Feb 3, 2023 21:02

the model is u'' + u = f, and we're trying to fit f

eyeballfrog

Feb 3, 2023 21:01

basically, there are two possibilities here

eyeballfrog

Feb 3, 2023 21:01

is the natural frequency of the oscillator a free parameter?

eyeballfrog

Feb 3, 2023 21:00

and you want to find the f that produces that wave

eyeballfrog

Feb 3, 2023 21:00

or, is the wave u

eyeballfrog

Feb 3, 2023 20:59

so that wave is the source term in a driven harmonic oscillator

eyeballfrog

Feb 3, 2023 20:59

oh yes

eyeballfrog

Feb 3, 2023 20:57

however, the norms will definitely be different

eyeballfrog

Feb 3, 2023 20:57

then c_n is a (probably) l1 representation of L1