Mathematics - 2020-02-29

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12:16 AM

@TedShifrin The formula is definitely true. Earlier it made me run into the integral $\int_{t_0}^t \sinh(t-s)\sin^2(s)ds$

Which I, of course, got wrong.

That's why I made all this fuss, how silly..

6 hours later…

maths student

6:02 AM

Let $X_1 & X_2 $ be independent random variable that take values from C = { 1, 2, 3} and D = {1,4} respectively with probability $P(X_1) $ = { 0.2 , 0.4, 0.4} and $P(X_2)$ = {0.7,0.3}

Now define another random variable X as X = X_1 with probability 0.8 and X = X_2 with probability 0.2 .Then we have to find H(X) . But I don't understand what symbol H(X) represents. Anyone know what is meaning of this?

6 hours later…

Mats Granvik

11:52 AM

The Riemann hypothesis is to show that the row sums of the lower triangular part of the matrix above is bounded by Sqrt(n)*Log(n)^2

Or at least if the partial sums of the Möbius inverse of the Harmonic numbers minus n are bounded by the same bounds as the partial sums of the von Mangoldt function minus n.

Row sums of the matrix above converge to the Chebyshev function when the size of the matrix goes to infinity.

The overall visible pattern resembles the familiar divisor pattern, which is bounded by two times a square root when taking sign(divisor)

oeis.org/A309229

1 hour later…

skullpatrol

1:14 PM

February 29th is to the leap year as i is to the imaginary number.

Happy "i day" everybody :-)

ABC

1:50 PM

Someone can help me to prove this equality?

Leaky Nun

$z=e^{it}$, $z^2 = e^{2it}$, $\sin(2t) = (z^2-z^{-2})/(2i)$

looks like complex analysis to me

ABC

perfect thanks you!

Leaky Nun

LHS = $\displaystyle \int_C \frac{3z9(z^2-z^{-2})(2i)^{-1}}{9z^2+4} 3\ \mathrm dz = \int_C \frac{81z(z^4-1)}{2iz^2(9z^2+4)}\ \mathrm dz$

then there's the 81/2

$= \displaystyle \frac{81}2 \int_C \frac{z^4-1}{iz(9z^2+4)} \mathrm dz = \frac{81}2 \int_C \frac{e^{4it}-1}{ie^{it}(9e^{2it}+4)} ie^{it} \ \mathrm dt = \frac{81}2 \int_C \frac{e^{4it}-1}{9e^{2it}+4} \ \mathrm dt$ = RHS

2 hours later…

adesh mishra

3:53 PM

@LeakyNun Is curl the measure of “how much twist” is there ?

Leaky Nun

4:12 PM

sure

geocalc33

4:45 PM

one notion of "twist" is called (torsion) @adeshmishra

3 hours later…

loch

7:59 PM

@mathsstudent probably entropy

Bob

8:30 PM

good afternoon

2 hours later…

geocalc33

10:09 PM

good evening

Q: Dynamical system of interacting fuzzy truth values yielding classical truth values?

Consider a finite set of fuzzy (partially true) truth values $T=\{t_0,t_1,t_2,...,t_n\}$ with a Partial map $\psi:T\to J.$ Where $0\le t_0,t_1,...\le1.$ "Equip $\psi$ with a driving function $\zeta(t)=\kappa B(t).$" Basically I want the map to be dynamic, in the sense that at any time, $t,$ ther...