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Sort an array of large numbers (each programming language) [on hold]

enter an array of array elements (unlimited number of array elements). Then sort gradually. If you meet -1, keep the -1 position

Multiplication of the square root of complex numbers

$$
sqrt {-21 + 20i}. sqrt {-21-20i} = pm (2 + 5i). pm (2-5i) = pm29
$$
But why it's not
$$
sqrt {-21 + 20i}. sqrt {-21-20i} = sqrt {(- 21 + 20i) (- 21-20i)} = sqrt {| z | ^ 2} \
= sqrt {441 + 400} = sqrt {841} = 29
$$

Does this have anything to do with $ sqrt {-a} sqrt {-b} = – sqrt {ab} neq sqrt {ab}, ; a, b in mathbb {R} $ ?

Where do all these rules come from?

What is the logic of finding all the possible numbers that add to the number requested?

But note that the number on the left is always greater than the number on the right. I've already tried it in several ways, but I think I did not understand the logic to create an algorithm that does it, so I always fail in numbers greater than 8.

Example: 8
8

7 + 1

6 + 2

6 + 1 + 1

5 + 3

5 + 2 + 1

5 + 1 + 1 + 1

4 + 4

4 + 3 + 1

4 + 2 + 2

4 + 2 + 1 + 1

4 + 1 + 1 + 1 + 1

3 + 3 + 2

3 + 2 + 2 + 1

3 + 2 + 1 + 1 + 1

3 + 1 + 1 + 1 + 1 + 1

2 + 2 + 1 + 1 + 1 + 1

2 + 1 + 1 + 1 + 1 + 1 + 1

1 + 1 + 1 + 1 + 1 + 1 + 1 + 1

How can I make sure that a page pops in record numbers greater than 1 in laravel – dompdf

It overflows and is lost during the page break

[GET][NULLED] – Numbers: Registration and connection to a WordPress Mobile number v6.4.1

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[GET][NULLED] – Numbers: Registration and connection to a WordPress Mobile number v6.4.1

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[GET][NULLED] – Numbers: Registration and connection to a WordPress Mobile number v6.4.1

prime numbers – Additive and multiplicative convolution deeply bound in modular forms

Because of the spaces of modular forms are finite dimensions, the decomposition into eigen forms of Hecke, and the formula of duplication for $ Gamma (s) $ there are a lot of identities mixing additive convolution $ oplus $ and multiplicative convolution $ otimes $, the basic example being $ 1 _ { mathbb {Z} ^ 2} ast 1 _ { mathbb {Z} ^ 2} = 1 otimes chi_4 $

It's in algebra $ M, $ oplus generated by modular forms for $ Gamma_0 (N) $ there will be a lot of identifying involving $ otimes $. How do Hecke's algebras and their representations come into play? Is there a more precise formulation, using higher things like automorphic representations?

Do you have heuristics making visible what property of prime numbers says this connection between additive convolution and multiplicative convolution? (if the heuristic implies HR, that's good too)

Assuming that prime numbers have been wildly distributed, can we still expect these identifiers to be identified? The evidence of multiplicative additive identifiers based on the functional equation and the product of Euler L functions, so I ask if it is plausible that the PNT and the HR still play a role. What about convolutions inverses, convolution with modular functions like $ 1 / d (z) $, additive convolution with $ mu (n) $ ?

time – How to display week numbers on the axis of a chart

I have chronological data visualized on a line chart and I wish to format the x-axis accordingly. I'm wondering what format to choose in English if the data is aggregated weekly. In German, I would use "KW x".

I found this discussion, this suggests "Week x" if you want to translate the German word "Kalenderwoche (KW) x" into English.
Also google calendar uses this format.
Moment.js offers other options

Anyone else has an entry one that?

complex numbers – If $ z = sqrt {3} + i = (a + ib) (c + id) $, search for $ tan ^ {- 1} dfrac {b} {a} + tan ^ {- 1} dfrac {d} {c} $

Yes $ z = sqrt {3} + i = (a + ib) (c + id) $then $ tan ^ {- 1} dfrac {b} {a} + tan ^ {- 1} dfrac {d} {c} = $ _______________

$$
arg z = theta, quad arg (a + ib) = theta_1, quad arg (c + id) = theta_2 implies arg (z) = theta = theta_1 + theta_2 = frac { pi} {6} \
tan theta = tan frac { pi} {6} = frac {1} { sqrt {3}}, quad tan theta_1 = frac {b} {a}, quad tan theta_2 = frac {d} {c} \
implies theta_1 = n pi + tan ^ {- 1} frac {b} {a}, quad theta_2 = m pi + tan ^ {- 1} frac {d} {c}
theta = frac { pi} {6} = (n + m) pi + tan ^ {- 1} frac {b} {a} + tan ^ {- 1} frac {d} { c} \
tan ^ {- 1} frac {b} {a} + tan ^ {- 1} frac {d} {c} = frac { pi} {6} – (n + m) pi = frac { pi} {6} + k pi
$$
Is this the right way to prove it? $ tan ^ {- 1} dfrac {b} {a} + tan ^ {- 1} dfrac {d} {c} = n pi + dfrac { pi} {6} which is what is given in my reference.

combinatorial – Prove that among any set of 34 different positive integers that are at most 99, there is always a pair of numbers that differs by at most 2.

Okay, so I'm pretty new to this. I think this should be a pretty simple solution, but I do not know how to start.

So here's where I was going

if we have a set {1,2, … 99}, then I start creating groups that have an integer pair other than 2 at most.

group 1 = (1,2)

group 2 = (3,4)

…

group 49 = (97.98)

Now, I know I do not use 99 but it says at most 99. Another thing I could do is make sure that the groups

group 1 = (1,3)

group 2 = (2,4)

group 3 = (5,7)

group 4 = (6.8)

…

group 48 = (94.96)

group 49 = (95.97)

group 50 = (98.99)

and I think it works better.

Now, from here, I do not know how to progress, in fact, I think it's a bad way to go, but I do not know how to solve this problem differently. Maybe I could use the pigeon trap principle but I do not know how.