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In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are three theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist for groups, rings, vector spaces, modules, Lie algebras, and various other algebraic structures. In universal algebra, the isomorphism theorems can be generalized to the context of algebras and congruences.
== History ==
The isomorphism theorems were formulated in some generality for homomorphisms of modules by Emmy Noether in her paper Abstrakter...
In mathematics, the Lucas–Lehmer–Riesel test is a primality test for numbers of the form N = k ⋅ 2n − 1, with k < 2n. The test was developed by Hans Riesel and it is based on the Lucas–Lehmer primality test. It is the fastest deterministic algorithm known for numbers of that form. For numbers of the form N = k ⋅ 2n + 1 (Proth numbers), either application of Proth's theorem (a Las Vegas algorithm) or one of the deterministic proofs described in Brillhart-Lehmer-Selfridge 1975 are used.
== The algorithm ==
The algorithm is very similar to the Lucas–Lehmer test, but with a variable starting point...
In mathematics, more specifically in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the polynomials that give rise to them via Galois groups is called Galois theory, so named in honor of Évariste Galois who first discovered them.
For a more elementary discussion of Galois groups in terms of permutation groups, see the article on Galois theory.
== Definition ==
Suppose that
E
{\displaystyle...
user434058
Q: Is this an example of an equation that generates solutions? And is everything below correct?
Part $1:$
Suppose that $x$ and $z$ are independent of each other.
$$\phi(x,z)= e^{\frac{z^2}{\ln(x)}}=x,$$
where $x,z\in\Bbb C.$
$x=e^{z},e^{-z}.$
With this information I rewrote the equatio...
7
One approach is to integrate $e^{-z^2}$ around a rectangle in the complex plane with vertices at $R,R+ia,-R+ia$ and $-R$.
Another approach is to expand $\cos ax$ in a Maclaurin series and then switch the order of integration and summation. You'll end up with a constant times the Maclaurin series...
2
You can integrate
$$f(z)=e^{-z^2},z\in\mathbb{C}$$
On the boundary of the rectangle
$$\{z\in\mathbb{C}:\Re(z)\in[-R,R]\wedge \Im(z)\in[0,a/2]\}$$
Given $a>0,R>a/2.$
Being $f$ entire, this integral is equal to zero.
Using the fact that
$$\int_0^\infty{e^{-x^2}dx}=\frac{\sqrt\pi}{2}$$
And observin...
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