## search engines – isomorphic (universal) web application and SEO

I am developing an isomorphic web application using React (SSR). However, I need to render some of my components on the client side. Therefore, the things that a Google bot sees may be different from the things that users see. The components of which I speak are not important for my SEO concerns and I try to show all the SEO related content on the server. Is this difference important to Google and can it have any impact on the ranking of my website?

## 3 sat – How many isomorphic 3SAT formulas?

For a 3SAT formula with $$n$$ variables and $$m$$ clauses, I am interested in counting the number of isomorphic formulas (isomorphic in the sense that they are logically equivalent Y have the same number of variables and clauses). I guess we can reverse the meaning of any variable (exchange everything $$x_i$$ with $$lnot x_i$$) and swap any of the variables. The first one must give $$2 ^ n$$ and the second one must give $$n!$$.

Is this all isomorphisms, or am I missing something?

If this is correct, I assume that the number of different 3SAT formulas with $$n$$ variables and "different clauses" equals $$frac {2 ^ m} {2 ^ nn!}$$, where $$m$$ is the number of 3 clauses on $$n$$ Variables in which all the variables are different. I assume that $$m = 2n (2n-2) (2n-4)$$. That sounds good?

## graphic isomorphism – Are the lambda expressions isomorphic (without type) semantically equivalent?

"Isomorphic" is defined as having the same form of syntax trees and the same variable links. However, the names of the variables can be completely different. In other words, it is to say that we have a graphical isomorphism between the two expressions (the graph is the syntax tree with the variable link included as edges).
The intuition behind this is that assuming we are given the isomorphism, it is always possible to change the name of the variables and get the other expression.

## General topology: attaching the map or two squares to a sphere in isomorphic to the union of sphere and an ellipsoid.

Consider the topological spaces. $$X = partial ([0, 1]^ 3) subset mathbb {R} ^ 3$$ (The limit of the unit cube in $$mathbb {R} ^ 3$$) and Y the unitary sphere $$mathbb {S} ^ 2 subset mathbb {R} ^ 3$$. Leave $$A subset X$$ be the union in the sacrares $$[0,1] mbox {x}[0,1] mbox {x} {0}$$ Y $$[0,1] mbox {x}[0,1] mbox {x} {1}$$, and consider the attached map $$f: A rightarrow Y$$ Sending the first place to the north pole and the second to the south pole. Show that:
$$X space space U_ {f} space Y cong Z$$
Where $$Z subset mathbb {R} ^ 3$$ It is the union of the unitary sphere and the ellipsoid E (2,2,1).

## abstract algebra: direct sum of \$ n \$ (infinite) isomorphic cyclic groups to direct the sum of \$ n \$ copies of \$ mathbb {Z} \$?

I am currently studying a bit of algebra and I am currently covering the various equivalent definitions of free abelian groups. However, to understand why these definitions are really equivalent, this question arose and I have not been able to resolve it on my own. The real question is:

Assume that I have $$n$$ infinite cyclic groups $$langle a_1 rangle, langle a_2 rangle, …, langle a_n rangle$$

Do you have that? $$bigoplus_ {i = 1} ^ n langle a_i rangle cong bigoplus_ {i} ^ n mathbb {Z} (n text {copies of} mathbb {Z})$$?

If anyone has a recommendation for a good book on this subject, I would also appreciate any recommendation.

Thanks for any help.

## Are the isomorphic polytopes having the same extension complexity?

Is there a simple proof of the fact that isomorphic polytopes have a complexity of similar extent?

## dg.differential geometry – Reference request: isomorphic stacks are given by Moro Moro groupoids equivalent of Morita

Leave $$mathcal {G}, mathcal {H}$$ The groupoids are lying. Leave $$B mathcal {G}$$ denotes the main stack $$mathcal {G}$$ packages and $$B mathcal {H}$$ denotes the main stack $$mathcal {H}$$ bunches Then, we have the following result.

The groupoids $$mathcal {G}$$ Y $$mathcal {H}$$ Morita is equivalent if and only if the differentiable batteries $$B mathcal {G}$$ Y $$B mathcal {H}$$ they are isomorphic

The only place where I could find proof of this is (theorem $$2.26$$) the batteries and gerbes of Paper Differences of Kai Behrend and Ping Xu. I understand the test. I think it is not clear (the notion that something is clear depends on the level of experience) and needs some more lines in the test. I have written in detail the proof. It is a long test.

Is there any other place where this result has been tested so that I can quote that document? I have seen three articles that mention this result without any proof and all refer to the article by Kai Behrend and Ping Xu. I'm not asking for proofs. I'm just asking for reference.

## Show that a finite abelian group \$ G \$ is not cyclic if and only if it contains a subgroup isomorphic to \$ mathbb {Z} _p times mathbb {Z} _p \$

Prove that a finite abelian group. $$G$$ it is not cyclic if and only if it contains an isomorphic subgroup $$mathbb {Z} _p times mathbb {Z} _p$$.

I am aware that there is an answer here. I have been trying to review the test step by step and I have some problems.

The reverse case is easy. Clearly, if $$G$$ contains a subgroup $$H$$ isomorphic to $$mathbb {Z} _p times mathbb {Z} _p$$it turns out that $$G$$ it can not be cyclical, since each subgroup of a cyclic group is cyclical, and $$H$$ it can not be cyclic if it is isomorphic $$mathbb {Z} _p times mathbb {Z} _p$$. The other direction is proving to be a great challenge for me.

Suppose that $$G$$ It is not cyclical. As $$G$$ is finite, it is generated finitely and, therefore, it is isomorphic for a group of the form $$mathbb {Z} _ {p_1 ^ {r_1}} times mathbb {Z} _ {p_2 ^ {r_2}} times … times mathbb {Z} _ {p_n ^ {r_n}} .$$ $$p_i = p_j$$ for some $$i, j$$ with $$i neq j$$, since otherwise $$mathbb {Z} _ {p_1 ^ {r_1}} times mathbb {Z} _ {p_2 ^ {r_2}} times … times mathbb {Z} _ {p_n ^ {r_n}}$$ it would be cyclical (and so too $$G$$ for the existence of an isomorphism between the two). Without loss of generality, I assume that $$i = 1, j = 2$$. I'm having trouble finding an isomorphic subgroup for $$mathbb {Z} _ {p_1} times mathbb {Z} _ {p_2}$$. Clearly the set of elements of the form. $$(a_1, a_2,0, …, 0)$$ with $$a_1 Y $$a_2 it is an isomorphic subgroup $$mathbb {Z} _ {p_1 ^ {r_1}} times mathbb {Z} _ {p_2 ^ {r_2}}$$, through isomorphism. $$phi: mathbb {Z} _ {p_1 ^ {r_1}} times mathbb {Z} _ {p_2 ^ {r_2}} to mathbb {Z} _ {p_1 ^ {r_1}} times mathbb {Z} _ {p_2 ^ {r_2}} times {0 } times … times {0 }$$, with $$phi: (a, b) mapsto (a, b, 0, …, 0)$$. However, I'm not sure how to use this kind of logic to find an isomorphic subgroup for $$mathbb {Z} _ {p_1} times mathbb {Z} _ {p_2}$$.

A push in the right direction would be much appreciated.

## abstract algebra – What is \$ mathbb {R}[x]\$ quoted by a polynomial \$ f (x) in mathbb {R}[x]\$ isomorphic for?

For the CRT, we have to: $$mathbb {R}[x]/ (x ^ 2-2) simeq mathbb {R}[x]/ (x- sqrt {2}) times mathbb {R}[x]/ (x + sqrt {2}) simeq mathbb {R} times mathbb {R}.$$

Considering the evaluation map. $$mathbb {R}[x] a mathbb {C}$$ to $$2i$$, it is also clear that:
$$mathbb {R}[x]/ (x ^ 2 + 2) simeq mathbb {C}.$$

Now, I wonder if it's true that
$$mathbb {R}[x]/ (x-2) ^ 2 simeq mathbb {R}[x]/ (x) ^ 2,$$ And if that is how I would try it. My idea was to simply consider the quotient explicitly as a set of the form $${a_0 + a_1x + f (x) (x ^ 2) }$$ and identifying it with $${a_0 + a_1x + g (x) (x ^ 2 – 4x + 4) }$$. But if that works, what prevents me from using this same method to show that this quotient is isomorphic to the previous quotients, which is obviously false?

## Real analysis: Is each finite graph isomorphic to the proximity graph of some \$ S subseteq mathbb {R} ^ n \$?

This is the question I should have asked before asking this previous question.

Yes $$(X, d)$$ is a metric space, we associate with it a simple, unguided graphic, called its proximity chart $$G (X, d)$$ given by $$V (G (X, d)) = X$$ Y $$E (G (X, d)) = big { {x, y }: x neq y in X text {y} d (x, y) leq 1 big }.$$

As noted by the user of MO @YCor in a comment to a recently deleted question, given any simple graph (not necessarily finite) not addressed $$G = (V, E)$$, the map $$d: V times V to mathbb {R}$$ given by $$d (v, v) = 0$$ for $$v in V$$, $$d (v, w) = 1$$ yes $${v, w } in E$$ Y $$d (v, w) = 2$$ otherwise it gives a metric in $$V$$ such that $$G cong G (V, d)$$.

Question. Yes $$G = (V, E)$$ is a finite graph, there is a positive integer $$n en mathbb {N}$$ and a finite subset $$S subseteq mathbb {R} ^ n$$ such that $$G cong G (S, || cdot ||),$$ where $$|| cdot ||$$ denotes the Euclidean metric that $$S$$ inherit from $$mathbb {R} ^ n$$?