When a search engine returns its search **results**, it gives you two types: organic and**paid**. Organic Research **results** are the lists of web pages that best match the search query of the user based on its relevance. Also called "natural" search **results**, high-end in the bio **results** what is it **SEO** is all about.

# Tag: general

## General topology – On various problems arising from the study of series in analytic number theory I: counting functions and irrational numbers

After studying (1) (in Spanish), I asked myself questions about the next two questions. The first seems to me just a question about the reference request, and for the second question (if I managed to write a question that has a complete mathematical meaning in this situation to exploit the beautiful reference (1)), it is necessary to have knowledge on the topology.

For the first question, which is related to the statement in (1), I am also inspired by a statement due to J. R. Chen that I read in the MathWorld encyclopedia, I say the article *Twin Awards*and I wonder about it, I need to know if the answer to question 1 is in the affirmative, to know if I can apply the statement that shows (1) in the last paragraph.

Question 1.We consider the sequence $ a_n $ prime numbers $ a_n = p $ such as $ p + 2 $ has at most two factors. I would like to know if this sequence has arbitrarily large differences, that is, if for the sequence $ a_n $ In connection with the work of J.R. Chen, we can find gaps as large as we wish. If you know it in the literature, please add a comment in your answer by adding what the paper is, and try to find and read the statement of the literature.Thank you so much.

Secondly, we consider the set of all these infinite sequences of positive integers (our integers, of its corresponding sequence, written in ascending / increasing order) which have arbitrarily large deviations, and we designate it by $ G $. On the other hand, we consider the counting function for each of the sequences $ g in $ G, noted as $$ pi (n, g) = # {g_n leq x text {such that} g = (g_n) _ {n geq 1} text {is an infinite suite} $$ $$ quad quad quad quad quad quad quad quad quad quad quad quad quad quad quad {text {positive integers with large arbitrary gaps} } tag {1} $$

as it has been said, our sequences have an infinity of terms written in ascending order as the sequence of prime numbers $ p = (p_n) _ {n geq 1} $, which are positive integers, and our sequences enjoy the same property as the sequence of prime numbers because, by definition, we can find spaces as large as we want. So as an example $ p in G $.

For each integer $ b> $ 1 we chose one of our series, tell us $ g $, and we define all of these sums $$ sum_ {m = 1} ^ infty frac { pi (m, g)} {b ^ m}, tag {2} $$

this set will be designated by $ mathcal {G} (g) $, just to emphasize the whole $ mathcal {G} (g) $ is the set of real numbers of the form $ (2) $, when the $ b $ passes on all positive integers greater than $ 1 $.

Finally, we define the union when $ g $ works on all of these sequences $ g in $ G, that's what we define $$ mathbb {G} = cup_ {g in G} mathcal {G} (g). tag {3} $$

We have that $ sum_ {n = 1} ^ infty n2 ^ {- n} $ so I'll ask my question for the whole $ (0.2) $ instead of $ (0.1) $ (but by the nature of my question, it is superfluous since $ (0,1) sub-set ($ 0.2). We work with the usual topology.

Question 2.Prove or throw it away, let the whole $ mathbb {G} $ is dense in a subset $ I sub-set ($ 0.2).Thank you so much.

## References:

(1) Javier Cilleruelo, *A series that converges to irremational número*, Miniaturas matemáticas, Miniatures matemáticas, La Gaceta of the RSME

Flight. 18 (2015), Núm. 3, Pág. 568.

## My 1,000,000 Sats trip (and General Beermoney's blog)

__My main goal:__Make 1,000,000 satoshis (0.01 BTC) in one year.

__Why I want to achieve this goal:__

- The first time, I locked 700k sats behind an address for which I no longer have a private key. Which is zero, since I lost 8 months to get them. I almost gave up because of that.
- Reaching any type of goal, big or small, is motivating for me.

** How will I achieve this goal:**…

My 1,000,000 Sats trip (and General Beermoney's blog)

## 5th dnd – In general, what do you think about it when you are about to create a character?

**This question is not just about the system, after all, creativity is also complicated for some people and there is nothing better than reading the reviews of someone who reads it often.**

In principle, I will say that I am completely new to D & D and, like most people in this situation, I have no creativity to create my initial character.

• What motivates you when you create your character? If you are generally inspired by a character created in books or movies, can you tell which one?

• When will you choose a race or class, what do you think at the moment? Often completing the lack of certain things in your party is what forces you to make such a choice?

• Do you have a favorite race or do you create according to the world proposed by DM?

• Do you have a favorite class or do you create for the needs of the party?

• Are you trying to create a more human and defective character or Deus Ex Machine?

• Are you criticizing yourself for creating the so-called "perfect characters"? (as a half-elf being a cleric, a bard, a warlock, or a paladin because of his racial bonuses)

*Notes: I do not speak English fluently, which limits me a lot to learn from the Player's Handbook. I therefore apologize for having to deal with such a topic on this subject and hope that it does not go against the rules of the site. If there are several grammatical errors, please ignore them (and blame Google Translate for its incorrect translations, I'm kidding)*

## Additional entity ID for networking instead of general EID?

A UUID occupies 128 bits.

Your own identifier can be as small as `ceil(log_2(n))`

parts; where n is the number of entities you want to distinguish.

Obtaining the entity from the UUID requires a hashmap. An intelligently designed identification system will simply allow you to index in a table, which is slightly faster.

You can also embed some type information in the ID by assigning ranges to each type. This allows you to avoid a downcast / typecheck in the network code because you can have separate containers for each type.

## General Topology – $ Cover Size[0,1]$

Equip $ [0.1] $ with its usual metric topology inherited from $ mathbb {R}. $ It is well known that the coverage dimension of $ [0.1] $ is $ 1 $.

**Question**: Given a fixed number $ n $, can I build a finished open blanket $ U_1, U_2, cdots, U_n $ of $ [0.1] $ such as any finite open refinement of the multiplicity $ 2 $ must consist of at least k $ elements for an arbitrary number k $?

For example, can I build an open cover of $ [0.1] $ consisting of $ 3 $ elements such as any open refinement of the multiplicity $ 2 $ must have at least $ 7 $ elements?

In general, given a compact space $ X $ coverage size $ s $ and a fixed number $ n $, can I build a finished open blanket $ U_1, U_2, cdots, U_n $ of $ X $ such as any finite open refinement of the multiplicity $ s + 1 $ must consist of at least k $ elements for an arbitrary number k $?

Any help would be greatly appreciated.

Definitions:

**Cover size**: A non-empty topological space $ X $ is said to have the coverage dimension $ n $ if $ n $ is the smallest non-negative integer with the property such that each open cover over $ X $ has a finite open refinement of the multiplicity at most $ n + 1 $.

**Multiplicity of a blanket** A cover of a topological space $ X $ has multiplicity $ n $ if and only if it is the smallest non-negative integer such that each point $ x $ of $ X $ belongs to the most $ n $ elements of the cover.

## reference request – From Laplacian case to general divergence form

What are some tips that can be used to generalize results for elliptic / parabolic equations in a related domain $ Omega $ with lapacian $ Delta u $ in the form of general divergence $ text {div} (A (x) nabla u) $.

**Case 1:** $ A $ is a constant matrix. In this case, we can use the change of variable $ v (x) = u (Ax) $ ($ A $ symmetrical and uniformly elliptical) to obtain the case of Laplacian. But this change sends $ Omega $ at $ A Omega $ and we can miss the regularity properties of $ Omega $.

**Case 2:** $ A $ is not constant. In this case, the previous variable change does not work here.

For Case 1, how can we reserve the regularity of $ Omega $ under $ A $?

In case 2, is there any trick to directly obtain the general result? Sometimes we have to impose restrictions on the ellipticity constant of $ A $ (eg higher or lower than a given number). Is there a trick to remove these restrictions in this case?

Thank you for any suggestions or references.

## General topology – closed in $ mathbb {R} ^ {2} $ closed in $ mathbb {R} $

Define the following subsets of $ mathbb {R} $ and $ mathbb {R} ^ {2} $ respectively:

$ A = left { frac {1} {n}: n in mathbb {N} right $

$ B = left {(x, frac {1} {x}): x in mathbb {R} smallsetminus left {0 right } right $

I know the first subset, $ A $, is neither open nor closed, and the second subset, $ B $, is closed.

However, the whole $ B $ is exactly $ A $ but "seen" in $ mathbb {R} ^ {2} $, so why $ B $ is neither open nor closed $ mathbb {R} ^ {2} $?

## General topology – Can the use of continuous fractions give a homeomorphism $ mathbb {Q} ^ + rightarrow ( mathbb {Q} ^ +) ^ 2 $?

Let $ mathbb {Q} $ to be the set of rational numbers and let $ mathbb {Q} ^ + $ be the set of positives ($ x> 0 $) rational.

I am looking for a simple construction of a homeomorphism $ phi: mathbb {Q} rightarrow mathbb {Q} ^ 2 $ (do not use an abstract result). In a previous article, it was suggested to use continuous fractions, but this poses problems with negative numbers. My idea is that we can first make a homeomorphism $ f: mathbb {Q} ^ + rightarrow mathbb {Q} $ via $ f (x) = (x-1) ^ 3 / x $ and then we can just worry about $ mathbb {Q} ^ + $.

Then the homeomorphism

$ phi: Q ^ + rightarrow (Q ^ +) ^ 2 $ would just like

$ (a_0, a_1, …, a_n) rightarrow ((a_0, a_2, …), (a_1, a_3, …)) $

this seems to be obviously bijective, and I think it is bicontinous only because two continuous fractions are close if and only if enough of their initial terms match (which seems plausible).

Does it make sense? Or am I missing something? Thank you for your help.