## General Topology – Does the use of continued fractions work to give a homeomorphism? \$ mathbb {Q} ^ + rightarrow ( mathbb {Q} ^ +) ^ 2 \$?

To let $$mathbb {Q}$$ Let the set of rational numbers and let $$mathbb {Q} ^ +$$ Be the set of positive ($$x> 0$$) rationals.

I'm looking for a simple construction of a homeomorphism $$phi: mathbb {Q} rightarrow mathbb {Q} ^ 2$$ (without using an abstract result). In an early post, it was suggested to use continued fractions, but this has problems with negative numbers. My idea is that we can make a homeomorphism first $$f: mathbb {Q} ^ + rightarrow mathbb {Q}$$ about $$f (x) = (x-1) ^ 3 / x$$ and then we can only worry $$mathbb {Q} ^ +$$,

Then the homeomorphism
$$phi: Q ^ + rightarrow (Q ^ +) ^ 2$$ just would go like that

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

this obviously seems to be bijective, and I believe it is bicontinuous, based only on the idea that two continued fractions are close to each other if and only if enough of their original concepts match (which sounds plausible).

Does it make sense? Or do I miss something? Thanks for your help.

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## Calculating a Continued Fraction – Mathematica Stack Exchange

I want to build this infinite continued faction

$$F_ {n} (x) = frac {1} {1-x frac {(n + 1) ^ 2} {4 (n + 1) ^ 2-1} F_ {n + 1} (x )$$

what gives for $$n = 0$$

$$F_ {0} (x) = dfrac {1} {1- dfrac {(1/3) x} {1- dfrac {(4/15) x} {1- dfrac {(9 / 35) x} {1- ddots}}}}$$
I was inspired by it
Post (@Michael E2), the problem is that I transform it as a list view

``````{b0, {a1, b1}, {a2, b2}, ...}

clear[F2,iF2];

IF2[0]= 0;
IF2[1]= {1,1};
IF2[2]= {- x / 3,1};
IF2[n_]: = {- x (n + 1) ^ 2 / (4 (n + 1) ^ 2-1), 1};
F2[n_]: = Table[iF2[k], {k, 0, n}];
``````

I can not find all terms, so I'm looking for 5 terms

``````block[{n=5},F2[n]](* {0, {1,1}, {- x / 3,1}, {- 16x / 63,1}, {- 25x / 99,1}, {- 36x / 143,1}} *)
``````

it lacks after$$-x / 3.1$$} the terms {$$-4x / 15.1$$} and {$$-9x / 35.1$$}

What is wrong, please?

## Print – The printer continued to work after the upgrade to 18.04 and then stopped

My HP 1320n printer worked long after the upgrade to 18.04
stopped working:
teun @ lucas: ~ \$ lpq
The HP LaserJet 1320 series is ready
No entries
teun @ lucas: ~ \$ lpr test file
teun @ lucas: ~ \$ lpq
The HP LaserJet 1320 series is ready
Rank Owner Job File (s) Total Size
1st teun 575 testfile 1024 bytes
teun @ lucas: ~ \$ lpq
The HP LaserJet 1320 series is ready
Rank Owner Job File (s) Total Size
1st teun 575 testfile 1024 bytes
teun @ lucas: ~ \$
I tried to restart cups. It does not work.

## SQL Server – ms SQL Querying Multiple Table Columns as Identical Tag (continued)

Expand this original question

I am using SQL Server Express 2017

To pretend this is my source table:

``````+ ------------- + --------------------------- +
| ID | Name | Age | HairColor | HasCar |
+ ------------- + --------------------------- +
| 1 | Bert | 25 | blonde | 1 |
| 2 | Agnes | 61 | white | 0 |
| 3 | Diane | 43 | brunette | 1 |
+ ------------- + --------------------------- +
``````

How can a result look like this?

``````

``````

Due to the Group attribute in MyGroupingTag I can not use this

SELECT … FOR XML PATH (& # 39; MyTag & # 39;), ROOT (& # 39; MyGroupingTag & # 39;)

The group itself is a simple number of lines.

## Mining Theory – How has the hash rate for Bitcoin continued to rise despite the drop in prices and the miners' failure?

Despite the price decline in 2018, the bithrine hashrate trebled from around 15 EH / s to 45 EH / s. Why is that? I am aware that network difficulties are adjusting, which means that miners will jump off and become more profitable for others, but what are the key factors in the increase in hashrates? Is it related to rising ledger, increased numbers of miners, and / or other factors?

## Do these series converge or diverge? Continued faction series

Do these series converge or diverge?
A. Σ {1 /[n+Σ(1/n)] }
Eg Σ {1 /[n + Σ (1 /[n + Σ (1 /[n + Σ (1 / …}
Where all Σ are in terms of n from 1 to infinity

Would it make a difference if the inner sums only went from 1 to this term for n and not to infinity? (i.e., A. would look like 1 / (1 + 1) + 1 / (2+)[1/1+1/2]) + … instead of 1 / (1 + Σ (1 / n)) + 1 / (2 + Σ (1 / n)) + 1 / (3 + Σ (1 / n)) …)

Out of curiosity, both series have a name or do they have any meaning in mathematics?

## pathfinder – Dinosaur Damage Dilemma (continued)

In my last question, the Dinosaur Damage Dilemma, I talked about damage to an Ankylosaurus, and I realized that while the dinosaur may numb, the DC would be different for an animal companion. It says it's based on strength, but I did the math and that did not make sense.

So the question is how to calculate and how Stun DC is calculated by Ankylosaurus.