Please allow me to talk a little bit about my research. I am interested in certain sums of unit fractions, like 1/2 + 1/3 + 1/4 + 1/5. More precisely, for positive integers a and b write the sum 1/a + 1/(a+1) + .. + 1/b as one fraction u(a,b)/v(a,b) (where numerator and denominator have no common factors). For example, v(2,5) = 60, as 1/2 + 1/3 + 1/4 + 1/5 = 77/60. Then, if we fix a and view the denominator v(a,b) as a function of b, then this function often grows quite quickly, roughly exponentially fast. To give you an example, let's say a = 100 and let b run from 100 to 105. Then we get the following sums and look at the denominators on the right-hand side:

1/100 = 1/100
1/100 + 1/101 = 201/10100
1/100 + 1/101 + 1/102 = 15301/515100
1/100 + 1/101 + 1/102 + 1/103 = 2091103/53055300
1/100 + 1/101 + 1/102 + 1/103 + 1/104 = 67632503/1379437800
1/100 + 1/101 + 1/102 + 1/103 + 1/104 + 1/105 = 188463347/3218688200

Fast-growing indeed!

But the (to me) interesting thing is that v(a,b) is not a monotonically increasing function. Even though it's a fast-growing function, sometimes if we increase b it gets smaller! As we noticed, v(2,5) = 60, whereas 1/2 + 1/3 + 1/4 + 1/5 + 1/6 = 29/20, so that v(2,6) = 20 which is indeed smaller than v(2,5).

Now, let b(a) be the smallest integer b > a such that v(a,b) is smaller than v(a,b-1). That is, for fixed a, b(a) is the first violation of monotonicity. Then it can be shown that b(a) exists and is finite for every a and, more precisely, b(1) = b(2) = b(3) = 6, b(4) = 18 and b(a) < 4.38a for all a > 4. On the other hand, there are lower bounds for b(a) as well and this is where we get to the title of this post. Because in a certain sense, b(a) can not be too small; if you start with b = a and then let b grow, then for the first few values of b, v(a,b) will actually be monotonically increasing. To be exact, there is an absolute constant c > 0 such that for all positive integers a we have b(a) > a + c * log(a). Rewriting this inequality we see that (for a > 1) the quantity f(a) = (b(a) - a)/log(a) is bounded from below. In particular, I can prove that for all large enough a we have f(a) > 0.54. So if a is such that f(a) is near this value of 0.54, then this means that v(a,b) stops being a monotonically increasing function of b more or less as quickly as theoretically possible. Furthermore, there do exist some a that actually violate the inequali

https://twitter.com/SteveDuin/status/1452974790889836546 ​ >This from @redrock_bball at BasketballMonster: "There is conjecture out there that Lillard is suffering through a decently significant injury, that worsened in Tokyo and probably should've had surgery on his abdomen. He is struggling to get lift on his 3s or to finish at the rim"

​ https://twitter.com/SteveDuin/status/1466295726514733063
>Josh Lloyd nailed the @Dame_Lillard injury five weeks ago

Lillard said before that he's been dealing with an abdomen issue for three and a half or four seasons. According to Chauncey Billups, there’s been no talk of surgery at this point.

continued from here

ty tnf for the run+assist

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next get is 664 (664 | 0) schedule

I took Real Analysis this semester so I had series and sequences on my mind. I had the random idea to look at the sum of the reciprocals for the Hailstone sequence of some numbers. So, taking 5 and applying the Collatz function iteratively you would get,

[5, 16, 8, 4, 2, 1]

Wherein I would stop. Then I would look at,

1/5 + 1/16 + 1/8 + 1/4 + 1/2 + 1/1 = 2.1375

I plotted this number with respect to its initial value (I started calling these seeds just to help me out).

Here are some of my findings.

Fig 1. Hailstone Sequences, Length of Sequence, Sum of Reciprocals

This first image was to get my head around everything. The first graph is of the hailstone sequences for the first 1000 integers. You can see how all the lines eventually fall down to 1.

The second graph plots the length of the sequences, from the starting seed to 1, against the seed's value. It seems to grow sort of logarithmically which is neat.

The third plot is my work. The sum of reciprocals for the sequences. The first thing to notice is that strange band at the bottom, followed by a gap, followed by a cluster of points. Very interesting. I'll comment more on the bottom band later. Second thing of note is how slowly it increases. This makes sense because the reciprocal of large numbers are small, so we have to go out very far to see any noticeable change. We know the harmonic series diverges and I am just taking different subsequences of the harmonic sequence. If there was some upper bound on these sums of sequences that could say something interesting. But I don't know.

Fig 2. Sum of Reciprocals up to 10,000

Here is a graph of just the sums of reciprocals but up to 10,000 integers. The interesting thing here is that you can see more pronounced bands forming. The lowest band around 2.0, a thin band about 2.3-2.4, a dense band around around 2.5, a thick band from 2.1 to 3.0, and then an upper band about 3.2. The bands themselves are interesting but what is even more interesting to me is the gaps between the bands.

Fig 3. Sum of Reciprocals up to 10E5

This figure is just to

Opinions are like assholes. Fuck em. Music is literally the #1 most subjective thing in the entire world. You will NOT agree with everyone and everyone will NOT agree with you. Chill the fuck out people!

Just a thought on my day off. It's conventional wisdom that MLK St is always in a crappy part of town. For all of Austin's well known racist history, they gave MLK blvd a place of respect.

What do we think?

I was personally hoping for a little clearer resolution that we're going to cantha, similar how other seasons ended with pretty clear directions, but i guess EoD is gonna have to give us that prologue on it's own.

Weird how the DSD is still a topic of speculation, i couldve sworn it's existence was already confirmed? like. wasn't mordremoth already supposed to be the "secret" sixth elder dragon? i thought we just didn't know /what/ exactly it is and the full name, not whether it existed at all. There's the entire storyline of all the aquatic races having been displaced by it's awakening, and also every single elder dragon simulation tami has ever run already included six dragons? i really don't know whats up with this mystery storyline.

who do we think raided taimi's rata sum lab after the S4 replay? im guessing it's either the jade brotherhood or maybe even jun. can't wait to get the resolution of that in eod.

Personally, I’m surprised at how well the mormon church has done in light of their past doctrine and history. As I began to open my mind to teachings beyond the white-washed version the church produces, I was surprised to learn that the church of today is vastly different than the one started by Joseph Smith. TBM’s explanation for this is the idea of an ongoing restoration. Somewhat ironically, even that thought is relatively new to mormonism.

I now have a different opinion to explain the church's success. It has been my experience that many of the sunday school and EQ lessons I’ve been involved with throughout my life has included conjecture, assumptions, ignorance, and complicated explanations. I feel these four things has allowed a good number of the church membership in the past to maintain a testimony of a complicated and IMO problematic religion.

I’ve noticed how many members are willing to interpret the gospel to fit into an ever changing culture and society. For example, by-in large, the consensus I’ve noticed about how members view the LGTBQ community is vastly different than those members just a short couple of years ago. Conjecture seems to allow for the drift in an eternal gospel of divine principles.

There are assumptions made about the validity of claims the church or its leaders make. There is an absolutism among TBM’s that gives a tremendous amount of grace and reverence to what the prophets, apostles or other leaders say and many times critical thinking is set aside under assumptions that the church’s leaders are simply honest, correct and guided by the influence of the Holy Ghost.

Many members I still encounter are under an umbrella of ignorance which I’m still struggling to get out from under. There is so much about church history and past doctrines which members simply don’t know about, or only have a passing understanding about.

I was once taught that the gospel was beautifully simple and simply beautiful. As I engage with current apologists more and more, I’m noticing that many of them are beginning to go down paths of complicated explanations which I get lost in and I usually come away with more questions when things wind down.

Let's suppose that Scion's never sent on his rampage, he just keeps drifting about doing good deeds.

The Endbringers are manifestations of Eidolon's power, and his power is running out at an insane rate. So at some point in the probably fairly near future, the Endbringers will run out of power and the whole slow motion apocalypse that Earth Bet is suffering will just... end. Not before wreaking a lot more damage, of course, possibly enough to wreck human civilization anyway, but they will run dry at some point. How long might that take, and what state do you think human civilization will be in by that time?

TL;DR Warfare is the last content update(s) for SE, as Keen will switch focus to a new game.

I just want to get out of the way first that this post is entirely hypothetical. I have no insider sources that confirm what I am about to suggest. Keep this in mind if you are already angrily typing up a reply about how Keen's abandoning yet another game.

We all know what happened to Medieval Engineers. A big update was promised, it kind of fizzled out, there were expectations of continued support which did not happen, etc etc. ME was abandoned, to put it kindly.

I honestly believe that Keen has learned their lesson on that fiasco.

I found that Keen registered a trademark for Roman Engineers, located here on the European IPO website: https://euipo.europa.eu/eSearch/#details/trademarks/018411268.

Yesterday there was a Devs Lost in Space livestream, which was a Modders roundtable on Warfare 2 and the coding changes being enacted. You should watch it, there is a lot of good stuff coming in the Warfare 2 update.

It got me thinking. It sounds (if you take the livestream at face value) like Keen has gone out of their way to include some of the top modders in their decisions for Warfare 2 and Space Engineers in general going forward.

While I am not sure of the exact numbers, but Keen has about 100 people (or less) working for them, and I can only assume that a largish portion of that are working on Space Engineers. I mean, there have to be administrative assistants, coordinators, social media people and so on, so, I am guessing 50 or so people working on active development for SE. I am simply pulling numbers out of my behind, but I don't think I am far wrong.

If they are going to work on a new game, which, despite what Kienata asserted in the livestream, I think Keen is gearing up to do. They've registered a trademark for Roman Engineers, which I think is the game which will be the first to use a new version of their VRAGE engine.

If they are going to switch focus to developing a new engine, and a new game, they are going to have to either hire about 30-ish new people, which would obviously be noticed and we'd know what was going on... or they are going to switch support from SE to this new project.

Now, if as I suggest, Keen has learned from what happened with ME, Keen is not going to cast SE adrift. It does, if the install numbers from various Steam watching sites suggest, make a decent amount of money. They al

It's simple, really.

We have:

-5=>-14=>-7=>-20=>-10=>-5

So we have a cycle starting from -5. QED.

I want to start this post by emphasizing that I am not claiming a proof of the conjecture, I am just an undergraduate and solving such a long standing problem is hopelessly unlikely. The purpose of this post is share with you some thoughts I had regarding this problem, and learn from you why these thoughts don't make up a complete proof.

I was reading the wikipedia article regarding the conjecture when I found this paper cited in it: https://arxiv.org/pdf/1708.01434.pdf. I do not have enough expertise to understand everything in it, but my surface understanding (maybe misunderstanding?) of the result is that it implies the following: given a sufficiently large constant k, we can find an integer k'>k such that any family of size k' satisfies the union-closed sets conjecture.

This sparked an idea: Forward-Backward induction.

Assuming every family of size n+1 satisfies the conjecture, let F be a family of size n. Let x be an integer that doesn't belong to any set in F, and A the union of all sets of F (which btw is a set in F), the family F' = F ∪ (A ∪ {x}) is a union-closed set of size n+1, so it satisfies the conjecture. The exists an element a such that at least half the sets of F' include it, that is (n+1)/2 - 1 = (n-1)/2 of the sets of F once we remove A ∪ {x}. If n is even, we are done. Else, look at the sets of minimum cardinality, if all such sets contain a, then all sets of F contain a, and we are done, else there exists such a set not containing a, in this case, remove that set from F, we get F'' another union-closed family (thanks to the minimum cardinality condition), F'' is a family with n-1 sets at least (n-1)/2 of them containing a, so it satisfies the condition. If we do that to every family of size n, we get that every family of size n-1 satisfies the conjecture, take any set of minimum cardinality and containing a, remove it from F, we get a union-closed family F''' of size n-1, it satisfies the conjecture, thus adding that set back, we deduce that F has at least (n+1)/2 sets containing a. The Backward step in the induction is complete, combined with the forward step implied by the paper, and the initial case provided by the family of one set, we are done.

Thank you for taking your time to read my post!

I'm pretty new to WoT (about 250 pages into The Shadow Rising), and fantasy for that matter, But one of the things I've enjoyed the most about the Comere since I've caught up with it is the conjecture. The questions people ask Brando Sando and the theories I see on the various Sanderson subreddits/17th shard are incredibly exciting to me. I feel that one of the things that I'm missing from WoT is the conjecture.

When there's a Min reading, or a dip into the various prophecies, or even the rules of the magic systems/dream world, that's the stuff that I'm unable to engage with very well, but that I love hearing what others think, which often gets my brain moving and lets me start to create theories.

The problem is of course, that I'm incredibly wary of delving into anything because of spoilers and a lack of understanding when it comes to what will appear in the books. For example, I've heard multiple times about how the world of WoT is ours in the future, and for a while I felt like I'd been spoiled, but at this point it seems more like something that Robert Jordan never officially canonized in the books, rather he hinted at it throughout (Mercedes logo, Rand discovering the strange names like Arthur at the end of The Great Hunt). Which leads me to believe that there are a lot of things which I might not even recognize as being or, more importantly, not being spoilers.

So are there any resources to attempt to engage with the discourses that happened while books were releasing? Cause if I'm able to spend a week in between books catching up on the biggest Word of Brandon equivalents and conjecture, I feel it would have me even more enthralled with the series than I already am.

Let’s discuss.

continued from here ty everyone whos counted the past month and /u/TheNitromeFan for the assist

The rule is that if you have an odd number, multiply 3 and add 1, and if it's even, divide by 2, until you hit 1.

The format is

[current number] ([starting number] | [steps])

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The next get is at 632 (632 - 0). Schedule

At the start of the video, it is claimed that Gödels theorem implies that problems may not be provable: namely, the Twin Prime Conjecture.

This is incorrect, though, right?

I just came out of Computer Science Theory this semester, and individual study has led be down the rabbit hole of Gödels incompleteness and I’m trying to wrap my mind around the concept of the formal system. My understanding is not what is expressed in their video.

Claim: Gödels incompleteness implies that the twin prime conjecture may be unsolvable.

My claim(?): Gödel’s incompleteness implies that the twin prime conjecture may not be solvable with the formal system (horrifically incorrectly) described a sort of union (z, +) U (z, *) groups. Assuming this arithmetic system with associativity, commutativity, etc; equipped with +, ..., can not prove the twin prime conjecture, this does not mean set theory can’t do it.

Now, this is assuming that using set theory or something for proving things about integers is not fundamentally equivalent to the classical arithmetic formal system. Is the twin prime conjecture only an artifact of this one formal system?

Conjectures which have very large counterexamples like the one with Polya Conjecture.

I would like to know about some other conjectures...

Below is an analytical proof of the Collatz conjecture. The conjecture is proven true for all positive integers.

I got new ideas while I was watching (for the 3rd time) the movie "Proof" where Gwyneth's character proves some theorem using new methods she had devised.

I realized I should devise a new method, at the same time preserving, for as long as possible, the only symmetry I could see; that is every other number is a single divider: when Collatz transform is applied to it, the resulting even number is divisible by 2 only once.

Collatz transform = take an odd number n. Calculate 3n+1, an even number. Divide the number by 2 one or more times until you get an odd number.

1. Let's consider a set of odd numbers 2n+1, n=0,1,2,3....

   1,3,5,7,9,11,13,15,17,19...

   We can subdivide it into 2 subsets:

   A. a subset of single dividers, or numbers divisible by 2 only once upon using the Collatz transform. Their format is 4n+3. Example:

   3,7,11,15,19,23,27,31,35,39,43... and

   B. a subset of multiple dividers, or numbers divisible by 2 two or more times, format 4n+1. Example:

   1,5,9,13,17,21,25,29,33,37,41,45...

2. 4n+1 numbers (multiple dividers) convert to 1 or 4n+3 numbers (single dividers) when a Collatz transform is applied (one or several times), so only 4n+3 numbers have to be proved.

3. The Collatz transform is applied to 4n+3 numbers only. This yields a mix of single and multiple dividers. Example:

   3, 7,11,15,19,23,27,31,35,39,43,47,51,55,59... after a Collatz transform turn into

   5,11,17,23,29,35,41,47,53,59,65,71,77,83...

   Multiple dividers are removed because we handled them in step 2. This yields the format 12n+11. Example:

   5, 11,17,23,29,35,41,47, 53, 59, 65, 71,77,83... after removing multiple dividers turn into

   11,23,35,47,59,71,83,95,107,119,131,143...

4. Another Collatz transform is applied. Example:

   11,23,35,47,59, 71, 83, 95,107,119,131,143... after a Collatz transform turn into

   (17),35,(53),71,(89),107,(125),143,(161),179,(197),215... Multiple dividers are enclosed in parentheses.

   The multiple dividers removed in step 4. are: 17,53,89,125,161,197,233,269,305,341,377,413,449,485,521,557,593,629,665,701,737,773,809,845,881,917,953,989,1025... Their format is 36n+17.

   All these numbers have the format 18n+17.

   Multiple dividers have the format 36n+17, or 4(9n+4)+1.

   Single dividers have the f

In the trailer it says "The attacks were all on people related to the original killers". That doesn't necessarily mean that all the victims in this are related to previous killers. But I would imagine that at least some of the cast are.

Vince in punk clothes with his long dark hair slicked back sure looks similar to Billy. And something about Wes' hair and clothing reads very similar to Stu to me.

So who do we think are related to whom?

Yesterday, the ongoing distributed computing project for Convergence verification of the Collatz problem by David Barina (u/lord_dabler) verified the convergence of all numbers below 2^(69). This is the same project that did it till 2^(68) last year.

The verification to numbers from 2^(68) to 2^(69) which is 2^(68) numbers took 19 months of computation which means more that 5.85 trillion (2^(42.4)) numbers per second.

LINKS:
Track the progress: collatz-problem.org

Path records: collatz-problem.org/table

Source code: github.com/xbarin02/collatz

Original paper: rdcu.be/b5nn1

Has anyone tried testing this yet? Just wondering.

Why is meme so important?

In Keith Stanovich's book "Robot Rebellion" has this statement.

>Man is a robot of genes and meme, genes and meme are the masters, man is the machine, the carrier. And what is needed for a robot rebellion is reason.

In China, there is also a psychologist named Yang Zhiping who expresses it like this.

>Man will shift from gene-guided evolution to evolution guided by memes

Of course, this phenomenon can be seen everywhere in our society nowadays, and this view has been authenticated time and again in these phenomena, such as religions, political parties, and various corporate cultures with considerable influence, and to a lesser extent, various social media netizens, all of which can be categorized as modalities, and it is they who have turned themselves into a meme leading their followers towards unknown distances.

Those who believe in science explore the universe and interstellar travel, those who believe in the idea of political parties build an ideal society with various experiments of trial and error, those who believe in net celebrities imitate their lifestyles, and those who believe in celebrities pay for their works ......

The origin of R.I.P

"Anima eius et animae omnium fidelium defunctorum per Dei misericordiam requiescat in pace."

Rest in peace (RIP), is a phrase from the Latin requiescat in pace, Church Latin. It is sometimes used in traditional Christian services and prayers, as in Catholic, Lutheran, Anglican, and Methodist denominations, in the hope that the souls of the dead will find eternal rest and peace.

It became ubiquitous on tombstones in the 18th century and is widely used today when referring to someone's death, regardless of religion.

So after so many years of development and precipitation, what R.I.P represents is actually what is common to all human beings revealed.

Love, peace and hope.

Understanding of life and death

Religious beliefs can make people act crazy and make people better, different people will have different understandings and perceptions of the same content, as the saying goes, a thousand readers have a thousand Hamlets.

But the same thing is that life and death will be explained

People in the West always approach the issue of life and death from an individual perspective, while people in the East mostly approach it from a group perspective. Westerners have a strong sense of individuality, but of course they also have a sense of community, but this communit

En Español:

Uno de los problemas matemáticos tratados como insoluble, lo he resuelto con un método considerado básico por la comunidad, obviamente, no mostró el resultado sin tener testigos de que la solución la encontré yo.

Lo único que voy a revelar es que la hipótesis de Terry Tao es correcta, los posibles resultados de estos son ciclos positivos o negativos.

In English:

One of the mathematical problems treated as insoluble, I have solved it with a method considered basic by the community, obviously, it did not show the result without having witnesses that the solution was found by me.

The only thing I'm going to reveal is that Terry Tao's hypothesis is correct, the possible outcomes of these are positive or negative cycles.

https://phys.org/news/2021-11-riemann-conjecture-unveiled-physics.html

" A mystery of mathematics that has remained unsolved for more than 150 years can be unraveled thanks to a completely unexpected approach coming from statistical physics. "

I am a math enthusiast but I am not even remotely qualified enough to realize whether this piece of news is complete bullshit as usual, or there is some substance to it. Perhaps someone can help to clarify?

Continued from here. Thanks to /u/atomicimploder for the finish!

The next get is at 650 (650 | 0).

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