#Mobius inversion

[informatikerin-freyja]

I feel like its a real disappointment that Mobius inversion is yet to become a fixture of my life, both professionally and personally. Especially this weird subset convolution variant:

There has to be a good use for this. Not just in terms of deriving these exact exponential algorithms as a theoretical feat, but in terms of building practical real world solvers. I think with a naive implementation the exponential cost is too high for real world use, but surely there is a way to avoid paying that full cost at least for well structured real world problem instances.

[nostalgia-tblr]

The Odin-v-Lauffey Discourse and the Mobius Discourse are in some ways the same in that they both boil down to "this character did some obviously Bad things, therefore I am convinced that they can only do Bad things, and if they appear to have done something Not Bad then it's because everyone else has interpreted it wrong."

[iratusmus]

honestly its a real shame we didnt get to explore moebius as a concept more because it has like an incredible amount of potential to be a really interesting deconstruction of the "alternate universe where all the moralities are FLIPPED!!!" type of stories. like i was talking about it with @/goosewhisker and she brought up a bunch of super interesting points like. okay see the main question that most of this springs off of is... what, exactly, is the relationship between mobius and moebius.

i think its really worthwhile to go ahead and put in this question ian answered on his now-defunct forum that has frankly haunted me ever since i first found it. i've transcripted it below for easier reading.

"Question #4 wrote: When we talk about anti-Mobius, is this an evil version of Mobius really? They seem flipped around, but Knuckles, for example, is a good guy really in either dimension, and anti-Sonic… well, there's some potent foreshadowing in #172. But if anti-Mobius isn't an evil Mobius, in what sense is it the anti-Mobius?" In terms of Anti-Knuckles, at the time he was the antithesis. Knuckles at that point was still a nebulous hero - distrusting, hermit-like, and ultimately heroic in that he helped against the greater evil. Anti-Knuckles also was very open and telling about himself and his world, the exact opposite of Knuckles who sequestered himself away and generally kept everything to himself. Our Knuckles has grown to be more open, trusting, helpful, and truly heroic. One would assume the Anti-Knuckles wouldn't be nearly as approachable nowadays. I think calling the Antiverse the "evil Mobius" is too easy. Originally that's exactly what it was, but as its slowly built up over the years there's more to it. Mobius is a world under one dark cloud as the Eggman Empire seeks to conquer everything. Despite this there are pockets of hope, of heroism, and selflessness that fight back. On Anti-Mobius, the one beacon of hope is Dr. Kintobor's hospital and what he's managed to rescue in the Great Forest. The world is in a perpetual state of chaos as pockets of selfishness, ruthlessness, and misery sustain each other. Mobius is an example of hope enduring despite all odds. Anti-Mobius is a sad look at inevitable doom. I'll have much more to say on this later on.

so as ian himself says, moebius is no longer just the "evil mobius", but at the same time it can't really escape the shackles of being the inverse of prime mobius (see: knuckles vs anti-knuckles). so is this connection one way? do the inhabitants of moebius truly have any free will, or are they simply bound to be the opposite of their mainline counterparts?

i think scourge is definitely an interesting way of exploring this concept. as a character he's concepted on the idea of somebody who hates the idea of being considered a derivative of somebody else. he's not his own person - he's simply a discount version of the "real" him. his identity isn't his own. in his desperation to become anything that isnt sonic, ironically he ends up leaning into... well. still being a derivative of sonic. namely, what he is. anti-sonic. so even in his attempt to break free of the place that the universe has assigned him as a "sonic", he ultimately becomes exactly that.

does he really have free will, or is he simply confined to his lot in life? alternatively, is it that his natural inclinations/personality/choices are exactly that of his assigned place, which is how he ended up as the "anti-sonic" in the first place?

there's also another aspect to consider here - zonic says that scourge's "mutation" has made him "something of a wild card in the grand scheme of things". a line which is... never explained. but fascinates me nonetheless.

what is scourge's mutation, then? is it the master emerald incident, or is it that he rejected(?) his assigned place in the universe? if its the latter, then what does that mean for the rest of moebius, who have all followed in his footsteps (willing or not) to take on new, personal identities that are allegedly separate from their mobian counterparts?

buns' joining of kintobor is also definitely worth mentioning here - at the point that she switches sides, bunnie is still an unwavering freedom fighter. morally, both the mobian and moebian are on the same side. given that after the antoine incident, bunnie was going to join eggman's side to get legionized, what does that mean for the moebius/mobius connection?

is it possible that when scourge made them all take on new identities, he made the connection twofold? can changes in a moebian now affect their mobius counterpart? ironically, again, could scourge's attempt to break the ties between mobius and moebius only brought them closer together, now that both sides could potentially affect each other?

on a different meta note, i really would've liked to see more of moebius just because like... if mobius is a world where people are generally default "good", then is moebius a world where people are generally default "evil"? are they literally set up to fail because of the world they were created into?

frankly, going back to ian's commentary on moebius posted earlier... its no wonder that scourge (+ by extension the rest of the ss) ended up like that. the entire thing is so screwed from top to bottom and i wouldve killed to see it explored more.

[primewritessmut]

H, K, N, T, and W <3

H - What is your favorite source text for fandom stuff?

Oof. Um? Idk that I care so much about the source material as I do about the characters. I can make any source material work if I’m obsessed enough.

That being said, I like the vaguer boundaries (for lack of a better word) around movies and comics. There’s a lot more room to play with a characters internal life and thoughts bc those things aren’t as clear in a more visual medium like a movie or comic. Shaping that for an existing character is probably the closest I’ll ever get to writing an OC.

K - What character has your favorite development arc/the best development arc?

You know I’m going to say Loki bc you got me hooked in the show. And Mobius, too. I think they’re sort of inverse arcs. Loki finds his purpose and Mobius loses his, and it’s so well-done and tragic and fascinating.

Also, I read a veritable fuckton of books and I know I’ve read some amazing character arcs in those but they tend to blur together. The one that sticks out to me rn is Evemer in A Touch of Gold and Iron bc it’s a FANTASTIC representation of “enemies” to “I worship you, I would die for you, use me like an attack dog” and I know you know how much I love that.

N - Name three things you wish you saw more of in your main fandom (or a fandom of your choice).

(1) More deeply considered kink scenes. This applies to most fandoms I’m a part of, I think. You and I have talked about this a lot. I just… I get tired of reading kink relationships that are just some iteration of a submissive wet hole versus dominant fuck machine. Like yeah, okay, I read smut for the smut but also the relationship. Call me a romantic or whatever, I guess.

(2) Eddie grappling with the symbiote’s need to eat brains. I think I’ve mentioned this before, but I feel like it gets left out of fic a lot. For obvious reasons, probably. If you’re writing about a guy getting dicked down by his alien goo parasite maybe you don’t want to deal with the in-world consequences of all that entails. But I think it’s really rich soil to dig through and I wish more fics did it.

(3) Smut scenes that aren’t just penetrative sex. Not every likes or wants penetrative sex. Or has penetrative sex every time they fuck. What can I say? I like a variety.

T - Do you have any hard and fast headcanons that you will die defending?

That I would die for? Yeesh. I think Peter is a lot angrier than people let him be or give him credit for. Wade enjoys his job and would be sad (not to mention a massive fucking nuisance) if he gave it up. The body horror of both Venom and Deadpool are an integral part of their characters. (Damn. I should have put that in my above answer.)

But would I die for those? Maybe the body horror one. You know how much I love body horror.

W - A trope which you are virtually certain to hate in any fandom.

Teacher/student. (And usually boss/Secretary or employee.) I’m not going to get into why just, BLECH. No thank you. Next.

[pico-digital-studios]

Into, Across and Beyond! Cast: Nitro/Christopher Smith

Replaces: Insomniac Spider-Man

(Art by BroadwayBash123)

"Since I'm following in my hero's footsteps, I know there's still a lot to learn, even after I've gotten past my depression and fought off all sorts of multiversal-threatening creeps. But one thing's for sure; no matter what happens, I'll always have my friends there by my side!"

Nitro the Hedgehog/Christopher Smith is my own original character based within his own Sonic universe. However, his journey didn't solely start there.

In his youth, he was an actual human, just like you and me. He had a great life going for him; a great education, a loving family, fun friends to hang out with, and even a girlfriend! However, after he got separated from his family in a heartless attack, he ended up in Mobius, where he was found and taken in by Tails and Cosmo.

As the days went on, he aspired to be a hero of his own to help others where he couldn't help his family, which was helped with training from both Sonic and Honey. During this time, he also fell in love with Amy Rose, who reciprocated his feelings sometime after that universe's War for Mobius.

One monster that gave him the worst trouble, though, was Toxinfect, who terrorised Nitro at every opportunity and tried everything to make his life a living hell, even causing a two-month-long apocalypse with him as one of the remaining survivors.

He did manage to turn things around, though, when he discovered a special artefact known as the Sacred Ring, which was way more powerful than Toxinfect and reversed all the damage he did, therefore kickstarting the monster's downfall and descent into pure insanity.

As the next years went on, Nitro met other heroes from a variety of universes and made good friends with them. His siblings, Evo and Rosie, alongside his human girlfriend, Amelia Walten, and mother, Susan, even returned to him over that amount of time. He even gained the ability to switch between his hedgehog and human selves at any time following his first trip outside of Mobius in a while.

But during that time, Nitro was hiding the fact he was suffering from long-term depression, which he finally revealed when a smaller "version" of him from another universe pressed him about the matter. It took a lot of therapy and support for him to overcome it, but he was able to do so in time for a surprise birthday party his friends and family had prepared for him, which he was overjoyed about.

Come More than One Universe, and he, alongside his Errorverse companions (including those from Sonic, Freedom Planet, Undertale, FNF and DDLC), were brought in to help put a stop to the madness going on within the OMT universe.

During his time there, Nitro got to know more about the Blur Gang, even offering to teach CU!Sonia some tricks on how to save the day without getting her hurt. He even got to meet his D-Sides counterpart, Hefty the Armadillo. Alongside them, he helped with curing the high level of multiversal threats that were flocking in.

After everything was said and done, he joined the Blur Gang as its official twelfth member, and even outside of it, he, OMT!Tails and OMT!Mina have since been known to work in a three-member team of their own whenever they can.

I'd say it's a pretty ironic inversion of Insomniac, in a way. While Insomniac Spidey came from a video game and knows it, Nitro's self-awareness comes from being a part-human within the Sonic world.

He's also one of those to have made it through the Errorverse's rewrite with all of his memories intact, allowing him and Amelia to change things up when the times were right. And because of it, their universe became better than it once did, while they continued to lend their support to the Blur Gang as needed.

In terms of romantic relationships, he's got Amy, Amelia Walten and his version of Mina, though he's also found a preference for women older than him to actively avoid any squicky stuff, hence why he's comfortable being around ladies such as Milly in the stories.

Oh, yeah! You guys are welcome to ask me anything you'd like about Nitro, even if it's unrelated to Into, Across and Beyond!. Be it his childhood friends, about his family and other original character companions, etc.

[addysfandomdump]

I've mentioned my Inverse AU a couple times in the past but never went into depth about it. This post will change that.

What's the Inverse AU? It's basically an AU inspired by the anti-personas in Archie Sonic, like Scourge, Rosy the Rascal, etc., but in a different direction.

Basically, almost every part of a Sonic character is the opposite. Sonic, the fast, agile, snarky, rebellious Hero of Mobius, is now Static, the slow, stiff, stone-cold, authoritarian Dictator of Suibom.

More info about other characters under the cut 👇

Static, as mentioned before, is the Dictator of Suibom. She spends most of their time cooped up in her castle, ruling Suibom with an iron fist from the comfort of their bedroom. She is cold, cruel, and indifferent to the suffering of their people; selfish, only caring about her own comfort at the exploitation of others.

Ivy Plantkin, (-Ivo Robitnik) known to the underground as the Chicken Woman (-Eggman), is the leader of the Chicken Coup, a powerful army of nature-loving mages fighting to reclaim land from Static's rule. Ivy is the most brilliant of those mages, both kind and competent.

The Meteres Twins (kilo "meter"), Key and Loe ("ki-lo" meter), known collectively as the Heads (-Tails) are living proof that two heads are not always better than one. If anything they're even stupider when they're together. They're Static's personal guards, two weirdly tanky 8 year olds that Static finds sooo goddamn annoying.

Annie Thorn (-Amy Rose) is the estranged wife of Static. She was forced into a marriage with Static for purely transactional reasons. She fucking hates it.

Anklette (-Knuckles) is just Rouge the Bat in Knuckles' body. Sly, smooth, sociable, and selfish, with a penchant for stealing.

Shimmer (-Shadow) is the cheeky and charming host of "Shimmer's Spotlight," Suibom's number one TV show. He's also Suibom's number two assassin, who's down horrendous for Static and does pretty much all his missions for her. Born in the countryside to the Plantkin family, a terrible accident led him to discover a thirst for blood that hasn't been quenched since.

Tired of writing detailed character bios. Speed round! Rosary (-Rouge) is a modest, plainly-dressed nun with no interest in material possessions and hates lying. Iota* (-Omega) is a peaceful nature spirit who's basically Baymax. Blizzard (-Blaze) is a chaotic terrorist with ice powers. S.O.O.T. (-Silver) is a weapon of mass destruction buried 200 years ago, only to be reawakened again.

And of course. Small the Cat. A vicious predator with an 99.99% kill rate. He's Suibom's number one assassin. Toady is the one victim he's never been able to kill, preventing him from having a 100% kill rate.

*Not really satisfied with this name for Inverse!Omega. Prone to change.

[jonathankatwhatever]

I think I just found CM1, that it translates into the Mobius function as the 1 over the zeta series. You can see why this escaped me until now: it’s a double penetration in which the first is within the zeta series itself and then the zeta series is embedded within CM1.

This followed an interesting discussion. It’s 4 July 2024. I have been having large numbers of already seen experiences. Like in the bathroom. Like on the sidewalk. Those occur as incidentals, meaning to me they’re moments when I’m relaxed and open and what pops into view is future relaxed and open moments. So that’s how I remember meetings those dogs and relaxing into conversations with people I’ve never met.

That’s actually the basis of confidence, and of a confidence interval meaning chances over iterations, because already seen fits, in part, to the counts, the gsPrime counts of course because n-1 counts gsPrimes for the conception of n-1 to work. That makes this another example of reflection, which we can now see occurring over CM1 through this inversion.

This is truly beautiful work. So, I relaxed a bit, and I asked Sri for help seeing what he saw, the correspondences in the identity he found and which appears in the various Ramanujan sums, functions and in the conjecture. It isn’t all making sense yet, but it’s getting there.

Then it morphed into me as the mom and you as the dad of this baby. That came out of realizing this is a form of infancy. And I’m the mom and you’re the dad, which connects you to the ancient conception of how the spirit flows into the literal male.

———-

Have more to add but I need to eat.

Inherent in the conceptions, like in the Ramanujan function, is the layering and the definition of a Thing and End enabling and responding to different perspectives. That means we can define problem solving at a very low level, and we can then use that to explain why P does not equal NP in a nice manner: searching for the matching perspective means searching for a pairing, which is reflective, which is the concept of ‘over CM1’.

[worstloki]

WAIT WAIT WAIT SO WHEN TONY WAS MAKING THE MODEL THING FOR TIME TRAVEL HE TOLD JARVIS TO MAKE IT IN THE SHAPE OF A MOBIUS STRIP OR SOMETHING AND NOW A GUY NAMED MOBIUS. M. MOBIUS WORKS FOR THE TIME VARIANCE AUTHORITY WHICH IS LITERALLY A BUNCH OF PEOPLE RUNNING AROUND IN UGLY UNIFORM FIXING THE TIMELINE MXÆDÖDÞÆDÖSLDÆDÐSKDMDOÐFÖSMSÐSKDMDÐSÐÆSÞSÖSÐÐÆZÞSÖAÆSÞSÆ

yes that is correct

[pushussmollworld]

Inktober 2021 day 19: loop

[daggerlove]

So the other day I was curious about which MCU pairings and dynamics people (the wider, general audience) search up on Google the most. Some of the results are kinda interesting.

Note:

- I didn't put ship names because general audience doesn't care about ship names, or they simply aren’t aware of them. If they are interested in the characters and their dynamic, they simply google "x and y"  (I have to admit I am guilty of doing that as well and I consider myself being well inversed in fandom) - I also used 'worldwide' filter and customized time range depending on when that pairing started appearing in the MCU

I focused on:

1. Most searched canon romantic pairings per Google and Youtube 2. Most searched non canon pairings 3. Most searched Loki-centric pairings 4. Most searched new MCU pairings of 2021 5. Most searched ‘Loki’ show related pairings

So without further ado:

1. Most looked up canon romantic pairings per Google, or those who produced the most interest are by far Wanda and Vision, Loki and Sylvie and Peter and MJ. These three pairings are the most notable which is visible given that they managed to garner the highest spikes of interest

Wanda and Vision is a juggernaut thanks to their show.

Loki and Sylvie had by far the highest debut which is outstanding for a new ship. 

Peter and MJ have lots of strong potential so they better reunite them at some point.

Youtube however indicates that the pairing that managed to get the most attention at one point was Peter and MJ, followed by Loki and Sylvie and then Wanda and Vision. In terms of interest Jane and Thor got replaced by Peggy and Steve thanks to their Endgame dance.

2. Most looked up non-canon pairings per Google: as expected it's led by Steve and Bucky in terms of 'spikes' and overall longevity. I'm not even surprised that Sam and Bucky got the 2nd highest spike there followed by Kate and Yelena. I also added Loki and Strange for comparison.

Steve and Bucky are understandably quite a juggernaut when it comes to non-canon dynamics and are rightfully looked up on Google by audience a lot.

However, I wanted to see how this, arguably most popular non-canon pairing compares to one of the canon ones and the results are kinda surprising.

Pretty neat tbh. It really kind of puts things in perspective.

Meanwhile this is the same, but Youtube version which shows a bit different picture

according to Youtube clicks, Sam and Bucky generated the most clicks followed by Kate and Yelena.

Now to compare these non canon pairings to the canon one again:

3. The most looked up Loki-centric pairings. As expected the most looked up constant is Loki and Thor. Has there ever been any doubts it'd be otherwise? No. People clearly love the Asgardian bros.

However, Loki and Sylvie peaked the all-time highest, overtaking Loki and Thor.

Youtube shows a different story and Strange got replaced with Tony in terms of interest. Loki and Thor and Loki and Sylvie are tied for that #1 in terms of highest spike, followed by Loki and Tony, Mobius and then Wanda.

However if we remove Thor and Loki, the only more notable pairing here, is Loki and Sylvie.

and the same goes for Youtube:

4. The new pairings of 2021. 2021 gave us some interesting new dynamics and according to this, Loki and Sylvie were by far the most looked up *new* pairing of 2021, followed by Sersi and Ikaris, Yelena and Kate and then Druig and Makkari followed by Mobius and Loki.

and according to Youtube searches, Loki and Sylvie still lead, followed by Yelena and Kate, Ikaris and Sersi, Druig and Makkari and then Loki and Mobius.

5. Just Loki show related pairings and dynamics (note: I only included 4 because for any other pairing it was an instant 0) Loki and Sylvie lead by a large margin both in terms of spikes and longevity. Unfortunately Ravonna and Kang is still nearly nonexistent but I suppose that will change with season 2

And Youtube says pretty much the same thing.

Note: The reason why I'm not using fandom created ship names is for multiple reasons: 

- "x and y" shows more broader interest in the dynamic beyond stan interest  ---.General audience who watch the shows/movies and like a certain pairing and their dynamic will look up "x and y" because they don't care about ship names, nor are they aware of ship names. Like I said, I personally prefer to Goggle "x and y" over ship names tbh. I only use ship names to tag posts & sometimes use them in tweets bc of character limit.  - some of the pairings don't have fandom created ship names - some ship names you literally can't look up, like for example "pepperony" or "stony" without Google  completely misinterpreting what you mean - some ship names cannot be used because they are affected by certain other events. Like for example "Lokius". Lokius is both a soundtrack as well as the name of some random famous dude on TikTok. 

For example:

you wonder what caused that September huge spike for Lokius because honestly, as far as I'm aware nothing fandom-significant happened and then you look up and find out the spike was caused because a random TikTok star named "Lokius" got arrested, it made some news and it all makes sense then

And even the July spike was influenced by some other event:

it was also the release date of a popular audio equalizer.

How do we know these events influenced the ‘Lokius’ search query? Well, the dates match (July spike for the audio equalizer and September spike for that dude), as well as this:

If you look up ‘lokius’ the most related queries are for that audio thingy and the TikToker. And only then followed by fandom related stuff

So yeah, in short:

ship names are irrelevant for multiple reasons. The General audience doesn’t use them - it’s a stan thing. Some ship names cannot even be used for various reasons so if you want more accurate results regarding what dynamic the vast, general public is more interested in, using their separate names, aka  "x and y" is by more reliable. 

with that being said, this made me realize some things: 

- marvel would be dumb not to reunite Thor and Loki at some point considering how much the GA loves the brothers 

- Yelena and Kate have A TON of potential because the GA are interested in them 

- considering they are the pairing with the second highest peak of all time interest, have the highest debut and have a strong staying power Loki and Sylvie are far more popular than what fandom sometimes wants you to think 

- they are also by far the most searched dynamic of the show. In terms of interest Marvel really lucked out with them.

[informatikerin-freyja]

Multiplicative Functions

A multiplicative function is a complex-valued function defined for non-negative integers which satisfies a certain multiplicative identity. Many functions of number-theoretic interest are multiplicative functions, such as the function containing the number of divisors of a given number. In this post, we examine the algebra of multiplicative functions.

Convolutions of Multiplicative Functions

A function \(f : \mathbb{N} \to \mathbb{C}\) is multiplicative iff whenever \(n\) and \(m\) are numbers with \(\gcd(n,m) = 1\) it follows that \(f(nm) = f(n) f(m)\). In the case \(f(nm) = f(n) f(m)\) for arbitrary numbers \(n\) and \(m\), we say that \(f\) is completely multiplicative. Perhaps the simplest example of a completely multiplicative, and hence multiplicative, function is the constant unit function \(\mathbf{1}\) defined by \(\mathbf{1}(n) = 1\) for all \(n \in \mathbb{N}\). The Euler totient function \(\phi : \mathbb{N} \to \mathbb{N}\) is defined by setting \(\phi(n) = \#\{k \le n \mid \gcd(n,k) = 1\}\). Theorem: Euler's totient function is multiplicative. The Dirichlet convolution of functions \(f : \mathbb{N} \to \mathbb{C}\) and \(g : \mathbb{N} \to \mathbb{C}\) is the function \(f \star g : \mathbb{N} \to \mathbb{C}\) defined by setting

\[ (f \star g)(n) = \sum_{k|n} f(k) g(nk^{-1}). \]

Theorem: If \(f\) and \(g\) are multplicative functions then \(f \star g\) is a multiplicative function.

Proof: Suppose \(\gcd(n,m) = 1\). Whenever \(k_1 | n\) and \(k_2 | m\) it then follows that \(\gcd(k_1, k_2) = 1\). Hence, we can argue as follows:

$$\begin{align*} (f \star g)(nm) &= \sum_{k|nm} f(k) g\left(nmk^{-1}\right)\\\\ &= \sum_{k_1|n} \sum_{k_2|m} f(k_1 k_2) g\left( nmk_1^{-1}k_2^{-1} \right)\\\\ &= \sum_{k_1|n} \sum_{k_2|m} f(k_1)f(k_2) g(nk_1^{-1}) g(mk_2^{-1}) \\\\ &= \left( \sum_{k_1|n} f(k_1) g(nk_1^{-1}) \right) \left( \sum_{k_2|m}f(k_2) g(mk_2^{-1}) \right) \\\\ &= (f \star g)(n) (f \star g)(m). \end{align*}$$ Hence, it follows that \(f \star g\) is an arithmetic function.

QED. Corollary: If a function \(f : \mathbb{N} \to \mathbb{C}\) is multiplicative then the function \(F(n) := \sum_{k|n} f(k)\) is multiplicative. Proof: Observe that \(F = f \star \mathbf{1}\). QED. Corollary: The divisor function \(d : \mathbb{N} \to \mathbb{N}\) defined by setting \(d(n)\) to be the number of divisors of \(n\) is multiplicative. Proof: Observe that \(d = \mathbf{1} \star \mathbf{1}\). QED. The set of multiplicative functions equipped with convolution becomes a monoid whose identity element is given by the function \(\epsilon\) given by \(\epsilon(n) = 1\) if \(n = 1\) and \(\epsilon(n) = 0\) if \(n \not= 1\). Theorem: If \(f\) is a multiplicative function then \(\epsilon \star f = f\) and \(f \star \epsilon = f\). Proof: \[ (\epsilon \star f)(n) = \sum_{k|n} \epsilon(k) f(nk^{-1}) = f(n). \] \[ (f \star \epsilon)(n) = \sum_{k|n} f(k) \epsilon(nk^{-1}) = f(n). \] QED.

The Möbius function \(\mu\) is defined by setting \(\mu(n) = (-1)^k\) if there exist distinct primes \(p_1, p_2, \dots, p_k\) with \(n = p_1 p_2 \cdots p_k\), and \(\mu(n) = 0\) otherwise. Theorem: \(\mu \star \mathbf{1} = \epsilon.\)

Proof: In the case \(n = 1\) this follows immediately. Suppose \(n = p_1^{e_1} p_2^{e_2} \cdots p_r^{e_r} \ge 2\) where each of the primes \(p_i\) is distinct. Let \(P := \{p_1, p_2, \dots, p_r\}\). As \(\mu(k) = 0\) if \(k\) is not squarefree, we can argue as follows: $$\begin{align*} (\mu \star \mathbf{1})(n) &= \sum_{k|n} \mu(k) \mathbf{1}(n k^{-1}) \\\\ &= \sum_{k|n} \mu(k) \\\\ &= \sum_{J \subseteq P} \mu\left( \prod_{p \in J} p \right) \\\\ &= \sum_{k = 0}^{r} {r \choose k} (-1)^{k} \\\\ &= (1 - 1)^r = 0 = \epsilon(n). \end{align*}$$ Hence, \(\mu \star \mathbf{1} = \epsilon\). QED. Mobius Inversion Theorem: If \(g = \mathbf{1} \star f\) then \(f = \mu \star g\). Proof: Under the assumption \(g = \mathbf{1} \star f\) it follows that \(\mu \star g = \mu \star (\mathbf{1} \star f) = (\mu \star \mathbf{1}) \star f = \epsilon \star f = f\). QED. Multiplicative functions, other than the constant zero function, constitute a group under the Dirichlet convolution.

The von Mangoldt function \(\Lambda : \mathbb{N} \to \mathbb{C}\) is defined by setting \[ \Lambda(n) := \begin{cases} \log p &\text{if}\;n = p^k, \\\\ 0 &\text{otherwise}. \end{cases} \] Theorem: \((\mathbf{1} \star \Lambda)(n) = \log(n)\).

[nostalgia-tblr]

I think some of the loki fans that hate sylvie Is also because they wish they could kiss tom/loki

Is that reaching

i actually don't think it's that. i know "ur just jealous" is a go-to answer for these things but a) Loki remains fictional and so you can't kiss him anyway and b) most of the complaints I've seen tend to focus on her either taking screentime from Loki or being "abusive" to him and with both of those I can see why people would be annoyed. I don't agree with them but I can see where they're coming from. Well, much less so on the abusive thing but female characters being called all sorts of things because they did anything that might possibly upset Blorbo is hardly new is it?

I think some amount of it is fear of her displacing Loki from his rightful place as The Best Person Ever. People get SUPER MAD about Mobius calling her "the Superior Loki" even though he's just talking shit to get a reaction, because he's saying the exact thing they don't want to hear anyone say. All these years of suffering and woe and now our poor downtrodden prince isn't even allowed to be the best version of his own self?!

Also "ur just jealous" is basically impossible for the accused to engage with in any way. If you deny it that's just taken as PROOF of your envy and even if you manage to express an opinion or two in response you will inevitably just hit the wall of "JEALOUS!!!" again within about a paragraph.

This is just the inverse of "you only like Sylvie because you project yourself onto her so you can kiss Loki" isn't it? And nobody enjoys that take either.

Sorry, Anon, I disagree with your thesis :(

[twh-news]

Masterpost | 'Loki' series interviews with Tom Hiddleston

This post includes interview videos/audio/articles specifically with Tom.

You'll find all articles about the show and interviews with the creators and additional cast under the tag Loki Series.

VIDEOS

All videos available on Youtube are collected to this playlist. interviews available on other platforms marked with *.

Tom's Tumblr Answertime videos: @tomblr *

Playlist of interviews on Youtube Playlist includes all video interviews linked below (that are on YouTube). However, not all interviews on the playlist are listed in this post - so check the playlist first 😊

Tom on The Tonight Show with Jimmy Fallon | 12 July 2021

Tom Hiddleston breaks down his career, from ‘The Avengers’ to 'Loki’ | Vanity Fair

“Buckle up!” Tom Hiddleston on Loki’s death, return and… signature!? | BBC Radio 1

Tom Hiddleston Talks Marvel’s ‘Loki’ | MTV News

Ashish meets Loki

Tom Hiddleston Isn’t Sure If Mephisto Will Appear In Loki | MTV Movies

Tom Hiddleston’s Oral History of Loki, the MCU’s Villainous Trickster | Rotten Tomatoes

Many sides of Loki. Marvel Studio's Loki Cast and Creators | Marvel

Jimmy Kimmel Live: Tom Hiddleston on Loki Premiere, meeting Chris Hemsworth & Matt Damon stealing his part | Jimmy Kimmel Live | 8 June 2021

Tom Hiddleston talks about new Disney+ series, ‘Loki’ l Good Morning America

Tom Hiddleston is the God of Word Association | Marvel Studios’ Loki | Disney+ Hotstar Premium

Tom Hiddleston Delves Into the Character of “Loki” | E! Red Carpet & Award Shows

Tom Hiddleston was asked why he thinks people are such big fans of his #Loki. | Phase Zero *

‘Loki’: What made Tom Hiddleston laugh in this interview | Black Girl Nerds

Tom Hiddleston has not shown Owen Wilson his impression of him | ET Canada

Tom Hiddleston Reveals Loki's Final Thoughts in Avengers: Infinity War | ComicBook.com

Tom Hiddleston Talks LOKI Series, Fan Crushes and the MCU Moment He WISHES He Was In | Jake's takes

TOM HIDDLESTON on emailing Chris Hemsworth while making LOKI, Owen Wilson & ‘cameo theories’! | Tara Hitchcock

Tom Hiddleston On Why ‘Loki’ Is Always A Riddle To Solve But Fascinating To Watch | Black Film and TV

Tom Hiddleston at the 2021 MTV Movie & TV Award presenting an exclusive scene from Loki Series | MTV Movie & TV Awards *

Announcement of Loki's premier date change | Marvel Studios' Loki | Marvel

AUDIO:

Happy Sad Confused: Tom Hiddleston, Vol II | Happy Sad Confused

Episode #22: Loki 1x01 Breakdown; Tom Hiddleston, Kate Herron Interviews | Listen to on Apple Podcasts or on Spotify | Phase Zero

The Zoe Ball breakfast show - Tom Hiddleston 10 June 21

Still Watching: Loki - “Glorious Purpose” with Tom Hiddleston | Vanity Fair Podcast

ARTICLES:

‘Loki’s Tom Hiddleston Teases Marvel Series Finale, What It All Means & Is There More Of The God Of Mischief To Come? | Deadline | 13 July 2021

Loki's knives are out: Tom Hiddleston explains the trickster's affinity for daggers | Entertainment weekly | 8 July 2021

A Very Loki Affair – Our Chat With Tom Hiddleston | M2 Jul/Aug 2021 Issue | 4 July 2021 (article scans)

Loki: Tom Hiddleston on Episode 2's Surprise Character Reveal | Den of Geek | 17 June 2021

Tom Hiddleston on The Evolution of Loki: From Villain to Hero and Back | Den of Geek | 13 June 2021

We spoke to Tom Hiddleston about Loki, PowerPoint presentations and free will | Whatsnew2day | 10 June 2021

Tom Hiddleston on Loki’s “glorious purpose,” quest for power | AV Club | 9 June 2021

Tom Hiddleston on Mobius, Loki and queer identity | Vanity Fair | 9 June 2021

Is Loki genderfluid? "It's always been there," Tom Hiddleston tells Inverse | Inverse | 9 June 2021

Loki’s ‘vulnerability’ and ‘spontaneity’ make him relatable, says Tom Hiddleston | The Independent | 8 June 2021

Tom Hiddleston Teases That ‘Loki’ Will Explore the Meaning of His Iconic Horns | Murphy's Multiverse |

Tom Hiddleston, Owen Wilson, Kevin Feige & creative team talk Loki | The TVA, D.B. Cooper & the god of mischief | The Gate | 7 June 2021

Near Myth | Emmy Magazine (article scans)

‘Loki’ star Tom Hiddleston secretly watches Marvel movies as a fan in theaters | Emmy Magazine

Loki: Tom Hiddleston Reveals the MCU’s Future Is Predetermined By the TVA | The Direct | 25 May 2021

Loki: Tom Hiddleston Reveals Why Owen Wilson Is So Brilliant In His Marvel Role | The Direct | 25 May 2021

Loki takes over: Tom Hiddleston on his new TV series and a decade in the MCU | EW.com | 20 May 2021

Tom Hiddleston breaks down a decade of Loki's most memorable scenes | EW.com | 20 May 2021

The Mischief Maker | Empire Magazine (article scans)

[idw-sonic-fan-blog]

The Mandates

Just wanted to share my thoughts on the pro-ported mandates because they cast a shadow on this comic.

“Game characters cannot have relatives unless they were estabilished in the game canon, i.e. Cream and her mother.”

This one is understandable and you can blame Penders for this. Mind you that most licensed comics of gaming franchises don’t actually delve too much in personal family relationships or expand on them. So this is expected and honestly Sega should have put the screws on Archie decades ago about this.

“Game characters can not die. There are workarounds for this, such as being Mistaken for Dying or "Mistaken For Dead”

Again. Yes. Not a big deal.

“Game characters cannot have wardrobe changes unless approved. Chao Races and Badnik Bases has some characters (mainly the female game characters) wear different clothes for extreme conditions. Male characters remain the same.”

This is a useless rule but whatever. I mean Sega, you are the ones putting bad wardrobe choices on the characters so again it’s whatever.

“Sonic can't be shown getting too emotional (i.e;cry)”

This is one that it complained about because it really wouldn’t matter unless it is called attention to. A lot of superheroes don’t cry. But that doesn’t prohibit them from expressing themselves. IDW Sonic has been sad. He has been pissed. He has been furious.

Is this not too emotional?

Is he not expressing himself appropriately?

I don’t even know why this is brought up. When in this comic has Sonic not been expressive or displaying the appropriate amount of emotion? When did Sonic needing to cry be necessary?

“Game characters cannot enter in a relationship.”

Oh GOD YES. Don’t threaten me with a good time.

“All major Character Development must be approved by SEGA.”

Yeah, of course. Let me remind you that Penders and Archie ruined any strand of trust Sega could have in comic media. They played loose at first and all of the sudden, they are involved in a lawsuit about characters in a Sonic comic that they didn’t even know about. They probably lost a video game business relationship because of it. If they want to be involved in the comics, fine. That means that they are now forced to World Build. They have to invest in it now and not just be like Lucas Films and let anybody do anything with their flagship title.

“Much like the post-reboot of the Archie comic, the words "Mobius" is banned—the planet is simply called "Sonic's World". Unlike the Post-Boot, which allowed the names "Mobian" and "Mobini", anything related to Mobius is banned in this comic.”

…Of course but how about you throw the writer’s a bone and I don’t know, name the fucking planet. If it is not Earth, give it a name.

“Sonic must always win at the end. Even if he and his friends are at the losing end in an overarching story (the Metal Virus arc, for example), they must come out on top when it concludes.”

I don’t even get this rule and the knee jerk hatred for it. Why even have it? Why even share the existence of this rule? Archie Sonic didn’t really lose too bad. It’s more on how you frame a victory. The fact of the matter is that Eggman is still actively trying to conquer the planet. Sonic stops him but Eggman still has control of land and has military installations all over.

This rule is offset by this. While Sonic can’t lose, Sonic can’t completely win.

“Characters and material from other licensed properties (Sonic the Comic, Sonic the Hedgehog (Archie Comics), Adventures of Sonic the Hedgehog, Sonic the Hedgehog (SatAM)', Sonic Underground, the OVA, Sonic X and the Paramount films cannot be used. This rule extends to characters and redesigns done by the current writers. The only exception is Sticks from Sonic Boom, and that's because she was created by SEGA themselves and showed up in non-Boom media, but any ideas regarding her use still need to be okayed by SEGA.”

First off I am glad that Sticks was spared by this rule and I look forward to her eventual inclusion. Second, again, this is not much of a big deal as it was expected. Sorry Freedom Fighter fans but honestly deal.

“Male characters, sans Eggman, can't wear pants, which was also a thing in the Post-Reboot, albeit never explicitly stated. The inverse is also true; female characters have to have some form of lower clothing.”

Okay this is a pedantic rule. It is so weird with how precise it is. Like…huh?

“Classic characters such as Mighty, Ray, Nack/Fang, Bean, and Bark won't appear in non-Classic issues, as Sega doesn't want Classic and Modern Sonic to mix.”

One of the most bullshit mandates fueled by the nostalgia boner fans created. Like this is stupid because Archie Modern Sonic has added more character and depth to all of these mentioned characters than any of the Sega Sonic games they appeared in which only amounts to 1 or 2 at most. Why neuter your own potential stories with this stupid limitation?

“According to Ian Flynn, a specific incident involving Shadow's characterization when he's exposed to the Zombot infection was written in a specific way because of Sega mandating that he be written as an "overconfident asshole rival" character, similar to Vegeta. He later followed up with an explanation that out of every character, Shadow has the most mandates and notes attached to how he's portrayed. According to the podcast, Sega says that Team Dark is no longer a thing. The three members are not a team and they have never worked for G.U.N.; Shadow also doesn't even consider them friends.”

This is my opinion is the worst rule. First it’s contradictory to the character Sega introduced us to. Stop trying to be like Dragon Ball for once and actually be your own thing. It’s one thing if we are changing it because Shadow was unpopular because of his personality. But no one likes this Shadow. People miss the somber but reserved Hedgehog that continued to fight in spite of the world betraying him. Hothead Shadow is a cheap Knuckles. And I don’t even understand why Shadow even has so many mandates when he wasn’t the most egregious offender. Knuckles was.

Also, Team Dark aren’t a thing and Shadow doesn’t even consider them to be his friends. First off that doesn’t even fly in your own games. Who outside of Sonic does Shadow interact the most? Rouge. They have teamed up and were a packaged duo since their inception. When Shadow appeared, Rouge appeared right next to him. If Rouge was in a game, so was Shadow.

Team Dark or just Rouge has fought alongside Shadow in every game they appeared in. Who else does Shadow talk to if not Rouge?

“Sega has stated to Flynn that only male hedgehogs are allowed to go Super with the Chaos Emeralds.”

Except in Sonic Mania.

“Ian isn't allowed to directly reference a game, since the comic is supposed to be its own thing.”

Okay. Not only is this rule stupid. But it’s untrue.

This references the end of Sonic Forces.

The first page of comic.

It has referenced Sonic Adventure, SA2, Sonic Generations , and Sonic Unleashed.

This referencing Shadow the Hedgehog.

I don’t believe this rule exists and even if it did, it is dumbest rule since the whole point of this comic is to base it off the games more. The dumbest rule.

“Knuckles is not allowed to leave Angel Island unless he has a very good reason to.”

For decades, people have complained that Knuckles routinely leaves the island. For decades. Now does this mean Sega is going to 1. Use Knuckles and 2. Amplify the importance of Angel Island and the Master Emerald? No. Again, this criticism should be levied at Sega because they often conveniently forget Knuckles purpose and just hand wave it instead of giving Knuckles more to do on the island like I don’t know, have other entities invested in attacking him.

In summary, here is what I think is going on. Do I think most of these mandates are real? Yes. Given what happened to Archie, I do think Sega is doing some brand alignment. I think they got the clamps on.

But what I think is going on is a Japanese cultural thing called Power Harassment. It is normalized abuse of power. Sega of Japan is normally laxxed about their brands. They don’t mind blatant rip-offs of their mascot nor do they get stiff about fandom creations or mods. The comic division, however, is getting tough love because not only did it cost them a publishing deal, but ruined a relationship with a high end developer. So the IDW writers and staff are being subjected to intentionally hypocritical rules and strict mandates that they know don’t make sense until they’ve shown to be obedient.

A lot of the mandates aren’t strict. But some are so asinine that I don’t think they aren’t aware with how stupid they sound imposing those rules. Like Shadow is the most narratively complete Sonic character and yet, Sega puts this tight mandate as if Archie Shadow was the most egregious thing. Archie Shadow was overpowered. He wasn’t out of character like Sonic, Knuckles, and Tails were. They can’t be that stupid or be that intentionally dense. So they want to see if the writing crew can follow orders. That’s it.

But that’s just my take.

[chao-chachaslide]

Saying random shit that could/should/won't happen in the Sonic 2 film trailer (and actual film)

Either Knuckles is gonna be shown in the trailer several times, or he shows up right at the end and says one line.

Sonic narrates the events of the previous film in a 10-15 second intro

Tails shows up near the beginning of the trailer and either just rocks up at Sonic's house or Sonic finds him first. Somehow.

Or there's a been a timeskip from the end of the last film

Tails briefly explains why he was looking for Sonic

G.U.N is in at least one shot

"somehow robotnik has returned"

There are at least two references to Sonic 2 the game

There's a reference to the first Sonic movie trailer. Yeah, that one.

Robotnik does some cool shit (exist)

We get to see why they were filming in Hawaii

It's inverse Sonic movie 1: we're going to not-allowed-to-call-it-mobius instead of Earth

Tikal

Where the fuck do they get the Tornado from??

Bonus points if Amy shows up right at the end of the film or any reference to Shadow or SA2 is made

[addysfandomdump]

Do you have anything else from the reversed au? And or the Sonic Kryztallized au? If so can you share them?

FUCK I COMPLETELY FORGOT TO POST ABOUT THOSE.

Okay okay so I don't really have any art but!! I do have rambles.

So the Kryztallized AU is basically the AU where I rewrite so many aspects of the original franchise to my own personal tastes that it basically becomes a whole other universe. It doesn't really follow a set theme or main schtick, it's just a combination of my personal headcanons plus worldbuilding.

Reversed/Inverted AU does have a schtick, though— everyone is a complete inverse of themself, from their personality, to role, to the powers they have, even their origin story to flipped to reflect the opposite of their counterpart.

For example: Inverted Sonic is named Static, since unlike her speedy and energetic counterpart, this AU Sonic is slow and lazy. They are cold, calculating, cruel, and controlling, does not care for friendship, is only interested in others if it furthers her own agenda, and can be downright abusive to those close to them at her worst.

(Btw, both Sonics are transfem and use she/they pronouns, just wanted to clarify.)

They are the Dictator of Suibom (Inverted Mobius), ruling the planet with an iron fist. She loves the water, frequently seen floating around on private beaches and taking lots of hot baths. They wear lots of pants, but hates wearing shirts. As mentioned before, she is slow, lazy, and lethargic, and physically very weak. They rely on her army for defense and manpower. They are a master of manipulation, compensating for being a weak ass bitch physically by making her adversaries think they're incredibly strong, and are willing to do anything to ensure her reign and keep the status quo.

I have more info on other characters (Shimmer, Heads, Annie Thorn, Rosary, Miss Plantkin aka Chickenwoman, etc.) if you want to know more ^^ I just figured to put them in their own post to make it easier to organize.

Thank you so much for letting me ramble!! I really appreciate you asking about my AUs a lot!!