Weird Shit with Numbers

[valtsv]

basically the best thing any character can do is decide they don't want to be afraid anymore - in fact they never want to be afraid of anything ever again - and take action so drastic they fail to realise that this too is a decision motivated by fear. or to account for the Consequences of that.

[valtsv]

[with obvious perverted intent] hey. don't you want to release the safety catches on that character. don't you want to flip off all the switches holding them back and let the control rods go.

[itsalwaysdark]

you have to be careful reading too many things that are good/smart/well-written bc then you encounter something that isnt and you get confused like ? why didnt they just make this good ? were they stupid

[weirdshitwithnumbers]

Webfiction that begs — begs — for an editor. Like I can see the potential! You just completely flubbed the delivery! Go back and edit some prior chapters, position a revelation at a more opportune moment, add a chapter on that critically important moment of character development instead of skipping it!

[autistobrat]

scientists are experimenting on cross-breeding a crab and a cheetah; things could go sideways real fast

[necarion]

They should just bite the bullet and make a female James Bond. Hot, athletic, suave. She wears tuxedos with a somewhat feminine cut, drinks vodka martinis, drives sports cars, and goes by "James", because why not.

Also, because this is incredibly important to Bond for some reason, she needs to be an incredibly predatory, womanizing lesbian. Some perfectly happy married straight woman needs to become gay by the end of the movie.

We live in the future, and we can admit that all of the cool things that a Male James Bond can do are things a Female James Bond can do. But at all costs, we need to avoid making this thing feel "woke" of self-aware. If Female Bond is not exactly as toxic and awesome as any of the male ones, we will have failed, and might as well be making another franchise.

[pico42]

Lilly's interesting general topology facts

I have recently learned that it is quite common for product topologies to inherit properties which are possessed by each of its coordinate spaces. Hausdorff-ness is a good example; the inverse of the projections used in constructing the product topology can be easily shown to preserve Hausdorff-ness (in that any product of Hausdorff spaces is itself Hausdorff).

However, does this apply to other topological invariants, or just to separation axioms?

[ilovecoelacanths]

went to a new optometrist today wearing my squid facts ‘save our freaks’ shirt from @sarahmackattack that has a strawberry squid on it. and i wasn’t even thinking about it but the optometrist walked in and he was like ‘oh what does your shirt say’ so i showed him and he was like ‘oh that’s neat!’ and then i thought he might like to know about strawberry squid eyes since they have weird eyes and he is an optometrist and all. so i was like ‘yeah it’s actually a real kind of squid called a strawberry squid, their eyes are really cool because they have one big yellow-green one and one small blue one’ and he kind of gasped and went ‘oh my god that’s so interesting i wonder why they have that. do you know what their retina composition is like?’ and i watched as he minimized my chart on the computer and started looking up images of strawberry squid and then he googled ‘strawberry squid retina composition’ and he was like ‘sorry we’ll get to your eye exam in a moment i just really want to find out’ LMAO 10/10 optometrist experience will be returning

[sarahmackattack]

Hell yeah

[shock-micro]

don't cry because it's over. attack. attack everything around you. do spin attacks

[sqrt-73]

Interesting math fact of the day #503:

The expected distance from a randomly selected point in a unit equilateral triangle to a fixed vertex is (4 + 3ln3)/12

[dycefic]

The Statues That Do Not Weather

For Sas, another story about helping.

#

There is a statue on the cliffs overlooking the harbour, of a man shading his eyes with one hand and looking out over the sea.

They say that when invaders came, a man went up to the cliffs, and prayed to the gods. He offered them his own life to save his people. The gods accepted his sacrifice, and a great fire burned across the water, sinking all the ships. The man became stone, and ever since then he has stood on the cliffs, looking out at ships that sank long ago.

There is a statue that stands in the center of the town, of an old woman with both hands held up before her, palm out.

They say that when invaders came again, a woman stood in the middle of the square, and ordered them to halt. She reminded them of the great fire that sank the ships years before, and called on the gods to strike down any man who took one more step, though it cost her life. The gods accepted her sacrifice, and the invaders who stepped forward became water, running back down the hill towards the sea and soaking the boots of the men behind them. The survivors fled in fear, and the woman became stone, her feet set among the cobbles, her hands raised to stop invaders long gone.

There is a statue that stands by the road that runs past our village, of a young woman holding a basket.

They say that when brigands came upon the village in the teeth of a hard winter, starving and desperate, a woman saw them coming and offered them the food in her basket. They mocked her, saying that so little would not feed them for a day. She, too, called on the gods, and she, too, was answered. She made a bargain with their leader, that every man would turn back when he had all the food he could carry. From that one basket, she filled every bandit’s hands and sacks with food until he could carry no more. When she had filled even the leader’s hands, she bowed her head and became stone, her basket empty at last. The bandits kept to their bargain, and never troubled the village again.

We all know these stories. We all know why those people became stone, stone that does not weather.

Keep reading

[compositionalotterfolk]

Have you ever wondered what a "half composition" of a function would be?

If we say a function "f" is composed with itself "k" times, we can intuitively interpret that as f∘f∘...k times (ie, the kth composition of f), so long as k is an integer; but what about when k isn't an integer?

We could interpret the "a/b th composition of f," as some function "g", such that, when g is composed "b" times with itself, it is equal to f composed "a" times with itself. While this interpretation is sound it still begs a few questions: is there an intuitive way to think of "real" compositions, what about complex?

These questions are interesting, however, I feel they are dwarfed by one you could be considering right now:

If we have some smooth function f, can we find functions that are these "exotic composites" of f? Can we do so for any smooth function?

Recommended preliminaries:

A good understanding of taylor expansions is very necessary.

Familiarity with ODEs would be helpful, but they are not used directly.

I had initially planned to include alt text for formulas but after much trial and error, I have given up. If anyone finds this interesting but was unable to engage with the content properly due to the lack of alt text I would be (genuinely) happy to converse about the subject via dm.

We will be breaking this into two steps:

Determining the a first order approximation of the desired function.

Finding higher order derivatives of such a function function.

We will be using the following notation to denote multiple compositions, as well as preserving the following homomorphism:

Theorem 1:

given a smooth function f:A->A, that fixes for some value x_f in A (ie, f(x_f)=x_f), then:

proof: If we repeatedly apply chain rule we can form the following product:

Clearly, this product can't be made to work for all values of s naturally, as that would inherently require some non-integer composition, but what is well understood is extending the product of a constant.

If we choose to evaluate this at x=x_f a peculiarity arises. Compositions of fixed points do not change, that is f(f(x_f))=f(x_f)=x_f. This allows the product to be simplified as follows:

we can then allow s to vary beyond integer values. QED

Now we could generate an approximation of our desired function nearby the fix point, however, this is fairly pointless.

one could notice that by repeatedly differentiating the initial product, you can actually continue to get higher order approximations, by applying the definition of the geometric series whenever nessesary. However, this endeavor is Extremely computationally intense; and no obvious pattern immediately discern itself (at least for me, I would invite anyone with blatant disregard for their own mental well being to try to find one directly).

Instead, we will be constructing an alternate method of improving our approximation.

Theorem 2:

For a given smooth function f:A->A, with some fix point x_f, and for some integer m>0, the following holds:

where B_{m,k} denote bell polynomial coefficients. (Just to be clear, the "mathcal B" defined for shorthand, it is not related to the statement directly.)

proof: Fa di Bruno's theorem is a generalization of the chain rule to higher order derivatives. It states the following:

where B_{m,k} are bell polynomial coefficients with the derivatives of g as "inputs".

However, rather than let this define the mth derivative of a composition of different functions, we instead use it to take the mth derivative of the s+1th composition of f with itself:

Note that the final term of this sum contains the mth derivative of the sth composition of f, making this expression recursively defined. We rectify this by subtracting this problem term from both sides (also, we begin to evaluate at the fix point, as it allows some slight clean up of the expression).

While this does get all unknowns on the same side of the expression, this is still unsolved.

To begin to solve this we will attempt to factor the "left side." There is a linear operator T_s called the "forward operator" defined so that T_sF(s)=F(s+1). We will simply take it for granted that this operator commutes with differential operators, but there are many proofs of this fact. This allows us to modify the above to a more neat form (the "right side" is unchanged):

our motivation for this simplification will be apparent momentarily. First, we will be applying the inverse of T_s to both sides (ie, the "backwards operator" T_s^-1F(s)=F(s-1)). We can directly apply it to the right by replacing all s with s-1.

(Here we use a slight abuse of notation, as the "1" on the left should in reality be the identity operator "I," however it is not of consequence.)

Note that the expression on the "left" consists of a linear operator L=(1-[f(x_f)]^mT_s^-1) next to an expression. Now, if we where to naively attempt to invert this operator we would have the reciprocal expression in the form 1/(1-L). This is where we use the definition of the geometric series.

We say this series approaches the desired operator so long as it converges (the convergence condition is noted below it). If this is unfamiliar, I would recommend really trying to grasp it (it's a rather interesting concept; try to apply this to (1-L)g(s) and see what happens I would also recommend replacing the main in the sum with L^l).

If we apply this inverse operator to both sides we can reach a formula explicitly for the derivative, assuming it converges.

Since m is finite we can exchange these sums without issue. I'm going to avoid specifics, but we can arrange to reach the final formula. introducing the shorthand.

Substitute the expression on the left side into the convergence condition found before to reach the above convergence condition. note that for any chosen m this expression only references derivatives less than m, which allows us to recursively apply this without issue.

To reach the expression in the beginning simply replace m with m+1 in this formula. QED

This formula allows us to find all derivatives of the desired function so long as the expression converges and the function fixes. The convergence condition is a little rough to determine, however from what I've found it is sufficient that |f'(x_f)|<1 (that proof was more arduous than the one above, so I have omitted it, more for my sake than yours); which is rather restrictive, however, if this condition is not fulfilled, you may simply invert the function on a nearby interval and the derivative must necessarily have a reciprocal magnitude by the inverse function theorem; leaving out only the case where |f'(x_f)|=1

Using theorems 1 and 2, we can obviously create a taylor expansion. Letting the following hold (yes I'm restating the short hand again; sue me):

"Most" differentiable functions defined on f:C->C, have at least one fix point, so as long as you aren't "unlucky" this should work.

A few notes I think are important:

is this the best way to do this?

Probably not. I was looking at math overflow for information, and I heard vague talk about a professor that had solutions to this very problem that worked for |f'(x_f)|=1, but I couldn't find anything substantial at the time. From the discussion I saw I was getting the vibe it was never published so idk.

I couldn't even begin to consider how my formula fairs in terms of convergence speed.

2. Is this even right?

Uhhh, I think; but idk I guess. I don't see much why it isn't correct, and like, I did just go over the proof, so if you think I'm wrong just like... idk what to say (⊙ _ ⊙ ) sry. The formula is disastrously hard to test so I just haven't tested it; if anyone wants to test it feel free. Just know if it isn't working you're actually wrong and just need more terms (ᵕ • ᴗ •) yeah, yeah...

If you do find out I'm wrong I actually would like to know.

3. How long did it take to find this solution?

... I don't wanna talk about that... then why did I bring it up??

[weirdshitwithnumbers]

One issue here I can see is that you assume that if you have a formula for f*s(x_f) and its derivatives at integer values of s that can be extended to the reals, that it does extend to the reals. Unless you prove continuity with respect to s the non integer values could be unrelated to the integer values. Assuming you do (you probably can), you need to prove that f*s(x_f) is analytic as a function of s (you cannot prove this by the formula for integer values being analytic when extended to reals) in order to say that the analytic continuation of the integer values is equal to the true function.

You probably can do so, but you’re missing important details.

[cloudsandstormsanddust]

My dash did a thing

[weirdshitwithnumbers]

Posts in the genre of the first can kinda get it wrong sometimes, but this one I like. “Something we were not designed to understand”.

[fudamnshi]

”time heals all wounds” WRONG. it merely allows for infection.. it is Too late for me

[goldensunset]

*guy stuck in a timeloop voice* well the years stop coming and they don't start coming

[official-time-loop-posts]

Official Time Loop Post

[phantomrose96]

Gosh I’d love to give it a try. It’s just so hard to find the time with all the *describes a daily schedule that is 65% dilly-dallying*

[succubusboyfriend]

are you “adaptable” or are you just willing to subject yourself to existing in low key background-level ambient misery

[inganikki]

First tweet: “I saw a piece of fabric? on the floor of my room so I went to pick it up but it was actually the moonlight that entered through the window lmao”

Quote RT: “1000 years ago this would have been a poem, not a tweet”

[fishing-princess]

Okay fuck me I guess