iiinova — where your digital self keeps evolving

Lilly Astar @nasaexploration.space · 12/02/2026

I am over New York City for a week and I hopped in the Stavros Niarchos Foundation Library to read some math books. I read a bit of « Advanced Calculus » by Widder (code 515W) and got a bit confused at some point so I added a few annotations of my own. If you go check them out, let me know :D

A photo of books in one of New York Public Libraries

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Lilly Astar @nasaexploration.space · 23/09/2025

This wolf needed a home, I gave him one… hope he’ll be happy ! Thanks @andre.moraph.art for the amazing merch :D

The André Moraph’s plush toy saying hi up close.

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Lilly Astar @nasaexploration.space · 27/12/2024

I saved some of @andre.moraph.art’s arts on my phone to show them to a good friend and my phone made a freaking video on it ! I found this so funny

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Lilly Astar @nasaexploration.space · 10/12/2024

Hey @andre.moraph.art out of curiosity, I have three questions for ya :D 1 - Is the text at 1:09:10 in the left corner intentional? 2- You switch between a beautiful script and cursive, is it your handwriting ? 3- At 4:24 in the vocal it's To linger in the what? Just need that to complete the sub :D

A small text appearing at 01:09:10 that last 12 frames

Mama look at that cursive writing. Even the script is amazing :D

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Lilly Astar @nasaexploration.space · 01/12/2024

@andre.moraph.art and @luiroi.bsky.social YOU BOTH GOT A PLACE AT FAUNTASTIC. Jeez 34 seconds I couldn’t even grab my phone TwT. Hopefully next week I can grab a residential pass…

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Lilly Astar @nasaexploration.space · 27/11/2024

:3c The Speckely Goober just arrived. Wonder if more of my French fellows received theirs :D

The Niko’s gang. Composed of a Niko plush, a Tetra plush, a turtle plush and a new addition of a Speckel !

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Lilly Astar @nasaexploration.space · 22/11/2024

Everyone it’s happening !!! It took about 10 days of transit to finally reach France ! I don’t know what took it so long when some French lads already have it it’s so weird ! It might come on my birthday that’s approaching very soon hehe

A plush of a silly YonK arriving to France

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Lilly Astar @nasaexploration.space · 16/11/2024

I think it was just the france side being completely incompetent. Stayed a week in the port of arrival if I understand the tracking correctly

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Lilly Astar @nasaexploration.space · 08/11/2024

I feel. Last year it wasn’t rare for me to do do 4 hours nap from 6 to 10 pm. I truly feel. You got this I am sure *pets* Here is my cat resting :D

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Lilly Astar @nasaexploration.space · 31/10/2024

I dwon’t seeeeee what ywourrr twalking about uwu furry talk 👉👈 (Yes I am bad at it but shushhhhh)

Groupie staring at you
Sticker from the popular Discord Server KennYon, a fan server around KennYoung and YonKaGor

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Lilly Astar @nasaexploration.space · 29/10/2024

Yesterday I saw a wild fox ! In the middle of the Pyrénées Mountains in France ! I know the pictures aren’t great but I was too busy yelling ‘Renard’ (Fox in French). They even sat for us for a few minutes. My life goal have been achieved !

A small Fox spotted in the Pyrénées Mountains.
Captured by Lilly Castella on the 28th of October 2024 at 7:23 PM

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Lilly Astar @nasaexploration.space · 28/10/2024

I saw two small kittens today that were found by an old man who posted an article to found them a new house. Sadly I couldn’t take one due to my personal situation but my sister’s friend did take one :D Small thought of @bremeows.bsky.social who call themselves silly kitten… much love Bre :3c

Two small kittens found by an old men who were offering to take them to a new a better place for them to grow

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Lilly Astar @nasaexploration.space · 23/10/2024

The red function with rectangles, anthony-mansuy.fr/TP25-ECE.pdf The subdivision x_0 to x_7, fsm.rnu.tn/useruploads/... Math equation, me with TeXit on discord That’s it. Thanks for reading ! Don’t hesitate to share !

Here the rectangles in yellow will count in negative towards the blue one. Recall ? Its a algebraic value :3

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Lilly Astar @nasaexploration.space · 23/10/2024

So there we go we built integral and even found a way to calculate it. This was my favorite lesson last year and I hope you enjoyed reading this. If you have any questions don’t hesitate to reach out to me ! In the next post you will find credits and a more elaborate function and triangles ! Thanks!

The notation we use for both integral and the limit where n tends to infinity.

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Lilly Astar @nasaexploration.space · 23/10/2024

We would get this formula. It’s called Riemann’s Sum. But then comes the magic. If we take a n larger and larger we add more and more triangles so we approach the value of the integral more and more. And that’s the magic, the sum will rigorously converge to the integral of f from a to b.

Here we simplified the x_(k+1) - x_k into (b-a)/n and imputed the x_k into the function.

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Lilly Astar @nasaexploration.space · 23/10/2024

If we create a nice even subdivision of length n where the step between each point of it is exactly (b-a)/(n-1), we can try to plug it into our sum that I showed earlier. What would it makes ? See in the picture the value of the subdivision on its k-th point.

Here we have the k-th point on the subdivision. We can note that x_0 = a and x_(n-1) = b. Also as (b-a)/(n-1) is positive so adding one at each point does make it greater so x_(n+1) > x_n.

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Lilly Astar @nasaexploration.space · 23/10/2024

What if now we take a subdivision finer and finer. If we make the distance between two subdivisions smaller and smaller won’t we approach the value of the integral more and more ? On the image below you can see how the triangles fits under the curve more and more.

Here on the same function with 25 triangles we already get an approximation of the integral from a to b. If we take more and more triangles will approach the integral nearly perfectly !

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Lilly Astar @nasaexploration.space · 23/10/2024

Then we simply add the area of each triangles in between every subdivision. Here we have the sum of every triangles area (the multiplication) and we take the value on the left of the subdivision. Try it yourself ! Draw a line and draw each triangles on a random subdivision you made.

Here we have the sum of every element where k will evolve from 1 to n-1. We would have (x_1 - x_0)f(x_0) + (x_2 - x_1)f(x_1) + … + (x_n - x_(n-1))f(x_(n-1)).

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Lilly Astar @nasaexploration.space · 23/10/2024

Then how could we calculate it ? Introducing the rectangle method ! We will use the subdivision to draw rectangles in between each subdivision that will approach the value of the area under the graph. Let’s go over the details.

The triangle method on a function where the subdivision contains 5 subdivisions.

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Lilly Astar @nasaexploration.space · 23/10/2024

So now you might wonder what are integral ? They are quite simple actually ! It’s simply the area under the curve of a function. Recall our function earlier ? Well the part in grey under the curve is actually what the integral of the function between the point a and b would be.

The part in grey here would be the integral of the function from a to b.

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Lilly Astar @nasaexploration.space · 23/10/2024

Now what if we apply the subdivision to a function, maybe when we draw the graph of a function you have to lift your pen but only for a point, then what if we find a subdivision where the function is continuous on every point exact the one in the subdivision ? That’s called by part continuity !

Here we have a function that’s obviously not continuous. Though it’s by part continuous on the segment [-3, 3] because you can find the subdivision (-3, -1, 1, 3) where the function is continuous on the segment [-3, -1[, ]-1, 1[ and ]1, 3]. This makes it by part continuous. Visually we can see this as having to lift the pen but only for the exact point.

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Lilly Astar @nasaexploration.space · 23/10/2024

We say that a subdivision delta = (a_1, a_2, …, a_n) is a subdivision if a_1 < a_2 < … < a_n where a_1 = a and a_n = b. While this might seems complicated, what it actually says is that it only grows and the starting and end point is the start and end of your subdivision. See picture for reference.

Here (x_0, x_1, …, x_6, x_7) is a subdivision because the first, x_0 is a, the last one, x_7 is b and the x grows, aka x_0 < x_1 < x_2 < … < x_7. This makes the set of points a subdivision of the segment [a,b]

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Lilly Astar @nasaexploration.space · 23/10/2024

We will say a function is continuous when you can draw the graph of the function without lifting your pen. For example here is the graph of a continuous function between the point a and b. See how you can draw it without lifting the pen. For the curious you can characterize it using limits.

A graph of a continuous function from the point a to the point b. The graph in red can be drawn without lifting your pen, « making » the function behind it continuous

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Lilly Astar @nasaexploration.space · 21/10/2024

@dynastylobster.bsky.social how would you get out of this sticky situation ? Art by @projjonmo117.bsky.social all credit goes to them :3

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