#induced geometry

[aviancataclysm]

said i'd be working on X but secretly practicing VeritY instead (it's the most doable medium demon for me rn from vibes since it feels easier to get consistent than HeLL, will just take more atts than easy demons............. aside from B we do NOT talk about B) although what the HELL IS THAT SHIP/UFO DUAL???? AT THE VERY END TOO????????? my worst skill is dual if we don't count swing so this is going to be FUN

[apolaskiart]

Frost and Gricko's tattoos for the iasuw au! Moving on to the kinda complex designs that made me have a bit of a crisis (legit), these ones are just as fun to make but relatively challenging to the previous one in terms of "what do I even put?".

I'll be yapping on my thoughts/headcannons regarding these below, so feel free to dive in! A warning that it is quite lengthy because of said crises-induced reflecting session (a proud citizen of yap city indeed)

First off, we have to talk about the technicalities of Frost's tattoos. It actually put me in a crisis when making this because the first question that came to mind when finding inspo was was "how the hell do you put tattoos on a tiger?"

Aside from the issue of tattooing onto fur, there is the presence of stripes that make it difficult to create a design in the first place. So for this a.u's sake (and my well being) we can treat fur like skin where tattoos can easily be applied and expressed, but would have a shorter lifespan, aka fade quicker, as a con so retouching would be more often. It can also overlap the patterns of the natural fur if designed as such (as I did with Frost).

With this, characters who have fur (e.g. Jornir) could possibly be given tattoos designs in the future while side stepping the technical issues to avoid putting to much focus on it. This could apply to other characters who may have leathery or feathery skin types.

Now of to the designing part. Unlike Gideon and Torbek, the aesthetic/style of the tattoo was not apparent at first and took a bit of a backseat to the actual design elements for Frost. Originally, using a purely geometrical style did not feel satisfying. After revisiting his canon for some ideas, an illustrative style was added to contrast the geometry which then seemingly worked out.

"Frost's design is built mostly of geometry (circles, squares and triangles) to reflect the motion of balance between familiarity and change, logic and unpredictability, comfort and discovery. Strict in pattern and position, most are simple as to take into account the stripes of his fur.

"The only complicated design, a dragon circling a tower and followed by a koi fish, was inspired by the legend of a koi travelling an upstream waterfall to turn into a dragon, signifying strength and perseverance. It is also a sign, that behind a rigid demeanor is a fiery passion waiting to be unleashed." -> The connected yarn was to give the idea that Frost can weave this path of his, but a connection as well to my other hc that it is his main reminder of his home in Yulong.

Things also mostly come in eight (resembles infinity as well as signs for wealth and success)! It was quite fun determining how to add this in as well :D

For the actual main inspo, IVE's "HEYA" has been on the forefront. Combined with another inspired by the phrase "When tigers used to smoke…" (which is the literary equivalent of "once upon a time…" and what has "once upon a…" in their name?), I must say that culture took the reigns in directing Frost's design and imho I would say that, compared to others, his was more appropriate to have strong semblances of his home as a remembrance (e.g. yarn, temple) during his travels.

Also I used a green coloring scheme because literally all other colors did not look as good as i intended. Fate really wanted him to have a green-orange scheme.

Now we got Frost out of the way, its Gricko's turn! Despite being challenging as well, Gricko's design had assistance from punk/heavy metal aesthetics. Though the only idea that I really had mostly was the logo of different band names

I will gladly argue on why Gricko's college band name should be "Goblin Deez..." Imagine, you can say nuts if you are being funny or hands if you want a fight. Anything really, which is liberating at its finest.

From here though, the main idea really that bloomed after this was that most of his tattoos were going to be personal too him!

"Aside from the egregious tattoo of his band name, the tattoos all over him speak of his journey. A pinup of a lady troll and a crown from the hit tv series game of chairs, for his childhood dreams of being a king and bagging a hot troll. A scale and guitar for his college pol. sci days. Others speaking of his collected hobbies and knick- knacks throughout a particularly challenging adulthood. Yet on his chest is his most cherished, his center and everything, which is none other than Hootsie."

The covered half sleeve tattoos on his arm ? That's to cover up the name of his exes (Headcanon that Gricko was just as much a womanizer as Gideon, but mostly in his college to early adult life. He slowed down once Hootsie entered the picture). Also its just more badass looking with designs of the monsters his canon self uses lol.

ALSO ALSO PAINTED NAILS, I'LL ARGUE WITH THE WHOLE COURT ROOM THAT HE AND HOOTSIE PAINT EACH OTHERS NAILS AS A HC!!!!

End of yapping session. Meeting adjourned!

[your-yandere-bestfriend]

　　　　　ₓ˚. ୭ ˚○◦˚.Wet dreams.˚◦○˚ ୧ .˚ₓ

Todoroki Shoto

Based off the song Wet Dreamz by J. Cole.

(Guys do I drop this after a 2 year hiatus or no )

。。。 Though it was the second day of school already, there had already been talk of a new student—most likely gossip overheard from the U.A. staff. Typically a new student wasn’t of much appeal, this was a huge school after all, even though it was prestigious, but what made the U.A. students so chatty about this new student was the fact that she was a foreigner. There had only even been three American students in the long history of U.A. high school, so getting a transfer was not only impressive, due to the fact that she wasn’t required to participate in the entrance exams, and also exciting.

The bell chimed for second blocks end, and next for the students of class 3.A., was the drowse inducing subject. Math. The teachers filtered out in order to teach their proper subject, after all just because this was a hero school didn’t mean that they were excused from engaging in normal, more uneventful subjects. Todoroki sat with perfect posture in the back of the room, awaiting the teachers entrance. His ears perked up when he heard Midoriya speak of the new girl who might’ve been making her first entrance today, though he didn’t quite care for a new student, it honestly had nothing to do with him, he hadn’t found a need for more friends, Midoriya, Uraraka, Iida, and Floppy felt plentiful.

The door slid open, the chattering of the class going still. In walked Cemento, but it wasn’t just him, it was also the new girl, and now that he has seen her with his own eyes, he could understand what the excitement was for. She had mesmerizing, smooth skin, she was delicate looking, with the such pretty eyes, they put his heterochromia ones to shame. Along with her pretty eyes and gorgeous figure her hair was long, around 30 inches, and red just like his left side, Todoroki couldn’t care less if it was real or not. He was sure his mouth was keenly open, but he knew he wasn’t alone, even the girls in the class were shocked at her appearance.

Cemento guided her to the front of the large classroom, in front of the expensive smart board that wasn’t yet displaying geometry problems, thankfully.

Cemento was preparing to give her the go ahead to start speaking but she already took that initiative for herself.

“Hi everyone it’s an absolute pleasure to meet you all! I’m … and I look forward to getting to know each and every one of you,” she beamed.

Sure she was already beautiful but todoroki could feel fire coming out his right side when she smiled revealing a straight line of pretty, white teeth. His multicolored eyes glanced around his classroom, and even the notorious Bakugou was in awe, a sight he has never yet seen.

“It is an absolute honor to have you here … ! Your Japanese is impressive!” Iida stood up moving his arm in the odd robotic like motions, but todoroki doubted anyone missed the telltale pink blush flushed across his pale face.

… bowed in response, traditional Japanese etiquette that she had obviously studied along with the language.

His eyes followed her long legs as she walked to the only empty seat in the room. Right next to him.

[bogleech]

All my life I’ve been unable to understanding finding an animal “gross” just for its natural appearance and I still can’t accept that anybody really means it. “Gross” is something an animal might DO or a condition it might be in, like when a dog rolls around in roadkill, right? I can get finding some animals “scary” for how they look or how they move, but....gross? Like an insect, all by itself, is some sort of unclean filthy gag-inducing “gross” to some people? How? What??? You’re saying there are basically unsanitary SHAPES? Specifically for animals? A creature can have a geometry that makes some people nauseated?

Or is it like some people have difficulty contextualizing insects or worms as individual animals and instead hold them in the same mental category as almost like a “substance” or a “texture,” like living “dirt?” Is THAT what it is?

[duckprintspress]

Celebrate Native American Heritage Month with 7 Queer Books We Love

November is National Native American Heritage Month! We’re celebrating with books (as always, lol). We asked our rec list contibutors for their favorite queer books either by Native American authors or starring Native American characters. Most of these books (maybe all, I couldn’t confirm for all the authors) are both! Contributors to the list are Nina Waters, hullosweetpea, D.V. Morse, Shea Sullivan and an anonymous contributor.

-

Indiginerds edited by Alina Pete

First Nations culture is living, vibrant, and evolving…

…and generations of Indigenous kids have grown up with pop culture creeping inexorably into our lives. From gaming to social media, pirate radio to garage bands, Star Trek to D&D, and missed connections at the pow wow, Indigenous culture is so much more than how it’s usually portrayed. These comics are here to celebrate those stories!

Featuring an all-Indigenous creative team, INDIGINERDS is an exhilarating anthology collecting 11 stories about Indigenous people balancing traditional ways of knowing with modern pop culture.

-

Postcolonial Love Poem by Natalie Díaz

Postcolonial Love Poem is a thunderous river of a book, an anthem of desire against erasure. It demands that every body carried in its pages – bodies of language, land, suffering brothers, enemies and lovers – be touched and held. Here, the bodies of indigenous, Latinx, black and brown women are simultaneously the body politic and the body ecstatic, and portrayed with a glowing intimacy: the alphabet of a hand in the dark, the hips’ silvered percussion, a thigh’s red-gold geometry, the emerald tigers that leap in a throat. In claiming this autonomy of desire, language is pushed to its dark edges, the astonishing dune fields and forests where pleasure and love are both grief and joy, violence and sensuality.

Natalie Diaz defies the conditions from which she writes, a nation whose creation predicated the diminishment and ultimate erasure of bodies like hers and the people she loves. Her poetry questions what kind of future we might create, built from the choices we make now – how we might learn our own cures and ‘go where there is love’.

-

A Snake Falls to Earth by Darcie Little Badger

Nina is a Lipan girl in our world. She’s always felt there was something more out there. She still believes in the old stories.

Oli is a cottonmouth kid, from the land of spirits and monsters. Like all cottonmouths, he’s been cast from home. He’s found a new one on the banks of the bottomless lake.

Nina and Oli have no idea the other exists. But a catastrophic event on Earth, and a strange sickness that befalls Oli’s best friend, will drive their worlds together in ways they haven’t been in centuries.

And there are some who will kill to keep them apart.

-

Never Whistle at Night: An Indigenous Dark Fiction Anthology edited by Shane Hawk and Theodore C. Van Alst Jr.

Many Indigenous people believe that one should never whistle at night. This belief takes many forms: for instance, Native Hawaiians believe it summons the Hukai’po, the spirits of ancient warriors, and Native Mexicans say it calls Lechuza, a witch that can transform into an owl. But what all these legends hold in common is the certainty that whistling at night can cause evil spirits to appear–and even follow you home.These wholly original and shiver-inducing tales introduce readers to ghosts, curses, hauntings, monstrous creatures, complex family legacies, desperate deeds, and chilling acts of revenge. Introduced and contextualized by bestselling author Stephen Graham Jones, these stories are a celebration of Indigenous peoples’ survival and imagination, and a glorious reveling in all the things an ill-advised whistle might summon.

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The Witch King (Witch King series) by H.E. Edgmon

Wyatt would give anything to forget where he came from–but a kingdom demands its king.

In Asalin, fae rule and witches like Wyatt Croft…don’t. Wyatt’s betrothal to his best friend, fae prince Emyr North, was supposed to change that. But when Wyatt lost control of his magic one devastating night, he fled to the human world.

Now a coldly distant Emyr has hunted him down. Despite transgender Wyatt’s newfound identity and troubling past, Emyr has no intention of dissolving their engagement. In fact, he claims they must marry now or risk losing the throne. Jaded, Wyatt strikes a deal with the enemy, hoping to escape Asalin forever. But as he gets to know Emyr, Wyatt realizes the boy he once loved may still exist. And as the witches face worsening conditions, he must decide once and for all what’s more important–his people or his freedom.

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Elatsoe (Elatsoe series) by Darcie Little Badger

Imagine an America very similar to our own. It’s got homework, best friends, and pistachio ice cream.

There are some differences. This America been shaped dramatically by the magic, monsters, knowledge, and legends of its peoples, those Indigenous and those not. Some of these forces are charmingly everyday, like the ability to make an orb of light appear or travel across the world through rings of fungi. But other forces are less charming and should never see the light of day.

Elatsoe lives in this slightly stranger America. She can raise the ghosts of dead animals, a skill passed down through generations of her Lipan Apache family. Her beloved cousin has just been murdered, in a town that wants no prying eyes. But she is going to do more than pry. The picture-perfect facade of Willowbee masks gruesome secrets, and she will rely on her wits, skills, and friends to tear off the mask and protect her family.

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Black Sun (Between Earth and Sky series) by Rebecca Roanhorse

A god will return When the earth and sky converge Under the black sun

In the holy city of Tova, the winter solstice is usually a time for celebration and renewal, but this year it coincides with a solar eclipse, a rare celestial event proscribed by the Sun Priest as an unbalancing of the world.

Meanwhile, a ship launches from a distant city bound for Tova and set to arrive on the solstice. The captain of the ship, Xiala, is a disgraced Teek whose song can calm the waters around her as easily as it can warp a man’s mind. Her ship carries one passenger. Described as harmless, the passenger, Serapio, is a young man, blind, scarred, and cloaked in destiny. As Xiala well knows, when a man is described as harmless, he usually ends up being a villain.

What are your favorite queer books with Native American representation?

Want to chat your favorite reads with us? Join our Book Lover’s Discord server!

Update your Goodreads TBR with any of these books by visiting our queer Native American books shelf  on Goodreads!Shop books with Native American rep using our rec list on our Bookshop.org affiliate page!

[algebraic-dualist]

Universal Algebraic Geometry, Part 1

In this post I recall a number of basic ideas from universal algebra in order to start setting up a framework for studying the algebraic geometry of arbitrary algebraic objects.

For our purposes, an algebra is just a set equipped with some operations. We assume that the set of operations has been fixed. Given an algebra A alongside an operation ♡ we write ♡ᴬ for the definition of that operation on the algebra A, or just ♡ when ambiguity does not result. Our notion of a homomorphism is as follows:

Definition: A homomorphism f from an algebra A to an algebra B is a function f : A -> B such that for each n-ary operation ♡ alongside elements x₁, ..., xₙ ∈ A it follows that f(♡ᴬ(x₁, ..., xₙ)) = ♡ᴮ(f(x₁), ..., f(xₙ))

Kernels and Congruences

Definition: A congruence θ on an algebra A is an equivalence relation such that for any (x₁, y₁) ∈ θ, ..., (xₙ, yₙ) ∈ θ and n-ary operation ♡ it follows that (♡ᴬ(x₁, ..., xₙ), ♡ᴬ(y₁, ..., yₙ)) ∈ θ. We write Cgr(A) for the set of congruences on A.

Example: The maximal congruence ∇ on A is just A × A.

Example: The minimal congruence ∆ on A is just { (x, x) | x ∈ A }.

Example: Any arbitrary intersection of congruences on A is a congruence on A. In particular, we can form the intersection of all congruences on A containing any arbitrary subset R of A × A in order to obtain the minimum congruence containing R. In particular, choosing R to be the union of two congruences, we get a meet of congruences.

In the case of a group, it follows that (x, y) ∈ θ if and only if (x - y, 0) ∈ θ hence a congruence is completely determined by the set { x | (x, 0) ∈ θ }. In other words, by a normal subgroup. So congruences are just the data of a normal subgroup. Similar remarks apply to ideals of rings.

Kernels of homomorphisms are a particularly important source of congruences on algebras:

Definition: The kernel of a homomorphism f : A -> B is denoted by ker(f) and defined by setting ker(f) := { (x, y) | f(x) = f(y) }.

Lemma: Given a homomorphism f : A -> B it follows that ker(f) is a congruence on A. Proof:

Given x ∈ A it follows that f(x) = f(x) hence (x, x) ∈ ker(f).

Given x, y ∈ A such that (x, y) ∈ ker(f) it follows that f(x) = f(y), which is to say f(y) = f(x) hence (y, x) ∈ ker(f).

Given x, y, z ∈ A it such that (x, y) ∈ ker(f) and (y, z) ∈ ker(f) it follows that f(x) = f(y) = f(z) hence (x, z) ∈ ker(f).

Given (x₁, y₁) ∈ ker(f), ..., (xₙ, yₙ) ∈ ker(f) and n-ary operation ♡ it follows that f(xₖ) = f(yₖ) for each k, thus by the homomorphism property f(♡ᴬ(x₁, ..., xₙ)) = ♡ᴮ(f(x₁), ..., f(xₙ)) = ♡ᴮ(f(y₁), ..., f(yₙ)) = f(♡ᴬ(y₁, ..., yₙ)) showing that (♡ᴬ(x₁, ..., xₙ),♡ᴬ(y₁, ..., yₙ)) ∈ ker(f).

QED.

Theorem: Given a homomorphism f : A -> B alongside a congruence θ on B it follows that the preimage f*(θ) := { (x, y) ∈ A | (f(x), f(y)) ∈ θ } is a congruence on A. Proof: 1. Given x ∈ A, as θ is reflexive it follows that (f(x), f(x)) ∈ θ hence x ∈ f*(θ). 2. Given (x, y) ∈ f*(θ) it follows that (f(x), f(y)) ∈ θ. As θ is symmetric it follows that (f(y),f(x)) ∈ θ, thus (y, x) ∈ f*(θ). 3. Given (x,y) ∈ f*(θ) and (y,z) ∈ f*(θ) it follows that (f(x), f(y)) ∈ θ and (f(y), f(z)) ∈ θ. As θ is transitive it follows that (f(x), f(z)) ∈ θ hence (x, z) ∈ f*(θ). 4. Given (x₁, y₁) ∈ f*(θ), ..., (xₙ, yₙ) ∈ f*(θ) alongside an n-ary operator ♡ it follows that (f(x₁), f(y₁)) ∈ θ, ..., (f(xₙ), f(yₙ)) ∈ θ, hence as θ is a congruence (♡(f(x₁), ..., f(xₙ)), ♡(f(y₁), ..., f(yₙ))) ∈ θ. Being that f is a homomorphism, this means (f(♡(x₁, ..., xₙ)), f(♡(y₁, ..., yₙ))) ∈ θ showing that (♡(x₁, ..., xₙ), ♡(y₁, ..., yₙ)) ∈ f*(θ). QED.

Therefore, every homomorphism f : A -> B induces a map f* : Cgr(B) -> Cgr(A). In fact, congruences organise themselves into a lattice, and this map f* is a homomorphism of lattices. But for the moment, we will not worry about that.

Composite Kernel Lemma: Given homomorphisms f : A -> B and g : B -> C it follows that

ker(g ∘ f) = f*(ker(g)).

Proof:

Observe that (x, y) ∈ ker(g ∘ f) if and only if g(f(x)) = g(f(y)), which is the case if and only if (f(x), f(y)) ∈ ker(g), which is the case if and only if (x, y) ∈ f*(ker(g)).

QED. Corollary: ker(f) = f*(∆). Proof: ker(id ∘ f) = f*(ker(id)) = f*(∆). QED.

Quotients of Algebras

Given a congruence θ on an algebra A it is the possible to construct the quotient algebra A/θ whose elements are equivalence classes of elements of A under the congruence θ. Sending every element to its equivalence class under θ gives a homomorphism π : A -> A/θ which we refer to as the canonical projection of θ.

Theorem: Given a congruence θ on an algebra A it follows for every homomorphism f : A -> B with θ ⊆ ker(f) that there exists a unique injective morphism g : A/θ -> B such that f = g ∘ π.

Proof: Observe that π(x) = π(y) iff (x, y) ∈ θ, which as θ ⊆ ker(f) implies f(x) = f(y). Therefore, it is well-defined to construct g by setting g(π(x)) = f(x). One can check that this works. QED.

Homomorphism Theorem: Given a surjective homomorphism f : A -> B it follows that there exists an isomorphic g : A/ker(f) -> B such that g ∘ π = f.

Proof:

Applying the prior Theorem there exists a unique injective morphism g : A/ker(f) -> B such that g ∘ π = f. Observe that as f is surjective, it then follows that g is surjective.

QED.

Hereditary Classes of Algebras

Recall that the prime ideals 𝔭 of a ring R are exactly those ideals 𝔭 such that R/𝔭 is an integral domain. Notice that being an integral domain amounts to satisfying a universal axiom, namely xy = 0 implies x = 0 or y = 0. Hence, integral domains are closed under moving to subalgebras, and the property of being an integral domain is preserved by isomorphisms.

Definition: A class K of algebras is hereditary iff it is closed under subalgebras and isomorphisms. Given a universal class K we say that a congruence θ on an algebra A is a K-congruence iff the quotient algebra A/θ belongs to the class K.

Theorem: Given a hereditary class K alongside a homomorphism f : A -> B it follows for every K-congruence θ on B that f*(θ) is a K-congruence on A.

Proof:

Consider a homomorphism f : A -> B alongside a K-congruence θ on B. It follows that f*(θ) is a congruence on A, so it suffices to demonstrate that it is a K-congruence. Recalling our canonical projection π : B -> B/θ and our Composite Kernel Lemma, it follows that ker(π ∘ f) = f*(ker(π)) = f*(θ). Therefore, the morphism π ∘ f : A -> B/θ factorises through a unique injective morphism g : A/f*(θ) -> B/θ. This means that A/f*(θ) is isomorphic to a subalgebra of B/θ. As θ is a K-congruence it follows that B/θ belongs to K, which as K is a hereditary class then implies that A/f*(θ) belongs to K, so that f*(θ) is then a K-congruence.

QED.

Corollary: Preimages of prime ideals are prime ideals.

[spacetimewithstuartgary]

Gravity from entropy: A radical new approach to unifying quantum mechanics and general relativity

In a new study published in Physical Review D, Professor Ginestra Bianconi, Professor of Applied Mathematics at Queen Mary University of London, proposes a groundbreaking new framework that could revolutionise our understanding of gravity and its relationship with quantum mechanics. The study, titled Gravity from Entropy, introduces a novel approach that derives gravity from quantum relative entropy, bridging the gap between two of the most fundamental yet seemingly incompatible theories in physics: quantum mechanics and Einstein’s general relativity.

The challenge of quantum gravity

For decades, physicists have struggled to reconcile the principles of quantum mechanics with those of general relativity. While quantum mechanics governs the behaviour of particles at the smallest scales, general relativity describes the force of gravity on cosmic scales. Unifying these two frameworks has been one of the most elusive goals in modern science. 

Professor Bianconi’s work offers a fresh perspective by treating the metric of spacetime, a key concept in general relativity, as a quantum operator. This innovative approach uses quantum relative entropy, a concept from quantum information theory, to describe the interplay between spacetime geometry and matter. 

The role of entropy and the G-field

The study introduces a new entropic action, which quantifies the difference between the metric of spacetime and the metric induced by matter fields. This approach leads to modified Einstein equations that, in the low coupling regime i.e. low energies and small curvature, reduce to the classical equations of general relativity. However, the theory goes further, predicting the emergence of a small, positive cosmological constant – a value that aligns with experimental observations of the universe’s accelerated expansion much better than for other pre-existing theories. 

A key feature of the theory is the introduction of the G-field, an auxiliary field that acts as a Lagrangian multiplier. The G-field not only plays a crucial role in the modified equations of gravity but also opens the door to new interpretations of dark matter – a mysterious substance that makes up a significant portion of the universe’s mass but has yet to be directly observed. 

The implications of this research are profound. By linking gravity to quantum information theory, the study provides a potential pathway to a unified theory of quantum gravity. Moreover, the G-field could offer new insights into the nature of dark matter, a long-standing puzzle in cosmology. 

“This work proposes that quantum gravity has an entropic origin and suggests that the G-field might be a candidate for dark matter,” explains Professor Bianconi. “Additionally, the emergent cosmological constant predicted by our model could help resolve the discrepancy between theoretical predictions and experimental observations of the universe’s expansion.”

While further research is needed to fully explore the implications of this theory, the study represents a significant step forward in the quest to understand the fundamental nature of the universe.

Professor Bianconi’s work challenges conventional wisdom and opens up exciting new avenues for exploration. By treating spacetime as a quantum entity and leveraging the power of entropy associated to the spacetime metrics, this research could pave the way for a deeper understanding of gravity, quantum mechanics, and the cosmos itself. 

IMAGE: Diagrammatic representation of the entropic quantum gravity action. The action for gravity is given by the quantum relative entropy between the metric of the manifold and the metric induced by the matter field and the geometry. Credit Ginestra Bianconi

[materialsscienceandengineering]

Excited state dynamics are essential for understanding fluorescence properties in molecules, impacting their application in technologies. Recent research at Shinshu University explores how molecular structure and geometry influence light emission in aggregation-induced emission molecules. The study reveals that changes in molecular shape affect emission behavior in both solution and solid states. These insights are crucial for advancing applications like organic light-emitting diodes and bioimaging, enabling innovations in material design and energy interactions. Light emission from molecules, particularly fluorescence, has fascinated scientists for over a century, revolutionizing areas like imaging, sensing, and display technologies. Recent advancements have brought attention to aggregation-induced emission (AIE) -- a unique phenomenon where molecules emit light more efficiently when in a solid or aggregated state. Studying the reaction dynamics underlying this phenomenon, is thus, important for understanding the molecular structural changes.

Read more.

[hexheathen]

SCP - Unknown

Item #: SCP-█████ Object Class: Keter

Special Containment Procedures: SCP-█████ is to be contained in a reinforced subterranean containment zone measuring [REDACTED] meters in diameter. The chamber walls must be coated in a proprietary alloy developed under Project ████████ to mitigate SCP-█████’s anomalous effects. Access is restricted to personnel with Level 4 clearance or higher, and all interactions must occur remotely via drone-operated systems.

Any personnel entering SCP-█████’s containment chamber must wear full-spectrum visors and Level III hazmat suits to counteract its [REDACTED] visual distortions and biological hazards. In the event of a containment breach, Site ██ is to initiate Protocol “Oblivion Net”: total lockdown with on-site warheads primed for immediate detonation. Under no circumstances are any reflective surfaces or biological samples to be introduced into SCP-█████’s vicinity.

Description:

SCP-█████, referred to colloquially as "The Unknown," is a humanoid entity measuring approximately 7 feet in height when fully upright, though it often adopts a hunched or semi-quadrupedal posture. Its physiology is highly mutable, oscillating between tangible flesh and [DATA EXPUNGED], rendering traditional analysis nearly impossible.

Key features include unnaturally elongated limbs, sharp, angular protrusions along its vertebrae, and a head encased in a [REDACTED] that refracts light in anomalous ways. Notably, SCP-█████ lacks visible eyes, yet it demonstrates acute environmental awareness, responding to stimuli with unnerving precision. Its face appears to shift subtly, giving the impression of watching without any defined gaze.

SCP-█████ currently resides within what has been described as a “skin shell,” a body once belonging to a deceased D-Class personnel, designated [REDACTED]. This shell, while outwardly human, is notably altered:

The skin appears pallid and waxy, with deep scar tissue radiating across the chest and abdomen.

Movements are disjointed and irregular, often accompanied by audible cracking sounds.

SCP-█████ utilizes this host as a façade, manipulating it with inhuman precision while maintaining a semblance of humanity.

Within the skin shell, thermal imaging reveals SCP-█████ itself as a dense, shifting mass of [DATA EXPUNGED], suggesting its true form is compacted or compressed to inhabit its host. Attempts to extract SCP-█████ from the skin shell have resulted in failure and increased aggression from the entity.

Its most anomalous property, “UVX Distortion,” involves manipulating light and sound to create persistent illusions of serpentine forms and fractal geometries. Victims of UVX Distortion often become disoriented and herded to the SCP’s location. After that, they report hearing phrases such as “cry, cry,” and “your fear makes you whole” before succumbing to [REDACTED].

UVX Distortion and Biological Toxins

SCP-█████’s most anomalous property is classified as “UVX Distortion,” a multifaceted phenomenon involving biological, psychological, and visual effects:

UVX Toxin Production: SCP-█████ produces an airborne neurotoxin designated UVX-13, which binds to synaptic pathways in the brain. Victims exposed to UVX-13 experience disorientation, paranoia, and sensory distortion. Prolonged exposure results in:

Blurred Vision: Perception of shifting shapes and serpentine forms.

Auditory Hallucinations: Persistent whispering, sobbing, and rhythmic ticking.

Cognitive Impairment: Difficulty forming coherent thoughts or distinguishing reality from illusion.

UVX Biological Impact:

Studies of UVX-13 samples reveal it mimics specific neurotransmitters, inducing a temporary state of hyper-awareness followed by severe cognitive shutdown.

The toxin causes cellular degradation in nervous tissue, leaving survivors with long-term psychological trauma and reduced motor function.

Victims report an intense sensation of "being watched" and recurring dreams of [REDACTED].

Visual Anomalies: SCP-█████ manipulates light to create persistent optical illusions resembling fractal geometries, shadow-like tendrils, and serpentine entities. This effect, combined with the toxin, overwhelms the victim's senses, making escape nearly impossible.

Behavioral Control: Individuals exposed to UVX-13 demonstrate unusual patterns of compliance with SCP-█████. These subjects have been observed walking directly toward SCP-█████ despite signs of fear or pain.

Acquisition:

Following multiple disappearances, SCP█████ was first encountered near [REDACTED] Greenville Theater. Mobile Task Force Theta-9 (“Trailblazers”) was deployed. SCP-█████ displayed hostility, resulting in █ fatalities and ██ injuries, before being subdued with experimental [REDACTED] weaponry.

Incident Report ████-A:

On ██/██/20██, SCP-█████ breached containment during routine maintenance. Security footage shows SCP-█████ phasing through [REDACTED]. Surviving personnel reported hearing voices resembling deceased loved ones and experiencing severe emotional distress. The breach was resolved after [DATA EXPUNGED], though ██ staff were declared MIA.

Post-incident analysis strongly suggests that SCP-█████ possesses highly dangerous cognitohazardous properties capable of [REDACTED]. This revelation underscores the critical importance of our ongoing research and the need for immediate containment measures.

Experiment Log █████-C: Test ███-07 Objective: Determine limits of UVX Distortion and toxin spread. Method: Remote drones equipped with Class-IV visual inhibitors and toxin detectors.

Results: Drone operators experienced hallucinations after [REDACTED] minutes, reporting visions of [REDACTED]. Biological analysis of air samples revealed UVX-13 concentration exceeded safe levels within 1.2 seconds of SCP-█████'s presence. a small .3 mm hole was found. Test aborted due to equipment failure and psychological strain on operators. Hole has been fixed.

Addendum █████-B:

Field Research Notes:

SCP-█████ shows fixation on individuals with high emotional distress. Patterns suggest it selects victims based on [DATA REDACTED].

SCP-frequently exhibits a deeply unsettling personal interest in those it encounters, often recounting specific, deeply personal details about their lives. This behavior is theorized to increase victim vulnerability, causing significant discomfort and unease among those affected.Unverified accounts suggest SCP-█████ has engaged in cryptic dialogues about “the Fog” and “the End.”

Interview Log █████-01

Interviewer: Dr. Amelia Crowe

Subject: SCP-█████

Date: ██/██/20██

[BEGIN LOG]

Dr. Crowe: SCP-█████, can you hear me?

SCP-█████: (A deep, layered voice, fragmented and uneven, responds after a long pause) Yes. Yes. I hear the... tick... tick… ticking. Your time runs thin.

Dr. Crowe: My time? What do you mean by that?

SCP-█████: Do you not feel it? The weight. The crushing tide of the Fog. It watches. It... [REDACTED]... waits.

Dr. Crowe: The Fog? Is this an external entity you serve?

SCP-█████: Serve? No. It is me. I am it. You are all drowning in its shadows.

Dr. Crowe: What do you want from us?

SCP-█████: I want to see. To know. To... unravel. Your chains tighten as you beg for them.

Dr. Crowe: Chains? Elaborate.

SCP-█████: You carry them. Wound tight. Around your soul.

*[SCP-█████ leans toward the microphone. Static overtakes the audio.]

Post-Incident Report:

Personnel present during the interview with SCP-reported experiencing recurring nightmares and severe emotional distress for several weeks. This psychological strain underscores the need for caution and the suspension of further direct communication until containment protocols are revised.

[positively-knotted]

Dissertationposting 2: Curvature & Hypersurfaces

(Note: I'm not gonna say anything about what I mean by "curvature", as if you haven't already seen it in some context I don't think you're gonna get much out of these posts :/. But for clarity, I'll only ever talk about scalar curvature and call it R. If you're used to Gaussian curvature written K, this is the same except R = 2K. Oh also all my manifolds are oriented.)

It may not be immediately obvious why we would want to consider hypersurfaces when trying to understand curvature - after all, R³ is flat, but contains surfaces with all kinds of curvature. So a motivating example may be in order. This is in fact the example that spurred all of the developments we'll talk about, and is of particular importance in theoretical cosmology! Maybe I'll say more about that later.

Consider a manifold homeomorphic to the n-torus Tⁿ - or more properly, Tⁿ with a choice of geometry (Riemannian metric). Intuitively, this shouldn't be able to have R > 0 everywhere, in the same way that T² can't. Proving this is surprisingly hard, but one sensible approach would be to try induction. After all, T³ is just T² × S¹, and S¹ is easy to understand. Thought of the other way around, we want to find nice hypersurfaces and do inductive descent until we reach dimension 2, then apply Gauss-Bonnet.

But if any old hypersurface doesn't say much, what about particularly nice ones? The obvious candidates are so-called stable minimal hypersurfaces. These are minimal points for area, in the sense that if you perturb them very slightly, their area must increase (e.g. the meridian of a standard torus in the motivating example). The big result about these is as follows:

Lemma 1.

If Σ is a stable minimal hypersurface in a manifold (M, g) and f is any function on Σ, then [1]

Here the left side is the gradient of f (in the normal calculus sense), and \II is the second fundamental form, a number that tells us something about how Σ sits inside M. You can think of it as a kind of error term, but the important bit is that it's squared, so non-negative. R is the scalar curvature of the ambient manifold M, and R_\Sigma is the induced scalar curvature of Σ.

You've probably never seen this before, even if you've done differential geometry. I don't know why this formula isn't better known, it's super useful, as with clever choices of f it gives big results. Just taking f=1 shows that for a stable minimal hypersurface, the intrinsic curvature has to be greater on average than the external curvature. This is a pretty big step forward!

Ok, so we have our choice of useful hypersurfaces. Now we just need to show that they always exist. And they do! In fact, they exist for each homology class:

Lemma 2.

Let (M, g) be a closed n-manifold, 3 ≤ n ≤ 7. Then for any class α ∈ H_{n−1}(M), we can find a closed stable minimal hypersurface Σ such that [Σ] = α in homology.

Let's unpack this a bit. We can think of hypersurfaces as living in a particular homology class in the usual geometric way. It's a standard result from differential topology [2] that we can in find a hypersurface living in any homology class, and that slightly perturbing it won't change that class. The interesting bit here is that we can always take a stable minimal representative, in sufficiently small dimensions. [3] Annoyingly, this means everything from here on out only applies in dimensions less than 8, even though there's no good topological reason why it isn't true more generally. (It definitely is true more generally, we just need a stronger version of this lemma and that makes it an analysis question.)

Next time, we'll use this to show that Tⁿ doesn't admit positive scalar curvature! Notes below the cut.

[1] Proof of Lemma 1. Start with the usual Gauss equation. Take the trace to pass to Ricci curvature, then again to pass to scalar curvature. It turns out that the irritating terms from varying dimensions cancel out! More detail in the screenshots below:

[2] Remember that there is a bijection between H¹(M) and homotopy classes of maps M -> S¹, pairing [f] with f^*(ω), where ω generates H¹(S¹). We can assume f is smooth, choose a regular value t, and set Σ = f^{-1}(t). Then [Σ] is Poincaré dual to f^*(ω).

[3] The actual argument here is pretty standard for the analysis I'll be omitting. In general, if we want to show that an object in space X with property P exists, we'll pass to a bigger space Y where standard results give us an object with that property; then somehow argue that P implies that it's actually of type X. Here, we pass from submanifolds to currents, and minimise area for currents in a given homology class. By measure theoretic magic, this minimisation forces the singular set of the current to have codimension at most 7, so in dimensions less than 8 it is in fact a submanifold.

[softrobotcritics]

Photoresponsive Shape Morphing

Movement with light: Photoresponsive shape morphing of printed liquid crystal elastomers

Michael J. Ford1 ∙ Dominique H. Porcincula1 ∙ Rodrigo Telles3 ∙ … ∙ Shu Yang2 ∙ Elaine Lee1 lee1040@llnl.gov ∙ Caitlyn C. Cook1,5 krikorian3@llnl.gov … 

Progress and potential

Soft matter that can adapt in response to a stimulus like light holds immense promise for various applications, such as biomedical devices and soft robotics. One example of adaptive soft matter is liquid crystal elastomer composites, which incorporate a functional additive and change shape through a phase transition. The combination of the material composition, the printed geometry of the material, and the localization of the stimulus can enable novel movement and reaction to light, as we demonstrate in this paper. Our results mark a significant advancement toward creating complex, 3D-printed, intelligent materials that pave the way for developing next-generation adaptive machines and devices that can transform in response to specific stimuli.

Highlights

•

Optimized inks for additive manufacturing of a liquid crystal elastomer composite

•

Developed spatiotemporal control during printing for complex three-dimensional structures

•

Demonstrated unique combinations of complex three-dimensional photoresponsive actuation

•

Controlled novel modes of actuation with computer vision techniques

Summary

Soft machines will require soft materials that exhibit a rich diversity of functionality, including shape morphing and photoresponsivity. The combination of these functionalities enables useful behaviors in soft machines that can be further developed by synthesizing materials that exhibit localized responsivity.

Localized responsivity of liquid crystal elastomers (LCEs), which are soft materials that exhibit shape morphing, can be enabled by formulating composite inks for direct ink writing (DIW). Gold nanorods (AuNRs) can be added to LCEs to enable photothermal shape change upon absorption of light through a localized surface plasmon resonance.

We compared LCE formulations, focusing on their amenability for printing by DIW and the photoresponsivity of AuNRs. The local responsivity of different three-dimensional architectures enabled soft machines that could oscillate, crawl, roll, transport mass, and display other unique modes of actuation and motion in response to light, making these promising functional materials for advanced applications....

Soft machines could enable new breakthroughs in technologies related to human-machine interactions, remote exploration in difficult-to-reach spaces, and individually tailored health care. These machines will require soft materials that exhibit a diverse range of functionalities, including actuation for movement, conductivity for sensing and signal processing, stimuli-responsivity, self-healing, and reprocessability.1,2,3The demonstration of such a diverse range of functionalities results in a profound outcome where “the material is the machine.”4,5 That is, by taking advantage of behaviors like self-assembly and phase transitions, these materials as machines can replace traditional sensors, transducers, gears, levers, and electromagnetic motors to enable perception, responsivity, and motion without engineered complexity.2,4

Liquid crystal elastomers (LCEs) that are pre-programmed to change shape in response to external stimuli are considered useful for soft machines.6,7 The shape morphing is induced by heat, electricity, and light.8,9Light may be useful to stimulate localized actuation and does not require physical contact with the shape-changing material, as wires that transmit electrical power might require.10,11 Localized actuation using light could also allow for unique modes of actuation.12 For example, asymmetric illumination of photoresponsive LCEs led to twisting and rolling motions.13 Peristaltic motion that resembles the movement of biological organisms has been demonstrated by using localized impingement of different patterns of light upon an LCE.14 To extend this work, the programmed order of the liquid crystal (LC) domains could be controlled and modified....

[rock-swag-tournament]

pyrite propaganda: pyrite is so amazing i’m not even kidding. in my mothertongue, instead of being called fool’s gold, it’s called cat’s gold and we all love cats right?

also, fossils can become pyritised, meaning that we have literal DINOSAUR FOSSILS that are made of pyrite!!! beautiful shiny golden pyrite!!!

plus pyrite has a lot of chemically similar minerals that are awesome that it can occur with as well. i love chalcopyrite because i love copper (CuFeS2 if im not mistaken) and arsenopyrite is haha funny because its chemical composition is literally FeAsS and it has arse in the name as well (sorry).

it also forms in cubes and im p sure the other shape is a dodecahedron (sorry im not good at remembering geometry and im writing this in a pyrite-induced feverish haze so im not gonna google that rn) and we love 1. minecraft (cubes) and 2. dnd (dodecahedron)

i literally love this rock so so so much i will cry forever if it doesnt win (real)

YES TO ALL OF THIS! And thank you especially for shouting out chalcopyrite i love chalcopyrite.

[dollking081]

I feel like a lot of fanmade geometry dash levels are less about level design and more about inducing seizures

[queenofcandynsoda]

Carmine the Bloodshed

(Thank you @jknerd~)

TW; Self-harm

Carmine the Bloodshed

Other Names: Carmine the Bloodshed (カーマイン・の・さっしょう Chimamire no Carumin), The Psycho’s Son, The Red Devil, Carmine Spini, R-O-001

Age: 7

Affiliation: Deepground; Shinra Electric Power Company (Formerly), Weiss’ Army

Occupation: Child Soldier, Sniper, Assassin (trainee)

Residence: Deepground Offspring Nursery (formerly), None/Nomadic (currently)

Family: Rosso the Crimson (mother), Cerise (younger sister), Eerie (younger sister), Cordovan and Maroon (younger twin brothers), Jasper (younger brother), Peach (youngest sister), Madder (younger brother), Cardinal (younger brother), Chili (youngest brother), The Restrictor (biological father)

Abilities: Firearm Proficiency (Rifle, Sniper Rifle, Shotgun, Handgun, Assault Rifle), Supernatural Gunmanship (Supernatural Accuracy, Supernatural Reflexes, Supernatural Dexterity, Quick Draw), Glowing Eyes, Knife Proficiency (Switchblade, Razor, Scalpel, Clever, Kitchen Knife), Superior Human Physiology (Flash Step, Supernatural Stealth, Afterimage Creation, Bulletproof Durability, Self-Sustenance, Disease Resistance, Enhanced Intelligence, Supernatural Strength, Healing Coma [after blood loss], Mental Regeneration), Blood Manipulation (Bleeding Inducement, Flammable Blood, Corrosive Blood, Haemokinetic Combat, Blood Bullets), Adaptive Regeneration [after MANY attempts to euthanize him] (Acid Immunity, Poison/Venom Immunity, Water Immunity, Fire Immunity, Explosion Immunity, Blunt Force Immunity, Cutting Immunity), Feign Death, Supernatural Cells, Killing Mastery, Killing Instinct (reluctant), Anatomical Mastery, Weakness Detection, Humanity Retainment, Indomitable Loyalty (to his mother and Weiss), Mathematics Mastery (Geometry, Shatterpoint, Numerical Precision, Calculation, Ricochet), Temporal Sense, Combat Adaptation

Likes: His family, Freedom, Daylight, Collecting souvenirs, Girl-in-the-Picture (Vermillion), Guns, Hot Chocolate, Burgers, Pizzas, Loyalty to Weiss, Target Practice

Dislikes: THE RESTRICTOR, Shinra Company, Professor Hojo, Doctor Fiorenzo Rosales, His Murderous Thoughts, Being controlled, Intense Training, Greedy People, Spoiled Brats

Origins: Carmine is a member of the Off-Color Tsviets and the first son and child of Rosso the Crimson. His biological father is the Restrictor, the cruel overseer of Deepground. From birth, Carmine was forced to train to be the most effective killer that could surpass his mother’s abilities. His body has been pushed beyond its limits to the point it is a wonder how Carmine lived past the age of five. After his birth, his mother attempted to hide him but he was quickly taken away and placed in the Offspring Nursery. Since then, he was subjected to torturous experiments as Rosso gave birth to his siblings due to her body being modified to be pregnant for three months. The mother and son would communicate at every opportunity, even having the chance to see his younger siblings. 

Due to the brutal experience he suffered, Carmine has become mentally unstable. He has become hostile to “Shinrites,” meaning people who work for Shinra”, and would aim his rifle at them, believing they intend harm regardless. He tends to be vulgar whenever he feels defensive, especially when it comes to his siblings and mother. This would also lead to profanity and threats of violence. 

Despite these frightening traits, he does his best to create and maintain his humanity. He would make himself bleed daily to remind himself that he is still a living person, not a weapon. Carmine had developed a reputation for trying to escape since he was about four years old. His common method is to use the air vent. This led him to enter various places in the Shinra Tower, such as the Materia Research Facility, SOLDIER Floor, Visual Entertainment Hall, the President’s Office, executives' offices, Hojo's laboratory, and the Skyview Hall, his favorite place. He would collect items, such as office supplies, Shinra/SOLDIER merch, souvenirs, and human teeth from Hojo’s lab. He also managed to collect Lazard’s pens, the Turks’ weapons, Palmer’s luxury eating utensils, Heidegger’s medals, Hojo’s lab equipment, Scarlet’s picture of her daughter, Vermillion, Rochelle’s makeup, Rufus’ coins and shotgun bullets, Junio’s handbags, and President Shinra’s car keys to one of his sports cars. He would then give most of them to the other children in Deepground, such as his siblings and his peers like Fuchsia. There are even times that he took jewelry and fur coats to Rosso in secret. 

When Sonon and Yuffie got captured, Carmine was one of the first to interact with the former. Though their first interaction was brief, the ninja intrigued the young boy. Once they get to meet more formally, Sonon is assigned to fight him, later, he tells him about the “Outside”. Sonon said how Wutai was a great nation before Shinra ruined it and that there was so much freedom. Carmine was amazed by this and would repeat it to the other children. 

Once Yuffie becomes pregnant, Carmine overhears conversations about a potential dissection of the baby, which would lead to its death as they are deemed “disposable”. The boy immediately told Sonon, who then told Nero, Weiss, and Yuffie. The Pure White Emperor then created an escape plan that would lead them into Wutai. Days later, Carmine hit his head hard enough that he got a concussion during training. In the infirmary, when Shelke is assisting in examining him with the doctors, she discovers that Carmine's “obedience” chip has been destroyed. This allows him to go against the Restrictor and gain free will. From there, Weiss sees him as essential to the escape. Soon after, Weiss secretly told the young boy how he could help. 

There are three tasks he needs to do once he escapes;

Seek out Viceroy Sarruf for Weiss to remove him from power for the Kisaragi Family to reclaim control over Wutai

Search for Wutai’s materia, especially the Leviathan materia

Return to help with the breakout once he gets Shelke’s message 

As a way to escape, Carmine forced his body into “Bleeding Sleep” to give the appearance of being dead as blood came out of his mouth and his heart stopped. Several Deepground children, including some of his siblings, pretend to mourn for him. The Restrictor, disgusted by Carmine’s “corpse”, threw him away in the trash truck. Carmine uses “Bleeding Sleep” long enough to wake himself and crawl out of the trash. He looks out of the trash truck to see the sun for the first time, amazed by the sight of freedom. Using this as an opportunity, he jumped out and headed to the nearest town outside of Midgar, Kalm. From there, he learns more about 

Along with getting a red hoodie, black pants, and slightly ragged white sneakers, he also gets a small rifle and stolen materia for any self-defense. 

Now, with all of his equipment, Carmine hopes to complete his tasks to ensure the escape from Deepground and the rescue of his family. As he travels, he learns to become more human and gain experience. 

[vampirevirus]

☠ intro post ☠

✶⋆.˚ hello, i'm diskey! i'm a 15 year old artist. i go by they/he/she/it + neos. (you can also find the other names i go by there!)

this is my more personal/casual sideblog. i also still post art here! this is mainly just a blog to more mindlessly ramble and be both cringy and more blunt about shit that Bugs me .. also vent from time 2 time .. warning that i can get quite rude-sounding/critical here! ..

you can refer to my main blog for more info abt me:

♥ .. (has my byi, dni, kinlist, and more!) .. ♥

╰┈➤ https://www.tumblr.com/dallydillying ࣪ ✶⋆.˚

𖦹・゜!! fandoms/special interests !!・゜𖦹

most active/hyperfixated on are in red !!

✶⋆.˚ sorted by mediums ✶⋆.˚

╰┈➤ books/movies: warrior cats, electric dreams, wonder (both the book and movie)

╰┈➤ web series/online media: osc [bfdi, ii, tdos, hfjone, ppt2, etc] eddsworld, happy tree friends, dick figures

╰┈➤ shows: spongebob, ginga, oobi, chalkzone, moral orel, loiter squad, home movies, rick and morty, aqua teen hunger force, 12 oz mouse, superjail, lazor wulf, henry danger (hear me out okay), blues clues/blues room, team umizoomi, max & ruby, wonder pets, tickety toc, fanboy and chum chum (i am So sorry), pocoyo, mlp:fim, wow wow wubbzy, unikitty, yo gabba gabba, peep and the big wide world, wild kratts, nature cat, storybots, ruby gloom, doug [nickelodeon since i haven't seen the disney iteration], making fiends, rocko's modern life, robot jones, mao mao: heroes of pure heart, sheep in the big city, foster's home for imaginary friends, robotboy

╰┈➤ games: cookie run kingdom, sally face, touhou, corpse party, detroit become human, undertale/deltarune, wobbledogs, portal, night in the woods, the walking dead games [telltale], danganronpa, omori, geometry dash, bully [video game], the i became a dog games, mouthwashing, henry stickmin, fnf, ddlc, animal crossing

╰┈➤ multi-section media/misc: pokemon, spy vs spy/mad, lalaloopsy, hoops and yoyo, the kagerou project, vocaloid, blaseball, razia's shadow [musical], batman

╰┈➤ bands/artists: the beatles, system of a down, tyler the creator, car seat headrest, kmfdm, korn, malice mizer, sade, esthero, erykah badu, billie holiday, imogen heap, death grips, underscores, ween, pearl jam, say anything, sublime, cocteau twins, red hot chili peppers, aphex twin, tally hall, weezer, streetlight manifesto, ludo, the cure, the smiths, kendrick lamar, lemon demon/neil cicierega, radiohead, glass beach, panchiko, tennis, linkin park, 2pac, static x, they might be giants [WILL FOR SURE BE UPDATED. i definitely forgot some.]

𖦹・゜!! TAGS !!・゜𖦹

CHANGELOG (1/2/25): #vanityvague and #vanityvents are now #vaguevampire and #vampirevents!

______________________________________________________________

╰┈➤ #diskey discussions: talking, hyperfixation-induced rambling and theorizing, basically just normal posts really

╰┈➤ #diskey drawings: art i decide 2 post here

╰┈➤ #diskaroke: me quoting song lyrics

╰┈➤ #diskuote: me quoting media i like

╰┈➤ #diskourse: involves blunt gripes/complaints i have with fandoms, also talking about controversial subject matters

╰┈➤ #vaguevampire: vagueposting

╰┈➤ #vampirevents: vent posts

☣ stamp dump below [no visuals in this post made by me] ☣

....... [don't feel like tagging] .......

[algebraic-dualist]

Just recording a few notes on Sheaves and Duality by Gehrke et J. v. Gool for later incorporation into my universal algebraic geometry blog post series.

Let T be a monad on Set.

Definition: A T-presheaf F on a topological space X is a functor F : (ΩX)* -> Alg(T). It is a sheaf of T-algebras iff for every open cover {Uᵢ}ᵢ of X alongside elements {sᵢ}ᵢ with each sᵢ ∈ F(Uᵢ) such that sᵢ|Uᵢ∩Uₖ = sₖ|Uᵢ∩Uₖ for all i and k, it follows that there exists a unique element s ∈ F(X) such that s|Uᵢ = sᵢ for each i.

Definition: A T-bundle over a topological space X consists of a topological space E alongside a local homeomorphism p : E -> X such that for every point p ∈ X setting Eₓ = p*({x}) it follows that we have a T-algebra (Eₓ, eₓ), and the partial morphism e : T(E) -> E induced by the eₓ is continuous where it is defined. We call Eₓ the stalk of the bundle at x.

For every open subset U of X a local section of p is defined as a morphism s : U -> E such that ps = 1. We write Γ(U) for the set of local sections of the bundle. Note that it is a subalgebra of the direct product ∏ x ∈ U : Eₓ This induces a sheaf Γ : (ΩX)* -> Alg(T) where restriction acts in the obvious manner.

Conversely, given a sheaf F : (ΩX)* -> Alg(T) it is possible to construct an etale space which gives a bundle inducing the sheaf.

Definition: A geometrisation of a T-algebra A is a sheaf F : (ΩX)* -> Alg(T) such that F(X) is isomorphic to A.

Question: When does a T-algebra admit a geometrisation? What do the stalks of a geometrisation reveal about the algebra?