#matrice 4

[eltristanexplicitcontent]

Introducing DJI Matrice 4 Series: The Age of Intelligent Flight

[jyunism]

i want an A for maths 😞

[liamphotoatl]

(Liam Photography Podcast)

[masquedrones]

DJI Matrice 4: La última apuesta de la marca china en el mercado profesional

DJI ha presentado su última apuesta en el sector de drones para empresas con la serie Matrice 4, que incluye los modelos Matrice 4T y Matrice 4E. Estos drones compactos están diseñados para una amplia variedad de aplicaciones, empleando tecnología avanzada que abarca capacidades de inteligencia artificial y sistemas de sensores robustos. El objetivo es optimizar las operaciones aéreas en…

[yearningforevermore]

Hell hath no fury like a person wearing braces for the second time.

[sreepadamangaraj]

In this webinar excerpt, join Enterprise UAS EVP Randall Warnas and Parrot Drones Sales Manager François-Xavier Charbonnel as they discuss the security features of the Parrot ANAFI Ai.

[corniart]

Happy wip whenever!!! Have more wizard losers and because I think it would be fun:

Sweet Bite of the Endless Night: a salacious rare gem of a novel by Matilda Merceria. Details a hot whirlwind of a romance between a bastard’s son of an empire royal sent away to the dynasty and taken as a ward to a member of a semi notable fictional den. Far less racist and surprisingly well informed, this novel does not many copies in print and banned by the empire, and not well known in the dynasty. Caleb found it in a smut bookstore in Port Damali

Practical Applications of Advanced Transmutative Matrices Vol 4-6: A collection of journals compiled by the Aydinlan Seminary. Very Dense. Very Dry. Only two wizard nerds could find this sexy.

An Iconoclast’s Treatise: The Necessity of Objectivity in Potentiality: authored by one E.Thelyss. It was his first solo publication, he published it in common rather than undercommon to make a statement. In Esseks opinion he was young, brilliant, and thought himself invincible. This publication however, taught him just how much he was under his Umavi’s thumb and by extension the dynasty. It was promptly banned from the Marble Tomes and largely buried from the dynasty’s record. It’s a bad face for the Shadowhand after all. However he still has fondness for that old work, it was well written after all.

Have a suggestion? Put them in the tags! (I love y’all’s tags. I see all of them 🫶🫶)

[alwayswriting101]

Can you recommend some Florelia fanfics please?

So this is actually a terrible question to ask me because I think all Florelia fics are amazing just by the nature of having my OTP 😅.

But here are my general recommendations:

Literally anything and everything by @florelia12. All of her work is fantastic x1000, But Moonlight & Monsters has literally changed my life. I love this fic (and all her fics) soooo much. She is also a (obviously) a massive Florelia stan. AO3 page

The Florelia chapters in Winx Club One Shots by SeaEmerald (chapters 2, 5, and 6). Oh my God is their writing incredible. Very highly recommend all of their fics, if you're interested in other couples/fandoms.

A Natural Fit: Soulmates AU. Also anything by QueenofHufflepuff. I love their writing style to the point where it's inspired my own fics!) They also go under Matrices and Vectors on ffn.

All the Petals Fall by @lynpheas. I swear to God this fic made me quit writing because like what point is there in writing when a fic this good exists 😭😭😭. Also recommend all of @lynpheas works but not all of them are Florelia. AO3 page

Black Sky by SilverSongstress. I cannot emphasize how epic this story is. There's lots of Florelia conflict (that doesn't really get resolved by the end (at least to my memory)) and I was okay with that! There are some issues re: formatting but it's truly a very strong story.

Lunar Rainbow and Under the Cherry Blossoms by Roxy Fan 4 Ever. Literally love these fics so much. So romantic!

The Neighbor by libra986. Now I actually don't remember shit from fuck from this fic, but I remember being OBESSED with it...do with that what you will...

High Temperatures . I like this one.

Six Years, This one is just so nostalgic for me, I think it was one of the OG Florelia fics 😭. I can't tell you how much it PAINS me the sequel is unfinished.

I Promise, Courage from the Heart, Save Me, My Turn to Save You, and Remember Me by Lill2350. Again very nostalgic fics! I think they have a few more Florelia fics as well.

Living a Dream. Very Cute fic.

Chastity Slightly NSFW so don't open if you're at work or underage. But I thought this was a really cute/romantic fic.

Style by KinkyWings. I think this fic is really, really cute. The author has other fics based on more...untraditional pairings (and fics that are definitely NSFW), but I really like this one.

Waned Worries. Why did this person orphan this account 😭? Really good oneshot.

Kamilia has written a bunch of Florelia stories, too many to list and many of them NSFW in one way or another, but check them out if you'd like!

Flora's Christmas. Just a cute Christmas Time Oneshot

As an aside: here are some fics that I tend to really enjoy that focus on Flora or (mostly) Helia as individuals and not as a couple.

Retired, Falling, and The Moonlight Project by @redemptionarcsucker. Their characterizations of Helia are unmatched. And their writing is impeccable! AO3 page

Also @floralovebot has really great Flora/Helia/Florelia headcanons/characterizations on their page.

A Conversation Between... Literally, love this fic with every beat of my heart since I read it years ago. This fic actually focuses mostly on Stormy but Flora has a lot of characterization through her.

The Meaning of the Color Black and Inspired by Kikurukina Bal Des'cagel. Fantastic Helia centric stories by a fantastic author. They have also written on other pairings.

Names. A random, unromantic one shot but I really like the concept.

Never Have I Ever and A Specialist's Journey by QueenofHufflepuff. I've already raved about them in the rec above. Really amazing work. The former is Flora based and the latter mostly Helia based (from what I can remember).

The Shimmering Sunflowers. I am a big fan of fics where Flora is a big sister to Miele.

And finally, I always recommend The Great Witch Rebellion, Brothers at Arms, Band of Brothers, and Bonds of Brotherhood by tearsinrain These fics are focused on the group as a whole and are basically rewrites of the main storyline, but these fics are fantastic. I would bind them into a book if I could. The characterization of Helia (who unfortunately is not really featured in Brothers at Arms), is amazing. And the Florelia scenes that are available are really amazing.

I am a big, big fan of Florelia. If I could fit every fic featuring them onto this ask, I would, but unfortunately, I can't! But if anyone has any additional recommendations please feel free to add! I'm sure there are some stories/authors whom I loved but might have missed, but this is a very general list.

[liacademia]

day 64/100

it was nice yesterday, i did pretty well, but we could be better (also i have a habit of adressing myself as "we" because it's me and my inner demons, we're tight like that)

also, i was wondering if i am an emotionally masochistic human, since i read angst and cry hard but then do it again, what fetish is this 😞

well, here goes the to-do for the day:

day 4 lectures on determinants and matrices (2 hrs) + problem sheet

day 1 lectures on alcohol, phenol and ether + problem sheet

haloalkanes and haloarenes live lecture

kinetic theory of gases lecture

solve more problems on waves and sounds

[physalian]

The Fantasy Language Translation Matrix

Whether you intend to write your own full-blown lexicon with different verb tenses and formal vs informal language, need unique words for spellwork, or just need new names for all your foreign places, behold… the Physalian patented Fantasy Language Translation Matrix.

(I kid. I have no idea if I’m the first to come up with this)

**Disclaimer!** After rolling out your fresh new vocab off the word assembly line, make sure you google it and that it doesn’t already exist and mean something you don’t intend.

Step 1: Pick your Derivative

You can make it sound completely foreign and like total gibberish, but I find it easier for you and other people to read if they have some real-world reference to compare it to, and so they have a clue for which pronunciation rules to rely on. For example: I did not know who René Descartes was my freshman year of high school. His last name was in my algebra book, and I, thinking he was Greek like so many other ancient mathematicians, pronounced his name as if he were Greek “Des-kart-ees.” I got made fun of.

Spare your readers the humiliation.

So say I want a vaguely… Russian/Latin/Italian influence. As opposed to French. Cool. That’s my starting point.

Step 2: Reorder the most common letters from English to your new language

In English, the average use of the standard alphabet by letter in order is this:

Ignore your vowels for a second. I don’t use charts like this on the regular, I use the Wheel of Fortune method and focus on RSTLNE, then go from there. I also want to make sure this isn’t a complete 1:1 ratio so it’s not super obvious I’m just juggling letters around, so I’ll knock out some “duplicate” letters and swap out singular letters for specific sounds.

The goal of this isn’t to stare at two existing language matrices and perfectly match them up, it’s to take the most common sounds and letters in English and make them new, common sounds in your new language, to sound more uniform and like you have a real etymology.

And I end up with this:

This might look a little confusing on how I got from A to Z so the basics:

All my vowels remain in the same place, they just get juggled around so I don’t end up with 8 consonants next to each other and word garbage

My “duplicate” letters are combined so I have more room for the new sounds, like c/k, f/ph/gh, h/wh, s/z. The new sounds then get the spare letters I had left over

Common english suffixes get reduced down so the pattern isn’t as obvious

If you want to include accent marks, this is your chance

I wanted to really emphasize the long “e” and long “i” sounds, so those got extra attention

Step 3: Translating

Oftentimes this is not perfect, or you end up with a word that just doesn’t fit the rest of your new vocabulary, because English is the bastard lovechild of German, Latin, Danish, and French.

I start with English, usually, but if the English word is too short or too long, I translate it first into another language, like Spanish, and go from there. Like “bus” vs “autobus”.

Using your matrix, go one by one. Let’s use a word like “letter”.

English: L-E-T-T-E-R

New: T-A-C-C-A-Z

Step 4: Polishing

So now I have my new word: “Taccaz”

Which is serviceable. I can throw an accent on either A or fiddle with the Z. I can start with “carta” instead and end up with “kizci”. The matrix is just a starting point. It’s designed to streamline the process when I’m otherwise feeling uncreative and in a rush, and it moves very quickly when I need to come up with full phrases and sentences that someone would actually say.

Step 5: Full sentences

This is only if you’re really digging deep and not coming up with the occasional fantasy curse word or new name for your fantasy land/realm/noun etc.

For this you’re going to need lots of tables. I based mine off romance languages because I know Spanish and romance languages make sense. This is where you decide how many pronouns, if any, you’re going to use, how the infinitive changes based on past, present, or future tense, how many nouns the word references, etc.

This is… a lot. Way more than you’d ever need for your manuscript. Ever. But I did it just for my own sake. Does it get long? Yes. Does it get tedious? Yes. The point here is to have little pre-manufactured word bytes you can plug and play with, with as little mental effort as possible so you can save it for the rest of your work.

I also came up with very common words already conjugated, like “to be” so I can just glance and type without having to remember to take “is” and go through the process over and over again.

Which means that I can take an entire sentence and translate it to my new language in about two minutes.

English: The payoff is worth it, this is so satisfying. New, roughly: Nu kioyb ela fyzip ne, iski ela valo nicenbalaev.

Of course, you can keep tinkering until you get something that’s easier on the eyes (I’ve been working with this language for years so I can read it pretty well), but not all languages are smooth and pretty and simple.

To be frank: Most readers will just gloss over this stuff anyway, but it shows that you put in the effort and it enhances the lore and the immersion when you do this. At least in the written medium. You can’t ignore it if this is meant to be in a screenplay.

Is this what a language professor would do or recommend? Probably not, I have no idea. Does it work? Yes. I have a fully functioning grammatical system where any input can give me a legible output.

To make this yourself, just change the order of the letters around, adjust your shortcuts, and come up with your own common sounds for those last two rows. The conjugation matrix is where you can really make it distinct, assuming you are basing yours off a romance language, which you don't have to.

—

And there you have it!

Don’t forget to vote in the dialogue poll before it closes!

[al-hekima-madara-blog]

Yin Yang Generality (Part 1/2)

My goal was initially to talk about the Yin/Yang concept in Naruto, but I realise I have first to explain a little bit what I mean by that so we're all in the same page. kudos to @shisui-uchiha-rp for inspiring me this double post.

Disclaimer : Usually people reading me are quite smart but who knows some people don't get use to my style. There is no meliorative or derogative values behind Yin/Yang or Female/masculine if you feel that one is better than the other. IT'S ON YOU. The author is not your mum, not your therapist, not your teddy bear to vent upon. If you don't feel comfortable and triggered here are my solutions : move on from my post, unfollow, or even block. But if you're high IQ, open minded, not self-centred or just generally curious and well-behaved then welcome and keep reading.

The duality is probably one of the oldest concept in metaphysic. It's presented in many cosmologies to explain the origin of the world. After the Unicity or primordial chaos comes the duality of opposite forces: Positive/Negative Light/Darkness, Up/Down, Heaven/Earth. A bit like the cells division in biology. You don't go from 1 to 8. You go first 1 splitting in 2 splitting in 4 splitting in 8... So duality is the matrice of any system. It's the fundamental for the law of nature, the balance in the universe and its complex beauty. It's not Yin against Yang, or Yin>Yang or Yang>Ying. It's about how Yin and Yang balance each other to create stability, order and peace.

What's Yang?

I like to represent it as a centrifugal force. A cyclonic energy expanding outside of itself. And by expanding , it takes place and when you take place, it grows. You create motions and frictions that create disruption and may lead to violence. Going outside can also means transcending an initial state. Looking for answer outside of oneself. Exploration of the unknown. To pierce and penetrate others. You've probably heard that sky crappers or obelisks are sometimes referred as phallic symbol, it's due to their shape that represent the will to project power vertically out and above others. Being the number one. The process of individuation, ego, independence, but in positive way, being able to assert one's will and not fearing judgement.

Yang is often associated with light because it's also acting in the open, unmasked, direct and loud language, explosive presence, rational.

An other thing coming to my mind, Yang energy is like an arrow or a missile going straight to the point.

In the negative, it can also being associate with escapism, distance, scattering when the ambition goes to far, separation, division, detachment, dryness. Ultraviolence, megalomania, destruction.

What's Yin?

The opposite is the centripetal force. Think about a spinning vortex, a black hole. Or just water whirling in your sink. The centripetal motion brings both the idea of cycle and gravity and when you extend the move to its maximum, the centre reaches immobility. And if you go even further you have high pressure exerts on a small spot.

In nature, do you know what's happened when carbon is submit to an extreme pressure in the depth of Earth's mantle? It crystallises into a diamond. And the diamond's attribute, except its beauty, has the highest hardness and thermal conductivity of any natural material. It's indestructible. It facilitates vibration. It's solid!

Hence why someone known for an extreme mastery of Yin energy is said to be intense. Because the spiritual core of this person is so strong, that the mere presence into a room is enough to create "gravity" toward them. It doesn't matter if the person is attractive or not, good or mean. Yet they have this thing...this magnetic charisma that hypnotises people.

Yin energy refers to obscurity, what is hidden, invisible, cold, passive, receptive, and wet. Observe again nature : a DNA nucleus, the pit of a fruit, seed, or the star in a solar system is the most important part. It's the roots. the raw material from whom everything else proceed and rotate. It is extremely powerful because it contains all potentials.

Something you can't see always seem more powerful and menacing that something that you can observe and measure. Imagination is always infinite compare to reality and its limitation. For instance : there is a lion in front of you, you might have a tiny chance to survive if you run away. But if I tell you there is a lion...somewhere around you. It seems suddenly the animal is everywhere and you can't prepare yourself from something you can't see. The paranoia will imagine that the slightest wind is the beast ready to jump on your throat. But maybe it's not true, maybe the lion doesn't exist at all and yet in your imagination it was very real!

Contrary to Yang which is direct and straight. Yin is curvy, suggestive, indirect, like a fog, slow like a sugar soaked by water. We can call it the boiling frog method. You slowly raise the temperature so nobody notice. When it starts to be too hot it's too late, the frog has been weakened for hours and don't have the strength anymore to escape the trap. In a Yang manner, if it has been directly from room temperature to 100°C, it would have jumped out immediately.

In negative a bad Yin can create obsession, brooding over and over the same thing. A sensation of inertia, mental instability, self-destruction. Implosion when the core is falling into itself due to too much external pressure or internal fragility. Narcissism and not being able to see the bigger picture outside of oneself. Also manipulation, lies, victimhood mentality, passivity to act, delusion.

Yin/Yang in the human species?

From a simple observation of anatomy it's quite obvious who's Yin and who's Yang.

Men are generally taller than women, broader shoulders and rib cage which aloud them to have a deeper and louder voice. the muscle mass is also higher with less body fat. The sexual organ is external and I don't need to go in detail on how it erects and projects to accomplish its mission... Also the limbs and the hips are straight. the pelvis is narrow, brow bone more proeminent.

Women in contrary are in general smaller than men. With a tendency to keep more fat in the lower part, than muscle. narrow shoulder, narrow torso, wider hips and generally curvier limbs : hips, breasts, rounded face, brighter skin, bigger eyes. And of course the sexual organ is internal and everything about pregnancy is hidden inside the darkness of the womb.

In term of character There is a saying that : Men kill you, Women drive you crazy.

Take in a positive light, Men can build new worlds with their hands, and women can help to civilise and heal the world. Would you live in a high tech city where people behave like cavemen? and in reverse would you be with kind and compassionate people in the wild with no protection, no home, no technology, in open air?

A lot of psychology studies has shown that from childhood, girls have a tendency for emotional connexion (networking, relationship) while boys are interested by things (how to built, how to work, how to perform)

But does it mean that a man is 100% yang and a woman 100% yin? No it's absurd. I think you kind of notice I tend to have a yang way of expressing myself. I don't really care if my posts hurt someone's feeling. I don't care about trigger warning, I don't care about politically correctness, I don't write to protect harmony among the group but to express what is in my mind, now. but does it nullify my womanhood card? Does that make me a man? Absolutely not, I just have my own and unique temperament and everyone is different. Yet I'm still deeply Yin. And if we observe humanity as a group, it's obvious that the most enchanting embodiment of Yin energy are Women and the most magnificent embodiment of Yang energy are Men.

Yin/Yang in other aspects

I like this term better than female/masculine because it's less restrictive than just humans. It can apply to many aspect of life

A student is yin when he receive knowledge from his teacher in a yang position.

The same teacher is yin when he's under the authority of the university director yang.

In the military field a samurai has a yang culture while shinobis by acting in shadow and using deception way have a yin way of fighting.

The soul is yin, the body is yang

Moon reflect the sunlight (yin), the sun is the source of light (yang)

A mother is yang toward her children (yin)

In a production the famous actress is yang while producers, and staff are yin (they work in the back sometimes no one know their face or their name). but in the same production who has the actual power? The production pouring the money and writing the script? the famous actress that attract people due to her popularity? The success of the whole film is a mix of both energy.

In the end Yin/Yang both are important and both are complementary. Yin can't live independently from Yang and Yang can't survive without Yin. Light produce shadow and shadow is enhanced by light. You can't appreciate shadow without light and you can't see the beauty of light without shadow.

Part 2 : Yin/Yang in Naruto, the case of Madara and Hashirama

[unnatural-transformations]

My Favorite Mathematical Proofs [2 of n]

For any convex polyhedron with V vertices, E edges and F faces it is always true that V-E+F=2. This result is known as Euler's formula, named after the prolific Swiss mathematician Leonard Euler. In more modern terminology, we can also express this fact by saying that a convex polyhedron has Euler characteristic equal to 2.

However, the story of this formula goes back to well before Euler, and it would be decades after Euler before any really satisfactory proofs were discovered.

Ancient Greek mathematicians had proved the existence of exactly five convex, regular polyhedron [also known as the Platonic solids]. Recall that these are:

The tetrahedron, which has 4 vertices, 6 edges and 4 [triangular] faces

The cube, which has 8 vertices, 12 edges and 6 [square] faces

The octohedron, which has 6 vertices, 12 edges and 8 [triangular] faces

The dodecahedron, which has 20 vertices, 30 edges and 12 [pentagonal] faces

The icosahedron, which has 12 vertices, 30 edges and 20 [triangular] faces

[The images above are from the Polytope Wiki and were created using Robert Webb's Stella Software, http://www.software3d.com/Stella.php]

In the 16th century, the Sicilian mathematician Francesco Maurolico observed that all five Platonic solids satisfy the equation V-E+F=2. A couple of centuries later, Euler -- who probably wasn't aware of Maurolico's earlier work -- discovered that the same formula seems to hold for convex polyhedron more generally [i.e. without the assumption of regularity]. However, Euler himself did not publish a convincing proof. (It feels a bit surprising that it took mathematicians this long to notice what now seems such an obvious fact. In fact, in the 17th century René Descartes had stated a result about the face angles of a polyhedron which implies Euler's formula, but Descartes doesn't seem to have explicitly recongized the connections between face angles and edges.)

Imre Lakatos' (excellent) book Proofs and Refutations uses a (deliberately very simplified) version of the history of attempts by mathematicians to investigate this formula as a way of discussing Lakatos's thoughts on the role of proof-attempts in developing mathematics. Rather than describe the actual history, Lakatos presents a discussion of Euler's claim in a classroom setting, with the various students (named Alpha, Beta, Gamma and so on) variously attempting to prove, falsify, defend or expand the scope of the original formula. Reading this book as a teenager made a huge impact on me, and I'd recommend it to anyone interested in the philosophy of mathematics who hasn't read it before (just don't take the historical footnotes too literally...).

I was particularly impressed by the final proof Lakatos presents (through the student named Epsilon), which is actually due to Henri Poincare (who is really the first mathematician to establish topology -- or 'analysis situ', as he called it -- as a branch of mathematics in its own right).

Epsilon's proof works by first translating the geometric definitions of polyhedron (or more generally any n-dimensional polytope) into purely combinatorial language. A polyhedron is a collection of objects called vertices, edges and faces, all of which can be related to each other in terms of incidence matrices. These incidence matrices let us translate our new combinatorial terms into the language of linear algebra. The vertices, edges and faces of our polyhedron become the basis of vector spaces over the field with two elements. The geometrically intuitive notion of a 'boundary' (of a given polygonal face, say) corresponds to a linear map between such vector spaces, which sends a given face to the sum of its boundary elements (a polygon is mapped to the formal sum of the edges that bound it, an edge is mapped to the sum of its two vertices, and so on).

Finally Euler's original claim about polyhedrons becomes a claim about the properties of this map, which we can prove directly using the rank-nullity theorem.

For the simplest possible example, consider the tetrahedron $P^T$.

(Obviously in any worked example like this when we can simply read off the number of faces, edges and vertices such a proof seems redundant. The point is that we can show that this approach will work for any convex polyhedron, or indeed convex polytope, not just a given one.)

The faces $F^T_i$, edges $E^T_j$ and vertices $V^T_k$ are related as shown in the Hasse diagram below.

We associate a vector space to each row of the diagram, with basis given by the number of nodes on that row. The boundary maps δn map each basis element to the [basis element corresponding to the] sum of the nodes incident to this element on the row below. This then extends linearly to a map defined on the whole vector space.

It can be checked directly from the diagram that the boundary maps are as given below:

Indeed, working over the field with two elements (so that addition is modulo 2), we can explicitly calculate the kernel and image of all four of these maps. The results are summarzied in the table below.

Note that the image of δn is always equal to the kernel of δn-1. Epsilon argues -- or asserts, anyway -- that this is necessarily true for any simply connected polyhedron [and in particular any convex polyhedron]. A polyhedron is simply connected if it contains no 'holes'; and a hole in a polyhedron would correspond exactly to an element of the kernal of δn that was not an element of the image of δn-1.

Assuming this to be true -- and noting that the images of both δ3 and δ0 are always one dimensional for any single polyhedron, we have, using the rank-nullity theorem and the fact that F, E and V are by definiton the dimensions of the vector spaces V2, V1 and V0:

The proof as presented in Proofs and Refutations is not quite as rigorous as Epsilon claims (and indeed the slightly longer version that appears in Coexter's Regular Polytopes suffers from some of the same flaws). Nonetheless, I found it pretty amazing when I first saw it. It would be a few more years before I knew anything about algebraic topology or category theory [even basic linear algebra was quite new to me at the time], and this idea of solving a problem in one mathematical field by transforming it into something that seemed to be completely unrelated kind of blew my mind at the time.

As always, more details below the cut, where I try to sketch a version of the proof that holds for n-dimensional convex polytopes.

We start by defining a convex polytopes, which are a generalization of two-dimensional polygons and three-dimensional polyhedra to any n dimensions.

This definition doesn't mention vertices explicitly, but these arise as particular points in the convex hull. For example, the tetrahedron is the convex hull of its four vertices in three dimensional space, while the (regular) pentagon is the convex hull of its five vertices in two dimensional space.

However note that two different sets may have the same convex hull. For example, if a and b are points and we add a new point 0.5a + 0.5b this clearly does not change the convex hull. More generally if any point p can be written as the convex combination of a set of points S then the convex hull of S ⨃ {p} is just the convex hull of S.

The vertices (or extreme points) of a convex set are exactly those elements that cannot be written as a convex combination of any other points.

Definition 1 also makes no explicit mention of 'faces' or 'edges' but these concepts can be recovered. It is conventional to use the more general term 'face' to refer to both edges and (two-dimensional) faces and their higher dimensional generalizations. (Some authors use 'facet' to specifically refer to the faces of codimension 1.)

[As @bubbloquacious pointed out in the replies, the version of this definition I stated originally didn't correctly define the dimension of a face. This definition has now been updated.]

The set of all faces has the structure of a particular type of partially ordered set. We recall some related terminology:

We can now introduce the face lattice of a polytope. This is a purely combinatorial object that (we claim) captures all the information about a convex polytope we will need for our proof of Euler's (generalized) formula.

See the texts by Grünbaum or Ziegler below for more details. The geometric properties of a polytope are often easier to reason about when translated into the languages of lattices. To begin to do this, we need a few more definitions.

The face lattice of a polytope has many of these properties. We will not use them all in the proof but list them anyway for background. The key property we want to make use of is the so-called diamond property.

For example, the dual of the face lattice of a cube is the face lattice of the octohedron while the dual of the face lattice of the dodecahedron is the face lattice of the icosohedron. The tetrahedron's face lattice is self-dual.

We will not prove Proposition 6 by the interested reader is again encouraged to check the references, particularly the book by Ziegler.

We will try to be a little more careful than Epsilon in establishing that the boundary maps have the properties we want. In words, we want to show that 'boundaries have no boundaries' and 'all cycles are boundaries'.

We just need one more result before stating our main theorem.

A proof of this result can be found in any linear algebra textbook.

Now we are ready to state and prove our result.

References & Further Reading

H. S. M. Coxeter, Regular Polytopes (Methuen, 1947)

Branko Grünbaum, Convex Polytopes (John Wiley & Sons, 1967)

Imre Lakatos, Proofs and Refutations (Cambridge University Press, 1976)

Richard Stanley, Enumerative Combinatorics Volume 1 (Wadsworth & Brooks/Cole, 1986)

Gunter M. Ziegler, Lectures on Polytopes (Springer-Verlag, 1995)

[shankhachil]

Countdown to JEE (Main): Week 2/33

I've done a lot this week! Hit almost all the topics I wanted to. You'll notice that there are some topics I've covered but not solved questions for — these are the ones I studied for the first time and/or made notes for.

Other than that, I have my school unit tests from next Wednesday, so there's that. Sadness.....

Test results:

Test at physics tuition center: 116/120, rank 1/67! Yay!

Aryabhatta National Mathematics Competition: score unknown, result expected on 10/07/2024.

Topics covered:

Physics: Potential and Capacitance; Electromagnetic Waves, Waves on a String; Modern Physics; Gravitation; Electromagnetic Induction (6/3)

Chemistry: Chemical Thermodynamics; Atomic Structure; Chemical Equilibrium; Halogen Derivatives; Solid State (5/3)

Mathematics: Differential Equations; Area Under a Curve; Applications of Derivatives; Determinants; Complex Numbers; Ellipses; Hyperbola; Binomial Theorem; Functions (9/3)

Questions solved:

Physics: - FIITJEE Electrostatics* module, Assignment section — 56 questions, 52 correct - Allen Potential and Capacitance module, O1 and O2 — 88 questions, 77 correct -FIITJEE JEE (Main) archives, Electromagnetic Waves — 12 questions, 10 correct - Allen Waves on a String module, O1 and O2 — 67 questions, 59 correct - FIITJEE JEE (Main) archives, Modern Physics — 40 questions, 34 correct - Allen Gravitation module, O1 and O2 — 61 questions, 55 correct Total: 383/60 questions, 287 correct *FIITJEE includes Potential and Capacitance under Electrostatics, while Allen does not.

Chemistry: - Allen Chemical Thermodynamics module, S1 and S2 — 40 questions, 37 correct - R. N. Sarin, Atomic Structure — 18 questions, 18 correct - R. N. Sarin, Chemical Equilibrium — 11 questions, 11 correct - Allen Halogen Derivatives module, JEE (Advanced) archives — 24 questions, 22 correct Total: 93/60 questions, 88 correct

Mathematics: - Allen Differential Equations module, O1 — 30 questions, 25 correct - Allen Area Under a Curve module, Do Yourself 1, 2, 3, 4, 5 and O2 — 51 questions, 46 correct - FIITJEE JEE (Main) archives, Applications of Derivatives — 45 questions, 39 correct - FIITJEE JEE (Main) archives, Determinants — 22 questions, 17 correct - Yellow Book, Complex Numbers, single-choice questions — 20 questions, 17 correct - Pink Book, Ellipses, single-choice questions — 16 questions, 14 correct - Yellow Book, Binomial Theorem, single-choice questions — 15 questions, 12 correct - Pink Book, Hyperbolas, single-choice questions — 21 questions, 18 correct - Allen Functions module, O1 and JEE (Main) archives — 48 questions, 44 correct Total: 268/60 questions, 232 correct

GRAND TOTAL: 744/400 questions, 607 correct

Upcoming tests:

23/06/2024 (next Sunday) — Allen monthly test. Topics: Kinetic Theory of Gases; Physical Thermodynamics; Electrostatics; Potential and Capacitance; Current Electricity; Electromagnetic Induction; Alternating Current; Electromagnetic Waves; Waves on a String; Sound Waves; Ray Optics; Wave Optics; Circle; Functions; Differentiation; Applications of Derivatives; Indefinite Integrals; Definite Integrals; Area Under a Curve; Differential Equations; Matrices; Silicates; Molecules That Do Not Exist; Coordination Chemistry; Metallurgy; Electronic Displacement Effects; Halogen Derivatives; Atomic Structure; Chemical Equilibrium; Solid State; Solutions; Chemical Kinetics; Chemical Thermodynamics. Yeah. I've no idea how I'm going to finish the syllabus in time, since I haven't even started Coordination Chemistry, Metallurgy or Alternating Current — they finished the classes in Allen before I even joined the batch. Anyway, let's see.

That'll be all till next week — see you again!

[then-ponder]

So I was talking about adding dissimilar matrices together: think 3 × 3 and 4 × 2. And it occurred to me that the main problem with those things is that ultimately we assumed the matrixes to represent the same space and not different spaces. For instance ℝ^3 is not necessarily a subset of ℝ^4. It could even be true that ℝ^4 ∩ ℝ^3 = ø.

I don’t think is a problem created by a closed operation as that’s my usual approach to generalizing problems, but I think this approach could be useful. More research needed.

[ballata]

Soffiano venti di Guerra? Russia Ucraina e la situazione critica nel Medio Oriente, combinati con l'instabilità profonda nell'africa subsahariana, le tensioni sempre maggiori nell' Indopacifico, mettono in luce un quadro con prospettive cupe. Ci davanti scenari estremamente fluidi e instabili. Anche se può sembrare che questi conflitti siano geograficamente distanti, sono direttamente collegati a noi con dinamiche profonde e interconnesse. In questo contesto riconosciamo una matrice comune, ossia uno scontro tra due visioni del mondo fortemente incompatibili.

L'ordine globale è minacciato da un nuovo acronimo, CRINK che racchiude le iniziali dei regimi coinvolti nelle crisi Cina, Russia, Iran, Corea del nord. I 4 costringono la Nato a rafforzare le relazioni con gli stati dell'Indo-Pacifico che condividono gli stessi timori. Australia, Nuova Zelanda, Corea del sud e Giappone hanno preso parte per la 1 volta ad un incontro a Bruxelles. Tuttavia, la Nato non è unita nell'espandere i legami con l'est. La Francia è stata la forza trainante nel bloccare il progetto della Nato per l'istituzione di un nuovo ufficio a Tokio, insistendo sul fatto che l'Organizzazione del Trattato Nord Atlantico si concentri sulla sua regione d'origine.

I Paesi dell'Europa centrale confinanti con la Russia vogliono che la Nato si concentri sul preparare la loro regione ad un'eventuale guerra con Mosca - e che lasci l'Indo-Pacifico agli Stati Uniti. La Cina, intanto, da anni avverte l'Alleanza di non avvicinarsi troppo alle 4 demi razie dell'Indo-Pacifico.

[smath-or-pass]

Rules/info

Any math concepts are acceptable, be they integers, other numbers, variables, operators, techniques, equations, formulae, matrices, etc

Geometric shapes and solids count I guess

Uhhhh if I think of anything else I'll just edit this

I'm the only mod and I'm starting a 4-week stats class (for context semesters at my college are 16 weeks and quarters are 8 weeks) so posting may be chaotic