Topological Invariant Puns

In on 2nd year of graduation, and I recently got in touch with a professor that works on a field that interests me. He told me that to get into his work, I should start taking a look in the concept of topological invariants. But he also told me that I shouldn't bother yet with the mathematical aspect of it, and instead look for the application in Physics, to get a good sense of the meaning. As an example, he showed me this part in a text:

"To get a feel for what a topological invariant is, consider Gauss’ law, in its integral form:

If S is a surface enclosing N electrons, the integral of the electric field over S is independent of the configuration of the electrons, or even the shape of the surface, provided the electrons remain inside the volume enclosed by this surface. This is the essence of a topological invariant, which in this example, is simply a count of the fundamental charges."

Where could I look for such topic and examples? I already found about it concerning the quantum hall effect.

Will someone knowledgeable in math please answer me? Why do you take the vertices and then subtract the edges and then add the faces to get the "difference"? Why not do any other combination of operations? For example, 12 - 6 +8 = 14, etc. That doesn't equal 2. Why isn't that the "difference"?

Sphere: V - E + F Torus: V - E + F

Abstract. We show that the number of gods in a universe must equal the Euler characteristics of its underlying manifold. By incorporating the clas- sical cosmological argument for creation, this result builds a bridge between theology and physics and makes theism a testable hypothesis. Theological implications are profound since the theorem gives us new insights in the topo- logical structure of heavens and hells. Recent astronomical observations can not reject theism, but data are slightly in favour of atheism.

Theorem 1. The number of gods in a universe equals the Euler characteristics of the underlying manifold,

Θ (U ) = χ (MU ) .

The formula spells trouble for all theologies which are based on a cyclic conception of time, which is widespread in India (Veda) and among native American religions. Since χ S1 = 0 there are no gods in such a universe,

Θ M ⊗ S1 = 0.

Correct me if I'm wrong, for I am only a lowly physicist.

The Chern number of a manifold can be computed as C \propto \int R_{abcd} *R^{abcd} where *R is the dual to the Riemann tensor. The Euler characteristic can be computed as \chi \propto \int *R_{abcd} *R^{abcd} .

Besides these two, what other combinations of curvature tensors (and other local geometric quantities?) can be integrated over a manifold and are topological?

Edit: and if not, what is the required cardinality of such a set?

new versions of knots and link invariants.

I was wondering if anyone knew how Homology and Topological Invariance were used in Topological Data Analysis. I've been reading up on the topic and I think I understand all the maths behind it but no where have I found how these features are used in TDA. Every source I have looked at has mentioned them in one way or another but none have actually stated how they are used to extract insights from the data sets.

I have already asked a similar question in /r/askmath but it must have been a slow day on that subreddit because there were a few upvotes to my post but no replies. I was hoping /r/math could help with my confusion. Thanks!

I'm trying to prove this using singular homology. I'm trying to show that the n dimensional closed upper half space minus the origin has trivial (n-1)^th homology group. Not sure how to do this.

edit/solved: Ok I thought there was some trick to this, but in the end I just wrote out the Mayer Vietoris sequence. Tell me if I'm right.

U = the open unit half ball

V = H^n - origin

Then {U, V} is an open cover of H^n which is contractible. Part of the sequence is

Hp(U int V) => Hp(U) X Hp(V) => Hp(H^n )

U, U int V, and H^n are all contractible, so the sequence reduces to

0 => Hp(V) => 0

Which says that Hp(V) is 0.

Confronted with the scale of the anti-efficiency and lack of scalability of the Bitcoin network (and other similar blockchains with proof-of-work consensus), its defenders frequently bring up so-called "Layer 2" solutions, which defer transactions to an off-chain system on top of the original blockchain which can then later be consolidated and settled as a single transaction on the original blockchain. By using these, Bitcoin advocates purport to gain greater efficiency while retaining the supposed advantages of a blockchain over conventional centralised, distributed databases.

The most commonly-discussed Layer 2 solution with respect to Bitcoin is the Lightning Network. Enormous advantages are claimed using the Lightning Network over the conventional Bitcoin network; claims have been made of transaction rates of 1 million transactions per second (tps) in a channel. However, like everything relating to cryptocurrency, it is worth investigating these claims with substantial skepticism, whereupon they do not appear to live up to the hype.

The Lightning Network: a case of "might, not is"?

A draft version of the Lightning Network was released in 2015. More than six years later, the Lightning Network is still far from the seamless, worldwide payment network that it promises, with poor adoption compared to its purported benefits. It's easy to make claims when arguing from hypotheticals, but when comparing them to worldwide payment networks which already exist practically, those claims have a bad habit of being aspirationalist wishful thinking.

The claims of the Lightning Network carrying out a million transactions per second may be true - in a very limited and artificial situation involving a single channel. But shuffling numbers between a set of simple ledgers is trivial. What's important is what the numbers look like in reality with an actual working network, not just marketing copy.

Bitcoin ultimately bottlenecks the Lightning Network

One of the fundamental problems with the Lightning Network is that it is tied to the Bitcoin network; creating a Lightning channel requires a Bitcoin transaction, while settling the channel afterwards requires another one. What's more, the Lightning Network does not do anything by itself to improve the scalability of the base layer and therefore, the rate of adoption is ultimately still bottlenecked by the t

This recurring thread will be for general discussion on whatever math-related topics you have been or will be working on this week. This can be anything, including:

• math-related arts and crafts,
• what you've been learning in class,
• books/papers you're reading,
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• giving a talk.

All types and levels of mathematics are welcomed!

If you are asking for advice on choosing classes or career prospects, please go to the most recent Career & Education Questions thread.

Well, the title says it all. What opinion of yours makes other people react this way?

This is a judgement free zone! Here you can admit that you don't believe Yalgeth's constant is smaller than 3, traverse detoid monofilaments kata-wise, and would really love if the Π-series were revived (although, maybe don't praise the Π-series too much).

>The inhuman in human beings: that is what the face is from the start. […] To the point that if human beings have a destiny, it is rather to escape the face, to dismantle the face and facializations, to become imperceptible[…]

Logic is compulsion, whether it's the (thematic) logic of who someone is or the (causal) logic of what something does. Maybe you've seen a rebuttal begin with, "By your logic[…]"; that is an attempt to compel someone into an interpretation of their own words, typically one that would undo their words by revealing a thread they hadn't seen. In the realm of human cognition, compulsion is both narrative and neuropsychological; a narrative might supervene on (as in vertical reduction), build to (as in neuropsychology itself), or flee from (as in strong emergence) neuropsychology. The metalogic of the zoo of possible relations—conditions for and consequences of—between them relies on their reciprocal presupposition: there is no narrative without the constraints of neural architecture, and these constraints lead to their self-discovery.

A culture is defined by its ecology and hierarchy of narratives, and typically the dominant narrative is central to, or even centralizes, that culture and analyses of it both within and without—even though the swarms of minor narratives are essential and constitutive too. In the plateau Year Zero: Faciality, Deleuze & Guattari articulate the logic of facialization: the substance, sustenance, and subsistence of dominant narratives. All dominant narratives are some mixture of authoritarian or despotic, and that mixture has two poles: objective destiny and subjective destiny.

Objective destiny imprisons us on a vast white sur-face on which o(ra)cular black holes sub-ject us to a world-tree of branching choices, choices which choose us, choices which compel us, choices which facialize us: "You don't so much have a face as slide into one." Your objective characteristics, such as socioeconomic status, lineage, physiognomy, dialect, sex, and so on determine your access, what doors are closed or disclosed to you, and therefore how far you fall along the tree before you find your niche, your role, that you might rationalize as your calling. This surface, this plan(e), stretches beyond sight, and its momentum only strengthens as it expands. Every branch is binary, and every ambiguity (e.g. homonymy) or non-binarity turns "X or Y? Neither!" into "(X or Y) or Z?" Even roles are binary with other rol

I don't want to step on anybody's toes here, but the amount of non-dad jokes here in this subreddit really annoys me. First of all, dad jokes CAN be NSFW, it clearly says so in the sub rules. Secondly, it doesn't automatically make it a dad joke if it's from a conversation between you and your child. Most importantly, the jokes that your CHILDREN tell YOU are not dad jokes. The point of a dad joke is that it's so cheesy only a dad who's trying to be funny would make such a joke. That's it. They are stupid plays on words, lame puns and so on. There has to be a clever pun or wordplay for it to be considered a dad joke.

Again, to all the fellow dads, I apologise if I'm sounding too harsh. But I just needed to get it off my chest.

I ask this because it seems that when we can compute homotopy groups of some class of spaces (like products of projective space or tori) they are sometimes called ‘nice’ or ‘good’ spaces. I know this depends on ones definition of ‘niceness’ but is there something about classifying homotopy groups of topological spaces that makes those topological spaces easier to understand? What’s the intrinsic difference between a ‘bad’ topological space (one we can’t compute homotopy groups on easily as far as we know) and ‘nice’ ones? Or is this just a matter of a lack of tools to do so?

Alot of great jokes get posted here! However just because you have a joke, doesn't mean it's a dad joke.

THIS IS NOT ABOUT NSFW, THIS IS ABOUT LONG JOKES, BLONDE JOKES, SEXUAL JOKES, KNOCK KNOCK JOKES, POLITICAL JOKES, ETC BEING POSTED IN A DAD JOKE SUB

Try telling these sexual jokes that get posted here, to your kid and see how your spouse likes it.. if that goes well, Try telling one of your friends kid about your sex life being like Coca cola, first it was normal, than light and now zero , and see if the parents are OK with you telling their kid the "dad joke"

I'm not even referencing the NSFW, I'm saying Dad jokes are corny, and sometimes painful, not sexual

So check out r/jokes for all types of jokes

r/unclejokes for dirty jokes

r/3amjokes for real weird and alot of OC

r/cleandadjokes If your really sick of seeing not dad jokes in r/dadjokes

Punchline !

Edit: this is not a post about NSFW , This is about jokes, knock knock jokes, blonde jokes, political jokes etc being posted in a dad joke sub

Edit 2: don't touch the thermostat

Do your worst!

Not a hole, the empty space inside is not connected to the space outside.

Ants don’t even have the concept fathers, let alone a good dad joke. Keep r/ants out of my r/dadjokes.

But no, seriously. I understand rule 7 is great to have intelligent discussion, but sometimes it feels like 1 in 10 posts here is someone getting upset about the jokes on this sub. Let the mods deal with it, they regulate the sub.

They were cooked in Greece.

I'm surprised it hasn't decade.

Don't you know a good pun is its own reword?

Two muffins are in an oven, one muffin looks at the other and says "is it just me, or is it hot in here?"

Then the other muffin says "AHH, TALKING MUFFIN!!!"

For context I'm a Refuse Driver (Garbage man) & today I was on food waste. After I'd tipped I was checking the wagon for any defects when I spotted a lone pea balanced on the lifts.

I said "hey look, an escaPEA"

No one near me but it didn't half make me laugh for a good hour or so!

Edit: I can't believe how much this has blown up. Thank you everyone I've had a blast reading through the replies 😂