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11h
comment What is the different semantic of existential and universal qualifier between classical and intuitionistic logic?
@JamesWood You can use Heyting algebras to show it is wrong (like here). It should have been $$\neg (p \land q) \iff \neg\neg(\neg p \lor \neg q),$$ I have no idea what made me write such nonsense, sorry for that.
11h
revised What is the different semantic of existential and universal qualifier between classical and intuitionistic logic?
deleted 147 characters in body
1d
answered How to denote an average of data satisfying a given condition
1d
comment Is there an established notation for this “replacement” operation?
There is a notation for subterm substitution, among others $[y/x]M$, $M[x := y]$, $M[x \to y]$ or $M[x \mapsto y]$, but your case is different, and for that particular reason I would create my own notation.
2d
comment For points A, B, does there there a billiard such that any trajectory from A will reflect twice and then reach B?
@PaulWright My intuition tells me that it could be possible only for $A = B$ for a constant and even number of reflections (of course, intuitions are frequently wrong).
2d
comment For points A, B, does there there a billiard such that any trajectory from A will reflect twice and then reach B?
@Blue I wonder if gluing infinite parabolas would be a valid solution ;-)
2d
comment For points A, B, does there there a billiard such that any trajectory from A will reflect twice and then reach B?
Does it have to work for all the trajectories? For example two parabolas may constitute some start (you can make the "wrong" region arbitrarily small).
2d
revised Is it true that: $|a_{n+1} - L| < |a_{n} - L| \forall n \in \mathbb{N} \implies \lim \limits_{n \to \infty} a_{n} = L ?$
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2d
answered Is it true that: $|a_{n+1} - L| < |a_{n} - L| \forall n \in \mathbb{N} \implies \lim \limits_{n \to \infty} a_{n} = L ?$
May
25
comment Matrix irreducibility. Strongly connected graph
@Rodrigo Check out the link from Git Gud, there's a really good explanation there.
May
25
comment Matrix irreducibility. Strongly connected graph
@Rodrigo See here. Matrix is reducible if it can be represented in block-upper-triangular form. The condition with $A^m$ is about irreducible matrix (no block-upper-triangular form).
May
25
answered Matrix irreducibility. Strongly connected graph
May
25
comment Serialize Node Graph to integer
Whatever you can represent in a file on a computer disk can be though of as a big number in 256-ary system. However, these serializations usually won't have properties like matching subgraphs as substrings. One important detail is the order of vertices, with fixed order things get much easier. Take a look at canonical forms and graph canonization.
May
24
comment Is university math all about proofs?
One thing that takes a lot of time is finding what really you want to prove. Hard conjectures are usually manifestations of much deeper dependencies and finding these may constitute majority of your research effort (but then you can write that some hard conjecture is "an easy corollary").
May
24
comment (Soft) What maths should I concentrate on at 16-18 years old?
BTW, the longer quote goes like this: Don't just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?
May
24
comment (Soft) What maths should I concentrate on at 16-18 years old?
@GPerez The belly is probably the first Bianchi identity (which follows from Jacobi identity). As for fighting, I think that P. Halmos urges us to be deeply (personally) engaged, as if our life was on the line, to be active instead of passive, to have a kind of intensity which is impossible when you "just read" math.
May
17
comment (Soft) What maths should I concentrate on at 16-18 years old?
@MarkK 1. If you do not enjoy MO problems, then forget it and do whatever else you feel better about. 2. Many MO problems are about proofs and for anything you wanted to learn instead, there still will be a time for you to learn it. 3. "Why not just go on to harder material" To be a mathematician you have to produce new math - even if you were to learn all the great theories and all the useful techniques, you won't be any good if you don't produce anything. 4. It's not enouh to be able to do something, you have to be efficient at it, that is, you should practice using and producing math.
May
17
comment (Soft) What maths should I concentrate on at 16-18 years old?
@KonstantinosGaitanas Sentence "Try to understand the proofs you read and not only memorize the theorems." suggests you think that in general doing MO problems will promote memorization over understanding. I've certainly met people like these, but there are others, perhaps you have just had bad experience? There is a significant difference between doing problems to get points (the more the better) and to understand what is really going on there (what causes the answer to be like that). There's more on this in one paragraph of my post.
May
17
revised (Soft) What maths should I concentrate on at 16-18 years old?
added 4979 characters in body
May
16
answered (Soft) What maths should I concentrate on at 16-18 years old?