I'd like to apologise in advance for the mixture of different notations.
This is Exercise I.8 of Mac Lane and Moerdijk's, "Sheaves in Geometry and Logic [. . .]". According to Approach0, this is new to MSE.
The Question:
Consider a small category $\mathbf{C}$. For each object $B$ of $\math...
Hey guys I wondering if the following statement $(\forall x \in R)(\forall y \in R)[((x+y \leq 7) \wedge (xy = x)) \Rightarrow (x < 7)]$ would always be true?
Meine Frage war ob man Fouriertheorie so verallgemeinern kann, dass sie sich auf Gruppentheorie reduziert. In dem Sinne dass die Darstellungen beliebiger Gruppen als Fourierbasen interpretiert werden könnten.
Hi chat! Just a small question: I proved that $\lim_{p\to\infty}\|x\|_p=\|x\|_{\infty}$ by showing the inequality: $$\|x\|_\infty\leq\left(\sum_{j=1}^n |x_j|^p\right)^{1/p}\leq n^{1/p}\|x\|_\infty$$ And then using squeeze theorem, I wonder if this proof doesn't have any holes.
@LukasHeger we have that X^k = X mod rs , where r, s are primes ,but we only know their product, how can we find r and s ? k is such that gcd(k , (r-1) , (s-1) ) = 1
@LukasHeger we have that X^k = X mod rs , where r, s are primes ,but we only know their product, how can we find r and s ? k is such that gcd(k , (r-1) , (s-1) ) = 1
Good day. I have done studying Linear Algebra this Semester and now i am preparing for my exam. I have pretty solid idea of what is the subject and how do they connect, nevertheless i have a small inquiry. Feel free to answer with depth since i will understand most likely. How can i think of Vectorspaces and Fields in an intutive way. What is the relationship between those two.
I know that you use numbers from the field to scale the vectors. But further than that i do not build a strong connection . Are vectorspaces a subgroup of that Field taken to the cross to itself (thus building the Vectors).
@MadSpaceMemer i mean the only way to answer your question is to actually look at the things in common and that differ between vector spaces and fields
I have to prove $$ \left(A \cap B \right) \cup C = \left(A \cup C \right) \cap \left(B \cup C\right)$$
My attempt is the usual one :
$$ \textrm{Let}~ x \in \left[
\left (
A \cap B
\right)
\cup C
\right] $$
$$\implies x \in \left(A \cap B \right) ~\textrm{or}~ x \in C $$
$$\implies \left( x ...
@MadSpaceMemer i feel you, but in this context (and i may be wrong) i don't see how else you can get any intuition other than by manipulating the axioms and using certain examples to get a feeling of how they differ
Well Can you answer this @StupidQuestionsInc for a vectorspace L on the field S. Does it mean that for each vector $l \in L$ Such that $l:=(l_1,l_2)$ is then $l_1 , l_2 \in S$?
@adeshmishra to prove things in naive set theory you usually need a good knowledge of basic propositional logic, i would recommend that you learn those few rules and then go back to proving things in naive set theory
:53549296 You mean elements of $L$ are of the form $(l_1,l_2)$ ? If so then it depends on how $L$ is defined, but there's nothing that necessitates that $l_1,l_2$ be elements of the field $S$
@StupidQuestionsInc The problem is whenever I start propositional logic I always end up giving up. There are reasons for that: first is that books explain things too much in the beginning that I already know (around 40-50 pages are devoted just for developing some terminologies) , second I never got any good book on that subject, third no one likes teaching Logic the way books teach.
@BalarkaSen @MikeMiller I've got a random topology idea motivated by algebra, but I'm not sure how to make it work. Here's the motivation: If $L/K$ is a (say finite for simplicity) Galois extension, then one can form the category of $L$-vector spaces with a semilinear $\mathrm{Gal}(L/K)$-action, where semilinear action is defined as a group action such that $\sigma(v+\lambda w)=\sigma(v)+\sigma(\lambda)\sigma(w)$.
we have that X^k = X mod rs , where r, s are primes ,but we only know their product, how can we find r and s ? k is such that gcd(k , (r-1) (s-1) ) = 1
This category turns out to be equivalent to $K$-vector spaces via the functors $W \mapsto W^G$ and $V \mapsto L \otimes_K V$ (with the action coming from the action on $L$)
@LeakyNun to the category of finite discrete spaces with a transitive continuouus $\widehat{\Bbb Z}$-action
the idea is this: let $p:X \to Y$ be a normal covering, then we can define a functor $\mathrm{Top}/Y \to \mathrm{Top}/X$ via taking pullbacks along $p$: $Z \mapsto Z \times_Y X$ which comes with a projection to $X$ and an action of $\mathrm{Aut}(p)$ on the left component
Look at a covering map $X \to Y$ and the category of vector bundles $\{E \to X\}$ equipped with a $\text{Gal}(X/Y)$-action on both the spaces $E$ and $X$ such the projection is equivariant under that action
This is equivalent to the category of vector bundles over $Y$
@adeshmishra yeah but you still have to go through it, there some things that few people enjoy but you really need them in order to do other more enjoyable things
If we have something like this $$ \left (\mathbf A \cdot \nabla \right) \mathbf B$$ then should we take the dot product first or operate $\nabla$ on $\mathbf B$ first?
Suppose each $X_i$ is first countable. Let $(p,i_0) \in \coprod_{i\in I}X_i$. Since $X_{i_0}$ is first countable, there exists a countable local basis $\mathbb{B}_p$ for $p$. I claim that the following set: $\mathbb{\sigma_{i_0}(\mathbb{B}_p})$ $=$ $\{$ $\sigma_{i_0}(B)$ $:$ $B\in \mathbb{B}_p$ $\}$ is a countable local basis for $(p,i_0)$. Clearly, it is a collection of open sets in the disjoint union. If $U$ is an open set of $(p,i_0)$ in the disjoint union, then $\sigma_{i_0}^{-1}(U)$ is open in $X_{i_0}$. But, $\sigma_{i_0}^{-1}(U)$ contains some $B\in \mathbb{B}_p$ which owns $p$, and …
Hello! I had a question just like this one: math.stackexchange.com/questions/1454363/… decided to search before asking myself. However I still can’t seem to understand the answer. 1. How is H a plane? 2. Why is it that v1 can’t be expressed by H?
Hi chat! Just a small question: I proved that $\lim_{p\to\infty}\|x\|_p=\|x\|_{\infty}$ by showing the inequality: $$\|x\|_\infty\leq\left(\sum_{j=1}^n |x_j|^p\right)^{1/p}\leq n^{1/p}\|x\|_\infty$$ And then using squeeze theorem, I wonder if this proof doesn't have any holes.
hello, I saw a function $f\,:\, \mathbb{N}\to \mathcal{\mathbb{N}}$. What does $\mathcal{\mathbb{N}}$ mean? Is it a set of all subsets of natural numbers or the set of the natural numbers?
Think of the list of coefficients as a (long) vector, starting with the constant coefficient. The fact that the polynomial has finite degree means that after some point, all the entries in the vector are $0$.
As an example, the polynomial $X^2+X+1$ belongs to the map $\mathbb{N_0}\rightarrow K$ that sends $0\mapsto1$, $1\mapsto1$, $2\mapsto1$ and $n\mapsto0$ for $n\ge3$
I see So the $ X^n + X^{n-1}..........$ is merly a notation to express the "long vector" which in itself is a "sequence". This sequence is "a function from the naturals to a field"
Yeah, it's good to know that this is the formal definition, but you should not start thinking about polynomials as sequences, cause that usually isn't helpful
I (if you are interested in my opinion) think this is actually extremely important for understanding. Just by understanding the connections, now I read the notations and don't scratch my head and act like it is magic. Now I actually understand why it is written f(X) for a polynom since obviously its a function . If I ever be a professor I will truly be extremely regirious as possible since sofar it has been the most amazing method to understanding stuff and building connections.
Possibly. Beginning teachers are over-zealous thinking they know how to do everything right, better than all their teachers had been. It turns out ... they realize they don't know.
No, writing $f(X)$ has nothing to do with the sequence definition of polynomials. A sequence is a function on $\mathbb{N_0}$, but $X$ is not a natural number. $f(X)=f$ has something to do with evaluating polynomials.
The thing to remember is that a polynomial is not a function, since a polynomial is not determined by its evaluation on all points in its domain, but if that is clear there's no need to be excessively formal
You have a map $K[X]\mapsto\mathrm{Map}(K,K)$ sending each polynomial to its polynomial function. This map is injective iff $K$ is infinite and surjective iff $K$ is finite.
I mean Stone-Weierstrass is already way more general than it is usually presented
If $X$ is (locally) compact Hausdorff and $P$ is a subset of $C(X)$ (complex valued continuous functions) which separates points, then the $\ast$-subalgebra of $C(X)$ generated by $P$ is dense
I'm trying to prove that any abelian group $G$ can be decomposed as $D\oplus N$ where $D$ is divisible and $N$ has no nontrivial subgroups. Having a little bit of trouble sinking my teeth into it, though
Maybe I could set up an exact sequence and show that it splits
Well, I know that $D$ has to contain all divisible elements of $G$, so I've shown that the set of all divisible elements of $G$ is a subgroup
"If $G$ is an abelian group, let $D\subseteq G$ be the set of all elements $g\in G$ such that, for every $n\in\mathbb{Z}$, there exists a $g'\in G$ such that $ng'=g$. We note that, if $e$ is the identity of $G$, then $ne=e$ for all $n\in\mathbb{Z}$, and so $D$ is not empty. Further, if $g,h\in D$ then there exist $g',h'\in G$ such that $g-h=ng'-nh'=n(g'-h')$ for all $n\in\mathbb{Z}$. So, $D$ is a subgroup of $G$. Clearly, $D$ is divisible."
Well, in the first, $\Bbb Q$ is divisible and $\Bbb Z$ has no non-trivial divisible subgroups. In the second, $\Bbb Z\cong\{0\}\oplus\Bbb Z$ and $\{0\}$ is divisible, by my understanding.
So you should probably go back to thinking about whether there's a way to split $G\to G/D$.
It's pretty clear that $G/D$ meets the criterion, I guess.
user447585
Does someone thinks that the conjecture that every prime $p\geq 7$ can be represented as $p=qr+s$ where $q,r,s$ are primes is harder to settle than conjecture of Goldbach?
@IshPaladin not exactly what you are asking, but it is known that every sufficiently large natural number (not just prime) is of that form or the sum of two primes
is it realistic to study some aspects of the representation thy of some (not so trivial) lie groups?
like the Poincaré group?
or would studying SO(n) be challenging enough?
I'm trying to find a topic for my bachelor's thesis, and I would really like to do sth that has to do with symmetries and possibly the standard model
but I'm just not sure about what is reasonable
also, stuff like Yang-Mills gauge theory sounds conceptually graspable, but I'm not sure how much I would be able to do
I have two supervisors (a math and physics professor), and they're both somewhat leaving me in the dark as far as choosing a specific topic goes, so I'm trying to think of as many ideas as possible to propose to them coming week
I really dislike doing research and finding a phrase that I don't understand. Like "It follows that $\phi$ factors through the quotient $B/\text{Im}(\alpha)$." What does "factors through," mean?
$\phi$ is a map $B\to C$, and there is a map $\psi\colon B/\mathrm{Im}(\alpha)\to C$ such that the diagram commutes
That is $\psi\circ\pi=\phi$
(where $\pi\colon B\to B\mathrm{Im}(\alpha)$ is the usual projection)
If you think about the functions you're composing as "factors" you are factoring $\phi$ by going through the quotient instead of taking the direct route
If $f_0, f_1 : S^1 \to \Bbb R^2$ are two immersions, then they are smoothly homotopic through immersions iff $\deg G(f_0) = \deg G(f_1)$ where $G(-)$ indicates the Gauss map.
I have three boards to work with, I plan to draw lots of pictures on 1.5 of the boards and talk about the mathematical details on the other 1.5
Hopefully I can strike the right balance between pictorial arguments and formal math. Leaning too much on either side will make one half of the audience completely lost
A little nervous, never really gave a topology talk to a wider audience before.
I also bet with a friend that in the middle of the talk I will loudly announce "Gromov is God" as a non-sequitur
Should sleep now though, gotta wake up early tomorrow.
@Sha: Some representation theory should be reasonable. Check out Fulton/Harris's book Representation Theory. It has lots of concrete examples in it. Yang-Mills requires a lot of differential geometry and functional analysis, so I'm not sure if that's reasonable. Plus the people advising you have to be semi-competent on the material you're doing.
lol, looks like AP style has it different than Chicago
AP style says spaces before/after, Chicago says no
ah, style
(the issue is that, to have a Google Docs autosubstitute "--" with an em dash, requires you to type "--" and hit space. you can't just type "--a" and have it change to an em-dash immediately. but then it always introduces the space. I guess one solution to that, though, is to always go left one character immediately after doing doing "-- ")
I'd like to apologise in advance for the mixture of different notations.
This is Exercise I.8 of Mac Lane and Moerdijk's, "Sheaves in Geometry and Logic [. . .]". According to Approach0, this is new to MSE.
The Question:
Consider a small category $\mathbf{C}$. For each object $B$ of $\math...
I know how to find tangent plane but I don't know about how to find osculating paraboloid and then approximate N.
Can somebody help me out?
I have to prove $$ \left(A \cap B \right) \cup C = \left(A \cup C \right) \cap \left(B \cup C\right)$$
My attempt is the usual one :
$$ \textrm{Let}~ x \in \left[
\left (
A \cap B
\right)
\cup C
\right] $$
$$\implies x \in \left(A \cap B \right) ~\textrm{or}~ x \in C $$
$$\implies \left( x ...
