Is there any procedure for moving questions from math.se to mathoverflow? For example, this question math.stackexchange.com/questions/2763434/… has received zero answers, despite having had a bounty and multiple up votes. I wonder then if mathoverflow might be a better place for it.
Because addition is only defined between pairs of vectors or pairs of scalars. Trying to add a vector to a scalar is nonsense, because + doesn't have any meaning between a vector and a scalar.
Yeah. You could just make up an addition law between vectors and scalars, but for it to be consistent, it would really just boil down to adding a multiple of some vector, and vector addition already covers that. There's no reason to add more structure.
@TheMaskedRebel Something that that definition lacks is "basis independence"
Suppose I have two arrows in 3-space. You don't know how my coordinate system is set up; you don't know in what direction the x-axis is or in what direction the y-axis is. But you can compute the vector sum anyway, using the parallelogram rule.
On the other hand, say I tell you to compute $(0,0,0)+1$. You have no way of figuring out what $(1,1,1)$ is without knowing anything about the coordinate system.
If I'm letting $m : G \times G \to G$ be the multiplication in a group, does showing that $G$ is a topological group amount to showing that $m^{-1}(U) \subset G \times G$ is open for all open sets $U \subset G$ (and for $i : G \to G$ by $g \mapsto g^{-1}$)? Is this a particularly practical way of doing things or is there some more workable trick that is more useful?
In mathematics, a paratopological group is a topological semigroup that is algebraically a group. In other words, it is a group G with a topology such that the group's product operation is a continuous function from G × G to G. This differs from the definition of a topological group in that the group inverse is not required to be continuous.
As with topological groups, some authors require the topology to be Hausdorff.
Compact paratopological groups are automatically topological groups.
== References... ==
In section 2.1 of this paper there is an explicit example:
take $G= (\mathbb{Z},+)$ as the group, and a basic neighbourhood of $0$ looks like $U_n(0) = \{0\} \cup \{k \in \mathbb{Z}: k \ge n\}$, $ n \in \mathbb{N}$. At $p \in \mathbb{Z}$ we of course get the shifted neighbourhoods $U_n(p) = \{p\...
@Leaky Nice, I just wrote a big report on like.. classical ANT and now that that's finished I'd like to look at local fields which I avoided due to having no topology
@Xander Conchita Wurst was on every Austrian billboard for about a year after Austria won Eurovision
In geometry, the minimum or smallest bounding or enclosing box for a point set (S) in N dimensions is the box with the smallest measure (area, volume, or hypervolume in higher dimensions) within which all the points lie. When other kinds of measure are used, the minimum box is usually called accordingly, e.g., "minimum-perimeter bounding box".
The minimum bounding box of a point set is the same as the minimum bounding box of its convex hull, a fact which may be used heuristically to speed up computation.
The term "box"/"hyperrectangle" comes from its usage in the Cartesian coordinate system, where...
i find that i get way more satisfaction typing up torus than typing up weil group
just look at the definition of cogroup
class cogroup :=
(comul : alg_hom A (tensor_a R A A))
(comul_assoc : (alg_hom.tensor_assoc R A A A).comp
((tensor_a.map R _ _ _ _ comul (alg_hom.id A)).comp comul)
= (tensor_a.map R _ _ _ _ (alg_hom.id A) comul).comp comul)
(coone : alg_hom A R)
(comul_coone : (alg_hom.tensor_ring _ _).comp
((tensor_a.map R _ _ _ _ (alg_hom.id A) coone).comp comul)
= alg_hom.id A)
(coone_comul : (alg_hom.ring_tensor _ _).comp
((tensor_a.map R _ _ _ _ coone (alg_hom.id A)).comp comul)
= alg_hom.id A)
In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by
Spec
(
R
)
{\displaystyle \operatorname {Spec} (R)}
, is the set of all prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space. A locally ringed space of this form is called an affine scheme.
== Zariski topology ==
For any ideal I of R, define
V
I
{\...
oh it seems like this did not link to wha ti wanted it to link to
but anyway there's some motivation from linear algebra/functional analysis
im not sure if this is the historic reason why it's called spectrum though
@LeakyNun if you want a group scheme such that the "group scheme multiplication" reflects the multiplication in G, you should take the dual of R[G], not R[G] itself, then G doesnt need to be abelian either
I have just started learning category theory and I am trying to get an understanding of how to think about the Yoneda lemma. Obvious applications are clear to me (Yoneda embedding is full and faithful, for example), but I want to understand how one uses it in practical mathematics. In particular...
Let $X = C[0, 1]$ be the space of all real-valued continuous functions on $[0, 1]$. Let me know a Cauchy sequence from $X$ that does not converge in $(X,d)$ where $d(f,g)=\int_0^1|f(x)-g(x)|\,dx$.
@Silent $\varphi(100)=40$, so for the exponent, only the residue class mod $40$ matters, so compute $9^9 \pmod{40}$ first and then 9 to the power of that result mod 100
Let $G\subset \mathbb{R}^n$ be open, bounded and $f:\overline{G}\rightarrow \mathbb{R}^n$ a continuous and open map. Then $\|f\|$ gets its maximum on the bounder of $G$.
@MatheinBoulomenos i don't think it's fair to say algebraic topology is a subfield of category theory lol (and of course algebraic topology is not dead)
@Zee im not sure if people would agree with that - eg im not sure if someone cares about homotopy groups of spheres would consider their field as only about infinite categories
@MatheinBoulomenos If $T$ is an affine group $F$-scheme such that $T \times_F \overline F = GL(n,\overline F)$, then is it true that $T \times_F F^{sep} = GL(n, F^{sep})$?
@LeakyNun I'm 90% sure this works, but I need to check again: $K/F$ is separable iff for every reduced commutative $F$-algebra $A$, $K \otimes_F A$ is reduced
Another question: Do other algebraic structures (like monoids or rings and so on) also have a representation theory? And if whats the peculiarity with groups there?
I think this is in Bourbaki - Algebra, but I don't know which chapter
@Rudi_Birnbaum there is representation theory for Lie algebras, algebras over a field and some objects which aren't even algebraic structures like quivers (basically graphs)
and then there's representation theory for groups with additional strucure, like Lie groups, topological groups or algebraic groups