@LeakyNun I don't know zeta2. But the problem amounts to showing that $\lim_{x \to 0^+}\sum_{n=1}^\infty \frac{\sqrt{x}}{1+n^2x}=\frac{\pi}{2}$? How do we do this?
@TedShifrin I think that $\frac{\sqrt{x}}{1+n^2x} \to 0$ uniformly, so we can exchange the limit and the sum. As a result, we would end up getting 0. Where am I going wrong?
if $H$ is a char. subgrp. of $G$ then we have a short exact sequence $1 \to \operatorname{Aut}(G,H) \to \operatorname{Aut}(G) \to \operatorname{Aut}(H)$
@TedShifrin I don't really have that much work to do. The integral test for convergence gives us that $\frac{\pi}{2}-\arctan{\sqrt{x}} \leq \sum_{n=1}^\infty \frac{\sqrt{x}}{1+n^2x} \leq \frac{\sqrt{x}}{1+x}+\frac{\pi}{2}-\arctan{\sqrt{x}}$ and so as $x \to 0^+$, we have $\sum_{n=1}^\infty \frac{\sqrt{x}}{1+n^2x} \to \frac{\pi}{2}$.
@Leaky: I think of $S_4$ as the symmetries of the cube, and, no, the symmetries of the square is not a normal subgroup. There are $3$ conjugate subgroups.
Guys, how do you from equation 2 to equation 3 in this answer? Maybe I'm retarded but I can't seem to factor this bad boy... math.stackexchange.com/a/2265930/12631
gap> for n in [1..30] do for G in AllSmallGroups(n) do for u in NormalSubgroups(G) do if IsCharacteristicSubgroup(G,u) then if Size(AutomorphismGroup(G)) mod Size(AutomorphismGroup(u)) <> 0 then Print(StructureDescription(G),"; ",StructureDescription(u),"\n"); fi; fi; od; od; od;
C4 x C2; C2 x C2
(C4 x C2) : C2; C2 x C2 x C2
(C4 x C2) : C2; C2 x C2
C4 : C4; C2 x C2
C8 x C2; C2 x C2
C8 : C2; C2 x C2
QD16; Q8
C4 x C2 x C2; C2 x C2 x C2
C2 x D8; C2 x C2
C3 x S3; C3 x C3
C12 x C2; C6 x C2
find H char G such that the map Aut(G) -> Aut(H) is not surjective
@MatheinBoulomenos also, suppose $P$ is an $R$-module and $M, N \le P$ are submodules such that $M_\mathfrak m = N_\mathfrak m \le P_\mathfrak m$ for each maximal $\mathfrak m \triangleleft R$. Show that $M = N \le P$.
I can't find any example of order <= 100 other than... C2
gap> for n in [1..100] do for g in AllSmallGroups(n) do if Size(Centre(g)) > 1 then if Size(InnerAutomorphismsAutomorphismGroup(AutomorphismGroup(g))) = Size(AutomorphismGroup(g)) then Print(StructureDescription(g),"\n"); fi; fi; od; od;
C2
gap>
is $(R_\mathfrak m)_\mathfrak n = R_{\mathfrak m \cap \mathfrak n}$?
On gambler's fallacy and the nature of probability:
Most events are independent events, thus say there is 50% probability that a Russian roulette will land on a red space (ignoring the green space for now for simplicity)
Gambler's fallacy says that if you keep on landing on a sequence of blacks, then the probability of landing on red will increase in order to balance out
The reason why this is a fallacy has to do with the independence of the events:
Because each event is independent, the probability of landing at e.g. 50 regardless of what happens before does not change. This also means that you can land on 50 again
If Gambler's fallacy is to be true, then it means the events are dependent in the following way:
Sorry typo, I mean Roulette, not Russian roulette
Let $x_i$ be a random variable on what number will be landed on the Roulette after the ith game. Then Gambler's fallacy is true if for all $i,j$:
$$\text{Pr} (x_i|x_j) < \text{Pr} (x_i)$$
This means, the probability of landing in the same place again will diminish over time, and hence by the conservation of probability $\int \text{Pr}(x) dx = 1$ the complementary outcomes has to increase
The most extreme of these cases coincides with the typical ball without replacement problem, the case where $\text{Pr}(x_i|x_j) = 0$ thus an outcome will not be landed twice
Now as for the nature of probability. What probability told us is given any n trials, how likely is for some event x to occur (Or for the Bayesian interpretation, first make a guess on how much you put your belief that x happens, and then refine your guess iteratively as more evidence became available).
By itself, $\text{Pr}(x)$ does not give you any information on whether the event will happen on the next trial, or the next next trial and so on. This is how it captures one aspect of unpredictability
If you want to say predict how likely that $x$ happens after say, n trials, then what you need is really all the conditional probabilities:
$$\text{Pr}(x|y_1,...,y_n)$$
If that list that follows after the "given" is ordered, then it tell us something about the system. A computation of all of these thus gives you some idea on the ordering that a sequence of events happens
Therefore, if a process is deterministic, it basically means something like this:
Put it in another way, let $\mathbf{x}$ be an ordered sequence of events. Let $x_0 \in \mathbf{x}$. Then for every $x_0$, known as the initial condition, there is a unique $\mathbf{x}$ such that $\text{Pr}(\mathbf{x})=1$ and all the rest are zero
Technically, one needs a topology to talk about continuity, thus when something is discontinuous at a point, it usually means the neighbourhood of the point has a different value than the point itself
Thus arbitrarily close to the point, the value is always different from that of the point itself. If I recall stuff from nonstandard analysis properly, it means there are points yinfinitesimally close to some point x such that |f(x)-f(y)| is not infinitesimal
so in a sense, you can say "f(x+dx)" is technically discontinuous, though one cannot really wrote x+dx as an argument rigourously
I am reading a text on non-standard analysis. I need to prove the following: Suppose that $f$ is non-decreasing on the real interval $[a,b]$ and that $f$ satisfies the intermediate value property. Then $f$ is continuous on $[a,b]$.
I need to show this by showing if $y-x$ is infinitesimal, the...
I'm trying to decide whether the line with two origins over $k$ algebraically closed is a regular scheme or not, but I'm confused by its structure sheaf and the stalks of the latter
this is not just intution, f you want to make that precise: if $X$ is a topological space and $\mathcal F$ is a sheaf on $X$ and $U \subset X$ is an open subset and $x \in U$, then $\mathcal F_x={\mathcal F_{\mid U}}_x$
Let $X$ be the line with two origins, obtained by glueing together two affine lines $Y_1$ and $Y_2$. A section of $\mathcal O_X(U)$ looks like a section of $\mathcal O_{Y_1}(U\cap Y_1)$ and a section of $\mathcal O_{Y_2}(U\cap Y_2)$ agreeing on $Y_1\cap Y_2\cap U$ (there's a fibered product of sets hidden in here that is too painful to type out in latex)
@MatheinBoulomenos Oh, yeah, of course, if you restrict to an open set you have a cofinal family in the directed system defining the stalks
this means any property of schemes which is defined in terms of local rings can be checked "locally", i.e. by showing that any point has an open neiborhood for which the property holds
So I'm looking at $\pi:X=\mathrm{Spec}\Bbb Z[i]\to\mathrm{Spec}\Bbb Z$ induced by $\Bbb Z\to\Bbb Z[i]$ and I'm asked to determine $\pi_\ast\mathcal O_X$ and whether it is locally free
So I want to describe $\pi_\ast\mathcal O_X$ on the standard basis for the Zariski topology on $\mathrm{Spec}\Bbb Z$
if $f:R \to S$ is a ring morphism and $\varphi:S \to R$ is the associated morphism of schemes and $M$ is a $S$-module, then we have $\varphi_*(\widetilde{M})=\widetilde{f_*M}$
But I'm not seeing this by looking explicitely at the sheaf. $\pi_\ast\mathcal O_X(D(p))$ is by definition $\mathcal O_X(\pi^{-1}(D(p)))$, and $\pi^{-1}(D(p))$ should be $\mathrm{Spec}\Bbb Z[i]$ minus $p$ if $p$ is $3$ mod $4$ or minus two points if $p$ is $1$ mod $4$, right?
Also since $A=\Bbb Z[i]\simeq\Bbb Z[x]/(x^2+1)$ the module of Kähler differentials $\Omega_{X/\mathrm{Spec}\Bbb Z}$ should just be the $A$-module generated by $dx$ with relation $2xdx$
I mean geometrically it's like a rank 2 vector bundle. Which makes sense since $\Bbb Q(i)$ has dimension $2$ over $\Bbb Q$ (that's the stalk at the generic point)
@AlessandroCodenotti right
more generally if $A=R[x]/(f)$, then $\Omega_{A/R}=R[x]/(f,df)$
Yeah, this can be generalized to whenever $A$ is a f.g. $R$-algebra
Anyway I'm leaving now to think about another exercise which is really an algebra exercise, thanks a lot for your help! Working out an explicit example was rather iilluminating!
That is the curve r=θ, what are the limits if we are finding the area? I thought it had to be split up to avoid double counting but I was told you can just integrate from 0 to pi/2. Who is correct?
I'm thinking about supports of modules (actually supports of quasicoherent sheaves on affine schemes, but there's no difference)
Quick sanity checks: If I have $A$ as an $A$-module then $\mathrm{Supp}(A)=\mathrm{Spec}(A)$. If I have $A/I$ as an $A$-module then $\mathrm{Supp}(A/I)=\mathrm{Spec}(A/I)$
@TedShifrin Wait why do you get a single point? I'm thinking of multiplication by $6$ in $\Bbb Z$, the cokernel should be supported in $2$ points as a $\Bbb Z$-modules if I'm not doing stupid mistakes
Even though I'm not algebraic, @loch, I love how the geometry of a symmetric space can be read off all the Lie algebraic data and basic representations.
Or how you give the tangent bundle of $G/H$ by the adjoint representation, etc.
Is there any standard notation to write that f is a function to B, that is, f: dom(f) -> B but dom(f) also has some a set A as a subset. the "f:A -> B" does not work since that means that A = dom(f) and I want to have A subset dom(f)
@famesyasd: I'm just saying you should define the domain of $f$. But then you need to restrict that domain to $A$. It's no longer the same function $f$. You have to write $f|_A$ or some similar notation.
The professor mentioned effective Cartier divisors extremely briefly when talking about invertible sheaves but they should be explained in detail in AG II
@TedShifrin yes but I do not want to take restrictions I just want to highlight that the very same function "f" also has "A" as a subset of its domain. like when you write that "f" is a surjection above the arrow or replace staright arrow with some fancy arrow to denote that "f" is a surjection from A to B.
Isn't looking at the cokernel of multiplication by $x$ more like looking at $k=k[x]/(x)$ as a $k[x]$-module, whose support is then $\mathrm{Spec}(k)$ so a single point?
So now that I know 3 basic facts about the support of modules it's time to start playing the big kids games and start computing supports of sheaves of differentials (which is really the same thing, but anyway)
I am reading a text on non-standard analysis. I need to prove the following: Suppose that $f$ is non-decreasing on the real interval $[a,b]$ and that $f$ satisfies the intermediate value property. Then $f$ is continuous on $[a,b]$.
I need to show this by showing if $y-x$ is infinitesimal, the...
