algebraic geometry – Commutation of integral closure and group invariance

We work over $mathbb{C}$. Let $G$ be a reductive group acting on a normal affine variety $X$: we have an induced $G$-action on $k(X)$ given by
$$gcdot p(x)=p(g^{-1}cdot x).$$
Thanks to the GITwe know the good quotient $Y$ is affine and given by $Y=text{Spec} k(X)^G$.

I would like to prove that $Y$ is still normal, that is $k(X)^G=k(Y)=overline{k(Y)}=overline{k(X)^G}$ (where the closure is taken in the field $k(X)$, that is, the operation of integral closure and $G$-invariance is commutative

The inclusion $k(X)^Gsubset overline{k(X)^G}$ is obvious, so let us prove the reverse one. By the normality, it suffices to show that

$$overline{k(X)^G}subset overline{k(X)}^G,$$
I take $fin overline{k(X)^G}$, so that I can write

$$f^n(x)+a_{1}(x)f^{n-1}(x)+ldots+a_{n-1}(x)f(x)+a_n(x)=0$$
with $a_iin k(X)^G=overline{k(X)}^G$. The $G$-invariance allow me to see that the above formula works also
$$f^n(g^{-1}cdot x)+a_{1}(g^{-1}cdot x)f^{n-1}(g^{-1}cdot x)+ldots+a_{n-1}(g^{-1}cdot x)f(g^{-1}cdot x)+a_n(g^{-1}cdot x)=0$$
for any $gin G$, but now I’m a bit stuck as I don’t know how to continue (I can subsract them both but I don’t see what this imply).

pr.probability – How to prove the coupling version of the Donsker’s Invariance Principle?

Donsker’s invariance principle:
Let $X_1,X_2,…$ be i.i.d. real-valued random variables with mean 0 and variance 1. We define $S_0=0$ and $S_n= X_1+ … + X_n$ for $n geq 1$. To get a process in continuous time, we interpolate linearly and define for all $t geq 0$
$$
S_t = S_{(t)}+ (t-(t))(S_{(t)+1}- S_{(t)}).
$$

Then we define for all $t in (0,1)$
$$
S^*_n(t)= frac{S_{nt}}{sqrt{n}}.
$$

Let $C(0,1)$ be the space of real-valued continuous function defined on $(0,1)$ and endow space with the supremumnorm. Then $(S^*_n(t))_{0 leq t leq 1}$ can be seen as a random variable taking values in $C(0,1)$. Now let $mu_n$ be its law on that space of continuous functions and let $mu$ be the law of Brownian motion on $C(0,1)$. Then the following holds:

Theorem (Donsker): The probability measure $mu_n$ converges weakly to $mu$, i.e. for every $F: C((0,1)) rightarrow mathbb{R}$ bounded and continuous,
$$
int F dmu_n rightarrow int F dmu
$$

as $n rightarrow infty$.

But for a two-dimensional case, the ‘coupling version’ is as following.

$textbf{‘coupling version’}$: Fix a square $S$ of size $s$. Fix $xin nS$. Let $X$ be a random walk starting from $x$ until it exits the square $S$ and let $B$ be a Brownian motion until it exits $S$. For $forall epsilon>0$, then there exists $N>0$ such that $nge N$, one can couple $X$ and $B$ so that
$$d(X,B)le epsilon n.$$

My questions:
(1) Can we extended the classical Donsker’s to the two-dimensional case?

(2) Is there any reference for the proof of the ‘coupling version’?

ag.algebraic geometry – An invariance property of rational singularities

Let $X$ be a normal variety over a field of characteristic zero with rational singularities.

If $pi:Y to X$ is a birational proper morphism with $Y$ also normal, then does $Y$ also have rational singularities?

It is easy to see that this is true if $dim(X) = 2$, but the higher dimensional case seems more difficult and perhaps it is even false. If true, I would also be interested in analogous results in positive or mixed characteristic e.g., for pseudo-rational singularities.

Loop invariance insertion sort algorithm

I have the following pseudo code for a insertion sort algorithm

INSERTION-SORT

1 for j = 2 to A.length
2    key = A(j)
3    // Insert A(j) into the sorted sequence A(1..j-1)
4    i = j -1 
5    while i > 0 and A(i) > key
6        A(i+1) = A(i)
7         i = i -1
8    A(i+1) = key

I am trying to convert it into executeable code written in Python

def main():
    A = (5,2,4,6,1,3)
    for j in range(1,len(A)):
        key = A(j)
        i = j - 1
        while i >= 0 and A(i) > key:
            A(i + 1) = A(i)

            i = i - 1
        A(i + 1) = key
        print A(0:j) #LOOP INVARIANCE A(indexstart .. j - 1)
    return A
 
main()

is this a correct translation ? Im not sure if i messed something up with the indexes.Furthermore im going to be testing correctness with initialization, maintenance and termination.

I added the line print A(0:j) to show initialization and maintenance but im not sure if it should be print A(0:j-1) because in my book it says print A(0:j-1)

Invariance of generalized eigenspace – Mathematics Stack Exchange

I have a lemma saying that each of the generalized eigenspaces of a linear operator $T$ is invariant under $T$. This means that if $E_j$ is a generalized eigenspace then $T:E_j rightarrow E_j.$

The proof of this goes like this.

Take a $vin E_j$ so that $(T-lambda_j I)^{n_{j}}v=0$.

Now we are going to show that the same holds for $T(v)in E_j$.

$(T-lambda_j I)^{n_{j}}T(v)=(T-lambda_j I)^{n_{j}-1} T(T-lambda_j I)v=T(T-lambda_j I)^{{n_j}-1}v= T(0)=0$.

Can someone help me with the proof?

ds.dynamical systems – Proving positive invariance

I need to prove that set $D$(A picture for Set $D$) given by
$$D={(x,y):0leq xleq L_0,~0leq yleq X_0,~0leq x+y leq R_0}subseteq mathbb{R}_+^2$$ of the system:
$$dot{x}=k_1(R_0-x-y)(L_0-x)-k_{-1}x,~dot{y}=k_2(R_0-x-y)(X_0-y)-k_{-2}y$$
is positively invariant. I thought of checking the direction of the vector fields on the boundary of $D$ (set $D$ is a trapezium as shown in the picture) and I showed that all the vector fields point inwards except at the corner (denoted $C$) where $$dot{x}=0,~dot{y}<0, text{for } x=0,~y=R_0 $$. Here, the vector field points downwards tangential to $D$. To determine what happens at this point, I thought of checking the sign of the dot product between the vector field and the normal’s field. I chose the normal such that it points inside $D$ by considering $vec{n}_C=(1,-1)$ such that:
$$(1)cdotdot{x}+(-1)cdot{y}$$
and by substituting $x=0,~y=R_0$, I got $$(1)cdotdot{x}+(-1)cdot{y}=k_{-2}y>0$$ which is positive. The positive value means that the vector fields and the normals’ point in the same direct.

Is my approach fine? If it is correct, I would like to back my reasoning with some literature/theorem. Can someone help me which theorem can I use to back up my reasoning? Is there another way in which I can show that the vector fields point inwards? I appreciate in advance.