rt.representation theory – Under what conditions representations of reductive Lie group in Banach space and in its Garding space have the same length?

Let $G$ be a real reductive Lie group (e.g. $G=GL(n,mathbb{R})$). Let $rho$ be a continuous representation of $G$ in a Banach space $V$. Let $V^inftysubset V$ be the subspace of smooth vectors equipped with the Garding topology. Let $rho^infty$ be the natural representation of $G$ in $V^infty$.

Under what precise technical conditions the representations $rho$ and $rho^infty$ have the same length?

A reference would be very helpful.

galois theory – If all roots of $f$ generate a splitting field, is $f$ irreducible?

Recently I had to prove the existence of some irreducible polynomial. I wanted to use the following statement, but I do not know if it is true:

The Statement:

Let $F$ be a field. If $fin F[X]$ is such that for all roots $alpha$ of $f$ $F[alpha]$ is a splitting field for $f$, then $f$ is irreducible.

My Question:

Is this true? Cyclotomic polynomials have this property and are irreducible. Would the answer change depending on the characteristic of the field?

probability theory – CDF of $S_{N_{t}}$ where $S_{N_{t}}$ is the time of the last arrival in $[0, t]$

enter image description here

I am confused on this problem. My professor gave this as the solution:

$S_{N_{T}}$ is the time of the last arrival in $(0, t)$. For $0 < x leq t, P(S_{N_{T}} leq x) sum_{k=0}^{infty} P(S_{N_{T}} leq x | N_{T}=k)P(N_{T}=k) = sum_{k=0}^{infty} P(S_{N_{T}} leq x | N_{T}=k) * frac{e^{- lambda t}*(lambda t)^k}{k!}$.

Let $M=max(S_1, S_2, …, S_k)$ where $S_i$ is i.i.d. for $i = 1,2,.., k$ and $S_i $~ Uniform$(0,t)$.

So, $P(S_{N_{T}} leq x = sum_{k=0}^{infty} P(M leq x)frac{e^{- lambda t}*(lambda t)^k}{k!} = sum_{k=0}^{infty} (frac{x}{t})^k frac{e^{- lambda t}*(lambda t)^k}{k!} = e^{- lambda t} sum_{k=0}^{infty} frac{(lambda t)^k}{k!} = e^{- lambda t}e^{- lambda x} = e^{lambda(x-t)}$

If $N_t = 0$, then $S_{N_{T}} = S_0 =0$. This occurs with probability $P(N_t = 0) = e^{- lambda t}$.

Therefore, the cdf of $S_{N_{T}}$ is:
$P(S_{N_{T}} leq x) =
begin{array}{cc}
{ &
begin{array}{cc}
0 & x < 0 \
e^{- lambda (x-t)} & 0leq xleq t \
1 & x geq t
end{array}
end{array}$

complexity theory – Classes of Functions Closed Under Polynomial Composition – Papadimitriou Exercise 7.4.4

I am studying Computation complexity using Papadimitrious’s book: “Computational Complexity”.

I am trying to do Problem 7.4.4:

“Let $C$ be a class of functions from nonnegative integers to nonnegative integers. We say that $C$ is closed under left polynomial composition if $f(n) in C$ implies $p(f(n))=O(g(n))$ for some $g(n) in C$, for all polynomials $p(n)$. We say that $C$ is closed under right polynomial composition if $f(n) in C$ implies $f(p(n))=O(g(n))$ for some $g(n) in C$, for all polynomials $p(n)$.

Intuitively, the first closure property implies that the corresponding complexity class is “computational model-independent”, that is, it is robust under reasonable changes in the underlying model of computation (from RAM’s to Turing machines, to multistring Turing machines, etc.) while closure under right polynomial composition suggests closure under reductions (see the next chapter).”

Which of the following classes of functions are closed under left polynomial composition, and which under right polynomial composition?

(a) – ${n^k: k > 0 }$

(b) – ${k cdot n: k > 0 }$

(c) – ${k^n : k > 0 }$

(d) – ${2^{n^k} : k > 0 }$

(e) – ${log^k n: k > 0 }$

(f) – ${log n}$

After understanding the definition of closed under left/right polynomial composition, I think I was able to solve items (a), (b), (c) and (f). However, I was not able to solve items (d) and (e).

My solution for item (a):

Closed Under Left Polynomial Composition: consider an arbitrary $f(n) in C$ and an arbitrary polynomial $p(n)$. Then, $f(n)$ is of the form $n^{k’}$, for some $k’ > 0$ and therefore $p(f(n))$ is a polynomial. Let $k”$ be the degree of the polynomial $p(f(n))$. Take $g(n) = n^{k”} in C$ and we have $p(f(n)) = O(g(n))$.

Closed Under Right Polynomial Composition: same reasoning.

My solution for item (b):

Not Closed Under Left Polynomial Composition: consider as a counterexample $f(n) = n in C$ and $p(n) = n^2$. Then, $p(f(n)) = n^2$. For every $g(n) = k n in C$ we have $O(g(n)) = O(n)$. Since $n^2 neq O(n)$ we conclude.

Not Closed Under Right Polynomial Composition: the previous counterexample applies.

My solution for item (c):

Closed Under Left Polynomial Composition: Consider an arbitrary $f(n) = k_1^n$ and a polynomial $p(n)$. Notice that $p(f(n))$ is a polynomial in $k_1^n$. For sufficiently large $n$, there exists some $k_2$ such that $p(n) leq n^{k_2}$ and therefore $p(f(n)) leq (f(n))^{k_2} = (k_1^{n})^{k_2} = (k_1^{k_2})^n$. Therefore, taking $g(n) = (k_1^{k_2})^n in C$ we obtain that $p(f(n)) = O(g(n))$.

Not Closed Under Right Polynomial Composition: Consider as a counterexample $f(n) = 2^n$ and $p(n) = n^2$. Then, $f(p(n)) = 2^{n^2}$, which is greater than $g(n) = k^n$, for every fixed value of $k$, if $n$ is sufficiently large. Therefore, $f(p(n)) not in O(g(n))$.

My solution for (f):

Not Closed Under Left Polynomial Composition: Consider as a counterexample $f(n) = log n$ and $p(n) = n^2$. Then, $p(f(n)) = log^2 n$. Also, $g(n) in C$ implies that $g(n) = O(log n)$. We have $log^2 n not in O(log n)$.

Closed Under Right Polynomial Composition: If $f(n) in C$ then $f(n) = log n$. Given an arbitrary polynomial $p(n)$, we have that there exists some $k’$ such that, for sufficiently large $n$, $p(n) < n^{k’}$. Then, for sufficiently large $n$:
$ f(p(n)) leq f(n^{k’}) = log n^{k’} = k’ log n = O(log n) = O(g(n)).$

Can anyone help me with items (d) and (e)?

Thanks in advance. Of course, corrections/comments on the other items are also welcomed.

homotopy theory – Identifying discrete points in derived hom spaces

Let M be a model category presenting an ∞-category $mathcal{M}$, and let $f : X to Y$ and $g : Y to Z$ be arrows of M. Consider the following propositions:

  1. The connected component of $f$ in $mathcal{M}(X,Y)$ is contractible
  2. $mathcal{M}(X, Y) xrightarrow{g_*} mathcal{M}(X, Z)$ restricts to a homotopy equivalence between the connected components of $f$ and $gf$

When can these propositions be expressed in an elementary way from the model structure on M?

I’m interested in the second proposition (it relates to homotopy uniqueness for properties expressed by arrows and extensions along arrows). But in the case $Z$ is fibrant, it can be reduced to the the first question for $f in mathbf{M}_{/Z}(gf, g)$. Conversely, the first proposition is the $Z=1$ case of the second.

Given a simplicial model category, when $X$ is cofibrant and $Y,Z$ are fibrant — or a general relative category by first computing a simplicial localization — one could answer these questions by appealing to the corresponding questions of simplicial sets.

However, I’m hoping there’s a useful way to express these propositions somewhat more directly in terms of the the model structure on M rather than having to appeal to more elaborate consturctions.

gr.group theory – Rack cohomology as derived functor cohomology

Let $X$ be a rack and $A$ be an $X$-module. By this paper, p. 33, we can associate a cochain complex $C^bullet(X,A)$ to the pair $(X,A)$. This complex is explicitly defined by a differential $d$. I wonder if the cohomology $H^bullet(X,A)$ of the complex has an interpretation as derived functor cohomology. What functor from $X$-modules to $X$-modules do we have to derive? And how to show then the equivalence of the two definitions? I think the analogy to group cohomology is not very helpful, or can we somehow define the invariants of an $X$-module and make it fit?

analytic number theory – What’s the average order of the reduction of a section of an elliptic curve

Suppose $E$ is an elliptic curve over $mathbb Q$ and $x in E(mathbb Q)$ is not torsion. We can reduce $x pmod p$ for a prime $p$ of good reduction and it will have some order $n_p$ in the group $E(mathbb F_p)$. Has there been any work on the asympotitcs of the average of $n_p$ for $p < X$ as $X to infty$?

More generally, suppose $x,y in E(mathbb Q)$ are two linearly independent sections and let them generate subgroups $G_x(p),G_y(p) subset E(mathbb F_p)$ for a prime of good reduction. Have the asymptotics of the average of $G_x(p)cap G_y(p)$ been studied?

This question seems tangentially related.

ct.category theory – Freeness of the action of the ground monoid in a monoidal category

Let $(mathcal{C}, otimes , 1)$ be a monoidal category, and let $mathrm{End}_{mathcal{C}} (1)$ be the ground monoid of $mathcal{C}$ – which is a commutative monoid. If $r_X : X otimes 1 to X$ denotes the right unit constrain of $mathcal{C}$, then $f cdot alpha := r_Y (f otimes alpha) r_X^{-1}$ defines a right action on $hom_{mathcal{C}} (X;Y)$.

Question 1. When is this action free? That is, under what conditions $f cdot alpha = f$ implies $alpha = mathrm{id}_1$?

Question 2. Is it possible to loosen the conditions on $hom_{mathcal{C}} (1;Y)$?

For instance, in the category of $k$-modules this action is free when $k$ is a field, but for a general ring there is torsion and this doesn’t have to be true.

rt.representation theory – The product of $Z(mathfrak{g})$-finite functions is also $Z(mathfrak{g})$-finite?

Let $G$ be a classical group defined over $mathbb{Q}$.

Let $mathfrak{g}$ be the Lie algebra of $G(mathbb{R})$ and $U(mathfrak{g}_{mathbb{C}})$ its universal enveloping algebra of $mathfrak{g}_{mathbb{C}}$.

Let $Z(mathfrak{g})$ be the center of $U(mathfrak{g}_{mathbb{C}})$. We regard the elements of $U(mathfrak{g}_{mathbb{C}})$ as differential operators on $C^{infty}(G)$, the space of smooth functions on $G(mathbb{R})$, acting by right infinitesimal translation.

Let $f,g in C^{infty}(G)$ be $Z(mathfrak{g})$-finite. (I.e. $<zcdot f | zin Z(mathfrak{g})>, <zcdot g | zin Z(mathfrak{g})>$ are finite dimensional vector space.)

Then I am wondering whether $f cdot g$ is also $Z(mathfrak{g})$-finite.

Any comments are appreciated!