I am trying to get the hang of the available software for computing automorphism groups of plane curves over finite fields. I am using this Magma code to test it out on $y^2 = x^3 + x$ over $\mathbb{F}_3$, which we know is of order 12 by (pg. 410, Prop 1.2(c)) Silverman's *Arithmetic of Elliptic Curves*.

The following Magma code gives me as output that the order is 24. Moreover, it also does this for seemingly every other plane curve I can think to put in.

I wrote this code according to the exposition here on the Magma webpage. Something terribly wrong is going on, likely I am misunderstanding something fundamental.

A<x,y> := AffineSpace(FiniteField(3),2); f := y^2 - x^3 + x; C := Curve(A,f); G := AutomorphismGroup(C); Order(G);Assume that $\mathcal{F}$ is a foliation of a $3$ dimensional compact Riemannian manifold $M$ which is invariant under flow of a transversal non vanishing vector field $X$. Let $\Delta_{\mathcal{F}}$ be the Laplace operator on each leaf which comes by restriction of Riemannian metric of $M$ to leaves.

Let $A=\{f\in C^{\infty}(M)\mid \Delta_{\mathcal{F}}f=X.f\}$

Is $A$ invariant under the derivation $g\mapsto X.g$? Under which condition the later is the case? What is an example of this situation, invariance of $A$, with extra condition that the codimension of the derivation operator $(g\mapsto X.g)|A$ is a finite number?

This question is some how motivated by the following two posts(An indirect motivation not a direct motivation):

Elliptic operators corresponds to non vanishing vector fields

I asked this question on Mathematics Stackexchange, but got no answer.

Let $\mathcal A$ and $\mathcal B$ be nonempty categories whose categories $\mathcal A^{\mathcal A}$ and $\mathcal B^{\mathcal B}$ of endofunctors are equivalent.

Are $\mathcal A$ and $\mathcal B$ necessarily equivalent?

Implicit assumption: we are working in ZFC, and we assume that ZFC is consistent.

If my understanding is correct, this post of Joel David Hamkins implies that one cannot prove that the answer is Yes, so that either the answer is No, or the question is undecidable. (I think that the answer is No.)

[Reminder 1: Categories are generalized sets in the following sense. Given a set $S$ let $\mathcal C(S)$ be the category whose objects are the elements of $S$ and whose only morphisms are the identity morphisms. Then the assignment $S\mapsto\mathcal C(S)$ commutes with exponentials in the obvious sense.]

[Reminder 2: Infinite cardinals $\kappa$ satisfy $\kappa^\kappa=2^\kappa$. Indeed $\kappa^\kappa\le(2^\kappa)^\kappa=2^{\kappa\kappa}=2^\kappa\le\kappa^\kappa$.]

Fundamental results in percolation theory are all based on the assumption that the system sizes are infinite, as the spanning/percolating cluster is by definition an infinitely sized cluster that connects the entire system (e.g. end-to-end). However, in reality or at least in simulations, one is often dealing with finite systems, where all percolation related transitions are *smeared out.* So in order to determine the percolation threshold, one performs a finite size scaling analysis, where for instance the percolation strength (or probability) is computed as a function of occupation probability and repeats this for a range system sizes. Then, it is expected that if the studied systems have been *large enough* their curves will cross at a common point, which can be taken as an accurate estimate of the threshold in the limit of infinite system sizes.

These systems could be:

- Site or bond percolation: e.g., in 2D where a grid (often a lattice) is chosen where sites (or bonds) are occupied with a fixed probability $p.$ Two or more of such occupied sites that happen to be neighbours are then deemed to be a cluster. Then by computing the average of the maximum cluster size normalized by the number of sites as a function of $p,$ and repeated for various grid sizes $L,$ one can estimate the percolation threshold $p_c$ as their intersection point (example).
- As an example of a widely different system: suppose a suspension of spheres or cylinder-like objects in a 3D box. Then similarly, pairs of connected particles form clusters. These clusters grow in size with either increasing connectivity range or increasing density/volume-fraction. Therefore, in order to estimate the threshold in terms of the two aforementioned quantities, one computes the percolation probability as a function of them and across varying system sizes (box sizes here) and considers the so-obtained intersection as a the threshold. (example)

Example taken from these lecture notes:

Clearly, these systems have quite different details, but the fact that the same estimation method for determining $p_c$ is expected to hold generally, is unclear to me. I understand that if we take the comparison of system size $L$ and the correlation length $\xi,$ then there are two important regimes:

$L\gg \xi:$ the relevant length scale remains the correlation length and we expect the known scaling laws for infinite systems to hold, namely $\xi \propto |p-p_c|^{-\nu}$ and the average cluster size $S\propto |p-p_c|^{-\gamma}.$ In this regime (corresponding to $p$ much larger than $p_c$), everything scales with $\xi,$ i.e., independent of system size $L.$

But when $p$ approaches $p_c$ from the left, finite size effects are visible as the percolation probability, strength or average cluster size scales with $L.$

Therefore, there is a crossover from the dependence on $L$ to $\xi$ when going from $p<p_c$ to $p>p_c.$

**Questions:**

Why this crossover suggests/implies that there should be a common intersection point for the curves of varying system sizes? Is this really expected to be the case irrespective of the details of the system (such as earlier examples)? Since in some cases, there may be additional relevant length scales, e.g. in the case of spheres or cylinders, we have not only $L$ (box size), but also the particles diameter (or aspect ratio) and the relative aspect ratio of the formed clusters with of the containing box. How can one rigorously assess all these length scales w.r.t to $L$ in order to make sure the systems are large enough and the curves will in fact intersect at same point?

How does one judge the system is

*large enough?*Are there methods that allow us to test whether for a given system the method of intersecting curves of different $L$ is going to be valid? Because how we define $\xi$ in general may change from system to system.

*tdlr: My attempt is to simply understand why this method works for estimating $p_c$ of so many widely different systems and what implies the intersection point, or at least how can we judge if it is going to work for a chosen problem. In many works, one often read "if the systems are large enough..." but I don't understand how large is supposed to be large enough.*

Pick three $a,b,c$ vectors in $\mathbb Z^n$ uniformly with $\max(\|a\|_\infty,\|b\|_\infty,\|c\|_\infty,)<T$ and a $\gamma\geq2$.

What is the probability that there are integers $A,B\asymp T$ with $\|Aa+Bb-c\|_\infty<\gamma\cdot T^{(n-1)/n}$?

At $n=1$ this is roughly $\frac1{\zeta(2)}=\frac{6}{\pi^2}$ even at $\gamma=2$.

Assume $a$ and $b$ are coordinatewise coprime. Then do we always have such $A$ and $B$ at general $n$ and if not then what is the correct probability asymptotics?

The unpointed version is easy: the model $X = EG \times X \to (EG \times X)/G = X^{un}_{hG}$ is a fibration with fiber $G$. But when we go pointed, $X = EG_+ \wedge X \to (EG_+ \wedge X) / G = X_{hG}$ is no longer a fibration: its fiber changes from $G$ over non-basepoints to $\ast$ over the basepoint.

Of course, these two cases are still closely related. So perhaps there's some hope of understanding the pointed case.

**Questions:** Let $G$ be a finite group (or perhaps something a bit more general), and let $X$ be a pointed $G$-space.

Is there a good way to understand the homotopy fiber $F$ of the map $X \to X_{hG}$ (where these are

*pointed*homotopy orbits)?In particular, does the homotopy fiber sequence $F \to X \to X_{hG}$ deloop to a homotopy fiber sequence $X \to X_{hG} \to BF$ (as it does in the unpointed case)?

Let $X$ be a $G$-space. There are a number of places in the literature where one can find the claim that under certain conditions rationalization and taking homotopy fixed points with respect to a finite group action commute, that is, the map $X^{hG} \to (X_{\mathbb{Q}})^{hG}$ is a rationalization or equivalently $(X^{hG})_{\mathbb{Q}} \to (X_{\mathbb{Q}})^{hG}$ is a weak equivalence.

I know how to prove a number of special cases of this (e.g. when $X$ is based with $G$-fixed base point and nilpotent with finitely many non-zero homotopy groups), but a quite general result is claimed in the 1989 thesis of Goyo (he uses $(-)^{(G)}$ as notation for homotopy fixed points):

However, I don't understand his proof (it seems to use the claim that an infinite limit of rationalizations is a rationalization, which is not true). So I am looking for a reference which gives an answer to the following question:

When exactly does rationalization commute with homotopy fixed points?

$k$ is a field. Let $X$ be a connected pointed $CW$-complex such that the homology $H_{n}(X;k)$ is a finite dimensional $k$-vector space for any $n\in \mathbb{N}$. Suppose that we have continuous mapping $$r: X\rightarrow K(\pi_{1}(X),1) $$ from the space $X$ to the Eilenberg-MacLane space $K(\pi_{1}(X),1)$ inducing an isomorphism on fundamental group $\pi_{1}$. Suppose also that there is a mapping $$ i: K(\pi_{1}(X),1) \rightarrow X $$ such that:

- $r\circ i= id$
- the homotopy cofiber of $i$ is homotopy equivalent to a finite $CW$-complex.
- and $H_{n}(r;k): H_{n}(X;k)\rightarrow H_{n}(K(\pi_{1}(X),1);k)$ is an isomorphism for any $n\in \mathbb{N}$.

My question is the following: Is the homology of $\tilde{X}$ (the universal covering of $X$) finite dimensional? i.e. $H_{n}(\tilde{X};k)$ is a finite dimensional $k$-vector space for any $n\in \mathbb{N}$?

I see two "obvious" classes of nonzero finite $E_\infty$ ring spectra $R$:

$R = \Sigma^\infty_+ (S^1)^{\times n}$

$R = D\Sigma^\infty_+ X$ ($X$ a finite space)

**Questions:**

Are there any others?

In all the above examples, the unit map $\mathbb S \to R$ splits off. Is this always the case?

In the second class of examples, I believe all elements of $\pi_\ast R$ not in the image of the unit $\pi_\ast \mathbb S \to \pi_\ast R$ are nilpotent. How generally is this true? Is it true for all examples not in the first class of examples?

How does the answer change if we localize at a prime, or perform some more drastic localization?

Let $X=\lambda_0u_0v_0^T\in\mathbb{R}^{n\times n}$ be a rank 1 matrix where $\lambda_0\in\mathbb{R}$, $u_0,v_0$ are of unit Euclidean norm. What is the solution of the following problem? $$\hat{X}=\arg\min_{\substack{Y:Y=aU\\U\in O(n)\\a\in\mathbb{R}}}\|Y-X\|_F^2$$ where $U$ is orthogonal and $\|\|_F$ is the Frobenius norm. In other words, what is the best scaled orthogonal approximation of rank 1 matrix?

Consider the finite field extension $\mathbb{F}_{{q}^{d}}$ over $\mathbb{F}_{q}$, where $q=p^a$ for some prime $p$. We assume $d\geq 2$. Let, $$ S=\{ \alpha \in \mathbb{F}_{q^d}\hspace{0.1 cm} | \hspace{0.1 cm} \mathbb{F}_{q}(\alpha)=\mathbb{F}_{q^d} \}$$

In other words, $S$ consists of all those elements in $\mathbb{F}_{q^d}$, whose minimal polynomial over $\mathbb{F}_{q}$ has degree $d$. Let, $S^m= \{ s^m | s\in S\} $, where $m$ is a positive integer $\geq 2$. Then, $$ |S\cap S^m|=?$$

I calculated, this for $m=2$. The answer depends on whether $d$ is odd or even. We have,

$$ |S\cap S^2|= \begin{cases} \frac{|S|}{2} & if \hspace{0.2 cm} d \text{ is odd}\\ \frac{1}{2}[|S|-\frac{(q^{d/2}-1)}{d}] & if \hspace{0.2 cm} d \text{ is even} \end{cases} $$

And, $|S|=dM(d,q)$, where $M(d,q)$ denote the number of irreducible polyomials of degree $d$ over $\mathbb{F}_{q}$.

I couldn't generalize my method to $m>2$. My, idea is that this problem seems to have been well studied in the literature of finite fields. So, I am hoping for some kind of help or suitable references in this case.

Thank you!

The theory of $\theta$-cycles (due to Tate, I think) and filtrations is to me a very beautiful and powerful tool in proving many statements about mod $p$ modular forms in more explicit and elementary manner. Is there an analogue of $\theta$-cycles for mod $p$ automorphic forms of higher rank groups? If so, is there any occasion where such theory is used in proving some interesting statements?

I'm trying to use FLINT (Fast Library for Number Theory) to calculate the Legendre Symbol of the following:

$$\left(\frac{n! + 1}{p}\right)$$

In my case, $p$ is a positive, odd prime (specifically $1,000,000,000,039 $), so I should be able to use the Jacobi symbol in its place when attempting to compute it.

How do I simplify the *numerator* if $n$ is a very large number, specifically $208,463,325,489$?

My current thought is that I would need to calculate n! mod p (which I believe is just a running product modulo p) and then add 1 before computing the symbol.

The value of n! mod p that I computed using FLINT is $133,008,788,325$, but I'm not sure if that's the correct value that I should be using in place of n! when computing the symbol.

Is it possible to simplify this mathematically so that I can verify that my computation is correct?

In the well known book by Littlewood (Mathematician's Miscellany, or the later edition called Littlewood's Miscellany) there is a remark made in the chapter 'A Mathematical education', the meaning of which has been complete mystery to me since I read it some years ago and still wonder what Mr. Littlewood meant:

I will say, however, that for me the thing to avoid,
for doing creative work, is above all Cambridge life, with the constant
bright conversation of the clever, the wrong sort of mental stimulus, **all
the goods in the front window**.

**Question:** what kind of goods does he mean exactly?

Is this some subtle hint at his drinking problem and he means that he should avoid alcoholic beverages? But then why does he say all the goods. This doesn't make any sense. Can anybody shed some light on this question?

- Is 'Given two codes with alphabet in $\mathbb F_2$ with Generator matrices $G_1$ and $G_2$ do they have the same minimum distance?' in $NP$ or is it in $coNP$ (I can see it in $P^{NP}$)?

If $G_1$ is known to give minimum distance $d$ then the problem 'is the minimum distance with $G_2$ less than $d$?' is $NP$-complete.

- Is 'Given two codes with alphabet in $\mathbb F_2$ with Generator matrices $G_1$ and $G_2$ do they have the same number of minimum distance code words?' in $PP$ (I can see it in $P^{PP}$)?

If $G_1$ is known to give $N_d$ number of minimum distance $d$ then the problem 'is the number of minimum distance codewords with $G_2$ greater than $N_d$?' is $PP$-complete.

i solved the problem (1/2-x)^2 with the quadratic formula and got the answer 1/4-x+x^2. But when i checked in the back of the book it said the answer was 1/x^2-2+x^2. Anybody know what im doing wrong?

Remember that a Bernstein set is a set $B\subseteq \mathbb{R}$ with the property that for any uncountable closed set, $S$, in the real line both $B\cap S$ and $(\mathbb{R}\setminus B)\cap S$ are non-empty. Some known results are the following: Bernstein sets are Baire spaces, also the Banach-Mazur game played in a Bernstein set is indeterminate. My question is: if $B, D\subseteq\mathbb{R}$ are Bernstein sets then $B\times D$ is a Baire space?

Thanks

How can define Grassmannian of regular subbundle of vector bundle $E$?

Everything in category of smooth vector bundle, and I don't want to use scheme theory.

I've been working with the following optimization problem:

$$ \max \int \left(\frac{1}{2}\left\| x \right\|^2 + \tilde{u}(x)\right) \,ds(x) + \int \left(\frac{1}{2}\left\| y \right\|^2 + \tilde{v}(y) \right) \,dt(y) $$ s.t. $\forall x,y,\quad \tilde{u}(x)+\tilde{v}(y) \leq -\langle x,y\rangle.$

Currently, I'm trying to understand stability/sensitivity of optimal solutions to small changes in $s$ and $t$.

I haven't found anything similar in the literature that covers this, so I'm wondering where I can begin. If you know a good reference, feel free to send that my direction.

Thanks!

Let $(X,d,m)$ and $(Y,\rho,\mu)$ be doubling metric measure spaces. The seminal work of Korevaar and Schoen discussed generalizations of $L^p$ spaces for maps from $X$ to $Y$. Standard results from Analysis show that when $X=\mathbb{R}^d$ and $Y=\mathbb{R}^D$ then $C_c(\mathbb{R}^d,\mathbb{R}^D)$ (continuous functions with compact support) is dense in the Bochner-Lebesgue space $L^1(\mathbb{R}^d,\mathbb{R}^D)$.

I'm wondering, if this is true for the $L^p(X,Y)$ spaces of of Korevaar and Schoen and more general if there are extensions of $L^{p(x)}$ spaces, Musielak-Orlicz spaces, between $(X,d,m)$ and $(Y,\rho,\mu)$ for which $C(X,Y)$ is dense in those spaces.