gr.group theory – Stabilizers in the action of $mathrm{GL}(n, mathbb Z)$ on $mathbb Z^n$

How can we calculate effectively the subgroups of $G: = mathrm{GL}(n, mathbb Z)$ which fix pointwise a given submodule $S$ of $mathbb Z^n$ in the action of $G$ on $mathbb Z^n$ by left multiplication?

Note: Suppose $S$ is freely generated by $xi_1, xi_2, cdots, xi_s$ then the clearly the problem is equivalent to describing the unimodular matrices fixing each of the tuples $xi_1, cdots xi_s$. If the $xi_j$s have a simple form, e.g., $xi_j$s are the standard basis vectors $e_1, cdots, e_n$ of $mathbb Z^n$ then the form of matrices fixing each $xi_j (j = 1, cdots, s)$ is easy to determine.
But in general the method to obtain the stabilizer is not clear.

real analysis – Does the continuous function $ H ^ 1 ( mathbb {R} ^ 2) $ have to tend towards zero in infinity?

Here, $ H ^ 1 ( mathbb {R} ^ 2) $ is the standard Sobolev space for $ L ^ 2 ( mathbb {R} ^ 2) $ Functions whose weak derivation is one of them $ L ^ 2 ( mathbb {R} ^ 2). $

My question in the title comes from the variational calculation. It is usually the case that a minimizer of a particular energy function is defined $ H ^ 1 ( mathbb {R} ^ 2) $ is known as continuous (or even continuous) $ C ^ 2 ( mathbb {R} ^ 2)) $. I want to know the behavior of this minimizer in infinity.

If $ u in L ^ 2 ( mathbb {R} ^ 2), $ then is known $ liminf_ {| x | to infty} u (x) = 0. $ But it cannot say $ limsup_ {| x | to infty} u (x) = 0 $ because counterexamples exist.

If we accept $ u in H ^ {1+ epsilon} ( mathbb {R} ^ 2) $ for some $ epsilon> 0, $ then the inequality of the classic Morrey can imply a uniform older continuity of $ u. $ So we can close $ limsup_ {| x | to infty} u (x) = 0 $ by proof by contradiction.

So my problem is the case $ epsilon = 0. $ That is when
$$ u in H ^ 1 ( mathbb {R} ^ 2) cap C ( mathbb {R} ^ 2) $$
(or $ u in H ^ 1 ( mathbb {R} ^ 2) cap C ^ 2 ( mathbb {R} ^ 2)), $ is it true that
$$ limsup_ {| x | to infty} u (x) = 0? $$

With evidence of contradiction, I think this should be true. Here is my non-rigorous evidence.

Let's not assume there is $ epsilon> 0 $ and $ x_n in mathbb {R} ^ 2 $ so that $ | x_n | to infty $ and $ | u (x_n) | geq 2 epsilon. $ Because of the continuity there is $ r_n> 0 $ so that $ | u (x) | geq epsilon $ for all $ x in B (x_n, r_n). $
Since $ u in L ^ 2, r_n to 0 $ how $ n to infty. $
I don't think so rigorously
$$ int_ {B (x_n, r_n)} | nabla u | ^ 2 gtrsim int_ {B (x_n, r_n)} ( frac { epsilon} {r_n}) ^ 2 = epsilon ^ 2 $ $
for big $ n $ and
$$
int _ { mathbb {R} ^ 2} | nabla u | ^ 2 geq sum_ {n , text {is large}} int_ {B (x_n, r_n)} | nabla u | ^ 2.
$$

So they imply a contradiction $ int _ { mathbb {R} ^ 2} | nabla u | ^ 2 = infty $

I am happy about every discussion.

elliptic curves – why $ E_1 ( mathbb {Q} _p) approx mathbb {Z} _p $

I have read an article that says: $ E_1 ( mathbb {Q} _p) approx mathbb {Z} _p $

Where $ E $ is an elliptic curve over $ mathbb {Q} _p $ and $ E_1 ( mathbb {Q} _p) = {P in E ( mathbb {Q} _p): tilde {P} = tilde {O} } $.

The author says the proof is in "Silverith Arithmetic of Elliptic Curves" on page 191, but it says here:

If $ E $ is an elliptic curve over $ mathbb {Q} _p $ and $ has {E} $ is the formal group, then:

$$ E_1 ( mathbb {Q} _p) approx hat {E} (p mathbb {Z} _p) $$

So I don't know a good reference for proving $ E_1 ( mathbb {Q} _p) approx mathbb {Z} _p $.

Limits – Prove that $ frac {f (x) – (f * K_t) (x)} {t} bis – Delta f (t bis 0) $ for $ f in C_0 ^ { infty} ( mathbb {R} ^ n) $

Describe
$$ K_t (x) = frac {1} {(4 pi t) ^ { frac {n} {2}}} exp (- frac {| x | ^ 2} {4t}) $$
the heat core, $ f in C_0 ^ { infty} ( mathbb {R} ^ n) $. It is known that $ f * K_t to f (t to 0) $ point by point and $ L ^ p (p ge1) $. How can we prove it? $$ frac {f (x) – (f * K_t) (x)} {t} bis – Delta f (t bis 0) $$
holds ($ Delta $ denotes the Laplace operator)? I got a proof with the spectral resolution, but I wonder if there is a simpler proof with the basic analysis. Thanks a lot.

Convexity – convergence of the Steiner symmetrization of a compact convex subset of $ mathbb R ^ n $ to a $ ell_1 $ sphere

To let $ p in (1, infty) $ and let $ | cdot | _p $ be the $ ell_p $-norm on $ mathbb R ^ n $, and let $ B_ {n, p} (r): = {x in mathbb R ^ n mid | x | _p le r } $ be the $ ell_p $-Balls of $ mathbb R ^ n $ of the radius $ r ge 0 $. Remember it . To let $ mathbb K_n $ be the set of all compact convex subsets of $ mathbb R ^ n $ and let $ K in mathbb K_n $. For any vector other than zero $ u $ in the $ mathbb R ^ n $you can see $ K $ as a family of line segments parallel to $ u $. Slide each line segment so that it is symmetrical to the hyperplane $ u ^ perp $. Designate the amount so obtained as $ S_uK $. Note that $ S_uK in mathbb K_n $.

Question. Is it always possible to find $ r ge 0 $ and a sequence $ u_1, u_2, ldots, u_N, ldots $ of nonzero vectors in $ mathbb R ^ n $ so that you have (for example in the Hausdorff topology)
$$
lim_ {N rightarrow infty} S_ {u_N} S_ {u_ {N-1}} ldots S_ {u_2} S_ {u_1} K = B_ {n, p} (r) ; ?
$$

  • The Euclidean case $ p = 2 $ is solved by Theorem 9.1 of "Convex and Discrete Geometry" (Peter M. Gruber).
  • I am particularly interested in the cases $ p = 1 $ and $ p = infty $.

ag.algebraic geometry – relationship between quasi-coherent sheaves and $ mathbb A ^ 1 $ -fpqc modules over an fpqc stack

In the following, assume several universes for the sake of simplicity.

To let $ X $ be a stack in groupoids at the fpqc site of small affine schemes $ mathbf {Aff} _ { text {fpqc}} $. We can define $ mathbf {QCoh} (X) $ formally taking into account the representable stack $$ mathbf {Hom} _ { operatorname {Stk} _ { text {fpqc}} ( mathbf {Aff})} (-, mathbf {QCoh}) $$
and restriction to the full 2 ​​subcategory $ operatorname {Pcs} ^ { text {Gpd}} _ { text {fpqc}} ( mathbf {Aff}) ^ { text {op}} $.

We can define the fpqc topos from $ X $ be the slice category $ operatorname {Shv} _ { text {fpqc}} ( mathbf {Aff}) / X $and we can ring this topos locally by pulling it back $ mathbb {A} ^ 1_X = X times _ { operatorname {Spec} ( mathbb {Z})} mathbb {A} ^ 1. $

Since $ mathbb {A} ^ 1_X $ If there is a local ring object in the topos, we can then define a category of $ mathbb {A} ^ 1_X $Modules in $ operatorname {Shv} _ { text {fpqc}} ( mathbf {Aff}) / X $. According to the definition of $ mathbf {QCoh} (X) $we can write down a forgetful functor $ U $ in the category of $ mathbb {A} ^ 1_X $Modules by sending a quasi-coherent sheaf $ F $ on $ X $ to evaluate his withdrawal, d. H $ f: operatorname {Spec} (R) to X $, we have
$$ U (F) (f) = Gamma ( operatorname {Spec} (R), f ^ ast F), $$
of course that's one $ mathbb {A} ^ 1_X $-Module.

Question: Is the forgetful functor $ U $ defined above true? If so, we can identify its main picture as the full subcategory of $ mathbb {A} ^ 1_X $Modules that allow a presentation like we can for programs? Will any of these statements be true or will it remain true if we infer everything? What if we limit ourselves to the case where $ X $ is an artin stack?

Galois Theory – Application of the method to determine the number of monically irreducible factors from $ X ^ {255} -1 $ in $ mathbb {F} _2 $ to another case

I was shown how to find the number of monically irreducible factors from $ X ^ {(2 ^ 8-1)} – 1 $ in the $ mathbb {F} _ {2} $ and I'm struggling to use the method to find the number of monically irreducible factors of $ X ^ {(5 ^ 6-1)} – 1 $ in the $ mathbb {F} _5 $.

The method for the first question is as follows:

$ 255 = 2 ^ 8-1 $ is the order of the multiplicative cyclic group $ mathbb {F} _ {256} ^ * $. To the $ alpha in mathbb {F} _ {256} $All of its conjugates are the roots of the same minimal polynomial $ mathbb {F} _2 $ and this minimal polynomial appears as an irreducible factor for $ X ^ {255} -1 $.

The subfields of $ mathbb {F} _ {256} $ are $ mathbb {F} _ {2} $, $ mathbb {F} _ {4} $, $ mathbb {F} _ {16} $ and $ mathbb {F} _ {256} $.

There is $ 1 $ element $ alpha in mathbb {F} _2 ^ * $ is that $ mathbb {F} _2 ( alpha) = mathbb {F} _4 $, she has $ 1 $ conjugate

There is $ 4-2 = $ 2 elements $ alpha in mathbb {F} _4 – mathbb {F} _2 $ so that $ mathbb {F} _2 ( alpha) = mathbb {F} _ {4} $, they have $ 2 $ Conjugates (they are conjugates of each other)

There is $ 16-4 = $ 12 elements $ alpha in mathbb {F} _ {16} – mathbb {F} _4 $ so that $ mathbb {F} _2 ( alpha) = mathbb {F} _ {16} $, they have $ 4 $ Conjugates

There is $ 256-16 = $ 240 elements $ alpha in mathbb {F} _ {256} – mathbb {F} _ {16} $ so that $ mathbb {F} _2 ( alpha) = mathbb {F} _ {256} $, they have $ 8 $ Conjugates

Hence the solution $ 1/1 + 2/2 + 12/4 + 240/8 = $ 35 monic irreducible polynomials.

Here is my attempt at the second question:

$ 5 ^ 6 – $ 1 is the order of the multiplicative group $ mathbb {F} _ {5 ^ 6} ^ * $.

The subfields of $ mathbb {F} _ {5 ^ 6} $ are $ mathbb {F} _ {5} $, $ mathbb {F} _ {5 ^ 2} $, $ mathbb {F} _ {5 ^ 3} $ and $ mathbb {F} _ {5 ^ 6} $.

There is $ 4 $ elements $ alpha in mathbb {F} _5 ^ * $ is that $ mathbb {F} _5 ( alpha) = mathbb {F} _ {5} $, they have $ 1 $ conjugate

There is $ 25-5 = $ 20 elements $ alpha in mathbb {F} _ {5 ^ 2} – mathbb {F} _5 $ so that $ mathbb {F} _5 ( alpha) = mathbb {F} _ {5 ^ 2} $, they have $ 2 $ Conjugates

There is $ 125-25 = $ 100 elements $ alpha in mathbb {F} _ {5 ^ 3} – mathbb {F} _ {5 ^ 2} $ so that $ mathbb {F} _5 ( alpha) = mathbb {F} _ {5 ^ 3} $, they have $ 3 $ Conjugates

There is $ 5 ^ 6-5 ^ 3 = $ 15500 elements $ alpha in mathbb {F} _ {5 ^ 6} – mathbb {F} _ {5 ^ 3} $ so that $ mathbb {F} _5 ( alpha) = mathbb {F} _ {5 ^ 6} $, they have $ 6 $ Conjugates

Which gives $ 4/1 + 20/2 + 100/3 + 15500/6 $ Polynomials.

The problem

Obviously I misunderstood the method because $ 3 $ does not share $ 100 $. What I think is likely is that I have not understood why these elements have the number of conjugates they have, or I have not understood how to properly count the number of such elements.

If anyone could give an insight into what I did wrong or to demonstrate how the method would work on the second question, I would be very grateful.

Continuity – $ mathbb {N} $ with the continuous cofinite topology

To let $ mathbb {N} $ be equipped with the co-finite topology.
I want to explain why the function $ f $ :: $ mathbb {N} $$ mathbb {N} $, $ n $$ n ^ 3 $
is continuous by using the definition of continuity that says a function $ f $: X → Y is continuous if the inverse image of an open set in Y is an open set in X. In other words, I want to be able to use the model. I know that $ mathbb {N} $ is associated with the co-finite topology and therefore its image is linked under a continuous function, but I'm not entirely sure where to go from here.

nt.number theory – does this series, which refers to the Hasse / Ser series for $ zeta (s) $, converge for all $ s in mathbb {C} $?

I asked this question when swapping math stacks, but it didn't get traction. Still curious about the answer.

Numerical evidence suggests that:

$$ lim_ {N to + infty} sum_ {n = 1} ^ N frac {1} {n} sum_ {k = 0} ^ n (-1) ^ k {n choose k } frac {1} {(k + 1) ^ {s}} = s $$

or equivalent

$$ lim_ {N to + infty} H left (N right) + sum _ {n = 1} ^ {N} left ({
frac {1} {n} sum _ {k = 1} ^ {n} { left (-1 right) ^ {k} {n select k} frac {1
} { left (k + 1 right) ^ {s}}}} right) = s $$

With $ H (N) $ = the $ N $-th harmonic number.

Convergence is fairly slow, but is much faster for negatives $ s $. Also the calculations for non-integer values ​​of $ s $ require high precision settings (I used Maple, pari / gp and ARB).

According to Mathematica, however, the series deviates through the "Harmonic Series Test", even though it was recorded $ s $ as an integer, it agrees with convergence.

This series converges for $ s in mathbb {C} $ ?

Some numerical results below:

s=0.5
0.497702121, N = 100
0.499804053, N = 1000
0.499905919, N = 2000

s=-3.1415926535897932385   
-3.14160222, N = 100
-3.14159284, N = 1000
-3.14159272, N = 2000

s=2.3-2.1i
2.45310498 - 1.94063637i, N = 100
2.33501943 - 2.09308517i, N = 1000
2.31996958 - 2.09923503i, N = 2000