How to get the Cartesian product of two intervals in $ mathbb R^1$?

How to get the Cartesian product of $a_1times a_2$ where $a_1=[0, 1]$, $a_2=[1,2]$. Note that both $a_1$ and $a_2$ are intervals in $mathbb R^1$ and I would like the result to be a set to support further manipulations.

ag.algebraic geometry – Degrees of syzygies of points in $mathbb P^2$

Let $X$ be a collection of points in $mathbb P^2$ over the complex numbers. Let $I_X$ be the defining ideal. I am interested in knowing when:

The syzygies of $I_X$ contains no linear forms. Since we are in $mathbb P^2$, this just says that the Hilbert-Burch matrix contains no (non-zero) linear entries. $(*)$

One obvious case when this happens is when we take two general curves $F,G$ of degrees $a,bgeq 2$ and let $X=V(F)cap V(G)$. Then the syzygies is just the Kozsul relation and has degrees $a,b$. I don’t know further examples and would like to know if there are interesting geometric conditions that would imply $(*)$.

One obvious necessary condition is that the generators of $I_X$ have degree at least $2n-2$, where $n$ is the number of generators.

Also, perhaps if you fix the degree $d$ of $X$, then $(*)$ defines a closed subscheme (? I am not sure about this) of the Hilbert scheme $mathbb P^{2(d)}$. If so, then knowing it’s dimension would be nice.

complex analysis – Information/references/examples on fields $mathbb R^3 to mathbb C^3$ with divergence and curl free real and imaginary parts

In the course of some physical considerations I came across a complex vector field
$$ mathbf u = mathbf v + i mathbf w, $$ with
begin{align}
mathbf v:& mathbb R^3to mathbb R^3\
mathbf w:& mathbb R^3to mathbb R^3
end{align}

the special propert, that it has a divergence-free imaginary and
curl free real components, that means

begin{align}
vec{nabla}times mathbf v & = mathbf 0 tag{1}\
vec{nabla}cdotmathbf w & = 0 tag{2}
end{align}

In my attempts to better understand and interpret the quantitiy described by this field, I started to wonder if:

Question 1: This property has a special name/term in complex vector analysis.

Question 2: If there are any important prominent examples of such fields.

Question 3: If there are other properties that follow as a consequence of (1) and (2).

Question 4: If anyone can hint me at literature where such fields are investigated.

I would be very grateful for any hints at this. I would hope for answers like those this this question about answers similar

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

I read an article where it 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 "Arithmetic of Elliptic Curves" by J.Silverman, on page 191, but here it says:

Yes $ E $ is an elliptic curve over $ mathbb {Q} _p $ and $ hat {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 testing $ E_1 ( mathbb {Q} _p) approx mathbb {Z} _p $.

abstract algebra – Calculation of cohomology groups of $ S_3 $ with coefficients in $ mathbb {Z} $

I'm trying to do a group cohomology exercise from Brown's Book to calculate $ H ^ n (S_3, mathbb {Z}) $exercise $ 1 $ page $ 85 $.

We know by theorem $ 14 $ that $ H ^ n (S_3, mathbb {Z}) cong H ^ n (S_3, mathbb {Z}) _ {(2)} bigoplus H ^ n (S_3, mathbb {Z}) _ {( 3)} $. We calculate first $ H ^ n (S_3, mathbb {Z}) _ {(2)} $, we note that this sylow subgroup is not normal but we will have it in $ H ^ n ( mathbb {Z} _2, mathbb {Z}) $ the set of $ G $-the invariant elements will be the whole set, so we get $ H ^ n (S_3, mathbb {Z}) _ {(2)} cong H ^ n ( mathbb {Z} _2, mathbb {Z}) $ and this is the cohomology of a cyclical group, which we have already calculated before.
Now here comes the hard part, we know that the Sylow subgroup associated with $ H ^ n (S_3, mathbb {Z}) _ {(3)} $ it's normal then we know that $ H ^ n (S_3, mathbb {Z}) _ {(3)} cong H ^ n ( mathbb {Z} _3, mathbb {Z}) ^ { mathbb {Z} _2} $and so we need to understand how $ mathbb {Z} _2 $ acts on $ H ^ n ( mathbb {Z} _3, mathbb {Z}) $, and this is the part that I am having difficulties, we know that the action of $ G $ in $ H ^ n (H, M) $ is given by $ (x rightarrow f (x)) rightarrow (x rightarrow gf (g ^ {- 1} x)) $, and from $ H $ it is normal this induces an action on $ G / H $, now I don't see what this is going to induce in the action of $ mathbb {Z} _2 $ in $ H ^ n ( mathbb {Z} _3, mathbb {Z}) $ because when we calculate these cohomology groups we are doing the identification $ Hom _ { mathbb {Z} ( mathbb {Z} _3)} ( mathbb {Z} ( mathbb {Z_3}), mathbb {Z}) cong mathbb {Z} $. Then any help is appreciated.

elementary set theory – Explanation test: $ X, Y $ with $ | X | = | And | = m in mathbb {N} $ exactly $ m exists! = 1 cdot 2 cdot ldots cdot m $

Motto For sets $ A, B $ with $ | X | = | And | = m in mathbb {N} $ exists exactly $ m! = 1 cdot 2 cdot ldots cdot m $ distinct bijective maps in pairs between $ X $ and $ Y $.

The test is left as an exercise.


It looks like the test is all about combinatorics, but I don't know how to make a fist assumption / statement.

Things we can conclude:

  1. We have $ n $ elements from which we want to know the number of ways to choose $ k $ elements, therefore, the size of our two sets $ X $ and $ Y $ is $ binom {n} {k} $.
  2. The rule of symmetry for the binomial coefficient states that $ binom {n} {k} = binom {n} {n-k} $. Therefore, we could say that $ | X | = binom {n} {k} $ and $ | And | = binom {n} {n-k} $

I don't know if that is useful and if so where to go from there.

linear algebra: why is $ mathbb {w} $ orthogonal to both SVM boundary lines?

See http://web.mit.edu/zoya/www/SVM.pdf

"Choose an arbirary point $ x1 $ lie online $ w ^ Tx + b = −1 $. Then the closest point online $ w ^ Tx + b = 1 $ to $ x1 $ is the point $ x2 = x1 + λw $ (since the closest point will always be on the perpendicular; remember that the vector $ w $ is perpendicular to both lines). "

Why should it be perpendicular to both lines? What is the math behind this?

ag.algebraic geometry – Relationship between quasi-inherent pulleys and modules $ mathbb A ^ 1 $ -fpqc on a fpqc stack

In what follows, assume multiple universes for simplicity.

Leave $ X $ be a stack in groupoids on the fpqc site of small related schemes $ mathbf {Aff} _ { text {fpqc}} $. We can define $ mathbf {QCoh} (X) $ formally considering the representable stack $$ mathbf {Hom} _ { operatorname {Stk} _ { text {fpqc}} ( mathbf {Aff})} (-, mathbf {QCoh}) $$
and restricting it to subcategory 2 complete $ operatorname {Stk} ^ { text {Gpd}} _ { text {fpqc}} ( mathbf {Aff}) ^ { text {op}} $.

We can define the fpqc topos of $ X $ be the cut category $ operatorname {Shv} _ { text {fpqc}} ( mathbf {Aff}) / X $, and we can make this polka dot locally ringed by pulling back $ mathbb {A} ^ 1_X = X times _ { operatorname {Spec} ( mathbb {Z})} mathbb {A} ^ 1. $

As $ mathbb {A} ^ 1_X $ is a local ring object in the moles, so we can define a category of $ mathbb {A} ^ 1_X $-modules in $ operatorname {Shv} _ { text {fpqc}} ( mathbf {Aff}) / X $. By the definition of $ mathbf {QCoh} (X) $we can write a forgetful functor $ U $ to the category of $ mathbb {A} ^ 1_X $-modules sending a quasi-inherent sheaf $ F $ in $ X $ to the evaluation of its decline, that is, to $ f: operatorname {Spec} (R) to X $, we have
$$ U (F) (f) = Gamma ( operatorname {Spec} (R), f ^ ast F), $$
which is naturally a $ mathbb {A} ^ 1_X $-module.

Question: Is the functor forgetful? $ U $ defined above totally true? If so, can we identify your essential image as the full subcategory of $ mathbb {A} ^ 1_X $-modules that support a presentation the way we can for schematics? Do any of these statements become true or remain true if we go on to derive everything? What if we limit ourselves to the case where $ X $ What is an Artin Stack?

abstract algebra – Galois group of $ x ^ {28} -1 $ over $ mathbb {Q} $

Galois's group $ x ^ 28 -1 $ finished $ mathbb {Q} $ is $ mathbb {Z} / 28 mathbb {Z} ^ times cong C_2 times C_6 $. But without knowing a priorihow can i say that $ mathbb {Z} / 28 mathbb {Z} ^ times $ is isomorphic to $ C_2 times C_6 $? I know the question of determining the group structure of $ mathbb {Z} / n mathbb {Z} ^ times $ it's difficult in general, but is there any clever trick I can do, in a test for example to easily verify that $ C_2 times C_6 $ it's really correct, and it's not something like $ C_ {12} $? I know being abelian reduces it a lot, but is that the best we have? Is there any other property that I can exploit?

T / F: If $ f (x), g (x) in mathbb {Q}[x]$ are irreducible polynomials that have the same division field, then deg $ f = $ deg $ g $.

This is a true or false problem, but I'm not sure how to deal with it. Thanks for any help / advice.