## The Galois Group \$Gamma(mathbb{Q}(sqrt[n]{2}):mathbb{Q})\$ is trivial if \$n\$ is odd.

The full question is: Given a group $$F_n=mathbb{Q}(sqrt(n){2})$$ then prove that:

• (A) If $$n$$ is odd then $$Gamma(F_n:mathbb{Q})$$={$$rm{id}$$} and
• (B) If $$n$$ is even then $$Gamma(F_n:mathbb{Q})congmathbb{Z_2}$$

I am confused as to how to find either, I know that $$F_n:mathbb{Q}$$ is a non-normal extension. We were given the hint:

"For any $$tauinrm{Aut}_mathbb{Q}(F_n)$$ we must have that $$tau(sqrt(n){2})in F_nsubseteqmathbb{R}$$ is again a real $$n$$-th root of 2"

## fitting – (Non)LinearModelFit hangs on “trivial” linear fit (Fit succeeds)

A simplified version of my function and data:

``````    myfunc(x_, a_) = 8*a*Exp(-8/Sqrt(x))/(Sqrt(x)*Abs(WhittakerW(E*I/Sqrt(x), 1/2, 8/10*I*Sqrt(x)*a))^2)
mydata = {{4.5, 195}, {2.9, 175}, {2.1, 95}}
``````

`myfunc` also needs a global rescaling. If I want to fit both parameter `a` and the scale, NonLinearFitModel works great out of the box and quickly:

``````    NonlinearModelFit(mydata, scale*myfunc(x, a), {scale, a}, {x})
``````

If, however, I only want to find the best scaling for an `a` value of my choice, this:

``````   NonlinearModelFit(mydata, scale*myfunc(x, 4), {scale}, {x})
``````

takes forever in my computer (I aborted after a while, I don’t know whether it would succeed at some point or not). I am surprised: why the “complicated” fit works so easily, and this “trivial” linear one not?

`LinearModelFit` is supposedly the right tool here, but:

``````   LinearModelFit(mydata, {myfunc(x, 4)}, {x}, IncludeConstantBasis -> False)
``````

hangs in the same way. There must be something wrong I am doing.
I tried using 4.0 instead of 4 for `a`, enclosing with an `N()` or putting all constants , adding `WorkingPrecision -> MachinePrecision`, but I get the same result.

NonlinearModelFit works adding `Method -> "NMinimize"`, although it is slow, and raises a warning “You asked for a non-linear method, note that the model is linear”. I do not really know what `NMinimize` is doing here, I guess it applies the “right” simplification to myfunc, so the computation is easier?

Finally, after wasting one day, `Fit` just works:

``````   Fit(mydata, {myfunc(x, 4)}, {x})
``````

This reinforces my impression that I just need to apply some magical simplification, that `Fit` always employs, that `NonlinearModelFit` uses only for the two-parameter fit, and that `Method -> "NMinimize"` will cause. Is `LinearModelFit` evaluating `myfunc` much more times than nedeed? I am clueless.

`LinearModelFit` has several niceties that I would rather not give up for what appears to be such a stupid issue. Furthermore, I am probably going to learn a new important Mathematica quirk from this. Thus:

Why the above code, using (Non)LinearModelFit for a simple `scale*f(x)` model, hangs, and how should I modify it to make it work?

## group theory – Non trivial homomorphisms \$mathbb{Z}/3mathbb{Z} rightarrow text{Aut}(mathbb{Z}/7mathbb{Z})\$

I have to find non trivial homomorphisms $$varphi$$, $$varphi’$$: $$mathbb{Z}/3mathbb{Z} rightarrow text{Aut}(mathbb{Z}/7mathbb{Z})$$. We know that Aut $$(mathbb{Z}/7mathbb{Z}) cong (mathbb{Z}/7mathbb{Z})^times cong mathbb{Z}/6mathbb{Z}$$. So we have to look order of elements of $$mathbb{Z}/6mathbb{Z}$$ that divide $$3$$. These elements are $$(2)_6$$ and $$(3)_6$$. In corrections these homomorphisms are defined as: $$Big(varphi((1)_3)Big)((i)_7) = (2i)_7$$ and $$varphi’$$:$$Big(varphi((1)_3)Big)((i)_7) = (4i)_7$$. The thing that i don’t understand, is what $$(i)_7$$ represents. For me, if we generalize for example $$varphi$$, we got $$varphi((r)_3) = (2^r)_7$$. I read a lot of articles on ”research” of homomorphisms, and i feel confused right now and can’t get an intuition for this. Thanks in advance for help.

## real analysis – Nontrivial signed measure on Lebesgue measurable sets being trivial on Borel sets

Let $$mathfrak{L}(mathbb{R})$$ be the collection of Lebesgue measurable sets and $$mathfrak{B}(mathbb{R})$$ be the Borel sets.

Question: Is there a nontrivial signed measure on $$mathfrak{L}(mathbb{R})$$ that is trivial on $$mathfrak{B}(mathbb{R})$$?

Obviously, any positive measure that is trivial on $$mathfrak{B}(mathbb{R})$$ is also trivial on $$mathfrak{L}(mathbb{R})$$, since any Lebesgue measurable set is a subset of a Borel set.

For the signed case, I have tried doing Jordan decomposition but it doesn’t seem work. It is hard (if ever possible) to show $$(mu|_{mathfrak{B}(mathbb{R})})^+ = mu^+|_{mathfrak{B}(mathbb{R})}$$ and $$(mu|_{mathfrak{B}(mathbb{R})})^- = mu^-|_{mathfrak{B}(mathbb{R})}$$.

Background: I am trying to prove (or disprove) that if $$mu$$ and $$lambda$$ are measures on $$mathfrak{L}(mathbb{R})$$, then $$mu|_{mathfrak{B}(mathbb{R})} = lambda|_{mathfrak{B}(mathbb{R})}$$ implies $$mu = lambda$$.

## Is any periodic function trivial

Say I define

xi(s) = xi(1-s)

Since this function is just a lagged copy of itself, over some period, isnt anything which holds for a period true of all other periods?

## ag.algebraic geometry – When is a twisted form coming from a torsor trivial?

Consider a sheaf of groups $$G$$, equipped with a left torsor $$P$$ and another left action $$G$$ on some $$X$$. Form the contracted product $$P times^G X := (P times X)/sim$$ where $$sim$$ is the antidiagonal quotient: $$(g.p, x)sim (p, g.x)$$.

Q1: When is $$Ptimes^G X$$ trivial? I.e., when do we have an isomorphism $$P times^G X simeq X$$?

Partial answer: $$P times^G X simeq X$$ over $$(X/G)$$ iff $$P times (X/G)$$ is a trivial torsor over the stack quotient $$(X/G)$$.

Proof: We can rewrite $$P times^G X$$ as a contracted product of two torsors $$(P times (X/G))times^G_{(X/G)} X$$. Then we contract with “$$X^{-1}$$” — the inverse to contracting with $$X$$ as a torsor over $$(X/G)$$ and we win. (as in B. Poonen’s Rational Points on Varieties, section 5.12.5.3)

Am I allowed to do this? This argument probably shouldn’t have to appeal to algebraic stacks and may be somewhat dubious.

Q2: If I have one isomorphism $$P times^G X simeq X$$, can I choose another one that lies over $$(X/G)$$? Or at least is $$G$$-equivariant?

Q3: Is there a natural way to write the triviality of such a twisted form?

I first thought $$P times^G X simeq X$$ iff $$P$$ was trivial, which is clearly false for trivial actions on $$X$$. Then I was excited to have the pullback $$* to BG$$ represent triviality of the twisted form $$P times^G X$$ as well as the torsor $$P$$. Is there a natural representative of the sheaf of isomorphisms between $$P times^G X$$ and $$X$$?

These can all be sheaves, although I’m primarily interested in $$G = GL_n, PGL_n, SL_n$$, etc. acting on $$X = mathbb{A}^n, mathbb{P}^n$$ as appropriate. More ambitious is $$G = text{Aut}(X)$$ for even simple $$X$$. I’d be happy with answers in any level of generality.

Due Diligence Statement: I’m a novice in the area of “twisted forms” of varieties, so I apologize if the above is evident or obtuse. I checked all the “similar questions” listed here and couldn’t find an answer.

## ordinary differential equations – \$lambda\$ values of \$y”-(frac{1}{4}+frac{lambda}{x}).y=0\$ such that there is a non trivial solution

What are the $$lambda$$ values such that $$; y”-(frac{1}{4}+frac{lambda}{x}).y=0 ;;; 0 $$y(0)=0 ; ; ; lim_{xrightarrowinfty}y(x)=0$$
have a non trivial solution.

Trying to use the Frobenius method, I found that $$; (s^2+s-lambda).a_1=0;$$ and the recurrence relation $$; a_k=frac{-1}{k.(k-1)}.(frac{1}{4}.a_{k-2}+lambda.a_{k-1});$$. These results that I got are correct ? How can I get the answer by these results ?

## Complex conjugation inducing a trivial map on the fundamental group

Let $$V$$ be a smooth projective complex variety defined over the rationals such that $$G=pi_1(V)$$ is a non-abelian finite simple group. Can the map $$Gto G$$ induced by complex conjugation be trivial?

## memory – Is it trivial to protect from double free just by LD_PRELOADing a custom malloc/calloc and free?

Can’t one just implement a malloc/calloc wrapper that adds the returned pointer address to a global hash table prior to returning, and then a free wrapper that checks for the presence of the pointer in the table prior to freeing (returning early if it isn’t present), and then LD_PRELOAD these malloc/calloc and free functions with a program like Firefox, in order to protect from double frees? Is there a reason why the standard malloc/calloc and free functions don’t use such a technique, or why there isn’t a secure variant that is suggested similarly to how strcpy_s is suggested in place of strcpy?

## Generation of trivial cofibrations of the Bousfield localization

Assume $$mathfrak {M}$$ is a category of appropriate celluar model left and $$S$$ is a set of cofibrations $$mathfrak {M}$$. What are the trivial cofibrations that generate $$L_S mathfrak {M}$$? Are they $$J cup S$$, or $$J$$ is all about generating trivial cofibrations of $$mathfrak {M}$$?