## Can a non-inner automorphism map every subgroup to its conjugate?

Let $$G$$ be a finite group. Can a non-inner automorphism map every subgroup to its conjugate? Namely, can there be a non-inner automorphism $$alpha$$ that, for every $$Hle G$$, there exists some $$g$$ in $$G$$ such that $$alpha(H)=H^g$$?

## personas – How to start a user journey map for a new product and how does it translate to a product backlog?

I’m working on a new product that’s a talent marketplace for the events and entertainment space. The client has a rough idea (like 20%) of what the user journey would be like.

I’m curious how should i go about setting up the foundations for a user journey map. Right now i’ve identified 3 personas that would use the product for different reasons and i’m guessing there will be 3 user journeys.

Should i brainstorm with the client to build this journey? If so what would be a suitable brainstorming method? Also how could i validate this journey with real users?

Lastly, as the user journey gets more detailed, how this would connect to the product roadmap or product backlog?

## react – Primeiro valor em uma função map no javascript

Eu tenho uma função map do JavaScript onde eu retorno todas as fotos de um certo registro. Dentre essas fotos, uma delas tem que receber um texto, que no meu caso é Foto principal, que sempre será o primeiro elemento do array. A dúvida é que o texto sempre vai para todas as imagens, e não faço ideia de como colocar o texto apenas na primeira foto.

Segue abaixo o código em questão.

``````<ul>
{fotoApi?.fotos?.map((foto, index) => (
<li key={index} style={{ listStyle: "none" }}>
<img
style={cssListImg}
src={fotoApi.fotos(index)}
onClick={() => {
setFile(fotoApi.fotos(index));
setImg(null);
setI(index);
setImgSel(foto);
}}
/>
<br />
{/* fotoApi.fotos(0) ? <>Foto principal</> : <>""</> */}
{/* A linha acima é o que eu tentei, mas sem sucesso */}
</li>
))}
</ul>
``````

``````const (fotoApi, setFotoApi) = useState({fotos: (), caminhoimagem: ""});
``````

## 8 – Map array of node IDs to bundle types without loading all the nodes

For complicated reasons I need to map an array of node IDs to a list of node bundles. That is, for a given list of node IDs, I need to get all the different bundles (node types) of the nodes the IDs belong to. The simple approach would be to load all the nodes, map that to their bundles and using array_unique to get the final list. Unfortunately, this is not a good solution because there may be hundreds of IDs in the array and loading all those nodes will result in a bad performance hit and/or go over the memory limit.

What is the best way to map an array of node IDs to (unique) node bundle types? Is there something built-in or does this require a custom database query? If so, I would be grateful for a dynamic query I could use for this (we’re using MySQL). Thanks!

## How do you map a Salesforce master detail field to a field in Drupal? I’ve tested all fields none work

I have successfully mapped all fields except company name which is a master-detail field in Salesforce. This field is required to make a record in salesforce. What field would be used to map to this type of data? Thank you

## dg.differential geometry – Are a map with constant singular values and its inverse always conjugate through isometries?

Let $$U subseteq mathbb R^2$$ be an open, connected, bounded subset. Fix $$0, such that $$sigma_1 sigma_2=1$$.

Suppose that there exist a diffeomorphism $$f:U to U$$ such that the singular values of $$df$$ equal $$sigma_1,sigma_2$$ everywhere on $$U$$.

Question: Do there exist smooth isometries $$phi_1, phi_2:U to U$$ such that $$phi_1 circ f circ phi_2=f^{-1}$$?

We must have $$phi_i in operatorname{O}(2)$$ (isometries are affine), but the point here is that I want them to map $$U$$ into $$U$$.

Note that $$df^{-1}=(df)^{-1}$$ has singular values $$sigma_1,sigma_2$$, the same as $$df$$.

Here are two examples where this phenomena happens:

1. Affine maps on ellipses:

Let $$0, $$ab=1$$, and let
$$U=U_{a,b}=biggl{(x,y) ,biggm | , frac{x^2}{a^2} + frac{y^2}{b^2} < 1 biggr}.$$

Take $$f(x,y)=Apmatrix{x\y}$$, where begin{align*} & A=A(theta)= begin{pmatrix} a& 0 \ 0 & b end{pmatrix} begin{pmatrix} costheta & -sintheta \ sintheta & cos theta end{pmatrix}begin{pmatrix} 1/a& 0 \ 0 & 1/b end{pmatrix}= begin{pmatrix} costheta & -frac ab sintheta \ frac ba sintheta & cos theta end{pmatrix} end{align*}.

Then $$A(theta)^{-1}=A(-theta)=JA(theta) J$$, where $$J=begin{pmatrix} 1& 0 \ 0 & -1 end{pmatrix}$$ is the reflection around the $$y$$ axis.

2. Non-affine maps on the disk:

Let $$U=Dsetminus{0}$$ where $$D subseteq mathbb R^2$$ is the unit disk.

$$f_c: (r,theta)to (r,theta+clog r )$$. Then we have $$f_{c}^{-1}=f_{-c}=Jf_{c}J$$.

Note that
$$(df_c)_{{ frac{partial}{partial r},frac{1}{r}frac{partial}{partial theta}}}=begin{pmatrix} 1 & 0 \ c & 1end{pmatrix},$$
so the singular values of $$f_c$$ are constants which depend on $$c$$.

Now, in general there are many local solutions to the PDE $$sigma_i(df)=sigma_i$$, so I don’t expect such a special relation between $$f$$ and its inverse in general. But I don’t have a counter-example yet.

## personas – How to kick off a user journey map for a new product and how does it translate to a product roadmap

I’m working on a new product that’s a talent marketplace for the events and entertainment space. The client has a rough idea (like 20%) of what the user journey would be like.

I’m curious how should i go about setting up the foundations for a user journey map. Right now i’ve identified 3 personas that would use the product for different reasons and i’m guessing there will be 3 user journeys.

Should i brainstorm with the client to build this journey? If so what would be a suitable brainstorming method? Also how could i validate this journey with real users?

Lastly, as the user journey gets more detailed, how this would connect to the product roadmap?

## When creating an application, when should I use the map satellite view instead of the plain one?

From UX Perspective, If I create a map-based mobile application why should I not use the map satellite view instead of the vector one?

Is it because it is crowded with data and the user will not be able to distinguish the roads as easily, or is there another reason that it has to do with the performance?

## linear algebra – Let \$theta>0\$, find the matrix associated with the identity map, and rotation of basis by angle \$-theta\$

Let $$theta>0$$, find the matrix associated with the identity map, and rotation of basis by angle $$-theta$$.

So it changes basis from $$(1, 0), (0, 1)$$ to $$(costheta, -sintheta), (sintheta, costheta)$$ associated with the identity map.

Then
$$(1, 0)=a_1(costheta, -sintheta)+a_2(sintheta, costheta)$$
$$(0, 1)=b_1(costheta, -sintheta)+b_2(sintheta, costheta)$$
$$a_1=costheta, a_2=sintheta, b_1=-sintheta, b_2=costheta$$
i.e. the matrix is $$begin{equation*} A_{m,n} = begin{pmatrix} costheta & -sintheta \ sintheta & costheta \ end{pmatrix} end{equation*}$$
But the solution says it’s
$$begin{equation*} A_{m,n} = begin{pmatrix} costheta & sintheta \ -sintheta & costheta \ end{pmatrix} end{equation*}$$
Where I did wrong?

## ag.algebraic geometry – Exponential Map of Moduli Space

Let $$mathcal{M}$$ be a compact moduli space over $$mathbb{C}$$ (the specific example I have in mind is the moduli of curves $$mathcal{M}_g)$$. Consider a point $$(X)in mathcal{M}(mathbb{C})$$. Then via deformation theory in algebraic geometry, we say that the tangent space $$T_{(X)}mathcal{M}$$ consists of lifts (or rather isomorphism classes of lifts) of $$X$$ to $$mathbb{C}(epsilon)/(epsilon^2)$$. This is naturally a $$mathbb{C}$$-vector space. My question is in two parts:

1. Does this construction match with the tangent space of the analytification $$mathcal{M}^{an}$$ at $$X$$? (This question doesn’t need compact, I think)

2. Since I assume that $$mathcal{M}$$ is compact, by Hopf-Rinow, the exponential map $$T_{(X)}mathcal{M}rightarrow mathcal{M}$$ is well-defined (if we assume that $$(X)$$ does not lie in the boundary of $$mathcal{M}$$. If part 1) is true, the tangent space corresponds to infinitesimal lifts of $$X$$. Can we explicitly say what the image of an such lift is under the exponential map?

If this isn’t possible in general, is there perhaps a way to do it for the moduli of curves, moduli of abelian varieties or the Hilbert scheme?