riemannian geometry – Critical points of the area functional restricted to CMC embeddings

For a fixed closed smooth manifolds $M^n$ and $N^{n+1}$, two $C^{k,alpha}$ embeddings $f, f’ : M to N$ are said to be equivalent if there exists $varphi in operatorname{Diff}(M)$ such that $f’ = f circ varphi$. Let $mathcal{M}$ be the set of pairs $((f), g)$, where $(f)$ is an equivalence class of CMC embeddings into the Riemannian manifold $(N,g)$ and $g$ belongs to a fixed open set $Gamma$ of $C^q$ metrics. Brian White showed that $mathcal{M}$ is a $C^{q-j}$ Banach manifold modelled on $Gamma$.

Now let $mathcal{A} : mathcal{M} to mathbb{R}$ be the area functional. I am trying to find its critical points. To this end, let $((f),g)$ be given and consider a curve $((f_t), g_t)$ in $mathcal{M}$ such that $((f_0),g_0) = ((f),g)$ and $d/dt ((f_t), g_t) = (X, h)$ at $t = 0$, where $X$ is a vector field along $f$ and $h$ is a symmetric 2-tensor. Then, by the usual variation formulae for the area and volume elements, we arrive at

$$left. frac{mathrm{d}}{mathrm{d}t} rightvert_{t=0} mathcal{A}((f_t), g_t) = – int_M H_{f} g(X, nu) , mathrm{d}A_{f^ast g} + int_M operatorname{tr}_g(h) , mathrm{d}A_{f^ast g},$$

where $H_f$ is the mean curvature of $f : M to N$, $nu$ is a unit normal for $M$ along $f$ and $operatorname{tr}_g(h)$ is just the trace of $h$ with respect to the metric $g$. How do I conclude? What are the critical points of $mathcal{A}$?

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unit testing – How to make mock in python restricted to one input value?

I have a scenario where I need to mock the boto_utils.client function in the python unittest. The problem I’m facing is as follows consider the following piece of code :

 @mock.patch('utils.boto_utils.client')
 def test_random(self, mock_client):
    stubbed_client = boto3.client('quicksight')
    stubber = Stubber(stubbed_client)
    stubber.activate()
    mock_client.return_value = stubbed_client
    cl = boto_utils.client('s3')
    ## Now we see cl is a quicksight client even though we expect a S3 client

How to make sure that the second call to boto_utils.client with ‘s3’ resource makes a normal external API call? Additionally I need to extend this functionality to other external API calls. How to design this functionality in a general way?

algorithms – Feedback Vertex Set restricted to planar graphs

In Feedback Vertex Set, we are given an undirected graph $G$ and $k in mathbb{N}$, and the objective is to decide whether there exists a subset $S subseteq V(G)$ of size at most $k$ such that $G-S$ is a forest. How can I solve it in running time complexity of $k^{O(sqrt{k})} n^{O(1)}$ when restricted to planar graphs?

functions – restricted access of PDFJS viewer to logged-in users

I’m using PDFJS Viewer plugin for WordPress. for security purpose i’m looking for a way to make restricted access of the viewer only to logged-in users. I want to make the no logged-in users redirected to 404 page if they try to consulte the viewer URL.

example :

If the user is logged-in and type url :

http://website.com/wp-content/plugins/pdfjs-viewer-shortcode/pdfjs/web/viewer.php?file=/wp-content/uploads/2021/06/example.pdf&dButton=false&pButton=false&oButton=false&sButton=true#zoom=auto&pagemode=none —> see the content

If logged-out user type the URL on the browser —-> redirected to 404

thanks in advance for your reply.

Regards.

jsom – Parse restricted csv file into CEWP

I have a page with CEWP. I am parsing a csv file (with js) from Documents library and displaying selective data in CEWP.

CEWP works only if I give permissions to read the csv file or else it returns blank. I want the csv file to be restricted but the data should be available when parsed in CEWP.

How can I retrieve data from csv file into CEWP with the csv file being restricted to view.

algorithms – Finding the distance from a point to a $R^m$ restricted area in $R^n$

The problem is described below:

enter image description here

When m=2 and n=3, it is basically finding the distance between a point and a line segment in $R^3$.

But when both m and n are larger, do I have to use a generic optimizer to solve this, or this problem can be precisely solved with mathematics, like the case when m=2 and n=3?

What I have done for now:

Approach A:

I tried to solve it with Gram–Schmidt process and projection but got stuck.

For example, the following R code:

P <- c(1,1,1,1)

m <- rbind(c(1,-1,1,2)*1/3,c(1,2,1,1)*1/5)

m2 <- qr.Q(qr(t(m)))

P2 <- P%*%m2(,1)*m2(,1)+P%*%m2(,2)*m2(,2)

It does not take into account the restriction $w_1+w_2=1$

Approach B:

Tried to solve it with lagrangian optimization, but also got stuck there.

list manipulation – Question about a restricted rewrite rule

I just started learning Mathematica and I’m not sure why a function I defined doesn’t work.

So I’m trying to write a function that takes a list of vectors and eliminates all zero vectors (with respect to an inner product). I manage to do it the following way:

    noZeros1(list_List, innerp_) := DeleteCases(list, x_ /; innerp(x, x) == 0.) 

It works fine if I try it:

In: noZeros1({{1, 1}, {0, 0}, {1, 0}, {0, 0}, {1, 2}, {0, 0}}, Dot)
Out: {{1, 1}, {1, 0}, {1, 2}}

However, I also tried the following approach:

noZeros(list_List, innerp_) :=list //. {a___, x_, b___} :> {a, b} /; innerp(x, x) == 0.

If I try that:

In: noZeros({{1, 1}, {0, 0}, {1, 0}, {0, 0}, {1, 2}, {0, 0}}, Dot)
Out: {{1, 1}, {1, 0}, {0, 0}, {1, 2}, {0, 0}}

It only deletes the first zero vector it encounters. This is weird to me, because it has the same structure as the following function:

maxima(list_List) :=list //. {a___, x_, y_, b___} /; y <= x :> {a, x, b};

Which is a small function that rewrites a list eliminating elements that are smaller than the previous ones. That’s essentially what I’m trying to do: a small function that rewrites a list eliminating elements that satisfy a condition.

complexity theory – Inapproximability of graph problems on a restricted setting

I am considring the following problem $mathcal{P}$.

$mathcal{P}$: Given an undirected graph $G$, and an integer $k$, find a set of vertices $S subseteq V(G)$, with $|S| = k$, such that the number of edges in the subgraph induced by $S$ is minimized.

which is clearly NP-hard, as the answer is 0 iff there is an independent set of size $k$ in $G$. So I am interested in studying whether the problem can be approximated assuming $mathrm{P} neq mathrm{NP}$ (assumption implicit from now on).

Now, for the restricted setting, imagine $c > 0$ is an arbitrary constant, and then define $cmathcal{P}$ as the same problem but with the restriction that $|V(G)| geq c cdot k$. The question I am wondering is whether, given the following claims 1) and 2), it is true that 1) implies 2).

  1. There is a constant $rho > 1$ such that $mathcal{P}$ cannot be approximated by a factor better than $rho$.
  2. There is a constant $rho’ > 1$ such that for any constant $c > 0$ the problem $cmathcal{P}$ cannot be approximated by a factor better than $rho’$.

I would appreciate any help, or pointers to problems when something like that is proven.