## 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 $$mathcal{A}$$?

## OnlineNIC service is restricted | Web Hosting Talk

Hello,

For the past few days, I cannot add funds into my reseller account on OnlineNIC. The error is “system under maintenance”.
Is anyone experience the same issue?

I heard they suffered a legal case, and another company is going to take over, so financial function is restricted for now, but not sure how long it’s going to take.
I have to renew some expired domains, but my account balance is too low. Transfer out is also impossible for expired domains. I’m not sure what to do now.

## 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

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:

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.

## artificial intelligence – Using Restricted Boltzmann Machines for clustering data

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## 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.