## Manipulating lists based on conditionals and 3D plotting the smooth surface that bounds a list of points

First of all I want to create a list of evenly spaceed points in a cubic region in $$mathbb{R}^3$$. Then I want to keep the points in this list which satisfy a certain condition, i.e. a function that takes these points in $$mathbb{R}^3$$ and outputs true or false depending on the details of my problem. Then I need to plot these points using some sort of a interpolation function, which will be able to show the surface that bounds the volume that the points in my list are located in.

The selection criterion is the following:

There is a matrix $$mathcal{M}=mathcal{M}(x,y,z)$$ defined for every point in this list. Say for the point $$P= (1,4,5)$$, the matrix $$mathcal {M}(1,4,5)$$ is Positive Semi-Definite, then I want to KEEP the point $$(1,4,5)$$. If at that point the matrix is nott Positive Semi-Definite, then I want to DISCARD that point.

My problem is

1. I could not create a list comprised of evenly spaced points in $$mathbb{R}^3$$.
2. I could not discard or keep the elements of a given set based on the output that an element of the list gives when inputed into a function.
3. I could not create a smooth surface which bounds a list of points in $$mathbb{R}^3$$.

Basically If at a point the matrix is positive semi-definite, then I want to keep that point. If at a point the matrix is not positive semi-definite, then I want to discard that point.

If needed I can also provide my function, which is the criteria for keeping or discarding certain points, but I am afraid it will only serve to clutter the discussion.

## ag.algebraic geometry – The smooth completion of a curve

Let $$C$$ be a smooth geometrically integral affine curve. This question concerns the smooth completion $$C_1$$ of $$C$$, both defined over a number field $$k$$.

We know that given any smooth projective geometrically integral curve $$X$$, and for a finite number of points $${p_1,…,p_r} subset X$$, the curve $$Xbackslash {p_1,…,p_r}$$ is affine. However, is $$C_1 backslash C$$ always a finite number of points?

Furthermore, we have the inclusion of adelic points $$C(mathbb{A}_k) subset C_1(mathbb{A}_k)$$. Is it possible that this inclusion is an equality? Or is it possible that this inclusion is in fact false to begin with? The latter question arised because I was worried about the archimedean places, even though I’m not certain if that will be a problem.

## reference request – Equivariant smooth approximation

Suppose we have a compact manifold $$M$$ with the action of a compact group $$G$$. Consider the space of $$C^l$$ $$G$$-equivariant diffeomorphisms $$text{Diff}_G^{l}(M)$$ with the $$C^l$$ topology and the space of $$C^infty$$ $$G$$-equivariant diffeomorphisms $$text{Diff}_G^{infty}(M)$$ with the $$C^infty$$ topology.

Then is the inclusion $$text{Diff}_G^{infty}(M) hookrightarrow text{Diff}_G^{l}(M)$$ a homotopy equivalence? If so what is a good reference where the proof is spelt out?

## dg.differential geometry – Local factoring of smooth maps

I apologize in advance if the question is vague or naive.

Let $$f:Mrightarrow N, g:Mrightarrow P$$ be smooth maps between boundaryless manifolds. Let $$x$$ be a point in $$M$$. I’m interested to know if there exists a smooth map $$h:Prightarrow N$$ such that $$f,hg$$ agree on a sufficiently small open set containing $$x$$.

A necessary condition for this to happen is that $$Ker(D^kg|_x)subseteq Ker(D^kf|_x)$$ for every positive integer $$k$$. Is this condition sufficient? I suppose bump functions could give us a counterexample, but perhaps there will not be a counterexample in the analytic case ? Are there known necessary and sufficient conditions in the smooth case in the literature ?

Notation: $$Df:TMrightarrow TN$$ is the derivative of $$f$$, $$D^2f:T(TM)rightarrow T(TN)$$ is the derivative of the derivative of $$f$$ and so on….

I also identify any manifold with the image of the zero section of it’s tangent bundle in a natural way.

## smooth manifolds – Does constant rank level set theorem imply regular level set theorem?

This might be a stupid question. The constant rank level set theorem says that if I have a smooth map $$f:Mto N$$ and a point $$pin M$$, then if on $$U_p$$ the rank of the differential is constant, then the preimage of $$p$$ is a submanifold. The regular level set theorem only demands that the differential with respect to $$p$$ is surjective. For the constant rank level set theorem to imply regular level set theorem, I need to know the surjectivity on the point $$p$$ implies the constant rank on some neighborhood of $$p$$. But I think this is not generally true. I’d appreciate any help!

## ag.algebraic geometry – On degree and section of a line bundle on a smooth plane quintic

Let $$X$$ be a smooth plane projective quintic curve (over $$mathbb C$$). Then we know that it has gonality $$4$$. Assume that it has genus $$g(X)=6$$. Then my question is the following:

Is it necessarily true that for every line bundle $$A$$ on $$X$$ with $$h^0(A) geq 2$$ one has $$text{deg}(A)geq h^0(A)+2$$?

Gonality $$4$$ means minimum degree of line bundles with atleast $$2$$ sections is $$4$$. On the other hand we have for any line bundle with atleast $$2$$ sections and with $$h^1(A) geq 2$$, $$text{deg}(A) geq 2h^0(A)-2$$. But then it’s not quite clear to me that how these two facts on gonality and clifford index (and may be Riemann-Roch) give us an affirmative answer to the question. May be I’m missing something obvious. Does there exist a more direct proof in the literature?

Any help from anyone is welcome

## dg.differential geometry – Existence of a non-vanishing smooth vector field on a given set of points

Given a set of pairwise distinct points $$A={p_1,p_2,⋯,p_n}$$ on some regular surface $$S$$. Does there exist some smooth vector field $$sigma: S rightarrow TS$$, such that $$sigma$$ is non-vanishing with respect to $$A$$ (i.e, $$sigma(p_{i}) neq 0$$ for all $$1 leq i leq n$$)?

## javascript – Smooth sidebar toggle animation with vuejs and tailwind

I’m making a slide sidebar with vuejs and tailwind. It works but feels kind of sluggish. Is there a way to make it smoother ?

working example: https://codepen.io/tuturu1014/pen/oNzRXeW

``````<button @click="isOpen = !isOpen" class="bg-blue-200 p-5">
<span v-if="isOpen">Open</span>
<span v-else>Close</span>
</button>
<div class="flex flex-row max-w-7xl mx-auto min-h-screen">
<transition name="slide">
<div class="flex flex-col w-64  shadow-xl sm:rounded-lg bg-blue-200" v-if="isOpen">
<div class="min-h-screen">sidebar</div>
</div>
</transition>

<div class="flex w-full  min-h-screen bg-red-400">
content
</div>
</div>
<style>
.slide-enter-active {
animation: slideIn 1s ease;
}
.slide-leave-active {
animation: slideIn 1s ease reverse;
}
@keyframes slideIn {
0%   {max-width: 0%;}
50%   {max-width: 50%;}
100% {max-width: 100%}
}
<style>
``````

## at.algebraic topology – Structures between PL and smooth

Let $$X$$ be a topological manifold of dimension at least five. The Kirby-Siebenmann invariant $$ks(X)in H^4(X,mathbb{Z}_2)$$ of is an obstruction to the existence of a PL structure on $$X$$. If it vanishes, the set of PL structures up to concordance is $$H^3(X,mathbb{Z}_2)$$. This can be phrased as the homotopy equivalence $$TOP/PL=K(mathbb{Z}_2,3)$$.

Now, let $$X$$ be a PL manifold. It is known that $$X$$ admits a smooth structure if its dimension is seven or less, and that this smooth structure is unique if the dimension is six or less. But $$PL/Oneq K(pi,7)$$ for some group $$pi$$—the homotopy groups are complicated, and depend on homotopy groups of higher dimensional spheres.

But perhaps there is some natural class $$C$$ of manifolds sitting between PL and smooth for which $$PL/C=K(pi,7)$$? Very simple ideas like taking piecewise quadratic functions don’t work because such functions aren’t invertible within the given class. I’m not really looking for a “formal solution” about the abstract existence of such a class $$C$$, but rather for some concretely defined geometric structure.

## photo editing – Is there a better way to “smooth out” CMYK halftones other than just blurring?

I’m trying to get the best possible scan of a CD cover (although this would apply to any print media using CMYK halftoning).

With the highest setting on my scanner (3200dpi) I quickly run into the DPI of the print itself.

Here’s a small portion for an example:

I can certainly blur the image, but there doesn’t seem to be a good compromise for the blur radius between still having some grid artifacts and loosing too much detail. This is my best attempt:

Is there a better way to do this? Surely there’s some filter that has more “knowledge” about halftones and can make a better result because of it.

The bluring above was done in GIMP, and while I’d love a GIMP solution, I’d appreciate any tool that can get the job done.