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:

Lots of CMYK dots making up a small piece of a guitar

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:

A blured picture of the same guitar, with visible grid patterns

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.