Given a compact Hausdorff space $X$, let me denote by the same symbol the corresponding condensed set. Similarly, given an open subset $U\subset X$, let me denote by the same letter $U$ the corresponding condensed set.

Given a condensed set $S$:
if $S$ is quasi-separated, then for every compact Hausdorff space $X$, every open dense subset $U\subset X$, and every map of condensed sets $f:U\to S$ there exists at most one extension $\tilde{f}:X\to S$ of $f$.

Question:
If a condensed set $S$ has the property that for every compact Hausdorff space $X$, every open dense subset $U\subset X$, and every map of condensed sets $f:U\to S$ there exists at most one extension $\tilde{f}:X\to S$, does it then follow that $S$ is quasi-separated?

Let $S^1 := \{z \in \mathbb{C} \;|\; \left\lvert z\right\rvert = 1\}$ — the standard unit circle which is also a topological group. Surely, there are continuous homomorphisms $z \mapsto z^n$ for each $n \in \mathbb{Z}$. It seems that there are no other. How can I prove that those are the only continuous homomorphisms $S^1 \to S^1$? And is similiar statement for arbitrary homomorphisms correct? If so, how to prove it also?

I tried to think on this problem for a day maybe, but I have only very vague ideas. Maybe one can approximate with finite cyclic groups and take a limit. Maybe it is fruitful to use topological covering $\mathbb{R} \to S^1$ to uplift homomorphism to a map $\mathbb{R} \to \mathbb{R}$ with property $\varphi(a + b) - \varphi(a) - \varphi(b) \in \mathbb{Z}\cdot 2\pi$. But even if those worked, case of arbitrary homomorphisms still completely evades me.

I'm interested in the affine Weyl group $W$ of type $\tilde A_{n-1}$, but the question makes sense in any type. The structure of this group is that it sits in a split short exact sequence $1\to \mathbb{Z}^{n-1}\to W \to \Sigma_n\to 1$, where $\Sigma_n$ is the symmetric group. The presentation is $$W=\langle w_0,\dots,w_{n-1}\mid w_i^2=1,\ w_iw_{i+1}w_i=w_{i+1}w_iw_{i+1},\ w_iw_j=w_jw_i\ (|i-j|\geqslant 2)\rangle,$$ where the subscripts are read modulo $n$.

The Question is this: Given $x$, $y\in W$, does there exist $z \in W$ such that the length of $xzy$ is the sum of the lengths of $x$, $y$, $z$? Here, the length function is in terms of the $n$ generators $w_0,\dots,w_{n-1}$.

If it helps, I can reduce to the case where $x$ and $y$ are in the normal subgroup $\mathbb{Z}^{n-1}$, but after that, I seem to be stuck.

$\DeclareMathOperator\Mfld{Mfld}\DeclareMathOperator\Disk{Disk}\DeclareMathOperator\Emb{Emb}\DeclareMathOperator\Map{Map}$I am reading "Factorization homology on topological manifolds" by John Francis and David Ayala, where in definition 2.7 and 2.9, they define the symmetric monoidal $\infty$-category $\Mfld_n^B$ and $\Disk_n^B$:

And

First, for the 'symmetric monoidal $\infty$-category $\Mfld_n^B$", let's use $\Mfld_n^B$ to denote its underlying $\infty$-category and $\Mfld_n^{B,\otimes}$ to denote its corresponding $\infty$-operad, which structure if we forget the $B$-framed structure, is just taking disjoint union of the underlying manifolds.

Now comes to the symmetric monoidal $\infty$-category $Disk_n^B$, I am a bit confused on how do we take the 'full subcategory":

One way is we take the full subcategory of the underlying category $\Mfld_n^B$, then we get a category $Disk_n^B$, which objects are in the form of (finite) disjoint unions of disks, thus is the same as the objects in $Fin$, while for the mapping space, if we take $B=*$, then $\Map_{\Disk_n^B}(\coprod_m R^n, R^n)=\Emb^{fr}(\coprod_m R^n, R^n)$ is the space of framing embeddings from $m$ $n$-disks to one $n$-disks. This is the multimorphism space between $\langle n\rangle $ and $\langle 1\rangle $ in the $\infty$-operad $\mathbb{E}_n^{\otimes}$ instead of the whole mapping space. Although it's close, but still the $\infty$-category $\Disk_n^{B=*}$ is not the same as the $\infty$-operad $\mathbb{E}_n^{\otimes}$, thus not what we want.

Another way is we take the full subcategory of the $\infty$-operad $\Mfld_n^{B,\otimes}$ spanned by the disjoint unions of $B$-framed n-dimensional $n$-disks, which is not very reasonable, since for $B=*$, we hope to recover the $\infty$_operad $\mathbb{E}_n^{\otimes}$, while the underlying category should only contains one object instead of all the disjoint unions.

So I guess the definition actually refers to " the full suboperad of $\Mfld_{n}^{B,\otimes}$ consisting of $B$-framed disks (instead of taking all its disjoint unions), and probably this is what the authors actually mean?

In Kazuo Okamoto and Yousuke Ohyama's paper "Mathematical works of Hiroshi Umemura", Annales de la faculté des sciences de Toulouse Mathématiques, XXIX, no. 5 (2020) pp. 1053-1062, there is a reference to a meeting at the IHES in 2010 between Umemura, Pommaret, and Malgrange. This is described as follows:

Umemura, Pommaret and Malgrange met at IHES in the fall 2010. Three researchers have their own differential Galois theory. After a stormy discussion, Umemura and Pommaret did not achieve consensus on differential Galois theory. This is the last time that Umemura met Pommaret.

What were the issues / differences in approach that led to this "stormy discussion"?

Edit: the paper by Jean-Pierre Ramis, "Hiroshi Umemura et les mathématiques françaises", Ibid p. 1007-1052 provides an account of the development of Umemura's ideas on differential Galois theory, and their relationship with Malgrange's DGT.

Let me give you some context first: just a few days ago I found some intriguing references to Ricci flows in the setting of directed graphs.

There are at least two versions of Ricci curvature in the discrete realm (one being the Ollivier-Ricci curvature, the other the Forman-Ricci, see here for reference (*)), and as it turns out, they are both useful in graph analytics.

To be a tad more specific, one application leads to a new method for determining communities (the so-called Ricci communities, for the interested ones there is even a github Python implementation which can easily be used for hands-on explorations ), whereas another quite useful one is used to get rid of "bottlenecks" in graph messaging ( thereby solving some critical issue in Graph deep learning see picture below).

https://towardsdatascience.com/over-squashing-bottlenecks-and-graph-ricci-curvature-c238b7169e16

Now, if I understand them correctly, the associated Ricci flow, just like in the differentiable realm, acts as a kind of "curvature heat-like operator", a diffusion which tends to smoothen out the curvature across the underlying geometrical object.

Perhaps naively, it occurred to me this:

why confining ourselves to diffusion? (note: I am aware of the centrality of the Ricci flow in the proof of the Poincare conjecture)

Could one replace the Ricci flow with some kind of PDE (or a difference equation in the finite setting) for the curvature change modeled on completely different PDEs?

For instance, what about a kind of wave equation?

Now the questions (and I apologize if this is too naive, I am coming from the data science world, my knowledge of Riemannian geometry does not go beyond standard grad courses):

1. Have such curvature flow involving non-heat-like PDEs been investigated in the world of Riemannian geometry? I would think the answer is in the affirmative, but I just do not happen to know it.
2. Are there any references for generalized curvature flows in discrete metric spaces and particularly in weighted directed graphs?

Any help is most welcome.

(*) actually in the referenced article there are three discrete Ricci curvatures, but I haven't wrapped my mind around the third one yet.

Let $T$ be an operator on simple functions on (say) $\mathbb{R}$.

The Riesz-Thorin interpolation theorem, in one form, says that the Riesz type diagram of $T$ is a convex subset of $[0,1]\times[0,1]$.

My question is about the converse: Is every convex subset of $[0,1]\times[0,1]$ the Riesz diagram of such an operator? Does the answer change if we are allowed to replace $\mathbb{R}$ with a general measure space $X$?

It is not clear to me how to find such a $T$ even for very simple regions, such as convex polygons or circles.

In some recent reading, I was reminded of the following (trimmed) quote from Terry Speed (from Cumulants and partition lattices, Australian Journal of Statistics 25(2) (1983), 378–388.)

In a sense which it is hard to make precise, all of the important aspects of distributions seem to be simpler functions of cumulants than of anything else [ ... ]

Now, without seeking to criticise this claim or otherwise, it did give rise to a few thoughts. I'm often interested in concentration and deviation inequalities for random variables, and my experience has been that cumulants per se are only used quite sparingly for these purposes.

Chebyshev's inequality certainly counts, but I don't know so many other examples. Of course, a preëminent approach to exponential inequalities is based on the cumulant-generating function, but I would say that the cumulants themselves are not necessarily the star of the show in this case (indeed, cumulants of all orders need to exist for this method to be viable, which is certainly a restriction).

Now, supposing that I had access to the first $p$ cumulants of a random variable, and I wanted to use them to estimate tail probabilities, I confess that I would probably proceed by converting them into the first $p$ moments of that variable, and then apply some basic version of Markov's inequality. While this would technically 'only involve using cumulants', this seems to not really reflect the idea that cumulants are central in their own right (i.e., I don't imagine that Speed's quote is motivated by the idea that cumulants are in bijection with moments).

With this preamble out of the way, my question is then:

Are there good examples of { concentration / deviation } inequalities which are most naturally stated in terms of cumulants?

I would already be interested to see other useful results which are based on a fixed number of cumulants (cf. Chebyshev, which uses only the first and second), but I would be even more interested to see families of results which apply to arbitrary sequences of cumulants (cf. how Markov's inequality is useful given any absolute moment of a random variable).

Let $B$ be a cyclic upper-triangular nonnegative matrix,

$$B = \begin{bmatrix} 0 & b_1 & 0& \dots &\dots &0 \\ 0 & 0 & b_2 & 0& \dots & 0\\ \vdots &\vdots&\vdots &\ddots &\ddots &\vdots \\ 0& 0 & 0 & 0 &\dots & b_{n-1}\\ b_n& 0 & 0 & 0 &\dots & 0 \end{bmatrix}$$

with $b_i \in [0,1]$. For any $\alpha > 0$, let us define the matrix

$$C(\alpha)=\alpha B +\alpha^{-1}B^T,$$

where $\alpha$ controls the degree of asymmetry, $\alpha=1$ giving a symmetric $C(\alpha)$ .

I am interested in controlling the localization of the normalized eigenvalues of $C(\alpha)$ on the complex plane. Formally, given any $C(\alpha)$ with eigenvalues $\{\lambda_1,\dots,\lambda_n\}$, let us define the normalized eigenvalues as $$\bar \lambda_i=\frac{\lambda_i}{\max_i |\lambda_i |}$$ By definition, the normalized eigenvalue fall within a circle of radius $1$ on the complex plane. Moreover, when $\alpha=1$ and $C(\alpha)$ is symmetric, the spectrum is real so the normalized eigenvalues fall on the real axis in the interval $[-1,1]$. For larger but finite $\alpha$, I have found numerically that the normalized eigenvalues fall within an ellipse of height $\tanh(\ln \alpha)$. This ellipse is achieved when all $b_i$ are equal so that $C(\alpha)$ is a circulant matrix (as pointed out in the partial answer by @Charr).

Any ideas for how to prove this for a general $B$?

Here's a plot of normalized eigenvalues from many random matrices (random $b_i$), $\alpha=1.2$, with the ellipse shown:

Edit: Let $v$ be the eigenvector associated with the largest (in magnitude) eigenvalue $\lambda^*$ of $C(\alpha)$. By Perron-Frobenius, we may assume $\lambda^*$ and $v$ are real-valued and nonnegative, so \begin{align}\lambda^* = v^T C(\alpha)v = (\alpha + 1/\alpha) v^T B v \end{align} Then, consider any other eigenvalue $\lambda_i$ with right eigenvector $u$ ($\Vert u \Vert =1$). It may be helpful to express $\bar{\lambda_i}$ as $$\begin{align} \bar{\lambda}_i = \frac{1}{\lambda^*}{u^* C(\alpha) u} &= \frac{1}{\lambda^*}\left[(\alpha + 1/\alpha) \,u^*\frac{B+B^T}{2}u + (\alpha -1/\alpha) u^*\frac{B-B^T}{2}u\right]\\ &= \mathrm{Re} x + \mathrm{i} \tanh(\ln \alpha)\mathrm{Im}\, x \end{align}$$ where for convenience I defined $x= u^* B u/v^T B v$. Thus, the problem reduces to showing that \begin{align}\vert x\vert =\left\vert \frac{u^* B u}{v^T B v}\right\vert \le 1 \end{align} where $u,v$ are two eigenvectors of $C(\alpha)$, with $v$ being the largest (Perron-Frobenius) one.

Suppose that $X$ is a centered, $\mathbf{R}^d$-valued random variable such that for all $t \in \mathbf{R}^d$, there holds the bound $$\log \mathbf{E} \left[ \exp \langle t, X \rangle \right] \leqslant \Psi \left( \| t \|_2 \right)$$ for some well-behaved function $\Psi$.

My question: What are the { simplest, sharpest } { MGF bounds, concentration inequalities, etc. } which are available for $\| X \|_2$, ideally phrased in terms of $\Psi$?

I acknowledge that some conditions on $\Psi$ will inevitably be necessary (probably e.g. control on growth, smoothness, maybe monotonicity / convexity, etc.), but I emphasise that I would still like relatively general results, e.g. having a result for only quadratic $\Psi$ would not be fully satisfactory. A result for only polynomial-type $\Psi$ (i.e. $\Psi: t \mapsto t^\alpha$ for $\alpha$ in some nontrivial interval) would be somewhat narrow, but still valuable to me, since my experience is that such results are likely to generalise well.

I also acknowledge that the "{ simplest, sharpest }" in the framing of the question perhaps induces some tension / subjectivity; I included these qualifiers just to give a sense of what sort of results I seek, rather than to penalise people for giving non-sharp bounds, or similar.

Let $\Omega$ be an open set, and $f: \Omega \to \mathbb R$ be a Lipschitz map. We define the local Lipschitz constant $Lf$ of $f$ at $x \in \Omega$ by

$$Lf(x) := \lim_{r \to 0_+} \text{Lip}(f, B_r (x)),$$

where $\text{Lip}(f, U) := \sup_{y,z \in U} \frac{|f(y) - f(z)|}{|y - z|}$ denotes the Lipschitz constant of $f$ on the set $U$.

Let $u: \bar \Omega \to \mathbb R$ be infinity harmonic on $\Omega$. If $g: \partial \Omega \to \mathbb R$ is differentiable everywhere, $g_{|\partial \Omega} = u_{|\partial \Omega}$ and $Lg = Lu$ everywhere, do we have $g = u$?

I am working on a bilinear inverse problem arising in multi-channel signal processing. My problem background is to reconstruct a certain one-dimensional information $\mathbf{w} $ of an object from observation signals of multiple channels. In practice, the gain of each channel $g_i$ may deviate from the theoretical value, and the impact of the deviation on the reconstructed $\mathbf{w}$ needs to be considered when solving.

The model is $ \mathbf{y}\approx(\mathbf{g}\otimes\mathbf{1}_L) \odot \mathbf{A} \mathbf{w} $, leading to the optimization problem: \begin{equation} \min \limits_{\mathbf{w},\mathbf{g}}\frac{1}{2}\|(\mathbf{g}\otimes\mathbf{1}_L) \odot \mathbf{A} \mathbf{w}-\mathbf{y}\|^2 +\lambda \|\mathbf{w}\|_1 \end{equation} where:

$\mathbf{y}$ is the complex observation vector with length $KL$.

$\mathbf{g}\in \mathcal{R}^K$, is a real, non-negative gain vector. The operator ($\mathbf{g}\otimes\mathbf{1}_L)$ creates a vector of length $KL$ by repeating each element of $\mathbf{g} $ $L$ times. This models a block-constant gain across $K$channels, each with $L$ measurements.

$\mathbf{w}$ is the complex vector with length $N$.

$\mathbf{A} \in \mathcal{C}^{KL\times N}$, and $\mathbf{A}_{i,j}=\exp(j \varphi_{i,j})$.

$\lambda $: regularization parameter.

$\|\cdot\|$: $l_2$ norm, $\|\cdot\|_1$: $l_1$ norm.

In my scenario, $KL$ is usually around 500, and $N$ is usually around 2500.

My purpose: Under sparse constraints on $\mathbf{w}$, I want to solve for both $\mathbf{g}$ and $\mathbf{w}$ simultaneously so that $\mathbf{A} \mathbf{w}\approx \mathbf{A} \mathbf{\hat{w}}$. The core challenge is the bilinear coupling between $\mathbf{g}$ and $\mathbf{w}$, which makes $F(\mathbf{g}, \mathbf{w})$ non-convex.

My previous attempts: assuming the first element in $\mathbf{g}$ is known. The obtained solution algorithm is as follows:

This solving method has randomness and can sometimes be solved, but there are often significant errors.

Specific Questions for the Community:

Beyond fixing $g_1=\text{Constant}$, what other constraints could be added to $\mathbf{g}$ to better regularize the problem and improve convergence without overly restricting the solution?

Given the block-constant gain structure, what algorithms are better suited than BCD?

If I want to find the exact value of $\mathbf{g}$ and make $\mathbf{A} \mathbf{w}\approx \mathbf{A} \mathbf{\hat{w}}$, what constraints should I add to $\mathbf{g}$ ?

Any criticism, guidance, reference, or algorithmic suggestions would be greatly appreciated. Thank you very much!

I'm wondering if anyone has shown Fatou's lemma for sums when the limits are taken over nets.

That is, has anyone proved the following?

Let $\{x_{k,\alpha}\}_{k\in \mathbb N,\alpha \in \mathcal A}\subseteq \mathbb R_+$ with $\mathcal A$ a directed set. Then

$$\sum_{k=1}^{\infty} \sup_{\bar \alpha}\inf_{\alpha\geq \bar \alpha} x_{k,\alpha}\leq \sup_{\bar \alpha}\inf_{\alpha\geq \bar \alpha} \sum_{k=1}^{\infty} x_{k,\alpha}.$$

This result is not true if the sum were replaced by a general measure.

Consider a Hamilton–Jacobi equation: $$u_{t} + f(u_{x}) = 0 \quad (x,t) \in \mathbb{R}\times [0,+\infty)$$ with two possible initial conditions $u(x,0) = g_{i}(x)$ for $x \in \mathbb{R}$ and $i=1,2$. The question is pretty much straightforward: if $g_{1} \le g_{2}$, do we have $u_{1} \le u_{2}$?

When $f$ is convex, I know the answer is yes and the proof is an easy consequence of the Hopf–Lax formula. However, I would like to know conditions for this to hold when $f$ is not convex. I am particularly interested in the case $f(x) = -\frac{1}{2}x^{2}$.

I did a little research before coming here, but all references I know give special attention to Hamilton–Jacobi equations in which $f$ is convex and satisfy $\lim_{p\to\infty}f(p)/p \to +\infty$. The only source I found which differs from this setting is Imbert - Convex Analysis techniques for Hopf-Lax formulae in Hamilton-Jacobi equations, which states that the Hopf–Lax formula holds when either $f$ or the initial condition is convex, a claim I found nowhere else. In this case, if both $g_{1}$ and $g_{2}$ are convex and satisfy $g_{1} \le g_{2}$, it seems that the previous argument using the Hopf–Lax formula applies here and one would have $u_{1} \le u_{2}$ indeed.

Let $B(H)$ be the space of all bounded linear operatores on a Hilbert space $H$. For two operators $T,S \in B(H)$, we say that $T\sim S$ if $uu^*\sim_h vv^*$ where $T=u|T|, S=v|S|$ be the polar decompositions of operatores $T,S$. Here $\sim_h$ means the usual homotopy equivalent on the space of projections.

For two elements $a,b$ in an arbitrary $C^*$ algebra $A$ we say that $a\sim b$ if $\pi(a)\sim \pi(b)$ for every faithful representation $\pi:A \to B(H)$.

So we may apply the standard methods of K theory and form the inductive limit of $M_n(A)$.

Does this process leads us to triviality?

What about if we replace "faithfull" with "irreducible" representation $\pi:A \to B(H)$?

BTW, what can be said about the equivalent classes of $B(H)$ when we restrtrict ourselve to non invertible operatores?

The Levy area of a $C^1$ curve $f:[0,\infty)\to \mathbb R^2$ is defined to be $$L_f(t):=\int_0^t (f_1(s)f_2'(s)-f_2(s)f_1'(s))ds. $$ It is called Levy area because by Green's theorem, it is twice the signed area enclosed by the curve with respect to the chord connecting the beginning and end points. I am interested in maximizing the asymptotic Levy area over curves with a given arc length. Set $$S:=\left\{g\in C^1([0,\infty),\mathbb R^2)\colon \int_0^t |f'(s)|ds=t\ \,\forall t\ge0\right\}.$$ What are the curves $f\in S$ so that for any other $g\in S$ we have $$\limsup_{t\to\infty}\frac{L_g(t)}{L_f(t)}\leq 1?$$

Note that if we are interested in simple curves at a fixed time, by the isoperimetric inequality we should get half circles. Similarly, if we are interested in simple closed curves at a fixed time, we should get circles.

It seems that (op)lax phenomena are receiving increasing attention recently, for example in papers on $(\infty,2)$-topoi (see arXiv:2410.02014), categorical spectra and stable Gray categories (see e.g. https://nmasuda2.github.io/notes/thesis_stable.pdf, arXiv:2505.22640), and in derived algebraic geometry (see e.g. https://people.mpim-bonn.mpg.de/Book/).

The reason for this interest is straightforward: lax phenomena are governed by the (op)lax Gray tensor product, whose right adjoint yields the important object $\text{Fun}^{\text{(op)lax}}(-,-)$. The main significance of this construction is not merely that its objects are strong functors with (op)lax natural transformations as 1-morphisms (and (op)lax k-transformations as k-morphisms), but rather that its associated arrow category construction detects higher morphisms. For instance, the projection to source and target $\text{Fun}^{\text{oplax}}([1], C) \to C\times C$ provides a higher-dimensional analogue of a two-sided fibration that precisely encodes the Hom-functor of $C$. The functor $(\infty,\infty)$-category $\text{Fun}(-,-)$ (obtained by means of cartesian enrichment) is insufficient for this construction since the associated arrow category cannot see higher morphisms — the strong arrow category "bifibration" only encodes the underlying hom-space functor rather than the hom-$(\infty,\infty)$-category functor. Thus $\text{Fun}^{\text{(op)lax}}$ provides the fundamental fibrations needed to establish category theory (e.g., the Yoneda Lemma).

My point is that even if lax phenomena are not your primary focus, developing $(\infty,n)$-category theory requires this perspective because non-trivial $(\infty,n)$-category theory demands a notion of fibrations and an associated higher-dimensional Straightening-Unstraightening equivalence (i.e., a Grothendieck construction).

Given this context, it strikes me as odd that I haven't encountered any papers of enriched lax phenomena in the following sense:

Suppose we have a monoidal $(\infty,1)$-category $\mathscr{V}$ and consider $\mathscr{V}$-enriched $(\infty,1)$-categories. Assume further that $\mathscr{V}$ admits a $2$-categorical enhancement - note that I'm not requiring $\mathscr{V}$ to be a monoidal $(\infty,2)$-category, but simply that there exists an $(\infty,2)$-category $\tilde{\mathscr{V}}$ such that $(\tilde{\mathscr{V}})^{\leq 1} = \mathscr{V}$, where $(-)^{\leq 1}$ denotes the right adjoint to the inclusion of $(\infty,1)$-categories into $(\infty,2)$-categories (though additional compatibility properties may be required).

Under these assumptions, it should be possible to define lax $\mathscr{V}$-enriched natural transformations. For example, if $F,G\colon \mathscr{C} \to \mathscr{D}$ are $\mathscr{V}$-enriched functors, then a $\mathscr{V}$-enriched lax natural transformation would consist of a family of morphisms $\alpha_c \colon 1_\mathscr{V} \to \mathscr{D}(Fc,Gc)$ together with a non-invertible $2$-cell

(+ higher coherences)...

Question: Has this been studied before (maybe in the classical literature)? In particular, lax enriched natural transformations (whenever they are definable, i.e. for $\mathscr{V}$ nice enough) should be governed by a biclosed $\mathscr{V}$-Gray tensor product - have people studied something of that kind before?

This thread follows Does there exist a simple closed curve in ????^3 whose projections down onto the three coordinate planes are simply connected with the aim to check if the clue below is original.

Let us consider the following “Rickard curve” in $[0,2]^3$ (closed walk; start=end): $\Omega_{0} := (0,0,0)\to(0,1,0)\to(2,1,0)\to(2,0,0)\to(1,0,0)\to(1,0,2)\to(2,0,2)\to(2,0,1)\to(2,2,1)\to(2,2,2)\to(2,1,2)\to(0,1,2)\to(0,2,2)\to(1,2,2)\to(1,2,0)\to(0,2,0)\to(0,2,1)\to(0,0,1)\to(0,0,0).$

The idea is to give a very simple proof that there exist infinitely many (non-isomorphic) simple closed curves in $\mathbb{R}^3$ whose projections onto the three coordinate planes are trees.

For this purpose, apply a tiny local modification on the two $z$–parallel edges $E_1:=(2,0,2)\to(2,0,1)$ and $E_2:=(2,2,1)\to(2,2,2)$, with midpoints $m_1=\big(2,0,\frac{3}{2}\big)$ and $m_2=\big(2,2,\frac{3}{2}\big)$.

Fix $n\ge 3$. Place two (small) regular $n$-gons $P^{(1)}_n$ and $P^{(2)}_n$ of side-length equal to $\frac{1}{n}$ in the plane $x=2$, centered at $m_1$ and $m_2$, respectively, and choose one side $s^{(1)}$ collinear with $E_1$ and one side $s^{(2)}$ collinear with $E_2$, each of them centered at the corresponding midpoint (i.e., $m_1$ and $m_2$). Now, delete $s^{(1)}$ and $s^{(2)}$, and replace the corresponding length-$\frac{1}{n}$ subsegments of $E_1$ and $E_2$ by the open chains $P^{(1)}_n - {s^{(1)}}$ and $P^{(2)}_n - {s^{(2)}}$ (since with side $a=\frac{1}{n}$ the circumradius is $R=\frac{a}{2 \cdot \sin\big(\frac{\pi}{n} \big)}=\frac{1}{2 \cdot n \cdot \sin\big(\frac{\pi}{n} \big)}\le\frac{1}{4}<\frac{1}{2}$, all vertices satisfy $\lvert z-z(m_1)\rvert\le R$ (and also $\lvert z-z(m_2)\rvert\le R$ by symmetry), so the detour’s $z$–projection stays within the hosting unit edge). Lastly, denote the modified curve by $\Omega_n$.

Therefore, each $\Omega_n$ has projections on the $XY$-plane, $XZ$-plane, and $XZ$-plane, simultaneously satisfying the required properties. Since each replacement turns one edge into $(n - 1)$ edges from the $n$-gon plus $2$ short straight pieces (total $n+1$), the net gain is $n$ edges per modified edge; starting from $\Omega_0$ (which consists of $18$ edges), we obtain $\lvert\text{edges of }\Omega_n\rvert=18+2 \cdot n$. Varying $n \geq 3$ yields $\aleph_{0}$ pairwise non-congruent examples ($\Omega_3,\Omega_4,\Omega_5,\ldots$).

Figures: I reused the images from Nikita Gladkov’s answer 2 and overlaid the $n$-gons to illustrate $\Omega_{3}$.

This question is somewhat related to my previous question and is also inspired from this other question concerning the credibility of extensive computations (although from a different perspective).

In the course of preparing a paper, I verified a conjecture computationally for a large range of cases.

For small cases, I was able to run the computations on my laptop, since they required little RAM. However, at some point this was no longer feasible. At my university we have access to a high-performance computing (HPC) cluster, which allowed me to perform the larger calculations.

This raises a concern: when the paper is submitted, the referee may wish to check the correctness of these computations. But what if they do not have access to an HPC cluster? In that case, they can only verify a portion of the claims.

Is it sufficient to provide the raw output together with the code on GitHub/Zenodo, or is this not considered credible? After all, the raw output will consist of a .txt file that could, in principle, be tampered with (although doing so would clearly be dishonest, against all academic standards, and would eventually be detected by the community). My concern is precisely how referees can reasonably test such results in order to avoid this potential issue.

One way to write the Courant-Fischer theorem is the following: given symmetric $A$, $$\sigma_k(A)=\sup_{U\in\mathbb C^{n\times k},U^*U=I}\sigma_k(U^*AU)$$ where $\sigma_k$ is the $k$th largest singular value. Is there a version of this statement that holds for any matrix? I.e. $$\sigma_k(A)\le c_{n,k}\sup_{U\in \mathbb C^{n\times k},U^*U=I}\sigma_k(U^*AU)$$ for some constants $c_{n,k}$?

When $k=1$, one can show $c_{n,1}=2$ both suffices and is necessary (by the polarization identity). But I'm looking for references for higher $k$.