Let $Q_1,Q_2(z)$ be a pair of polynomials of the same order with complex coefficients. Let $S$ be a closed curve that passes through all roots of both polynomials. For a $t\in S$ define polynomial

$$ Q_t(z) = Q_1(z) Q_2(t) - Q_1(t) Q_2(z) $$

When there is such an $S$ that for any $t \in S$ all roots of $Q_t(z)$ lie on $S$?

I am new to the theory of currents and pursuing this note-"https://webusers.imj-prg.fr/~tien-cuong.dinh/Cours2005/Master/cours.pdf". My concept of product of positive currents is limited to that note and based on that I am stuck in computing the following two $(2,2)$ currents(distributions) on $\mathbb{C}^2$ (Exercise 6.3.5 and 6.3.6):

- $dd^{c}\log|z_{1}|\wedge dd^{c}\log|z_{2}|$ and
- $(dd^{c}log\|z\|)^{2}$.

My feeling is that both of them will be dirac-delta at the origin but unable to show it precisely.

What are the best references for finding explicit formulas for Belyi maps for rational dessin d'enfants?

I am most interested in a formula for the Belyi map that corresponds to a specific rational dessin d'enfant: it is clean, with four black vertices, each of valency 3, and with ramification indices above the infinity of 6,3,2,and 1. (I believe that there is only one such dessin d'enfant). Ideally, it would be great to have tables or software for other similar examples.

I am wondering which journal are considered good to publish at in the areas of Dynamical Systems (Hamiltonian Dynamics) and Complex Dynamics.

I have a tendency to ignore completely the journal in which the papers I am reading were published (and in most cases I just get the arxiv copies) and I cannot really make sense of all the different ranking tables.

EDIT: I was not completely clear. I meant to ask about 2 separate categories:

- Dynamical Systems
- Complex Dynamics

I have a good result in Hamiltonian systems that I need to publish and I think I can get a good (related but separate) result in Complex Dynamics.

did you know Daniel Barreto? I'm Luis Enrique Torrealba, his friend. If you are, i would like to talk .

Can $n$ circles on a plane generate $m$ intersection points where at least $k$ circles intersect?

For $k = 2$ the answer is obvious since we can always place circles so that every one of them intersects every other, generating in total at most $2 {n\choose{2}}$ intersection points of $2$ circles.

What can we say for $k = 3$? In particular I am interested in $n = 7$, $m = 12$.

It is known(see Figure attached) that $8$ circles can generate $12$ intersection points of at least $3$ circles.

The question is if we can generate $12$ intersection points of at least $3$ circles using $7$ circles in total.

Given an algebra B isomorphic to the trivial extension of a pathalgebra of dynkin type with n simples. What is the largest dimension of an indecomposable module depending on n? What is the largest entry in a dimension vector for an indecomposable representation depending on n? Not sure if this is written down somewhere. I need this to make a faster programm to test for finite representation type for certain algebras.

In *$D$-Modules, Perverse Sheaves and Representation Theory* from R. Hotta, K. Takeuchi and T. Tanisaki, I found the following statement (in section 8.2, the lines before Definition 8.2.2):

Setting: Let $X$ be an analytic space, $U\subset X$ Zariski-open, $j\colon U\hookrightarrow X$ the open embedding, $D^b_c(U)$ the (complexes of) sheaves with constructible cohomology, and $F\in D^b_c(U)$. Then there is stated basically the following:

If $\sqcup _{\alpha\in A}X_\alpha$ is a stratification of $X$ such that $F$ and $\mathrm{D}_U(F)$ (the dual of $F$) have locally constant cohomologies on each stratum $X_\alpha$, and one assumes that the given stratification satisfies the Whitney-conditions, then $Rj_\ast (F)\vert_{X_\alpha}$ and $j_!(F)\vert_{X_\alpha}$ have locally constant cohomologies again, so in particular $Rj_\ast(F),j_!(F)\in D^b_c(X)$.

Unfortunately, I don't see why this would be obviuos, so my question is:

Could anyone tell me a reference for this? As this is actually the first time I heard of the term "Whitney stratification", I would appreciate very much a reference that goes into some details.

Otherwise, if the statement should be trivial (sorry for my question in this case), any explanation on that would help me out a lot as well.

Thank you very much in advance!

I am trying to understand the coniveau spectral sequence for the cohomology of a "big" regular scheme over a field. This involves cohomology with support at points, and I am getting some strange results when trying to compute this.

To fix ideas, assume that $X$ is a regular local scheme over the spectrum of complex numbers (say, something like the spectrum of a ring of formal power seties), and the residue field of $X$ at the closed point $0$ is $\mathbb{C}$. I would like to compute the relative $\mathbb{Z}/l\mathbb{Z}$-etale cohomology for the pair $(X-\{0\},X)$.

Now, $X$ can be presented as the inverse limit of smooth $\mathbb{C}$-varieties $X_i$. Moreover, we can assume that $0$ lifts to $X_i$. Is it true that the relative ($\mathbb{Z}/l\mathbb{Z}$-etale) cohomology $(X-\{0\},X)$ is the limit of that for $(X_i-\{0\},X_i)$? Anyway, the latter pairs form an inductive system, and so one can pass to the limit in the corresponding long exact sequences (cf. Étale cohomology with support and functoriality). However, the relative cohomology of $(X_i-\{0\},X_i)$ is concentrated in the degree $2codim_{X_i}{0}$ according to the Gysin long exact sequence; hence the limit appears to be zero if the codimensions do not "stabilize" and to be $\mathbb{Z}/l\mathbb{Z}$ in the degree $2codim$ if they do. This conclusion seems to be very strange to me, so I would like to know whether my argument contains any errors and what is the correct answer.

Suppose $A\subset\{0,1\}^d$ for some $d\geq 1$. Then how large must $A+A=\{a+b:a,b\in A\}$ be?

Suppose $U$ is an open subset of $\mathbb{R}^n$, and $f:U\to \mathbb{R}$. When $f$ is $C^2$ we know that the mixed partial derivatives are symmetric, i.e. $\partial_i\partial_jf= \partial_j\partial_if.$ But as it is famous the continuity of the 2nd order partial derivatives is not necessary for this to happen. For example if $\partial_if$, $\partial_jf$ exist on $U$ and they are both differentiable (in the sense of Fréchet) at some point $a\in U$ then $$\partial_i\partial_jf(a)= \partial_j\partial_if(a).$$

Now for the 3rd order partial derivatives we can obtain the symmetry if we assume that the 1st order partial derivatives of $f$ are differentiable on $U$ and its 2nd order partial derivatives are differentiable at $a$. Let me explain the proof for the particular case $$\partial_3\partial_2\partial_1f(a)= \partial_2\partial_1 \partial_3f(a).\tag{*}$$ First as $\partial_1 f$ has 1st order partial derivatives in $U$ and they are differentiable at $a$ we have $$\partial_3\partial_2\partial_1f(a)= \partial_2\partial_3 \partial_1f(a).\tag{1}$$ Then since the 1st order partial derivatives of $f$ are differentiable in $U$ we have $\partial_3\partial_1f(x)= \partial_1\partial_3 f(x)$ for all $x\in U$. Hence we can differentiate to obtain $$\partial_2\partial_3\partial_1f(a)= \partial_2\partial_1 \partial_3f(a).\tag{2}$$ By combining (1) and (2) we get (*).

As you can see the full force of differentiability of the 1st order partial derivatives of $f$ on all of $U$ is only used for the equality of the 3rd order partial derivatives appeared in (2). So my question is

**Question:** Can we prove the symmetry of 3rd order mixed partial derivatives of $f$ at $a$ by merely assuming that the 1st and 2nd order partial derivatives of $f$ exist on $U$ and they are all differentiable at $a$? If not, can you provide a counterexample?

This comes purely out of curiosity and experiments. I'm not sure if the literature has any coverage.

Let $p(n)$ be the number of integer partitions of $n$. Then, we have the well-known generating function $$\sum_{n\geq0}p(n)x^n=\prod_{k=1}^{\infty}\frac1{1-x^k}.$$

**Question.** For each fixed $k\in\mathbb{N}$, is the following set finite?
$$\mathcal{A}_k:=\{(n,m)\in\mathbb{Z}_{\geq0}^2: p(n)+k=m^2\}.$$

It seems that there are different conventions in the literature as to what is a locally compact space (when the space is not supposed Hausdorff).

The two main non equivalent definitions I've seen are :

(LC1) every point has a compact neighborhood

(LC2) every neighborhood of any point contains a compact neighborhood of the point.

I am wondering if there is a reason as to why one might prefer one definition to another. More precisely, in practice, what definition is really useful (yields interesting results), or are they are both important in their own right ? In that case why not give a different name to those definitions ?

Naively, when one looks at the definition of locally connected space, one does not use the version (LCn1)"every point has a connected neighborhood", but always (LCn2) "every neighborhood of any point contains a connected neighborhood of the point". Is there a deep reason as to why LCn1 is never considered, but LC1 is ?

I am aware that for Hausdorff space, LC1 and LC2 are equivalent, and since LC1 is easier to check, one might prefer that as a definition, but this argument is unconvincing if LC2 is actually more useful for non-Hausdorff space.

What are known examples of two smooth, closed, oriented Manifolds $M,N$ of the same dimension that are simple homotopy equivalent, but not homeomorphic ?

It is well-known that the homotopy type of a given such $2$-manifold is the same as its homeomorphism type, so there won't be any such easy examples. Moreover (and this is where for me, the question becomes really interesting), I've heard at a recent conference that the *simple* homotopy type of a closed, oriented $3$-manifold is the same as its homeomorphism type, so any known examples must be in even higher dimension. Also, as the Borel conjecture is still open, any known such example must consist of non-aspherical manifolds.

A student asked me this, and I can't believe I never knew the answer to this.

Let $R$ be a commutative ring, and $M$ be an $R$-module.

If $M$ has a set of $n$ linearly independent vector for each $n\in\mathbb{N}$, does that necessarily imply that $M$ has an infinite set of linearly independent vectors?

More generally, if $\kappa$ is the minimum cardinal such that there is no set of cardinality $\kappa$ of linearly independent vectors. Must $\kappa$ be always a successor cardinal?

Thank you.

Does anyone know what is the fewest-piece dissection of the surface of a regular tetrahedron to the surface of a cube (of the same area)?

It is well-known that the volume of a regular tetrahedron cannot be dissected to the volume of a cube, because their Dehn invariants differ. But their surfaces have a dissection, by applying the Boyai-Gerwien theorem. Applying that theorem in one way (among several options) leads to an $31$-piece dissection (if I calculated correctly), illustrated below. But this is surely very far from optimal (and hardly aesthetically pleasing). Likely the question has been explored, but I have not found any literature. It could make a pleasing contrast to the impossibility of a volume dissection.

Let the cube edge length be $1$, so its surface area is $6$. A tetrahedron edge length of $L= 2^{\frac{1}{2}} 3^{\frac{1}{4}} \approx 1.81$ leads to a surface area of $6$.

Surface dissection of regular tetrahedron to cube.

Related: "Covering a Cube with a Square."

I would like to know a reference for Grothendieck duality in a resolution of singularities. More precisely, let $Y$ be a normal, Gorenstein variety with finite quotient singularities, and suppose that $f\colon X \to Y$ is a crepant resolution of singularities, meaning that $f^*\omega_Y \cong \omega_X$. In particular, since $Y$ is normal, we know that $Rf_*\mathcal{O}_X \cong \mathcal{O}_Y$.

By general results valid for any projective morphism, there is a functor $f^!\colon D^b(Y) \to D^b(X)$ between the derived categories of $X$ and $Y$ which gives an isomorphism

$$ Rf_*R\mathcal{Hom}(F, f^! G ) \cong R\mathcal{Hom}(Rf_*F,G) $$ for any $F \in D^b(X)$ and $G \in D^b(Y)$

My question is: is there nice description of $f^{!}$ in this situation?

I guess it should be related to $Lf^*$ and the relative canonical, but I don't know much about this topic.

Let $\mathfrak{g}$ be a semisimple Lie algebra (over $\mathbb{C}$), $\mathfrak{b}$ a Borel subalgebra of $\mathfrak{g}$, and $L$ a semisimple element contained in $\mathfrak{b}$.

I know that $L$ is contained in a Cartan subalgebra (CSA) of $\mathfrak{g}$. Is it true that $L$ is contained in a CSA of $\mathfrak{b}$? If so, why?

$Ax=0$, $A$ has $m$ rows and $n$ columns, $m \le n$, all entries of $x$ are non-negative.

What should $A$ satisfy to guarantee the equation set have only zero solution?

Given a field $\mathbb{F}$ and a consistent underdetermined system $Ax=b$ over $\mathbb{F},$ $A\in \mathbb{F}^{m \times N}$ and $b \in \mathbb{F}^m,$ finding a vector $z \in \mathbb{F}^N$ such that $Az=b$ and $\|z\|_0 \leq \|x\|_0$ for all $x \in \mathbb{F}^N$ such that $Ax=b$ is NP-hard.

I have been working on this question for a while, and I think I have reached a solution via modifying a proof from Foucart's Compressive Sensing text.

This question is cross-posted in StackExchange-Theoretical Computer Science as well, so if that is not allowed, let me know and I will take one of the posts down.

To the proof: we reduce from 3-SET-COVER, which asks whether there exists a (nonoverlapping) partition $C_j, j \in J$ within a given collection $\mathbf{C}$ of three-element subsets of some set $S,$ where $S$ has cardinality $|S|=m=3k$ for some natural number $k$ (hence without loss of generality we can refer to $S$ as the set $[m]=\{1, \ldots, m\}$ ).Papadimitriou and Steiglitz have shown that 3-SET-COVER is NP-Complete by reduction from 3-D MATCHING.

Here, $\|x\|_0$ simply means the number of nonzero components in the vector $x.$

For a given set $S$ and collection $\mathbf{C_i}, i \in [N]$ of three-element subsets of $S$ as described above, construct a matrix $A \in \mathbb{F}^{m \times N}$ with columns $a_i$ such that $a_{i,j}=1$ if j $\in C_i$ and $a_{i,j}=0$ if $j \not \in C_i.$ Additionally, let $b=\mathbf{1},$ the vector in $\mathbb{F}^m$ whose components are all equal to $1 \in \mathbb{F}.$ It is obvious this system can be constructed in polynomial time on $m$ and $N.$

Let $z$ be such that $Az=b$ and $\|z\|_0 \leq \|x\|_0$ for all $x$ such that $Ax=b.$ If $\|z\|_0 < \frac{m}{3}$ then $\|Az\|<3\cdot\frac{m}{3}=m=\|b\|_0,$ a contradiction since $\|b\|_0=m.$ Hence $\|z\|_0 \geq \frac{m}{3}.$

Run an algorithm solving for a minimal support solution of $Ax=b$ with output $z.$ There are two cases remaining: first, if $\|z\|_0 = \frac{m}{3},$ then $C_j, j \in supp(z) $ covers $[m]$ exactly. Second, if $\|z\|_0 > \frac{m}{3}$ then the collection $\mathbf{C}$ contains no subcollection covering $[m]$ exactly; if such a subcollection $C_j \subset \mathbb{C}$ existed, the $\frac{m}{3}$ corresponding columns $a_j$ add to $\mathbf{1},$ contradicting the initial assumption for the second part.

This means any algorithm solving the $\ell_0$ minimization problem solves any 3-EXACT COVER problem through a polynomial transformation, and the proof is complete.

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Is this proof valid? My main worry is that if $\mathbb{F}$ is a finite field, then the cyclic nature of addition could produce a cover, but the cover could not be exact, since this minimal support solution involves columns of $A$ corresponding to overlapping subsets within $\mathbf{C}$ in this case.

Thanks for any help ahead of time!