A family of residue classes $a_i (\mod n_i)$ with $2\leq n_1\leq\cdots\leq n_r$ is called a covering system of congruences if every integer belongs to at least one of the residue classes, that is, every integer satisfies at least one of the congruences $a_i (\mod n_i)$. The known examples are:

$0 (\mod 2),\ 0 (\mod 3),\ 1 (\mod 4),\ 5 (\mod 6),\ 7 (\mod 12)$

$0 (\mod 2),\ 0 (\mod 3),\ 1 (\mod 4),\ 3 (\mod 8),\ 7 (\mod 12),\ 23 (\mod 24)$

The proof of that the above families are each a covering system of integers are not difficult.

My question is the other side, i.e., that how can we construct a covering for integers from the given numbers for example $2,3$ with their multipliers as moduli?

**The question in a nutshell:**

Let $f:\mathbb{R}^d \to \mathbb{R}^d$ be a differentiable map. Under some natural conditions, the minors of degree $k$ of $df$ uniquely determine $df$, in a smooth way. Now suppose these minors have some "good" regularity/integrability property. Does $df$ have this property as well?

*The details:*

Let $\Omega \subseteq \mathbb{R}^d$ be an open bounded domain. Fix some integer $1<k<d$ and some $k<p<d$. Suppose that $k,d$ are not both even.

Let $f \in W^{1,p}(\Omega;\mathbb{R}^d)$ be a continuous map with $\det df > 0$ a.e. Denote by $f^i$ the $i$-th component of $f$.

Suppose that all the $k$-minors $$df^I:=df^{i_1} \wedge df^{i_2} \wedge \dots df^{i_k} \in W^{1,p}_{loc}(\Omega,\bigwedge^k (\mathbb{R}^d)^*),$$

for every increasing multi-index $I=(i_1,\dots,i_k)$.

Is it true that $f \in W^{2,q}(\Omega;\mathbb{R}^d)$ for some $q>1$?

A-priori the fact $f \in W^{1,p}$ only ensures $df^i \in L^p$, so $df^I \in L^{\frac{p}{k}}$. (You can think on $df^I$ as $\binom{d}{k}$ scalar functions). However, we are now told that in fact the minors $df^I$ have an improved regularity- they are weakly differentiable with (locally) controlled integrability.

The question is whether we can push this regularity back to $f$.

The "$k$-minors map" $\psi: A \to \bigwedge^k A$, considered as a map $\text{GL}(\mathbb{R}^d) \to \text{GL}(\bigwedge \mathbb{R}^d)$ is **smoothly invertible** from its image (which is a closed embedded Lie subgroup).

The injectivity of $\psi$ uses the assumption $k,d$ are not both even; in general $\bigwedge^k A=\bigwedge^k B$ implies (for invertible elements $A,B$) $A=\pm B$. If $k$ is odd, then of course $A=B$. If $d$ is odd, then assuming $\det A>0,\det B>0$ we again deduce $A=B$.

So, since $\det df>0$ a.e. we can "almost everywhere" invert the collection of minors $$\{df^I\}_I \stackrel{\psi^{-1}}{\to} df.$$

So, $df$ is obtained from the $(\binom{d}{k})^2 \, \,$ $k$-degree minors, which are in $W^{1,p}_{loc}$, via a composition with a smooth operator. Thus, I expect there is a chance that this "composition" (with $\psi^{-1}$) won't reduce the regularity too much- so we will have $df \in W^{1,q}_{loc}$.

It's known (due to Perelman) that in class of Alexandrov spaces of fixed dimension and bounded from below curvature Gromov-Hausdorff distance separates homeomorphism types — every $\epsilon$-close to $X$ space will be homeomorphic to $X$ for some $\epsilon$.

Well, if we have some finite metric space $X_{\delta}$ which is $\epsilon/2$-close to $X$, then $(X_{\delta}, n, C)$ define homeomorphism type of, say, compact Riemannian manifold, where $n$ is dimension and $C$ is lower curvature bound.

Now let's fix $C$ once for all (take $-1$, for example) and call finite metric space $X_{\delta}$ *a model* of a manifold $X$ if for some $\epsilon$ the only manifold $\epsilon$-close to $X_{\delta}$ is $X$ with some metric with curvature bounded below by $-1$. We can define two functions on homeomorphism (diffeo, if dim > 4, thanks to Grove-Peterson-Wu) classes of $n$-dimensional manifolds: $min \, |X_{\delta}|$ and $min \, k: X_{\delta} \to \Bbb R^k$ for isometric embedding into real space with some norm, where minimum is taken over all models. It seems appropriate to me to call first one metric complexity $mCom(X)$ and second one — essential dimension $edim(X)$.

Can $edim(X)$ be strictly less than dimension of $X$?

Are there some bounds on $mCom$ in terms of something like LS category or topological complexity (i. e. minimal cardinality of open cover over which $eval: X^I \to X \times X$ has local sections?

What is, for example, $mCom(S^1 \times S^1)$ — or something else $\geq 2$-dimensional — and what is the model? (I guess that for all surfaces answer should be derivable from known results about triangulations et cetera).

(I'm totally not an expert in this area, so maybe those questions are either very easy or hopelessly hard; if it's so, I'll gladly accept as an answer putting them into one of these two categories.)

For any (finite) group $G$ its *length* $l(G)$ is the length of maximal chain of proper subgroups (it's known and pretty widely used invariant). But we can also define *width* function $w_G(k)$ in such fashion: $w_G(k) := \#\{H<G: l(H) = k\}$. Then we can do some adjustments — make it a function $W_G:[0, 1] \to \Bbb R$ by setting $W_G(k/l(G)) = w_G(k)$, interpolating linearly and then maybe normalizing by setting integral over $[0, 1]$ to $1$. For example, $W_{\Bbb Z/n}$ is constant and $W_{\Bbb (Z/p)^n}$ is $p$-binomial distribution.

So, my question is

What is limit of $W_{S_n}$ for large $n$ — is there some "central limit theorem"? Is it dominated by $W_{Syl_2(S_n)}$?

(Exact length of $S_n$ is known (Cameron-Solomon-Turull, 1989) and asymptotically equal to length of 2-Sylow.)

What is $v_n(x)$, s.t.

$\int_{-1}^{+1} v_n(x) u_n(x) dx = \delta_{nm}$

or

$\int_{-1}^{+1} v(k', x) u(k, x) dx = \delta(k-k')$,

with $u_n(x) = (a-b x)^{c_n}$, $c_n$ discrete in the first, continuous in the second case?

This question is very similar to "Functions orthogonal to x^n" and I am sure the answer by Robin Chapman is easy to extend to my case, but I am not sure.

Specifically I am looking at eigenfunctions of the operator $L[f(x)] = \partial_x [(a-bx)f(x)]$. So $c_n = {-\frac{\lambda_n}{b} -1}$.

I recently ran into a 30+ years old literature by Andersen and Jantzen on some calculations on cohomology of flag varieties (Cohomology of Induced Representations for Algebraic Groups). Here is the setting:

$G$ complex simple Lie group, $B = HN$ a Borel subgroup corresponding to the positive roots of $\mathfrak{g}$, and $\mathfrak{n} = Lie(N)$. Theorem 3.6(a) of [Andersen-Jantzen] says that for all $i > 0$ and $n \geq 0$,

$$H^i(G/B, S^n(\mathfrak{n}^*)) = 0.$$

I tried an example for $G = SL(3)$ and $n = 2$, where $\alpha_1 = (1,-1,0)$ and $\alpha_2 = (0,1,-1)$, $\alpha_1 + \alpha_2 = (1,0,-1)$ are the positive roots in $\mathfrak{h}^*$. The weights of $S^2(\mathfrak{n}^*)$ are given by $$S^2(\mathfrak{n}^*) = \mathbb{C}_{(-2,2,0)} \oplus \mathbb{C}_{(0,-2,2)} \oplus \mathbb{C}_{(-2,0,2)} \oplus \mathbb{C}_{(-1,0,1)} \oplus \mathbb{C}_{(-2,1,1)} \oplus \mathbb{C}_{(-1,-1,2)}.$$

Then I tried to apply Bott-Borel-Weil: $$H^i(G/B, \mathbb{C}_{\lambda}^*) = \begin{cases} V_{\mu}^* &\text{if }w(\lambda + \rho) = \mu + \rho\ \text{for some}\ w \in W\ \text{with}\ l(w) = i, \\ 0 & \text{otherwise.} \end{cases} $$ Here $\mu$ must be a dominant weight of $G$.

For instance, if $\mathbb{C}_{(-2,2,0)} = \mathbb{C}_{(2,-2,0)}^*$, then $$(2,-2,0)+\rho = (2,-2,0) + (1,0,-1) = (3,-2,-1).$$

Let $w \in W$ be the transposition of the last two entries with $l(w) = 1$, then $w((2,-2,0)+\rho) = (3,-1,-2) = (2,-1,-1) + \rho$.

Does it imply that $H^1(G/B,S^2(\mathfrak{n}^*))$ contains a copy of $V_{(2,-1,-1)}^* = V_{(1,1,-2)}$, which is non-trivial? Any insight would be appreciated.

Let $G$ be a locally compact Hausdorff (second countable) groupoid with Hausdorff (second countable) unit space $X$.
Assume $G$ is étale, i.e., the source and range maps of $G$ are local homeomorphisms.
We say that $G$ is *proper* if the map $(s,r)\colon G \to X\times X$ is proper, i.e., the preimage of a compact subset is compact.

I think the following theorem is true, but I can't find a proof. I'd greatly appreciate any pointer.

$G$ is proper if and only if each $x\in X$ admits an open neighborhood $U\subseteq X$ equipped with an action of $G_x^x$ (the automorphisms group at $x$) such that the restriction $G|_U$ is isomorphic to the action groupoid $U\rtimes G_x^x$ and the map (whenever it is defined) $$ G\times_{G_x^x}U \to X$$ sending $[g,x]$ to $gx$ is a $G$-equivariant homeomorphism onto an open neighborhood of (the orbit of) $x\in X$.

Let $E/F$ be a quadratic extension of number fields and $\chi$ is a unitary automorphic character of $E^{\times}$.

Let $\pi$ be an automorphic representation of $U(n)(F)$ associated to $E/F$, which has a base change $BC(\pi)$ to $GL_n(E)$.

Then what is the automorphic representation of $U(n)(F)$ whose base change to $GL_n(E)$ is equals to $BC(\pi)\otimes (\chi\circ \det)$?

I suppose it should be $\pi \otimes \chi_1^{\frac{1}{2}}$ where $\chi_1$ is the restriction of $\chi$ to $E^1=\{x\in E \ | \ Norm(x)=1\}$.

Is this right? Because $BC(\chi)(x)=\chi(\frac{x}{\bar{x}})$ for $x\in E$.

If there is some wrong, any comments will be appreciated.

Thanks in advance.

The question is in the title:

Is there a topological space $X$ containing a copy of the real line and having the property that all the nonempty open subsets of $X$ are homeomorphic?

Let us say that $X$ is a *homeomorphic open set space*, or a *hoss* for short, if all the nonempty open subsets of $X$ are homeomorphic.
Such spaces were asked about here, and N. de Rancourt's answer shows that if $D$ is discrete then $D^\omega$ is a hoss. It follows that every ``ultrametrizable'' space embeds in a hoss. (A space is called *ultrametrizable* if it is homeomorphic to an ultrametric space. Spaces of the form $D^\omega$ are themselves ultrametrizable, and every other ultrametrizable space embeds in one of this form.) This is just about all I know about hosses and spaces that embed in them -- any other information or insight is welcome.

Familiar examples of hosses include the space $\mathbb Q$ of rational numbers and the space $\mathbb R \setminus \mathbb Q$ of irrational numbers.

Let $A$ be a complex unital Banach algebra. Let exp$(A)$ denote the range of the exponential function on $A$. Now exp$(A)$ lies in the set of all invertible elements of $A$ (denoted by $G(A)$). Can you give an example of an element belonging to $G(A)\setminus$ exp$(A)$?

Consider the affine hyperquadric $Q:=\biggl\{(z_1,...,z_{n+1})\in\mathbb{C}^{n+1}\biggl|\sum_{i=1}^{n+1}z_i^2=1\biggr\}\cong TS^n$.

What is a reasonable Kähler metric for $Q$ (induced by the pullback of the metric from the ambient space $\mathbb{C}^{n+1})$? Furthermore, how do we explicitly calculate the curvature form $\Omega$ on $Q$? Hence, compute the Chern classes of $Q$. Given this, how do we find $\chi(Q,\mathcal{O}_Q)$?

If you down-vote, please explain why so I can improve the question. Thanks in advance!

Say that two triangles are *incommensurate* if they do not
share an edge length or a vertex angle, and their areas differ.
Suppose you'd like to tile the plane with pairwise incommensurate triangles.
I can think of at least one strategy.

Spiral out from an initial triangle in the pattern depicted below, with the red extensions chosen to avoid length/angle/area coincidences with all previously constructed triangles. It seems clear that this approach could work, although it might not be straightforward to formalize to guarantee incommensurate triangles. Which brings me to my question:

** Q.**
What is a scheme that details a

This requires a more explicit design that effectively describes the triangle corner coordinates in a way that makes it evident that no lengths/angles/areas are duplicated. Without such a clear description, it is not even immediately evident (to me) that it is possible.

The same question may be asked for incommensurate simplex tilings with vertices in $\mathbb{Z}^d$.

See also: Tiling the plane with incongruent isosceles triangles.

Let us say a metric probability space $(X,\rho,\mu)$ has property (*) if: the support of $\mu$ is contained in a separable subspace of $X$.

Questions: 1. Is there a standard name for this property?

Is it true that continuum+choice implies property (*) -- is there a reference?

Is it true that if (either? both?) continuum+choice don't hold, (*) can fail? Again, reference please!

Let $S$ be a maximal split torus of a connected, reductive group $G$. Let $P_0$ be a minimal $k$-parabolic containing $S$, $T$ a maximal torus of $P_0$ which is defined over $k$ and contains $S$, and $B$ a Borel subgroup contained in $P_0$ and containing $T$.

The choice of $P_0$ and $B$ determine simple roots $_k\Delta$ and $\Delta$ for $_k\Phi = \Phi(G,S)$ and $\Phi = \Phi(G,T)$.

For each $a \in \space _k\Delta$, the set of $\alpha \in \Delta$ which restrict to $a$ form an orbit under the $\ast$-action of $\operatorname{Gal}(k_s/k)$. If $G$ is quasisplit, then the $\ast$-action is just the usual Galois action on characters. This is explained in section 12 of Brian Conrad's notes on reductive groups over fields.

What if we take an arbitrary $a \in \space _k\Phi$? Do the set of roots in $\Phi$ which restrict to $a$ also form a Galois orbit?

I know by Van Kampen's Theorem that we can obtain $\pi_1(S_1 \vee S_1) = \mathbb{Z} * \mathbb{Z}$, so I am wondering if we can construct a surface or 3-manifold whose fundamental group is $\mathbb{Z}_n * \mathbb{Z}_2$ or even $\mathbb{Z}_m \rtimes \mathbb{Z}_n$.

This might be an interesting problem because I have written semidirect product $\rtimes$ rather than the free product $*$. A torus knot $K$ is defined in Hatcher as the image of an embedding of a map $f : S^1 \to S^1 \times S^1 \to \mathbb{R}^3 \subset S^3$ given by $z \mapsto (z^m, z^n)$ then the fundamental group $\pi_1(\mathbb{R}^3 - K)$ is $Z_m \ast Z_n$ possibly up to some number-theoretic conditions. Hatcher doesn't quite give you the answer.

I think the semidirect product $\mathbb{Z}_m \rtimes \mathbb{Z}_n$ is unique. We have to specify $\mathbb{Z}_m \lhd G$ and then $G = \mathbb{Z}_m \ltimes \mathbb{Z}_n$.

Consider an infinite graph that satisfies the following property: if any finite set of vertices is removed (and all the adjacent edges), then the resulting graph has only one infinite connected component.

So, obviously, the Cayley graph for the group $\mathbb Z \times \mathbb Z$ w.r.t. the standard generating set is an example. Obviously, the Cayley graph for a free group is not an example.

I have a question: what is the name of such a property? Has it been studied?

And the next question: which are the Cayley graphs with this property?

Let $X$ be a smooth projective complex algebraic variety. Let $V_i$, for $i=1,\dots, n$, be a collection of (smooth) connected hypersurfaces such that, for all $I\subseteq [n]$, the intersection $\cap_{i \in I} V_i$ is smooth.

Is the intersection $\cap_{i=1,\dots, n} V_i$ equidimensional?

Added later: If we take $V_i$ to be smooth effective (possible non ample) divisors, is the intersection $\cap_{i=1,\dots, n} V_i$ equidimensional?

Let $S$ be a smooth projective surface, say over $\mathbb{C}$.

We have the Hilbert schemes $S^{[n]}$ and $S^{[n+1]}$ classifying length $n$-subschemes (resp. length $n+1$) together with the flag Hilbert scheme $S^{[n,n+1]}$ classifying pairs $(Z,W)$ with $Z\in S^{[n]}$ and $W\in S^{[n+1]}$ such that $Z\subset W$. There are projections $S^{[n]}\xleftarrow{p}S^{[n,n+1]}\xrightarrow{q} S^{[n+1]}$.

Let $\varphi:S^{[n+1]}\rightarrow S^{(n+1)}$ be the Hilbert-Chow morphism to the symmetric product.

Now pick a reduced $Z_0\in S^{[n]}$, that is $Z_0=\{p_1,\ldots,p_n\}$. Define $X:=p^{-1}(Z_0)$ and $Y:=q(X)$ then $X\cong Y \cong Bl_{Z_0}(S)$.

$\textbf{Question:}$ Can one describe the differences (if there are any) between $Y$ and $\varphi^{-1}(\varphi(Y))$?

From the view of closed points, $X$ is classifying all pairs $(Z,W)$ with $Z\subset W$ and $Z=Z_0$, so $Y$ should classify all length $n+1$ subschemes $W$ in $S$ containing $Z_0$. Such a $W$ is therefore given by adding one point $p\in S$ to $Z_0$. If $p\notin Z_0$ the $W$ is just the reduced point $W=\{p_1,\ldots,p_n,p\}\in S^{[n+1]}$. If $p\in Z_0$ the we get something non-reduced (that's why we blow up $Z_0$ in $X$).

So all $W$ which are reduced should land in the smooth locus of $S^{(n+1)}$ using $\varphi$. The $W$ with non-reducedness (we have $n$ of them) should land in the singular locus, but in the somehow good singular locus, because $W$ still contains $n-1$ simple points and one non-reduced length 2-subscheme.

So if one looks at the preimage, almost all $W$ are in the isomorphism locus of the Hilbert-Chow morphism, and over the $n$ non-reduced $W$ we get a $\mathbb{P}^1$ since we blow up the singularity, as $\varphi$ is a resolution of singularities of $S^{(n+1)}$.

So it seems to me, that we should have $Y\cong \varphi^{-1}(\varphi(Y))$. Is this geometric picture correct? What about the scheme structure in this situation? Do we have $Y\cong \varphi^{-1}(\varphi(Y))$ as schemes? Or is there something more happening, which is not so obvious, since $S^{(n+1)}$ is singular or because of bad properties of $\varphi$?

Can we do the same thing if we start with a more complex $Z_0\in S^{[n]}$?

I remember coming across this result some time ago but I am having trouble finding a reference for it. It goes something like this:

Let $p$ be a(n odd?) prime, then the $p$-primary component of $\pi^S_k$ is $\Bbb Z_p$ when $k=2l(p-1)-1$ for $l=1,\dots,p-1$ and is trivial for all other $k<2p(p-1)-2$.

This is what I have written down on the back of an envelope. I checked this with Wikipedia's table and it seems to be true.

What is the reference in which it is proved? And if its simple could you overview it as an answer here?

My guess is that it is proven by Toda but his papers are difficult to search through.

(**Note:** I've migrated this question from math.stackexchange, as the lack of answers there made me believe it was perhaps too advanced for that forum.)

Consider the one-dimensional heat equation $$\partial_t u(t,x)=\frac12\Delta u(t,x),\qquad t\geq0,~x\in I$$ on some interval $I=(-a,a)$, with some initial condition $u(0,x)=f(x)$ and boundary condition on the interval $I$.

Suppose that we define the following processes

- $(K_t)_{t\geq0}$, a Brownian motion
*killed*on the boundary of $I$; - $(R_t)_{t\geq0}$, a Brownian motion
*reflected*on the boundary of $I$; - $(P_t)_{t\geq0}$, a
*periodic*Brownian motion in the interval $I$.

I know from a mix of intuition or folklore that we have the following probabilistic representations the solution $u(t,x)$ for different choices of boundary conditions on $I$: $$u(t,x) =\begin{cases} \mathbb{E}\big[f(x+K_t)\big]&\text{for Dirichlet boundary};\\ \mathbb{E}\big[f(x+R_t)\big]&\text{for Neumann boundary};\\ \mathbb{E}\big[f(x+P_t)\big]&\text{for Periodic boundary}. \end{cases}$$

**Question.** Are there any references out there with *full proofs* of these facts?

I've tried extensive researching on this subject to no avail.