parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
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https://mathoverflow.net/questions/302557 | 1 | We say that a topological space $(X,\tau)$ is *point-removal insensitive* if for all $x\in X$ we have $X\cong X\setminus \{x\}$.
If $X,Y$ are point-removal insensitive, does this imply that $X\times Y$ with the product topology is point-removal insensitive?
| https://mathoverflow.net/users/8628 | Product of point-removal insensitive spaces | Let $X = {\mathbb R} \setminus {\mathbb Z}$, that is, $X$ is the free union of countably many open intervals. Then $X$ is point-removable insensitive because the removal of any point leaves a free union of countably many open intervals. However, $X \times X$ is not point-removal insensitive because every component of $... | 7 | https://mathoverflow.net/users/89233 | 302623 | 132,524 |
https://mathoverflow.net/questions/302606 | 15 | I'm really enjoying the AMS column "What is ..." (<http://arminstraub.com/math/what-is-column>) and The Princeton Companion to Mathematics.
I am looking for something similar. I'd like to acquire some intuition behind different subjects and the general overview rather than digging into details and technical proofs.
... | https://mathoverflow.net/users/nan | Where can I read reviews of mathematical theories? | On the popular level, there is an AMS feature column:
<http://www.ams.org/samplings/feature-column/fc-current.cgi>
and a series of 10 volumes "What's happening in mathematical sciences":
<https://bookstore.ams.org/HAPPENING>,
also by AMS.
On a higher level, there are journals which publish surveys, addressed to
the g... | 16 | https://mathoverflow.net/users/25510 | 302625 | 132,525 |
https://mathoverflow.net/questions/302604 | 7 | 1) Suppose that I have a sifted diagram of categories $\mathcal{C}\_i$, another of the same shape $\mathcal{D}\_i$, and that I have a system $F\_i:\mathcal{C}\_i\to\mathcal{D}\_i$ commuting with the morphisms in each diagram. Then I get a functor $$(\operatorname{colim} F\_i):\operatorname{colim} \mathcal{C\_i} \to \op... | https://mathoverflow.net/users/1040 | Are sifted (2,1)-colimits of fully faithful functors again fully faithful? (And a de-categorified variant) | Reflexive coequalizers are examples of sifted colimits in a 1-category, and groupoids are examples of reflexive coequalizers. But quotients don't preserve monomorphisms in general.
This gives a counterexample to question (2) in the category Set:
Let $V = S \rightrightarrows S$ denote the trivial groupoid acting o... | 5 | https://mathoverflow.net/users/7762 | 302630 | 132,527 |
https://mathoverflow.net/questions/302626 | 6 | In [OEIS A003436](https://oeis.org/A003436), it is written that the number of inequivalent labeled Hamilton Cycles of an n-dimesnional [Octahedron](https://en.wikipedia.org/wiki/Octahedron) is the same as the number of Perfect Matchings in a the complement of the [Cycle Graph](https://en.wikipedia.org/wiki/Cycle_graph)... | https://mathoverflow.net/users/63938 | Why is the number of Hamiltonian Cycles of n-octahedron equivalent to the number of Perfect Matching in specific family of Graphs? | The $n$-dimensional analogue of the octahedron is the complement of a perfect matching of its vertex set. (Every vertex is joined to every other vertex except its antipode.) If you take a Hamilton cycle in the $n$-dimensional octahedron then you can think of that as a labelling of the $2n$-cycle, and the non-edges of t... | 9 | https://mathoverflow.net/users/124004 | 302632 | 132,528 |
https://mathoverflow.net/questions/301722 | 6 | Language: first-order logic
Primitives: $=, S, \in $ (the first denotes identity, the second denotes “is a successor of”, and the third denotes membership relation).
Axioms: those of identity theory +
**Existence:** $\exists y, x (y \ S \ x)$
Define: $y \text{ is a successor } \iff \exists x (y \ S \ x)$
... | https://mathoverflow.net/users/95347 | What is the consistency strength of this theory? | $\let\itp\vartriangleright\def\mr{\mathrm}\DeclareMathOperator\dom{dom}\let\bez\smallsetminus\let\sset\subseteq$The weakened theory is much weaker than bounded arithmetic.
Let me denote the theory as $\mr{WT}$. Notice that the weakened comperehension schema is simply equivalent to the schema
$$\tag{$V\_n$}\forall w\_... | 11 | https://mathoverflow.net/users/12705 | 302634 | 132,530 |
https://mathoverflow.net/questions/302631 | 8 | I may be wrong, but we should be able to write the Bag monad in a polynomial form. The bag monad, is exectly the multiset monad whose category of algebras are the commutative monoids. Another name for the bag monad is a container.
Containers are synonymous with polynomail functors. The data that defines a container a... | https://mathoverflow.net/users/10007 | What is the polynomial functor for the Bag monad | **The bag monad is not polynomial.**
Any polynomial endofunctor must preserve pullbacks: $f^\*$ and $\Pi\_g$ preserve all limits since they’re right adjoints, while $\Sigma\_h$, being just the forgetful functor from a slice category, is well known (and easily seen) to preserve all connected limits.
However, the bag... | 14 | https://mathoverflow.net/users/2273 | 302638 | 132,533 |
https://mathoverflow.net/questions/302406 | 5 | The integral
$$Z\_3(\lambda\_1,\lambda\_2,\lambda\_3)=\frac{1}{2}\int\_{-1}^1 I\_0\left[\tfrac{1}{2}(\lambda\_1-\lambda\_2) (1-x)\right] I\_0\left[\tfrac{1}{2} (\lambda\_1+\lambda\_2)(1+x)\right]\,e^{\lambda\_3 x}\,dx,$$
with $\lambda\_1,\lambda\_3,\lambda\_3\in\mathbb{R}$ and $I\_0$ a Bessel function, arises in the ev... | https://mathoverflow.net/users/11260 | Invariance of an integral over SO(3) under permutation of parameters | Using the variable change $x=2t-1$, the integral $Z\_3(\lambda\_1,\lambda\_2,\lambda\_3)$ can be put into the form
\begin{equation}
Z\_3(\lambda\_1,\lambda\_2,\lambda\_3)=\int\_{0}^1 I\_0\left[(\lambda\_1-\lambda\_2)
(1-t)\right]
e^{-\lambda\_3(1-t)}I\_0\left[(\lambda\_1+\lambda\_2)t\right]\,e^{\lambda\_3 t}\,dt
\end{... | 7 | https://mathoverflow.net/users/46744 | 302646 | 132,536 |
https://mathoverflow.net/questions/301504 | 2 | Let $k$ be a field of characteristic zero (I do not mind to assume that $k=\mathbb{C}$, if things are easier in this case).
[Lüroth theorem](https://en.wikipedia.org/wiki/L%C3%BCroth%27s_theorem) says that a field $L$, $k \subset L \subset k(x)$ containing a nonconstant polynomial over $k$ is equal to $k(h)$ for some $... | https://mathoverflow.net/users/72288 | Lüroth theorem for $k \subset k(f,g) \subseteq k(x)$ | This answer is just repeating comments, above, about a negative answer to conjecture (2). I'm unable to say anything for question (1).
The conjecture (2) is not correct. As you note, $k(x) = k(x^3,x^6+x^2)$; and then
$$
k(x) = k(x^3 \cdot (x^6+x^2), x^6+x^2) = k(x^9+x^5, x^6+x^2),
$$
where $\deg(f)=9$ and $\deg(g)=6... | 2 | https://mathoverflow.net/users/88133 | 302648 | 132,537 |
https://mathoverflow.net/questions/302645 | 7 | I am aware of the classical work by Smale and Barden computing the diffeomorphism type of smooth simply connected 5-manifolds in D. Barden, [Simply connected five-manifolds](https://www.jstor.org/stable/1970702), Ann. of Math. (2) 82 (1965), 365–385 and S. Smale, [On the structure of 5-manifolds](https://www.jstor.org/... | https://mathoverflow.net/users/21985 | Five-dimensional manifolds fibering over a fixed hyperbolic surface | When you say "fibering" do you means as a smooth fiber bundle? Smooth fiber bundles with fiber $F$ and base $B$ are classified by homotopy classes of maps from $B$ to the classifying space $B\mathrm{Diff}(F)$. If you case $F$ is a $3$-manifold so $\mathrm{Diff}(F)$ is somewhat better understood, see [Hatcher's survey]... | 10 | https://mathoverflow.net/users/1573 | 302650 | 132,538 |
https://mathoverflow.net/questions/302450 | 8 | Define the functions
$$F\_n(q)=\frac1{(1-q)^{2n}}\sum\_{k=0}^n(-q)^k\frac{2k+1}{n+k+1}\binom{2n}{n-k}
\prod\_{j=0,\,j\neq k}^n\frac{1+q^{2j+1}}{1+q}.$$
The numbers $\frac{2k+1}{n+k+1}\binom{2n}{n-k}$ belong to a family of Catalan triangle of which the special case
$k=0$ yields the Catalan numbers $C\_n=\frac1{n+1}\bino... | https://mathoverflow.net/users/66131 | Prove that these are polynomials | First notice that $\frac{2k+1}{n+k+1}\binom{2n}{n-k} = \frac{2k+1}{2n+1}\binom{2n+1}{n-k}$.
To prove that $F\_n(q)$ is a polynomial it is enough to show that
$$\sum\_{k=0}^n(-1)^k(2k+1)\binom{2n+1}{n-k}\frac{q^k}{1+q^{2k+1}}$$
has the zero $q=1$ of multiplicity $2n$.
Plugging $q=e^t$, one needs to show that
\begin{sp... | 14 | https://mathoverflow.net/users/7076 | 302652 | 132,539 |
https://mathoverflow.net/questions/302607 | 3 | [Recall the Bishop-Phelps Theorem](https://en.wikipedia.org/wiki/Bishop%E2%80%93Phelps_theorem).
>
> **Bishop-Phelps Theorem:** Let $B\subseteq E$ be a bounded, closed, convex subset of a real Banach space $E.$
> Then the set
> $$\{e^\*\in E^\*: e^\* \text{ attains its supremum on } B \}$$
> is norm-dense in th... | https://mathoverflow.net/users/42411 | Does Bishop-Phelps Theorem hold for extreme points (slightly different version)? | If $B$ is a bounded convex set that doesn't contain $0$, the Hahn-Banach separation theorem says there are $e^\* \in E^\*$ with $\| e^\*\| = 1$ and $\epsilon > 0$ with $e^\*(e) < -\epsilon$ for all $e \in B$; moreover, this is true (maybe with a smaller $\epsilon$) in a neighbourhood of $e^\*$. If $E^\*$ is strictly co... | 5 | https://mathoverflow.net/users/13650 | 302656 | 132,540 |
https://mathoverflow.net/questions/302675 | 7 | As you maybe remember, Isbell duality is an adjunction
$$\mathcal O : [A°,Set] \leftrightarrows [A,Set]° : {\cal S}pec$$
as defined [here](https://ncatlab.org/nlab/show/Isbell+duality); since every functor $f : A\to B$ defines both
1. a functor $f^\* : B\to [A°,Set]$ by $$(a,b)\mapsto B(fa,b)$$ and
2. a functor $f... | https://mathoverflow.net/users/7952 | Functors in Isbell duality exchange $f^*a$ and $f_*a$ | Let's give names to these two separate conditions:
\begin{align}
Nat(B(b,f-),hom(a,-)) &\cong B(fa,b) \tag{1} \\
Nat(B(f-,b),hom(-,a)) &\cong B(b,fa) \tag{2}
\end{align}
Neither of these must be true in general, though they hold under some natural conditions. I will treat (2) first, since (1) is completely dual.
As ... | 3 | https://mathoverflow.net/users/1015 | 302680 | 132,548 |
https://mathoverflow.net/questions/302679 | 1 | Hironaka's theorem states that for any algebraic variety (analytic space) $X$ there exists a smooth variety (complex manifold) $X'$ and a morphism $f : X' \rightarrow X$ such that $f$ restricted to $X \backslash \, f^{-1}X$ is an isomorphism of $X' \backslash \, f^{-1}X$ with the non-singular locus of $X$ and $f^{-1}X$... | https://mathoverflow.net/users/91572 | Hironaka's theorem and smooth completion | Let $\bar{X}$ be any projective completion of $X$ ($\bar{X}$ is projective and $X$ is contained in $\bar{X}$ as a dense Zariski open subset). Now $\bar{X}\backslash X$ is a closed subvariety of $\bar{X}$ so you can resolve it into a simple normal crossing divisor while $X$ stays unchanged (by a finite sequence of blow-... | 3 | https://mathoverflow.net/users/15124 | 302683 | 132,549 |
https://mathoverflow.net/questions/302684 | 4 | Does there exist a quasisimple group $G$ and an odd prime $p$ such that $G$ has cyclic Sylow $p$-subgroups and a weakly real element of $p$-power order?
From [Strongly real elements of odd order in sporadic finite simple groups](https://mathoverflow.net/questions/224519/strongly-real-elements-of-odd-order-in-sporadic... | https://mathoverflow.net/users/20764 | Quasisimple group with cyclic Sylow p-subgroup and weakly real p-elements? | ${\rm SL}(2,q)$ for odd $q$ has a unique element of order $2$, which is central. So for any odd prime $p$ dividing $q-1$ or $q+1$ it has a cyclic Sylow $p$-subgroup $P$ and a generator of $P$ is conjugate to its inverse but not by an involution. The smallest example is ${\rm SL}(2,5)$ with $p=3$, and $p=5$ also works i... | 6 | https://mathoverflow.net/users/35840 | 302687 | 132,550 |
https://mathoverflow.net/questions/302688 | 3 | Does there exists a smooth projective surface $X$ which contains a projective line $L$ and a smooth conic $C$ such that $L\cap C=$empty?
| https://mathoverflow.net/users/nan | Intersection of a line and a conic in a surface | Yes. This holds for any cubic surface over an algebraically closed field $k$.
Let $S$ be such a surface. Let $L'$ be a line. The pencil of hyperplanes containing $L'$ forms a conic bundle $\pi: S \to \mathbb{P}^1$. This conic bundle has $5$ singular fibres, which are each a pair of lines meeting in a point. Take $C$ ... | 5 | https://mathoverflow.net/users/5101 | 302690 | 132,552 |
https://mathoverflow.net/questions/302673 | 1 | I'm looking for an example of a non-compact spin manifold $M$ and a compact subset $K\subseteq M$ such that $\partial K$ is a compact hypersurface in $M$ with $\hat{A}(\partial K)\neq 0$.
(At first I thought that $M=\mathbb{CP}^3$ and $\partial K = $ the surface defined by $x^4+y^4+z^4+w^4=0$ might be an example, but... | https://mathoverflow.net/users/78729 | Example of a certain partitioned manifold | Assuming that you want $K$ to be a compact manifold with boundary, no such example exists. This is because $\partial K$ is orientedly nullcobordant ($K$ is a cobordism between $\partial K$ and $\emptyset$), so its Stiefel-Whitney and Pontryagin numbers vanish. But the $\hat{A}$ genus is just a rational linear combinati... | 3 | https://mathoverflow.net/users/21564 | 302692 | 132,553 |
https://mathoverflow.net/questions/279753 | 6 | See <https://en.wikipedia.org/wiki/Nakayama_algebra> for the definition of Nakayama algebras and define the permanent of such an algebra to be the permanent of its Cartan matrix.
(all algebras are assumed to be connected and finite dimensional)
Conjecture:
Let $X$ be the set of Nakayama algebras of finite global dimens... | https://mathoverflow.net/users/61949 | Permanent of Nakayama algebras | Mare's answer reduces the problem to showing that the permanant of the $n \times n$ matrix
$$\begin{bmatrix}
1 & 1 & 1 & \cdots & 1 \\
1 & 2 & 2 & \cdots & 2 \\
1 & 1 & 2 & \cdots & 2 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
1 & 1 & 1 & \cdots & 2 \\
\end{bmatrix}$$
is the number of weak orders on $[n]:=\{ 1,2... | 6 | https://mathoverflow.net/users/297 | 302706 | 132,557 |
https://mathoverflow.net/questions/302709 | 8 | It is not hard to construct such curves explicitly, e.g. my favourite example is a curve $U$ singular at $(3:4:5i)$ and also passing through $(1:0:0)$, $(0:1:0)$, $(0:0:1)$, $(1:1:0)$, $(1:0:1)$, $(0:1:1)$.
(Amusingly, one more real point on $U$ is also rational, namely $(3^4:4^4:5^4)$;
two more are quadratic irratio... | https://mathoverflow.net/users/11100 | projective plane cubics with exactly 9 real points | Let $P\_1,\ldots,P\_8$ be "random" real points (which could even be rational).
Then the space of cubic polynomials vanishing on all $P\_i$ has dimension
$10 - 8 = 2$. Let $(C\_1, C\_2)$ be a basis of this real vector space.
Then the cubic curves $C\_1=0$ and $C\_2=0$ meet at $P\_1,\ldots,P\_8$
and at some ninth point ... | 14 | https://mathoverflow.net/users/14830 | 302710 | 132,559 |
https://mathoverflow.net/questions/302699 | 1 | I am trying to compute the Chern classes of the restriction of a rank two vector bundle on $\mathbb{P}^3$, denoted by $E$, with fixed Chern classes, $c\_1(E) = c\_1$ and $c\_2(E) = c\_2$, to a hyperplane $H \subset \mathbb{P}^3$. For this let $s$ be the equation of the hiperplane $H$ and consider the exact sequence:
... | https://mathoverflow.net/users/43027 | Restriction of vector bundles | OK, I'll write my comment as an answer. The computation of the OP using the multiplicative property of exact sequences gives the Chern classes of the coherent sheaf $i\_\*(E\_{|H})$ on $\mathbb{P}^3$, where $i:H\hookrightarrow \mathbb{P}^3$ is the inclusion map. They are different from the Chern classes of the vector b... | 5 | https://mathoverflow.net/users/40297 | 302713 | 132,561 |
https://mathoverflow.net/questions/293758 | 2 | I once read a short paper on the following subject.
Fix a graph $H$ with fractional coloring number $c$ and let $G$ be a graph with $n$ vertices and $e$ edges, with $e$,$n$ large. In terms of $n$, $e$, and $c$, we want to bound from above and below the number of homomorphisms from $H$ to $G$.
I cannot seem to find th... | https://mathoverflow.net/users/104594 | Number of homomorphisms from graph $H$ to $G$ , bounds that have to do with fractional coloring number? | You are referring to [E. Friedgut and J. Kahn, On the number of copies of one hypergraph in another](https://link.springer.com/article/10.1007/BF02780332), Israel Journal of Mathematics 105 (1998), 251–256.
For graphs this result is the very first paper of Noga Alon, [On the number of subgraphs of prescribed type pf g... | 3 | https://mathoverflow.net/users/1532 | 302720 | 132,564 |
https://mathoverflow.net/questions/302719 | 9 | Suppose I give a list of vertices $(v\_1, v\_2, ..., v\_n)$, and a list of "adjacencies", i.e. pairs of vertices $(v\_i,v\_j)$. Does it exists a unique polytope that has this vertices and realises the adjacencies with 1-dimensional faces?
I do not know if that is important, but each and every vertex has exactly d adja... | https://mathoverflow.net/users/48526 | Minimal combinatorial data needed to define a polytope | I don't know conditions of existence ($d$-connectivity is necessary), but the uniqueness was proved by Blind and Mani in 1987, see also the 1988 article "A simple way to tell a simple polytope from its graph" by Gil Kalai. Kalai's proof is also presented in Guenter Ziegler's book "Lectures on polytopes".
The simplici... | 13 | https://mathoverflow.net/users/98590 | 302723 | 132,566 |
https://mathoverflow.net/questions/302704 | 1 | A *digraph* (direct graph) consists of a set $V$ of vertices and a set $E$ of directed edges $v\to v'$. A *multidigraph* is a digraph in which $E$ is a multiset, so edges may appear multiple times in $E$, or equivalently, $E$ is a set of directed edges that are assigned multiplicities.
Is there a standard name for a ... | https://mathoverflow.net/users/11926 | Is there a standard name for this type of multidigraph? | I haven't seen the multigraph version, but non-multi directed graphs with at most one outgoing neighbor per vertex have been called [directed pseudoforests](https://en.wikipedia.org/wiki/Pseudoforest), and with exactly one outgoing neighbor they are also called functional graphs or maximal directed pseudoforests.
| 5 | https://mathoverflow.net/users/440 | 302725 | 132,567 |
https://mathoverflow.net/questions/299307 | 8 | A *homogeneous space* $(X,\tau)$ is a topological space such that for all $x,y\in X$ there is a homeomorphism $\varphi:X\to X$ such that $\varphi(x)=y$. As [a previous question](https://mathoverflow.net/questions/299163/chains-of-homeomorphic-subspaces) implies, the union of an ascending chain of homogeneous spaces nee... | https://mathoverflow.net/users/8628 | Spaces without maximal homogeneous subspaces | **Theorem.** The topological sum $X=\bigoplus\_{n\in\omega}\ell\_2(\aleph\_n)$ of Hilbert spaces of density $\aleph\_n$ does not contain maximal homogeneous subspaces.
*Proof.* Let $H$ be a non-empty homogeneous subspace in $X$.
Then for some $k\in\omega$ the intersection $H\cap\ell\_2(\aleph\_k)$ is not empty and he... | 5 | https://mathoverflow.net/users/61536 | 302729 | 132,569 |
https://mathoverflow.net/questions/302697 | 6 | **Edit: 28 January 2023** I just realized that this metric is frequently used in this paper
<https://hal.science/hal-01382281/document>
Let $$M=\{(x\_1,x\_2,\ldots,x\_n)\in \mathbb{R}^n\mid x\_i>0,\;i=1,2,\ldots,n\}$$
For $X=(x\_1,x\_2,\ldots,x\_n)\in M$ put $|X|=\sum\_{i=1}^n x\_i$.
We consider the Shahshahani Rie... | https://mathoverflow.net/users/36688 | The group of isometries of Shahshahani metric | The point is that, if you set $x\_i = {u\_i}^2$ where $u\_i>0$, this becomes a diffeomorphism of $M$ with itself with the property that, in the $u$-coordinates, the Shahshahani metric becomes
$$
g = 4({u\_1}^2+\cdots+{u\_n}^2)\bigl({\mathrm{d}u\_1}^2+\cdots + {\mathrm{d}u\_n}^2\bigr).
$$
Clearly, this metric is just th... | 6 | https://mathoverflow.net/users/13972 | 302736 | 132,573 |
https://mathoverflow.net/questions/302738 | 5 | In Kottwitz's 1985 Compositio paper,
[Isocrystals with additional structure](http://www.numdam.org/item/CM_1985__56_2_201_0), first page, paragraph 4:
>
> Let $\mathbb{D}$ be the diagonalizable pro-algebraic group over $\mathbb{Q}\_p$ with character group $\mathbb{Q}$
>
>
>
What is this $\mathbb{D}$?... I com... | https://mathoverflow.net/users/125609 | Diagonalizable pro-algebraic group in Kottwitz's 1985 Compositio paper | One just take the inverse limit of $\mathbb G\_m$ under the inverse system of maps $\mathbb G\_m \to \mathbb G\_m$ by raising to a natural number power. Or, equivalently, the sequence $\dots \mathbb G\_m \to \mathbb G\_m \to \mathbb G\_m \to \mathbb G\_m$ where the $n$th-to-last map is raising to the $n$th power.
The... | 6 | https://mathoverflow.net/users/18060 | 302739 | 132,574 |
https://mathoverflow.net/questions/302728 | 9 | The question is in the title. For Easton's theorem see [Wikipedia](https://en.wikipedia.org/wiki/Easton%27s_theorem). Loosely speaking we can use forcing to manipulate the powerset function on regular cardinals as much as we like given we satisfy the following restrictions: (a) $\kappa<\lambda$ implies $2^\kappa\le 2^\... | https://mathoverflow.net/users/13694 | Does Easton forcing preserve measurable cardinals? | There are a variety of positive and negative results on this topic, depending on the Easton function and the set-theoretic background.
The Kunen-Paris theorem, for example, provides a large number of positive instances.
**Theorem.** If $\kappa$ is a measurable cardinal and $2^\kappa=\kappa^+$, then for any Easton ... | 9 | https://mathoverflow.net/users/1946 | 302757 | 132,580 |
https://mathoverflow.net/questions/302758 | 2 | Let $X$ be a smooth complex projective variety of dimension $n$, and let $\mathcal{F}$ be a globally generated rank $n$ vector bundle on $X$. Let $D$ be a smooth divisor on $X$.
Is it true that there is a dense subset $U\subset H^0(\mathcal{F})$ such that for all $s\in U$, the zero set of $s$ does not intersect $D$?... | https://mathoverflow.net/users/64302 | Set of sections whose zeroes avoid a given divisor is (Zariski) dense? | Yes. Consider the incidence variety $Z\subset D\times \mathbb{P}(H^0(E))$ of pairs $(x,[s])$ with $x\in D$, $s\in H^0(E)\smallsetminus \{0\} $ and $s(x)=0$. Let $p,q$ be the projections from $Z$ to $D$ and $\mathbb{P}(H^0(E))$. For $x\in D$, the fiber $p^{-1}(x)$ is the subspace of $\mathbb{P}(H^0(E))$ formed by sectio... | 3 | https://mathoverflow.net/users/40297 | 302760 | 132,581 |
https://mathoverflow.net/questions/302743 | 39 | Sometimes it is not easy to formulate a correct question. Here is a better version of [this](https://mathoverflow.net/questions/302367/translating-first-order-statements-about-symmetric-groups-into-the-language-of-n) question (I still do not know if it is optimal, but it is better than the previous one).
We say that ... | https://mathoverflow.net/users/nan | The symmetric group theory of natural numbers | It's possible to embed within the theory of the permutation group $S\_n$ the theory of second-order arithmetic for numbers between $0$ and $n-1$. Using this we can construct propositions corresponding to any property of $n$ that can be determined by a Turing machine with resource bound that roughly corresponds to a max... | 22 | https://mathoverflow.net/users/41947 | 302771 | 132,583 |
https://mathoverflow.net/questions/302761 | 1 | Let $R$ be the ring of distributions $T\in \mathcal{D}'(\mathbb{R})$ with support in $[0,\infty)$ and with the operations of pointwise addition and multiplication taken as convolution, and $I$ be the ideal in $R$ with support in $(0,\infty)$. Is $I$ maximal in $R$?
| https://mathoverflow.net/users/124733 | A question arising in the distribution theory of L. Schwartz | The set $J=\{u\in\mathscr D'(\mathbb R):$ supp$u \subseteq [0,\infty)$ and singsupp$u \subseteq (0,\infty)\}$ is a strictly bigger ideal.
To see that it is an ideal decompose such a $u$ by multiplying with a cut-off function as $u=\varphi + v$ where $\varphi \in \mathscr D([0,\infty))$ and $v\in I$.
| 4 | https://mathoverflow.net/users/21051 | 302772 | 132,584 |
https://mathoverflow.net/questions/302773 | 3 | Suppose that $G$ is a regular feebly compact Moore quasitopological group. Must $G$ be a topological group? This was previously posted [here on MathSE](https://math.stackexchange.com/questions/2817752/a-question-on-quasitopological-group) also.
A semitopological group $G$ is a group $G$ with a topology such that the ... | https://mathoverflow.net/users/39873 | A question on quasitopological group | Yes. The paper [KKM] contains more general results. In particular, by Corollary 1 each regular semitopological group which is a cover semi-complete Baire space is a topological group. It remains to note that each $p-\sigma$-fragmentable space (see [Bou] for the definition) (in particlar, each $p$-space, so each Moore s... | 3 | https://mathoverflow.net/users/43954 | 302779 | 132,586 |
https://mathoverflow.net/questions/302750 | -1 | Suppose we have two linear programs $Ax\leq b$ and $Bx\leq c$ is there a way to combine them into one program of possibly a larger dimension $Cy\leq d$ such that projection of vectors $y$ into a subspace of dimension $length(x)$ yields feasible points of union of original programs?
$C$ could have exponentially many c... | https://mathoverflow.net/users/10035 | On OR condition in Linear Programming with exponentially many constraints | The answer is: This can be done if and only if the union is convex.
Indeed, let $P:=\{x\colon Ax\le b\}$, $Q:=\{x\colon Bx\le c\}$, and $R:=P\cup Q$. The necessity of the convexity of $R$ was already pointed out by Robert Israel.
Now suppose that $R$ is convex. Note that $P$ and $Q$ are convex polyhedra. Also, an... | 1 | https://mathoverflow.net/users/36721 | 302782 | 132,587 |
https://mathoverflow.net/questions/302640 | 1 | Let $A$ be a C\*-algebra. Suppose that every cyclic representation of $A$ is finite dimensional.
>
> Q. Is $A$ finite dimensional?
>
>
>
| https://mathoverflow.net/users/84390 | finite dimensional C*-algebras | The answer is yes. To prove it, assume first that $A$ is separable. In
this case, by Theorem 7.10 in John B. Conway's book "A course in
Operator Theory", we may choose a faithful representation $\pi$ of $A$
on a separable Hilbert space $H$. Choose an orthonormal basis
$\{e\_n\}\_{n\in{\mathbb N}}$ for $H$ and define
$... | 5 | https://mathoverflow.net/users/97532 | 302789 | 132,589 |
https://mathoverflow.net/questions/302510 | 3 | Let $G$ be a locally compact topological group. A continuous bounded function $f$ on $G$ is called (weakly) almost periodic if the set $L\_Gf$ of left translates is relatively compact in the (weak) norm topology. Then $f$ is uniformly continuous with respect to the left and right uniform structures on $G$ and all (weak... | https://mathoverflow.net/users/90755 | About understanding manifold structure on WAP compactification of $\Bbb{C} \rtimes \Bbb{T}$ | Let me recall that the WAP compactification of a locally compact group is a semi-topological compact monoid ("semi-topological" means that both left and right multiplications are continuous, but maybe not jointly) which is universal as such.
To be precise, given a locally compact group $G$ there exists a semi-topolog... | 4 | https://mathoverflow.net/users/89334 | 302792 | 132,591 |
https://mathoverflow.net/questions/302790 | 11 | The $j$-ivariant has the following Fourier expansion
$$j(\tau)=\frac 1q +\sum\_{n=0}^{\infty}a\_nq^n=\frac{1}{q}+744+196884q+21493760q^2+\cdots.$$
Here is $q=e^{2\pi i \tau}$.
Is there some simple **effective** bound on the coefficients $a\_n$?
**Backround.**
This question comes from [On the “gap” in a theorem o... | https://mathoverflow.net/users/122104 | Effective bound on the expansion of the $j$-invariant | Once you know that the coefficients are all positive (see postscript),
it's easy to get an effective upper bound that grows as $\exp(4\pi \sqrt{n})$,
which is within a factor $O(\sqrt n)$ of the correct order of growth.
Start from the inequality
$$
a\_n = q^{-n} (a\_n q^n) < q^{-n} \sum\_{k=-1}^\infty a\_k q^k = q^{-n... | 12 | https://mathoverflow.net/users/14830 | 302799 | 132,595 |
https://mathoverflow.net/questions/302802 | 0 | Let
\begin{equation\*}
X\_t=xe^{-\lambda t}+\sigma e^{-\lambda t}\int\_0^t e^{\lambda s} dB\_s
\end{equation\*}
be the solution of Ornstein-Uhlenbeck equation where $B$ is Brownian motion, and $x,\sigma,\lambda$ are all constants. Compute
\begin{equation\*}
\liminf\_{t\rightarrow \infty}\frac{X\_t}{\sqrt{\log t}}\qu... | https://mathoverflow.net/users/78326 | Limit distribution of Ornstein-Uhlenbeck equation | $\newcommand{\al}{\alpha}
\newcommand{\de}{\delta}
\newcommand{\De}{\Delta}
\newcommand{\ep}{\varepsilon}
\newcommand{\ga}{\gamma}
\newcommand{\Ga}{\Gamma}
\newcommand{\la}{\lambda}
\newcommand{\si}{\sigma}
\newcommand{\Si}{\Sigma}
\newcommand{\thh}{\theta}
\newcommand{\om}{\omega}
\newcommand{\R}{\mathbb{R}}
\newcomma... | 2 | https://mathoverflow.net/users/36721 | 302810 | 132,599 |
https://mathoverflow.net/questions/302807 | 8 | Here $\omega\_1$ is the first uncountable ordinal, and $\mathcal{P}(\omega\_1)$ denotes the power set of $\omega\_1$. Separable means countably generated as a $\sigma$-algebra.
| https://mathoverflow.net/users/67151 | Is the measurable space $(\omega_1,\mathcal{P}(\omega_1))$ separable? | Whether $\mathcal P(\omega\_1)$ is separable is independent of ZFC.
If $2^{\aleph\_0} \neq 2^{\aleph\_1}$ (which is consistent with ZFC -- it is implied by CH for example), then $\mathcal P(\omega\_1)$ (which has size $2^{\aleph\_1}$) is larger than any countably generated $\sigma$-algebra (which has size at most $2^... | 11 | https://mathoverflow.net/users/70618 | 302817 | 132,604 |
https://mathoverflow.net/questions/302777 | 4 | Let $X$ be a (infinite) separable topological space and consider $C\_p(X)$, the space of continuous functions on $X$ endowed with the point-wise convergence topology.
>
> Q. I am looking for topological properties on $X$ which make $C\_p(X)$ hereditary Lindelöf.
>
>
>
$$X=?\implies C\_p(X)=\textrm{Hereditary L... | https://mathoverflow.net/users/84390 | Hereditary Lindelöfness in $C_p$-spaces | If $X^n$ is hereditarily separable for each $n \in \mathbb{N}$ then $C\_p(X)$ is hereditarily Lindelof by Zenor-Velichko's theorem. It is consistent with ZFC that this is also a necessary condition and it was an open problem in the 80's to find a consistent counterexample. I don't know the status of this problem (I'm a... | 4 | https://mathoverflow.net/users/17836 | 302828 | 132,607 |
https://mathoverflow.net/questions/302724 | 6 | A surjective homeomorphism $h:X\to X$ is *minimal* if $$\overline{\{h^n(x):n\in \mathbb N\}}=X$$ for every $x\in X$. In other words, the orbit of each point is dense.
**Does either of the Erdös spaces $\mathfrak E$ or $\mathfrak E\_c$ have a minimal homeomorphism?**
The Erdös spaces are defined as:
$\mathfrak E=\... | https://mathoverflow.net/users/95718 | Transitive homeomorphisms of Erdős spaces | The answer to both questions is affirmative.
**Theorem 1.** The complete Erdos space $\mathfrak E\_c$ has a self-homeomorphism whose every orbit is dense in $\mathfrak E\_c$.
*Proof.* We use a known result of [Kawamura, Oversteegen and Tymchantyn,](http://matwbn.icm.edu.pl/ksiazki/fm/fm150/fm15021.pdf) that the com... | 7 | https://mathoverflow.net/users/61536 | 302839 | 132,612 |
https://mathoverflow.net/questions/302837 | 4 | Let $M$ be a matroid with ground set $E$.
Deletion and contraction in matroids commute with each other and with themselves, i.e. for all $e,f \in E$ one has
$(M/e)\setminus f = (M\setminus f)/e$, $\hspace{0.1cm}$ $(M\setminus e)\setminus f = (M \setminus f) \setminus e$ $\hspace{0.1cm}$ and $\hspace{0.1cm}$ $(M/e)/f ... | https://mathoverflow.net/users/125655 | Interchanging deletion and contraction in matroids | Uniform matroids are the only matroids with this property.
Assume we have a matroid $M$ of rank $k$ on a ground set $E$ with this property. Consider a base $B$ of the matroid and any $e \in B$, $f \not\in B$. If we cannot find such $e$ and $f$, then $M$ must be a uniform matroid. Now, $B \setminus e$ is a base of $(M... | 5 | https://mathoverflow.net/users/51668 | 302840 | 132,613 |
https://mathoverflow.net/questions/302734 | 3 | Internet searches haven't helped. Can you?
Let $\, f = \prod\_{i=1}^n (a\_i x + b\_i y + c\_i).$
Is each component of $\, f^{-1}(1)$ a convex curve?
I expect so, and can prove it for $n=2,$ but I'm hopeless beyond that. Thanks!
| https://mathoverflow.net/users/37002 | Is the level set of a product of affine linear functions comprised of convex curves? | fedja's solution: where $f>0$, $\log(f)$ is defined, with nonpositive Hessian. Thus $\log(f)$ is concave, hence has convex superlevel sets.
| 1 | https://mathoverflow.net/users/37002 | 302851 | 132,615 |
https://mathoverflow.net/questions/302859 | 0 | For any group $G$, we let $\text{Sub}(G)$ be the complete lattice of subgroups of $G$. Let $\text{Sym}(\omega)$ be the group of all bijections $f:\omega\to\omega$.
What is an element of $U\in\text{Sub}(\text{Sym}(\omega))$ such that for all $V\in \text{Sub}(\text{Sym}(\omega))$ such that the subgroup generated by $U... | https://mathoverflow.net/users/8628 | Complements in $\text{Sub}(\text{Sym}(\omega))$ | I understand the question as: "what is a subgroup $U\le G$ such that ($\*$) for every $V\le G$ such that $V\cap U=\{1\}$ we have $\langle U,V\rangle\neq G$"? I also understand "what is a subgroup $U$" as an awkward way to ask about the existence of such a subgroup.
Then the subgroup of finitely supported permutations... | 1 | https://mathoverflow.net/users/14094 | 302861 | 132,619 |
https://mathoverflow.net/questions/302784 | 7 | Consider the Markov chain $(\theta\_n, \phi\_n)$ on $S^1 \times S^1$ constructed in the following way. For $\xi\_n$ a sequence of i.i.d. normal random variables and $\kappa > 0$ a fixed number, we set
$$
\theta\_{n+1} = \theta\_n + \kappa \xi\_n\;,\qquad \phi\_{n+1} = \arg((3+2\sqrt 2)\cos \phi\_n,\sin \phi\_n) + \thet... | https://mathoverflow.net/users/38566 | An interesting Markov chain with uniform marginals | Since I was requested to elaborate, here goes. First, let's look at the automorphism of the unit circle induced by this mapping (written in the *least* revealing way). With $z=e^{it}$, as usual, we have $2\cos t=z+z^{-1}, 2i\sin t=z-z^{-1}$, so for positive $3+\sqrt 2$ (I absolutely loved this red herring) the directio... | 11 | https://mathoverflow.net/users/1131 | 302863 | 132,620 |
https://mathoverflow.net/questions/302864 | 2 | In an abelian category, each subobject $A \stackrel{f}{\to} X$ individuate an equivalence relation $R(f) \to X^2$ which is given by the equalizer of $$X^2 \rightrightarrows X \to \text{Coker}(f). $$
In that case, this correspondence $\text{Sub}(X) \to \text{EqRel}(X)$ is even injective.
I am wondering if one can a... | https://mathoverflow.net/users/104432 | An immersion of $\text{Sub}(X) \to \text{EqRel}(X)$ in a Malcev category | This can't be true, even in the category of groups. Indeed, the category of groups is exact, so every equivalence relation is the kernel pair of some regular epimorphism, and these coincide with surjective homomorphism; in particular, if $X$ is a simple group, its only quotients are $X$ itself and the trivial group, so... | 2 | https://mathoverflow.net/users/111486 | 302873 | 132,624 |
https://mathoverflow.net/questions/302783 | 4 | $\newcommand{\End}{\operatorname{End}}$
Let $V$ be a $d$-dimensional **real** vector space. ($d \ge 3$). Fix an **odd** $2 \le k \le d-1$. Define
$H\_{>k}=\{ A \in \End(V) \mid \operatorname{rank}(A) > k
\}$. $H\_{>k}$ is an open submanifold of $\End(V)$.
We also define, for a given number $s$, the open submanifol... | https://mathoverflow.net/users/46290 | Is the map $A \to \bigwedge^{k}A $ from matrices above rank $k$ proper? | The answer is negative:
Let $d=4,k=2$: Let $A\_n=\text{diag}(n,\frac{1}{n},\frac{1}{n},\frac{1}{n}) \in \text{End}( V)$. (We choose a basis and let $A\_n$ be diagonal w.r.t this basis).
Then $\bigwedge^2 A\_n =\text{diag}(1,1,1,\frac{1}{n^2},\frac{1}{n^2},\frac{1}{n^2}) \in \text{End}( \bigwedge^2 V)$ converges to ... | 2 | https://mathoverflow.net/users/46290 | 302887 | 132,628 |
https://mathoverflow.net/questions/302876 | 54 | When modular forms are usually introduced, it is by: "We have the standard action of $SL(2,\mathbb Z)$ on the upper half-plane, so let us study functions which are (almost) invariant under such transformations".
But modular forms were discovered in the 19th century, before group theory was available. Therefore, I'd ... | https://mathoverflow.net/users/114143 | How were modular forms discovered? | You don't need the *language* of group theory to talk about some aspects of groups. For example, number theorists going back to Fermat were studying the group of units mod $m$ (including things like the order of a unit mod $m$) and geometers were studying groups of motions in space long before anyone defined a "group".... | 58 | https://mathoverflow.net/users/3272 | 302890 | 132,631 |
https://mathoverflow.net/questions/302746 | 8 | Let $\sigma(n)$ denote the sum of divisors of $n$, that is,
$$
\sigma(n) = \sum\_{d | n} d.
$$
It is known that $\sigma$ can have values as large as order $n \log \log n$. However, obviously the sum is reduced when we restrict it to divisors which are smaller than some threshold $D$.
Question 1: How large can $D$ be ... | https://mathoverflow.net/users/46852 | Sum of divisors below threshold | Put
$$
B= C \frac{\log (10\sigma(n)/n)}{\log \log (10 \sigma(n)/n)}
$$
for a suitably large positive constant $C$. Then I claim that the desired inequality holds with
$$
D = \frac{n}{(\log n)^B},
$$
for all large $n$. Since $\sigma(n)/n \ll \log \log n$, it follows that one may always take
$$
D = n \exp\Big( -C ... | 11 | https://mathoverflow.net/users/38624 | 302899 | 132,632 |
https://mathoverflow.net/questions/302901 | 1 | Let $X$ be a metric continuum (compact + connected) which is the one-to-one continuous image of the interval $[0,\infty)$. Such an $X$ is called a linear continuum.
It seems like $X$ should be chainable (as defined in the 2nd paragraph [here](http://www.auburn.edu/~mincpio/pm/newextension.pdf)). Equivalently, for eve... | https://mathoverflow.net/users/91061 | Continuum image of line is chainable? | The circle is a linear continuum according to the definition provided in the question. But the circle is not chainable, by the Borsuk-Ulam Theorem.
| 2 | https://mathoverflow.net/users/61536 | 302908 | 132,635 |
https://mathoverflow.net/questions/302897 | 4 | Football (soccer) is arguably one of the most unpredictable sports. Countless variables play a role in determining the outcome of a certain football match. Due to the high complexity of the entire set of involved parameters, the results of football matches may seem pretty random at the first glance. However, there are ... | https://mathoverflow.net/users/82843 | On Mathematical Foundations of Football | Typing "mathematics of soccer" into the internet led to J.A. Tenreiro Machado, António M.Lopes, On the mathematical modeling of soccer dynamics, Communications in Nonlinear Science and Numerical Simulation, Volume 53, December 2017, Pages 142-153:
Abstract
This paper addresses the modeling and dynamical analysis o... | 6 | https://mathoverflow.net/users/3684 | 302914 | 132,637 |
https://mathoverflow.net/questions/302900 | 1 | Fix two $2^t$ length vector of form $p=\begin{bmatrix}u\_1&v\_1\end{bmatrix}\otimes\dots\otimes\begin{bmatrix}u\_t&v\_t\end{bmatrix}$ and $r=\begin{bmatrix}w\_1&z\_1\end{bmatrix}\otimes\dots\otimes\begin{bmatrix}w\_t&z\_t\end{bmatrix}$ where each $u\_i,v\_j,w\_{i'},z\_{j'}$ is a distinct prime.
Consider $2^r$ length ... | https://mathoverflow.net/users/10035 | On ranks of matrices with tensor structure | Clearly it can't exceed $2^r$ since each row is a linear function of the vector $q(...)$. To show that this upper bound is reached, we just have to show that we can take the $x,y$ such that
$$
Q =
\begin{bmatrix}
q^{[1]}\\ \vdots \\
q^{[2^r]}
\end{bmatrix}
$$
has full rank.
Take $[x\_i, y\_i] = [3^r,1]$ or $[1,3^r]... | 1 | https://mathoverflow.net/users/1898 | 302916 | 132,638 |
https://mathoverflow.net/questions/302919 | 3 | I am trying to obtain a PDF of the thesis of Zygmunt Janiszewski, which is titled ["Sur les continus irréductibles entre deux points"](https://books.google.com/books/about/Sur_les_continus_irr%C3%A9ductibles_entre_de.html?id=GZdiGQAACAAJ).
I have learned that this is also contained in a later publication ["Oeuvres ch... | https://mathoverflow.net/users/91061 | Need help obtaining the thesis of Zygmunt Janiszewski from 1911 | There is one Polish publication of Zygmunt Janiszewski available [online](http://matwbn.icm.edu.pl/ksiazki/pmf/pmf26/pmf2612.pdf), from which one learns that the paper in question was also printed in
Journal de l'École Polytechnique in 1912, and this is available [online](http://gallica.bnf.fr/ark:/12148/bpt6k4336477/f... | 8 | https://mathoverflow.net/users/11100 | 302920 | 132,639 |
https://mathoverflow.net/questions/302865 | 9 | Given a prime number $p$ and a primitive root $a$ modulo $p$, let
$\sigma\_{a,p}$ denote the permutation of the set $\{1, \dots, p-1\}$ which
maps $b$ to $a^b$ modulo $p$.
**Question:** Let $p$ be fixed. Does the following hold?:
* If $p$ is congruent to $1$ modulo $4$, then for precisely half of the
primitive roo... | https://mathoverflow.net/users/28104 | Sign of permutation induced by modular exponentiation | Your second guess is also correct.
At first, we write down the sign of the permutation $\sigma\_a$ as the product $\prod\_{1\leqslant i<j\leqslant p-1}\frac{a^j-a^i}{j-i}$ modulo $p$. The denominator equals $(p-2)!(p-3)!\dots 1!$, and denoting $p=2m+1$ ($m$ is odd) we write it as $m!\prod\_{j=1}^{m-1} j!(p-1-j)!=m!\... | 9 | https://mathoverflow.net/users/4312 | 302921 | 132,640 |
https://mathoverflow.net/questions/302906 | 5 | Let $\langle a, b \rangle = F\_2$ be a two-generator free group and $\hat{F\_2}$ be its profinite completion. Is there an element $c\in \hat{F\_2}$ such that $\langle a, b, c\rangle \le \hat{F\_2}$ is isomorphic to the 3-generator abstract free group $F\_3$?
I posted the same question on Math StackExchange([link](htt... | https://mathoverflow.net/users/125681 | Dense abstract free subgroups in a free profinite group | Yes. Here's a recipe to get such a group. Find a profinite group $K$ with three elements $a',b',c'$ such that $\langle a',b'\rangle$ is dense and $(a',b',c')$ is a free family. If so, we can "pull them back" to $\hat{F\_2}$. Namely considering the unique homomorphism $\hat{F\_2}\to K$ mapping $a\mapsto a'$, $b\mapsto b... | 3 | https://mathoverflow.net/users/14094 | 302922 | 132,641 |
https://mathoverflow.net/questions/302221 | 3 | I would like to prove (or find a counterexample to) the following statement:
Let $X$ be a complex analytic scheme and let $\pi: Y \to X$ be its universal cover. Let $F$ be a coherent sheaf on $X$ and suppose that $L\otimes F\cong F $ for all $L\in \operatorname{Pic}^0(X)$. Then there exists a sheaf $G$ on $Y$, essent... | https://mathoverflow.net/users/9617 | Criteria for a coherent sheaf pushing forward from the universal cover | Let $E$ be the elliptic curve. Let $E\_1$ and $E\_2$ be two different double covers of $E$, with $E = E\_1 /x\_1$ and $E=E\_2/x\_2$ for two-torsion points $x\_1,x\_2$.
Let $M$ be the minimal resolution of singularities of $ E\_1 \times E\_2 /\langle (a,b) \to (-a,-b), (a,b) \to (a+ x\_1, x\_2-b) \rangle $.
Then $M... | 2 | https://mathoverflow.net/users/18060 | 302932 | 132,642 |
https://mathoverflow.net/questions/301336 | 3 | Nakajima & Yoshioka [1] showed that
\begin{equation}
F^{inst}(\epsilon\_1,\epsilon\_2,\mathbf{a},\mathbf{q}) = \sum\_{n = 1}^\infty \mathbf{q}^nF^{inst}\_n(\epsilon\_1,\epsilon\_2,\mathbf{a}) := \epsilon\_1\epsilon\_2\log Z^{inst}(\epsilon\_1,\epsilon\_2,\mathbf{a},\mathbf{q})
\end{equation}
is regular at $\epsilon\_1... | https://mathoverflow.net/users/65854 | Nekrasov Partition Function: $F^{inst}(\epsilon_1,\epsilon_2,\mathbf{a},\mathbf{q})$ analytic at $\epsilon_1 = \epsilon_2 = 0$? | We meant that each coefficient of $\mathfrak q^n$ in $\varepsilon\_1 \varepsilon\_2 \log Z^{\mathrm{inst}}(\varepsilon\_1,\varepsilon\_2, \vec{a}, \mathfrak q)$ is regular at $\varepsilon\_1 = \varepsilon\_2 = 0$.
| 2 | https://mathoverflow.net/users/3837 | 302935 | 132,643 |
https://mathoverflow.net/questions/302936 | 2 | In an enumeration problem the sequence of number of Dyck paths semilength n having no UUDD's starting at level 0 with generating function $$\frac{2}{(1+2z^2+\sqrt{1-4z})}$$ showed up, see also <https://oeis.org/A114487>.
It is not really important but I wonder whether a nice explicit formula exists for this sequence.
... | https://mathoverflow.net/users/61949 | Explicit formula for a generating function | The coefficient of $z^n$ is
$$\sum\_{0\le k\le n/2} (-1)^k \frac{k+1}{2n-3k+1}\binom{2n-3k+1}{n-2k}.$$
To see this, let $C(z)$ be the Catalan number generating function,
$$C(z) = \frac{1-\sqrt{1-4z}}{2z}=\frac{2}{1+\sqrt{1-4z}}.$$
Then
$$
\frac{2}{1+2z^2+\sqrt{1-4z}} = \frac{C(z)}{1+z^2 C(z)}.
$$
It is well known th... | 9 | https://mathoverflow.net/users/10744 | 302947 | 132,648 |
https://mathoverflow.net/questions/302823 | 2 | Let $G=(V,E)$ be a simple, undirected and connected graph. We say that $S\subseteq V$ is a *cutting set* if $S\neq V$ and the induced subgraph on $V\setminus S$ is not connected any more.
If $S \subseteq V$ is a cutting set of $G$, is there a cutting set $S\_0\subseteq S$ of $G$ such that for all $x\in S\_0$ the set ... | https://mathoverflow.net/users/8628 | Minimal cutting sets in connected graphs | No. Here is a counterexample.
Let $V = \{ x\_n,y\_n : n \in \mathbb N \}$. Put $x\_n E y\_m$ and when $n \leq m$, and put an edge between any two $y\_n$'s. If $A \subseteq \mathbb N$ is cofinite, then the induced subgraph on $V \setminus \{ y\_n : n \in A \}$ is not connected, since there are no edges with endpoint $... | 2 | https://mathoverflow.net/users/11145 | 302957 | 132,650 |
https://mathoverflow.net/questions/302933 | 24 | We call an integer $k\geq 1$ *good* if for all $q\in\mathbb{Q}$ there are $a\_1,\ldots, a\_k\in \mathbb{Q}$ such that $$q = \prod\_{i=1}^k a\_i \cdot\big(\sum\_{i=1}^k a\_i\big).$$
Euler [showed](https://mathoverflow.net/a/268336/8628) that $k=3$ is good.
Is the set of good positive integers infinite?
| https://mathoverflow.net/users/8628 | Question on a generalisation of a theorem by Euler | I suspect that $k = 4$ is good, but am not sure how to prove it. However, every positive integer $k \geq 5$ is good. This follows from the fact (see [the proof of Theorem 1 from this preprint](https://arxiv.org/pdf/1607.01957.pdf)) that for any rational number $x$, there are rational numbers $a$, $b$, $c$, $d$ so that ... | 33 | https://mathoverflow.net/users/48142 | 302958 | 132,651 |
https://mathoverflow.net/questions/302898 | 22 | Let $F$ be a one-dimensional local field. Then Langlands conjectures for $GL\_n(F)$ say (among other things) that there is a unique bijection between the set of equivalence classes of irreducible admissible representations of $GL\_n(F)$ and the set of equivalence classes of continuous Frobenius semisimple complex $n$-d... | https://mathoverflow.net/users/nan | Langlands correspondence for higher local fields? | The Langlands correspondence for higher local fields is still at an early stage of development. I haven't really kept up with it, but here's some key points.
As the question stated, and Loren commented, the starting point is the $GL\_1$ case, which is class field theory for higher local fields. Local class field theo... | 11 | https://mathoverflow.net/users/3545 | 302959 | 132,652 |
https://mathoverflow.net/questions/302461 | 16 | One of the most interesting questions in Mathematics concerns the Mordell-Weil rank of the group of rational points on elliptic curves $E/\mathbb{Q}$, namely whether this quantity is bounded as one varies over all elliptic curves defined over the rationals (or some other number field $K$). It is known that there are in... | https://mathoverflow.net/users/10898 | Largest rank assumed by infinitely many elliptic curves | As suggested by @JoeSilverman, I am turning my comment into an answer.
According to the table maintained by Dujella at
<http://web.math.pmf.unizg.hr/~duje/tors/generic.html>,
the current record is **19**,
due to Noam Elkies in 2006. This is apparently obtained by a family of
elliptic curves over an elliptic curve ov... | 13 | https://mathoverflow.net/users/21146 | 302960 | 132,653 |
https://mathoverflow.net/questions/302961 | 10 | In the famous book 'Residues and duality', the author notes that one of the principal difficulties in constructing the exceptional inverse image functor $f^{!}$ is that the derived category of quasicoherent sheaves is not a local object. In order to resolve this difficulty, the author first defines it for residual comp... | https://mathoverflow.net/users/nan | Why does passage to DG categories cure non-locality of derived categories? | Just a quick answer :-)
Triangulated categories lack intrinsicness (is that even a word?), and also the 2-category of triangulated categories is extremely poorly behaved, whereas the 2-category of DG-categories behaves better under 2-categorical constructions (it admits several shapes of 2-co/limits and bi-co/limits)... | 4 | https://mathoverflow.net/users/7952 | 302963 | 132,654 |
https://mathoverflow.net/questions/302975 | 3 | My limited knowledge on hyperbolic geometry suggests me that the following proposition should be true (please correct me if I'm wrong):
**Proposition.** The convex core of a complete hyperbolic surface has finite area if and only if the surface is of finite type.
>
>
> >
> > **Question.** What is the best liter... | https://mathoverflow.net/users/17294 | Reference request: geometric finiteness of Fuchsian groups | Yes, this is a theorem of Siegel, and a reference with simple proof is
Theorem XI.12 in
M. Tsuji, Potential theory in modern function theory, Maruzen, Tokyo 1959 (there is an AMS Chelsea reprint, 1975).
| 6 | https://mathoverflow.net/users/25510 | 302985 | 132,661 |
https://mathoverflow.net/questions/302913 | 3 | This question is motivated by this [MO-problem](https://mathoverflow.net/questions/302724/automorphisms-of-erd%C3%B6s-spaces/302839#302839) asking if the Erdős spaces $\mathfrak E$ and $\mathfrak E\_c$ admit a self-homeomorphism with dense orbits of points.
The affirmative answer would follow from the affirmative ans... | https://mathoverflow.net/users/61536 | Has the Erdős space the structure of a monothetic topological group? | Let me try to supply some useful constructions, with a hope that they would help to solve the problems proposed by Taras (possibly, by an analogy rather than directly).
>
> $\mathbf{\ell^2}$**-product of arbitrary metric spaces**
>
>
>
The following construction must be classical. Let $\ (X\_n\ d\_n)\ $ be
met... | 2 | https://mathoverflow.net/users/110389 | 302987 | 132,662 |
https://mathoverflow.net/questions/302979 | 6 | In Ginzburg's notes [Lectures on Noncommutative Geometry](https://arxiv.org/pdf/math/0506603.pdf), he claim that the path algebra of a quiver is formally smooth (See Section 19.2).
I have two questions. First, how to show this claim and which criterion of formal smoothness we are going to check? Secondly, does this ... | https://mathoverflow.net/users/91245 | Path algebras are formally smooth | Assuming standard results on lifting idempotents, it's not hard to check that a path algebra $kQ$ satisfies the lifting property that Ginzburg uses to define formal smoothness in Definition 19.1.1.
If $B$ is an algebra with a nilpotent ideal $I$, we need to check that every homomorphism $\varphi:kQ\to B/I$ lifts to a... | 8 | https://mathoverflow.net/users/22989 | 302999 | 132,665 |
https://mathoverflow.net/questions/302945 | 3 | I am not really a professional, but this question has been asked on [Math.SE](https://math.stackexchange.com/q/2787474/169789) already and in spite of a bounty it was not answered.
That made me decide to give it a try here, and I hope that is acceptable.
---
It is clear to me that I can build a suitable underl... | https://mathoverflow.net/users/125223 | How to construct a Poisson process not based on Lebesgue measure? | Another construction, which does not use the structure of $\mathbb{R}$ and works for a sigma-finite measure $\nu$ on arbitrary measurable space $\Omega$, is as follows: let $\Omega=\bigcup\_i E\_i$ with $\nu (E\_i)<\infty$ and $E\_i\cap E\_j=\emptyset$ for $i\neq j$. For each $i$ independently, sample a Poisson random ... | 3 | https://mathoverflow.net/users/56624 | 303004 | 132,666 |
https://mathoverflow.net/questions/303007 | 4 | Let $X$ be a topological space and $\mathcal{E}$ be a topological base for $X$. Let us denote Bor$(\mathcal{E})$, by the smallest $\sigma$-algebra containing $\mathcal{E}$.
Q. Let $O$ be an open set contained in Bor$(\mathcal{E})$. Does there exists a sequence of
open sets $\{O\_n\}\subseteq \mathcal{E}$ with $O=\c... | https://mathoverflow.net/users/84390 | Nice arrangement of open sets in $\sigma$-algebras | Not necessarily. Let $X$ be an uncountable set with the discrete topology, and let $\mathcal{E}$ be the collection of singletons, which is a base for the topology, since every set is a union of singletons.
The $\sigma$-algebra generated by the singletons, what you call $\text{Bor}(\mathcal{E})$, is the algebra consi... | 6 | https://mathoverflow.net/users/1946 | 303009 | 132,667 |
https://mathoverflow.net/questions/302930 | 6 | Let $\mathbb{F}$ be a field and $V$ an $\mathbb{F}$-vector space. Let $\operatorname{T}\in\mathrm{End}(V)$ be an $\mathbb{F}$-linear operator. It is well known that if $\dim V<\infty$ then $\operatorname{T}$ has a Jordan canonical form, i.e., it is similar to the direct sum of a number of Jordan blocks. If a certain ei... | https://mathoverflow.net/users/89313 | Jordan form on an invariant vector subspace | The question can be restated as follows: let $F$ be a field $T$ be an endomorphism of a vector space over $F$. Suppose that it decomposes as a direct sum of finite-dimensional $T$-stable subspaces. Does the same property hold for every $T$-stable subspace?
In turn, it can be formalized as follows, in a pure language ... | 8 | https://mathoverflow.net/users/14094 | 303011 | 132,668 |
https://mathoverflow.net/questions/303013 | 7 |
>
> I am looking for a proof of the inequality as follow:
>
>
>
Let $n$ be an integer number $n \ge 2$ and $x\_1, \cdots, x\_n$ and $y\_1,\cdots, y\_n$ are nonegative real numbers such that $(x\_1,\cdots, x\_n)$ [majorizes](https://en.wikipedia.org/wiki/Majorization) $(y\_1,\cdots, y\_n)$; Let $0 \leq a\_1, a\_2... | https://mathoverflow.net/users/122662 | A Muirhead Like Inequality | $\newcommand{\al}{\alpha}
\newcommand{\be}{\beta}
\newcommand{\de}{\delta}
\newcommand{\De}{\Delta}
\newcommand{\ep}{\varepsilon}
\newcommand{\ga}{\gamma}
\newcommand{\Ga}{\Gamma}
\newcommand{\la}{\lambda}
\newcommand{\si}{\sigma}
\newcommand{\Si}{\Sigma}
\newcommand{\thh}{\theta}
\newcommand{\om}{\omega}
\newcommand{\... | 8 | https://mathoverflow.net/users/36721 | 303022 | 132,673 |
https://mathoverflow.net/questions/303014 | 3 | Let $(X,\tau)$ be a topological space. For a given topological base $\mathcal{E}$ for $\tau$, let us denote Bor$(\mathcal{E})$, by the smallest $\sigma$-algebra containing $\mathcal{E}$.
Q. Assume that Bor$(\mathcal{E})=$Bor$(\tau)$. Let $O$ be an open in $X$. Does there exists a sequence of
open sets $\{O\_n\}\sub... | https://mathoverflow.net/users/84390 | Nice representation of open sets in $\sigma$-algebras in certain circumstances | At least under MA+$\neg$CH the answer is negative. It is known that under MA+$\neg$CH the real line contains an uncountable set A such that every subset of A is Borel in A (moreover, it is of type $F\_\sigma$ and $G\_\delta$).
Now consider the product $X=A\times\mathbb Q$ and let $\mu$ be the (metrizable separable) ... | 3 | https://mathoverflow.net/users/61536 | 303024 | 132,674 |
https://mathoverflow.net/questions/268812 | 6 | Let $K$ be a field equipped with a non-Archimedean absolute value, let $\Gamma$ be a Schottky group in $PGL\_2(K)$, and let $X\_\Gamma$ be the associated Mumford curve, which is a proper smooth rigid analytic space over $K$, but also a proper algebraic curve (due to Mumford's theorem). Let $J\_\Gamma$ be the Jacobian v... | https://mathoverflow.net/users/2234 | Abel-Jacobi map for Mumford curves analytically | This is done in Manin and Drinfeld's "Periods of p-adic Schottky groups." Journal für die reine und angewandte Mathematik 0262\_0263 (1973): 239-247. You will also find it in Gerritzen and van der Put's book on Mumford curves.
| 6 | https://mathoverflow.net/users/4069 | 303030 | 132,677 |
https://mathoverflow.net/questions/303020 | 2 | Let $G$ be a (discrete) torsion free group with identity $e$. Recall that for an element $\alpha=\sum a\_gg$ in $\mathbb C[G]$ (complex functions on $G$ with compact support), $\alpha^\*$ is defined to be $\sum\bar{a\_g}g^{-1}$ and for $\beta=\sum b\_gg\in\mathbb C[G]$, we have the (convolution) product $\alpha\beta:=\... | https://mathoverflow.net/users/84700 | Does this sequence contain a nonnegative number? | No in general. If $G$ is an infinite cyclic group generated by $g$, and $\beta=1 +g+1/g$, the sum of coefficients of $\beta^{2 n}$ equals $9^n$, and the coefficient of 1 is $9^n$ times the probability that a symmetric random walk with steps $0,\pm 1$ ends up in the origin after $2n$ steps. This probability tends to 0. ... | 4 | https://mathoverflow.net/users/4312 | 303037 | 132,679 |
https://mathoverflow.net/questions/303046 | 5 | Let $A$ be a finite dimensional algebra over an algebraically closed field $K$. The Auslander-Reiten quiver $\Gamma\_A$ of $A$ is a means of presenting the category of finitely generated right $A$-modules. The Auslander-Reiten quiver is a locally finite quiver whose vertices are indecomposable modules (up to isomorphis... | https://mathoverflow.net/users/95742 | Given a representation-infinite algebra, when is every AR component infinite? | If $A$ is connected and has infinite representation type, then every component of its Auslander-Reiten quiver is infinite. See, for example, Theorem 5.4 in Assem, Simson and Skowronski’s *Elements of the Representation Theory of Associative Algebras*, Volume 1.
The example you give is not a complete component. It con... | 7 | https://mathoverflow.net/users/22989 | 303056 | 132,688 |
https://mathoverflow.net/questions/303062 | 2 |
>
> Is it consistent that there exists a nonzero atomless finite measure
> on some $\sigma$-algebra on a cardinal $\kappa$ satisfying $\kappa<\mathfrak{c}$? Can
> there be such a measure on $\omega\_1$ when $\omega\_1<\mathfrak{c}$?
>
>
>
I would generally like to know where one can find related results.
My ... | https://mathoverflow.net/users/35357 | How small can a set admitting a nonatomic finite measure be? | If $X \subseteq [0, 1]$ is a non Lebesgue null set of reals, then $B \cap X \mapsto \mu(B)$ (where $B$ is Borel and $\mu$ is Lebesgue measure) is an atomless measure on $X$. Conversely, if $m$ is an atomless probability measure on $\kappa$, then there is an inverse measure preserving map $f:\kappa \to [0, 1]$ - For eve... | 4 | https://mathoverflow.net/users/125750 | 303073 | 132,693 |
https://mathoverflow.net/questions/302923 | 5 | This is a follow-up to this earlier question on elliptic curves: [Largest rank assumed by infinitely many elliptic curves](https://mathoverflow.net/questions/302461/largest-rank-assumed-by-infinitely-many-elliptic-curves)
Let $g \geq 1$ be an integer. For each $g$, what is known about the largest positive integer $r(... | https://mathoverflow.net/users/10898 | Largest ranks achieved by abelian varieties of fixed dimension | Here is a construction (as far as I know, originally due to Mestre)
that gives a lower bound $r(g) \ge 4g + 5$.
Write
$$ \prod\_{i=1}^{4g+6} (x - t\_i) = h(x)^2 - f(x)\,, $$
where the $t\_i$ are independent indeterminates, $h$ is monic of degree
$2g+3$ and $f$ has degree $2g+2$. Then there are $4g+6$ points
$P\_i = (... | 9 | https://mathoverflow.net/users/21146 | 303078 | 132,696 |
https://mathoverflow.net/questions/303076 | 1 | Suppose $T\_{n}$ converging $T$ in vN algebra $M$ in weak operator topology, can we conclude $||T\_{n}||$ is uniformly bounded? Another question if a linear functional $\varphi$ is continuous in unit ball in $M$ in weak operator topology does it imply that $\varphi$ is normal that means in the predual?
| https://mathoverflow.net/users/125816 | On predual of von Neumann algebra | Suppose $M$ acts on a Hilbert space $H$, for a fixed $x\in H$, the set $\{\langle T\_nx,y\rangle: n\in\mathbb N\}$ is bounded for all $y\in H$, so $\{T\_nx: n\in\mathbb N\}$ is bounded by the principle of uniform boundedness, and again using principle of uniform boundedness we conclude that $\{T\_n: n\in\mathbb N\}$ is... | 3 | https://mathoverflow.net/users/84700 | 303083 | 132,697 |
https://mathoverflow.net/questions/303001 | 0 | Several places in "Optimal Stopping and Free-Boundary Problems" Peskir and Shiryaev make the assumption that a (Markov) process $X = (X\_t)\_{t\geq 0}$ has sample
paths which are right continuous and left continuous over stopping times.
What is the reason for making the distinction between continuity with respect t... | https://mathoverflow.net/users/124127 | Continuity w.r.t time vs Continuity w.r.t. stopping times | A process which is right continuous can still be left continuous *at a fixed time* almost surely. For example, a Poisson process $N=(N\_t)\_{t \ge 0}$ is right continuous, and for any (deterministic) time $t$ it holds that $\lim\_{s \uparrow t}N\_s=N\_t$ a.s., simply because $\mathbb{P}(N \text{ jumps at time } t)=0$. ... | 0 | https://mathoverflow.net/users/44169 | 303085 | 132,699 |
https://mathoverflow.net/questions/303041 | 13 | I remember seeing somewhere that for every connected compact set $\Omega$ in $\mathbb{R}^2$ with piecewise $C^1$ boundary we have
$$A(\Omega\_r)\leq A(\Omega)+L(\partial \Omega)r+ \pi r^2,$$
where
$$\Omega\_r=\{x\in \mathbb{R}^2: d(x,\Omega)\leq r\},$$
$A$ denotes the area and $L$ the length. I tried to find a referen... | https://mathoverflow.net/users/108630 | Steiner's inequality reference request | In the first note to section 4.2 of
*Schneider, Rolf*, Convex bodies: the Brunn-Minkowski theory, Encyclopedia of Mathematics and Its Applications. 44. Cambridge: Cambridge University Press. xiii, 490 p. (1993). [ZBL0798.52001](https://zbmath.org/?q=an:0798.52001).
Rolf Schneider cites several articles with general... | 9 | https://mathoverflow.net/users/98590 | 303090 | 132,703 |
https://mathoverflow.net/questions/301982 | 2 | Let $K$ be a field, preferably a function field of a variety $X$ over $\overline{\mathbb{F}}\_p$. I am looking for an answer or existing literature on the following question:
>
> What is known about the structure of the maximal pro-$\ell$-quotient of the absolute Galois group of $K$? Is it finitely generated? Torsi... | https://mathoverflow.net/users/39055 | maximal pro-l-quotients of absolute Galois groups | I assume your field has characteristic $p>0$.
Then the maximal pro-$l$ quotient of the absolute Galois group is torsion-free. Indeed, by results of E. Becker, *Euklidische Korper und euklidische Hullen von Korpern*, J. reine angew. Math. 278-269 (1974), 41-52. He shows that torsion elements in such quotients can only b... | 2 | https://mathoverflow.net/users/101929 | 303103 | 132,706 |
https://mathoverflow.net/questions/303005 | 6 | I call a $\mathbb{Z}$-graded (non-commutative, associative, unital) ring $A$ (left) graded-Noetherian if every homogeneous (left) ideal is finitely generated, and (left) Noetherian if it is (left) Noetherian as a ring.
In the commutative setting, I think I can prove that a graded-Noetherian ring is Noetherian. This b... | https://mathoverflow.net/users/111049 | Is every (left) graded-Noetherian graded ring (left) Noetherian? | The answer is **yes**, by Corollary 2.2 in C. Nastasescu, F. Van Oystaeyen, *Graded rings with finiteness conditions II,* Comm. Algebra 13 (1985), 605-618.
More generally, we have the following.
Let $G$ be a commutative group. Every epimorphism $\psi\colon G\twoheadrightarrow H$ of commutative groups gives rise to ... | 6 | https://mathoverflow.net/users/11025 | 303108 | 132,707 |
https://mathoverflow.net/questions/303113 | 8 | Does there exist a Riemannian metric on the $n$-sphere ($n > 2$) such that at each point some (but not every) sectional curvature is negative?
For $n=2$ it is easily seen that such a metric cannot exist.
| https://mathoverflow.net/users/48208 | Riemannian metric on the sphere with at least one negative sectional curvature at every point | Joachim Lohkamp has shown ([Annals of Mathematics, 1994](https://www.jstor.org/stable/2118620)) that any smooth manifold of dimension greater than two admits a metric of negative Ricci curvature. So it seems that the answer to your question is yes.
| 17 | https://mathoverflow.net/users/nan | 303116 | 132,709 |
https://mathoverflow.net/questions/303130 | 5 | This question is related to [How to make a sandwich from just one piece of bread?](https://mathoverflow.net/questions/262919), asked on Feb 23 '17 by erz, and it goes as follows:
>
> **Question.** If a convex closed and bounded region $C$ in the plane $\mathbb{R}^2$ can be cut along some straight line into two con... | https://mathoverflow.net/users/36904 | Cutting a convex body into two congruent pieces | Here is a counter-example (in the complex plane $\mathbb C=\mathbb R^2.)\ $ Let
$$\ P\ :=
\ \{ (x\ y)\in\mathbb C : 0\le x\le 1\quad\&\quad 0\le y\le 1-x^2\} $$
Then,
$$ C\,\ :=\,\ P\,\cup\, i\!\cdot\! P $$
The imaginary line is the requested cut.
| 7 | https://mathoverflow.net/users/110389 | 303133 | 132,716 |
https://mathoverflow.net/questions/302990 | 17 | [Bing gave](https://www.jstor.org/stable/1970322) a classical example of spaces $X, Y, Z$ such that $X \times Y = Z$, where $X$ and $Z$ are manifolds but $Y$ isn't. The space $Z$ in his example has dimension four. Is it known if this is best possible? In other words, if $X \times Y =Z$ where $X$ is a manifold and $Z$ i... | https://mathoverflow.net/users/110965 | Lowest Dimension for Counterexample in Topological Manifold Factorization | As asked Dusan Repovs (who is an expert in the theory of topological manifolds), and he sent me the following answer:
This is indeed best possible result, since whenever a product of two spaces is a topological manifold, both factors must be generalized manifolds - which in dimensions below 3 are topological manifold... | 9 | https://mathoverflow.net/users/61536 | 303145 | 132,719 |
https://mathoverflow.net/questions/303134 | 8 | In algebraic geometry, we are frequently interested in parametrizing geometric objects. Formally, parametrization of geometric objects having some property can be viewed as a functor $F:Sch\rightarrow Set$ from the category of schemes to the category of sets (which assigns to a scheme the set of families of geometric o... | https://mathoverflow.net/users/nan | Moduli 'space' of stacks? | Such a moduli problem for stacks is expected to be a $2$-stack.
For example, consider the stack of line bundles on $X$, whose objects are parameterized by $H^1(X, \mathbb{G}\_m)$. This is a (trivial) example of a $\mathbb{G}\_m$-gerbe on $X$; these in turn are parameterized by $H^2(X, \mathbb{G}\_m)$. The collection ... | 10 | https://mathoverflow.net/users/3847 | 303147 | 132,720 |
https://mathoverflow.net/questions/303119 | 1 | Consider, as a motivating example, the multiset $\left(\mathbb{P}\_n,2\setminus\mathbf{0}\right)$ consisting of the underlying set of polynomials of order $n$ and lower, all with multiplicity $2$—expect for the polynomial identically equal to $0$, $\mathbf{0}$. (Heuristically, this is intended to be what occurs when yo... | https://mathoverflow.net/users/125775 | Is there any meaningful extension of the notion of a vector space for multisets? | It is hard to posit more than one additive identity by the usual proof:
$e+f=f$ if $e$ is an additive identity and $e+f=e$ if $f$ is an additive identity hence $e=f$ if both are.
If the underlying field is $F=\{0,1\}$ then scalar multiplication is no issue and the vector space $W=F^{n+1}$ is essentially the double... | 4 | https://mathoverflow.net/users/8008 | 303148 | 132,721 |
https://mathoverflow.net/questions/302549 | 2 | I have a large matrix $A \in \mathbb{R}^{n \times m}$ and would like to subtract a sparse matrix $B \in \mathbb{R}^{n \times m}$ with less than $c (n+m)$ non-zero entries, where $c > 0$ is a constant one is free to choose, such that the singular values of $A-B$ decay as quickly as possible.
I have no idea how to app... | https://mathoverflow.net/users/75786 | Maximise singular value decay by sparse matrix approximation | Maximizing the "decay" of the singular values could be thought of as minimizing the (numerical) rank. Hence, I believe that the original problem could be rephrased as follows:
>
> Given $\mathrm A \in \mathbb R^{m \times n}$, find a **sparse** matrix $\mathrm X \in \mathbb R^{m \times n}$ such that $\mbox{rank} (\m... | 1 | https://mathoverflow.net/users/91764 | 303150 | 132,722 |
https://mathoverflow.net/questions/303149 | 28 | What is the probability that three pairs $(a,b) $ , $(c,d) $ and $(e,f) $ of integers generate $\mathbb Z^2$? As usual the probability is the limit as $n\to \infty$ of the same probability for the $n\times n$ square. It is well known that for $\mathbb Z $ the probability of two numbers to generate is $6/\pi^2$.
| https://mathoverflow.net/users/nan | Probability of generation of ${\mathbb Z}^2$ | According to Proposition 1 in the paper
G. Maze, Gérard, J. Rosenthal, U. Wagner: [Natural density of rectangular unimodular integer matrices](http://dx.doi.org/10.1016/j.laa.2010.11.015), *Linear Algebra Appl.* **434**, No. 5 (2011), 1319-1324, [ZBL1211.15044](https://zbmath.org/?q=an:1211.15044),
the probability... | 37 | https://mathoverflow.net/users/7460 | 303153 | 132,723 |
https://mathoverflow.net/questions/303111 | 11 | At [Hamkins - Different set theories are never bi-interpretable](http://jdh.hamkins.org/different-set-theories-are-never-bi-interpretable/), it is mentioned that different set theories extending ZF are never bi-interpretable.
Where different means "not theoretically equivalent", i.e. there must be a theorem that one ... | https://mathoverflow.net/users/95347 | Can two set theories extending Z be different and yet bi-interpretable? | **UPDATE (January 30, 2022):** The *first* question was answered (in the negative) by Hamkins and Freire for $\mathrm{Z}$ (Zermelo set theory) and $\mathrm{ZF}^{-}$ ($\mathrm{ZF}$ without powerset). Their paper "Bi-interpretation in weak set theories" was recently published in the Journal of Symbolic Logic. See [here](... | 9 | https://mathoverflow.net/users/9269 | 303157 | 132,725 |
https://mathoverflow.net/questions/303117 | 0 | Consider smooth positive solutions $u\_m$ of
$$-\Delta u\_m(x) = u\_m(x)^p \quad \mbox{ in } \Omega$$ with $u\_m=0$ on $ \partial \Omega$. My interest is in obtaining some sort of global integral estimates independent of $m$. Here $p>1$ and you can assume its close to $1$. The reason I am asking about integral estimat... | https://mathoverflow.net/users/66623 | boundary integral estimates for elliptic pde | OK, let me elaborate. We assume that $\Omega$ is bounded with smooth boundary (this can be relaxed a bit, but we still need something for the naive argument below to work). Let $v$ be the first eigenfunction of the Laplacian normalized by $\int\_\Omega v=1$. Then, integrating against $v$ and transferring the Laplacian ... | 2 | https://mathoverflow.net/users/1131 | 303165 | 132,730 |
https://mathoverflow.net/questions/302891 | 9 | Constructively, my only interest in regular cardinals is in terms of the “$\Sigma$-universes” they generate. By a *$\Sigma$-universe*, I mean a collection of triples $(X,Y,f: X \to Y)$ closed under base change, composition, and isomorphism – here $X,Y$ are sets and $f: X \to Y$ is a function between them. A $\Sigma$-un... | https://mathoverflow.net/users/2362 | Is every set smaller than a regular cardinal, constructively? | I think there are enough representable $\Sigma$-universes in any regular locally cartesian closed category with disjoint coproducts and $W$-types. One can show the category of sets has $W$-types in $\mathbf{ZF}$ and even $\mathbf{IZF}$. So I don't think any form of choice or existence of regular ordinal is necessary fo... | 5 | https://mathoverflow.net/users/30790 | 303170 | 132,732 |
https://mathoverflow.net/questions/303017 | 4 | Let $(X,\omega)$ be a Riemann surface of genus $g$ with holomorphic 1-form $\omega$ (or equivalently a translation structure). Let $\Omega\mathcal{T}\_g$ be the space of holomorphic 1-forms over genus $g$ surface. The famous $\mathrm{SL}\_2\mathbb{R}$ action on $\Omega\mathcal{T}\_g$ is defined by composing each coordi... | https://mathoverflow.net/users/125697 | Teichmueller disk and the $\mathrm{SL}_2\mathbb{R}$ action | I have found an answer to my question with which I am satisfied by studying the original paper of Veech: *Teichmuller curves in moduli space, Eisenstein series adn an application to triangular billiards*. He considered two actions on $\Omega \mathcal{T}\_g$. One is the $PSL\_2(\mathbb{R})$ action from left mentioned ab... | 0 | https://mathoverflow.net/users/125697 | 303171 | 132,733 |
https://mathoverflow.net/questions/291592 | 6 | Let $G$ be a simply connected semisimple group over a perfect field $k$ (at the moment I am interested in the case $k=\mathbb R$).
Then $G$ is an inner form of a quasi-split $k$-group $G\_{\rm qs}$:
there exists a quasi-split form $G\_{\rm qs}$ of $G$ and a 1-cocycle $c\in Z^1(k,\overline{G}\_{\rm qs})$ such that
$G=\,... | https://mathoverflow.net/users/4149 | The Tits classes of simply connected simple real groups | **Question 1:** I haven't seen an explicit table in the literature of the Tits classes for simple $R$-groups.
That said, such a table can be constructed from tables in the literature. Specifically, the Tits class is determined by the Tits algebras corresponding to the minuscule dominant weights by Proposition 7 in m... | 2 | https://mathoverflow.net/users/6486 | 303176 | 132,735 |
https://mathoverflow.net/questions/303033 | 8 | I am looking (for $n,k\in{\mathbb Z}$) for a presentation (in the best of all worlds concretely, as a list of relators) for the group ${\rm SL}\_n(R)$ for $R={\mathbb Z}[\frac{1}{k}]=\{\frac{a}{k^l}\mid a\in {\mathbb Z}\}$.
A search in MathSciNet found a paper of Behr and Mennicke(A presentation of the groups PSL(2,p... | https://mathoverflow.net/users/59303 | Presentation of special linear group over localizations of the integers | The question appears to be rather difficult. I'll give some literature references and discuss how it can be approached, and where the difficulties lie.
First, the results mentioned by Luc Guyot in the comments can also be found in Section II.1.4 of Serre's book "Trees". More precisely, there is an amalgam decompositi... | 4 | https://mathoverflow.net/users/50846 | 303177 | 132,736 |
https://mathoverflow.net/questions/303172 | 3 | I am trying to remember a result stating that under certain assumptions,
given a transitive smooth action of a (compact?) Lie group on a smooth manifold, also the action of the semisimple factor is transitive. It could be due to Borel, but I cannot remember or find the reference.
Does anybody know?
Edit: the man... | https://mathoverflow.net/users/15155 | The role of semisimple factors in transitive group actions on manifolds | Yes, it's true: whenever $G$ is a compact Lie group and $X$ a simply connected (topological) manifold, and $G$ acts transitively continuously on $X$, then $G'=[G^0,G^0]$ acts transitively.
Indeed, one can write $X=G/H$. The $G^0$-orbits being open, $G^0$ acts transitively, so we can suppose that $G$ is connected. We ... | 4 | https://mathoverflow.net/users/14094 | 303178 | 132,737 |
https://mathoverflow.net/questions/299942 | 3 | Let $m\ge 2$, and let $G={\rm SO}^\*(4m)$ denote the "quaternionic" real form of the special orthogonal group ${\rm SO}(4m,\mathbb C)$ of type ${\sf D}\_{2m}$.
Let $\tau\in{\rm Aut}\_{\Bbb R}(G)$ be a *real* automorphism of $G$, that is,
an automorphism defined over $\Bbb R$.
My Galois-cohomological calculations sugge... | https://mathoverflow.net/users/4149 | Real automorphisms of the "quaternionic" real group ${\rm SO}^*(4m)$ | No, there are no real outer automorphisms of $G = SO^\*(4m)$. Suppose, for sake of contradiction, that one exists and call it $\phi$. One of the half-spin representations $\rho \!: G \to GL(V)$ is real, and the composition $\rho \phi$ provides an irreducible representation defined over $\mathbb{R}$ of $G$ that is (by e... | 2 | https://mathoverflow.net/users/6486 | 303179 | 132,738 |
https://mathoverflow.net/questions/302611 | 3 | For any homogeneous polynomial $f \in \mathbb R [x,y]$, define the homogeneous polynomial
$$H(f) := \partial\_yf^2\partial\_x\partial\_xf-2\partial\_xf\partial\_yf\;\partial\_x\partial\_yf+\partial\_xf^2\partial\_y\partial\_yf$$
which is the Hessian of $f$ applied to the tangent of its level set. I came across it w... | https://mathoverflow.net/users/21179 | Factoring certain Hessians of real homogeneous bivariate polynomials | Let $f$ be a homogeneous polynomial of degree $n$.
Define:
$$Q:= f\_{xx} f\_y^2 - 2 f\_{xy} f\_x f\_y + f\_{yy} f\_x^2.$$
Consider $(n-1)^2 Q$ in the following way
$$(n-1)^2 Q = f\_{xx} \Big((n-1)^2 f\_y^2\Big) - 2 f\_{xy} \Big((n-1)f\_x\Big) \Big((n-1)f\_y\Big) + f\_{yy} \Big((n-1)^2 f\_x^2\Big)$$
By Euler's Lemm... | 1 | https://mathoverflow.net/users/125796 | 303189 | 132,740 |
https://mathoverflow.net/questions/303038 | 28 | Let $R$ be a commutative ring. Then there is a forgetful functor from the $\infty$-category of simplicial commutative $R$-algebras to the $\infty$-category of connective $E\_{\infty}$-algebras over $R$. It's well-known that this functor admits left and right adjoint. Moreover, it's an equivalence if $R$ contains the fi... | https://mathoverflow.net/users/nan | Spectral algebraic geometry vs derived algebraic geometry in positive characteristic? | I'll try to answer this question from the topological viewpoint. The short summary is that structured objects in the spectral setting have cohomology operations and power operations, which forces spectral algebraic geometry to be different from derived algebraic geometry.$\newcommand{\FF}{\mathbf{F}}\DeclareMathOperato... | 31 | https://mathoverflow.net/users/102390 | 303190 | 132,741 |
https://mathoverflow.net/questions/303137 | -1 | I am not very familiar with the notion of projective algebraic varieties, I work mostly from an algebraic topology/differential geometry point of view, but I am trying to find a prove for the following fact.
A non-singular complex projective algebraic variety $X$ admits a rational 2-form $\omega$ such that the multip... | https://mathoverflow.net/users/116775 | Kähler form on complex projective algebraic variety | To answer the original question, a non-singular complex projective variety has a canonical Kaehler manifold structure (given by the pullback of the Fubini--Study form and the standard complex structure on $\mathbb{C}P^n$). In the converse direction, a compact Kaehler manifold can be embedded into $\mathbb{C}P^n$ for so... | 2 | https://mathoverflow.net/users/nan | 303201 | 132,743 |
https://mathoverflow.net/questions/303202 | 3 | I'am try to work with Chern class of the coherent sheaves, in this sense. If I have a vector bundle $E$ of rank $r$ and $L$ a line bundle we have the Chern class property
$$c\_{r}(E\otimes L) = \sum\_{i = 0}^{r}c\_{i}(E)c\_{1}(L)^{r-i}.$$
I need a similar result for coherent sheaf.
If I have a coherent sheaf $\ma... | https://mathoverflow.net/users/69938 | How to calculate the Chern class of the tensor product of a torsion free sheaf with a line bundle | No, this is false. Take the simple example $\mathcal{F}=\mathcal{O}\_Y$, where $Y$ is a hypersurface in $X$. Using the resolution $0\rightarrow L(-Y)\rightarrow L\rightarrow L\otimes \mathcal{O}\_Y\rightarrow 0$, one gets $c\_1(L\otimes \mathcal{O}\_{Y})=c\_1(L)-c\_1(L(-Y))= [Y]$, so it is independent of $L$, while
yo... | 3 | https://mathoverflow.net/users/40297 | 303205 | 132,745 |
https://mathoverflow.net/questions/303193 | 21 | I've received conflicting messages on this point -- on the one hand, I've been told that "forming a natural home for algebraic $K$-theory" was one motivation for the development of motivic homotopy theory. On the other hand, I've been warned about the fact that algebraic $K$-theory isn't always $\mathbb A^1$-local. By ... | https://mathoverflow.net/users/2362 | Is algebraic $K$-theory a motivic spectrum? | Let me assume that $S$ is a regular Noetherian scheme (for example a field). Then algebraic K-theory is a motivic spectrum, and in fact it is represented by the $\mathbb{P}^1$-spectrum that is $BGL\_\infty\times\mathbb{Z}$ in each level (so it is a $\mathbb{P}^1$-periodic motivic spectrum).
This is theorem 4.3.13 in
... | 14 | https://mathoverflow.net/users/43054 | 303210 | 132,748 |
https://mathoverflow.net/questions/303212 | 1 | I've got a polynomial (which comes from solutions of the heat conduction PDE) which seems so simple I'm wondering if anyone recognizes it
$$f\_{m}=x^{m-1}
+(m-1)x^{m-3}+(m-1)(m-3)x^{m-5}
+(m-1)(m-3)(m-5)x^{m-7}
+\cdots$$
where the sum terminates when the exponent becomes negative
This can be written with double fac... | https://mathoverflow.net/users/125805 | do you recognize this polynomial with double factorials? | For $m$ an odd integer there is a closed-form expression in terms of the incomplete Gamma function$\Gamma(a,z)$,
$$f\_{m}=\sum\_{0\le k\le (m-1)/2} \frac{(m-1)!!x^{m-1-2k}}{(m-1-2k)!!}=2^{\frac{m-1}{2}} e^{\frac{x^2}{2}} \Gamma \left(\frac{m+1}{2},\frac{x^2}{2}\right).$$
I do not have a closed-form expression for $m$... | 5 | https://mathoverflow.net/users/11260 | 303215 | 132,751 |
https://mathoverflow.net/questions/303196 | 5 | In the following, I focus on trees of height $\omega\_1$: if there exists a nonspecial tree any of whose $\aleph\_1$-subtrees is special, must CH fail?
**Some neither consistent nor coherent thoughts**: Notice that the tree must have cardinality at least $\aleph\_2$ and this is a fragment of $MA\_{\aleph\_1}$. It is... | https://mathoverflow.net/users/119731 | Negation of CH implied by lots of special subtrees? | Your hypothesis is (implied by) the negation of Rado's conjecture, and this is known to be consistent with CH.
Rado's conjecture is a combinatorial statement about instances of compactness in chromatic numbers of certain graphs, but Todorčević proved that it is equivalent to the statement that any tree, all of whose ... | 6 | https://mathoverflow.net/users/1058 | 303216 | 132,752 |
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