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https://mathoverflow.net/questions/335357 | 0 | I'm working on a convex quadratic separable min-cost flow problem with the following structure:
$P = \{\min \frac{1}{2}x^tQx + qx : Ex = b, 0 \leq x \leq u\}$
But I'm stuck on deriving the KKT conditions to solve the problem.
Can someone help me with the computation?
| https://mathoverflow.net/users/142637 | KKT conditions for min-cost flow QP | The KKT condition is given by introducing relaxation parameters to the objective (derivative equals 0) so that whenever any element of $x$ is above $0$ or below $u$, parameters can increase to allow the system of equations to satisfy but at the same time cause penalties when x is not optimal.
So we have the following... | 0 | https://mathoverflow.net/users/136882 | 335389 | 143,280 |
https://mathoverflow.net/questions/335400 | 12 | We can define a very simple cardinal characteristic in the following way. Recall the relation $\leq^\*$ on $\omega^\omega$ defined by $x\leq^\* y$ iff $x(i)\leq y(i)$ for all but finitely many $i$. For $x,y\in\omega^\omega$, say that $x$ and $y$ are *comparable*, denoted by $x\parallel y$, if either $x\leq^\* y$ or $y\... | https://mathoverflow.net/users/67193 | A simple cardinal characteristic associated with $\omega^\omega$ | I claim that $\mathfrak{c}\mathfrak{p}= \mathfrak{d}$.
For any $x\in \omega^{\*\omega}$ define its "inverse" $x'$ by $x'(n) = \min \{k\mid\forall j\ge k : x(j)>n\}$. If $x$ grows very fast, then $x'$ grows very slowly.
In particular we have $x\le ^\* y \Rightarrow y'\le^\* x'$, and $x\le^\* x''$.
Define $x^+$ as... | 9 | https://mathoverflow.net/users/14915 | 335404 | 143,283 |
https://mathoverflow.net/questions/335397 | 10 | If $C$ is a symmetric monoidal category, then $BC$ is canonically an algebra over a certain $E\_\infty$ operad, but if $F: C \to D$ is a symmetric monoidal functor then (as far as I can see) $BF: BC \to BD$ is not a map of algebras over that operad (unless all the morphisms $(Fx) \otimes (Fy) \to F(x \otimes y)$ are id... | https://mathoverflow.net/users/142661 | Functoriality of infinite loop space machines? | Here is a nice gentle old-fashioned answer. Symmetric monoidal categories are functorially equivalent as symmetric monoidal categories to permutative (symmetric strict monoidal) categories, and those are functorially equivalent (essentially the same as) algebras over a certain $E\_{\infty}$ operad $\mathcal{P}$, known ... | 8 | https://mathoverflow.net/users/14447 | 335427 | 143,289 |
https://mathoverflow.net/questions/335419 | 3 | Good day to All.
Let $S\_{1,n} = \sum\_{i=1}^{n}\xi\_{i}$, where $(\xi\_{i})\_{i \in \mathbb{N}}$ be independent RV with values in some Banach space.
On pages 79-80 in [this book](https://www.springer.com/de/book/9783540603115) author provides an example that illustrates the fact that in the infinite dimensional s... | https://mathoverflow.net/users/115734 | Question on example 3.0.1 in Yurinsky's book "Sums and Gaussian vectors" | Yes we can. Indeed, with $\xi\_j= (\xi\_j^i)\_{i=1}^\infty$ and $\eta\_n=(\eta\_n^i)\_{i=1}^\infty:=\sum\_1^n\xi\_j$, we see that $\eta\_n^1,\eta\_n^2,\dots$ are iid random variables with values in $[-n,n]$ and $P(\eta\_n^1=n)=1/2^n>0$. Hence,
$$\|\eta\_n\|\_\infty=\sup\_{i=1,2,\dots}|\eta\_n^i|=n
$$
almost surely, whe... | 2 | https://mathoverflow.net/users/36721 | 335430 | 143,291 |
https://mathoverflow.net/questions/331050 | 6 | Let $(X,\mathcal{O})$ be a ringed space. Also assume that $X$ is nice, e.g. locally compact, Hausdorff, some type of finite dimension, ...
We can then consider $\mathcal{D}(\mathcal{O}\text{-}Mod)$. When is this a perfectly generated category?
Recall that a non-empty set of objects $S\subset Ob(\mathcal{C})$ for a ca... | https://mathoverflow.net/users/82627 | When is derived category of ringed space perfectly generated? | This can be found as Theorem 14.2.1 in Categories and Sheaves by Kashiwara and Schapira. One has to note that they use a slightly different notion for the second property.
* For a non-empty countable family of morphisms $(C\_i\to D\_i)$ such that $Hom(s,C\_i)\to Hom(s,D\_i)$ vanishes for each $i$ and $s$ then $Hom(s... | 0 | https://mathoverflow.net/users/82627 | 335431 | 143,292 |
https://mathoverflow.net/questions/335281 | 1 | Suppose $M$ is a line bundle on a smooth projective curve $C$ and $N$ is a vector bundle on $C$. Suppose the degree of $M$ is sufficiently big so that ${\rm Hom}(M, N)=0$ and ${\rm Ext}^1(N, M)=0$. Suppose $x$ is a point of $C$. Question: what are extensions of the form $0 \to M \oplus N \to X \to {\mathcal O}\_x \to 0... | https://mathoverflow.net/users/12395 | Parametrization of extensions of vector bundles | Since in the map $M\oplus N\to M\_1\oplus N$, one has no non-zero maps from $M\to N$, one must have an exact sequence $0\to M\to M\_1\to O\_x\to 0$. The rest should be easy.
This is just an elaboration of the above asked by the OP. Given an exact sequence $0\to M\oplus N\to M\_1\oplus N\to O\_x\to 0$ as in the questi... | 1 | https://mathoverflow.net/users/9502 | 335446 | 143,295 |
https://mathoverflow.net/questions/335445 | 6 | Let $f: \mathbb R^2 \rightarrow \mathbb R$ be a smooth strictly convex function with unique minimum at $0$ such that all level sets $A\_x:=\left\{z ; f(z) \le x \right\}$ are compact. Imagine something like $f(z)=\Vert z \Vert^2.$
Define the integral function
$$F(x):=\int\_{A\_x} g(z) dz$$
where $g$ is as smooth ... | https://mathoverflow.net/users/nan | Second derivative of integral function | Let $B\_u:=\{z\colon f(z)=u\}$ for $u>u\_0:=\min f$. Let $[0,2\pi)\ni t\mapsto(x\_u(t),y\_u(t))$ be any smooth parametrization of $B\_u$, so that
$B\_u=\{(x\_u(t),y\_u(t))\colon t\in[0,2\pi)\}$.
(For instance, one may take $(x\_u(t),y\_u(t))=(\rho\_u(t)\cos t,\rho\_u(t)\sin t)$, where $\rho\_u(t):=f\_t^{-1}(u)$ and... | 4 | https://mathoverflow.net/users/36721 | 335447 | 143,296 |
https://mathoverflow.net/questions/335428 | 2 | Tarski in the article ["WHAT IS ELEMENTARY GEOMETRY"](https://pdfs.semanticscholar.org/34a6/da709bbb7e99e52d3aa0d0a2c6588bffd040.pdf) describes four candidates ($\mathscr{E}\_2,\mathscr{E}'\_2,\mathscr{E}''\_2,\mathscr{E}'''\_2$) to be called "Elementary geometry". Here the name "elementary" for a geometry comes from f... | https://mathoverflow.net/users/73577 | Further study of "Elementary geometry" in the sense of Tarski | As you note, Tarski claimed that no finitely axiomatizable subtheory of $\mathcal{E}\_2$ is decidable. Since $\mathcal{E}\_2$ and the theory of real-closed ordered fields are bi-interpretable, Tarski's conjecture was essentially established by M. Ziegler (Einige unentscheidbare Körpertheorien Enseign. Math. (2) 28 (198... | 4 | https://mathoverflow.net/users/18939 | 335451 | 143,297 |
https://mathoverflow.net/questions/335438 | 1 | Consider the following definition in the second page of [this article](https://pdfs.semanticscholar.org/da0b/1a3430e4be45119921b8b8a6578b355bb2f6.pdf):
>
> For any two integers $k,n\ge 1$, there is a unique way of writing
> $$n=\binom{a\_k}{k}+\binom{a\_{k-1}}{k-1}+\dots+\binom{a\_i}{i}$$
> so that $a\_k > a\_{k-... | https://mathoverflow.net/users/140960 | Monotonicity of $M$-sequence | The reason is that this ordering is lexicographic. We may induct on $k$. The base $k=1$ is clear. Assume that $n>m$ and
\begin{align\*}
n=\binom{a\_k}{k}+\binom{a\_{k-1}}{k-1}+\ldots+\binom{a\_i}{i},a\_k>a\_{k-1}>\ldots>a\_i\geqslant i,\\
m=\binom{b\_k}{k}+\binom{b\_{k-1}}{k-1}+\ldots+\binom{b\_j}{j},b\_k>b\_{k-1}>\ldo... | 2 | https://mathoverflow.net/users/4312 | 335454 | 143,298 |
https://mathoverflow.net/questions/335443 | 1 | Let $(\cal F,\mu)$ be the stable measured foliation of a pseudo-Anosov on an oriented surface $S$. Can there be two non-isotopic multi-loops (collections of disjoint simple loops) $L\_1,L\_2\subset S$, both transverse to $\cal F$ with the same transverse measure, $\mu(L\_1)=\mu(L\_2)$?
If yes, how one can construct s... | https://mathoverflow.net/users/23935 | Two multi-curves in a surface with the same transverse measure | Yes. For example, by lifting - there are other constructions, as well.
---
Here are the details of a lifting construction. Suppose that $f$ is a pseudo-Anosov on a surface $T$. Pick a simple closed curve $\alpha$ in $T$. Choose a double cover $S$ of $T$ where the preimage of $\alpha$ has two components.
There ... | 3 | https://mathoverflow.net/users/1650 | 335455 | 143,299 |
https://mathoverflow.net/questions/335448 | 16 | A finite complex over a field $k$ is pretty simple: it's the direct sum of its homology with a split-exact complex. How complicated can a finite double complex be? Does it make a difference if $k$ is algebraically closed?
It seems as though quiver representation theory may be relevant here, but since a double complex... | https://mathoverflow.net/users/2362 | How complicated can a finite double complex over a field be? | It seems that [Jonas Stelzig](https://arxiv.org/abs/1812.00865) has answered your question completely, for the sake of applications to non-Kähler geometry.
$\require{AMScd}$
I will quote here Theorem A, which enumerates all of the isomorphism classes of double complexes as direct sums of certain standard double compl... | 14 | https://mathoverflow.net/users/40804 | 335459 | 143,301 |
https://mathoverflow.net/questions/318791 | 1 | In [posting](https://mathoverflow.net/questions/317658/what-is-the-strength-of-adding-limitation-of-size-and-a-simple-version-of-reflec) about a reflection principle coupled with a limitation of size axiom over Ackermann set theory, the answer is that the theory is blown up to a Mahlo cardinal.
I'm here just wonderin... | https://mathoverflow.net/users/95347 | What is the limit to iterating class comprehension, reflection and limitation of size? | I claim $K^+(V^\lambda)$, for any $\lambda$ (I switched the notation so not to be confused with the von Neumann universe) is equiconsistent with the schema "$ORD$ is Mahlo" (Not to be confused with full stationarity, which could be called "$Ord$ is Mahlo" if you really wanted to distinguish them). First off each $V^\la... | 2 | https://mathoverflow.net/users/141402 | 335465 | 143,304 |
https://mathoverflow.net/questions/335343 | 6 |
>
> Is there a notion of "cyclic element" in a simple Lie algebra? In particular, is it independent of the irreducible representation chosen?
>
>
>
Explanation.
An endomorphism A is called *cyclic* if there is a vector v which by the action of A generates the whole vector space (v is also called a cyclic vecto... | https://mathoverflow.net/users/142627 | Cyclic vectors in irreducible representations of simple Lie algebras |
>
> **Summary** The answer to the first question is affirmative and to the second question is negative, but for rather mundane reasons. In the simple Lie algebra case, cyclicity of ${\rm ad}\, a$ for some $a$ implies that $\frak{g}$ has rank $1$, in which case every non-zero element $x\in\frak{g}$ is cyclic in every ... | 5 | https://mathoverflow.net/users/5740 | 335474 | 143,306 |
https://mathoverflow.net/questions/335481 | 1 | Motivation:
-----------
This is [related to a different question I asked in April](https://mathoverflow.net/questions/328558/understanding-the-geometry-of-h-n-vecx-in-n-nn-sum-i-1n-x-i-0). It occurred to me while thinking about the sums of uniform random variables and it stuck in my mind because it's the special case... | https://mathoverflow.net/users/56328 | Given $H_{N}=\{\vec{x} \in [-1,1]^N:\sum_{i=1}^N x_i = 0\}$, what is the smallest subset $S \subset H_{2N}$ such that $\mathrm{conv}(S)=H_{2N}$ | It suffices to verify (see the comment by Duchamp Gérard H. E. and (<https://en.wikipedia.org/wiki/Extreme_point>)) that the set $V\_{2N}$ is precisely the set of extreme points ${\bf EP}(H\_{2N})$ of $H\_{2N}$. Clearly
$V\_{2N} \subset {\bf EP}(H\_{2N})$. To see the converse, suppose $x \in {\bf EP}(H\_{2N})$.
Let $\... | 5 | https://mathoverflow.net/users/7691 | 335490 | 143,309 |
https://mathoverflow.net/questions/83980 | 16 | Suppose $\mathbb{T}$ is a geometric theory, $\mathcal{E}$ is a topos, and $M$ is a model of $\mathbb{T}$ in $\mathcal{E}$. Is there any sort of elementary condition on $M$ and $\mathcal{E}$ (or, even better, on the geometric morphism $\mathcal{E}\to \mathbf{Set}$) which would allow us to recognize $\mathcal{E}$ as the ... | https://mathoverflow.net/users/49 | Recognizing classifying toposes | This is several years late, but it may be helpful nonetheless.
As alluded to by Buschi, Olivia has given an explicit answer to this in Theorem 2.1.29 of her monograph **Theories, Sites and Toposes**:
>
> Let $\mathbb{T}$ be a geometric theory, $\mathcal{E}$ a Grothendieck
> topos and $M$ a model of $\mathbb{T}$... | 4 | https://mathoverflow.net/users/113942 | 335495 | 143,311 |
https://mathoverflow.net/questions/335456 | 3 | Let $f\colon X\to Y$ be a birational map of complex algebraic varieties. Are there necessarily open immersions of $X$ and $Y$ into varieties $X’$ and $Y’$, resp., which admit a proper morphism $f’\colon X’\to Y’$ extending $f$?
| https://mathoverflow.net/users/36720 | Can a birational map be completed to a proper map? | This follows from Nagata's compactification theorem [Nag62] and its relative version [Nag63]. Indeed, one may compactify $Y$ to get an open immersion $Y \hookrightarrow Y'$ with $Y'$ proper [Nag62]. Replacing $Y'$ by the reduced structure on the closure of $Y$, we may assume $Y'$ is integral.
Now apply relative compa... | 4 | https://mathoverflow.net/users/82179 | 335499 | 143,313 |
https://mathoverflow.net/questions/335418 | 4 | Given $\Sigma$ a consistent finite first order theory in vocabulary $L$, one can consider the category of its models $\mathcal{M}(\Sigma)$, its objects are the models of $\Sigma$ and arrows are exactly $L$-structure homomorphisms.
After fixing a finite index category $J$ we define a decision problem
$$P(J)=\{\Sigma \... | https://mathoverflow.net/users/123891 | Algorithmically deciding existence of finite limits in a category | No - the undecidability of first-order logic makes it very hard to algorithmically decide *anything* about the class of models of an arbitrary first-order sentence. In this case, you can reduce the consistency problem for first-order sentences to a family of instances of your problem, just involving consistent sentence... | 4 | https://mathoverflow.net/users/2126 | 335507 | 143,315 |
https://mathoverflow.net/questions/311461 | 3 | For me, a *polytope* is the convex hull of finitely many points. It is said to be *vertex-transitive* / *edge-transitive* if its symmetry group acts transitively on its vertices / edges. Let's call a polytope *symmetric* if it is simultaneously vertex- and edge-transitive.
I wonder what are the symmetric polytopes wi... | https://mathoverflow.net/users/108884 | Maximal edge length of symmetric polytopes | The vertices of a symmetric (i.e. vertex- and edge-transitive) polytope form a spherical code, where the edge-length defines the minimal distance between two points. In fact, the same can be said about any polytope for which all vertices are on a sphere and all edges have a common length.
In the book
* Conway, John... | 0 | https://mathoverflow.net/users/108884 | 335513 | 143,316 |
https://mathoverflow.net/questions/334367 | 0 | The following system is quoted from Harvey Friedman's lecture notes. The language is first order logic with membership $\in$.
*Axioms:*
1. **Extensionality.** $(\forall x)(x \in y \leftrightarrow x \in z) \to (\forall x)(y \in x \leftrightarrow z \in x).$
2. **Pairing.** $(\exists x)(y,z \in x).$
3. **Union.** $(\e... | https://mathoverflow.net/users/95347 | Reference request: for a proof of a reflection [from transitive sets] based axiomatization of ZF\Reg.? | This was actually a result I used in a paper I am working on.
Let $M$ reflect $\exists x(x\in X\land\phi(x,y))$ (In this case $\phi(x,y)↔y=f(x)$). Then $\{y│(M,∈)\vDash\exists x(x\in X\land\phi(x,y))\}=\{y│\exists x(x\in X\land\phi(x,y))\}\cap M$ and $\{y│\exists x(x\in X\land\phi(x,y))\}\cap M=\{y│\exists x(x ∈ X\la... | 1 | https://mathoverflow.net/users/141402 | 335515 | 143,318 |
https://mathoverflow.net/questions/335476 | 7 | In
>
> Makkai, *A theorem on Barr-exact categories, with an infinitary generalization*
>
>
>
a definition of infinitary regular category is given: a complete regular category $C$ with the additional requirement
**(DC)** for every diagram $F : \alpha^{\text{op}} \to C$, $\alpha$ an ordinal, such that each $F(... | https://mathoverflow.net/users/97958 | Definition of infinitary regular category | **The two definitions are not equivalent:** The following counter-example might not be the simplest, and I mostly learned it from Christian Espindola. It is a nice counterexample to quite a lot of similar questions...
Let $I$ be the poset of rational number $0 \leqslant q \leqslant 1$, seen as a category with a uniqu... | 4 | https://mathoverflow.net/users/22131 | 335516 | 143,319 |
https://mathoverflow.net/questions/335506 | 4 | Let $L$ be a number field, let $p$ be a prime number, and let $I$ be a ideal of $\mathcal{O}\_L$ containing $p$. I am not assuming that $\mathcal{O}\_L$ or that $I$ is prime. The quotient ring $\mathcal{O}\_L/I$ has a natural structure of $\mathbb{F}\_p$-algebra.
**Question.** Do we have an isomorphism of $\mathbb{F}... | https://mathoverflow.net/users/36683 | Quotients of a ring of integers | The answer to your main question is no. Let $I = (p)$ where $p$ *splits completely* in $L$ and $r := [L:\mathbf Q] > p$. (Example: cubic field in which $p = 2$ splits completely.) Then $\mathcal O\_L/(p) \cong \mathbf F\_p^r$ but if $\mathcal O\_L/(p) \cong \mathbf F\_p[X]/(f)$ for monic $f$ in $\mathbf F\_p[X]$ then $... | 12 | https://mathoverflow.net/users/3272 | 335518 | 143,321 |
https://mathoverflow.net/questions/334977 | 9 | What are the examples of “tame” minimal Kan simplicial sets having finite number of simplexes in each dimension besides simplicial point and $B(G)\approx K(G,1) $ for a finite group $G$? I believe that Alain Connes’ simplicial circle is also minimal Kan.
| https://mathoverflow.net/users/2702 | On minimal Kan simplicial sets having finite number of simplexes in each dimension | A tame minimal Kan complex has finite homotopy groups. The converse is also true: if homotopy groups of a minimal Kan complex are finite, then there are finitely many $n$-simplices. The proof is by induction on $n$. First, $X\_0 = \pi\_0(X)$ is finite. If $n > 0$, there are finitely many possible choices of boundaries ... | 3 | https://mathoverflow.net/users/62782 | 335522 | 143,322 |
https://mathoverflow.net/questions/335526 | 2 | Let $C$ be a full-dimensional cone in $\mathbb{R}^{d}$, defined as the positive span of $c = {n \choose 3} \gg d$ vectors. $C$ is highly symmetric in the following sense: each such vector is labelled with a distinct unordered triple from $[n]$, and the automorphism group of $C$ is basically $S\_n$: for each permutation... | https://mathoverflow.net/users/25121 | Do highly symmetric cones have "small" supporting hyperplanes? | The prescribed group action allows to start from it and build the cones in question. It is well-known how the permutation representation of $S\_n$ on 3-subsets decomposes into irreducibles - there will be just 4 of these, one of them trivial 1-dimensional one. With the other ones (more precisely, subsets of these) you ... | 2 | https://mathoverflow.net/users/11100 | 335536 | 143,323 |
https://mathoverflow.net/questions/335412 | 5 | A binary relation $R$ on $\mathbb{N}$ is *$(n,k)$-Ramseyan* iff for every coloring $f:[\mathbb{N}]^n\rightarrow l$, there exists a subset $G$ of $\mathbb{N}$
such that $(G,R)$ is isomorphic to $(\mathbb{N},R)$ and $|f([G]^n)|\leq k$
(i.e., there exists a 1-1 mapping $h:G\rightarrow \mathbb{N}$ such that $R(x,y)$ if and... | https://mathoverflow.net/users/74918 | Ramseyan property of structure | This question is closely related to the problem of proving the existence of *finite big Ramsey degrees*. Namely, for a first-order language $L$ and an infinite $L$-structure $\textbf{B}$, we say $\textbf{B}$ has finite big Ramsey degrees if for every finite substructure $\textbf{A}$ there is an integer $T\_{\textbf{A}}... | 2 | https://mathoverflow.net/users/134207 | 335542 | 143,326 |
https://mathoverflow.net/questions/334095 | 9 | I'm trying to find lecture notes of Mahowald and Unell, titled "Lectures on Bott periodicity in stable and unstable homotopy at the prime 2". Does anyone happen to know if a copy exists online (and if it does, where I can find it)?
It is cited in Peter May's "Applications and generalizations of the approximation theo... | https://mathoverflow.net/users/102390 | Lecture notes by Mahowald and Unell | I scanned the notes (apologies for the delay). Thanks a lot to Peter May for lending me the notes and for letting me scan them! Here's the link: <http://www.mit.edu/~sanathd/mahowald-unell-bott.pdf>.
| 9 | https://mathoverflow.net/users/102390 | 335543 | 143,327 |
https://mathoverflow.net/questions/335544 | 3 | This question is about elliptic curves defined over the rational numbers $\mathbb{Q}$, though the question will make sense for any number field with class number one (due to the existence of unique minimal models).
Let
$$\displaystyle E\_{A,B} : y^2 = x^3 + Ax + B, A, B \in \mathbb{Z}$$
be a minimal Weierstrass... | https://mathoverflow.net/users/10898 | $S$-units which can be discriminants of elliptic curves over $\mathbb{Q}$ | The finiteness follows from Falting's theorem on finiteness of abelian varieties with good reduction outside a given set of primes (or some easier stuff too, probably).
The uniformity is false. Take an elliptic curve $E$ with bad reduction at primes $p\_1,\dots, p\_n$. Assume that at none of these primes does a quadr... | 7 | https://mathoverflow.net/users/18060 | 335546 | 143,328 |
https://mathoverflow.net/questions/335528 | 8 | It is easy to show that $\mathbb{N}$ with the cofinite topology is not path connected and that any set with cardinality $\geq 2^{\aleph\_0}$ equipped with the cofinite topology is in fact path connected.
what about cardinalities $\aleph\_0<\alpha<2^{\aleph\_0}$ (under the assumption that such exist obviously)?
If $... | https://mathoverflow.net/users/142734 | Is a cofinite topology for a set with cardinality between $\aleph_{0}$ and $2^{\aleph_{0}}$ path-connected? | A continuous non-constant function from $[0,1]$ into $X$ with the cofinite topology exists iff $[0,1]$ has a partition into $\le |X|$ many disjoint closed non-empty subsets.
[This question](https://mathoverflow.net/q/285780/2060) discusses the options for the cardinality of such a partition. One of the conclusions i... | 5 | https://mathoverflow.net/users/2060 | 335553 | 143,331 |
https://mathoverflow.net/questions/335564 | 7 | Let $\Bbbk$ be a field; I am interested in the following ring (which I suspect is a field). Its elements are formal expressions that look like
$$ \sum\_{n=0}^{\infty} a\_n x^{b\_n} $$
where $a\_n\in \Bbbk$ and $b\_n\in \mathbb{R}$, with $b\_n$ strictly increasing, and $\lim\_{n\to\infty} b\_n = \infty$. (Technicall... | https://mathoverflow.net/users/49 | A ring of generalized power series | I think your ring looks similar to the Novikov ring (see topology papers).
| 10 | https://mathoverflow.net/users/nan | 335573 | 143,340 |
https://mathoverflow.net/questions/335578 | 4 | Denote "the" category of sets and functions by $S$. The hom set of functions from set $X$ to set $Y$ is denoted by $S(X,Y)$.
If $C$ is a cartesian closed category denote by $C(x,y)$ the set of morphisms from $x$ to $y$ in $C$. In such a $C$ there exists a natural bijection between $C(x,y)$ and $C(1,y^x)$. In a sense... | https://mathoverflow.net/users/4945 | What is the relationship between external and internal composition in a cartesian closed category? | The internal composition $y^x\times z^y\to z^x$ induces, by composition, $C(1,y^x\times z^y)\to C(1,z^x)$. The domain of this morphism is naturally equivalent to $C(1,y^x)\times C(1,z^y)$, by definition of product in $C$. So we get a function $C(1,y^x)\times C(1,z^y)\to C(1,z^x)$, which, as you noted, is equivalent to ... | 10 | https://mathoverflow.net/users/6794 | 335583 | 143,343 |
https://mathoverflow.net/questions/335549 | 2 | Let $k$ be a field. Let $A=k[X,Y]/(Y^2)$ be the quotient of polynomial ring $k[X,Y]$. Let $\mathcal{C}$ be the category of finite-dimensional $A$-modules $M$ with the action of $X$ nilpotent, and of finite projective dimension. In fact, for any $M\in\mathcal{C}$, we have $\mathrm{proj.dim}\_A M\leq 1$. Then $\mathcal{C... | https://mathoverflow.net/users/142751 | Simple object of $k[X,Y]/(Y^2)$ | For any $\lambda\in k$, $A/(X-\lambda Y)$ is another example, I think.
| 6 | https://mathoverflow.net/users/22989 | 335584 | 143,344 |
https://mathoverflow.net/questions/335537 | 8 | I am reading the famous paper by Dold and Whitney "Classification of Oriented Sphere Bundles Over A 4-Complex".
I'll state their theorem for the case of SO(3) bundles
**Classification Theorem**:Let $B\_1, B\_2$ be principal $SO(3)$ bundles over a complex $K$ of dimension at most 4. let $h\_i:K\to G\_n\simeq BSO(3)$... | https://mathoverflow.net/users/99042 | Obstruction to homotopy, cohomology operations and Dold-Whitney theorem | Let me explain why the condition is not the triviality of the difference cocycle. Maybe an important point to note is that the condition with the difference cocycle is not about the vanishing of an obstruction, it is about the enumeration of possible lifts of maps from one Postnikov truncation to the next.
If we deno... | 9 | https://mathoverflow.net/users/50846 | 335587 | 143,345 |
https://mathoverflow.net/questions/334758 | 7 | My question is about the proof of Claim 1 in this paper: [Gillman (1993)](https://pdfs.semanticscholar.org/eae4/a397e716b677e79187ce51ab5dc1f6c3a34d.pdf).
At the end of the proof, the author says:
>
> The matrix product $U^\top\sqrt{D^{-1}}(P+(\mathrm{e}^x-1)B(0)-\mu I)\sqrt{D}U$, which is equal to $(D'-\mu I)(I+(D... | https://mathoverflow.net/users/128183 | Understanding Gillman's proof of the Chernoff bound for expander graphs | First, we'll explain why $U^\top\sqrt{D^{-1}}(P+(\mathrm{e}^x-1)B(0)-\mu I)\sqrt{D}U$ is sungular. It is enough to show that $P+(\mathrm{e}^x-1)B(0)$ has eigenvalue $\mu$. We know that $\mu$ is an eigenvalue of $P(x)$, so we'll show why $P+(\mathrm{e}^x-1)B(0) = P(x)$.
Note that $E\_r:=\operatorname{diag}(\mathrm{e}^... | 1 | https://mathoverflow.net/users/142777 | 335594 | 143,348 |
https://mathoverflow.net/questions/334326 | 11 | Let's say I have a combinatorially self-dual polytope $P\subseteq\Bbb R^d$, i.e., its face lattice is isomorphic to its dual (you reverse the direction of the lattice order).
>
> **Question:** Is it always possible to realize $P$ geometrically, so that $P$ and its polar
> $$P^\circ := \{x\in\Bbb R^d \mid \langle x... | https://mathoverflow.net/users/108884 | Geometric realization of combinatorial self-duality in polytopes | Alathea Jensen defines "self-polar":
>
> Self-polar polytopes are convex polytopes that are equal to an orthogonal transformation of their polar sets.
>
>
>
and writes some interesting things about self-polar polytopes here:
<https://arxiv.org/abs/1902.00784>
Your questions are partially answered there:
* ... | 3 | https://mathoverflow.net/users/39495 | 335595 | 143,349 |
https://mathoverflow.net/questions/334989 | 5 | Are there any purely combinatorial criteria which allow you to deduce that a spherical simplicial complex is polytopal (i.e., there exists a simplicial polytope whose boundary is isomorphic to it)?
For example, I was kind of hoping that shellability might be enough, but this is not true. All 3-spheres on up to ten ve... | https://mathoverflow.net/users/468 | Sufficient criterion for a simplicial sphere to be polytopal | I can think of a few purely combinatorial criteria, that allow to deduce realizability as a polytope.
1. All d-polytope with at most d+2 vertices is realizable
2. Stacked polytopes. (It can be easily combinatorially checked if a simplicial complex is stacked.)
Those and some more obscure criteria a mentioned for ex... | 3 | https://mathoverflow.net/users/39495 | 335598 | 143,352 |
https://mathoverflow.net/questions/335413 | 6 | Consider $\mathbb{R}^d$ with Gibbs measure $d\mu=Z^{-1}\exp(-V(x))dx$, where the potential $V(x)$ is strongly convex ($\nabla^2 V(x) \ge \lambda Id $). We can assume the regularity of $V$ is as good as we need, and the Hessian is also bounded above.
Now, for “Laplace equation” in $(\mathbb{R}^d, d\mu)$: $$\nabla^\* ... | https://mathoverflow.net/users/142670 | Elliptic Regularity with Gibbs Measure Satisfying Bakry-Emery Condition | As you observed, by Lax-Milgram it is easy to show well-posedenss in $H^1(d\mu)$ with a right-side in $H^{-1}(d\mu)$. The constant in the estimate should be given by whatever the optimal constant in the Poincar\'e inequality for the measure is. Now let's try to differentiate the equation. First, we can write $\nabla^\*... | 1 | https://mathoverflow.net/users/5678 | 335599 | 143,353 |
https://mathoverflow.net/questions/332950 | 5 | Let $A$ be a subhomogeneous C$^{\*}$-algebra (i.e., there is a finite upper bound on the size of the irreducible representations of $A$). Let $\hat{A}$ denote its spectrum. I heard of a result that states that
>
> If $(\pi\_{n})$ is a sequence in $\hat{A}$, then $(\pi\_{n})$ can
> converge to at most finitely man... | https://mathoverflow.net/users/126776 | A sequence of points in the spectrum of a subhomogeneous C$^{*}$-algebra can converge to at most finitely many points | See JMG Fell, The Dual Spaces of C$^\*$-Algebras, Trans. Amer. Math. Soc. 94 (1960), 365-403, Corollary 1 on p. 388:
Let $A$ be a C$^\*$-algebra with dual space $\hat{A}$. Let $T^i$ be a net of elements of $\hat{A}$, all of dimension equal to or less than the integer $n$; and let $S^1, \ldots S^r$ be distinct element... | 7 | https://mathoverflow.net/users/142780 | 335603 | 143,356 |
https://mathoverflow.net/questions/335572 | 4 | When I read the paper "[THE FARGUES–FONTAINE CURVE AND DIAMONDS](http://www.bourbaki.ens.fr/TEXTES/Exp1150-Morrow.pdf)" of Matthew Morrow, I have a question on page 11.
Question: The arthur said that the de Rham and crystalline period rings implicitly depended on having chosen an untilt of $F$, where $F=C\_p^b$ is th... | https://mathoverflow.net/users/nan | Fontaine-Fargues curve and period rings and untilt | As Daniel Litt says, the choice of $C$ is actually an untilt. The "classical" approach to period rings, which you might have in mind, was to start with a certain complete, algebraically closed field $C\_p$, then to construct $R$ out of the quotient $O\_{C\_p}/p$ and out of its Witt vectors to construct the period rings... | 2 | https://mathoverflow.net/users/18238 | 335617 | 143,360 |
https://mathoverflow.net/questions/335616 | 2 | Let $\Gamma=(\Gamma\_1\rightrightarrows \Gamma\_0), \Gamma’=(\Gamma’\_1\rightrightarrows \Gamma’\_0)$ be Lie groupoids and $\Gamma\_{\bullet} ,\Gamma’\_{\bullet}$ be the [simplicial manifolds associated to](https://arxiv.org/pdf/math/0401420v3.pdf) $\Gamma,\Gamma’$ respectively.
Question : If the simplicial manifold... | https://mathoverflow.net/users/118688 | Simplicial manifold associated to Lie groupoid | If the simplicial manifolds are isomorphic, then the groupoids are also isomorphic, since the nerve functor from groupoid objects in a category to simplicial objects in the same category is fully faithful.
So to answer your question: yes, but in a boring way.
| 6 | https://mathoverflow.net/users/4177 | 335624 | 143,362 |
https://mathoverflow.net/questions/335631 | 6 | [MOVED HERE FROM MSE.]
The statement of the Grothendieck-Lefschetz fixed point theorem is well-known. For a proper algebraic variety $X$ over $\mathbb F\_q$,
$$\#X(\mathbb F\_q) =\sum\_i (−1)^i Tr(Fr\_X, H^i\_c(X, \mathbb Q\_l)).$$
Also known is the version for general constructible l-adic sheaves $\mathcal F$:
$$\su... | https://mathoverflow.net/users/92131 | Generalized Behrend version for Grothendieck-Lefschetz trace formula | This is Theorem 4.2 of Shenghao Sun's paper *$L$-Series of Artin stacks over finite fields*, Algebra & Number Theory **6** (2012) pp 47–122, doi:[10.2140/ant.2012.6.47](https://doi.org/10.2140/ant.2012.6.47), arXiv:[1008.3689](https://arxiv.org/abs/1008.3689).
>
> Let $f:\mathscr X\_0\to\mathscr Y\_0$ be a morphism... | 10 | https://mathoverflow.net/users/18060 | 335633 | 143,364 |
https://mathoverflow.net/questions/335632 | 4 | I'm having difficulty finding this result in the standard texts.
>
> **Theorem.** Let $T$ be a theory in a language $\mathcal{L}$. TFAE:
>
>
> 1) $T$ has quantifier elimination,
>
>
> 2) Whenever $M, N$ are $\aleph\_1$-saturated models of $T$, $A \subset
> M$, $B \subset N$ are countable nonempty substructures... | https://mathoverflow.net/users/126815 | Reference request: quantifier elimination test | I'm not sure if this formulation with $\omega\_1$-saturated models is in any of the "main standard" texts. But there is a complete treatment in these [course notes](https://www3.nd.edu/~apillay/pdf/lecturenotes_modeltheory.pdf) by Pillay (see Proposition 2.29).
If you're looking for something in a standard published ... | 6 | https://mathoverflow.net/users/38253 | 335639 | 143,367 |
https://mathoverflow.net/questions/335557 | 5 | An [Odd Cycle Transversal](https://en.wikipedia.org/wiki/Odd_cycle_transversal) is a set of vertices that, when removed from a graph, renders it bipartite.
>
> **Question:**
>
>
> does the collection of "critical" sets of vertices, whose removal renders a graph bipartite, resemble a [Matroid](https://en.wikipedia... | https://mathoverflow.net/users/31310 | Do the Odd Cycles of a Graph Define a Matroid? | Greedy algorithm doesn't do that well in the worst case, even provided the odd cycle counting oracle.
Take large $n$, and consider complete bipartite $K\_{n, n}$ with parts $V\_0, V\_1$ together with an additional disjoint triangle $T$, and connect all vertices of $T$ with two vertices $v, u \in V\_0$ pairwise. Odd c... | 2 | https://mathoverflow.net/users/106512 | 335660 | 143,372 |
https://mathoverflow.net/questions/335650 | 2 | The treewidth is a parameter of the graph that describes its similarity to a tree. Treewidth is NP-hard to find. For the introduction please see [wikipedia](https://en.wikipedia.org/wiki/Treewidth)
The question is how to generate interesting graphs with a specified treewidth? I know that:
1) Any graph containing a $k... | https://mathoverflow.net/users/142806 | Is there a way to generate a graph of specified treewidth | This is an extended comment. Here are some classes of graphs that should be interesting:
* $k$-trees by definition have treewidth $k$; moreover, adding any edge to a $k$-tree increases treewidth. This should be an interesting example since $k$-trees are "borderline maximal" with treewidth $k$.
* Graphs with treewdith... | 0 | https://mathoverflow.net/users/106512 | 335664 | 143,373 |
https://mathoverflow.net/questions/335672 | 6 | Why does coherence begin to matter at the tricategorical level?
It is well known that every weak $2$-category is equivalent to a strict $2$-category, with the equivalence essentially given by (unless I'm mistaken) the $2$-Yoneda embedding for an arbitrary $2$-category into its strict $2$-category of $2$-presheaves in... | https://mathoverflow.net/users/92164 | Tricategorical coherence | The short answer is that what matters is the *combination* of strict interchange and strict units, because interchange and units are what go into the Eckmann-Hilton argument, and dimension 3 is the first dimension in which you can have weak commutativity (e.g. braided monoidal categories are degenerate tricategories). ... | 6 | https://mathoverflow.net/users/49 | 335673 | 143,377 |
https://mathoverflow.net/questions/335679 | -2 | We say that a function $f: \mathbb{Z}\times \mathbb{Z}\to \mathbb{Z}$ is *uniformly continuous* if there is an integer $K\geq 1$ such that
whenever $(x,y),(x',y')\in \mathbb{Z}\times \mathbb{Z}$ with $|(x,y)-(x',y')| = 1$ in the Euclidean distance, then $|f(x,y)-f(x',y')| < K$.
Is there an injective uniformly contin... | https://mathoverflow.net/users/8628 | Injective uniformly continuous function $f:\mathbb{Z}\times \mathbb{Z}\to \mathbb{Z}$? | No. A uniformly continuous function takes $O(N)$ distinct values on an $N\times N$ grid.
| 14 | https://mathoverflow.net/users/4312 | 335680 | 143,378 |
https://mathoverflow.net/questions/335532 | 5 | It is a standard exercise to show that if $X\subseteq\mathbb{R}^n$ has smooth boundary, then the characteristic function $1\_X$ has wavefront set $$\{(x,\xi)\in\partial X\times\mathbb{R}^n\setminus\{0\}:\xi\text{ is normal to }\partial X\}.$$ This is shown by locally "flattening" to reduce to the case of the upper half... | https://mathoverflow.net/users/142740 | Wavefront set of characteristic function of rough set | Let me first begin with an elementary example, taking $X=[0,1]^2$ in $\mathbb R^2$. It is then easy to see directly that the wave-front-set of $\mathbb 1\_X$ is everywhere the conormal bundle except at the four corners $c\_j$:
$$
WF\mathbb 1\_X=\cup\_{1\le j\le 4}(c\_j;\mathbb R^2\backslash {(0,0)})\cup\text{conormal ... | 3 | https://mathoverflow.net/users/21907 | 335695 | 143,381 |
https://mathoverflow.net/questions/335691 | 3 | If we remove the axiom that zero doesn't have a predecessor, and stipulate that every natural number has a predecessor, and that no number is the successor of itself. And keep all other axioms of $\small \sf PA$, and introduce a new primitive relation symbol $``<"$ to signify *strictly smaller than relation*, and axiom... | https://mathoverflow.net/users/95347 | Is cyclic PA interpretable in PA? | The theory cPA is interpretable already in Robinson’s theory $R$ by a result of Albert Visser [1], because it is recursively axiomatized and *locally finitely satisfiable*, meaning that every finite subtheory of cPA has a finite model. Indeed, a finite subtheory of cPA only mentions finitely many axioms 3, hence it is ... | 6 | https://mathoverflow.net/users/12705 | 335700 | 143,382 |
https://mathoverflow.net/questions/335713 | 3 | I am aware that there is a nice result where one uses curve shortening flow to prove the isoperimetric inequality on the Euclidean plane, but could not seem to find the original article where this was done and was wondering if someone could point me towards this? I believe it was first done by Gage?
| https://mathoverflow.net/users/119114 | Proof of Isoperimetric Inequality using Curve Shortening Flow | The final paper:
Grayson, Matthew A.(1-UCSD)
*Shortening embedded curves.*
**Ann. of Math.** (2) 129 (1989), no. 1, 71–111.
using results from:
Gage, Michael E.(1-CWR)
*An isoperimetric inequality with applications to curve shortening.*
**Duke Math. J.** 50 (1983), no. 4, 1225–1229.
Gage, M. E.(1-CWR)
*Curve sh... | 2 | https://mathoverflow.net/users/13268 | 335718 | 143,390 |
https://mathoverflow.net/questions/335723 | 10 | For a natural number $n$ let $F\_n$ be the free group with $n$ generators.
The group $F\_n$ is endowed with the discrete topology.
Given an increasing sequence $\vec p=(p\_k)\_{k\in\omega}$ of prime numbers, consider the Polish group $F\_{\vec p}=\prod\_{k\in\omega}F\_{p\_k}$.
>
> **Problem.** Let $\vec p,\vec q$... | https://mathoverflow.net/users/61536 | The rigidity of the countable product of free groups | Yes, and even in the group category. More generally, I claim that if $\prod\_{i\in I}G\_i$ and $\prod\_{j\in J}H\_j$ are isomorphic groups, for two sets $I,J$ and two families $(G\_i)$, $(H\_j)$ of groups that are center-free and directly indecomposable, then there is a bijection $f:I\to J$ such that $G\_i$ and $H\_{f(... | 13 | https://mathoverflow.net/users/14094 | 335725 | 143,392 |
https://mathoverflow.net/questions/335686 | 13 | This question is strongly related to [this question](https://mathoverflow.net/questions/334912/representing-j-mathcalo-u-as-filtered-colimit-of-perfect-complexes). However it seems to me sufficiently distinct to warrant asking it separately.
>
> Let $X$ be a quasi-compact, quasi-separated scheme. When is the ∞-cate... | https://mathoverflow.net/users/43054 | When is the non-negative derived category compactly generated? | This is true more generally for every qcqs spectral algebraic space (assuming that D(X) has the meaning that I think it does). This is proven in (the current version of) Spectral Algebraic Geometry as 9.6.1.2. Probably there is some more classical reference in the case of schemes (it follows by a mild variation on Thom... | 8 | https://mathoverflow.net/users/7721 | 335733 | 143,393 |
https://mathoverflow.net/questions/335683 | 3 | Let us recall that an operator $T$ from a Banach space $X$ to a Banach space $Y$ is called $p$-nuclear if $T$ can be written as $$T=\sum\_{n=1}^{\infty}x^{\*}\_{n}\otimes y\_{n},$$
where $\|(x^{\*}\_{i})\_{i=1}^{\infty}\| \_{p}:=(\sum\_{i=1}^{\infty}\|x^{\*}\_{i}\|^{p})^{\frac{1}{p}}<\infty$ and $$\|(y\_{n})\_{n}\|\_{q... | https://mathoverflow.net/users/41619 | $p$-nuclear operators from $C(K)$ to $L_{p}$ | With the usual definition of a $p$-nuclear operator (see comment above), $\nu\_p(P\_\tau)\le1$: Let $x\_i^\*(f)= \int\_{A\_i} f / \mu(A\_i)^{1/q}$ and $y\_i= \chi\_{A\_i}/ \mu\_(A\_i)^{1/p}$. Then $P\_\tau= \sum x\_i^\* \otimes y\_i$, $\|(x\_i^\*)\|\_p=1$ und $\|(y\_i)\|\_q^w\le1$.
| 3 | https://mathoverflow.net/users/127871 | 335744 | 143,397 |
https://mathoverflow.net/questions/335746 | 15 | Stephen Smale famously proved in [[Trans. Amer. Math. Soc. 90 (1959), 281-290](https://doi.org/10.1090/S0002-9947-1959-0104227-9)] that any two $C^2$ immersions $S^2\to\mathbb R^3$ are regularly homotopic. This is how we knew that one can do a [sphere eversion](https://youtu.be/wO61D9x6lNY) before ever constructing suc... | https://mathoverflow.net/users/1409 | Immersions of surfaces in $\mathbb{R}^3$ | (Based on the comment of Mariano Suárez-Álvarez, there was a false assumption in my original answer. This is an attempt to correct it.)
1) Let $M$ be a closed smooth manifold with $k < n$. According to Smale-Hirsch theory, the space of immersions
$M^k \to \Bbb R^n$ is homotopy equivalent to the space of tangent bundl... | 17 | https://mathoverflow.net/users/8032 | 335749 | 143,399 |
https://mathoverflow.net/questions/335752 | 5 | Let $(X, \omega)$ be a closed symplectic manifold of dimension $2n$ and $\pi: X \rightarrow Q$ a Lagrangian torus fibration. Let $F\_q$ denote the fiber at $q \in Q$. It is claimed in a paper of Abouzaid "The Family Floer Functor is Faithful" that the Arnol'd-Liouville theorem furnishes an isomorphism $T\_q Q \cong H^1... | https://mathoverflow.net/users/123002 | Lagrangian torus fibrations and Arnol'd-Liouville theorem | By the time I finished writing this answer someone has explained the whole idea in a comment, but I thought I'd post it anyway as there is some more detail here. I assume the version of Arnold-Liouville you're familiar with says something like this: of you have a complete system of commuting Hamiltonians $(H\_1,\ldots,... | 6 | https://mathoverflow.net/users/10839 | 335755 | 143,401 |
https://mathoverflow.net/questions/331446 | 2 | Fix a discrete addition subgroup in $\mathbb{R}^n$. Given a finite spanning set, how can one find a group basis?
| https://mathoverflow.net/users/10668 | Find a lattice basis given too many points | Fix a $d$-dimensional discrete addition subgroup $L\subset \mathbb{R}^n$. Call the spanning elements $v\_1,\dots, v\_k\in L$ and the matrix whose columns are these $v$'s as $V\in \mathbb{R}^{n\times k}$. We seek a group basis $b\_1,\dots,b\_d\in L$.
Two solutions. I have tried the first one since it is readily imple... | 1 | https://mathoverflow.net/users/10668 | 335758 | 143,402 |
https://mathoverflow.net/questions/335761 | 2 | Let $(M,g)$ be a closed(oriented) Riemannian 4 manifold. It is well-known that, if $scal^g\geq0$ and not identically zero, then $M$ admits a PSC metric by conformal transformation.
**Q** Is $T^4$ the only oriented closed 4 manifold, which admits metric with flat scalar curvature but not PSC metric?
| https://mathoverflow.net/users/95296 | Flat scalar curvature on 4 manifold | No : K3 surfaces admit zero scalar curvature metrics (in fact even Ricci flat ones) but no metric with positive scalar curvature. See for instance the lecture notes by Gromov ([Four lectures on scalar curvature](https://www.ihes.fr/%7Egromov/wp-content/uploads/2019/04/scalar-lectures-IHES-2019.pdf)) p. 18.
This comes... | 6 | https://mathoverflow.net/users/8887 | 335763 | 143,403 |
https://mathoverflow.net/questions/335520 | 6 | Recall the following concept due to Malcev and Gromov. Let $C$ be some class of groups. A group $G$ is said to be approximable by the class $C$ if for every finite symmetric subset $F\subset G$ containing 1, there is a group $H$ in $C$ and an injection $f:F\to H$ such that $f(1)=1$ and $f(x)f(y)=f(z)$ for every $x,y,z\... | https://mathoverflow.net/users/142729 | What can the approximation of a group by some class be used for? | Let $\mathcal{F}\_k$ be the class of free groups that can be generated by at most $k$ elements. In your terminology, *limit groups* are the groups that can be approximated by the groups in $\mathcal{F}\_k$. [This](https://arxiv.org/abs/math/0401042) article by Champetier--Guirardel develops the theory of limit groups f... | 3 | https://mathoverflow.net/users/1463 | 335768 | 143,406 |
https://mathoverflow.net/questions/335777 | 2 | Suppose that we have a faithful representation $\rm{G}\rightarrow\rm{GL}(V)$ of a semisimple linear algebraic group into a complex vector space $\rm{V}$ of dimension n. Suppose that we have a projective algebraic curve $\rm{X}$ (or just a projective algebraic variety) and a $\rm{G}$-principal bundle over it $\pi:\rm{P}... | https://mathoverflow.net/users/140062 | Natural morphism to the scheme of isomorphism | The definition of $\mathrm{Isom}(V\times X, E)$ should be relative to $X$, so that it is a bundle on $X$. Namely, $\mathrm{Isom}(V\times X, E) = \{(x, \phi)\;;\; x\in X, \phi: V\cong E\_x\}$. It is then easy to see that $P\to \mathrm{Isom}(V\times X, E)$ should be defined as $P\ni g\mapsto (\pi(g), (v\mapsto (g^{-1}, v... | 0 | https://mathoverflow.net/users/38052 | 335781 | 143,407 |
https://mathoverflow.net/questions/335774 | 3 | Let $k$ be a field, let $X/k$ be a stable curve. Is it always possible to find a deformation $\mathcal{X}/k[[t]]$ such that $\mathcal{X}$ is regular?
(Sorry for the confusion, this is a duplication of one of my previous post....The answer to this question is yes, by Theorem B.2 in Brian Conrad’s Appendix to “Special... | https://mathoverflow.net/users/nan | Deformation of stable curve with regular total space | Yes for $k$ algebraically closed.
First of all, for each singularity, we can choose a local deformation over $k[[t]]$ such that the total space around this singularity is regular. In some local coordinates it can be written as $k[[x, y, t]] / (xy-t)$. To get a global deformation, one uses the local-global principle (1... | 2 | https://mathoverflow.net/users/38052 | 335786 | 143,409 |
https://mathoverflow.net/questions/335775 | 1 | Let $ X \hookrightarrow Y$ be a closed immersion of (connective) spectral Deligne-Mumford stacks, is $ i\_\* : Qcoh(X) \rightarrow Qcoh(Y)$ conservative? Somehow I couldn't find the statement in SAG...
| https://mathoverflow.net/users/38075 | Is the pushforward of a closed immersion of spectral Deligne-Mumford stacks conservative? | First, recall this is true when $X$ and $Y$ are spectral affine schemes. Indeed by [SAG, Proposition 2.5.1.1], $i\_\*$ is t-exact so it suffices to show this on the hearts, and in particular you can reduce to the case where $X$ and $Y$ are 0-truncated/ordinary affine schemes, which is obvious.
The general case follow... | 1 | https://mathoverflow.net/users/85136 | 335795 | 143,410 |
https://mathoverflow.net/questions/335797 | 2 | Following up to the question [raised here](https://mathoverflow.net/questions/57410/rate-of-convergence-of-smooth-mollifiers), I am searching for a reference (or a simple argument) to establish (in the whole space) the following (suggested) equivalence :
>
> Given $N\_1$ and $N\_2$ two (homogeneous spaces semi-) n... | https://mathoverflow.net/users/27767 | Rate of convergence of mollifiers // Sobolev norms | You have, say with $\varphi\ge 0$ even, with integral 1,
$$
(f\ast \varphi\_\delta)(x) -f(x)=\int \bigl(f(x+\delta z)-f(x)\bigr)\varphi(z) dz.
$$
As a consequence, we get with Taylor's formula with integral remainder,
$$
(f\ast \varphi\_\delta)(x) -f(x)=\int \int\_0^1(1-\theta)f''(x+\theta \delta z)\delta^2 z^2\varphi(... | 1 | https://mathoverflow.net/users/21907 | 335800 | 143,412 |
https://mathoverflow.net/questions/335753 | 1 | Assume $\lambda\_1+\lambda\_2=1$ and both $\lambda\_1$ and $\lambda\_2$ are positive reals.
>
> **QUESTION.** What is the value of this limit? It seems to exist.
> $$\lim\_{n\rightarrow\infty}\int\_0^1\frac{(\lambda\_1+\lambda\_2x)^n-x^n}{1-x}\,dx.$$
>
>
>
| https://mathoverflow.net/users/66131 | Limits of a family of integrals | The limit is $-\ln(\lambda\_{2})$.
\begin{align\*}
\int\_{0}^{1}\frac{(\lambda\_{1}+\lambda\_{2}x)^{n}-x^{n}}{1-x}dx
&= \int\_{0}^{1}\frac{(\lambda\_{1}+\lambda\_{2}x)^{n}-1}{1-x}dx + \int\_{0}^{1}\frac{1-x^{n}}{1-x}dx \\
&= \int\_{0}^{1}\frac{(1-\lambda\_{2}t)^{n}-1}{t}dt + \int\_{0}^{1}\sum\_{k=0}^{n-1} x^{k-1}dx... | 9 | https://mathoverflow.net/users/50901 | 335809 | 143,414 |
https://mathoverflow.net/questions/335811 | 3 | Prove that$$\lim\_{n\to\infty}\frac1n\sum\_{i\_1,i\_2,...i\_k=1}^n\lambda\_1^{|i\_1-i\_2-s\_1|}\lambda\_2^{|i\_2-i\_3-s\_2|}...\lambda\_k^{|i\_k-i\_1-s\_k|}$$is equal to$$\sum\_{j=1}^k\lambda\_j^{S+k-1}\prod\_{l=1,l\ne j}^k\frac{1-\lambda\_l^2}{(\lambda\_j-\lambda\_l)(1-\lambda\_j\lambda\_l)}$$where all $\lambda$s are,... | https://mathoverflow.net/users/141969 | Prove an existing formula for a limit of a specific sum | Denote $$A=\sum\_{p\_1+\ldots+p\_k=S} \lambda\_1^{|p\_1|}\lambda\_2^{|p\_2|}\ldots
\lambda\_k^{|p\_k|}.\quad (1)$$
We prove two things:
1) Your limit equals $A$.
2) $A$ equals to what you write.
Start with 1). Note that any summand in your expression is of the form $\lambda\_1^{|p\_1|}\lambda\_2^{|p\_2|}\ldots
\... | 5 | https://mathoverflow.net/users/4312 | 335816 | 143,416 |
https://mathoverflow.net/questions/335810 | 9 | Given two monoidal categories $\mathcal{M}$ and $\mathcal{N}$, can one form their tensor product in a canonical way?
The motivation I am thinking of is two categories that are representation categories of two algebras $R$ and $S$, and the module category of the tensor product algebra $R \otimes\_{\mathbb{C}} S$.
A... | https://mathoverflow.net/users/125941 | The tensor product of two monoidal categories | The book [Tensor Categories](http://www-math.mit.edu/~etingof/egnobookfinal.pdf) discusses, with many variations, the details of Robert McRae's answer. Just like for vector spaces, there are a number of related but inequivalent "tensor products" of linear categories, with the choice dependent on the types of linear cat... | 10 | https://mathoverflow.net/users/78 | 335818 | 143,418 |
https://mathoverflow.net/questions/335803 | 6 | Let $S$ be a non-empty, possibly infinite, set of integers, all of which are greater than $1$. For a given group $G$, let $S[G]$ denote the collection of statements
$$
\forall (n \in S, a \in G, b\in g) \,\,(ab)^n=a^nb^n
$$
For some sets $S$, $S[G]$ is sufficient to prove $G$ is Abelian. For example, the set $\{2\}$ is... | https://mathoverflow.net/users/82067 | Classification of minimal sets of properties proving a group is Abelian | Here is an answer to the second question. Call $S$ an *abelian forcing set* if $S[G]$ implies that $G$ is abelian.
**Proposition.** If $S$ is an abelian forcing set then some finite subset of $S$ is an abelian forcing set. So there is no infinite minimal abelian forcing set.
**Proof.** Given $n>0$, let $\phi\_n$ ... | 8 | https://mathoverflow.net/users/38253 | 335823 | 143,419 |
https://mathoverflow.net/questions/335757 | 3 | If $G, H$ are simple, undirected graphs, we define the *exponential graph* $\text{Exp}(G,H)$ to be the following graph:
* the vertex set is the set of all maps $f:V(G)\to V(H)$
* two maps $f\neq g: V(G)\to V(H)$ form an edge if and only if whenever $\{v,w\}\in E(G)$ then $\{f(v), g(w)\}\in E(H)$.
If $G$ is any simp... | https://mathoverflow.net/users/8628 | Induced subgraphs of $\text{Exp}(G, K_2)$ | In the sequel we allow $\mathrm{Exp}(\cdot, \cdot)$ to have loops, then the only induced subgraphs of $\mathrm{Exp}(G', K\_2)$ are disjoint unions of isolated vertices, complete graphs with a loop on each vertex (denote such a graph $\overline{K\_n})$ and complete bipartite graphs.
First, consider a non-empty connect... | 3 | https://mathoverflow.net/users/106512 | 335824 | 143,420 |
https://mathoverflow.net/questions/335780 | 8 | It is well known that a subset $Y$ of a Polish space $X$ is completely metrisable iff it is a $G\_\delta$ subset. This relates a relative topological property of the subspace $Y \subset X$ to an absolute topological property of $Y$.
I wonder are there more general versions of this result where $G\_\delta$ is replaced... | https://mathoverflow.net/users/58082 | Do the higher levels of the Borel hierarchy correspond to absolute topological properties? | You can use complete Borel sets for this.
Recall that for Polish spaces $X,Y$ and subsets $A\subseteq X$ and $B\subseteq Y$, $A$ is *Wadge-reducible* to $B$ if there is a continuous $f:X\to Y$ such that $f^{-1}(B) = A$. It can be proved (for instance, see Theorem 22.10, and Exercises 22.11 and 24.20 in Kechris' *Clas... | 2 | https://mathoverflow.net/users/66044 | 335825 | 143,421 |
https://mathoverflow.net/questions/335790 | 5 | I've encounter a group with properties that are very familiar, but I cannot say what group is it.
Consider the variables $(t,x,y,z)$, an affine transformation $M \in A(3)$ on the last three variables can be represented as
$$ M =
\begin{pmatrix}
1 & 0
\\
\vec{a} & {R}
\end{pmatrix},$$
such that
$$
\begin{pmatrix}
t'
... | https://mathoverflow.net/users/25356 | Identification problem: Does this group have a name? | The group is a (homogeneous) Carroll group: see eq. 2.7 in <https://arxiv.org/abs/1405.2264>
| 5 | https://mathoverflow.net/users/32389 | 335838 | 143,426 |
https://mathoverflow.net/questions/335829 | 3 | The development of Fast Fourier transform is attributed to Cooley & Tukey, both have written a lot about it is historical development. Both Cooley and Tukey call it a re-discovery rather. However, I am searching for early publications which showed how people made the conversions in discrete frequency axis from time to ... | https://mathoverflow.net/users/142414 | Origin of the theorem related to the integral transform pair | I quote from [Gauss and the history of the fast Fourier transform](https://www.researchgate.net/publication/226049108_Gauss_and_the_history_of_the_fast_Fourier_transform) (1985) (DFT = Discrete Fourier Transform):
>
> [Alexis-Claude Clairaut](https://en.wikipedia.org/wiki/Alexis_Clairaut) (1713-1765) published in 1... | 5 | https://mathoverflow.net/users/11260 | 335842 | 143,428 |
https://mathoverflow.net/questions/331471 | 5 | If $A$ is the coordinate ring of a smooth variety over a finite field is it known whether the kernel of the determinant map $K\_1(A)\rightarrow A^{\times}$ is torsion or not?
| https://mathoverflow.net/users/127776 | Kernel of the determinant morphism from the first algebraic K-theory | I think the following affine deleted quadric is an example where the kernel of the determinant fails to be torsion. Assume we are working in odd characteristic. Then we have the smooth affine quadric $Q\_2$ defined by the equation $XY+Z^2=1$. We have $K\_0(Q\_2)=\mathbb{Z}\oplus\mathbb{Z}$ (either by direct computation... | 3 | https://mathoverflow.net/users/50846 | 335851 | 143,435 |
https://mathoverflow.net/questions/335484 | 19 | Gross-Hacking-Keel-Kontsevich (<https://arxiv.org/abs/1411.1394>) constructed a canonical basis (the so-called “theta basis”) for a cluster algebra, at least assuming it satisfies a certain combinatorial condition. This condition holds for the homogeneous coordinate ring of the Grassmannian (see e.g. <https://arxiv.org... | https://mathoverflow.net/users/25028 | Is it possible that the GHKK canonical basis for cluster algebras is the Lusztig/Kashiwara dual canonical basis? | I think there is good reason to think the answer is "no".
In rank 2, the theta basis agrees with the greedy basis ([arXiv:1508.01404](https://arxiv.org/abs/1508.01404)). Greedy basis elements are indecomposable positive elements (see [arXiv:1208.2391](https://arxiv.org/abs/1208.2391)) (i.e., they cannot be written a... | 10 | https://mathoverflow.net/users/468 | 335859 | 143,437 |
https://mathoverflow.net/questions/335856 | 19 | Is it constructively true that all (not necessarily finitary) equational theories $T = (\Sigma, E)$ have an initial model?
The usual proof for finitary equational theories I know constructs first from the signature $\Sigma$ the set $P$ of syntax trees/preterms. This set is by construction the initial model of the the... | https://mathoverflow.net/users/84063 | Constructive proof of existence of free algebras for infinitary equational theories | It was proved by Andreas Blass in [Words, free algebras, and coequalizers](https://www.impan.pl/en/publishing-house/journals-and-series/fundamenta-mathematicae/all/117/2/103865/words-free-algebras-and-coequalizers) that free infinitary algebras are not constructible neither in topoi nor in ZF. It is easy to see that th... | 16 | https://mathoverflow.net/users/62782 | 335860 | 143,438 |
https://mathoverflow.net/questions/335862 | 4 | In theses [these notes](https://www.dpmms.cam.ac.uk/~hjrw2/Notes/cubenotes.pdf), Example 5.6, it is said that there is a "symmetric tiling of a genus 2 surface by 8 right-angled hyperbolic pentagons".
Question 1: What does this tiling look like?
Question 2: Is it always possible to tile a genus $n$ surface by $f$ r... | https://mathoverflow.net/users/142946 | Tiling of genus 2 surface by 8 pentagons | A picture answering Question 1 is here: <https://mathoverflow.net/a/331408/1345>
Question 2 is a duplicate of [regular tiling of a surface of genus 2 by heptagons](https://mathoverflow.net/questions/198040/regular-tiling-of-a-surface-of-genus-2-by-heptagons), although as you point out the accepted answer there is uns... | 7 | https://mathoverflow.net/users/1345 | 335871 | 143,442 |
https://mathoverflow.net/questions/198040 | 12 | Let $S$ be a Riemann Surface of genus 2. Is there a picture in the literature for a regular tiling of $S$ by 12 heptagons (where three heptagons meet at each vertex)? Also, apart from the obvious restriction given by the Euler characteristic $2-2g=f-nf/2+nf/v$ (where $g$ is the genus, $f$ is the number of faces, $v$ is... | https://mathoverflow.net/users/19369 | regular tiling of a surface of genus 2 by heptagons | Addressing the second question positively, this is the Main Theorem of this paper:
*Edmonds, Allan L.; Ewing, John H.; Kulkarni, Ravi S.*, [**Regular tessellations of surfaces and (p,q,2)-triangle groups**](http://dx.doi.org/10.2307/2007049), Ann. Math. (2) 116, 113-132 (1982). [ZBL0497.57001](https://zbmath.org/?q=a... | 4 | https://mathoverflow.net/users/1345 | 335874 | 143,443 |
https://mathoverflow.net/questions/335875 | 2 | Let $\theta$ be the adjacency matrix of a simple graph (symmetric and zeros on the diagonal). What is the characterization of those $\theta$ which satisfy $$\theta^2 \equiv 0 \pmod{2}$$ i.e. which $\theta$ are nilpotent of order 2 over $\mathbb{Z}\_2$?
| https://mathoverflow.net/users/142955 | Characterization of nilpotent adjacency matrices | I do not think you will find a characterisation that is substantially better than the one suggested by LSpice, because this class includes other classes of graphs for which characterisations are unknown.
In particular, a $(0,2)$-graph is a graph such that each pair of distinct vertices has exactly 0 or 2 common neigh... | 3 | https://mathoverflow.net/users/1492 | 335892 | 143,448 |
https://mathoverflow.net/questions/335863 | 5 | Let $M$ be a Riemannian manifold with volume measure $\lambda$, let $f \colon M \to M$ be a continuous map, and let $\mu$ be a probability measure on $M$ with compact support.
>
>
> >
> > **Definition.** The measure $\mu$ is a *physical measure* of $f$ if there is a set $V \subset M$ with [PropertyX] such that fo... | https://mathoverflow.net/users/15570 | Is there a name for a "stable" physical measure? | I realize this doesn't directly answer the "reference request" part of the question, but I believe that if you require $V$ to be full (Lebesgue) measure in a neighborhood of the support of $\mu$, then
this definition is too strong to be satisfied by any of the usual examples that one would want, going beyond delta mea... | 1 | https://mathoverflow.net/users/5701 | 335900 | 143,453 |
https://mathoverflow.net/questions/335902 | 7 | To parametrize Teichmüller space, it suffices to measure the hyperbolic lengths of a finite number of curves. It is well-known that $9g-9$ curves suffice, by a standard pair-of-pants argument given in, for instance Fathi-Laudenbach-Poenaru.
I recall that you need exactly $6g-5$ curves: you cannot achieve it by $6g-6$... | https://mathoverflow.net/users/5010 | To find a point in Teichmüller space or measured foliation, how many lengths of curves do you need? | As indicated in my comments, the Teichmüller question is a duplicate of this [question](http://mathoverflow.net/q/243622/1345).
For the measured lamination case, the fact that $6g-5$ curves suffice was shown by Hamenstädt.
*Hamenstädt, Ursula*, [Parametrizations of Teichmüller space and its Thurston boundary.](htt... | 10 | https://mathoverflow.net/users/1345 | 335903 | 143,455 |
https://mathoverflow.net/questions/335799 | 3 | The Kähler Ricci flow on a compact Kähler manifold are formulated as $\frac{\partial}{\partial t}w(t) = -Ric(w)$, $w(0) = w\_0$, where $w(t)$ is a family of Kähler metrics and $w\_0$ is the initial Kähler metric. But it is not obvious to me in which space we differentiate $w(t)$. My guess is the vector space of the clo... | https://mathoverflow.net/users/142911 | In which space are we solving the Kähler Ricci flow? | The object $\omega(t)$ is not a cohomology class, it is a $(1,1)$-form. The time derivative of $\omega(t)$ is taken in each vector space $\Lambda^{1,1} T^\*\_m M$, pointwise. If $\omega(t)$ is $C^k$, then local coordinate calculation shows that $\operatorname{Ric}\_{\omega(t)}$ is $C^{k-2}$. If you vary the choice of a... | 2 | https://mathoverflow.net/users/13268 | 335908 | 143,457 |
https://mathoverflow.net/questions/335869 | 6 | Given a Dynkin quiver $Q$ and a field $K$.
>
> Question 1: For which such $Q$ are there only finitely many indecomposable representations over the dual numbers $K[x]/(x^2)$?
>
>
>
Note that those representations are exactly those of $KQ \otimes\_K K[x]/(x^2)$.
This is for example true for $Q$ being of type $A\... | https://mathoverflow.net/users/61949 | Representation-finite quivers over dual numbers | You can find this result in [Geiss, Leclerc, Schröer: Quivers with relations for symmetrizable Cartan matrices I: Foundations] as Proposition 13.1. They consider a more general class of algebras. The class of algebras you are considering is obtained by setting their parameters $(c\_1,\dots,c\_m)$ to all the same number... | 3 | https://mathoverflow.net/users/15887 | 335914 | 143,460 |
https://mathoverflow.net/questions/335916 | 4 | I follow Kobayashi "Differential Geometry of Complex Vector Bundles", pages 11-12, prop. 4.9. Given a rank-$r$ Hermitian holomorphic vector bundle $(E,h)$ over a complex manifold $M$, there exists a unique $h$-connection $D$ such that $D^{0,1}=\bar\partial$.
The proof works in a local holomorphic frame $s=(s\_1,...,... | https://mathoverflow.net/users/126993 | Confusion about complex differential forms | Forget about vector fields $\partial\_{z^{\mu}}$ and $\partial\_{z^{\bar\mu}}$. Just think about 1-forms: $\omega^j\_i = \Gamma^j\_{i\mu}dz^{\mu}$, with $C^{\infty}$ functions $\Gamma^j\_{i\mu}(z)$. Then it is clear why $\partial h$ is the $(1,0)$-part of $dh$, and so if we write out $h^{-1}dh=\omega+\bar\omega$ as $(1... | 4 | https://mathoverflow.net/users/13268 | 335919 | 143,461 |
https://mathoverflow.net/questions/335929 | 2 | On a Hilbert space $\cal H$, consider an essentially self-adjoint operator $A\colon Dom(A)\to {\cal H}$, and a vector $\psi\in\bigcap\_{n=1}^\infty Dom(A^n)$. Without further assumptions, can we say that the associated spectral measure $\mu\_\psi$ is the unique Borel measure satisfying
$$\langle \psi,A^n\psi\rangle = \... | https://mathoverflow.net/users/49288 | are spectral measures characterized by their moments? | The answer is no. There are random variables with all moments finite, whose distributions are not determined by their moments (e.g. the log-normal distribution).
Let $X$ be such a random variable and let $\mu$ be its distribution. Then the operator of multiplication by $x$ on the space $L^{2}(\mathbb{R},\mu)$ has $\p... | 3 | https://mathoverflow.net/users/24953 | 335930 | 143,464 |
https://mathoverflow.net/questions/335742 | 7 | There are lots of similar questions that have been answered on this topic (particularly [Homotopy limit-colimit diagrams in stable model categories](https://mathoverflow.net/questions/135462/homotopy-limit-colimit-diagrams-in-stable-model-categories)), but I have a specific question that I do not believe has been answe... | https://mathoverflow.net/users/nan | Homotopy pullbacks and pushouts in stable model categories | Many thanks and much credit to Dmitri Pavlov for clearing up my confusions. In the interest of having a definitive answer, I've organized his comments and my understanding into the following answer.
Let $\mathcal{C}$ be a stable model category; then (by definition, for Hovey) $Ho(\mathcal{C})$ is a triangulated categ... | 2 | https://mathoverflow.net/users/nan | 335933 | 143,465 |
https://mathoverflow.net/questions/335932 | 8 | I'm looking for a reference (or proof) for the statement given in the title: that when we have an adjunction between quasicategories in the sense of Riehl and Verity (defined e.g. in Section 4 of their paper "The 2-category theory of quasi-categories" or Definition 2.1.1 of their ongoing book project at <http://www.mat... | https://mathoverflow.net/users/132357 | Reference request: The unit of an adjunction of $\infty$-categories in the sense of Riehl-Verity is a unit in the sense of Lurie | Let $\mathcal{C},\mathcal{D}$ two $\infty$-categories and $f:\mathcal{C}\to\mathcal{D}$ and $g:\mathcal{D}\to \mathcal{C}$ two functors.
Recall (HTT.5.2.2.7) that a natural transformation $u:1\_{\mathcal{C}}\to gf$ is a *unit transformation* if the natural transformation
$$(\ast)\qquad\mathrm{Map}\_{\mathcal{D}}(f-,-... | 11 | https://mathoverflow.net/users/43054 | 335934 | 143,466 |
https://mathoverflow.net/questions/119464 | 26 | Almost 17 years ago, I asked the following question on USENET, motivated by a method in numerology (I kid you not).
Pick integers $n \ge 2$, $k \ge 1$. Toss $n$ $k$-sided dice. The sides of each die are numbered $0,1,\ldots,k-1$. The dice are unbiased and the tosses are independent.
**What is the probability $P(n,... | https://mathoverflow.net/users/9025 | probability of zero subset sum | First off, let me prove that all $P(n,k)=0$ for $n\geq k$, which follows from a simple lemma:
**Lemma.** In any sequence of $k\geq 1$ integers $m\_1, m\_2, \dots, m\_k$, there exists a subsequence summing to a multiple of $k$.
**Proof.** Define $s\_i := (m\_1+m\_2+\dots+m\_i)\bmod k$, including $s\_0=0$. By the pig... | 2 | https://mathoverflow.net/users/7076 | 335940 | 143,469 |
https://mathoverflow.net/questions/332436 | 3 | Let $R\_n$ be the integral polynomial ring $\mathbb{Z}[x\_1,x\_2,...,x\_n]$, let $A\_n$ be the group of ring automorphisms $\mathrm{Aut}(R\_n)$, and for $f\in R\_n$ let $\mathrm{Aut}(f)=\{\alpha\in A\_n\ |\ \alpha(f)=f\}$.
Define a polynomial $f\in R\_n$ to be *interesting* if $\deg\_{x\_i}(f)\geq 1$ for $1\leq i\leq... | https://mathoverflow.net/users/12218 | Infinite order automorphisms of planar polynomials | First, YCor answered the first question in the comments. Here is a summary:
Take any polynomial $f(x)$ in only one variable $x$ and apply any automorphism $\alpha$ to get $\alpha(f)$. Then $\alpha\circ T\circ \alpha^{-1}$ is an automorphism that fixes $\alpha(f)$ whenever $T$ is an automorphism that fixes $x$. In par... | 2 | https://mathoverflow.net/users/12218 | 335944 | 143,472 |
https://mathoverflow.net/questions/335948 | 4 | Let $f\in L^2(0,\infty)$ be a positive, decreasing function. Is it then true that
$$
\limsup\_{x\to\infty} xf(x) = \limsup\_{x\to\infty} \frac{1}{f(x)}\int\_x^{\infty} f^2(t)\, dt
$$
(and similarly for $\liminf$)?
This looks strange at first; for example, the quotient of the two quantities can easily become both larg... | https://mathoverflow.net/users/48839 | Comparing two limsup's | No, this equality does not hold in general. What is true is that rhs is at most twice as large as lhs and this is sharp.
For the sake of brevity I will only show an example where lhs is $1$ and rhs is greater than $1.9$. Choose a very fast growing sequence $a\_n \ge 1$. We will construct our function on each interva... | 7 | https://mathoverflow.net/users/104330 | 335956 | 143,476 |
https://mathoverflow.net/questions/335891 | 3 | Let $G$ be an adjoint Chevalley group. Are all of $G(\mathbb Z)$'s finite-index subgroups congruence subgroups?
I read a theorem that states: When $G$ is the universal Chevalley group and it's not of type $\operatorname{SL}\_{2}$ every finite index subgroup of $G(\mathbb Z)$ is a congruence subgroup (in [Matsumoto -... | https://mathoverflow.net/users/142244 | For $G$ an adjoint Chevalley group, are all of $G(\mathbb Z)$'s finite-index subgroups congruence subgroups? | Chapter VI of my old Springer Lecture Notes in Mathematics 789 *Arithmetic Groups* (1980) is in English and gives a version of H. Matsumoto's and C. Moore's arguments for the subgroups of $\operatorname{SL}\_n(\mathbb{Z})$ of finite index when $n \geq 3$. But this account also gives some clues to the general case, wher... | 7 | https://mathoverflow.net/users/4231 | 335958 | 143,477 |
https://mathoverflow.net/questions/335923 | 6 | Let $X$ be an affine scheme with an open affine subscheme $U\subset X$. Given an automorphism of $U$, we can glue $X$ with itself along $U$ to get a new scheme. Is there a description in terms of commutative algebra of automorphisms such that the resulting scheme is affine/separated?
If $U=\mathrm{Spec}\:B$, $X=\mat... | https://mathoverflow.net/users/nan | When does glueing affine schemes produce affine/separated schemes? | Here are some thoughts in the case of gluing a DVR along an automorphism of its fraction field:
**Setup**: Let $A$ be a DVR with uniformizer $\pi$ and fraction field $K$, and let $\varphi : K \to K$ be a ring automorphism. Let $S$ be the gluing of two copies of $\operatorname{Spec} A$ along the automorphism of $\oper... | 5 | https://mathoverflow.net/users/15505 | 335962 | 143,478 |
https://mathoverflow.net/questions/335952 | 2 | I'm trying to combine two ways of looking at the Laplacian $\Delta$ on $\mathbb R^n$ (and on other domains).
Firstly, it is well known that this operator is essentially self-adjoint on $C\_c^\infty(\mathbb R^n)$. Secondly, I know that for $f,g \in C^\infty\_c(\mathbb R^n)$ it holds that $\langle u, \Delta u \rangle ... | https://mathoverflow.net/users/104461 | Domain of definition of Laplace Operator on $L^2$ | Maybe this is the argument you're looking for.
Suppose you know that $\Delta$ is essentially self-adjoint on $C^\infty\_c(\mathbb{R}^n)$. This means $C^\infty\_c$ is a core for $\Delta$, so for any $u \in D(\Delta)$, there is a sequence $u\_n \in C^\infty\_c$ such that $u\_n \to u$ and $\Delta u\_n \to \Delta u$ in $... | 7 | https://mathoverflow.net/users/4832 | 335976 | 143,482 |
https://mathoverflow.net/questions/335889 | 10 | The universal enveloping algebra of a Lie algebra has a canonically defined Hopf algebra structure. Is the same true of the universal enveloping of a super Lie algebra? A presentation in terms of the universal properties would be of most interest.
| https://mathoverflow.net/users/125941 | Hopf structure on the universal enveloping of a super Lie algebra | No, it is not true unless the odd part is zero or the characteristic is 2. More precisely, there is no natural Hopf algebra structure outside these conditions, but there may be some odd examples. It is a Hopf superalgebra instead, which is what Peter May means in his answer.
You can see this on the easiest example: $... | 5 | https://mathoverflow.net/users/5301 | 335986 | 143,484 |
https://mathoverflow.net/questions/335963 | 6 | Let $G$ be a Lie group acting on a manifold $M$.
Consider the transformation groupoid $\mathcal{G}=(G\times M\rightrightarrows M)$. We have the notion of de Rham cohomology of a Lie groupoid by considering de Rham cohomology of simplicial manifold associated to it, denoted by $\mathcal{G}\_\bullet$.
Is there any r... | https://mathoverflow.net/users/118688 | De Rham cohomology of Lie groupoid | Proposition $13$ and Remark $16$ in page $10$ of [Cohomology or Stacks](https://www.math.ubc.ca/~behrend/CohSta-1.pdf) says that, there is a natural isomorphism
$$H^i\_G(X)\rightarrow H^i(X\times G\rightrightarrows X)$$
where, $H^i\_G(X)$ is the $i^{\text{th}}$ equivariant cohomology of $X$ with respect to action of... | 1 | https://mathoverflow.net/users/118688 | 335991 | 143,485 |
https://mathoverflow.net/questions/335966 | 8 | This question arose out of [this stack exchange post](https://math.stackexchange.com/questions/3287675/taming-mathbbr-diffeomorphisms). I am wirting a thesis about the $s$-cobordism theorem and Siebenmann's work about end obstructions. Combined they give a quick proof of the uniqueness of the smooth structure for $\mat... | https://mathoverflow.net/users/134967 | Wildness of codimension 1 submanifolds of euclidean space | For $n\geq 5$, such a diffeomorphism exists iff the complement of $S$ has an unbounded connected component. Indeed, the existence of such a diffeomorphism implies there is an unbounded component of the complement of $S$. Conversely, suppose that the complement of $S$ has an unbounded component. Choose a proper arc $\ga... | 7 | https://mathoverflow.net/users/35353 | 336002 | 143,488 |
https://mathoverflow.net/questions/336006 | 7 | Take a weak solution $u$ of the Poisson equation on $\mathbb{R}^d$
$$ \Delta u = f $$
By standard elliptic regularity theory we have (for some $\alpha\in (0,1]$) $f\in C^{0, \alpha}\_{\text{loc}}(\mathbb{R}^d)$, then $u \in C^{2, \alpha}\_{\text{loc}}(\mathbb{R}^d)$. Now my (naive) question is:
>
> Are there some w... | https://mathoverflow.net/users/91098 | Minimal assumptions such that the solution of Poisson equation is $C^2$ | *Dini continuity* may be what you are looking for: if $f = \Delta u$ is Dini continuous, that is, $$\int\_0^1 \frac{\omega\_f(t)}{t} \, dt < \infty,$$ then $u$ is $C^2$. This is a rather old result, but I do not know the reference. A quick Google search leads to [*Poisson's equation*](http://www.math.mcgill.ca/gantumur... | 9 | https://mathoverflow.net/users/108637 | 336015 | 143,491 |
https://mathoverflow.net/questions/324838 | 10 | Let $F$ be a field of characteristic not $2$. A well-known theorem of Pfister asserts that the torsion of $W(F)$, the Witt group of $F$, is $2$-primary.
Baeza [B, V.6.3] extended this result to Witt groups of semilocal (commutative) rings $A$ (assume $2\in A^\times$ for simplicity).
**Question:** Is it known whethe... | https://mathoverflow.net/users/86006 | Is it possible for the Witt group of a scheme to have non-trivial odd torsion? | Examples of smooth real varieties whose Witt group has odd torsion can be found in a paper of Jacobson:
* J.A. Jacobson. From the global signature to higher signatures. arXiv:1411.0993, <https://arxiv.org/pdf/1411.0993.pdf>
The idea is the following: from the Gersten conjecture for Witt groups, there is a spectral... | 4 | https://mathoverflow.net/users/50846 | 336021 | 143,495 |
https://mathoverflow.net/questions/335996 | 3 | Does there exist $p>1$ such that for all $n\geq 2$, if $(a\_{ij})$ and $(b\_{ij})$ are symmetric positive semidefinite $n\times n$ matrices and $a\_{ij}, b\_{ij}\geq 0$ then $\bigl(\|(a\_{ij},b\_{ij})\|\_p\bigr)=\bigl((a\_{ij}^p+b\_{ij}^p)^{1/p}\bigr)$ is also positive semidefinite?
Maybe, a simpler question: is it ... | https://mathoverflow.net/users/143037 | Hadamard $\ell_p$ sum of two symmetric positive semidefinite matrices | Looks like there any many counterexamples already when $n=3$.
Take $u=(0,1,1)$, and $v = (1,2,0)$. Consider rank-1 matrices $A=u^{T}u$ and $B=v^{T}v$. Then the $\ell^{p}$ Hadamard matrix is
$$
\begin{pmatrix}
1 & 2 & 0\\
2 & (1+2^{2p})^{1/p} & 1\\
0 & 1 & 1
\end{pmatrix}
$$
whose determinant is $(1+2^{2p})^{1/p}... | 5 | https://mathoverflow.net/users/50901 | 336028 | 143,499 |
https://mathoverflow.net/questions/336027 | 3 | I came across Marijn Heule and Oliver Kullmann's paper on recent techniques in highly efficient [SAT solvers](https://cacm.acm.org/magazines/2017/8/219606-the-science-of-brute-force/fulltext). In particular they describe the [Pythagorian Triple Problem](https://en.wikipedia.org/wiki/Boolean_Pythagorean_triples_problem)... | https://mathoverflow.net/users/63938 | Mapping problems to Boolean formulas for SAT solvers | I think it's very much specific to particular problems. The problem statement might be directly translatable into logic clauses, but for nontrivial problems it definitely helps if you formulate the problem in such a way that the SAT solver may handle it efficiently.
I might mention [this paper](https://doi.org/10.33... | 6 | https://mathoverflow.net/users/13650 | 336032 | 143,500 |
https://mathoverflow.net/questions/335994 | 0 | For any positive integer $[n]$, let $[n]=\{1,\ldots,n\}$. Let $r\in\mathbb{R}$. We define for every positive integer $n\in\mathbb{N}$ the *minimal difference from a rational with denominator $\leq n$ to* $r$ by $$\text{md}\_n(r)=\min\{|r-\frac{a}{b}|: a\in\mathbb{Z}, b\in[n] \},$$ and let $$d\_n(r) = \min\{b\in[n]: \ex... | https://mathoverflow.net/users/8628 | Denominator approximation sequence of a real number | Look up rational approximation in Wikipedia to find answers to your questions. It is pretty lovely.
This question does not seem well thought through. Why not just list $1,3,4,9,13$ instead of $1,1,3,4,4,4,4,4,9,9,9,9,13?$
You need only consider $0 \leq r \leq 1/2$ since the sequences for $r$ and $1-r$ have the sam... | 2 | https://mathoverflow.net/users/8008 | 336035 | 143,501 |
https://mathoverflow.net/questions/331367 | 7 | This is a technical problem about applications of Grothendieck topologies. In some recent works, the technique of $h$-topology and $h$-descent is very useful, for an introduction see <https://stacks.math.columbia.edu/tag/0ETQ>.
However, it seems that such technique is not well-known (for example, there are very few d... | https://mathoverflow.net/users/102104 | Applications of $h$-topology and $h$-descent | I guess a relevant part of the philosophy is that schemes are smooth locally in the $h$-topology. So, if we are in characteristic zero, we can use resolution of singularities to produce an $h$-hypercovering $U\_\bullet\to X$ of a scheme $X$. Then, if we have something like a cohomology theory, which is defined on smoot... | 10 | https://mathoverflow.net/users/50846 | 336055 | 143,506 |
https://mathoverflow.net/questions/336058 | 2 | I find it hard to find information on the so-called "Gaussian null coordinates", which [Wikipedia](https://en.wikipedia.org/wiki/Construction_of_a_complex_null_tetrad) says is used to describe "near horizon geometries". Can someone provide a reference where I can read about the basics and usage of them?
| https://mathoverflow.net/users/142501 | Gaussian null coordinates | The [M.Sc. thesis](http://www.theorie.physik.uni-goettingen.de/forschung/qft/theses/dipl/Morfa-Morales.pdf) of Eric Morales dicusses the basics of Gaussian null coordinates in appendix D and gives an application in chapter 3.
| 4 | https://mathoverflow.net/users/11260 | 336059 | 143,507 |
https://mathoverflow.net/questions/334790 | 5 | What corresponds to $\forall m\forall n(2m \neq 2n+1)$ or $\forall p\forall q(p^2 \neq 2q^2)$ in the monadic theory of the real line?
[Shelah (1975)](https://www.jstor.org/stable/1971037?seq=1#page_scan_tab_contents) proved that arithmetic can be reduced the monadic theory of the real line. The paper gives a procedur... | https://mathoverflow.net/users/nan | Translating basic number theory to the monadic theory of the real line | Consider the sentence $\theta$:
$$\exists m,n\, (2n=2m+1)$$
This is the negation of the first sentence in the above question. In relational form this is
$$\exists m,n,u,v\, (Smmu \wedge Snnv \wedge Ruv)$$
where $Sabc$ is the relation for $a+b=c$ and $Ruv$ is the relation for $u+1=v$.
**To find out whether $\theta$ is... | 1 | https://mathoverflow.net/users/nan | 336060 | 143,508 |
https://mathoverflow.net/questions/335931 | 6 | Is there a specific terminology for a symmetric monoidal category in which for any object $x$ the switch map $x\otimes x\to x\otimes x$ is the identity ? (Or alternatively the action of the symmetric group $\mathfrak{S}\_n$ on $x^{\otimes n}$ is trivial.)
Is there a paper or book I can cite where basic properties of ... | https://mathoverflow.net/users/68479 | Symmetric monoidal category with trivial switch morphisms | Coincidentally, terminology for such categories has been introduced very recently:
* John C. Baez and Jade Master. Open Petri Nets. Nov 2018. [arXiv:1808.05415](https://arxiv.org/abs/1808.05415)
More precisely, the authors refer to a strict symmetric monoidal category in which (not only the associator and unitors b... | 4 | https://mathoverflow.net/users/1015 | 336068 | 143,512 |
https://mathoverflow.net/questions/336078 | 12 | Are L-functions uniquely determined by their values at negative integers? In another words, is there a sequence of integers $a\_1, a\_2, a\_3, \cdots$ such that
* the corresponding L-function
$$L\_{\{a\_n\}}(s):=\sum\_{n=1}^{\infty}\frac{a\_n}{n^s}$$
converges well for $\text{Re}(s) > M$ for some $M\in \mathbb{R}$
*... | https://mathoverflow.net/users/45553 | Are L-functions uniquely determined by their values at negative integers? |
>
> Are L-functions uniquely determined by their values at negative integers?
>
>
>
No. The rescaled Riemann zeta function
$$
\zeta(2s) = \sum\_{m=1}^\infty \frac{1}{m^{2s}} =
\sum\_{n=1}^\infty \frac{a\_n}{n^s},
$$
corresponding to the coefficient sequence
$$
a\_n = \begin{cases}
1 & \textrm{if $n$ is a square}... | 28 | https://mathoverflow.net/users/78525 | 336081 | 143,515 |
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