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https://mathoverflow.net/questions/424142 | 1 | We work over $\mathbb{C}$. Let $X$ be a normal projective irreducible variety, and let $\mathbb{C}^\*$ act nontrivially on $X$. The fixed point locus of $X$, namely $X^{\mathbb{C}^\*}$, can be decomposed into a disjoint union of connected fixed point components, let us call them $F\_1,\ldots,F\_s$. Moreover, for any $k... | https://mathoverflow.net/users/481375 | Plus and minus Białynicki-Birula decomposition for normal variety | Yes, \eqref{star} is always a disjoint union (that's obvious). Moreover, each set $X^+(F\_k)$ is locally closed and the map $x\mapsto\lim\_{t\to0}t\cdot x$ induces an affine morphism $\pi\_k:X^+(F\_k)\to F\_k$. In general, the morphism $\pi\_k$ is not a fiber bundle anymore.
These assertions can be easily reduced to ... | 5 | https://mathoverflow.net/users/89948 | 424186 | 172,338 |
https://mathoverflow.net/questions/424189 | 1 | How can I evaluate ( estimate ) the sum $s(n)=\sum\_{k=1}^{n-1}\mu ^{2}(k(n-k))$ ($\mu$ is the Möbius function). Trivial estimate
$s(n)<\varphi (n)$ follows from the fact that $\mu (k(n-k))$ is zero if $k$ is not prime to $n$. It seem that in the case of $n$ - primorial, $s(n)$ is pretty close to $\varphi (n)$
| https://mathoverflow.net/users/169583 | Möbius function summation | Let $f\_n(x)=x(n-x)$. You count squarefree values of $f\_n(k)$ for $k$ between $1$ and $n$.
Given a prime $p$ let $a\_{p,n}$ be the number of solutions $k\in \mathbb{Z}/p^2\mathbb{Z}$ to the congruence $f\_n(k)\equiv 0\bmod p^2$.
The heuristic asymptotic answer is $n$ times the following infinite product over primes
... | 4 | https://mathoverflow.net/users/31469 | 424193 | 172,340 |
https://mathoverflow.net/questions/424171 | 4 | In Hirschhorn's "Model Categories and Their Localizations", section 15.7, there's the following corollary to the preceding Proposition:
$\mathbb{Corollary}$ 15.7.2 If $\mathfrak{M}$ is a cellular model category and $\mathfrak{C}$ is a Reedy category, then the cofibrations of the Reedy model category on $\mathfrak{M}^... | https://mathoverflow.net/users/482932 | Can Reedy cofibrations be monomorphisms? | I believe what you are after is the notion of "[elegant Reedy category](https://ncatlab.org/nlab/show/elegant+Reedy+category)"
This sort of things isn't true for a general Reedy category, but for an elegant one $R$ (see the link for the definition) if $\mathcal{E}$ is a Grothendieck topos (for e.g. simplicial preshea... | 4 | https://mathoverflow.net/users/22131 | 424195 | 172,342 |
https://mathoverflow.net/questions/424199 | 4 | Given a $\delta$-hyperbolic group $G$, I have been told that the Rips $n$-complex of $G$ is contractible for high enough $n$. The only proof I have found for this statement is in an [expository essay](https://warwick.ac.uk/fac/sci/maths/people/staff/lopezdegamiz/hyperbolic_groups.pdf) by Jone Lopez de Gamiz. I don't be... | https://mathoverflow.net/users/14257 | Looking for a citation: the Rips $n$-complex of a $\delta$-hyperbolic group is contractible for high enough $n$ | In his monograph *Hyperbolic groups* (1987), Gromov states and proves:
**Lemma 1.7.A.** *Let $X$ be a $\delta$-hyperbolic space
such that every $x\in X$ can be joined by a segment with
a fixed reference point $x\_0 \in X$. Then the polyhedron
$P\_d(X)$ is contractible for all $d \geq 4 \delta$.*
Here, the polyhedro... | 9 | https://mathoverflow.net/users/122026 | 424202 | 172,343 |
https://mathoverflow.net/questions/424210 | 4 | Consider the quantum anharmonic oscillator, with Hamiltonian $H=p^2/2+q^2/2+gq^4$ for some real $g\geq 0$, with $p$ and $q$ obeying the usual Heisenberg commutation relations. For $g=0$, the ground energy is equal to $1/2$. Suppose $g>0$ and $g$ is algebraic. I would guess that the ground state energy is then not an al... | https://mathoverflow.net/users/83174 | Ground state energy of anharmonic oscillator: algebraic or transcendental? | There is no exact expression for the ground state energy $E\_0$ for any nonzero $g$, but there are upper and lower bounds: for $g=1/2$ the upper bound for $2E\_0$ is 1.3923516415302918570 and the lower bound for $2E\_0$ is 1.3923516415302918502 , see [Upper and lower bounds of the ground state energy of anharmonic osci... | 2 | https://mathoverflow.net/users/11260 | 424212 | 172,347 |
https://mathoverflow.net/questions/424216 | 2 | Given a real $n \times n$ matrix $A$ (feel free to assume its entries are non-negative, if it helps), what is known about the problem of computing the quantity
$$
\max\_{\sigma \in S\_n} \left\{\sum\_{j=1}^na\_{j,\sigma(j)}\right\}?
$$
(Here $S\_n$ is the symmetric group, so we are maximizing over all permutations of t... | https://mathoverflow.net/users/11236 | Maximum permuted row/column sum of a matrix | This is the maximum weight matching problem in the weighted complete bipartite graph $K\_{n,n}$, also known as the [assignment problem](https://en.wikipedia.org/wiki/Assignment_problem). It does have a few polynomial-time algorithmic solutions.
| 10 | https://mathoverflow.net/users/7076 | 424217 | 172,348 |
https://mathoverflow.net/questions/424165 | 6 | I am currently reading Rolfsen's "Knots and Links". At page 82 Whitehead manifold $W$ is defined and an exercise asking to show that $W\times \mathbb{R}\cong \mathbb{R}^4$ is left. Reference in Wiki says that this can be proved using Generalized Schonflies theorem, can anyone give any hint or material about this? Appre... | https://mathoverflow.net/users/483712 | How to prove the product of Whitehead manifold and $\mathbb{R}$ is homeomorphic to $\mathbb{R}^4$? | I would suggest a classical proof showing that the one-point compactification of $W$ is a manifold factor. See [Wild wild whitehead manifold](https://www.ams.org/journals/notices/201904/rnoti-p581.pdf). The proof is originally due to J. Andrews and L. Rubin, Bull. Amer. Math. Soc. 71(1965), 675-677. Once you understand... | 4 | https://mathoverflow.net/users/114032 | 424221 | 172,350 |
https://mathoverflow.net/questions/424214 | 4 | Let $S$ be a compact complex surface. It is well-known that the following two facts are equivalent
1. $c\_1^2(S) = 3 c\_2(S)$ and $S \neq \mathbb{CP}^2$
2. The universal cover of $S$ is biholomorphic to the unit ball in $\mathbb{C}^2$.
The unit ball in $\mathbb{C}^2$ is biholomorphic the complex hyperbolic plane $\... | https://mathoverflow.net/users/25511 | Constructions of complex surfaces covered by the ball of $\mathbb{C}^2$ | Below are two book references, both originating in the 1983 paper by Hirzebruch:
*Hirzebruch, Friedrich*, Arrangements of lines and algebraic surfaces, Arithmetic and geometry, Pap. dedic. I. R. Shafarevich, Vol. II: Geometry, Prog. Math. 36, 113-140 (1983). [ZBL0527.14033](https://zbmath.org/?q=an:0527.14033).
*Ba... | 6 | https://mathoverflow.net/users/39654 | 424246 | 172,360 |
https://mathoverflow.net/questions/424271 | 1 | Let $\Omega \subset \mathbb{R}^2$ be a domain which is "well behaved" (has all "wishable" properties), so as its boundary. For every $u \in C^\infty(\Omega,\mathbb{R})$, I would like to express the following integral
$$\int\_{\partial \Omega} \frac{((Du)^\perp \cdot n)^2}{|Du|^3} dS,$$
where $dS$ is the Hausdorff measu... | https://mathoverflow.net/users/121671 | Expressing the integral over boundary of a domain as an integral over the domain | $\newcommand{\Om}{\Omega}$Let
\begin{equation\*}
I\_u(\Om):=\int\_{\partial\Om} \frac{((Du)^\perp\cdot n)^2}{|Du|^3}\,dS.
\end{equation\*}
Then
\begin{equation\*}
I\_u(\Om)=\int\_\Om \frac{I\_u(\Om)}{|\Om|}\,dz
\end{equation\*}
if $|\Om|:=\int\_\Om dz>0$.
This trivial representation of $I\_u(\Om)$ as "an integral... | 2 | https://mathoverflow.net/users/36721 | 424275 | 172,367 |
https://mathoverflow.net/questions/424269 | 2 | In the end of the Abstract of the paper [Minsky and Papert - Unrecognizable Sets of Numbers](https://doi.org/10.1145/321328.321337), the authors write "…for every
infinite regular set $A$ there is a nonregular set $A'$ for which
$$ \lvert\pi\_A(n)-\pi\_{A'}(n)\rvert\leq 1\text{",} $$
where $\pi\_A(n)$ is the counting f... | https://mathoverflow.net/users/159935 | A question on regular sets | The second question has a negative answer. The asymptotic behavior of $\pi\_B(n)$ and $\pi\_{B'}(n)$ would be the same, and if $\pi\_B(n)$ satisfies any of the criteria for non-regularity on page 283 of the paper by Minski and Papert, then $\pi\_{B'}(n)$ would satisfy the same, and thus, cannot be regular.
For the fi... | 2 | https://mathoverflow.net/users/483601 | 424278 | 172,369 |
https://mathoverflow.net/questions/424279 | 3 | I have a multivariate polynomial $P$ which is a product of $M$ low degree polynomials $p\_i$
$$P(x\_1, x\_2, \dotsc, x\_n) = \prod\_{i=1}^M p\_i(x\_1, x\_2, \dotsc, x\_n)$$
where the maximum degree of each $p\_i$ can be $4$. For example:
$$P(x\_1, x\_2, x\_3) = (1+x\_1x\_2^2x\_3)(x\_3^3+x\_1x\_2^3).$$
I need to... | https://mathoverflow.net/users/402086 | Checking presence of a specific term in product polynomial | You can solve the problem via integer linear programming as follows.
Let $a\_{ijk}$ be the power of $x\_j$ in term $k$ of $p\_i$, that is, $p\_i = \sum\_k \prod\_j x\_j^{a\_{ijk}}$. Let binary decision variable $y\_{ik}$ indicate whether term $k$ of $p\_i$ is used to form the target term $\prod\_j x\_j^{m\_j}$. The tar... | 4 | https://mathoverflow.net/users/141766 | 424282 | 172,370 |
https://mathoverflow.net/questions/413189 | 14 | Fixing a dimension $n \ge 4$, is the class of closed hyperbolic $n$-manifolds recursively enumerable?
Since hyperbolic manifolds are triangulable I can reformulate this in the following more explicit way: say $M\_1, \ldots, M\_m,\ldots$ is an enumeration of all triangulations of $n$-manifolds. Is there a Turing machi... | https://mathoverflow.net/users/32210 | Are hyperbolic $n$-manifolds recursively enumerable? | The class of closed hyperbolic manifolds is recursively enumerable. I’ll describe a terrible algorithm which nevertheless gives an enumeration.
A couple of basic facts: a hyperbolic $n$-manifold $M$ admits a triangulation by geodesic simplices, and the representation of the fundamental group into $PO(n,1)$ may be con... | 5 | https://mathoverflow.net/users/1345 | 424283 | 172,371 |
https://mathoverflow.net/questions/424254 | 2 | Assume that $Q=(q\_{ij})$ is a $k\times k$ with $q\_{ij}\in \{0, 1\}.$ The two side subshift of finite type associated to the matrix $Q$ is a left shift map $T:\Sigma\_{Q}\rightarrow \Sigma\_{Q}$, where
$$\Sigma\_{Q}:=\{x=(x\_{i})\_{i\in \mathbb{Z}} : x\_{i}\in \{1,...,k\} \hspace{0.2cm}\textrm{and} \hspace{0.1cm}Q\_{x... | https://mathoverflow.net/users/127839 | Union of admissible words are subshift of finite type | So, I probably did not initially understand you correctly. Let me analyze four interpretations of your construction; the first is what I thought first, the second gives something uninteresting, the third gives something uninteresting, the fourth is now my best guess of what you meant (you may want to jump there first t... | 1 | https://mathoverflow.net/users/123634 | 424290 | 172,374 |
https://mathoverflow.net/questions/424200 | 6 | Let $X$ be a Hausdorff compact space, and let $\mathrm {Ba}$, $\mathrm {Bo}$ be its Baire, respectively, Borel, $\sigma$-algebras. Let $\mu:\mathrm {Ba}\to[0,+\infty)$ be a finite Baire measure: it is well-known that it extends uniquely to a *regular* Borel measure $\tilde\mu:\mathrm {Bo}\to[0,+\infty)$ (since $\mu$ in... | https://mathoverflow.net/users/6101 | Extending a finite Baire measure to a regular Borel measure | Let $X=\{0,1\}^{\aleph\_1}$ and $x\in X$, and let $\mu$ be the Borel probability measure such that $\mu(\{x\})=1$. Since $\{x\}$ itself is not a Baire set, there is no Baire set $A\subseteq \{x\}$ such that $\mu(A)=1$. By taking the complements, we get answer **no** for the question.
For the second question: The usua... | 6 | https://mathoverflow.net/users/95282 | 424295 | 172,376 |
https://mathoverflow.net/questions/424314 | 0 | Let $L/\mathbb{Q}$ be a finite extension and $f\_{1},\dotsc,f\_{n}\in L[x\_{1},\dotsc,x\_{k}]$ be degree $d$ homogeneous polynomials. Is there a way to find homogenous degree $d’$ polynomials $g\_{1},\dotsc,g\_{n}\in L[x\_{1},\dotsc,x\_{k}]$ such that $f\_{1}g\_{1}+\dotsc f\_{n}g\_{n}\in \mathbb{Q}[x\_{1},\dots,x\_{k}]... | https://mathoverflow.net/users/483823 | Changing base field for sum of polynomials | Not always. Take $k=3$, $n=2$, $f\_1=\sqrt{2}x\_1^2+x\_2^2$, $f\_2=x\_3^2$.
| 1 | https://mathoverflow.net/users/4312 | 424315 | 172,383 |
https://mathoverflow.net/questions/424187 | 1 | Let $Z$ be a finite set and $\mathcal S$ be a finite set of vectors in $Z^n$.
For each $z\in Z$ let $x\_z$ be a formal variable. A **distribution** (please excuse abuse of language) $d \in \mathbf N^n$ is a vector with nonnegative integer entries, not all zero, such that the set $\{\sum\_{i=1}^n d\_i x\_{s\_i} | s\in... | https://mathoverflow.net/users/3032 | are minimal subdistributions determined by their support? | Extracted from the comments:
there are examples of the following type: $Z=\{0,1\}$, the set $\mathcal{S} $ consists of sequences $(t\_1,\ldots,t\_n)$ satisfying two different relations of the form $\sum c\_i t\_i=C$, where $c\_i$ are positive integers and $C$ is a positive integer (it is natural to choose it close to... | 1 | https://mathoverflow.net/users/4312 | 424321 | 172,385 |
https://mathoverflow.net/questions/424339 | 1 | Let $\lambda>0$ be given. Define
$$G\_{\lambda}(\xi) = \chi\_{\_{\lbrace |\xi|^{2} \leq \lambda \rbrace }}.
$$
and
$$
E\_{0}(\lambda)f = \mathcal{F}^{-1}[G\_{\lambda}(|\xi|^{2})\mathcal{F}(f)], \ \ f \in L^{2}(\mathbb{R}^{n})
$$
How do I show that
$$(E\_{0}(\lambda)f|f) = \|E\_{0}(\lambda)f\|^{2}\_{L^{2}}, \ \ f \in ... | https://mathoverflow.net/users/481556 | Spectral family associated with the Laplacian operator in $L^{2}(\mathbb{R}^{n})$ | Your identity amounts to
$$\frac12(E\_0(\lambda)^\*+E\_0(\lambda))=E\_0^\*(\lambda) E\_0(\lambda).$$
Since ${\cal F}^\*=\cal F$, this is equivalent to saying that $$\frac12(G\_\lambda+\overline{G}\_\lambda)=|G\_\lambda|^2,$$
which is true because $G\_\lambda(\xi)$ equals either $0$ or $1$.
**Edit**. To explain the fi... | 2 | https://mathoverflow.net/users/8799 | 424341 | 172,392 |
https://mathoverflow.net/questions/424340 | 1 | I have the following question, which I am thinking about for days now and can't get the answer right. I have a sequence of elements in this order $x\_{1},x\_{2},...,x\_{2n}$, $n \ge 1$ and then I perform a permutation $\pi$ on this set, so it becomes $x\_{\pi(1)},x\_{\pi(2)},...,x\_{\pi(2n)}$. Let us denote the sign of... | https://mathoverflow.net/users/235802 | Sign of the permutation which brings a subsequence back to its original form | Imagine a bubble sort where you bring each element to its original position. $x\_1$ would have taken $\pi^{-1}(1) - 1$ transpositions to bring it back to its original position. Let's perform those transpositions, even though we'll discount them in a moment. $x\_i$ would have taken $\lvert\pi^{-1}(i) - i\rvert$ transpos... | 4 | https://mathoverflow.net/users/2383 | 424344 | 172,393 |
https://mathoverflow.net/questions/424351 | 1 | I am reading Engelen´s paper and have trouble with this proof of Lemma 2.1 (a) (link is below).
>
> It is easily seen that any non-empty open subspace $U$ of $\mathbb{Q}^\infty$ can be
> written as an infinite disjoint union of non-empty basic clopen
> subsets; hence, $U = \mathbb{N} \times \mathbb{Q}^\infty \simeq... | https://mathoverflow.net/users/421149 | Why can any open subset $U$ of $\mathbb{Q}^\infty$ be written as disjoint union of basic clopen subsets? | We can first express the open set $U$ as a countable union of basic clopen subsets $A\_n=\{(q\_k)\_{k\in\mathbb{N}};i\_{n,k}<q\_k<j\_{n,k} \text{ for $k=1,\dots,n$}\}$, where $i\_{n,k}$ and $j\_{n,k}$ are irrational numbers. Some $A\_n$ may be empty.
To answer the question it is enough to express $A\_n\setminus\bigcu... | 4 | https://mathoverflow.net/users/172802 | 424361 | 172,396 |
https://mathoverflow.net/questions/424298 | 0 | Every weakly null sequence in a Banach space, as a subset, is clearly relatively weakly compact. To quantify the elementary fact, we need the following quantities:
$$\delta\_{0}((x\_{n})\_{n}):=\sup\_{x^{\*}\in B\_{X^{\*}}}\limsup\_{n}|\langle x^{\*},x\_{n}\rangle |$$ for a bounded sequence $(x\_{n})\_{n}$ of a Banac... | https://mathoverflow.net/users/41619 | Weakly null sequences in Banach spaces | I answer Question 1 by myself and I am sure my proof is correct.
Let $A=\{x\_{n}:n=1,2,\cdots\}$ and let $0<c<\operatorname{wck}\_{X}(A)$ be arbitrary. Then there exists a sequence $(z\_{n})\_{n}$ in $A$ so that $\textrm{d}(\textrm{clust}\_{X^{\*\*}}((z\_{n})\_{n}),X)>c$. Let $Y=\overline{\textrm{span}}\{x\_{n}\colon... | 0 | https://mathoverflow.net/users/41619 | 424383 | 172,401 |
https://mathoverflow.net/questions/424403 | 0 | Let $(a\_{n})\_{n}$ be a bounded real-valued sequence. Suppose that $(b\_{n})\_{n}$ is a sequence (not necessarily a subsequence) in the set $A:=(a\_{n})\_{n}$. Assume that the limit $\lim\limits\_{n}b\_{n}$ exists. Moreover, assume that $(b\_{n})\_{n}$ is not eventually constant (we say that a sequence $(c\_{n})\_{n}$... | https://mathoverflow.net/users/41619 | the limit of a scalar sequence in a sequence of scalars | Yes. Denote $B=\{b\_n\colon n=1,2,\ldots\}$. If $B$ is finite, then, since $\lim b\_n$ exists, the sequence $(b\_n)$ is eventually constant. If $B$ is infinite, there exists a subsequence $n\_1<n\_2<\ldots$ such that all $b\_{n\_j}$ are distinct. Denote $b\_{n\_j}=a\_{m\_j}$. All $m\_j$ are distinct, thus $\lim\_j a\_{... | 0 | https://mathoverflow.net/users/4312 | 424404 | 172,404 |
https://mathoverflow.net/questions/424415 | 5 | If we violate the partition principle and add to $\sf ZF$ the axiom that there exists a set $X$ that has a partition on it that is greater in cardinality than the set of singleton subsets of $X$.
>
> Can $X$ be a standard stage $V\_\alpha$ of the cumulative hierarchy?
>
>
>
>
> Can $X$ be a non-standard stag... | https://mathoverflow.net/users/95347 | Can a stage of the cumulative hierarchy violate the partition principle? | If you are asking whether or not a $V\_\alpha$ could violate the partition principle, the answer is easily yes.
As we all know, it is always the case that $\Bbb R$ can be partitioned into $\aleph\_1$ parts; but it is consistent that there is no injection from $\omega\_1$ into the reals. Next, observe that $V\_{\omega... | 4 | https://mathoverflow.net/users/7206 | 424417 | 172,406 |
https://mathoverflow.net/questions/424410 | 4 | Let $H^1$ be the Hardy space on $\mathbb{R}^n$, defined e.g. as the set of $u\in L^1$ such that $Ru\in L^1$, where $R$ is the Riesz transform on $\mathbb{R}^n$. It seems to me that simple functions with bounded support and average 0 are dense in $H^1$ (simple functions = finite linear combinations of characteristic fun... | https://mathoverflow.net/users/7294 | Dense subspaces of the Hardy space $H^1$ | It seems to be true. Due to the atomic decomposition, it is enough to approximate in $H^1$ any "$(1,\infty)$"-atom, that is, a function $a$ such that $$\mathrm{supp}\, a \subset Q,\quad \|a\|\_{L^\infty}\leq |Q|^{-1}, \quad \int\_Q a=0$$
for a cube $Q$.
In order to do this, we can cut $Q$ into $N=k^n$ smaller equal c... | 3 | https://mathoverflow.net/users/69086 | 424419 | 172,407 |
https://mathoverflow.net/questions/424374 | 9 | This question is a request for assistance in surveying the existing literature on applications of Lawvere's Fixed Point Theorem (LFPT).
Yanofsky [0] has demonstrated several applications of LFPT to prove various limitative results, in particular Goedel's First Incompleteness Theorem, and alludes to its applicability ... | https://mathoverflow.net/users/483796 | Has Goedel's Second Incompleteness Theorem been proven using Lawvere's Fixed Point Theorem? | As suggested by the OP, I'm turning my comment into an affirmative answer. WARNING: The presentation of Joyal's proof in the paper I cited contains an incorrect conclusion about Joyal's sentence (that it is undecidable from mere consistency). I have modified my answer accordingly.
Both Gödel's first and second incomp... | 12 | https://mathoverflow.net/users/12976 | 424422 | 172,408 |
https://mathoverflow.net/questions/424413 | 1 | If there are some square free numbers, a finite amount of them $\{t\_1,t\_2,...t\_k\}$ and we define the set $\mathcal{N}\_L=\{l\in\mathbb{N}/L-l^2\notin\bigcup t\_j\mathbb{Z}^2\}$, where $n\notin\bigcup t\_j\mathbb{Z}^2$ means that if we write $n$ as the product of its square part and square free part as $n=tm^2$ then... | https://mathoverflow.net/users/411616 | Justify that a certain set depending on a parameter is large | Let $p\_1,\dots,p\_s$ be the distinct primes in $t\_1,\dots,t\_k$.
We claim that, for any prime $p\_i$, there is $a\_i$ such that whenever $\ell \equiv a\_i \pmod{p\_i}$, we have $L - \ell^2 \not\equiv 0 \pmod{p\_i}$. This is clear, since there are at least two different squares modulo $p\_i$ (in other words, $a\_i$ ... | 1 | https://mathoverflow.net/users/483601 | 424425 | 172,409 |
https://mathoverflow.net/questions/423865 | 6 | Suppose we have a simplicial complex / poset / small category without loops $X$ equipped with a functor $F$ into the category of posets / small categories without loops. Suppose further that for each arrow / edge $e$ of $X$, the functor $Fe$ is an equivalence of categories. If we perform the Grothendieck construction o... | https://mathoverflow.net/users/135175 | When is the Grothendieck / category of elements construction a fibration on geometric realizations? | Just so this question doesn't sit around forever with no "answer" (even though it's perfectly answered in the comments), I am writing a CW expansion to Cisinski's comment, to help folks who don't have Quillen's lecture notes from 1973 on hand.
Definition: a commutative square of categories is **homotopy cartesian** i... | 4 | https://mathoverflow.net/users/11540 | 424429 | 172,411 |
https://mathoverflow.net/questions/424427 | 12 | Let R be a noetherian ring and V a valuation ring with maximal ideal $\mathfrak{m}\_V$. Does every morphism of rings $\varphi: R \rightarrow V$ factor through a discrete valuation ring?
One may consider the homomorphic image $S$ of $\varphi$ in V. This is a noetherian subring and we may even assume it to be local by ... | https://mathoverflow.net/users/111259 | Does every map from a noetherian ring to a valuation ring factor through a DVR? | The answer is no. I give two examples, which are standard non-dvr points on the Riemann-Zariski space of the plane.
(1) Let $R=k[x,y]$ and let $V\subseteq k(x,y)$ be the subring consisting of rational functions $f(x,y)$ which have non-negative valuation along $x=0$, and such that the restriction of $f$ to $x=0$ (whic... | 19 | https://mathoverflow.net/users/3847 | 424434 | 172,413 |
https://mathoverflow.net/questions/424447 | 2 | This question is related to my other question [Sign of the permutation which brings a subsequence back to its original form](https://mathoverflow.net/questions/424340/sign-of-the-permutation-which-brings-a-subsequence-back-to-its-original-form). Suppose I have a complete ordered set $\{a\_{1},\dotsc,a\_{2n}\}$ and take... | https://mathoverflow.net/users/235802 | Given $\pi$ permutation on $\{1,\dotsc,n\}$, what is the sign of a permutation of $\{2,\dotsc,\hat\jmath,\dotsc,n\}$? | $\DeclareMathOperator\sgn{sgn}$This is almost the same as your previous question, just with the order of the operations switched—whether you think of $\pi$ as ordering or disordering is just a matter of taking the inverse.
You previously needed $\pi^{-1}(1) - 1$ transpositions to bring the element $a\_1$ to its new p... | 5 | https://mathoverflow.net/users/2383 | 424449 | 172,416 |
https://mathoverflow.net/questions/424412 | 2 | Let $G$ be a finite abelian etale group scheme over a number field $k$. Let $E$ be an elliptic curve over $k$ and $C := E\backslash \{O\}$ its affine model of the same equation.
Recall that for a variety $X$, the pointed (etale cohomology) set $H^1(X,G)$ classifies all $X$-torsors under $G$.
Given a torsor $Y \righ... | https://mathoverflow.net/users/172132 | Torsors over elliptic curves | Indeed this map is an isomorphism. There is a diagram of five-term exact sequences, arising from the Leray spectral sequence for the maps $E\to \text{Spec}(k), C\to \text{Spec}(k)$, from
$0\to H^1(k, H^0(E\_{\overline{k}}, G))\to H^1(E, G)\to H^0(k, H^1(E\_{\overline{k}}, G))\to H^2(k, H^0(E\_{\overline{k}}, G))\to H... | 2 | https://mathoverflow.net/users/6950 | 424452 | 172,419 |
https://mathoverflow.net/questions/424444 | 11 | This question now has two sequels, [Pointless groups II](https://mathoverflow.net/questions/424456/pointless-groups-ii) (to which @R.vanDobbendeBruyn gave a [counterexample](https://mathoverflow.net/questions/424456/pointless-groups-ii#comment1091038_424456) for an infinite, imperfect field) and [Pointless groups III](... | https://mathoverflow.net/users/2383 | Pointless groups | Perhaps I'm missing something, but it seems to me that $k=\mathbb{F}\_2$, $G=\mathbb{G}\_{m, k}$ works, where $H$ is the trivial subgroup.
| 13 | https://mathoverflow.net/users/6950 | 424453 | 172,420 |
https://mathoverflow.net/questions/424456 | 7 | This question is a sequel to [Pointless groups](https://mathoverflow.net/questions/424444/pointless-groups), where I asked for a certain kind of counterexample. @DanielLitt [produced](https://mathoverflow.net/a/424453) an elegant and easy-to-understand counterexample, but also [suggested](https://mathoverflow.net/quest... | https://mathoverflow.net/users/2383 | Pointless groups II | One other type of example is constructed in Conrad–Gabber–Prasad's *Pseudo-reductive groups*, Example 11.3.2. The construction is kind of classic (and possibly predates the book), so let me recall it here:
**Example.** Let $k = \bar{\mathbf F}\_p(t)$, and let $G \subseteq \mathbf G\_a \times \mathbf G\_a$ be the subg... | 9 | https://mathoverflow.net/users/82179 | 424462 | 172,423 |
https://mathoverflow.net/questions/424438 | 7 | The bar construction is a functor $A\mapsto Bar(A)$ from the category of augmented differential graded algebras over a commutative ring $R$ to the category of chain complexes of $R$-modules. It sends an algebra $A$ to the derived tensor product $R\otimes^{L}\_AR$.
Assume that I have a map of augmented algebras $A\to ... | https://mathoverflow.net/users/10707 | Under which conditions is the bar construction a conservative functor? | $\vphantom{0pt}$ Hi Geoffroy. The natural way to approach this is by finding conditions under which the cobar functor preserves quasi-isomorphisms. If $A \to A'$ is a morphism of dg algebras, then you get a zig-zag
$$ A \leftarrow \Omega BA \to \Omega BA' \to A'.$$
The two counit maps here are always quasi-isomorphisms... | 5 | https://mathoverflow.net/users/1310 | 424467 | 172,426 |
https://mathoverflow.net/questions/424461 | 0 | This question is based on [this Math.SE answer](https://math.stackexchange.com/questions/4182260/on-the-function-n-mapsto-a-n-frac-1n-for-a-given-power-series-sum-n/4445420#4445420), so let's recall a few concepts dealt with there. If $\{a\_n\}\_{n\in\Bbb N}$ is the sequence of coefficients of a power series $\sum\_{n=... | https://mathoverflow.net/users/113756 | Reference(s) on the smallest concave majorant for the sequence of coefficients of a given power series? | A better reference on Valiron is
George Valiron, *Fonctions analytiques*, Presses universitaires de France, 1954, [MR0061658](https://mathscinet.ams.org/mathscinet-getitem?mr=MR0061658), [Zbl 0055.06702](https://zbmath.org/?q=an%3A0055.06702).
This smallest concave majorant is nothing but the Newton polygon general... | 1 | https://mathoverflow.net/users/25510 | 424485 | 172,430 |
https://mathoverflow.net/questions/424471 | 4 | Let $\chi$ denote the Legendre symbol of conductor $q$. A Siegel zero for the $ L $ series associated to $ \chi $, which we denote by $ L(s,\chi) $ is a real zero $ \sigma $ satisfying $ 1-\frac{c}{\log |q|} < \sigma < 1$ for some constant $c$.
I have read in many places (see for example the second page of the articl... | https://mathoverflow.net/users/148866 | Calculating the explicit constant – Siegel zeros and class numbers | One place to find this worked out in detail is the paper ["On the Siegel-Tatuzawa theorem"](https://www.impan.pl/shop/en/publication/transaction/download/product/102670) by Jeffrey Hoffstein (published in 1980 in Acta Arithmetica). Lemma 1 of that paper states that if $\chi$ is a quadratic Dirichlet character with cond... | 8 | https://mathoverflow.net/users/48142 | 424488 | 172,431 |
https://mathoverflow.net/questions/424489 | 12 | This question is a sequel to [Pointless groups](https://mathoverflow.net/questions/424444/pointless-groups), to which @DanielLitt produced an elegant and easy-to-understand [counter-example](https://mathoverflow.net/a/424453), and [Pointless groups II](https://mathoverflow.net/questions/424456/pointless-groups-ii), whe... | https://mathoverflow.net/users/2383 | Pointless groups III | Yes.
Let $|k| = q$. Let $T$ be an $n$-dimensional torus defined over $k$ and let $\overline{T}$ be the base change of $T$ to $\overline{k}$. Then the character group $\text{Hom}(\overline{T}, \mathbb{G}\_m)$ is isomorphic to $\mathbb{Z}^n$ and the Frobenius acts on $\text{Hom}(\overline{T}, \mathbb{G}\_m)$ by a matri... | 13 | https://mathoverflow.net/users/297 | 424493 | 172,432 |
https://mathoverflow.net/questions/424501 | 2 | I can deduce the following simple proposition, the definitions for $\sigma(x)$ the sum of divisors functions and $\varphi(x)$ the Euler totient function are assumed. After I present a conjecture that I've tested for the square $1000\times 1000$. (Please add feedback in comments if you think that the question can be imp... | https://mathoverflow.net/users/142929 | A conjecture concerning the equation $\sigma\left(\square\right)=\text{prime}$ | Your conjecture is true. Notice first that it is enough to show that $C=p^k$ for some prime $p$. Indeed, in this case
$$
B=\sigma(C)=\frac{p^{k+1}-1}{p-1},
$$
hence
$$
\mathrm{rad}(C)(B-C)=p\frac{p^k-1}{p-1}=B-1.
$$
This means that $\varphi(B)=B-1$, so $B$ is prime.
To show that $C$ must be a prime power, assume the ... | 7 | https://mathoverflow.net/users/101078 | 424507 | 172,440 |
https://mathoverflow.net/questions/424495 | 6 | For a hobby software project I am working with exact rational arithmetic, as it happens this produces numbers $\frac{n}{k}$ of huge size even after reducing them, I am searching for an efficient algorithm to "simplify" these rational with a specified error tolerance.
In my own working out the "simplicity function" I ... | https://mathoverflow.net/users/38349 | How can I efficiently find the "simplest" rational in an interval? | Here's a version of what has been said in the comments that's reformulated into an explicit algorithm, first assuming that $p$ and $q$ are *irrational* to avoid the edge case:
* Lazily compute the continued fraction expansions of $p = [a\_0; a\_1, a\_2, a\_3,\ldots]$ and $q = [b\_0; b\_1, b\_2, b\_3,\ldots]$: stop at... | 9 | https://mathoverflow.net/users/17064 | 424509 | 172,441 |
https://mathoverflow.net/questions/424364 | 2 | Consider the SDE
$$dX\_t =b(t)dt + a(t)dW\_t,\quad \forall t>0,$$
with $X\_0>0$ has a density function $\rho:\mathbb R\_+\to\mathbb R\_+$. Consider the probability $g(t):=\mathbb P[\inf\_{0\le s\le t}X\_s>0]$. Can we prove $g'$ exists and
$$-\frac{C}{\sqrt{t}}\le g'(t)\le 0,\quad \forall t>0,$$
where $C$ depend... | https://mathoverflow.net/users/nan | Is $g(t)=\mathbb P[\inf_{0\le s\le t}X_s>0]$ differentiable with respect to $t$? | **Yes**, $g(t)$ is differentiable with respect to $t$ under the assumption that $a(s) > 0 $ for $s \in [0,t]$.
---
*Proof*. Introduce the time-change $\tau(s) := \int\_0^s a(u)^{2} du$ and set $c(s) := b(\tau^{-1}(s)) a(\tau^{-1}(s))^{-2}$ for $s \in [0,t]$. In terms of this time change, the process $X\_t$ can be... | 0 | https://mathoverflow.net/users/64449 | 424512 | 172,442 |
https://mathoverflow.net/questions/424405 | 4 | In Mike Shulman's article [Brouwer’s fixed-point theorem in real-cohesive homotopy type theory](https://www.cambridge.org/core/journals/mathematical-structures-in-computer-science/article/abs/brouwers-fixedpoint-theorem-in-realcohesive-homotopy-type-theory/8270C2EAC4EE5D5CDBA17EEB3FF6B19E), the fundamental axiom adopte... | https://mathoverflow.net/users/483446 | Predicativity and axiom $\mathbb{R}\flat$ in real cohesive homotopy type theory | My suggestion would be to start from your intended semantics. First ask what kind of "predicative topos" you have in mind where the theory you're asking about would have a model, and construct a particular such category that you're interested in. Then once you have a model, ask what axioms that model satisfies. That's ... | 2 | https://mathoverflow.net/users/49 | 424520 | 172,445 |
https://mathoverflow.net/questions/424479 | 0 | Let $(x\_{n})\_{n}$ be a sequence in a Banach space $X$. Assume that the set $\{x\_{n}:n=1,2,\cdots\}$ is finite. Let $(f\_{m})\_{m}$ be a weak\*-null sequence in $X^{\*}$ satisfying the following conditions:
(1) the limit $a\_{m}:=\lim\limits\_{n}\langle f\_{m},x\_{n}\rangle$ exists for each $m$;
(2) the limit $a:... | https://mathoverflow.net/users/41619 | Weak*-null sequences in dual spaces | Let's write $\{x\_{n}:n=1,2,\cdots\}=\{y\_{1},y\_{2},\cdots,y\_{N}\}$. For each $i=1,2,\cdots,N$, we set $A\_{i}=\{n:x\_{n}=y\_{i}\}$. Then $\cup\_{i=1}^{N}A\_{i}=\mathbb{N}$. Then there exists $1\leq i\_{0}\leq N$ so that $A\_{i\_{0}}$ is infinite. Write $A\_{i\_{0}}=\{k\_{n}:n=1,2,\cdots\}$. Hence $x\_{k\_{n}}=y\_{i\... | 0 | https://mathoverflow.net/users/41619 | 424521 | 172,446 |
https://mathoverflow.net/questions/424518 | 2 | We all know that the famous Hypergeoemtric function $\_2F\_1$ has an integral form as follows:
$$\_2F\_1(a,b,c;z)=-\frac{e^{-i\pi c} \Gamma(c)}{4\Gamma(b)\Gamma(c-b)\sin\pi b\sin\pi(c-b)}\int\_P t^{b-1}(1-t)^{c-b-}(1-zt)^{-a}dt,$$
where $c\neq0,-1,-2,...$ and $P$ is Pochhammer contour. This integral form can give a... | https://mathoverflow.net/users/483996 | Does any Hyper-geometric function can be analytically continuated to the whole complex plain except $e^{\pm i\pi/3}$ | Continuation to the whole $z$-plane? As stated, not true. Example:
$$
{}\_2F\_1\left(-\frac12,1;1;z\right) = (1-z)^{1/2},\quad |z|<1,
$$
cannot be extended analytically to any neighborhood of $z=1$.
| 3 | https://mathoverflow.net/users/454 | 424542 | 172,452 |
https://mathoverflow.net/questions/415346 | 6 | I'm quite a newbe in the field of motives & A1 homotopy theory,
so please forgive me if the question is too elementary:
In the intro from [wikipedia on Nisnevish topology](https://en.wikipedia.org/wiki/Nisnevich_topology)
is remarked that it's founder Yevsey Nisnevich was originally motivated
by the theory of adeles.... | https://mathoverflow.net/users/108274 | Nisnevich topology inspired by Adeles | He used the topology to give a cohomological interpretation of the class group/set $c(G)$ of an affine algebraic group (which is defined in terms of adeles, see below), and then to partially prove the Grothendieck-Serre conjecture. I quote Nisnevich's summary of section 2 of his thesis.
2.1. The principal role in the... | 6 | https://mathoverflow.net/users/nan | 424543 | 172,453 |
https://mathoverflow.net/questions/420071 | 7 | Given a general topological space $X$ does the category $\mathbf{coShv}(X,\mathbf{Mod}\_R)$ have enough projectives ? I know that under some conditions this is true, for example if $X$ is a cell complex equipped with the Alexandrov topology and $R$ is a field (in the case of cellular sheaves of vector spaces). In this ... | https://mathoverflow.net/users/299054 | Does the category of cosheaves have enough projectives? | The answer is yes. A reference for the claim about Alexandrov spaces and cellular (co)sheaves is [Justin Curry's thesis](https://arxiv.org/pdf/1303.3255.pdf). When he proves Claim 7.1.9 he points out that the statement is true for all spaces $X$. It's also not essential that $R$ is a field. You can see directly that Ju... | 0 | https://mathoverflow.net/users/11540 | 424546 | 172,454 |
https://mathoverflow.net/questions/424545 | 20 | I sent a paper to an elite journal (the top in the field). Two weeks later I got a decision "reject" but the editors added that "we believe it should deserve a good publicity and publication".
The paper was described by the associate editor as "of very good quality" and the reason for the rejection is that "the techn... | https://mathoverflow.net/users/161837 | Rejection for a seemingly odd reason | When you submit to an elite journal, expect a rejection most of the time. Then submit to a less-prestigious journal. It is a waste of your time to attempt an analysis of the reasons given for rejection.
1. Yes, it is common for journals that receive far more submissions than they can publish to reject most of them — ... | 48 | https://mathoverflow.net/users/454 | 424547 | 172,455 |
https://mathoverflow.net/questions/424549 | 8 | We all take for granted the theorem that every ordinal $\alpha > 0$ has a Cantor normal form, and there are plenty of proofs of it, some of which are on this site. However, where was it proved? Was it actually proved by Cantor in his 1883 paper introducing ordinals? Or somewhere else?
| https://mathoverflow.net/users/473200 | Where was the Cantor normal form theorem first proved? | Cantor proved the normal form theorem in his 1895 paper [Beiträge zur Begründung der transfiniten Mengenlehre](https://eudml.org/doc/157768), which was his last paper on transfinite set theory.
| 12 | https://mathoverflow.net/users/11260 | 424550 | 172,456 |
https://mathoverflow.net/questions/424539 | 1 | I asked this question fourteen days ago on MathStackexchange (see [here](https://math.stackexchange.com/questions/4460652/star-autonomous-categories-are-categorifications-of-boolean-algebras)). I have not received any answers or comments until now. It seems to me that on MathStackexchange not many people are familiar w... | https://mathoverflow.net/users/160778 | Star-autonomous categories are categorifications of Boolean algebras? | The starting point for decategorification is the observation that a category in which any parallel arrows are equal must necessarily be a preorder.
Restricting to skeletal categories makes it a poset.
Thus, we might as well ask: what is a \*-autonomous category whose underlying category is a poset?
For the specific... | 4 | https://mathoverflow.net/users/402 | 424554 | 172,457 |
https://mathoverflow.net/questions/424457 | 3 | Do we have any examples of non-modular elliptic curves over number fields $K \neq \mathbb{Q}$?
In particular, I came across a paper by Freitas, Le Hung, and Siksek, "[Elliptic curves over real quadratic fields are modular](https://doi.org/10.1007/s00222-014-0550-z)", which shows that there are at most finitely many n... | https://mathoverflow.net/users/478525 | Non-modular elliptic curves | It is a widely believed conjecture that all elliptic curves, over any number field $K$, are modular (in the sense that there exists an automorphic representation [\*] $\pi$ of $\operatorname{GL}\_2 / K$ whose $L$-function is the same as that of $E$). No counterexamples are known, and it would be *extremely* big and dis... | 9 | https://mathoverflow.net/users/2481 | 424562 | 172,458 |
https://mathoverflow.net/questions/424551 | 3 | **The main references** for this question are
**1** : V.Voevodsky's paper [Triangulated categories of motives over a field](https://www.math.ias.edu/vladimir/sites/math.ias.edu.vladimir/files/s5.pdf)
**2** : the book "Lecture notes in motivic cohomology" written by Carlo Mazza, Vladimir Voevodsky and Charles Weibel... | https://mathoverflow.net/users/169282 | Motivic cohomology as $\mathit{Hom}$ in the category of geometric motives, with coefficient in a Chow motive | You can also define the motivic cohomology of $X$ as a $\mathrm{Hom}$ in the category $\mathrm{DM}(X)$ of motives over $X$, as defined by Cisinski-Déglise – see Example 11.2.3 in their book Triangulated categories of mixed motives (Springer, 2019).
Once in $\mathrm{DM}(X)$, you can use "motivic sheaves on $X$", for e... | 2 | https://mathoverflow.net/users/6506 | 424563 | 172,459 |
https://mathoverflow.net/questions/424455 | 6 | As discussed here, [Are monads monadic?](https://mathoverflow.net/questions/19906/are-monads-monadic), in "On the monadicity of finitary monads" by Steve Lack, the following is shown, the forgetful functor from $Mnd\_f(C) \rightarrow Endo\_f(C)$ is monadic, note the finitarity restrictions on both domain and codomain. ... | https://mathoverflow.net/users/129807 | Are infinitary monads monadic? | These question of existence of free monad are not "derailling" the discusion. They are the whole point of the discusion. Let me clarify :
If I'm not mistaken, we have the following:
**Theorem:** Let $V$ be a monoidal category. Let $A$ be the category of monoids in $V$, then the forgetfull functor $U: A \to V$ is mo... | 6 | https://mathoverflow.net/users/22131 | 424564 | 172,460 |
https://mathoverflow.net/questions/422925 | 3 | Let $\rho(A)$ denote the spectral radius of a square matrix $A$. Let $r,d$ be positive integers. Let $X\_1,\dots,X\_r$ be $d\times d$-real matrices. Then do we necessarily have $$\rho(X\_1\dots X\_r)^{2/r}\leq \frac{d}{r}\cdot\rho(X\_1\otimes X\_1+\dots+X\_r\otimes X\_r)?$$
If $X\_1,\dots,X\_r$ are $d\times d$-real m... | https://mathoverflow.net/users/22277 | Is $\rho(X_1\dots X_r)^{2/r}\leq \frac{d}{r}\cdot\rho(X_1\otimes X_1+\dots+X_r\otimes X_r)$ for $d\times d$-real matries $X_1,\dots,X_r$? | The answer to all these questions is **Yes**. We shall answer this question in the more general case when $X\_1,\dots,X\_r$ are complex matrices (the inequality for this question looks a little bit different in the complex case since we need to apply conjugations and transposes). This answer shall be in the context of ... | 0 | https://mathoverflow.net/users/22277 | 424572 | 172,462 |
https://mathoverflow.net/questions/424568 | 2 | Consider an $n \times n$ unitary $U$, drawn from the Haar measure. I'm trying to find the distribution for $U^{2}$. Is it true that $U^{2}$ is also Haar random?
Note that for any fixed unitary $V$, $VU$ and $UV$ are both Haar random unitaries, by the translational invariance of the Haar measure. But our case is sligh... | https://mathoverflow.net/users/166840 | Question about squaring a Haar random unitary | The probability distribution of $U^p$ for $U$ uniformly distributed with the Haar measure in $\text{U}(n)$ has been calculated by Eric Rains in [Images of eigenvalue distributions under power maps](https://arxiv.org/abs/math/0008079).
For $p=2$ and $n$ even the eigenvalue distribution of $U^2$ is obtained by taking t... | 5 | https://mathoverflow.net/users/11260 | 424573 | 172,463 |
https://mathoverflow.net/questions/424566 | 10 | Suppose $A=(a\_{ij})$ is a symmetric (0,1)-matrix with 1's along the diagonal, and let $A\_{ij}$ be the matrix obtained by removing the $i$-th row and $j$-th column.
Based on substantial numerical experimentation it seems that the bound $$\operatorname{Perm}(A) \geq 2 \operatorname{Perm}(A\_{ij})$$
holds for any $(i,... | https://mathoverflow.net/users/126170 | A bound for the permanent of a nonnegative matrix | Please check the details.
Below I denote $i=1,j=2$.
Expand ${\rm Perm}\,A$ as a polynomial in $a\_{11}$ and $a\_{22}$ as $a\_{11}a\_{22}X+a\_{11}Y+a\_{22}Z+T$. Then $$({\rm Perm}\,A)^2\geqslant 4a\_{11}a\_{22}(XT+YZ).$$
I claim that $XT+YZ$ is coefficient-wise not less than ${\rm Perm}\, A\_{12} {\rm Perm}\, A\_{... | 2 | https://mathoverflow.net/users/4312 | 424593 | 172,465 |
https://mathoverflow.net/questions/424567 | 2 | Let $N=\{1,\dots,n\}$. Let $M$ be the set of all binary trees $B$ formed from all elements of $N$ (i.e of size $n$). Let $P$ be a numeric property defined for all $B\in M$. Let $O\in M$ be the optimal $B$ that minimizes $P$.
Example: Let $P$ be the sum of the heights of all nodes of $B$. This measures the "balancing"... | https://mathoverflow.net/users/11504 | Finding the best binary tree with a general property | There are many ways to generate the binary trees systematically. An older example is in [this paper](https://watermark.silverchair.com/290171.pdf%5DADCCAqcGCSqGSIb3DQEHATAeBglghkgBZQMEAS4wEQQMrKjgZkHACgY74gsDAgEQgIICeMZ44ylFTEpgx8fPGdZC0E8nZO4Pmd0nc0uOXM3dRDI03jqH5OdyxRr6xUxItzDL8H88_Dxae1YoAzHc4UynsczKNba1D3k7jk7VGuk2... | 1 | https://mathoverflow.net/users/9025 | 424597 | 172,466 |
https://mathoverflow.net/questions/424602 | 3 | $\require{AMScd}$
Related to [this](https://mathoverflow.net/questions/424558/in-luries-higher-topos-theory-lemma-4-3-2-7), I have a question about the proof given in [Kerodon](https://kerodon.net/) of the following result:
>
> [**Proposition 7.3.7.1**](https://kerodon.net/tag/030V): Let $C$ be an $\infty$-catego... | https://mathoverflow.net/users/76636 | Question about the proof of Kerodon tag 030V (Proposition 7.3.7.1) | I think you are right that the cube is not a levelwise equivalence - I saw you already put a comment on the Kerodon page so you should get an answer soon.
Here's how I would fix the proof: call your first square $S$ and your second square $S\_0$ (for "restriction to $C^0$). Restriction is a map $S\to S^0$. Draw this ... | 4 | https://mathoverflow.net/users/102343 | 424606 | 172,469 |
https://mathoverflow.net/questions/424585 | 2 | The spherical derivative of a meromorphic function is defined as
$$f^\#(z):=\frac{|2f'(z)|}{1+|f(z)|^2}.$$
The motivation is that given a piecewise smooth curve $\gamma$ in the complex plane, the length of $f\circ \gamma$ in the Riemann sphere is given by $\int\_{\gamma} f^\#(z)d|z|$. When $f=az$, where $a$ is a nonzer... | https://mathoverflow.net/users/51546 | On a rigidity question related to spherical derivative of meromorphic functions | The other possibility is that $f(z)=e^z$. Explicit computation gives
$$\frac{d}{dr}f^\#(re^{i\phi})=2e^{r\cos\phi}\cos\phi(1-e^{2r\cos\phi}),$$
which is $\leq 0$ everywhere.
On the other hand, the exponential is the only exception among entire functions.
Indeed, if $f^\#$ is decreasing on each radius, then $f^\#$ h... | 5 | https://mathoverflow.net/users/25510 | 424609 | 172,470 |
https://mathoverflow.net/questions/424577 | 5 | I'm reading a paper in which the author use $(p,q)$-forms and currents on a complex analytic space.
My question is how to define $(p,q)$-current on complex space? Does it have similar properties like closeness, positivity and cohomology as in the smooth case? Could we define intersection theory or wedge product in th... | https://mathoverflow.net/users/167083 | How to define a current on a complex analytic space | The definition can be found in for example Section 3.3 of T. Bloom, M. Herrera: De Rham Cohomology of an Analytic Space (Inventiones math. 7, 275-296 (1969)) and Section 4.2 of M. Herrera, D. Lieberman: Residues and Principal Values on Complex Spaces (Math. Ann. 194, 259-294 (1971)) (in the more general setting of semi... | 7 | https://mathoverflow.net/users/49151 | 424610 | 172,471 |
https://mathoverflow.net/questions/424612 | 2 | Let $X$ be a smooth projective variety and $E$ a vector bundle of rank $3$ over $X$. Moreover let $L \in Pic(X)$ be a line bundle and $$q:S^2E \rightarrow L$$ a $L-$valued quadratic form.
Then we can consider the subvariety $C\_q \subset \mathbb{P}(E)$, where $\pi:\mathbb{P}(E) \rightarrow X$ is the projective bundle a... | https://mathoverflow.net/users/146431 | Class of the discriminant of a conic bundle | The map $q$ induces a morphism
$$
E \to E^\vee \otimes L
$$
and the discriminant is the zero locus of its determinant
$$
\det(E) \to \det(E^\vee) \otimes L^{\otimes 3}.
$$
Thus
$$
[D] = 2c\_1(E^\vee) + 3c\_1(L).
$$
| 4 | https://mathoverflow.net/users/4428 | 424614 | 172,472 |
https://mathoverflow.net/questions/303637 | 4 | In 2003 E. S. Croot [Ann. of Math. 157(2)(2003), 545-556] proved the Erdos-Graham Conjecture which states that if $\{2,3,\ldots\}$ is partitioned into finitely many subsets then one of the subsets contains finitely many distinct integers $x\_1,\ldots,x\_m$ satisfying $\sum\_{k=1}^m1/x\_k=1$.
Here I ask for the density... | https://mathoverflow.net/users/124654 | Density version of the Erdos-Graham conjecture | The answer is yes (even to the strong upper density version), as proved in this preprint <https://arxiv.org/abs/2112.03726>.
(I can give this answer with a high degree of confidence, since the proof is now completely formalised in Lean! This was a joint project by myself and Bhavik Mehta - see <https://github.com/b-m... | 9 | https://mathoverflow.net/users/385 | 424616 | 172,473 |
https://mathoverflow.net/questions/424605 | 5 | *Note: This question is closely related to an earlier question: [A large noise limit](https://mathoverflow.net/questions/417015/a-large-noise-limit).*
Let $W$ be a standard one dimensional Brownian motion.
For every $\varepsilon > 0$, let $A\_\varepsilon$ denote the event
$$\{\underset{0 \leq t \leq 1}{\text{max}... | https://mathoverflow.net/users/173490 | Endpoint of Brownian motion conditional on high maxima | $\newcommand{\ep}{\varepsilon}\newcommand{\vpi}{\varphi}\newcommand{\de}{\delta}$Yes, this is true:
By the reflection principle (see e.g. [Proposition 2](https://ocw.mit.edu/courses/15-070j-advanced-stochastic-processes-fall-2013/aca1518a09539a09ddd37428ab0d0268_MIT15_070JF13_Lec7.pdf), for $M:=\max\_{0\le t\le1}W\_t... | 7 | https://mathoverflow.net/users/36721 | 424617 | 172,474 |
https://mathoverflow.net/questions/424619 | 4 | Let $SL(n)$ be algebraic group defined over finite field $\mathbb{F}\_{p^n}$, $B$ be Borel subgroup consist of upper triangular matrices and $T$ be maximal torus consist of diagonal matrices. Let $W$ be Weyl group, defined by $N\_{G}T/T$. Let $G/B=\cup\_{w\in W} B\cdot e\_{w}$ be the Bruhat decomposition, where $e\_{w}... | https://mathoverflow.net/users/147080 | May Schubert cell intersection with opposite big cell polynomial count? | Yes, this is polynomial point count. An intersection of a Bruhat cell and an opposite Bruhat cell is called a "Richardson variety", and Richardson varieties come with decompositions known as Deodhar decompositions. Each piece of the Deodhar decomposition is of the form $\mathbb{G}\_m^{N-2k} \times \mathbb{A}^k$ where $... | 8 | https://mathoverflow.net/users/297 | 424626 | 172,476 |
https://mathoverflow.net/questions/422249 | 1 | Suppose that $V$ is a finite dimensional complex Hilbert space. Let $L(V)$ denote the collection of all linear mappings from $V$ to $V$. Let $A\_1,\dots,A\_r:V\rightarrow V$ be linear operators. Then define a completely positive superoperator $\Phi(A\_1,\dots,A\_r):L(V)\rightarrow L(V)$ by letting $\Phi(A\_1,\dots,A\_r... | https://mathoverflow.net/users/22277 | Are these $L_2$-spectral radii approximations strictly increasing? | **Yes.** If $C\_1,\dots,C\_r$ are the matrices in Example 4, then by [my answer to my other question](https://mathoverflow.net/a/424572/22277), $$\rho\_{2,d}(C\_1,\dots,C\_r)=\sqrt{d/r}$$
whenever $1\leq d\leq r$.
| 0 | https://mathoverflow.net/users/22277 | 424628 | 172,477 |
https://mathoverflow.net/questions/424613 | 19 | Mark Hovey maintains a [list of open problems in model category theory](https://www-users.cse.umn.edu/%7Etlawson/hovey/model.html). I think this list is quite old, and I don't know if Hovey is still updating it or not.
>
> My question is:
>
>
>
>
> i) which of the 13 problems in that list are still
> conside... | https://mathoverflow.net/users/139854 | Mark Hovey's open problems in the theory of model categories | I am a former student of Mark Hovey's, and during grad school, I wrote a document giving an update on the status of the 13 problems (as of 2012 or 2013, I guess). I just briefly went through it a moment ago to give some updates, but it's still in rough shape. I apologize for what I'm sure will be many omissions, as so ... | 38 | https://mathoverflow.net/users/11540 | 424641 | 172,481 |
https://mathoverflow.net/questions/424408 | 20 | **Question P.** *Can a polynomial $P(x)=\sum\_{n=0}^ma\_nx^n$ with coefficients $a\_n\in\{-1,1\}$ (and $P(1)=0$) have a multiple root in the interval $(\tfrac12,1)$?*
Also I am interested in a similar question for analytic functions.
**Question A.** *Let $f(x)=\sum\_{n=0}^\infty a\_nx^n$ a series with coefficients ... | https://mathoverflow.net/users/61536 | Multiple roots of polynomials with coefficients $\pm 1$ |
>
> **Question P.** *Can a polynomial $P(x)=\sum\_{n=0}^ma\_nx^n$ with coefficients $a\_n\in\{-1,1\}$ (and $P(1)=0$) have a multiple root in the interval $(\tfrac12,1)$?*
>
>
>
Yes. The following four Littlewood polynomials:
* $z^{27} + z^{26} + z^{25} + z^{24} + z^{23} - z^{22} - z^{21} + z^{20} + z^{19} + z^... | 14 | https://mathoverflow.net/users/46140 | 424650 | 172,484 |
https://mathoverflow.net/questions/424646 | 6 | Let $V$ be a (complex, finite-dimensional) vector space. Suppose that two (finite) groups $A$ and $B$ act on $V$. Furthermore, suppose that these actions commute, so that the direct product $A \times B$ also acts on $V$.
As a representation of $A$, we can decompose $V$ as the direct sum $$ V = \bigoplus\_i V\_i ,$$ w... | https://mathoverflow.net/users/73667 | If I know how to decompose a vector space in irreducible representations of two groups, can I understand the decomposition as a rep of their product? | To be totally clear: no, the decomposition as a representation of $A$ and the decomposition as a representation of $B$ separately don't determine the decomposition as a representation of $A \times B$, because this is not enough information by itself to determine which irreducibles of $A$ pair with which irreducibles of... | 13 | https://mathoverflow.net/users/290 | 424654 | 172,486 |
https://mathoverflow.net/questions/424430 | 10 | Consider the $N$ by $N$ matrix
$$M\_N= \begin{pmatrix} 1+3\lambda & -1-2\lambda & - \lambda & 0 & 0 &0 &0\\
-1-2\lambda & 2+3\lambda & -1 & -\lambda & 0 & 0 & 0\\
-\lambda & -1 & 2(1+\lambda) &-1 & -\lambda & 0& 0 \\
0 & \ddots & \ddots & \ddots & \ddots& \ddots & 0 \\
0 & 0 & -\lambda & -1 & 2(1+\lambda) & -1 & -\la... | https://mathoverflow.net/users/150549 | Support of eigenvectors | This looks to be true for $\varepsilon=1/4$.
First of all, the space of symmetric ($u\_i=u\_{N+1-i}$ for all $i=1,\ldots, N$) vectors is invariant for $M\_N$, and so is the space of antisymmetric ($u\_i=-u\_{N+1-i}$ for all $i=1,\ldots, N$) vectors. Since the sum of these two subspaces consists of all vectors, we con... | 2 | https://mathoverflow.net/users/4312 | 424655 | 172,487 |
https://mathoverflow.net/questions/424272 | 16 | Main Question
-------------
This question is about finding *explicit*, *calculable*, and *fast* error bounds when approximating continuous functions with polynomials to a user-specified error tolerance.
---
EDIT (Apr. 23): See also a [revised version of this question](https://mathoverflow.net/questions/442057/e... | https://mathoverflow.net/users/171320 | Explicit and fast error bounds for polynomial approximation | If $f, f', \dotsc, f^{(\nu-1)}$ are all absolutely continuous on $[-1, 1]$, and $f^{(\nu)}$ has bounded variation $V$ on $[-1, 1]$, where $\nu \ge 1$, then the polynomial interpolant $p\_n$ of degree $n > \nu$ through the Chebyshev points of the second kind,
$$ p\_n(x\_j) = f(x\_j), \quad x\_j = \cos \tfrac{j\pi}{n}, \... | 5 | https://mathoverflow.net/users/70005 | 424657 | 172,488 |
https://mathoverflow.net/questions/424683 | 9 | In a closed (say differentiable) Riemannian manifold you see only continuous features when looking at small neighbourhoods of points. From afar,
discrete features appear ((co)homology, closed geodesics, eigenvalues of the laplacian and so on).
In physics, I have the impression that more or less the opposite is going ... | https://mathoverflow.net/users/4556 | Why are discreteness and smoothness in physics inversed with respect to geometry? | The "manifold picture" can be applied to physics in the context of the [Brillouin zone](https://en.wikipedia.org/wiki/Brillouin_zone), see for example [On Brillouin Zones](https://www.math.stonybrook.edu/preprints/ims98-7.pdf). The reason that discreteness and smoothness appear inverted, is that the Brillouin zone desc... | 6 | https://mathoverflow.net/users/11260 | 424690 | 172,494 |
https://mathoverflow.net/questions/424686 | 2 | Let $(C, \|\cdot\|)$ be the Banach space of continuous paths $x: [0,1]\rightarrow\mathbb{R}^d$ starting at zero with sup-norm $\|\cdot\|$.
Let further $B\subset C$ be the subspace of $0$-started continuous paths of bounded variation, i.e.
$$B = \big\{x\in C \ \big| \ \|x\|\_1:=\sup\nolimits\_{(t\_\nu)}{\textstyle\s... | https://mathoverflow.net/users/472548 | Borel $\sigma$-algebras on paths of bounded variation | I think there is a problem with a mere semi-norm. The constant functions have variation distance $0$ from each other. Any variation-open set contains either all the constants, or none of them. Therefore, any variation-Borel set contains either all the constants, or none of them.
Choose $\mathbf u \in \mathbb R^n \set... | 2 | https://mathoverflow.net/users/454 | 424692 | 172,495 |
https://mathoverflow.net/questions/424694 | 20 | Let $p$ be a prime, and consider $$S\_p(a)=\sum\_{\substack{1\le j\le a-1\\(p-1)\mid j}}\binom{a}{j}\;.$$
I have a rather complicated (15 lines) proof that $S\_p(a)\equiv0\pmod{p}$. This must be
extremely classical: is there a simple direct proof ?
| https://mathoverflow.net/users/81776 | Analogue of Fermat's "little" theorem | Let $$P(x)=(1+x)^a-1-x^a=\sum\_{1 \le j \le a-1} \binom{a}{j}x^j.$$ Working in a field $F$ where $|\{\mu \in F: \mu^{p-1}=1\}|=p-1$ (roots of unity of order $p-1$ exist), we have
$$ \frac{1}{p-1}\sum\_{\mu^{p-1}=1}P(x \mu) = \sum\_{\substack{1 \le j \le a-1\\(p-1)\mid j}} \binom{a}{j}x^j.$$
We now specialize the field ... | 38 | https://mathoverflow.net/users/31469 | 424696 | 172,496 |
https://mathoverflow.net/questions/424699 | 2 | Let $X$ be a variety over $k = \mathbb{C}$ with an action of a finite group $G$. According to [this paper](https://arxiv.org/pdf/1711.07909.pdf) (Section 4), the induced action of $G$ on the cohomology of $X$ respects the mixed Hodge structure, allowing to define a $G$-equivariant $E$-polynomial
$$
e^G(X) = \sum\_{k,... | https://mathoverflow.net/users/484150 | Representation-induced relations in the Grothendieck of varieties | No.
Let $X$ be a curve of genus $6g+1$ for some $g>0$ with a free action of $S\_3$.
None of $X, X/\langle \tau \rangle$, $X/ \langle \rho \rangle$, and $X/ S\_3$ will be stably birational to each other, since they have genus, respectively, $6g+1$, $3g+1$, $2g+1$, $g+1$ by Riemann-Hurwitz and thus all have different... | 5 | https://mathoverflow.net/users/18060 | 424700 | 172,497 |
https://mathoverflow.net/questions/424672 | 3 | I'm trying to understand Berman's classic paper on the subject ("Local Nondeterminism and Local Times of Gaussian Processes"). In order to define local nondeterminism, he considers the ratio
$$
\frac{Var(X\_{t\_m}-X\_{t\_{m-1}}|X\_{t\_{m-1}}, X\_{t\_{m-2}}, \ldots X\_{t\_1})}{Var(X\_{t\_m}-X\_{t\_{m-1}})}.
$$
Then ... | https://mathoverflow.net/users/148035 | Local nondeterminism | If $X, Y\_1,\ldots Y\_m$ are jointly Gaussian variables over a probability space $\Omega$, think of them as vectors in the Hilbert space $L^2(\Omega)$. WLOG all these variables have mean zero, otherwise subtract the means without affecting variances. We can decompose $$X=Y+Z$$
where $Y\!\!\in $span$(Y\_1,\ldots Y\_m)$ ... | 4 | https://mathoverflow.net/users/7691 | 424706 | 172,498 |
https://mathoverflow.net/questions/424715 | 3 | Let $F$ be a free group and $w\in F$ a cyclically reduced word. Let $v$ be a non-trivial proper subword of $w$. Is it true that $v\notin \langle w^F\rangle$?
| https://mathoverflow.net/users/10482 | Is is true that a proper subword cannot lie in the normal closure of a word? | This is true. See Theorem 2 of [ON RELATORS AND DIAGRAMS FOR GROUPS WITH ONE DEFINING RELATION
BY
C. M. WEINBAUM](https://projecteuclid.org/journalArticle/Download?urlId=10.1215%2Fijm%2F1256052287&referringURL=https%3A%2F%2Fwww.google.com%2F&isResultClick=False).
| 5 | https://mathoverflow.net/users/15934 | 424716 | 172,502 |
https://mathoverflow.net/questions/424722 | 2 | Throughout, we denote by $\mu$ the Lebesgue measure on $[0, 1]$.
Let $f \in L^1([0, 1])$ be a nowhere zero function, that is, the set of all $x \in [0, 1]$ such that $f(x) = 0$ is empty.
Suppose there exists some $\varepsilon$ with $0 < \varepsilon < 1$ such that for every open subset $E$ of $[0, 1]$ with Lebesgue ... | https://mathoverflow.net/users/173490 | Is this function positive on a set of large measure? | I'm guessing your question means what it says literally, that $\int\_U f>0$ for every open set $U$ of measure *exactly* $\epsilon$. Is this right? (The answer even with this restriction is still positive).
Let $N=\{x\colon f(x)<0\}$ and suppose for a contradiction that $\mu(N)\ge\epsilon$. Let $N\_1$ be a subset of $... | 3 | https://mathoverflow.net/users/11054 | 424724 | 172,503 |
https://mathoverflow.net/questions/424721 | 0 | Suppose $W$ is a proper von Neumann Algebra contained in $B(H)$ and the identity in $W$ is the identity mapping of $H$ (namely, $W$ does not have non-trivial null space).
1. Given a self-adjoint $T\in W$, does there exists a projection $P$ that is not in $W$ such that one of $TP, PT, PTP$ is not in $W$? (we definitel... | https://mathoverflow.net/users/151332 | "Project" an operator outside of a von Neumann Algebra into it | These aren't research-level questions, but anyway. Let $W$ be a von Neumann algebra unitally contained in $B(H)$ and let $T \in W$ be nonzero (doesn't have to be self-adjoint). Suppose for every projection $P \in B(H)$ we have $PT \in W$. Taking linear combinations and weak\* limits, we get $ST \in W$ for all $S \in B(... | 1 | https://mathoverflow.net/users/23141 | 424728 | 172,504 |
https://mathoverflow.net/questions/424734 | 2 | Let $ \Omega = \mathbb{T}^d (1 \leq d \leq 3)$ be the $d$ dimensional torus and $ u \in H^2(\Omega) $ be a complex valued function. For some $ 0 < \alpha < 1 $, let $ g(u) = |u|^\alpha u $.
**My question is**: do we have some estimates for $ g(u) $ which can ensure that it belongs to some fractional Sobolev space $ H... | https://mathoverflow.net/users/484187 | $H^s$ norm of non-integer power of functions | In Christ-Weinstein, JFA 100 (1991) 87-109 you can find the fractional chain rule (Proposition 3.1)
$$
\|F(u)\|\_{\dot H^s\_r}\le C
\|F'(u)\|\_{L^p}\|u\|\_{\dot H^s\_q}
$$
where $s\in(0,1)$, $p,q,r\in(1,\infty)$, $1/r=1/p+1/q$, $F\in C^1$, and $\dot H^s\_r$ is the homogeneous Sobolev space with norm $\||D|^su\|\_{L^r}$... | 3 | https://mathoverflow.net/users/7294 | 424739 | 172,510 |
https://mathoverflow.net/questions/424747 | 1 |
>
> Is the topological group $(\mathbf{Q}\_p/\mathbf{Z}\_p)^{\oplus k}$, $k\ge 1$, a $\sigma$-compact topological group when endowed with its natural $p$-adic topology?
>
>
>
More generally, I'm looking for a criterion for locally compact Hausdorff topological groups to not contain a nested exhaustive sequence o... | https://mathoverflow.net/users/nan | $\sigma$-compactness of some locally compact Hausdorff topological groups | I agree with @MatthewDaws ... for example the real Lie group $(\mathbb R,+)$ with the usual topology is locally compact and sigma-compact but not compact.
Take any locally compact group $G$. There is a neighborhood $V$ of $e$ with compact closure. The subgroup $H$ generated by $V$ is an open, locally compact, sigma-c... | 1 | https://mathoverflow.net/users/454 | 424752 | 172,513 |
https://mathoverflow.net/questions/424746 | 2 | Let $A$ be a linear map over the finite-field vector space $(\mathbb F\_2)^n$, i.e., an $\mathbb F\_2$-valued $n\times n$ matrix, not necessarily symmetric. I'm interested in the sum
$$Z(A) = \sum\_{X\in \mathbb F\_2^n} (-1)^{X^T A X}\;,$$
where
$$x\rightarrow (-1)^x$$
should be thought of as a function from $\mathbb F... | https://mathoverflow.net/users/115363 | Sum over exponentiated bilinear form in finite-field vector space | This is a multidimensional Gauss sum, and can be handled by the same methods used to handle Gauss sums.
$Z(A)=0$ if and only if $X^T A X$ is nonzero for some $X \in \ker (A + A^T)$, and, if $Z(A) \neq 0$, then $$Z(A) = \pm 2^{ \frac{n + \dim ( ker (A + A^T))}{2}}$$ which implies your divisibility claim.
To prove th... | 5 | https://mathoverflow.net/users/18060 | 424769 | 172,518 |
https://mathoverflow.net/questions/424774 | 4 | I recall that there was a theorem mimicking the change of variables' integral formula. Surprisingly, I can't find it on the Fosco Loregian book. The change of variables formula states that, if $f: E \to B$ is a measurable mapping between measurable spaces and $\mu$ is a measure on $E$, then for all $g: B \to \mathbb{R}... | https://mathoverflow.net/users/140013 | Change of coordinates for coends | $\require{AMScd}$First of all, I suspect that "$p$ has set fibers and it is surjective" means that it is a discrete fibration. In that case, $p$ corresponds to a functor $G\_p : C \to Set$ (covariant if $p$ is an opfibration, contravariant if it's a fibration) defined precisely as $Gx = E\_x$. So, a better way to rephr... | 3 | https://mathoverflow.net/users/7952 | 424781 | 172,519 |
https://mathoverflow.net/questions/424775 | 1 | Suppose $p(x\_1,...,x\_n)$ is a polynomial in $n$ real variables whose Hessian is not identically zero. Can we say that there is a set $Z \subset {\mathbb R}^n$ of measure zero and a constant $M$ such that for all real numbers $c\_1,...,c\_n$ the cardinality of the set of $(x\_1,...,x\_n) \in {\mathbb R}^n - Z$ for whi... | https://mathoverflow.net/users/149955 | Is there a uniform bound on the number of solutions to ${\partial p \over \partial x_i} (x_1,...,x_n) = c_i$ outside a set of measure zero? | This follows from basic facts about polynomials, to your generalization of a polynomial map $F: \mathbb R^n \to \mathbb R^n$. Since the Jacobian determinant of $F$ does not vanish identically, its zero set $Z$ (the critical points of $F$) has measure 0, since $F$ is a polynomial. Away from critical values, the inverse ... | 3 | https://mathoverflow.net/users/5279 | 424782 | 172,520 |
https://mathoverflow.net/questions/424445 | 0 | **Inequality 1**
\begin{align}
\mathbb{P}\left(\frac{1}{n} \sum\_{i=1}^{n}\left(f\left(x\_{i}\right)-\mathbb{E}[f]\right) z\_{i} \geq \frac{\epsilon}{8}\right) \leq 2 \exp \left(-\frac{\epsilon^{2} n d}{9^{4} c L^{2}}\right)
\end{align}
Since we assumed that the range of the functions is in $[-1,1]$ we have $\mathb... | https://mathoverflow.net/users/nan | Union Bound of two events? | By the triangle inequality, the event
$$A=\left\{\exists f \in \mathcal{F}: \frac{1}{n} \sum\_{i=1}^{n} f\left(x\_{i}\right) z\_{i} \geq \frac{\epsilon}{4}\right\}$$
is contained in the union of the two events
$$B=\left\{\exists f \in \mathcal{F}: \frac{1}{n} \sum\_{i=1}^{n}\left(f\left(x\_{i}\right)-\mathbb{E}[f]\ri... | 0 | https://mathoverflow.net/users/7691 | 424786 | 172,522 |
https://mathoverflow.net/questions/424771 | 2 | Consider a random matrix $A \in \mathbb{R}^{m\times n}$ with i.i.d. entries, with mean zero and variance 1 and $m <n $. I am interested in the expectation of $$E\_{A}(\mathrm{Tr}( (A^T A + \lambda \mathrm{Id})^{-1})).$$ This question is similar to [Trace of inverse of random positive-definite matrix in high dimension?]... | https://mathoverflow.net/users/483386 | Trace inverse of random PSD matrix? | I think the product $A^\top A$ in the OP should read $AA^\top$, to avoid a trivial contribution from zero eigenvalues (assuming $A\in\mathbb{R}^{m\times n}$ and $m<n$).
For $m,n\gg 1$, and $m/n\equiv r\in (0,1)$ fixed, an integration over the [Marchenko–Pastur distribution](https://en.wikipedia.org/wiki/Marchenko%E2%... | 1 | https://mathoverflow.net/users/11260 | 424789 | 172,524 |
https://mathoverflow.net/questions/422922 | 3 | Tyma Gaidash has recently [posted](https://math.stackexchange.com/a/4442541/2513) solutions to some quintics in terms of Inverse Beta Regularized function. He also [found](https://math.stackexchange.com/questions/46934/what-is-the-solution-of-cosx-x/4389007#4389007) the closed form for the equation $\cos x=x$ using the... | https://mathoverflow.net/users/10059 | Can a general quintic be solved using inverse beta regularized function? | Yes, it seems, it can. Any general quintic can be reduced to the [Bring-Jerrard form](https://mathworld.wolfram.com/Bring-JerrardQuinticForm.html) $x^5+ax+b=0$.
Then Tyma Gaidash found a solution for the Bring-Jerrard form via Inverse Beta Regularized function:
$x=\frac{5b}{4a\left(\text I^{-1}\_{\frac{3125b^4}{256... | 2 | https://mathoverflow.net/users/10059 | 424790 | 172,525 |
https://mathoverflow.net/questions/424731 | 1 | Let $k=\mathbb F\_q\left(\left(\frac1T\right)\right)$, $\overline k$ be an algebraic closure of $k$ and $K$ be the completion of $\overline k$ for the $\frac1T$ valuation. Consider the morphism $\sigma:k\to L=K\left(\left(\frac1Z\right)\right)$ defined by $\sigma(\sum\_{n\ge m}a\_n\left(\frac1T\right)^n)=\sum\_{n\ge m}... | https://mathoverflow.net/users/33128 | Extension of morphisms in function fields | Not if $\alpha = \sqrt{T}$ since $\alpha^2 = T$ is sent to $Z$ which does not have a square root in $L$.
| 1 | https://mathoverflow.net/users/18060 | 424792 | 172,526 |
https://mathoverflow.net/questions/424756 | 1 | Let $X\neq \emptyset$ be a set. By $\text{Part}(X)$ we denote the set of all partitions of $X$ not containing $\emptyset$ as an element.
First, note that $\bigcup{\frak P}$ is the collection of subsets of $X$ that are a member of at least one partition in ${\frak P}$. We say that ${\frak P}\subseteq \text{Part}(X)$ i... | https://mathoverflow.net/users/8628 | Optimal partitions amongst a given set of partitions | Modifying your note at the end slightly, try the following.
Let $X=\{0,1,2,\ldots\}$ and let $P=\{\{0,1\},\{2,3\},\{4,5\},\ldots\}$. Let $\mathfrak{P}$ be all partitions of $X$ which refine $P$ and moreover contain only finitely many of the blocks in $P$ (thus for instance $P$ itself does not belong). Cover $X$ by se... | 2 | https://mathoverflow.net/users/25028 | 424793 | 172,527 |
https://mathoverflow.net/questions/424776 | 2 | Let $\Omega\subset\mathbb{C}^n$ be any $C^{\infty}$ bounded domain and let $H^2(\partial\Omega)$ denotes a holomorphic Hardy space which is a $L^2(\partial\Omega)$ closure of $A^{\infty}(\Omega)(=\mathscr{O}(\Omega)\cap C^{\infty}(\overline{\Omega})).$ Because $H^2(\partial\Omega)$ is a closed subspace of $L^2(\partial... | https://mathoverflow.net/users/482747 | Regarding basis of holomorphic Hardy space | One should be able to find a countable linearly independent subset of $A^\infty(\Omega)$ whose linear span is dense in $H^2(\partial\Omega)$ (it would suffice to show that the orthogonal complement of this subset inside $H^2$ is zero).
Once you have such a subset, order it in any ways as a sequence and then apply the... | 1 | https://mathoverflow.net/users/763 | 424800 | 172,528 |
https://mathoverflow.net/questions/424744 | 7 | Work in a theory with (deep breath) a countable number of primitives denoted with capital letters from the end of the alphabet with numerical subscripts $\{X\_n,Y\_n,Z\_n,\dots\}\_{n<\omega}$ indicating which kind of primitive they are together with primitive relations $\{\in\_n\}\_{n<\omega}$ with numerical subscripts... | https://mathoverflow.net/users/92164 | Consistency strength of an attempt at higher order set theory | I believe $ZFC$ plus the existence of a countable collection of strictly inaccessible cardinals is also a lower bound on the consistency strength of this theory, and am posting this as a CW answer to close the question.
For a proof sketch, define ordinals to be hereditarily membership transitive primitives, define ra... | 1 | https://mathoverflow.net/users/92164 | 424802 | 172,529 |
https://mathoverflow.net/questions/424783 | 3 | Let $k$ be a field, $A$ a $k$-algebra and $X$ a finite dimensional indecomposable $A$-module. Then $\text{End}\_A (X)$ is a local ring. Let $m$ be its maximal ideal. Can we say anything about the $k$-dimension of $\text{End}\_A (X)/m$? It always equals $1$ if $k$ is algebraically closed. Can we bound it if $k$ is not a... | https://mathoverflow.net/users/145920 | Dimension of division rings coming from indecomposable modules | Even if $A$ is a finite dimensional $k$-algebra, there may be no bound on the dimension of $\text{End}\_A(X)/m$.
Let $Q$ be the Kronecker quiver (i.e., the quiver with two vertices and two arrows from the first vertex to the second), and let $A=kQ$ be its path algebra over $k$.
Suppose $k(\alpha)$ is a simple algeb... | 7 | https://mathoverflow.net/users/22989 | 424815 | 172,530 |
https://mathoverflow.net/questions/424799 | 2 | Let $x\_1,...,x\_N$ be $N$ mutually distinct real numbers.
I wonder: How does the expression
$$ f\_x(N)=\sup\_{\lambda \in \mathbb R}\min\_{i \neq j} \min\_{n \in \mathbb Z}\left\vert \lambda (x\_i-x\_j)-n\right\rvert$$
depend on $N$? I acknowledge the answer might depend on $x$, but I am looking for a lower bound ... | https://mathoverflow.net/users/457901 | Maximal distance of set to integers | If I understood correctly, your question is a form of the [Lonely runner conjecture](https://en.wikipedia.org/wiki/Lonely_runner_conjecture). You can see several results and a summary on that Wikipedia link. An important special case is when the difference of the $x\_i$'s are irrational, in which case you can spread th... | 8 | https://mathoverflow.net/users/955 | 424816 | 172,531 |
https://mathoverflow.net/questions/424704 | 3 | I am looking for a reference. I hope that what follows is in some textbook.
Let $q$ be an odd prime power and let $\ell$ be a positive integer. Now, let $\mathfrak{q}:\mathbb{F}\_{q^\ell}^2\to\mathbb{F}\_{q^\ell}$ be a non-degenerate quadratic form. If we compose $\mathfrak{q}$ with the trace mapping $\mathrm{Tr}:\ma... | https://mathoverflow.net/users/45242 | Inclusions among finite orthogonal groups over finite fields | The answer will depend on $q$, $\epsilon$ and potentially also $\ell$. Instead of looking at spinor norms you can find an element $x$ such that $\mathrm{SO}\_2^\epsilon(q^\ell)=\langle \Omega\_2^\epsilon(q^\ell),x\rangle$. You then lift $x$ to $\mathrm{O}^\epsilon\_{2\ell}(q)$ and see what sort of element it is. When y... | 2 | https://mathoverflow.net/users/3214 | 424826 | 172,533 |
https://mathoverflow.net/questions/424777 | 7 | For a free group $F$, an element $w$ is primitive if it is part of some free basis for $F$.
Let $\pi:F[x\_0,x\_1,...,x\_n]\rightarrow F[x\_1,x\_2,...,x\_n]$ be defined $\pi (x\_0)=1$ and $\pi (x\_i)=x\_i$ for $i\geq 1$.
My question is:
Is there an example of a cyclically reduced primitive element $w$ in $F[x\_0,x... | https://mathoverflow.net/users/484234 | Primitive elements in a free group with trivial projection | No, there cannot be a primitive element $w \in \ker \pi$ that is not conjugate to $x\_0$ or $x\_0^{-1}$.
The map $\pi$ factors through $F[x\_0, x\_1, \dots, x\_n] / \langle \langle w \rangle \rangle$, which is isomorphic to $F\_n$ since $w$ is primitive. As the free group is Hopfian, this means the induced surjection... | 10 | https://mathoverflow.net/users/24447 | 424830 | 172,534 |
https://mathoverflow.net/questions/424827 | 10 | Consider the minimal class of (simple, undirected) connected graphs (strictly speaking, isomorphism classes of connected graphs) which contains a single vertex $K\_1$, and is closed under following operations:
1. add a copy $\tilde{v}$ of a vertex $v$ (neighbours of the new vertex are the same as of $v$) and do not j... | https://mathoverflow.net/users/4312 | "Gluing and copy" graphs | This is the same as the class of distance-hereditary graphs, which have received a fair amount of attention (see <https://en.wikipedia.org/wiki/Distance-hereditary_graph>).
| 11 | https://mathoverflow.net/users/385 | 424833 | 172,535 |
https://mathoverflow.net/questions/424829 | 3 | Let $L$ be a Galois extension of a number field $K$ with the Galois group $G$. Let $N$ be the smallest integer with the following property: For any conjugacy class $C$ of $G$ there exists an unramified prime $\mathfrak{p}$ of $K$ such that $\text{Frob}\_\mathfrak{p}\in C$ and $\text{Norm}\_{K/\mathbb{Q}}\mathfrak{p}\le... | https://mathoverflow.net/users/95601 | A variant of effective Chebotarev theorem | There are bounds on $N$, but whether they are good or not I leave up to you to decide.
First, conditional on GRH, Lagarias and Odlyzko proved a bound on $N$ in their 1977 paper ''Effective versions of the Chebotarev density theorem''. This was made explicit by Bach and Sorenson in 1996 (in "Explicit bounds for primes... | 7 | https://mathoverflow.net/users/48142 | 424837 | 172,537 |
https://mathoverflow.net/questions/424603 | 3 | Let $K$ be a compact subset of $\mathbb{R}^n$ and $\mathcal{U}$ be a finite collection of open subsets covering $K$ satisfying the minimality property: *for every $U\in \mathcal{U}$, the sub-collection $\mathcal{U}-\{U\}$ does not cover $K$.*
*Notation:* Given any subset $A\subseteq K$ denote the relative complement ... | https://mathoverflow.net/users/469470 | Lipschitz-regularity of partition of unity | For convenience write
$$ \phi\_U(x) = \| x - (K-U)\|. $$
Let $N = |\mathcal{U}|$ the number of open sets in the cover.
Define
$$ f\_1(x) = \frac{1}{N} \sum \phi\_U(x), \qquad f\_2(x) = \left( \frac1N \sum \phi\_U^2(x) \right)^{1/2} $$
Note that since $\mathcal{U}$ is a cover we have
$$ 0 < f\_1(x) \leq f\_2(x) \leq... | 4 | https://mathoverflow.net/users/3948 | 424862 | 172,541 |
https://mathoverflow.net/questions/421155 | 2 | Let $V$ be a complex finite dimensional inner product space. If $A\_{1},\dots,A\_{n}:V\rightarrow V$ are linear operators, then let $\Phi(A\_{1},\dots,A\_{n}):L(V)\rightarrow L(V)$ be the superoperator defined by letting $\Phi(A\_{1},\dots,A\_{n})(X)=A\_1XA\_1^\*+\dots+A\_nXA\_n^\*$. The completely positive superoperat... | https://mathoverflow.net/users/22277 | When does the Cauchy-Schwarz inequality for spectral radii of tensor products become equality? | **Yes.** We have a necessary and sufficient characterization for when the Cauchy-Schwarz inequality becomes equality.
For this post, $U,V,W$ shall denote finite dimensional complex Hilbert spaces. Suppose that $A\_1,\dots,A\_n:U\rightarrow U,B\_1,\dots,B\_n:V\rightarrow V$ are linear operators. Then define an operato... | 1 | https://mathoverflow.net/users/22277 | 424866 | 172,543 |
https://mathoverflow.net/questions/424760 | 8 | Disclaimer: This is a crosspost (see [MathStackexchange](https://math.stackexchange.com/questions/4466696/cartesian-monoidal-star-autonomous-categories)). Apologies if cross-posting is frowned upon. However, it seems that on Stackexchange there are not many people familiar with star-autonomous categories.
**1. Questi... | https://mathoverflow.net/users/160778 | Cartesian monoidal star-autonomous categories | [I'm going to assume $S' \cong S$, which holds in every symmetric monoidal $\*$-autonomous category. (See e.g. Lemma 5.6 of [this paper](http://arxiv.org/abs/1108.6020v2).) This applies here since cartesianness implies symmetry. Part of the proof also works with weaker assumptions, such as merely that the monoidal unit... | 6 | https://mathoverflow.net/users/27013 | 424871 | 172,544 |
https://mathoverflow.net/questions/424856 | 2 | Let $p$ be an odd prime and $F={\mathbb Z}/p{\mathbb Z}$. With $a,b,c\in F$, let $Q(x,y)=ax^2+bxy+cy^2$ where $p\nmid b^2-4ac$. I wish to prove that the number of solutions $(x,y)\in F^2$ of
$$ax^2+bxy+cy^2=u$$ is the same for all units $u\in F$.
For example, when $p=5$, the equation $x^2-2xy+3y^2=u$ has one solution... | https://mathoverflow.net/users/47804 | Are the number of solutions to $ax^2+bxy+cy^2\equiv u\pmod{p}$, $(x,y)\in\{0,\dotsc,p-1\}$, the same for all units $u$? | Let $F$ be a finite field, such as $\mathbf Z/(p)$ but it could be a more general finite field (including of characteristic $2$, and not necessarily $\mathbf Z/(2)$, so I won't make a change of linear variables using division by $2$ as in another answer). If both $a = 0$ and $c = 0$ then the equation becomes $bxy = u$,... | 5 | https://mathoverflow.net/users/3272 | 424877 | 172,548 |
https://mathoverflow.net/questions/424876 | 2 | Given a directed acyclic graph $G=(V,E)$ with a source node $s$ and a sink node $t$, and we have a weight function that is defined on $E\times E$, $f:E\times E\to R^{+}$. We want to find a $s$-$t$ path $P$ that maximizes the sum $\sum\_{\forall e\_i,e\_j\in P} f(e\_i,e\_j)$.
Does this problem have a polynomial soluti... | https://mathoverflow.net/users/481313 | Longest path on directed acyclic graph when the weight is defined on the pair of edges | The *quadratic shortest path problem* (QSSP) can be reduced to the problem. Because QSSP is NP-hard [1], the problem in the question has no polynomial-time solution unless P = NP.
To remove the restriction of the acyclicity of the graph, we can use the standard technique of using $V \times \{0,\dots,|V|\}$ as the new... | 2 | https://mathoverflow.net/users/476793 | 424887 | 172,551 |
https://mathoverflow.net/questions/424879 | 12 | Consider the two matrices with some parameter $s \in \mathbb R$
$$A\_1= \begin{pmatrix} s& -1 &0& 0 \\1&0 &0&0 \\ 0&0&1&0 \\0&0&0&1 \end{pmatrix}$$
and
$$A\_2= \begin{pmatrix} s& -1 &-1& 0 \\1&0 &0&0 \\ -1&0&s&-1 \\0&0&1&0 \end{pmatrix}.$$
I then noticed that the eigenvalues of arbitrary products of $A\_1$ and $A\_... | https://mathoverflow.net/users/150564 | Eigenvalues come in pairs | This follows from the identities
$$A\_1^{-1}=UA\_1U^{-1},\;\;A\_2^{-1}=UA\_2U^{-1},$$
$$A\_1^{\top}=VA\_1V^{-1},\;\;A\_2^{\top}=VA\_2V^{-1},$$
with
$$U=U^{-1}=\left(
\begin{array}{cccc}
0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0 \\
\end{array}
\right),\;\;V=V^{-1}=\left(
\begin{array}{cccc}
1... | 13 | https://mathoverflow.net/users/11260 | 424896 | 172,554 |
https://mathoverflow.net/questions/424895 | 5 | Consider the Sierpinski space $\text{S} = (\{0,1\}, \tau)$ where $\tau = \big\{\emptyset,\{0\}, \{0,1\}\big\}$. Endow $\text{S}^\omega$ with the product topology.
If $X, Y$ are topological spaces with $\text{S}^\omega \cong X \times Y$, does this imply that there are $\alpha, \beta \in \big(\omega\cup \{\omega\}\big)... | https://mathoverflow.net/users/8628 | Decomposing $\{0,1\}^\omega$ endowed with the Sierpinski topology | I think the answer is yes. I try to prove the stronger statement that every such homeomorphism is of the form $S^{\mathbb{N}}=S^A\times S^B$ for a disjoint union $\mathbb{N}=A\amalg B$.
Let me switch the roles of 0,1. This makes the following more easily readable.
We can introduce a order on $S^w$ by defining $x\le... | 2 | https://mathoverflow.net/users/3969 | 424918 | 172,558 |
https://mathoverflow.net/questions/424898 | 5 | Let $G$ be a finitely generated group, and let $F\_1, F\_2$ be two subgroups of $G$ which are free of finite rank at least 2. I am wondering what conditions can be placed on $G$ so that $F\_1\cap F\_2$ is finitely generated. For example:
* It is a classical result of Howson that if $G$ is itself free then $F\_1\cap F... | https://mathoverflow.net/users/6503 | Finite generation of intersections of free subgroups | The answer to both 1. and 2. is no.
Burns and Brunner showed that free-by-cyclic groups do not have the "finitely generated intersection property" (a.k.a. Howson property), which was recently generalized by Bamberger and Wise in [Failure of the finitely generated intersection property for ascending HNN extensions of ... | 9 | https://mathoverflow.net/users/24447 | 424920 | 172,559 |
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