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https://mathoverflow.net/questions/319530 | -2 | He folks, here's my problem:
Let $\mathbf{A}, \mathbf{B}, \mathbf{X}\_1, \mathbf{X}\_2, \mathbf{X}\_3, \mathbf{Y}\_1, \mathbf{Y}\_2, \mathbf{Y}\_3\in\mathbb{R}^{3\times 3}$ with determinante det()=+1. The matrices $\mathbf{A}$, $\mathbf{B}$, $\mathbf{X}\_1$, $\mathbf{X}\_2$ and $\mathbf{X}\_3$ are known and the relat... | https://mathoverflow.net/users/133906 | Decomposition of one Matrix into six matrices | If $X\_1, X\_2, X\_3$ are invertible, $Y\_1$ and $Y\_2$ can be any invertible matrices and $Y\_3 = (X\_1 Y\_1 X\_2 Y\_2 X\_3)^{-1} A B$.
| 0 | https://mathoverflow.net/users/13650 | 319535 | 138,061 |
https://mathoverflow.net/questions/319527 | 4 | Let $\mathcal{A}$ be a subalgebra of $M\_n(\mathbb{C})$. Is there a characterization of (or at least a name for) orthogonal projections $P \in M\_n(\mathbb{C})$ with the property that $PABP = PAPBP$ for all $A,B \in \mathcal{A}$? In other words, for which the compression map $A \mapsto PAP$ is a homomorphism.
The ort... | https://mathoverflow.net/users/23141 | A generalization of invariant and coinvariant subspaces | Since you are refering to subalgebras of $M\_n(\mathbb{C})$, i am not sure if this is what you are looking for, but if $S$ is a semigroup of operators on a Hilbert space $\mathcal{H}$, $\mathcal{V}\subseteq\mathcal{H}$ is a subspace of $\mathcal{H}$ and $P$ an orthogonal projection onto $\mathcal{V}$, satisfying $PABP=... | 4 | https://mathoverflow.net/users/85967 | 319540 | 138,062 |
https://mathoverflow.net/questions/319553 | 1 | Let $\zeta$ be the Riemann zeta function.
My question is: For fixed $\sigma<1/2$, how large can $|\zeta(\sigma+it)|$ be for $t\in \mathbb{R}$, even assuming zeta conjectures like the RH or the LH ?
My searches in relevant texts like Titschmarsh reveal that people seem to be focused on the case $\sigma= 1/2$, for w... | https://mathoverflow.net/users/133915 | How large can $|\zeta(\sigma + it)|$ be for $\sigma<1/2$? | I answer here for what is known for general $\sigma$, without assuming anything (in particular not LH). Let $s = \sigma + it$ where $s$ and $t$ are real numbers.
**If** $\mathbf{\sigma > 1}$, then the absolute convergence of the Dirichlet series yields
$$\zeta(s) \ll 1 \qquad(\sigma>1)$$
(here and below, the $\ll$-co... | 9 | https://mathoverflow.net/users/43737 | 319555 | 138,065 |
https://mathoverflow.net/questions/319192 | 5 | This is a refinement (perhaps a simpler version) of a question I asked here before and couldn't get an answer for.
Fix $\alpha \in (0,1]$ and a small constant $c>0$. For $x \in [0,1]$ and $N\in\mathbb{N}$, we denote by $S(x,N)$ the function that counts the number of pairs of integers $(p,q)$ such that $$|x-\frac{p}{... | https://mathoverflow.net/users/16040 | Counting primitive solutions to a diophantine inequality | An elementary way of looking at the problem gives the uniform bound you want - the trick is to look at the actual fractions.
Any two non-equivalent fractions with denominator $\le N$ have a difference in absolute value at least $\frac{1}{N^2}$. So the number of primitive pairs $(p,q)$ where $N \le q \le 2N$ satisfyin... | 0 | https://mathoverflow.net/users/43383 | 319556 | 138,066 |
https://mathoverflow.net/questions/319550 | 9 | Since etale fundamental group of a scheme $X$ is the group of natural automorphisms of the fibre functor of the category of finite etale covers of $X$, it comes with structure of a topological group. Is there some natural topology on higher etale homotopy groups (if we do not assume that $X$ is not geometrically unibra... | https://mathoverflow.net/users/132313 | Are higher etale homotopy groups topological groups in a natural way? | TL;DR The higher étale homotopy groups are the homotopy groups of the profinite completion of the shape of the étale topos. As such they are profinite groups. If you choose to see profinite groups as topological groups, group schemes or pro-systems is largely a matter of choice.
---
How is the étale homotopy grou... | 17 | https://mathoverflow.net/users/43054 | 319574 | 138,069 |
https://mathoverflow.net/questions/319562 | 1 | Let $C$ be a smooth projective curve and $X=\mathbb{P}\_C(E)$ be a ruled surface over $C$.
Let $x\_1,\ x\_2\in X$ be closed points and define $X\_1,\ X\_2$ to be elementary transforms of $X$ at $x\_1,\ x\_2$, respectively.
Then when $X\_1\cong X\_2$ holds? Even if $x\_1,\ x\_2$ are in the same fiber $\pi^{-1}(p)$ where... | https://mathoverflow.net/users/75699 | When are two elementary transforms isomorphic? | Let $c\_1,c\_2 \in C$ be a pair of points.
The existence of an isomorphism $X\_1 \cong X\_2$ (for some choices of points $x\_1$, $x\_2$ in $X$ over $c\_1$ and $c\_2$) over $C$ is equivalent to the existence of a line bundle $L$ on $C$ of degree 1 such that
$$
L^2 \cong O\_C(c\_1 + c\_2)
$$
and the existence of a morphi... | 2 | https://mathoverflow.net/users/4428 | 319577 | 138,071 |
https://mathoverflow.net/questions/319578 | 2 | Let $\pi:X \to Y$ be a $\mathbb{C}^\*$-fibration between complex manifolds in the sense that there exists a fixed integer $a$ such that for every $y \in Y$, $\pi^{-1}(y)=(\mathbb{C}^\*)^a$. Suppose further that $Y$ is compact. Does there exist a compactification $\widetilde{X}$ of $X$, which is a $(\mathbb{P}^1)^a$-fib... | https://mathoverflow.net/users/32151 | Naive compactification of $\mathbb{C}^*$-fibrations | No that is not true. There is a much simpler example than in the MO answer above. I might have written the following example in one of my earlier MO answers.
Let the proper target of the morphism be $\mathbb{P}^1\_k = \text{Proj}\ k[S,T]$. The domain of the morphism will be an open subset of $$\mathbb{P}^1\_k \times\... | 5 | https://mathoverflow.net/users/13265 | 319583 | 138,072 |
https://mathoverflow.net/questions/319559 | 24 | This question is inspired by article [Alexander Shen "Gauss multiplication trick?"](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=mp&paperid=941&option_lang=rus) (Russian, "Mathematical Enlightenment", 2019).
*Dasgupta, Papadimitriou, Vazirani, Algorithms* (2008) [Ch. 2:](https://people.eecs.berkeley.edu/... | https://mathoverflow.net/users/5712 | "Gauss trick" vs Karatsuba multiplication | I made an effort to trace the source of what [Wikipedia describes as:](https://en.wikipedia.org/wiki/Karatsuba_algorithm) *"The basic step of Karatsuba's algorithm is a formula that allows one to compute the product of two large numbers $x$ and $y$ using three multiplications of smaller numbers, each with about half as... | 25 | https://mathoverflow.net/users/11260 | 319589 | 138,073 |
https://mathoverflow.net/questions/314493 | 0 | what is written below is a conjecture that I posed , and I ask for a proof or a disproof of it .I have checked the conjecture from $n$=$1$ up to $n$=$10$ using Matlab, and all results were in agreement with the conjecture .
The conjecture is as follows :
assume $x$ is a positive real parameter that does not equal $1$ ,... | https://mathoverflow.net/users/113991 | Conjecture that relates matrix systems with some specific functions as solution sets | The affirmative answer and explicit solution to this question directly follows from [my answer](https://mathoverflow.net/q/281647) to the previous one by substituting there $u\_i:=X^{Yi+Z}+1$ and $x:=X^Y$. Here I use capital letters to refer to the variables in the present question and distinguish them from those in my... | 1 | https://mathoverflow.net/users/7076 | 319593 | 138,075 |
https://mathoverflow.net/questions/319598 | 5 | In HTT, given a inner fibration $p : X \rightarrow S$ of simplicial, an edge $f : x \rightarrow y$ of the simplicial set $X$ is said to be a $p$-Cartesian if the induced map
$$ X\_{/f} \rightarrow X\_{/y} \times\_{S\_{/p(y)}} S\_{/p(f)}$$ is a trivial Kan fibration.
In Remark 2.4.1.4 Lurie says that this definition ... | https://mathoverflow.net/users/131022 | Remark 2.4.1.4 Higher Topos Theory | Let's see what the data of a diagram
\begin{matrix}
\partial\Delta^{n-2}&{\to}&X\_{/f}\\
\downarrow &&\downarrow\\
\Delta^{n-2}&{\to}&X\_{/y}\times\_{S\_{/p(y)}}S\_{p(f)}\end{matrix}
translates to. I claim this is formally the same as a diagram
\begin{matrix}
\partial\Delta^{n-2}\star\Delta^{\{n-1,n\}}\sqcup\_{\... | 7 | https://mathoverflow.net/users/51424 | 319601 | 138,077 |
https://mathoverflow.net/questions/319580 | 15 | It is well known (e.g., [Reference for "lax monoidal functors" = "monoids under Day convolution"](https://mathoverflow.net/questions/130616/reference-for-lax-monoidal-functors-monoids-under-day-convolution/130619) ) that if $\mathcal C$ is a monoidal $\mathcal V$-enriched category, then a monoid in $[\mathcal C, \mathc... | https://mathoverflow.net/users/70015 | Monoidal functors $\mathcal C \to [\mathcal D,\mathcal V]$ are monoidal functors $\mathcal C \otimes \mathcal D \to \mathcal V$? | **Edit:** Alexander Campbell points out in the comments that a reference for this, in the case ${\cal V}=\rm Set$, is Claudio Pisani's paper [Sequential multicategories](http://tac.mta.ca/tac/volumes/29/19/29-19abs.html). Following is a sketch of the argument.
Consider the category $\mathcal{V}\text{-}\mathrm{Mult}$ ... | 9 | https://mathoverflow.net/users/49 | 319606 | 138,079 |
https://mathoverflow.net/questions/319609 | 11 | I am aware that at least for lower dimensions,
>
> "smooth manifolds iff triangulable manifolds"
>
>
> at least for dimensions below a certain critical dimensions D.
>
>
>
My question is that for
>
> * For orientable manifolds, in which lower dimensions $\leq$ D, such that
> "smooth manifolds iff triangu... | https://mathoverflow.net/users/27004 | Critical dimensions D for "smooth manifolds iff triangulable manifolds" | All smooth manifolds are triangulable, as you say. This follows from Morse theory, which dictates that you only need to know how to triangulate (PL) handle-attachments, which one can do by hand. The result, though not phrased then in terms of Morse theory, has been known since the 30s. So in my post I meant "topologica... | 25 | https://mathoverflow.net/users/40804 | 319611 | 138,081 |
https://mathoverflow.net/questions/319613 | 19 | Consider $SL\_2$ embedded into $SL\_3$ as upper left block matrices. The quotient $SL\_3/SL\_2$ is an affine variety, as is any quotient of reductive groups. How does one describe $SL\_3/SL\_2$? What are the equations for it in some affine space?
(One can also pose the same question more generally for $SL\_{n}/SL\_{n... | https://mathoverflow.net/users/130024 | What are the equations for $SL_3/SL_2$? | In general, $SL\_n/SL\_{n - 1}$ is isomorphic to an affine subvariety $X$ of $\mathbb{A}^{2n}$ with coordinate equations given by $\sum\_i x\_i y\_i = 1$.
The map $SL\_n/SL\_{n-1} \rightarrow X$ is given by: the coordinates $y\_i$ are given by the "last" vector (i.e. the last column), which is unaffected by $SL\_{n-1... | 27 | https://mathoverflow.net/users/44191 | 319616 | 138,082 |
https://mathoverflow.net/questions/319620 | 8 | Let $\eta=e^{\frac{2\pi i}n}$, an $n$-th root of unity. For pedagogical reasons and inspiration, I ask to see different proofs (be it elementary, sophisticated, theoretical, etc) for the following product evaluation.
>
> If $T(n)=\frac{(3n-2)(n-1)}2$ and $i=\sqrt{-1}$ then
> $$\prod\_{j<k}^{0,n-1}(\eta^k-\eta^j)=n... | https://mathoverflow.net/users/66131 | Different derivations of the value of $\prod_{0\leq j<k<n}(\eta^k-\eta^j)$ | We first find the norm; we then determine the argument.
Call the product you wrote $A\_n$. Then $A\_n^2 = \prod\_{j<k}^{0,n-1} (\eta^k - \eta^j)^2 = Disc(x^n - 1) = (-1)^{\frac{n (n -1)}{2}}Res(x^n - 1, n x^{n - 1})$
$= (-1)^{\frac{n(n-1)}{2}} n^n \prod\_{0 \leq i < n, 0 \leq j < n-1} (\eta^i - 0)$
All terms in t... | 10 | https://mathoverflow.net/users/44191 | 319624 | 138,083 |
https://mathoverflow.net/questions/319634 | 2 | Let $T$ be a bounded linear operator acting on a complex Banach space. Suppose that $T$ has spectral radius strictly less than $1$. If we introduce an analytic perturbation to $T$, $s\mapsto T\_s$ for $|s|<\epsilon$ (with $T\_0 = T$), then by upper-semicontinuity of the spectrum, assuming that $\epsilon$ is sufficientl... | https://mathoverflow.net/users/80930 | Uniform inequality for an analytic perturbation | Continuity of the perturbation (in the norm operator topology, is it what you mean?) is enough. Denote $T\_s=T+A$ where $\|A\|<\varepsilon$ (this is so for small enough $s$ by norm continuity.) Choose any $\rho$ strictly between spectral radius of $T$ and 1 and fix $n\_0$ such that $\|T^n\|\leqslant \rho^n$ whenever $n... | 2 | https://mathoverflow.net/users/4312 | 319636 | 138,087 |
https://mathoverflow.net/questions/319633 | -1 | A [hypergraph](https://en.wikipedia.org/wiki/Hypergraph) $H=(V,E)$ consists of an non-empty set $V$ and a collection $E\subseteq {\cal P}(V)\setminus \{\emptyset\}$ of non-empty subsets of $V$. A *transversal* of $H$ is a set $T\subseteq V$ such that $|T\cap e| = 1$ for all $e\in E$.
It is easy to see that transvers... | https://mathoverflow.net/users/8628 | Finding a good transversal basis | Let $V=\mathbb Z,E=\{\{m\in\mathbb Z:m\geq n\}:n\in\mathbb Z\}$. It's clear every transversal basis has at most one element, but no basis $\{k\}$ is good, since $\{k+1\}$ intersects more sets from $E$.
| 3 | https://mathoverflow.net/users/30186 | 319641 | 138,089 |
https://mathoverflow.net/questions/319644 | 2 | How to calculate easily the eigenmatrix of a 3D tensor.
I try immersing the tensor in a big matrix, in my case, the tensor is of nxnxn and I can build an n^2 x n^2 matrix that contains all the "coefficients" of my original tensor, but my calculations give me more eigenvalues that I have in my original problem (n^2).... | https://mathoverflow.net/users/133970 | Eigenvalue and Eigenmatrix of a 3D Tensor - How to calculate it? | You ask for the eigenvalues of an $m=3$-order $n$-dimensional tensor $M$. There is no unique definition of the "eigenvalue" $\lambda$ for $m\geq 3$. One frequently used definition is
$$\sum\_{i\_2,i\_3,\ldots i\_m=1}^n M\_{i,i\_2,i\_3,\ldots i\_m}x\_{i\_2}x\_{i\_3}\cdots x\_{i\_m}=\lambda x\_i^{m-1}, \;\;\text{for all}... | 5 | https://mathoverflow.net/users/11260 | 319647 | 138,090 |
https://mathoverflow.net/questions/319618 | 1 | I am reading about robust optimization and there is a claim:
$$
\max\_{\mu^-\leq\mu\leq \mu^+}
\ln \left(
\exp\left\{w+\ln\left(\frac{1+\mu}{2}\right)\right\}
+\exp\left\{-w+\ln\left(\frac{1-\mu}{2}\right)\right\}
\right)
$$
is equivalent to
$$
\max\_{-1\leq u\leq1} \{wu-\phi(u)\},
$$
with
$$
\phi(u)=
\begin{cases}
... | https://mathoverflow.net/users/127755 | Maximization of log-sum-exp function | This question is answered mostly by simple algebra and calculus, so here is a sketch of how the second expression is derived.
Set $y\_{1}=u$ and $y\_2=1-u$ in the conjugate function expression for the log-exp-sum. Simplify what is obtained after using the conjugate function expression to replace the log-exp-sum in th... | 1 | https://mathoverflow.net/users/118731 | 319659 | 138,095 |
https://mathoverflow.net/questions/319650 | 2 | Let $X$ be a compact Riemann surface, $u$ a meromorphic function on $X$ with divisor supported on a set of points $\{P\_1, ..., P\_n\}$, and $f$ a meromorphic function on $X$ such that $f$ has no pole or zeros on $\{P\_1, ..., P\_n\}$. Let $d\log(u)$ be the logarithmic differential form of $u$. Do we have the following... | https://mathoverflow.net/users/60519 | Extended Abel-Jacobi theorem | This is true. Given two points $a,b\in X$ with $u(a)$ and $u(b)\neq 0$, one can choose a path from $a$ to $b$ and a determination of $\log u$ along that path, so that $\exp\int^b\_a d\log u=u(b)/u(a)$. Thus, if $ \operatorname{div}(f)=\sum n\_jQ\_j$, the left-hand side is $\prod u(Q\_j)^{n\_j}$.
Recall that if $f,g$... | 4 | https://mathoverflow.net/users/40297 | 319661 | 138,097 |
https://mathoverflow.net/questions/319615 | 3 | I am a student of physics and, especially in quantum mechanics, we are presented with the Legendre equation:
\begin{eqnarray}
(1-x^2)y''-2xy'+l(l+1)y=0.
\end{eqnarray}
Doing some calculations, we conclude that $l\in\mathbb{Z}^+$. My question is to know why this happens in a geometric and / or qualitative way. In other ... | https://mathoverflow.net/users/133957 | Legendre equation: An interpretation | I am not sure if this is the qualitative/geometric interpretation -of the integrality of the $l$ parameter- you are looking for, but if the parameter $l$ is a non-negative integer then the Legendre polynomials $P\_l$ and the associated Legendre polynomials $P\_l^m$ are proportional to the matrix elements of special cas... | 3 | https://mathoverflow.net/users/85967 | 319666 | 138,099 |
https://mathoverflow.net/questions/319673 | 3 | What is the consistency strength of Basic Theory of Elementary Embeddings (BTEE) from [The spectrum of elementrary embeddings j : V → V](http://pcorazza.lisco.com/papers/spectrum_embeddings.pdf) by Paul Corazza?
BTEE uses the language of $(V,∈,j)$ and asserts:
ZFC (without separation and replacement for formulas u... | https://mathoverflow.net/users/113213 | Strength of BTEE | BTEE is conservative over the stationary reflection principle (SRP), i.e. ZFC + (schema) {there is $n$-subtle cardinal}$\_{n∈\mathbb{N}}$. Using $n$-ineffable in the schema is equivalent.
Note that we get conservativity and not just equiconsistency, and also that (without full conservativity) SRP shows up in a number... | 3 | https://mathoverflow.net/users/113213 | 319674 | 138,100 |
https://mathoverflow.net/questions/319679 | 4 | I am looking for a reference containing the following result:
>
> Let $a$ and $b$ be two elements of a right-angled Artin group $A$. Assume that $a$ and $b$ have minimal length (with respect to the canonical generating set of $A$) in their conjugacy classes. Let $a\_1 \cdots a\_n$ and $b\_1 \cdots b\_m$ be words of... | https://mathoverflow.net/users/122026 | Conjugacy in right-angled Artin groups | Look at Lemma 9 of <https://arxiv.org/abs/0802.1771> for what you want.
| 2 | https://mathoverflow.net/users/15934 | 319682 | 138,102 |
https://mathoverflow.net/questions/319693 | 4 | Say we have a Grothendieck fibration $p : E \to B$ and a monad $T$ on $B$ and a lift $T'$ of $T$ to $E$, i.e. a monad on $E$ such that $pT' = Tp$ and $p$ preserves $\eta, \mu$.
Then because the Eilenberg–Moore construction is functorial, we have a morphism $EM(p)$ from $T'\text{-}Alg$ to $T\text{-}Alg$. Is $EM(p)$ ge... | https://mathoverflow.net/users/82445 | Does the Eilenberg Moore Construction Preserve fibrations? | Let $C$ be a 2-category and $Mnd(C)$ the 2-category of monads in $C$. As explained by Street in [the formal theory of monads](https://doi.org/10.1016/0022-4049(72)90019-9), the Eilenberg-Moore construction is right 2-adjoint to the inclusion 2-functor $C \to Mnd(C)$ sending an object to the identity monad on it. Theref... | 7 | https://mathoverflow.net/users/2362 | 319696 | 138,105 |
https://mathoverflow.net/questions/319547 | 11 | Let $G$ be a transitive permutation group of degree $d$ having a cyclic regular subgroup $K = \langle k \rangle \cong C\_d$. Let $\pi(g) = |\mathrm{Fix}(g)|$ be the permutation character of $G$ and let
$$\pi = \mathbb{1} + \pi\_1 + \cdots + \pi\_\ell $$
be the decomposition of $\pi$ into irreducible characters of $... | https://mathoverflow.net/users/7709 | Permutation groups having a regular cyclic subgroup and a conjectured algebra of characters | The conjecture is true and holds, in a generalized version, whenever $K$ is an abelian regular subgroup. In fact by Theorem 1.9(d) in O. Tamaschke, [*Zur Theorie der Permutationsgruppen mit regulärer Untergruppe. I*](https://link.springer.com/article/10.1007/BF01162390), Math. Zeit. (1963) **80** 328–354, there is an e... | 7 | https://mathoverflow.net/users/7709 | 319706 | 138,109 |
https://mathoverflow.net/questions/319703 | 2 | I am (still) working through the paper *Singular semipositive metrics in non-Archimedean geometry* by Sebastien Boucksom, Charles Favre and Mattias Jonsson (J. Algebraic Geom. **25** (2016), 77-139, doi:[10.1090/jag/656](https://doi.org/10.1090/jag/656), arXiv:[1201.0187](https://arxiv.org/abs/1201.0187)).
Here are s... | https://mathoverflow.net/users/117432 | Definition of model functions and their density in $C^0(X^\text{an})$ | As alluded to in the question, the space $\mathcal{D}(X)\_{\mathbb{Z}}$ is indeed the space of model functions arising from integral divisors.
Now, in order to deduce that $\mathcal{D}(X)\_{\mathbb{Q}}$ separates points and is closed under max from the corresponding fact for $\mathcal{D}(X)\_{\mathbb{Z}}$, you can proc... | 3 | https://mathoverflow.net/users/47692 | 319711 | 138,111 |
https://mathoverflow.net/questions/319651 | 0 | I'm working in an algebra of rational functions in compact Riemann surfaces with arbitrary genus. The idea I'm struggling is how to count the number $n$ of poles for the rational functions defined in a coordinate ring $R$ of a curve $\Sigma$ with $R=\mathbb{C}[t,t^{-1},u]/\langle u^m-p(t) \rangle$.
I'm doing an exten... | https://mathoverflow.net/users/119743 | Counting the number of poles for rational functions in a coordinate ring of a curve | I think the best interpretation of the paper you link to is that they mean a Riemann surface is a nonsingular projective algebraic curve and they have simply forget to include the condition that $p(t)$ should be squarefree. For instance in Theorem 2.1 they use the algebraic de Rham theorem, which as stated is only righ... | 1 | https://mathoverflow.net/users/18060 | 319720 | 138,113 |
https://mathoverflow.net/questions/319657 | 6 | Let $E$ be a supersingular elliptic curve over $\mathbf{F}\_p$, and $H$ its endomorphism algebra $\text{End}(E)\otimes\_{\mathbf{Z}}\mathbf{Q}$, a quaternion algebra (non split at $p$ and $\infty$).
For every prime $\ell\neq p$, there is a faithful $\ell$-adic algebra-representation:
$$\rho\_{\ell} : H\to \text{End... | https://mathoverflow.net/users/nan | Quaternion algebra actions on $\ell$-adic cohomology | The fact that the action of $f+g$ by pullback on $H^1$ is equal to the action of $f$ plus the action of $g$ does not follow, as far as I can see, just from functoriality of the etale site, but it does follow from general properties of etale cohomology, in particular the Kunneth formula.
From the Kunneth formula we ge... | 2 | https://mathoverflow.net/users/18060 | 319722 | 138,114 |
https://mathoverflow.net/questions/319715 | 2 | This question is motivated by one that has been previously asked on this website: [Elliptic problem on a domain split in two subdomains](https://mathoverflow.net/questions/319667/elliptic-problem-on-a-domain-split-in-two-subdomains)
Consider an open domain $U$ split in two non-overlapping subdomains: $U = U\_1 \cup U... | https://mathoverflow.net/users/nan | Boundary condition for elliptic problems and domain decomposition | The place where $U\_1$ and $U\_2$ meet is known as an interface, and so this is a Poisson interface problem which can be read about in the paper *On the Existence and Uniqueness of Solutions of the Poisson Interface Problem* by D. P. Squier found [here](https://www.jstor.org/stable/2373213?seq=1#page_scan_tab_contents)... | 0 | https://mathoverflow.net/users/118731 | 319723 | 138,115 |
https://mathoverflow.net/questions/319728 | 4 | In thermodynamics, the physical meaning of free energy is the maximum amount of work that can be extracted from a system. Now if we take an Ising model on a graph, with interaction weights on each edge, we have a well defined notion of thermodynamical free energy. Can the interpretation of free energy in terms of work ... | https://mathoverflow.net/users/112954 | Interpretation of free energy for Ising models | A convenient way to think about this is in terms of the [Jarzynski equality.](https://en.wikipedia.org/wiki/Jarzynski_equality) Suppose you vary a control parameter $H$ from $H\_1$ to $H\_2$ and in this process the system does work $-W$ on the environment at inverse temperature $\beta$. Then the free energy difference ... | 3 | https://mathoverflow.net/users/11260 | 319730 | 138,117 |
https://mathoverflow.net/questions/319726 | 5 | My research is in analysis, but it moved to the area that requires algebraic topology. I have some working knowledge in that area, but I always feel that I am on a shaky ground and I need to go back and study algebraic topology again. However, that would also require refreshing my knowledge in algebra. My question is:
... | https://mathoverflow.net/users/121665 | Algebra for algebraic topology | If you need a concise but **very** clear book which covers a lot of Algebraic Topology and just the necessary algebra (spectral sequences as well) I think that *Differential Forms in Algebraic Topology*- Bott & Tu is the book you are looking for.
Edit: It seems that Bredon-Topology and Geometry is closer to that you... | 6 | https://mathoverflow.net/users/80084 | 319731 | 138,118 |
https://mathoverflow.net/questions/319733 | -3 | [EDIT] the prior question (see the second section below) was trivially false, however the intention is to *arrange* a possible world of such universes, in other words the question is about if it is possible to have a proper class $\mathcal {G}$ of Grothenderick universes that contains $V$ in them, such that every objec... | https://mathoverflow.net/users/95347 | Can there be elementary embedding between a universe and a universe inside it? | No, even the "weak" version (indeed, a weakening of that) is impossible.
Consider the sentence *(no parameters needed!)* $$\mbox{$\varphi\equiv$ "There is a largest inaccessible cardinal."}$$ If there is a proper class of inaccessible cardinals, then for proper class many inaccessible $\kappa$ we have $V\_\kappa\mod... | 7 | https://mathoverflow.net/users/8133 | 319734 | 138,119 |
https://mathoverflow.net/questions/319740 | 12 |
>
> Let $S$ be a finite set and $F$ is the free group on that set. Is there an algorithm which takes as input a sequence of $w,w\_1,\ldots,w\_k\in F$ and decides whether $w\in \langle w\_1,\ldots,w\_k\rangle$?
>
>
>
This question keeps appearing in some of my work. My intuition is that this has been solved somew... | https://mathoverflow.net/users/64294 | Is there an algorithm to decide if a word is in a finitely generated subgroup of a free group? | Let $T$ be a finite subset of the free group on a set $S$. Nielsen's original proof (described nicely in the beginning of Lyndon and Schupp's book) gives an algorithmic process to find a free generating set $T'$ for the subgroup generated by $T$ with the following very nice property: for a word $u$ in $T'$, the $T'$-le... | 17 | https://mathoverflow.net/users/317 | 319741 | 138,121 |
https://mathoverflow.net/questions/319245 | 4 | I hate to ask such a naive question, but here goes. Suppose $A$ and $B$ are rational Lie algebras, i.e. rational vector spaces together with a bracket. Then, $A\otimes\_{\mathbb{Q}} B$ is a rational vector space. Can it be endowed with a Lie bracket in such a way that the category of rational Lie algebras is a monoidal... | https://mathoverflow.net/users/11540 | Is the category of rational Lie algebras monoidal? | Andrew Salch wrote me a nice email about this question, and with his permission, I'm turning it into an answer. Any mistakes are my own. I've made this answer CW, so I don't get any points for Andrew's work. The following is Andrew's email, lightly edited.
"The Milnor-Moore theorem tells you that, if $k$ is a field o... | 3 | https://mathoverflow.net/users/11540 | 319750 | 138,124 |
https://mathoverflow.net/questions/319748 | 15 | What is known about spaces $X$ with the property that $K^\*(\text{point})\to K^\*(X)$ is an isomorphism?
The same question for $K$-homology $K\_\*(X)\to K\_\*(\text{point})$; I don't even know whether these conditions are equivalent.
Note that replacing $K$-theory with integral homology one gets very interesting (I... | https://mathoverflow.net/users/41291 | Which spaces have trivial K-theory? | I'll just give an answer for finite complexes $X$. The condition $\widetilde{K}^\*(X)=0$ only depends on the suspension spectrum of $X$ so this is naturally regarded as a question in stable homotopy theory. The condition also implies that $\widetilde{H}^\*(X;\mathbb{Q})=0$ and thus that $n.1\_X=0$ as a stable map for s... | 13 | https://mathoverflow.net/users/10366 | 319753 | 138,125 |
https://mathoverflow.net/questions/319751 | 1 | Let $X$ be a compact metric space, $A\subset X$ a closed subset and $f:A\to A$ be a continuous map.
Can $f$ be extended to a continuous map $X\to X$?
If so, is there an extension which is injective if $f$ is?
If not, are there handy additional conditions under which it holds?
| https://mathoverflow.net/users/nan | Extension of continuous map on metric space | Let $G = \langle a,b \mid R \rangle$ be a 1-relator group. To this presentation we obtain a twodimensional CW-complex $X$ by attaching a two cell to $S^1\vee S^1$ via the relator. Let $A=S^1\vee S^1$ be the one skeleton of $X$.
A continuous map $f:A\rightarrow A$ gives us two group elements in $\pi\_1(X)=G$; the imag... | 0 | https://mathoverflow.net/users/3969 | 319755 | 138,127 |
https://mathoverflow.net/questions/319752 | 3 | Given any digraphs $G$ and $H$ we say a surjection $f:V(G)\to V(H)$ reduces $G$ to $H$ if and only if it satisfies $(u,v)\in E(G)\iff (f(u),f(v))\in E(H)$. Where if there exists at least one surjection that reduces $G$ to $H$ then we refer to $H$ as a reduction of $G$ and write $H\preceq G$. Thus by this definition we ... | https://mathoverflow.net/users/38626 | Generalized digraph homomorphisms and graph cores | First I claim any reduction of $G$ is isomorphic to an induced subgraph. Indeed if $G$ maps onto $H$ by $f$ choose a minimal size induced subgraph $G'$ of $G$. If $f$ identifies $v,w$ then they have the same edges going into both $v$ and $w$ by your definition of reduction. So we can drop one of these vertices from $G'... | 3 | https://mathoverflow.net/users/15934 | 319762 | 138,130 |
https://mathoverflow.net/questions/319718 | 5 | Let $G$ be an algebraic group. We can try to reconstruct $G$ from its lie algebra $\mathfrak{g}$, but the best we get in general is a formal group scheme $\operatorname{Spf}(U(\mathfrak{g})^\*)$, where $U(\mathfrak{g})$ is the universal enveloping algebra of $\mathfrak{g}$.
However, I have heard that certain conditio... | https://mathoverflow.net/users/30211 | Reference for showing that $\mathcal{O}(G) \cong U(\mathfrak{g})^{\circ}$ when $G$ is connected and simply connected | In general (if the characteristic is not zero) you should use the "hyperalgebra" (or the "algebra of distributions" in modern terms) instead of U(g). The reference is Mitsuhiro Takeuchi's paper "On coverings and hyperalgebras of affine algebraic groups", Trans. AMS 211 (1975), 179-196. Possibly you can also find the re... | 3 | https://mathoverflow.net/users/5107 | 319764 | 138,131 |
https://mathoverflow.net/questions/319732 | 0 | Let $(X,\mathscr{B},\mu)$ be a $\sigma$-finite measure space. Let $\gamma$ be a probability measure on $L\_2(\mu)$ with $\mathrm{supp} \, \gamma = L\_2(\mu)$ and existing first moment. Then
$$
f \mapsto \|f\| := \int |\langle f,g\rangle|\,\gamma(\mathrm{d}g)
$$
is a norm on $L\_2(\mu).$ Does this norm have a name? ... | https://mathoverflow.net/users/134012 | Does this norm have a specific name? Banach space? References? | Let's write $H$ for $L\_2(\mu)$, since its Hilbert space structure is all we care about. I assume $\gamma$ is a Borel measure.
By duality, you can naturally identify $H$ with $H^\*$, and $H^\*$ is a set of real-valued continuous, hence $\gamma$-measurable, (linear) functions on $H$, which are integrable by your first... | 4 | https://mathoverflow.net/users/4832 | 319777 | 138,138 |
https://mathoverflow.net/questions/319667 | 2 | Consider the following elliptic problem in a split domain:
$$ (\ast) \quad\begin{cases} -\Delta u=f\_1 \quad &\text{ in } U\_1\\
-\Delta u =f\_2 & \text{ in }
U\_2\\
u=g & \text{ on } \partial U
\end{cases} $$
where $U = U\_1 \cup U\_2$ is an open domain.
Where can I find a proof of existence, uniqueness and regul... | https://mathoverflow.net/users/nan | Elliptic problem on a domain split in two subdomains | There are many different papers treating such Poisson interface problems. A couple sources have been mentioned in the comments (although the paper of Squier on regularity of solutions differs slightly from your problem and deals specifically with the problem in the plane). One source that is mostly self-contained which... | 0 | https://mathoverflow.net/users/118731 | 319779 | 138,140 |
https://mathoverflow.net/questions/319759 | 7 | In a [recent preprint](https://arxiv.org/abs/1812.10454), Adiprasito proves that if $\Delta$ is a simplicial complex of dimension $d$ that can be embdedded in a $2d$-dimensional homology sphere (say $\Sigma$) that satisfies *a version of the hard Lefschetz theorem*, then:
$$ f\_{d}(\Delta) \leq (d+2)f\_{d-1}(\Delta),$$... | https://mathoverflow.net/users/37214 | Inequality number of facets simplicial complex | I follow your notation rather than mine.
$A^\bullet(\Delta)$ is obtained as follows: You construct a linear system of parameters for $\Sigma$. If $\Sigma$ is of dimension $d-1$, then this is of length $d$. In fact, you can think of this linear system as a set of coordiantes for the vertices of $\Sigma$ in $\mathbb{R}... | 8 | https://mathoverflow.net/users/134042 | 319780 | 138,141 |
https://mathoverflow.net/questions/319789 | 5 | Given a positive integer $n$, the [Hamming distance](https://en.wikipedia.org/wiki/Hamming_distance) $d^H\_n(x,y)$ of $x,y\in \{0,1\}^n$ is defined by $$d^H\_n(x,y) = |\{k\in\{0,\ldots,n-1\}: x(k)\neq y(k)\}|.$$
We say that a positive integer $s$ is $n$-*spreadable* if there is $T\subseteq \{0,1\}^n$ with $|T|=n$ and ... | https://mathoverflow.net/users/8628 | Spreading $n$ points in $\{0,1\}^n$ as far as possible | If I understood the question correctly, what you're asking is related to the maximum distance of binary codes with large minimum distance $d\geq s$.
In coding theory, $A\_q(n,d)$ is defined as the maximum cardinality of a $q-$ary code with length $n$ and minimum distance $d.$
You can never have more than 2 codeword... | 11 | https://mathoverflow.net/users/17773 | 319790 | 138,145 |
https://mathoverflow.net/questions/319786 | 7 | Let $G$ be an algebraic group acting on an affine variety $X=\operatorname{Spec}A$ (all over $\mathbb{C}$). This gives an action of $G$ on the $\mathbb{C}$-algebra $A$, and an action of the Lie algebra $\mathfrak{g}$ of $G$ on $A$ by derivations.
If the action is not transitive, then $G$ will preserve some nontrivial... | https://mathoverflow.net/users/97652 | Lie algebra preserving ideal of functions | A counterexample is $X=\mathbb{A}^2\backslash\{y=0\}$, $A=\mathbb{C}[x,y,y^{-1}]$, and $\mathfrak{g}$ the span of the derivation $D(x)=1$, $D(y)=y$. Now $\mathfrak{g}$ is $1$-dimensional and $X$ is $2$-dimensional, so the map $\mathfrak{g}\to T\_x X$ is not surjective for any $x\in X(\mathbb{C})$. The integral curves o... | 6 | https://mathoverflow.net/users/5263 | 319795 | 138,149 |
https://mathoverflow.net/questions/319794 | 4 | Holomorphic functions are my muse. As my muse, I love drawing them different ways. Allow me to frame this as though an artist talking about his muse.
A holomorphic function $f$ on the unit disk $\mathbb{D}$ is completely determined (and not only determined, represented) by $\{f^{(j)}(0)\}\_{j=0}^\infty$.
$$f(z) = \... | https://mathoverflow.net/users/133882 | Compilation of representations of holomorphic functions | The distinction between "determined" and "represented" is not clear.
Consider a function $f$ analytic in domain $U$ containing, say, $0$.
The values of $f$ on a sequence $p\_n$ of nonzero points with limit $0$ determine $f$. If you want a "representation", you can represent
the coefficients $a\_k$ of the Maclaurin ser... | 6 | https://mathoverflow.net/users/13650 | 319796 | 138,150 |
https://mathoverflow.net/questions/319801 | 0 | I am currently reading the appendices of Higher Topos Theory, and I was puzzled by Lurie's proof of lemma A.2.6.7 (I can not make sense of the end of the proof.)
He uses this result to prove Jeff Smith's theorem, but the proof on the n lab (<https://ncatlab.org/nlab/revision/combinatorial+model+category/59>) does not... | https://mathoverflow.net/users/128857 | A question about combinatorial model categories | Lurie uses A.2.6.7 to prove the "easy" direction of Jeff Smith's theorem in A.2.6.8, namely that every combinatorial model category arises from the construction of the theorem. This part of the theorem is proven in the last sentence of the current revision on the nlab page ("To prove the converse,..."), where the facts... | 5 | https://mathoverflow.net/users/2362 | 319821 | 138,157 |
https://mathoverflow.net/questions/319441 | 9 | Let $G$ be a simple complex algebraic group. Let $V$ be a finite-dimensional algebraic representation of $G$. Thus, we can write $V=V\_1\oplus \cdots \oplus V\_n$ where $V\_i$'s are irreducible representations. Let $I:=\mathbb{C}[V]^G$ denote the space of invariant polynomials. We know that $I$ is a finitely generated ... | https://mathoverflow.net/users/41301 | Polynomial invariants for simple algebraic groups | Generally, one has $\dim V//G=\dim V-\dim G$ but there are exceptions. For simple $G$ the exceptions have been classified in
Èlašvili, A. G. Canonical form and stationary subalgebras of points in general position for simple linear Lie groups ([MSN](https://mathscinet.ams.org/mathscinet-getitem?mr=304554) [English art... | 8 | https://mathoverflow.net/users/89948 | 319822 | 138,158 |
https://mathoverflow.net/questions/319805 | 12 | First, I have to admit that I have already asked [the same question on MSE](https://math.stackexchange.com/questions/3055767/prove-sum-n-1p-1-np-1-equiv-p-1-p-pmod-p2-for-p-b) several days ago. If I am bending any rules, I apologize for that and moderator can delete or close this question without warning. The problem h... | https://mathoverflow.net/users/134054 | Prove that $\sum_{n = 1}^{p - 1} n^{p - 1} \equiv (p - 1)! + p \pmod {p^2}$ with $p$ being an odd prime | The result can be easily proved without using Bernoulli numbers. If $a$ and $b$ are integers not divisible by an odd prime $p$, then
\begin{align}(ab)^{p-1}-1=&b^{p-1}(a^{p-1}-1)+(b^{p-1}-1)
\\\equiv& (a^{p-1}-1)+(b^{p-1}-1)\pmod {p^2}.\end{align} Thus
\begin{align\*}\sum\_{n=1}^{p-1}(n^{p-1}-1)\equiv& \prod\_{n=1}^{p-... | 26 | https://mathoverflow.net/users/124654 | 319824 | 138,159 |
https://mathoverflow.net/questions/319825 | 1 | For digraphs $G$ and $H$ if we can partition $V(G)$ into a family $\{Q\_t\}\_{t\in V(H)}$ indexed by $V(H)$ such that $E(G)=\bigcup\_{(u,v)\in E(H)}Q\_u\times Q\_v$, then is every subgraph of $G$ isomorphic to $H$ induced in $G$ by some set of complete representatives for the partition $P=\{Q\_t:t\in V(H)\}$?
| https://mathoverflow.net/users/38626 | Enumerating isomorphic subgraphs | This is not true in general. Here is a non-trivial counter example: Let $H \sim K\_{1,2}$ and $G\sim C\_4$, with the vertices of $G$ labelled $a,b,c,d$ around the cycle. Then $\{a\},\{c\},\{b,d\}$ works as a partition, but the induced subgraph $G[a,b,d]$ is isomorphic to $H$.
| 3 | https://mathoverflow.net/users/69010 | 319832 | 138,161 |
https://mathoverflow.net/questions/319814 | 5 | I've been reading *Quantitative ergodic theorems and their number-theoretic applications* By Gorodnik and Nevo ([arXiv:1304.6847](https://arxiv.org/abs/1304.6847)). Early on, there is a comment on rates of convergence in the mean ergodic theorem which puzzles me.
Briefly, let $G$ be some (locally compact second count... | https://mathoverflow.net/users/100163 | Connection between rates of convergence in ergodic theorems and spectral gap property | The "in particular" refers to the previous paragraph. It is the almost invariant sequence of mean 0 functions (which is equivalent to amenability of the acting group) that guarantees the lack of uniform convergence in the ergodic theorem. You can see this because for any Folner set and any $n$, you have a function of n... | 2 | https://mathoverflow.net/users/11054 | 319833 | 138,162 |
https://mathoverflow.net/questions/319830 | -2 | Under Goldbach's conjecture, let $r\_{0}(n) : =\inf\{r>0,(n-r,n+r)\in\mathbb{P}^{2}\} $ and $k\_{0}(n) : =\pi(n+r\_{0}(n))-\pi(n-r\_{0}(n)) $. The PNT implies that one can expect to have $ \dfrac{2r\_{0}(n)-1}{k\_{0}(n)}\sim\log n $.
As for all $ n>1 $ one has $ \tau(n)\geq 2 $, the equality occurring exactly wheneve... | https://mathoverflow.net/users/13625 | Approximation for $ \inf\{r>0,(n-r,n+r)\in\mathbb{P}^{2}\} $ by minimizing a distance | Certainly not, if the twin primes conjecture is true. When $n$ is large and $n-1$ and $n+1$ are both primes, then $S\_1(n)$ is nearly exactly $\frac32$ and $(2\cdot1-1)/\log n$ is nearly $0$, and $r=2$ will already yield a smaller value of $|S\_r(n) - (2r-1)/\log n|$ than $r=1$ (we lose $2/\log n$ but gain at most $2\c... | 2 | https://mathoverflow.net/users/5091 | 319836 | 138,163 |
https://mathoverflow.net/questions/319496 | 1 | Let $G$ be a graph with total vertices $|V(G)|$. Let the maximum degree of the graph be $\Delta$. Let us assume the graph is total colourable( no adjacent vertices, adjacent edges and an edge and its incident vertices receive same colour) with $\Delta+1$ colours. Let the vertices be properly coloured with $\Delta+1$ co... | https://mathoverflow.net/users/100231 | On a theorem of Chetwynd and Hilton in Graphs | A graph $G$ is called *conformable* is there exists a proper vertex coloring using $\Delta + 1$ colors (really at most this many colors as color classes are allowed to be empty) such that
$$\sum\_{v \in V} (\Delta - d(v)) \geq \Delta - r + 1$$
where $r$ is as defined in the question.
In Lemma 1 of "Non-conformable su... | 1 | https://mathoverflow.net/users/51668 | 319839 | 138,166 |
https://mathoverflow.net/questions/290226 | 35 | I refer to this paper: de Silva, Vin; Morozov, Dmitriy; Vejdemo-Johansson, Mikael. *Dualities in persistent (co)homology.* Inverse Problems 27 (2011), no. 12, 124003, 17 pp. ([Journal link](https://iopscience.iop.org/article/10.1088/0266-5611/27/12/124003), [arXiv link](https://arxiv.org/abs/1107.5665)).
According to... | https://mathoverflow.net/users/83274 | Why is persistent cohomology so much faster than persistent homology | There are several factors contributing to the improved performance of the algorithm reported in the paper; the use of cohomology is one, but there is also a computational shortcut involved, and the top Betti number (more precisely, the dth Betti number of the (d+1)-skeleton) also plays a role.
The computation of pers... | 31 | https://mathoverflow.net/users/13357 | 319840 | 138,167 |
https://mathoverflow.net/questions/319852 | 2 | Let $R=\mathbb C[q^{\pm 1}]$ and let $A$ be a graded (possibly non-commutative) $R$-algebra, $A=\oplus\_{n=0}^\infty A\_n,$ where $A\_0=R$ and all $A\_n$'s are free $R$-modules.
Then $A'=A/(q-1)$ is a graded algebra over $R/(q-1)=\mathbb C$ and so
$A$ can be thought as a deformation of $A'.$ I couldn't find a definit... | https://mathoverflow.net/users/23935 | Finite generation of flat deformations of algebras | To say that a deformation is flat, you should assume that $A$ is flat over $R$, e.g. each $A\_n$ is a free $R$-module of finite rank.
The answer to the main question is no, even if $A$ is commutative. Consider $A$ where each $A\_n=R$ with basis element $x\_n$, with multiplication defined by the formula $$x\_n x\_m = ... | 2 | https://mathoverflow.net/users/3847 | 319858 | 138,173 |
https://mathoverflow.net/questions/319855 | 5 | Let $A \subseteq B$ be a ring extension where $A,B$ are both finitely generated $\mathbb C$-domain of the same Krull dimension. Also assume $A$ is regular (i.e. $A\_{ \mathfrak p}$ is regular local ring for every prime ideal $\mathfrak p$ of $A$ ).
If $P$ is a prime ideal of $A$ with finitely many prime ideals of $B... | https://mathoverflow.net/users/127118 | A question on dominant morphism of affine schemes | The comment of @MatthieuRomagny gives a link to some positive answers under additional hypotheses. Nonetheless, the answer is negative without further hypotheses. Before stating the negative counterexamples, let me state the positive result (that I suspect motivated this question).
**Zariski's Main Theorem (Original ... | 5 | https://mathoverflow.net/users/13265 | 319863 | 138,175 |
https://mathoverflow.net/questions/319870 | 17 | Let $A$ be a path-connected subset of $\mathbb R^2$ such that the removal of any singleton from $A$ splits $A$ into two open connected components, each of which is path-connected.
Is $A$ necessarily homeomorphic to $\mathbb{R}$?
| https://mathoverflow.net/users/132446 | Homeomorphic characterization of the real line? | Ward has given the following characterization of the real line: a connected, locally connected separable metric space in which each point is a cut point, i.e., its removal splits the space into two connected subsets (Proc. London Math. Soc. 1936). This implies a positive answer to your question, assuming the set has mo... | 21 | https://mathoverflow.net/users/131781 | 319872 | 138,179 |
https://mathoverflow.net/questions/319874 | 6 | I just came across a Wikipedia article [Nested Radicals](https://en.wikipedia.org/wiki/Nested_radical#Landau%27s_algorithm) that mentions [Landau's Algorithm](https://en.wikipedia.org/w/index.php?title=Landau%27s_algorithm&redirect=no) for deciding, whether a nested radical can be denested, but that Wikipedia article i... | https://mathoverflow.net/users/31310 | Relevance of Landau's Algorithm for Denesting Radicals | This 2017 article [Gkioulekas - On the denesting of nested square roots](https://www.researchgate.net/publication/314248287_On_the_denesting_of_nested_square_roots) summarizes the status of this topic. Landau's algorithm is not a "final" solution because it runs in exponential time with respect to the depth of the expr... | 4 | https://mathoverflow.net/users/11260 | 319877 | 138,181 |
https://mathoverflow.net/questions/319878 | 2 | Let $G=(V,E)$ be a simple, undirected graph. Suppose that ${\cal S}$ is a collection of non-empty, connected, and pairwise disjoint subsets of $V$. Let $G({\cal S})$ be the graph with vertex set ${\cal S}$; and $S\neq T\in {\cal S}$ form an edge if and only if if there are $x\in S, y \in T$ such that $\{x,y\}\in E$.
... | https://mathoverflow.net/users/8628 | Induced minors of $\{0,1\}^\omega$ | Let $G=(\omega,E)$ be an arbitrary graph. Let $S\_n=\{x\_n\}\cup\{y\_{nm}:\{n,m\}\in E\}$, where $x\_n$ is the characteristic function of $\{2n\}$ and $y\_{nm}$ is the characteristic function of $\{2m,2n+1\}$. The induced minor of $\mathcal S=\{S\_n:n\in\omega\}$ is clearly isomorphic to $G$, showing every countable gr... | 4 | https://mathoverflow.net/users/30186 | 319890 | 138,184 |
https://mathoverflow.net/questions/319884 | 2 | **TLDR:** trying to solve,
$$\int\_1^\infty \exp\left(-\frac{x^2}{2\omega^2}\right) \frac{1}{\sqrt{ax^2+bx-1}}dx$$
After doing some reading and looking at some other questions [1](https://math.stackexchange.com/questions/1188838/integral-of-gaussian-distribution-divided-by-square-root-of-x), [2](https://math.stackexc... | https://mathoverflow.net/users/134086 | Integrating nasty gaussian over square root | Let's factor the quadratic as $(x-c)(x-d)$, where I'll assume $|c|, |d| < 1$. For $x\ge 1$ we have
$$ \frac{1}{\sqrt{x-c}} = \sum\_{k=0}^\infty \frac{(2k)!}{k!^2} (c/4)^k x^{-(1/2+k)}$$
and similarly for $1/\sqrt{x-d}$. Thus
$$ \eqalign{\frac{1}{\sqrt{(x-c)(x-d)}}&= \sum\_{k=0}^\infty \sum\_{j=0}^k \frac{(2j)! (2k-2j... | 2 | https://mathoverflow.net/users/13650 | 319902 | 138,188 |
https://mathoverflow.net/questions/319699 | 6 | The late Vladimir Arnold, in
*Arnold, V.*, [**Arithmetics of binary quadratic forms, symmetry of their continued fractions and geometry of their de Sitter world**](http://dx.doi.org/10.1007/s00574-003-0001-8), Bull. Braz. Math. Soc. (N.S.) 34, No. 1, 1-42 (2003). [ZBL1044.11016](https://zbmath.org/?q=an:1044.11016).... | https://mathoverflow.net/users/469 | Name of a group-like structure | $n$-group property is not a good idea, because it's a substructure (and also because groups have inverses). One option is $n$-subsemigroup (or any obvious variant such as $n$-fold subsemigroup, $[n]$-subsemigroup, if any reason to do so)...
As mentioned in the comments, an $n$-subsemigroup with unit is just the same ... | 1 | https://mathoverflow.net/users/14094 | 319917 | 138,193 |
https://mathoverflow.net/questions/319843 | 8 | It is written (cf. [Moore 1980](https://philpapers.org/rec/GREBFL), page 100) that mathematical logicians (e.g. Peirce, Schröder, Hilbert) at the turn of the last century did not yet distinguish between syntax and semantics when formulating logical and logico-mathematical theories.
I am trying to understand how conf... | https://mathoverflow.net/users/116705 | Syntax/semantics conflation leads to infinitary logic | It sounds to me like this question is asking us to divine about the thinking of mathematicians in the early 20th century. Obviously, I can do no such thing, but perhaps I can explain how the early views of logic make very good sense from the point of view of modern logic.
In classical treatments of first-order logic ... | 8 | https://mathoverflow.net/users/1176 | 319921 | 138,194 |
https://mathoverflow.net/questions/319919 | 7 | Why is $\pi\_{-\*}F(H\mathbb{F}\_p, H\mathbb{F}\_p)$ the mod $p$ Steenrod algebra? (This is quite a common statement, seen, for instance, in EKMM.)
To be more precise, stable mod $p$ cohomology operations can be realized by maps of spectra (in particular, elements of $\pi\_{-\*}F(H\mathbb{F}\_p, H\mathbb{F}\_p)$), bu... | https://mathoverflow.net/users/128857 | Why is $\pi_{-*}F(H\mathbb{F}_p, H\mathbb{F}_p)$ the mod $p$ Steenrod algebra? | Recall that by representability of cohomology plus the Yoneda lemma, a cohomology operation $H^i→H^j$ is the same thing as a map
$$ K(\mathbb{F}\_p,i)→K(\mathbb{F}\_p,j)\,.$$
Moreover, the suspension isomorphism $\sigma:H^i(X)\cong H^{i+1}(\Sigma X)$ is implemented by the counit of the suspension-loopspace adjunction
$... | 12 | https://mathoverflow.net/users/43054 | 319923 | 138,195 |
https://mathoverflow.net/questions/319930 | 7 | I'm looking for a reference on how to sample uniformly (and preferably efficiently, elegantly, etc.) from the vertices of a polytope. I gather that [enumerating vertices is hard](https://doi.org/10.1007/978-0-387-87363-3_17). I also note the MO questions [Uniformly Sampling from Convex Polytopes](https://mathoverflow.n... | https://mathoverflow.net/users/1847 | Sampling uniformly from the vertices of a polytope | Here is one efficient approach, performing a random walk with a rapid mixing time, that has been implemented for a particular class of polytopes, but which might well be adaptable to a more general setting: [Random Walks on the Vertices of Transportation Polytopes](http://homepages.inf.ed.ac.uk/mcryan/cdms-jour.pdf) (2... | 5 | https://mathoverflow.net/users/11260 | 319932 | 138,199 |
https://mathoverflow.net/questions/319940 | 1 | We construct graph on the vertex set $\{0,1\}^n$ where $n$ is a positive integer. For $x,y \in \{0,1\}^n$ the [Hamming distance](https://en.wikipedia.org/wiki/Hamming_distance) of $x,y$ is the cardinality of the set $\{ i \in \{0, ..., n-1\} : x(i) \neq y(i)\}$ (i.e. we count the positions on which $x$ and $y$ do not a... | https://mathoverflow.net/users/8628 | Conjecture on representing graphs within $\{0,1\}^n$ | For $n\geq 8$, $K\_n$ with one edge removed is not such an induced subgraph. It's clearly not a subgraph of $H(n,1)$ since the latter is bipartite.
Suppose that graph is an induced subgraph of $H(n,2)$, say it's spanned on strings $x\_1,\dots,x\_n$ with $x\_{n-1},x\_n$ not connected. Observe we may assume $x\_1$ is t... | 3 | https://mathoverflow.net/users/30186 | 319945 | 138,203 |
https://mathoverflow.net/questions/319947 | 1 |
>
> A cardinal κ is a Berkeley cardinal, if for any transitive set $M$ with $κ∈M$ and any ordinal $α<κ$ there is an elementary embedding $j : M → M$ with $\alpha<\text{crit}(j)<\kappa$.
>
>
>
My question is about the restrictions involved in that definition of $\kappa \in M$? and of $M$ being *transitive*? and o... | https://mathoverflow.net/users/95347 | Why the restrictions in the definition of Berkeley cardinals? | Dropping transitivity doesn't actually add anything: given $M\supseteq\kappa$, just consider $\hat{M}=$ the Mostowski collapse of $M$. Given $\alpha<\kappa$ and $j:\hat{M}\rightarrow \hat{M}$ nontrivial elementary with $\alpha<crit(j)<\kappa$, $j$ lifts to a nontrivial elementary embedding $\hat{j}$ of $M$ into itself ... | 5 | https://mathoverflow.net/users/8133 | 319948 | 138,204 |
https://mathoverflow.net/questions/319951 | 4 | This is a follow up on an [older](https://mathoverflow.net/questions/319940/conjecture-on-representing-graphs-within-0-1-n) question.
We construct graph on the vertex set $\{0,1\}^n$ where $n$ is a positive integer. For $x,y \in \{0,1\}^n$ the [Hamming distance](https://en.wikipedia.org/wiki/Hamming_distance) of $x,y... | https://mathoverflow.net/users/8628 | Hamming representability of finite graphs | Yes. This is going to be very inefficient, but: Let $E$ be the number of edges and let $V$ be the number of vertices. I will embed $G$ into $H(|E|(|V|-1),\ 2|E|-2)$. To each vertex $v$ of $G$, we will associate an $|E| \times (|V|-1)$ matrix $M\_v$ with rows indexed by the edges of $G$. There will be a single $1$ in ea... | 3 | https://mathoverflow.net/users/297 | 319959 | 138,208 |
https://mathoverflow.net/questions/319955 | 10 | Much of the literature on analysis in metric spaces makes use of an assumption called *Ahlfors regularity* or *Ahlfors-David regularity*. Let $q>0$. A metric space $(X,d)$ is *Ahlfors(-David) $q$-regular* if there exists $C\geq 1$ such that $C^{-1}r^q \leq \mathcal{H}^q(B(x,r)) \leq Cr^q$ for all $x \in X$ and $r \in (... | https://mathoverflow.net/users/126691 | Origin of term Ahlfors-David regular | To answer the last question, Calderón's problem was a question regarding mapping properties of the Cauchy integral
$$ C\_{\Gamma}f(z)=\frac{1}{2\pi i} \int\limits\_{\Gamma} \frac{f(\xi)}{\xi-z} d\xi$$
namely, to determine the rectifiable Jordan curves $\Gamma$ for which $C\_{\Gamma}$ gives rise to a bounded operat... | 7 | https://mathoverflow.net/users/118731 | 319961 | 138,209 |
https://mathoverflow.net/questions/314994 | 4 | Let $S$ be a closed oriented surface of genus $g\geq 2$. Let $\mathcal{T}$ be the corresponding Teichmuller space. Given a free homotopy class of closed curve $[\gamma]$ we can define the length function $l\_{\gamma}$ on $\mathcal{T}$.
It is well known that there exist $[\alpha]\neq [\beta]$ such that $l\_\alpha=l\_\... | https://mathoverflow.net/users/37286 | Length functions on Teichmuller space with constant difference | Let me first prove an easier statement, namely that there are no loops $c\_1, c\_2$ on $S$ such that $tr(\rho(c\_1)) - tr(\rho(c\_2))=a$ for some nonzero constant $a$ and all discrete and faithful representations $\rho: \pi\_1(S)\to SL(2, {\mathbb R})$. Consider the representation variety
$$
Rep(S)=Hom(\pi\_1(S), SL(2... | 5 | https://mathoverflow.net/users/21684 | 319966 | 138,212 |
https://mathoverflow.net/questions/319963 | 6 | This should a be basic enough question, but I’m a little confused.
In proving that $H^\*(X,\mathbf{Q}\_{\ell})$ is functorial (in the sense of Weil cohomology theories: see axiom D2 [here](https://www.math.columbia.edu/%7Edejong/seminar/note_on_weil_cohomology.pdf)) as $X$ ranges over smooth projective varieties over... | https://mathoverflow.net/users/nan | Functoriality for $\ell$-adic cohomology - a question | As you say, one can talk about functoriality quite generally and obtain maps of the form
$$f^\* : H^n(Y, F) \to H^n(X, f^{-1} F),$$
for any ($\ell$-adic, if you want) sheaf $F$. This will be appropriately functorial. And then, if you have amorphism of sheaves on $X$, $\alpha: f^{-1} F \to G$, then you can append, as yo... | 1 | https://mathoverflow.net/users/18116 | 319997 | 138,226 |
https://mathoverflow.net/questions/289116 | 0 | I'm writing a [MIZAR](http://mizar.org/) article about foundations in graph theory e.g. constructing a supergraph from a given graph by adding a vertex to it. The main theorem of the article will be that any graph drawable by hand (i.e. with finitely many vertices and edges) actually exists mathematically and can be co... | https://mathoverflow.net/users/118059 | Literature about most basic existence proofs in graph theory | The article has been finished and since I couldn't find any reference to this basic research I guess I provided the first reference :-)
>
> Sebastian Koch, *About Supergraphs. Part I*, Formalized Mathematics, **26** (2) (2018) pp 101–124, doi:[10.2478/forma-2018-0009](https://doi.org/10.2478/forma-2018-0009)
>
>
... | 1 | https://mathoverflow.net/users/118059 | 320008 | 138,231 |
https://mathoverflow.net/questions/320018 | 2 | The Besicovitch-Federer structure theorem enables us to make the following definition:
>
> Suppose $k < n$, $E \subset \mathbf{R}^n$, and $0 < \mathcal{H}^k(E) < \infty$. We say $E$ is **purely unrectifiable**, provided that $\mathcal{H}^k(\pi\_K(E))= 0$ for almost every $k$-plane $K$.
>
>
>
Many examples of s... | https://mathoverflow.net/users/121486 | Generalizing a notion of purely unrectifiable sets | Actually quite a lot is known about such sets. You can find many results in Chapter 9 of Mattila's *Geometry of Sets and Measures in Euclidean Spaces*. Here are some examples:
>
> **Theorem 1.** If $0<d\leq 1$, then there is a set $F\subset\mathbb{R}^2$ such that $0<H^d(F)<\infty$, but projections on
> every line ... | 3 | https://mathoverflow.net/users/121665 | 320023 | 138,235 |
https://mathoverflow.net/questions/319849 | 7 | Why should we define the differential in Weil model as follows? I could understand $\sum\_{j,k} c\_{jk}^i \theta\_j \wedge \theta\_k$ plays a role in the formula because it is the dual of the structure map of $\mathfrak{g}$. The rest of formula looks mysterious to me.
[Cartan-Weil model for Equivariant Cohomology](ht... | https://mathoverflow.net/users/130879 | Differentials in Weil model for equivariant cohomology | Suppose $E \to B $ is a principal $G$-bundle with connection $\omega \in \Omega^1(E,\mathfrak{g})$, and corresponding curvature $\Omega \in \Omega^2(E,\mathfrak{g})$. Then $\Omega^\*(E)$ is a differential graded algebra, with the exterior derivative as the differential, and wedge product of forms being the multiplicati... | 5 | https://mathoverflow.net/users/40030 | 320027 | 138,238 |
https://mathoverflow.net/questions/319967 | 3 | This is Theorem 10.1.1 of Lind & Marcus's book, [An Introduction to Symbolic Dynamics and Coding](https://doi.org/10.1017/CBO9780511626302). They say that is "straightfordward" to go from
>
> Let $X$ a shift of finite type and $Y$ a mixing shift of finite type such that $\text{Per}(X)\hookrightarrow\text{Per}(Y)$ a... | https://mathoverflow.net/users/134135 | On Krieger's Embedding Theorem | the difference between the two statements is rather subtle. Of course proving the result with $X$ any SFT is more general than assuming $X$ to be irreducible, so there is nothing to do there. For $Y$, going from irreducible to mixing seems to be a stronger condition, however the structure threoy of (one-dimensional) SF... | 3 | https://mathoverflow.net/users/52920 | 320031 | 138,241 |
https://mathoverflow.net/questions/261846 | 0 | Assume that X and Y both have piecewise linear integral structures. Let f : X → Y be a piecewise linear homeomorphism such that
• V-1: f (t · x) = t · f (x), and
• V-2: x ∈ X is integral if and only if f (x) is integral in Y.
Then f is volume preserving.
how can i prove it? is it any classical proposition?
| https://mathoverflow.net/users/74907 | volume preserving homeomorphism between piecewise linear integral structures | To show that f is volume preserving, we only need to show that each of its linear pieces is volume preserving. So, it’s a property of linear maps.
If a linear map $f:R^n \to R^n$ sends integer points to integer points, it must have integer coefficients. Because of the “only if” part of the hypothesis, its inverse also ... | 0 | https://mathoverflow.net/users/74907 | 320035 | 138,243 |
https://mathoverflow.net/questions/320021 | 0 | Assuming one draws two points from a von Mises distribution on a circle, I am looking for the expected distance between two such points.
Given the pdf of a centered von Mises distribution
$$f\_X(t \mid \kappa) = \frac{e^{\kappa \cos(t)}}{2\pi I\_0(\kappa)} \cdot 1\_{[-\pi, \pi]}(t) \; ,$$
one can calculate the pdf th... | https://mathoverflow.net/users/111496 | Closed form of integration of modified Bessel function composed with trigonometric function times a linear term | $$I\_0(2\kappa \cos(t/2)) = \sum\_{k=0}^\infty \cos(t/2)^{2k} \frac{\kappa^{2k}}{k!^2}$$
and it seems to me that
$$\int\_0^\pi \cos(t/2)^{2k} t \; dt = {2k \choose k} 2^{-2k-1} \pi^2 - \sum\_{r=0}^{\lfloor (k-1)/2 \rfloor} {2k \choose k+2r+1} \frac{2^{2-2k}}{(2r+1)^2} $$
so that, unless I've made an error,
$$ \eqalign{... | 2 | https://mathoverflow.net/users/13650 | 320040 | 138,245 |
https://mathoverflow.net/questions/320051 | 15 | If $X$ is a spectrum with trivial (integer-valued) homology groups, does it have to be weakly-equivalent to a point?
This is easy to prove for connective spectrum, as a Hurewitz-type argument is then possible, but what about the general case?
Furthermore, if this is not the case, how should I think of the functor $... | https://mathoverflow.net/users/128857 | Is a spectrum with trivial homology groups trivial? | If $K(n)$ is the $n$-th Morava K-theory for $n>0$, then $K(n)\otimes H\mathbb{Z}=0$ because, via the 2 complex orientations of $K(n)\otimes H\mathbb{Z}$, there are two formal groups over the ring $\pi\_\*(K(n)\otimes H\mathbb{Z})$ and an isomorphism between them. But one has height 0 (additive formal group from $H\math... | 21 | https://mathoverflow.net/users/134186 | 320054 | 138,248 |
https://mathoverflow.net/questions/320007 | 6 | Here is an example which I'd like to have a name for.
Let $P$ be a compact smooth manifold of dimension $p$, possibly with non-empty boundary.
Define $E(k,P)$ to be the space of smooth (codimension zero) embeddings
$$
\coprod\_{k} D^p \to P \, ,
$$
that is the space of embeddings of $k$ disjoint $p$-disks in $P$, ... | https://mathoverflow.net/users/8032 | An operad-like structure, is there a name for it? | As others have said, this is precisely the structure of a right module over the operad $E(D^p)$.
Since it doesn't feel right to give such a short answer that's already in the comments, let me expand a bit; I don't claim I'm saying anything new, but hopefully, maybe some readers will find something interesting.
If $... | 6 | https://mathoverflow.net/users/36146 | 320055 | 138,249 |
https://mathoverflow.net/questions/320053 | 2 | Let $B\_b(E)$ be the space of bounded measurable functions on some Polish space $E$ endowed with the supremum norm. It seems quite classical that Markov semigroups $P\_t:B\_b(E)\to B\_b(E)$ are in one to one correspondence with Markov processes on $E$.
By Markov semigroup I mean a strongly continuous semigroup $P\_t:... | https://mathoverflow.net/users/62636 | Bounded-pointwise continuity of Markov operators / semigroups | Some continuity assumption is needed, as shown by the following "Markov" semigroup on the set of natural numbers $\mathbb{N}$.
Let $\omega$ be an ultrafilter in $\mathbb{N}$. Every element $f \in B\_b(\mathbb{N})$ (that is, every bounded sequence $f(n)$) has a finite "generalised limit" along $\omega$, that we denote... | 2 | https://mathoverflow.net/users/108637 | 320063 | 138,253 |
https://mathoverflow.net/questions/320061 | 7 |
>
> **Question:** What is the least number that is a sum of three squares of primes in exactly six ways?
>
>
>
... I know it is not research mathematics. Happy new year!
EDIT: Now that it is answered I should note that I learned this puzzle from a tweet by [Ed Southall](https://twitter.com/solvemymaths). I tho... | https://mathoverflow.net/users/8176 | Sums of squares of primes | the answer is 2019
$$2019=a^2+b^2+c^2,\;\;\text{with}\;\;(a,b,c)\in\{(7,11,43),(7,17,41),(13,13,41),(11,23,37),(17,19,37),(23,23,31)\}.$$
I think this was first noticed by [Ed Southall](https://www.theguardian.com/science/2018/dec/31/can-you-solve-it-2019-in-numbers)
| 3 | https://mathoverflow.net/users/11260 | 320065 | 138,255 |
https://mathoverflow.net/questions/319439 | 5 | Consider the usual simple random walk on $\mathbb{Z}$, taking steps of +1 or -1 with equal probability. Of course, each trajectory corresponds uniquely to an element of $\{-1,1\}^\infty$. Now, there is an obvious way to identify this with $\mathbb{Z}\_2^\infty$, and this is in fact a nice compact abelian group.
So, t... | https://mathoverflow.net/users/58551 | Fourier transform of a simple random walk | I doubt that this is of any use but the 1-element modes are not hard to compute. Let $X\_i:=\sum^i\_{k=1}\omega\_k$ be the random walk and $\tau=\min\{i:X\_i=a\text{ or }X\_i=-b\}$. We have
$$
\mathbb{E }(\mathbb{1}\_{X\_\tau=a}
\omega\_i)=\mathbb{E }(\mathbb{1}\_{X\_\tau=a}
\omega\_i\mathbb{1}\_{\tau<i})+\mathbb{E }... | 0 | https://mathoverflow.net/users/56624 | 320066 | 138,256 |
https://mathoverflow.net/questions/320067 | 2 | Suppose a continuous function $f:[0,1] \to \mathbb{R}$ satisfies the following equation for all $z \in \left(0,\frac{1}{2}\right)$,
$$\int\_z^{2z} [f(x)-f(z)] dx = 0.$$
It is clear that a constant function $f(x)=c$ satisfies it. I would like to prove that there are no other such continous functions.
Note: this is a m... | https://mathoverflow.net/users/49831 | Functional equation $\int_z^{2z} [f(x)-f(z)] dx = 0$ | Let $p$ be a zero of $2^{p+1}-p-2$ other than $-1$ and $0$ (e.g. one is approximately $2.54536493037426+10.7539751752688 i$). Then
the real and imaginary parts of $f(x) = x^p$ satisfy the equation.
Note that (with $f(0)=0$) $f$ is continuous on $[0,\infty)$ if $\text{Re}(p) > 0$.
| 5 | https://mathoverflow.net/users/13650 | 320074 | 138,257 |
https://mathoverflow.net/questions/320073 | 3 | Suppose we are given a contractible Kan complex $S$ and a map of simplicial set $f : S \rightarrow T$. Under what conditions can we say that $f$ factors through the largest Kan complex $Z$ contained in $T$?
I am asking this question because it pops up in Proposition 2.2.5.7 of *Higher Topos Theory*. In the proof we a... | https://mathoverflow.net/users/131022 | Factorization of a map from a contractible Kan complex through a Kan complex | First note that $\mathcal C^K$ is again an $\infty$-category. Then this follows from the fact that passing from an $\infty$-category to the largest Kan complex contained in it is a functor right adjoint to the inclusion of Kan complexes into all $\infty$-categories. This is proved in Prop 1.2.5.3 of HTT.
That is, adj... | 6 | https://mathoverflow.net/users/2362 | 320075 | 138,258 |
https://mathoverflow.net/questions/320081 | 4 | I've heard in informal conversations before the claim that:
"a generic singular hypersurface has a single singularity of type $\mathbf{A}\_1$".
What is the precise statement of this result? Where can I find a proof?
| https://mathoverflow.net/users/5101 | Generic singular hypersurface | Consider the space of all hypersurfaces of degree $d$ in $\mathbb P^n$, say, with $n \geq 1$ and $d \geq 2$. Consider the closed subspace of singular hypersurfaces. Then there is a dense open set of this closed subspace over which all hypersurfaces have a single singularity of type $\mathbf A\_1$.
The proof is to che... | 7 | https://mathoverflow.net/users/18060 | 320083 | 138,261 |
https://mathoverflow.net/questions/320089 | 14 | Is there a topological space $X$, which is not a singleton, and satisfies the following property?
For every continuous function $f: X\times S^2\to\mathbb{R}^2$ there exist a point $x\in S^2$ such that $f(t, x)=f(t,-x),\;\forall t \in X$?
Is there a classification of all spaces $X$ with such property?
| https://mathoverflow.net/users/36688 | A parametric version of the Borsuk Ulam theorem |
>
> **Theorem.** For a topological space $X$ the following conditions are equivalent:
>
>
> 1) for any continuous map $f:X\times S^2\to\mathbb R^2$ there exists a point $s\in S^2$ such that $f(x,s)=f(x,-s)$ for any $x\in X$;
>
>
> 2) any continuous map $f:X\to \mathbb R$ is constant.
>
>
>
Proof. (1) $\Righ... | 19 | https://mathoverflow.net/users/61536 | 320098 | 138,264 |
https://mathoverflow.net/questions/320110 | 0 | Let $M$ be a compact smooth manifold. And particularly I am interested in the case the torus $M=T^n$.
Consider the de Rham complex $(\Omega^\*(M), d)$ and the cochain complex
$$
C:=\mathrm{Hom} (\Omega^\*(M),\Omega^\*(M))
$$
with the differential map $\delta$ given by
$$
\delta(f)=d \circ f - f\circ d$$
>
> **Que... | https://mathoverflow.net/users/69190 | Compute the cohomology of $\mathrm{Hom} (\Omega^*(M),\Omega^*(M))$ | Since everything is linear over the field $\mathbb{R}$, $\Omega^\*(M)$ is chain homotopy equivalent to $H^\*\_{dR}(M)$ endowed with the trivial differential, hence the cohomology of your complex $C$ is just $\operatorname{Hom}(H^\*\_{dR}(M),H^\*\_{dR}(M))$. This identification is not natural in $M$ though.
| 3 | https://mathoverflow.net/users/12166 | 320114 | 138,271 |
https://mathoverflow.net/questions/320137 | 1 | Consider the Nisnevich site of a noetherian scheme $S$ of finite Krull dimension (the objects are schemes $U$ smooth and of finite type over $S$), let $A$ be a sheaf of abelian groups on this site. I want to know:
>
> Is the Nisnevich sheafification of the presheaf $$U\mapsto\mathrm{H}\_{\mathrm{Nis}}^{n}(U, A)$$
>... | https://mathoverflow.net/users/42571 | The sheafification of taking cohomology is trivial? | This is true (in any site), you are deriving the identity functor, which is exact, so you get zero.
More generally, for a map of sites $$ \varepsilon\colon C \to D $$
(which by definition is a functor $\varepsilon^{-1}$ in the opposite direction satisfying some properties), the higher direct image functors $R^i \vare... | 5 | https://mathoverflow.net/users/3847 | 320140 | 138,275 |
https://mathoverflow.net/questions/320019 | 27 | The question below was posted on [Mathematics Stack Exchange](https://math.stackexchange.com/questions/3040258/is-there-any-two-variable-function-which-has-no-representation-of-the-form-sum). It received no answer, and I do not expect any direct answer to it here. However, the question seems to me a natural one. Thus I... | https://mathoverflow.net/users/7458 | Is this a known question about the expression of a function on $\Bbb R^2$ as an infinite sum of products? | In ["Representation of functions of two variables as sums of rectangular functions, I"](http://matwbn.icm.edu.pl/ksiazki/fm/fm85/fm85118.pdf), Roy O. Davies shows that under the continuum hypothesis every function has a representation of this form.
More precisely, he shows that we can get a representation with the ad... | 24 | https://mathoverflow.net/users/60398 | 320146 | 138,276 |
https://mathoverflow.net/questions/320151 | 2 | Let G = $<a,b : a^2= b^n = 1 >$ be the (2,n,$\infty$)-triangle group. Define a map $\sigma:G \to Z\_2 \times Z\_n$ via $a \mapsto (-1,1), b \mapsto (1,[1]).$ The kernel H of $\sigma$ is then a torsion free subgroup. With a realization of $a$ and $b$ as elements of PSL(2,R) ($a$ as a rotation and $b$ also as a rotation ... | https://mathoverflow.net/users/105481 | a normal subgroup of a triangle group | The cusp in $G$ can be described as the conjugacy class of the infinite cyclic group generated by $b a^{-1}$. The intersection of this cyclic group with $H$ is described in two cases. When $n$ is odd then, as you say, the intersection is generated by $(ba^{-1})^{2n}$, and the conjugacy class of this cyclic group in $G$... | 2 | https://mathoverflow.net/users/20787 | 320155 | 138,278 |
https://mathoverflow.net/questions/320123 | 0 | $\newcommand\binorm[1]{\lVert#1\rVert}\newcommand\trinorm[1]{\lVert\lvert#1\rvert\rVert}$Consider the space $\ell ^{2}$ with the standard norm
\begin{align\*}
\binorm x\_{2} = \left( \sum \_{i =1} ^{\infty} x \_{i} ^{2} \right) ^{1/2}
\end{align\*}
and define the equivalent norm
\begin{align\*}
\trinorm x \_{\sqrt{2}}... | https://mathoverflow.net/users/134213 | $B _{\ell ^{2}} ^{+}$ with the norm $\lVert\lvert \cdot \rvert\rVert _{\sqrt{2}}$ doesn't have normal structure | For $x$ in $B^+\_{\ell\_2}$ one has $$\sup\_{y\in B^+\_{\ell\_2}}\|x-y\|\_2^2=\sup\_{y\in B^+\_{\ell\_2}}\Big( \|x\|\_2^2+\|y\|\_2^2-2(x\cdot y)\Big)\le \|x\|\_2^2+1,$$
and
$$\sup\_{y\in B^+\_{\ell\_2}}\sqrt{2}\|x-y\|\_\infty=\sqrt{2}\sup\_{y\in B^+\_{\ell\_2},\,n\in\mathbb{N}}\big| x\_n -y\_n \big|\le \sqrt{2},$$
be... | 1 | https://mathoverflow.net/users/6101 | 320158 | 138,279 |
https://mathoverflow.net/questions/320154 | 8 | Is a characterization known for the set of Laurent polynomials arising as the Jones polynomial of some knot? More generally, is such a characterization known for any of the famous knot polynomials?
| https://mathoverflow.net/users/29961 | Set of Jones polynomials as the knot varies | I believe your question is open for the Jones polynomial. However, it is solved for the Alexander polynomial. On page 171 of Rolfsen's Knots and Links, the following theorem appears.
**Theorem.** Let $p(t)$ be any Laurent polynomial satisfying:
1. $p(1) = \pm 1$, and
2. $p(t)=p(t^{-1})$.
There exists a knot $K$ w... | 7 | https://mathoverflow.net/users/21250 | 320165 | 138,280 |
https://mathoverflow.net/questions/320129 | 5 | I feel like there must be a classical result answering this question (or easily modified to do so) but a quick flip through Soare didn't produce anything so rather than waste time I figured I'd just ask.
Given incomplete r.e.sets $C \nleq\_T B$ must there exist a low r.e. set $A <\_T C$ with $A \nleq\_T B$?
I'm gue... | https://mathoverflow.net/users/23648 | Given B,C incomplete, incomparable r.e. sets must C compute low r.e. set avoiding cone below B? (ADDED: Uniformly?) | By Sacks’s splitting theorem, the degree of C is the join of the degrees of two low r.e. sets, both strictly below the degree of C. They can’t both be recursive in B, so at least one of them can serve as A. So, yes, there is such an A.
| 7 | https://mathoverflow.net/users/31026 | 320169 | 138,281 |
https://mathoverflow.net/questions/319707 | 2 | Let $u:\mathbb{R}^N \to \mathbb{R}^N$, $u \in BV(\mathbb{R}^N)$, be a function of bounded variation.
We have that the following holds
>
> $$(\ast) \qquad \frac{1}{|B\_r(0)|}\int\_{B\_r(0)} \frac{|u(x+z)-u(x)-Az|}{|z|} dz =0$$
> for the points where $Du$ is not singular, where $A$ is the approximate differential o... | https://mathoverflow.net/users/nan | Difference quotient for functions of bounded variation | In fact the following is true:
>
> **Theorem.** If $u\in BV(\mathbb{R}^N)$, then for almost all $x\in\mathbb{R}^n$: $$
> \frac{1}{|B\_r(0)|}\int\_{B\_r(0)}\left|\frac{u(x+z)-u(x)-[Du(x)]\_{ac}z}{r}\right|^{\frac{N}{N-1}}\,
> dz\to 0 \quad \text{as $r\to 0$.} $$
>
>
>
Here $[Du(x)]\_{ac}$ is the density of th... | 0 | https://mathoverflow.net/users/121665 | 320178 | 138,283 |
https://mathoverflow.net/questions/320186 | 10 | Henkin-style completeness proofs are founded on a few basic presuppositions, such as the assumptions that the language of a logical theory must be enumerable (or at least that the axiom of choice holds), and that it must contain a (not necessarily primitive) logical connective for negation.
One could fairly describe... | https://mathoverflow.net/users/58734 | Henkin-style completeness proofs for intuitionistic logic | What you are looking for is a proof of completeness for intuitionistic logic in a constructive metatheory. When one turns constructive, even the notion of completeness varies according to how it is formulated. For example, saying that a valid sentence is provable is not equivalent anymore to saying that a consistent th... | 15 | https://mathoverflow.net/users/12976 | 320187 | 138,286 |
https://mathoverflow.net/questions/320171 | 4 | In this question, we will follow the notations and definitions from the book "Automorphic Representations and L-Functions for the General Linear Group" by Goldfeld and Hundley. For simplicity, let's focus of $GL\_2$. The definition of an (cuspidal) automorphic representation with central character $\omega$ is given in ... | https://mathoverflow.net/users/87910 | To what extent does the $(\mathfrak{g},K_{\infty})$ module determines the automorphic representation? | 1) No, two distinct cuspidal automorphic representations can have the same underlying $(\mathfrak{g},K)$-module. In particular, if the $(\mathfrak{g},K)$-module is a weight $k$ discrete series, then the cuspidal automorphic representations corresponds to a holomorphic newform of weight $k$, and there are plenty of thes... | 8 | https://mathoverflow.net/users/3803 | 320195 | 138,290 |
https://mathoverflow.net/questions/320144 | 1 | Suppose we have a $C^\*$ algebra $A=\{(x\_n)\in \prod M\_n(\Bbb C),lim\_n tr\_n(x\_n^\*x\_n)=0\}$.
If $B$ is any nonzero $C^\*$-sub algebra of $A$,does there exist a finite dimensional irreducible representation of $B$?
| https://mathoverflow.net/users/63864 | irreducible representation of a $C^*$ algebra | Yes. Irreducible representations are the building blocks. If there is a finite dimensional representation, there is a finite dimensional irrep. Any C\*-subalgebra of $A$, indeed any C\*-subalgebra of $\prod M\_n$, has plenty of finite dimensional representations, namely evaluation on the $n$th factor, for any $n$.
| 1 | https://mathoverflow.net/users/23141 | 320196 | 138,291 |
https://mathoverflow.net/questions/318772 | 3 | I think that there is a typo in the paper referred to in the title of the question, available [here](http://www.ams.org/journals/bull/2018-55-01/S0273-0979-2017-01598-4/S0273-0979-2017-01598-4.pdf). While discussing Roth's proof of AP discrepancy, on page 2, the author states that
>
> Roth [20] established the exis... | https://mathoverflow.net/users/17773 | Typo in Soundararajan's *Bulletin of the AMS* article on Tao's resolution of The Erdos Discrepancy Problem? | I actually think there is a more significant flaw. A counterexample to the centered inequality is $\rho = \frac{1}{2}$ and $A = \{1,2,\dots,\frac{N}{2}\}$. Indeed, we always have $\sum\_{n \in A, n \equiv a \pmod{q}} 1 -\frac{N/2}{q} = O(1)$.
What he actually meant to say is what he ended up saying in words: "evenly... | 4 | https://mathoverflow.net/users/129185 | 320203 | 138,292 |
https://mathoverflow.net/questions/320179 | 11 | What is an effective way to organize collaborations with several people on the same paper? How do you arrange the $\LaTeX$ document, the shared (digital) papers library, and other aspects?
More in general, remarks on how to set up an effective collaborations are very welcome too.
| https://mathoverflow.net/users/nan | How to organize collaborations? Managing shared library and LaTeX document | For writing the LaTeX document, here are several options I've seen used:
* The "cleanest" and safest option is a git repository (on [github](https://github.com/) or [gitlab](https://about.gitlab.com/) or [bitbucket](https://bitbucket.org/) or, the good old way, on someone's server). [Example with 3 authors](https://g... | 13 | https://mathoverflow.net/users/2530 | 320207 | 138,294 |
https://mathoverflow.net/questions/320201 | 5 | I was wondering whether the cyclic group $\mathbb Z/4\Bbb Z$ acts freely on $S^{2k} \times \Bbb CP^n$ where $n>1$? It seems to me that it does not act freely. In case it acts freely then the induced action on cohomology must be non-trivial as the Euler characteristic is non-zero. I was trying to prove using Lefschetz f... | https://mathoverflow.net/users/134255 | Does the cyclic group $\Bbb Z/4 \Bbb Z$ acts freely on $S^{2k} \times \Bbb CP^n$? | Nothing really changes from Will Sawin's answer [here](https://mathoverflow.net/a/315731/40804).
Given a self-homeomorphism $\sigma: S^{2k} \times \Bbb{CP}^n$, you want to know what the induced map is on cohomology, written via the Kunneth decomposition as $\Bbb Z[c, S]/(c^{n+1}, S^2)$, where $|c| =2$ and $|S| = 2k$... | 6 | https://mathoverflow.net/users/40804 | 320211 | 138,296 |
https://mathoverflow.net/questions/320205 | 16 | Summary
=======
Famously, the categories of 4-dimensional smooth manifolds and 4-dimensional piecewise linear manifolds are equivalent. Is there a constructive proof for this theorem or does it depend on the axiom of choice?
Background
==========
A smooth manifold is a manifold with a smooth atlas, that is, an at... | https://mathoverflow.net/users/13767 | Is there a constructive proof that in four dimensions, the PL and the smooth category are equivalent? | In this, as in much constructive analysis/algebra, the issues fall into two aspects:
1. general formalities of how to set up the structures involved;
2. the specific constructive content of the problem at hand.
This is a half-answer, addressing just the first aspect. If you’re already experienced with constructive ... | 7 | https://mathoverflow.net/users/2273 | 320214 | 138,297 |
https://mathoverflow.net/questions/320173 | 4 | Let $\mathbb K$ be an algebraically-closed complete non-archimedean field whose absolute value is non-trivial. Consider the Tate algebra $T\_n=\mathbb K\langle X\_1,\dots, X\_n \rangle$ and fix $f\in T\_n$. We use $|\cdot|$ to denote the non-archimedean norm.
Then $f(x\_1,\dots,x\_n)$ converges for every $x=(x\_1,\do... | https://mathoverflow.net/users/69190 | Tate algebras and fundamental theorem of algebra | Yes, the argument matches exactly your edit. We can reason as follows. Let $f$ be a nonzero element of the Tate algebra. By definition, $|c\_\nu| \to 0$ as $|\nu|\to \infty$. Thus $|c\_\nu|$ attains a maximal value for some $\nu$. By dividing by $c\_\nu$, we may assume that the maximum value is $1$. Then the coefficien... | 1 | https://mathoverflow.net/users/18060 | 320234 | 138,301 |
https://mathoverflow.net/questions/320188 | 1 | **Update:** Since the question was put on hold, work on this will continue on this [notebook](https://beta.observablehq.com/@ishi/arithmetic).
*So long, and thanks for all the fish!*
First, let me state that I am not a formally-trained mathematician, therefore please forgive my approximations and help me correc... | https://mathoverflow.net/users/134241 | Is there any work related to Hypertype Theory? | **1. Where can we find prior work on the subject?**
The model of [exponential fields](https://en.m.wikipedia.org/wiki/Exponential_field) seems to be related, but these are defined with an exponential function instead of a commutative operator, and no inverse function. Furthermore, an exponential field defines a singl... | 4 | https://mathoverflow.net/users/134241 | 320238 | 138,302 |
https://mathoverflow.net/questions/283302 | 5 | At <https://dmishin.github.io/js-revca/index.html>, you can play around with reversible cellular automata. I noticed that on that site, that for the reversible linear cellular automata (which I have tested), there is always a small natural number $p$ ($p\leq 12$ in the cases which I have tested) where if the grid is a ... | https://mathoverflow.net/users/22277 | Why do some linear cellular automata over $Z_{2}$ on the torus have small order? | So I have a proof of the low period phenomenon that works for all cellular automata prime characteristics $p$ and all finite dimensions (i.e. for linear cellular automata whose alphabet is a vector space of characteristic $p$ over a finite $p$-group). The general result can quite easily be verified empirically. These c... | 0 | https://mathoverflow.net/users/22277 | 320240 | 138,304 |
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