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https://mathoverflow.net/questions/286527 | 2 | Let $A$ and $B$ be two full column rank real matrices of dimension $n \times m$, where $n \ge m$. Let $P$ be an $m\times m$ positive definite matrix.
>
> **Question:** Does there always exist a symmetric $n \times n$ matrix $X$ such that the following holds?
>
>
> $$\mathrm{tr}(P(A^\top XB+B^\top X A)) \ne 0$$
> ... | https://mathoverflow.net/users/62673 | Condition for non-vanishing trace | Your trace equals ${\rm tr} ((BPA^\top+APB^\top)X)$. This equals 0 for all symmetric matrices $X$ if and only if $C=BPA^\top+APB^\top=0$ (else take $X=C$, note that $C$ is symmetric). Of course, this is possible. For example, if $m=n$ and $P=\rm I$ is identity matrix, the product $BA^{\top}$ may be antisymmetric withou... | 3 | https://mathoverflow.net/users/4312 | 286532 | 126,498 |
https://mathoverflow.net/questions/286535 | 10 | Let $\{x\_{n}\}$ be a sequence in $\mathbb{N}$ with $x\_{1}=1$ such that for any prime $p$, the set $$A=\{x\_{1},x\_{2},\ldots,x\_{p}\}$$ forms a complete residue system $\pmod{p}$. Now is it true that $\lim\limits\_{n\to\infty}\frac{x\_{n}}{n}$ exists? If yes what is it's value?
| https://mathoverflow.net/users/117461 | Sequence $(x_n)$ whose first $p$ terms is a complete residue system: value of $\lim\limits_{n\to\infty}\frac{x_n}{n}$? | This problem is due to Imre Ruzsa who posed it at the 2015 Miklós Schweitzer Contest in Hungary: <https://mathproblems123.wordpress.com/2015/10/31/miklos-schweitzer-2015-problems/>
Here is a solution (thanks also to YCor for his comments and encouragement). It is easy to see that $x\_1=1$ and $x\_2=2$. We claim that ... | 21 | https://mathoverflow.net/users/11919 | 286543 | 126,501 |
https://mathoverflow.net/questions/286512 | 14 | Can anyone reference/disprove the theorem in the case where the embedded submanifold is merely $C^1$ instead of smooth? I have a compact $C^1$ embedded submanifold of $\mathbb{R}^n$ without boundary that I want to show there exists a tubular neighborhood of (with the radius of the tube not necessarily constant). Actual... | https://mathoverflow.net/users/117452 | Tubular Neighborhood Theorem for $C^1$ Submanifold | The answer to your question depends on whether you are looking for a tubular neighborhood in the general differential topological sense or the more restrictive geometric sense. The answer in the topological sense is **Yes**, but in the geometric sense the answer in general is **No**. These two conceptions may coincide ... | 13 | https://mathoverflow.net/users/68969 | 286545 | 126,502 |
https://mathoverflow.net/questions/286456 | 7 | Let $D(s) = \sum\_{n=1}^\infty a\_n n^{-s}$ be a Dirichlet series with $a\_n ≥ 0$ and abscissa of convergence $\sigma\_a = 1$. Further, we assume that $D(s)$ is holomorphic in each point $\Re(s) = 1$ except a pole of order $k > 0$ in $s = 1$ and that $D(s)$ possesses a meromorphic continuation on some half-plane $\Re(s... | https://mathoverflow.net/users/116289 | A question concerning Tauberian theory | The answer is no, even if we assume there are no other poles than $1$ in $\sigma > 1- \epsilon\_0$. I give an example below with $\epsilon\_0=1$. This is a variant of an example given by Karamata in $1952$.
Let $b\_n = 1 + \cos(\log^2(n)) \geq 0$ and consider $B(s) = \sum\_{n \geq 1} b\_n n^{-s}$. Let us write
$$
\su... | 5 | https://mathoverflow.net/users/21724 | 286546 | 126,503 |
https://mathoverflow.net/questions/286539 | 1 | (**Also in Mathematics stack Exchange:** <https://math.stackexchange.com/questions/2528216/polarization-operators-identity-and-gl-ell-mathbbr>)
Let $X$ be a matrix of variables $x\_{ij}$ of size $\ell\times n$:
\begin{equation\*}
X=\left(\begin{array}{cccc} x\_{11}&x\_{12} &\dots & x\_{1n}\\
x\_{21}&x\_{22} &\dots &... | https://mathoverflow.net/users/72331 | Polarization operators and the action of $GL_{\ell}(\mathbb{R})$ on $\mathcal{R}_{n}^{(\ell)}$ | I don't have the time to work out precisely how to derive Procesi's claim from the formula for $f(MX)$ but probably it is a cute exercise along the following lines: let $E\_{k,i}$ be the elementary matrix whose $(k,i)$-entry equals $1$ and all other entries are $0$. Now calculate $f(MX)$ for $M={\bf1}\_\ell+tE\_{k,i}$ ... | 2 | https://mathoverflow.net/users/89948 | 286549 | 126,505 |
https://mathoverflow.net/questions/286547 | 7 | In their 2008 paper "Torelli theorem for curves over finite fields" Bogomolov, Korotiaev and Tschinkel mention in the beginning of Section 9 that absolute Galois groups of curves over $\mathbb{F}\_p^{alg}$ are free profinite on countably many generators. However, they do not give reference, and after talking to some co... | https://mathoverflow.net/users/2234 | absolute Galois group of the function field of a curve over $\mathbb{F}_p^{alg}$ | The theorem is true. It seems to have been proved independently by Florian Pop and by David Harbater. In Pop's paper, it is the corollary on p. 556.
MR1334484 (96k:14011)
Pop, Florian
Étale Galois covers of affine smooth curves. The geometric case of a conjecture of Shafarevich. On Abhyankar's conjecture.
... | 8 | https://mathoverflow.net/users/13265 | 286551 | 126,506 |
https://mathoverflow.net/questions/286238 | 2 | Naimark's dilation theorem in papers and textbooks is usually stated as:
>
> Let $E$ be a regular, positive, $B(\mathcal H)$-valued measure on $X$. Then there exists a Hilbert space $\mathcal K$, a bounded linear operator $V: \mathcal H \rightarrow \mathcal K$, and a regular, self-adjoint, spectral $B(\mathcal K)$-... | https://mathoverflow.net/users/76593 | Original statement of Naimark's dilation theorem | I found [it](http://nauka1941-1945.ru/files/pdf/EP_1943_AKS_00000374.pdf). It's a (as far as I can make out) legitimate, non-paywall source and in English! Thumbs up for careful googling. It's in an online library called the "Scientific Heritage of Russia" which seems to be an archive of scientific papers from the year... | 5 | https://mathoverflow.net/users/76593 | 286557 | 126,507 |
https://mathoverflow.net/questions/286554 | 3 | Suppose $W\_r(\mu\_n,\mu)\to0$, where $\mu\_n$ and $\mu$ are discrete probability measures on some metric space $\Omega$, and that all measures have the same number of atoms $d$ (but not the *same* atoms):
$$\mu\_n = \sum\_{i=1}^d p\_{n,i}\delta\_{\theta\_{n,i}}, \quad \mu = \sum\_{i=1}^d p\_{i}\delta\_{\theta\_{i}}.... | https://mathoverflow.net/users/99132 | Wasserstein convergence of conditional measures | The edited version of the question requires that $\langle\theta\_{i,n}\rangle$ converges to $\theta\_i$ for all $i$. In that case, the result is true. It suffices to show that we also have $\langle p\_{i,n}\rangle$ converging to $p\_i$. This is easy to see when we note that the Wasserstein-metric metrizes a topology st... | 2 | https://mathoverflow.net/users/35357 | 286558 | 126,508 |
https://mathoverflow.net/questions/285945 | 2 | Let $\mathbb{N}$ denote the set of the positive integers. The *Golomb space* is a space ${\bf G} =(\mathbb{N},\tau)$ where a basis of $\tau$ is generated by
$$\big\{\{a+bn: n\in \mathbb{N}\cup\{0\}\}: a,b\in\mathbb{N} \text{ and } a,b \text{ relatively prime}\big\}.$$
Is there a continous map $f: {\bf G}\to {\bf G}$ th... | https://mathoverflow.net/users/8628 | Continuous self-maps in the Golomb space that are neither increasing nor decreasing | For polynomials with non-negative integer coefficients and no constant term, the following simple (but not obvious) fact was observed by [Paulina Szczuka](https://www.degruyter.com/view/j/dema.2013.46.issue-2/dema-2013-0454/dema-2013-0454.xml).
**Theorem.** Each polynomial $f:\mathbb N\to\mathbb N$, $f:x\mapsto a\_1x... | 3 | https://mathoverflow.net/users/61536 | 286575 | 126,510 |
https://mathoverflow.net/questions/286566 | 5 | Let us work in Kelly-morse set theory, so we can talk about $V$. For some model $M=(\mathbb N, \in\_M)$ that is elementary equivalent $(V, \in)$, we can have an oracle that corresponds to $(\mathbb N, \in\_M)$'s truth predicate. We will say that $M$ is a true countable model of set theory, and we will call the oracle $... | https://mathoverflow.net/users/65915 | Is there an oracle that can compute something iff it is computable in every countable model that is equivalent to $(V, \in)$? | Yes - this real is exactly the parameter-free theory of $V$ (or anything Turing-equivalent to it).
One direction is immediate: if $M$ is elementarily equivalent to $V$, then from the elementary diagram of $M$ we can compute the theory of $V$ (just look at the parameter-free sentences). It's the other direction that i... | 6 | https://mathoverflow.net/users/8133 | 286580 | 126,511 |
https://mathoverflow.net/questions/286522 | 1 | I should probably start with a warning that this is my first post in this board and that I am sorry, if it is not up to standards. It would be great, if you could let me know how to improve the post.
I am currently working towards my master thesis, which is mainly concerned with the following paper: [Chatterji, Iozzi... | https://mathoverflow.net/users/51356 | Conull subspace containing orbit of an (ergodically acting) group | Formally, your question was answered in the comments, and I will not repeat the answer here, as it simply indicates that your initial question was "wrong". Possibly you just missed some "up to null set" comment (or hidden assumption) somewhere in the text. Nevertheless, if you are genuinely interested in delicate and f... | 1 | https://mathoverflow.net/users/89334 | 286590 | 126,515 |
https://mathoverflow.net/questions/286583 | 1 | Let $X$ be a smooth, projective variety, $Y \subset X$ a smooth, projective subvariety of codimension $3$. Denote by $\pi:\tilde{X} \to X$ the blow-up of $X$ along $Y$ and by $E$ the exceptional divisor. We know, that $\pi|\_E:E \to Y$ is a $\mathbb{P}^2$-bundle. Let $\pi':F \to Y$ be a $\mathbb{P}^1$-sub-bundle of $E$... | https://mathoverflow.net/users/58203 | Gysin map for projective sub-bundles of exceptional divisors | OK, let me assume that your $F$ is a projective sub-bundle. Then $i\_\*$ is injective.
Note that the blowing up setting is completely irrelevant to the situation: you just have two projective bundles $p:E\rightarrow Y$, $q:F\rightarrow Y$ and an embedding $i:F\hookrightarrow E$ over $Y$. Now the Chow group of such bu... | 2 | https://mathoverflow.net/users/40297 | 286604 | 126,520 |
https://mathoverflow.net/questions/286610 | 2 | Let $X$ be a smooth projective algebraic variety over the complex numbers.
(a) Do there exist:
a smooth proper map $\pi : \mathcal{X}\to S$ of algebraic varieties over the complex numbers, such that $\mathcal{X}$ and $S$ are smooth, $S$ is connected, $X$ is isomorphic to the fiber of $\pi$ over some $\mathbf{C}$-po... | https://mathoverflow.net/users/nan | Smooth proper fibration of complex projective varieties | The answers to Questions (2) and (3) are positive, and the answer to Question (1) is negative. The positive answers follow by a version of the "Lefschetz principle", often also called "spreading out".
Let $X\subset \mathbb{P}^n\_{\mathbb{C}}$ be a closed subscheme. Denote by $\mathcal{I}$ the corresponding ideal she... | 3 | https://mathoverflow.net/users/13265 | 286617 | 126,522 |
https://mathoverflow.net/questions/286613 | 13 | I am writing a paper on the topology of [the Golomb space](https://dml.cz/bitstream/handle/10338.dmlcz/700933/Toposym_01-1961-1_41.pdf) and need a good (standard) reference to the following
**General Chinese Remainder Theorem.** For integer numbers $a\_1,\dots,a\_n$ and positive integers $b\_1,\dots,b\_n$ the inters... | https://mathoverflow.net/users/61536 | A good reference to the general Chinese Remainder Theorem | It seems that you are after this result which can be found, for example, as [Theorem 3.12](https://books.google.com/books?id=LN6PBAAAQBAJ&pg=PA60) in Gareth A. Jones, Josephine M. Jones: *Elementary Number Theory*, Springer-Verlag, London, 1998. Springer Undergraduate Mathematics Series. (It is in the section 3.5 entit... | 23 | https://mathoverflow.net/users/8250 | 286618 | 126,523 |
https://mathoverflow.net/questions/286609 | 3 | Let $X$ be a comapct Riemann surface of genus $g$ and let $J\: : \: X\to \mathbb{C}^{g}/\Lambda$ be the Abel-Jacobi map. This map is a smooth embedding. Let $p\in X$ such that $J(p)=\Lambda$ and consider
$$
J\: : \: X\setminus p \to (\mathbb{C}-\Lambda)/\Lambda.
$$
Then $J^\*$ induces a surjection in complex de Rham co... | https://mathoverflow.net/users/41970 | Jacobian and configuration space and massey products | The answer is yes.
First off, the map $\mathrm{Conf}\_l M \to M^l$ induces an isomorphism on $H^1$ if $M$ is an oriented manifold such that $\dim M > 2$, or such that $\dim(M)=2$ and $M$ has positive genus. See my answer to a previous question [fundamental group of configuration spaces of ordered points on open Riema... | 5 | https://mathoverflow.net/users/1310 | 286620 | 126,524 |
https://mathoverflow.net/questions/286597 | 6 | For a cardinal $\kappa$ such that $V\_{\kappa}$ satisfies Vopěnka's principle as a first-order axiom schema, am I allowed to say "first-order Vopěnka cardinal", or is there any kind of standard term for it?
| https://mathoverflow.net/users/15482 | Am I allowed to say "first-order Vopěnka cardinal"? | In my paper, [The Vopěnka principle is inequivalent to but conservative over the Vopěnka scheme](http://jdh.hamkins.org/vopenka-principle-vopenka-scheme/), I distinguish between the (second-order) Vopěnka principle and the first-order version, which I call the Vopěnka scheme, and prove that these principles are not equ... | 11 | https://mathoverflow.net/users/1946 | 286625 | 126,526 |
https://mathoverflow.net/questions/260729 | 3 | Let $\phi$ be an $L\_{\omega\_1,\omega}$ sentence. The *amalgamation spectrum* of $\phi$ is the set of all cardinals $\kappa$ such that the models of $\phi$ of size $\kappa$ satisfy amalgamation.
**Question**: Is there a known example where the amalgamation spectrum is right-open? E.g. of the form $[\kappa,\lambda)$,... | https://mathoverflow.net/users/13694 | Example with right-open amalgamation spectrum | In [this paper](https://arxiv.org/abs/1705.05821) we were able to prove that there exists some $\psi\in L\_{\omega\_1,\omega}$ and it is consistent that the amalgamation spectrum of $\psi$ is consistently equal to $[\aleph\_1,2^{\aleph\_1})$, where $2^{\aleph\_1}$ is weakly inaccessible. This (consistently) answers the... | 3 | https://mathoverflow.net/users/13694 | 286627 | 126,527 |
https://mathoverflow.net/questions/286598 | 1 | It seems I am too fast to ask the question without thinking carefully.
The equation I consider is find $w:\Omega \to R$ satisfying
$$\frac{Du}{\sqrt{1+|\nabla u|^2}}=Dw. \tag{\*}$$
Where $u:\Omega \to R$ is given and $u=0$ on $\partial \Omega$.
It is obvious this equation could not be solved for general $u$, the c... | https://mathoverflow.net/users/91939 | Could we solve the vector value ODE in a approximation way? | The Frobenius condition which is necessary and sufficient for local integrability is that $Du$ and $D(|\nabla u|^2)$ are parallel. So going back through [my answer to this question](https://mathoverflow.net/q/42617/3948) you see that $u$ must be a function whose gradient descent curves are geodesics.
This also tells... | 4 | https://mathoverflow.net/users/3948 | 286632 | 126,528 |
https://mathoverflow.net/questions/286631 | 1 | Is there a simple, undirected graph $G= (V,E)$ with $\chi(G) \geq \aleph\_0$, and if $M\subseteq E$ is a matching then $|M|<\chi(G)$?
| https://mathoverflow.net/users/8628 | Matchings in graphs with infinite chromatic number | No. Take a maximal matching $M$. Its complement is independent set, that allows to color our graph with $2|M|+1$ colors. This is either finite or $|M|$, thus $|M|\geqslant \chi(G)$.
| 2 | https://mathoverflow.net/users/4312 | 286633 | 126,529 |
https://mathoverflow.net/questions/265724 | 5 | Let $M/L/K$ be a tower of local fields such that $M/L$ is abelian with Galois group $G$. The Artin map $\psi\_{M/L}$ restricted to $K^\times$ is a continuous map to $G$ and thus corresponds to some abelian extension $T/K$ with an embedding $\operatorname{Gal}(T/K) \hookrightarrow G$. Challenge: Find $T$. (I am primaril... | https://mathoverflow.net/users/105625 | Artin map restricted to base field | I've finally found the answer to my question by perusing Serre's *Local Fields,* Ch.XIII, specifically Propositions 10-12, which contain the functorial properties of the Artin symbol used below.
We describe $\left.\phi\_{M/L}\right|\_K$ in terms of $\phi\_{\tilde M / K}$:
\begin{alignat\*}{5}
\text{Gal}(\tilde M / K... | 2 | https://mathoverflow.net/users/105625 | 286638 | 126,530 |
https://mathoverflow.net/questions/286626 | 19 | This is an immediate successor of [Chebyshev polynomials of the first kind and primality testing](https://mathoverflow.net/q/286304/41291) and does not have any other motivation - although original motivation seems to be huge since a positive answer (if not too complicated) would give a very efficient primality test (s... | https://mathoverflow.net/users/41291 | Is there an explicit expression for Chebyshev polynomials modulo $x^r-1$? | There's a rapid algorithm to compute $T\_n(x)$ modulo $(n,x^r-1)$. Note that
$$
\pmatrix{T\_n(x) \\ T\_{n-1}(x)} = \pmatrix { 2x & -1 \\ 1&0} \pmatrix{T\_{n-1}(x) \\ T\_{n-2}(x)} = \pmatrix { 2x & -1 \\ 1&0}^{n-1} \pmatrix{ x\\ 1}.
$$
Now you can compute these matrix powers all modulo $(n, x^{r}-1)$ rapidly by repeat... | 28 | https://mathoverflow.net/users/38624 | 286639 | 126,531 |
https://mathoverflow.net/questions/286654 | 2 | The $\phi$ entropy is defined as $\text{Ent}\_{\phi}[X]= \mathbb{E}[\phi (X)]-\phi(\mathbb{E}[ X])$ where $X$ is a random variabel and $\phi$ is a convex function ($\text{Ent}\_{\phi}[X] \geq 0$). By choosing $\phi(x)=x^2$ we get $\text{Ent}\_{\phi}[X]=\text{Var}(X)$. if $\phi(x)=x\log x$ and $X=\frac{dv}{d\mu} $ (rado... | https://mathoverflow.net/users/117055 | $\phi$ - Entropies - $\phi$ - Divergences and classical entropy recovery | The choice $\phi(x)=x\log(x)$ yields an entropy distinct from Shannon entropy. See p. 94 of Concentration Inequalities: A Nonasymptotic Theory of Independence by Boucheron, Lugosi, Massart:
<http://www.oxfordscholarship.com/view/10.1093/acprof:oso/9780199535255.001.0001/acprof-9780199535255>
The two entropies are rel... | 3 | https://mathoverflow.net/users/12518 | 286659 | 126,538 |
https://mathoverflow.net/questions/286652 | 4 | Let ${\bf S} = (S\_1,...,S\_d) \in \mathcal{L}(E)^d$. We recall that the norm of $\|{\bf S}\|$ is defined by
\begin{eqnarray\*}
\|{\bf S}\|
&=&\sup\left\{\bigg(\displaystyle\sum\_{k=1}^d\|S\_kx\|^2\bigg)^{\frac{1}{2}},\;x\in E,\;\|x\|=1\;\right\},
\end{eqnarray\*}
I want to show that if the operators $S\_k$ are commu... | https://mathoverflow.net/users/116483 | Why this equality holds? | As it was mentioned in the comments, this is basically writing down the definition.
Since all the $S\_i$'s commute with each other, then each component of $\boldsymbol{S}^n$ is of the form $(S\_1 \ldots S\_1)(S\_2 \ldots S\_2) \ldots (S\_d \ldots S\_d)$, where $S\_i$ apprears $\alpha \_i$ times and $\sum \_i \alpha \... | 4 | https://mathoverflow.net/users/117484 | 286661 | 126,539 |
https://mathoverflow.net/questions/286621 | 15 | Suppose that $I \subset \mathbb C[z\_1,\dots, z\_n]$ is a prime ideal. Consider the ideal $I\_{hol}$ in the ring of holomorphic functions $f: \mathbb C^n\to \mathbb C$ generated by polynomials from $I$.
Is $I\_{hol}$ prime?
| https://mathoverflow.net/users/41487 | Do prime ideals in polynomial ring generate prime ideals in the ring of holomorphic functions? | **Edit. I added some lemmas to address the issue raised by David Speyer and the OP.** The books by Grauert and Grauert-Remmert are wonderful sources. The proofs in those books are the "correct" arguments, using "sledgehammers" as little as possible. Even though it is a sledgehammer, Hironaka's Resolution of Singulariti... | 7 | https://mathoverflow.net/users/13265 | 286669 | 126,541 |
https://mathoverflow.net/questions/286665 | 3 | Let $X\_t$ be a real-valued stochastic process (if it helps, we can assume that it is a component of a multivariate diffusion or jump-diffusion process).
I'm looking for sufficient conditions under which $X\_t$ diverges in probability, i.e.
$$\forall a>0: \quad P(|X\_t|<a)\stackrel{t\rightarrow\infty}{\rightarrow} 0.$$... | https://mathoverflow.net/users/69603 | Sufficient conditions for divergence of continuous-time stochastic process | It appears you are looking for sufficient conditions in terms of expectations of functions of $X\_t$. If so, the following may be offered:
\begin{equation}
\text{$|X\_t|$ diverges to $\infty$ in probability iff $E\frac1{1+|X\_t|}\to0$;} \tag{\*}
\end{equation}
here everywhere the convergence is as $t\to\infty$.
... | 2 | https://mathoverflow.net/users/36721 | 286671 | 126,542 |
https://mathoverflow.net/questions/286655 | 7 | Let $\Lambda \subseteq \mathbb{R}^n$ be a full-rank lattice, i.e. $\Lambda = A \mathbb{Z}^n$ for some $A \in \mathrm{GL}\_n (\mathbb{R})$, and let $C \subseteq \mathbb{R}^n$ be a $0$-symmetric convex body. Then Minkowski's theorem asserts that
$$
\# | \Lambda \cap C | \geq \frac{|C|}{2^n | \mathbb{R}^n / \Lambda|}.
$$
... | https://mathoverflow.net/users/nan | Minkowski's theorem for non-0-symmetric sets | **The best reference I know on this question,** restricted to convex polytopes whose vertices are lattice points (symmetry not assumed), is:
Douglas Hensley, *Lattice vertex polytopes with interior lattice points*, Pacific Journal of Mathematics, Vol 101, No. 1, p. 183-191; MR0688412.
>
> **Author's Abstract.** Con... | 3 | https://mathoverflow.net/users/36904 | 286682 | 126,545 |
https://mathoverflow.net/questions/286641 | 13 | Let $R$ be a commutative ring. If I am not mistaken, there is the following fact:
>
> For a finitely generated abelian group $A$, the $R$-module $A\otimes R$ is free if and only if we can write the torsion part of $A$ as a direct sum of cyclic groups of the form $\mathbb{Z}/k$, where $k$ is invertible or zero in $R... | https://mathoverflow.net/users/2039 | When is $A\otimes R$ a free $R$-module? | Here is a 7-line proof of your statement.
>
>
> >
> > **Claim.** Let $R$ be a commutative ring with identity $1\_R$. Let $A \simeq \mathbb{Z}/d\_1\mathbb{Z} \oplus \cdots \oplus \mathbb{Z}/d\_k\mathbb{Z}$ be a finitely generated Abelian group given with its invariant factor decomposition, i.e., $d\_i \ge 0, d\_i ... | 19 | https://mathoverflow.net/users/84349 | 286690 | 126,548 |
https://mathoverflow.net/questions/286689 | 5 | What's a name for a general technique I've seen used many times?
Given any family $\mathcal{F}$ of functions such that $f:X\to Y$ for all $f\in \mathcal{F}$ when one wishes to study in general for an arbitrary $y\in Y$ what (if any) $x\in X$ satisfy $f(x)=y$ it seems often the approach taken is to first find some per... | https://mathoverflow.net/users/38626 | Classifying functions up to suitable pre-composition and/or post-composition | The emphasis on finding solutions to equations $f(x) = y$ is a red herring, and note that in your examples your symmetries aren't acting pointwise on $Y$ as in your general explanation (sometimes they are acting on $X$), but that's not particularly important.
More generally, suppose you want to understand anything a... | 3 | https://mathoverflow.net/users/290 | 286700 | 126,551 |
https://mathoverflow.net/questions/286693 | 25 | The "traditional" approach to the so-called "field with one element" $\mathbb{F}\_{1}$ is by using monoids, or, to put it in another way, by forgetting the additive structure of rings. In Deitmar's approach to the subject, $\mathbb{F}\_{1}$ is declared to be the trivial monoid $\{1\}$. Furthermore, $\mathbb{F}\_{1}$-mo... | https://mathoverflow.net/users/85392 | How is Borger's approach to $\mathbb{F_{1}}$ related to previous approaches (e.g. Deitmar's)? | Given a monoid $M$, the ring $\mathbb{Z}.M$ of $\mathbb{Z}$-linear combinations of elements of $M$ has a natural $\lambda$-ring structure given by setting the elements $m \in M$ to have rank $1$ (i.e. $\lambda^k(m)=0$ for all $k>1$). We then have functors
$$
\text{Monoids} \xrightarrow{\mathbb{Z}.-} \lambda\text{-rings... | 25 | https://mathoverflow.net/users/103678 | 286702 | 126,552 |
https://mathoverflow.net/questions/194642 | 3 | Given a poset $(P,\leq)$ the *interval topology* on $P$ is generated by
$$\{P\setminus\downarrow x : x\in P\} \cup \{P\setminus\uparrow x : x\in P\},$$
where $\downarrow x = \{y\in P: y\leq x\}$ and $\uparrow x = \{y\in P: y\geq x\}$.
Let $\{P\_i : i\in I\}$ be a family of posets such that the interval topology of ea... | https://mathoverflow.net/users/8628 | Product of posets with Hausdorff interval topology | The statement in the answer by Dominic that the interval topology of the product poset is the product topology of the interval topologies is incorrect. The argument that the product topology contains the interval topology is correct, but the one for the opposite containment is not, as shown in this [MO answer](https://... | 5 | https://mathoverflow.net/users/2926 | 286704 | 126,553 |
https://mathoverflow.net/questions/286707 | 1 | Suppose $D$ is a first-order differential operator on a manifold $M$ and that the inverse $(D+t)^{-1}:H^0(M)\rightarrow H^1(M)$ exists for all $t > 0$, where $H^i(M)$ is the $i^\text{th}$ Sobolev space.
Let $\psi\in C\_c^\infty(M)\subseteq H^0(M)$. Then in particular $(D+t)^{-1}\psi$ is smooth. Also, for any $\phi\i... | https://mathoverflow.net/users/78729 | Continuity of image of resolvent operator with respect to resolvent parameter | Yes, $((D + t)^{-1}\psi)(x)$ is continuous provided there is a $p > dim(M)$ such that
$$
(D + t)^{-1} : H^p(M) \to H^{p+1}(M),
$$
exists and is continuous as a map from $\mathbf R$ to $B(H^p, H^{p+1})$. This is true if $D$ is a first order differential operator with smooth coefficients and the resolvent is not just c... | 1 | https://mathoverflow.net/users/99924 | 286714 | 126,557 |
https://mathoverflow.net/questions/286699 | 0 | It is well known that given a function $f \in L^p(B\_R)$ such that $|\{x \in B\_R: f(x) = 0\}|>0$, the following Poincaré inequality holds:
$$ \int\_{B\_R} \left(\frac{|f|}{R}\right)^p \ dx \leq c \int\_{B\_R} |\nabla f|^p \ dx \, .$$
My question is: does something like this hold on annular regions? More specifically... | https://mathoverflow.net/users/100801 | Poincaré inequality on annular regions | No. Consider the positive part of a coordinate function on a large but thin annulus.
| 2 | https://mathoverflow.net/users/37103 | 286715 | 126,558 |
https://mathoverflow.net/questions/286716 | 0 | For any finite, simple, undirected graph $G=(V,E)$ denote by $G\_2 = (V\_2, E\_2)$ the graph, in which $V\_2$ and $E\_2$ are defined as follows:
1. $V\_2 = \big(V\times\{1\}\big) \cup \big(V\times \{2\}\big)$, and
2. $E\_2 = \big\{\{(x,i),(y,i)\} : \{x,y\} \in E \textrm{ and }
i \in \{1,2\}\big \} \cup \big\{\{(v,1)... | https://mathoverflow.net/users/8628 | Complete minors in a "redoubled" graph | There are more involved counterexamples than the one in the comments (an empty graph on three vertices with $n = 2$), in fact there are examples for any $n \in \mathbb{N}$. Let $G = (V, E)$ be a graph on $n + 1$ vertices with $E = {[n] \choose 2} \backslash \{12\}$. Clearly, $G$ has no $K\_n$ minor. On the other hand, ... | 2 | https://mathoverflow.net/users/106512 | 286724 | 126,562 |
https://mathoverflow.net/questions/286706 | 2 | Here is the question; it may seem very simple, but it is difficult (at least for me).
Let $f(x)$ be a continuous function on $R$ that is strictly increasing, and suppose $g(x)=f(x)-x$ is a periodic function with period 1.
Prove that for all $x\in R$, $\lim\_{n\to \infty}\frac{f^n(x)}{n}$ exists.
In an equivalent ... | https://mathoverflow.net/users/91939 | Does this limit always exist? | **EDITED: below I give a counter example in the case where $f$ is not monotonic. The question you’re asking is well known to be true in the monotonic case**
There's a counterexample (even if $f$ is highly regular). I'll give a piecewise linear counterexample: let
$$
f(x)=\begin{cases}4x&\text{if $x\in [0,\frac 12)$;... | 5 | https://mathoverflow.net/users/11054 | 286727 | 126,563 |
https://mathoverflow.net/questions/286582 | 9 | It is well known that [there exists a $C^1$ isometric embedding of flat torus into $\mathbb{R}^3$](https://mathoverflow.net/questions/31222/c1-isometric-embedding-of-flat-torus-into-mathbbr3), and that this embedding cannot be $C^2$.
Is there a $W^{2,2}$ isometric embedding? (i.e an isometric map $f \in W^{2,2}(\math... | https://mathoverflow.net/users/46290 | Is there a $W^{2,2}$ isometric embedding of the flat torus into $\mathbb{R}^3$? | This is just an expansion on my comment to @j.c. 's answer.
We want to use Pakzad's result cited in that answer in order to prove that there is no $f$ as in the original question.
My idea:
By Sobolev embedding theorems we have that $f\in C^{0,\alpha}$ for all $\alpha\in (0,1)$. In particular, the image $\Sigma=f(\m... | 3 | https://mathoverflow.net/users/26801 | 286728 | 126,564 |
https://mathoverflow.net/questions/286688 | 3 | In the paper (<http://www.sciencedirect.com/science/article/pii/S0022247X97953439>), Webster obtained a unique solution of the functional equation $f(x+1)=g(x)f(x)$
(where $f,g:\mathbb{R}^+\rightarrow \mathbb{R}^+$) under some conditions one of which is $\lim\_\limits{x\to \infty}\frac{g(x+w)}{g(x)}=1$ for all $w>0$.
... | https://mathoverflow.net/users/40520 | Solution of the functional equation $f(x+1)=g(x)f(x)$ | Such a function $g$ does not exist; that is, under the given conditions, for each real $w>0$ necessarily $\frac{g(x+w)}{g(x)}\to1$ as $x\to\infty$. In my previous answer, I apparently misunderstood the question, and so, gave an answer to a different, but related question, which I think may be of independent interest. T... | 1 | https://mathoverflow.net/users/36721 | 286729 | 126,565 |
https://mathoverflow.net/questions/286465 | 2 | Let $E$ be a complex Hilbert space. For $S=(S\_1,S\_2)\in \mathcal{L}(E)^2$, $W(S)$ is defined as
$$W(S)=\{(\langle S\_1 z\;,\;z\rangle,\langle S\_2 z ,\;z\rangle)\,;\,z \in E,\;\;\|z\|=1\}.$$
The pair $S=(S\_1,S\_2)$ satisfy the property $(^\*)$ if:
$\forall\,\lambda\_1=(\langle S\_1 x\; ,\;x\rangle,\langle S\_2 x\... | https://mathoverflow.net/users/113054 | Question related to the Toeplitz-Hausdorff Theorem | Your question is basically if $W(S\_1,S\_2)$ is convex then is $W(PS\_1P, PS\_2P)$ convex over the space $PE$ where $P$ is a rank 2 projection. This is not true as demonstrated in the following example.
Fact: There exist operators $T\_1, T\_2$ on a separable Hilbert space such that $W(T\_1, T\_2) = \mathbb D^2$, the ... | 2 | https://mathoverflow.net/users/76593 | 286730 | 126,566 |
https://mathoverflow.net/questions/286717 | 9 | Short version: if $G$ is a Coxeter group and $H \subset G$ is a parabolic subgroup, both acting on a space $V$, is it true that the invariant-coinvariant algebra $(S(V)\_G)^H$ has a natural bilinear form induced by taking coefficients in the top component?
Now the long, detailed version. Let $G$ be a Coxeter group ac... | https://mathoverflow.net/users/17353 | A duality result for Coxeter groups | Yes. By Chevalley-Shepard-Todd, $S(V)^G$ and $S(V)^H$ are polynomial rings. Let $S(V)^G=\mathbb{R}[g\_1, \ldots, g\_n]$ and $S(V)^H = \mathbb{R}[h\_1,\ldots, h\_n]$ where the $g\_i$ and $h\_i$ are homogenous. Then
$$S(V)\_G^H = \mathbb{R}[h\_1,\ldots,h\_n]/\langle g\_1,\ldots, g\_n \rangle.$$
Here the denominator is th... | 10 | https://mathoverflow.net/users/297 | 286744 | 126,572 |
https://mathoverflow.net/questions/286437 | 6 | Let $S\_1,S\_2,\dots,S\_k$ be subsets of the set $S=\{1,2,\dots,n\}$, not necessarily distinct. We will color each element of $S$ red, green, or blue. From this coloring, each set $S\_i$ will receive one or more color according to the following rule:
Let $r\_i,g\_i,b\_i$ denote the number of red, green, and blue elem... | https://mathoverflow.net/users/92373 | Fraction of the sets receive each color | The statement seems to follow from [Sperner's lemma](https://en.wikipedia.org/wiki/Sperner%27s_lemma). We will give such a coloring that the first few elements are red, the middle ones are green and the last few are blue (for any ordering of the elements).
We can represent the colorings of the $n$ elements with a sub... | 4 | https://mathoverflow.net/users/955 | 286754 | 126,582 |
https://mathoverflow.net/questions/286777 | 0 | Just having some difficulties with this system of inequalities...
We know *E* is a system of **m** linear inequalities of the form:
a1,1x1+ ··· +a1,nxn ≤ b1
...
am,1x1+ ··· +am,nxn ≤ bm
And *E'* an equivalent system, derived from *E*:
a'1,2x2+ ··· +a'1,nxn ≤ b'1
...
a'm,2x2+ ··· +a'm,nxn ≤ b'm
I have... | https://mathoverflow.net/users/117567 | Equivalent linear inequalities system - Coefficients bound? | There is no such bound. Let's consider the simplest case: a single inequality in one variable:
$$a x \le b$$
For any $c > 0$ this is equivalent to
$$ c a x \le c b$$
But since $c$ is arbitrary, there is no way to bound numerator or denominator of $|c a|$ or $|c b|$.
| 1 | https://mathoverflow.net/users/13650 | 286779 | 126,597 |
https://mathoverflow.net/questions/286742 | 8 | I'm searching for a proof of Witt's result that a biquadratic extension $K(\sqrt{a},\sqrt{b})/K$ extends to a Galois extension $L/K$ with quaternion group $Q\_8$ iff the quadratic forms $<a,b,\frac{1}{ab}>, <1,1,1>$ are equivalent iff $(a,b)(a,a)(b,b) = 0 \in Br(K)$.
I know there is a proof in his original paper "Kon... | https://mathoverflow.net/users/70019 | Proof of Witt's result about quaternion extensions | A complete proof can be found in the first few pages of
<https://mathscinet.ams.org/mathscinet-getitem?mr=977759>
Jensen, Christian U.(DK-CPNH); Yui, Noriko(3-TRNT)
Quaternion extensions. Algebraic geometry and commutative algebra, Vol. I, 155–182, Kinokuniya, Tokyo, 1988.
I will scan the relevant pages and post... | 7 | https://mathoverflow.net/users/2821 | 286781 | 126,599 |
https://mathoverflow.net/questions/286762 | 16 | I'm curious how much of homological algebra carries over to a constructive setting, like say HoTT (or some other variety of intensional type theory) without AC or excluded middle. There doesn't seem to be a lot of literature on this topic (or at least it's difficult for an outsider like me to find).
The category of a... | https://mathoverflow.net/users/56938 | Constructive homological algebra in HoTT | As regards HoTT, my own current opinion is that the best way to do "homological algebra" therein is by working directly with [spectra](http://ncatlab.org/nlab/show/spectrum).
With only a working mathematician's knowledge of homological algebra you may not know what a spectrum is. If you know the [Dold-Kan theorem](ht... | 21 | https://mathoverflow.net/users/49 | 286786 | 126,600 |
https://mathoverflow.net/questions/286770 | 4 | Let $X\_{1},...,X\_{d} \in \{-1,1\}^d$ be random variables, with $E[X\_j]=\mu\_j$. Having $n$ i.i.d. samples $x^{(i)}\_1,x^{(i)}\_2,....,x^{(i)}\_d$, $i=1,...,n $, let $\hat{\mu}\_{j}=\frac{1}{n}\sum^{n}\_{i=1}x^{(i)}\_j$ Then we would like to find an upper bound for $\text{Pr}[|\prod^{d}\_{i=1}\hat{\mu}\_{j}-\prod^{d}... | https://mathoverflow.net/users/117055 | Product of estimates of mean values - Concentration of measure inequality | The simplest idea is to estimate the variance. One has
$$
\left(\prod^{d}\_{j=1}\hat{\mu}\_{j} \right)^2=\frac{1}{n^{2d}}\sum^{n}\_{i\_1,...,i\_{2d}=1}\prod^{d}\_{j=1}x^{(i\_j)}\_j x^{(i\_{j+d})}\_j
$$
When $i\_1,\dots,i\_{2d}$ are all distinct the inner term has expectation $\left(\prod^{d}\_{j=1}\mu\_{j} \right)^2$, ... | 1 | https://mathoverflow.net/users/21724 | 286787 | 126,601 |
https://mathoverflow.net/questions/286792 | 2 | Let $\text{Cont}(\mathbb{R},\mathbb{R})$ denote the set of continuous self-maps of $\mathbb{R}$ and let $\mathbb{R}^\mathbb{R}$ denote the set of all self-maps of $\mathbb{R}$, endowed with the product topology. Is $\text{Cont}(\mathbb{R},\mathbb{R})$ dense in $\mathbb{R}^\mathbb{R}$?
| https://mathoverflow.net/users/8628 | Is $\text{Cont}(\mathbb{R},\mathbb{R})$ dense in $\mathbb{R}^\mathbb{R}$? | Yes.
Let $g: \mathbb{R} \to \mathbb{R}$ be an abritrary function.
Let $\mathcal{F}$ denote the set of all finite subsets of $\mathbb{R}$. We endow $\mathcal{F}$ with the order $\subseteq$, which renders it a directed set.
For each $F \in \mathcal{F}$, choose a continuous function $f\_F$ which fulfils $f\_F(x) = g... | 10 | https://mathoverflow.net/users/102946 | 286794 | 126,605 |
https://mathoverflow.net/questions/286803 | 5 | *Disclaimer*: Feel free to downvote or vote to close, if this is again trivial (I seem to have a [bad day](https://mathoverflow.net/questions/286792/is-textcont-mathbbr-mathbbr-dense-in-mathbbr-mathbbr) today; I promise that if this is again a bummer question, I will wait $\geq 1$ day before asking new questions).
Fo... | https://mathoverflow.net/users/8628 | Connected $T_2$-space with $\text{Cont}(X,X)$ not dense in $X^X$ | Pick $X$ such that there are path-components $Y \neq Z$, and $y \in Y$, $z\in Z$ with the conditions: $y$ belongs to the interior of $Y$, $z$ belongs to the interior of $Z$, and $Y$ is not a singleton.
Choose $x\in Y\smallsetminus\{y\}$.
Then we can't approach with continuous maps a map mapping $x \mapsto z$ and $y \... | 6 | https://mathoverflow.net/users/14094 | 286805 | 126,610 |
https://mathoverflow.net/questions/286681 | 2 | In a paper of Bloch (Torsion algebraic cycles and a theorem of Roitman) he claims the following:
Let $X$ be a projective smooth variety over an algebraic closed field. Now, let $Y$ be a general linear space section of large degree and dimension $2$. Then $\text{Alb}(X)\cong \text{Alb}(Y)$.
Why is this true? Is ther... | https://mathoverflow.net/users/108963 | An isomorphism between the Albanese varieties of a variety and a general linear space section | I am expanding on my comments above. I looked at Bloch's paper. He includes a further hypothesis on $Y$, namely that it contains a reducible curve whose irreducible components are smooth and whose union has only ordinary double points (nodes) as singularities. That does not really affect the claim.
Let $X$ be a proje... | 5 | https://mathoverflow.net/users/13265 | 286829 | 126,623 |
https://mathoverflow.net/questions/282237 | 4 | Let $S\_0=0$ and $S\_n = \sum\_{k=1}^n Z\_k$ with i.i.d real valued random variables $(Z\_n)$ with $E[Z\_1]=0$ and $P[Z\_1 \geq 1]>0$. Let furthermore $0<\alpha<1/2$. I'm interested in lower bounds for the probabilities
$$ p\_n:= P[\forall i=1,\ldots,n: S\_i \geq i^\alpha] $$
for $n \rightarrow \infty$. One might t... | https://mathoverflow.net/users/115138 | Lower bound for small deviation probability of driftless random walk | You can find the asymptotics for this probability.
For that let
$$
\tau:=\inf\{i\ge 1: S\_i\le i^\alpha\}.
$$
Then, if $Var(S\_1)<\infty$ then the asymptotics for
$$
p\_n=\mathbf P(\tau>n)\sim \frac{C}{\sqrt n},\quad n\to \infty,
$$
see <https://arxiv.org/abs/1403.5918> for random walks with i.i.d. increments, and <... | 1 | https://mathoverflow.net/users/85303 | 286834 | 126,625 |
https://mathoverflow.net/questions/286514 | 6 | By *abstract construction of a combinatorial model category*, I mean starting from a locally presentable category satisfying some assumptions, e.g. equipped with a cylinder or a cocylinder satisfying some special hypothesis, and from these data build a model category structure. The question now is:
>
> What are th... | https://mathoverflow.net/users/24563 | Construction of combinatorial model categories with all objects fibrant | There are a ton of papers about what you are asking. Another is the thesis of Richard Williamson (arXiv:1304.0867v1). Also, Valery Isaev has a paper that produces a model structure with all objects fibrant, given some cylinder or path object information (<https://arxiv.org/pdf/1312.4327.pdf>). The thesis of Remy Tuyere... | 2 | https://mathoverflow.net/users/11540 | 286837 | 126,627 |
https://mathoverflow.net/questions/286828 | 14 | For which morphisms of schemes $f : X\rightarrow Y$ do we have $f^{-1}\mathcal{O}\_Y = \mathcal{O}\_X$? (note that I don't mean $f^\*$, I really just mean the basic inverse image sheaf as a sheaf of rings/abelian groups/sets)
This is obviously true if $f$ is an open immersion, and intuitively I feel like this is the ... | https://mathoverflow.net/users/88840 | When is the inverse image of the structure sheaf the structure sheaf? | Here is a trivial lemma.
>
> **Lemma.** Let $f \colon X \to Y$ be a morphism of schemes. Then the natural map $f^{-1}\mathcal O\_Y \to \mathcal O\_X$ is an isomorphism if and only if for each $x \in X$ the natural map $\mathcal O\_{Y,f(x)} \to \mathcal O\_{X,x}$ is an isomorphism.
>
>
>
*Proof.* A morphism of ... | 11 | https://mathoverflow.net/users/82179 | 286838 | 126,628 |
https://mathoverflow.net/questions/286581 | 1 | **Definition:** A filtered space $X$ of formal dimension $n$ is *locally cone-like* if for all $i$, $0 \le i \le n$, and for each $x \in X^i - X^{i-1} = X\_i$ there is an open neighborhood $U$ of $x$ in $X\_i$, a neighborhood $N$ of $x$ in $X$, a compact filtered space $L$, and a homeomorphism $h:U \times cL\rightarrow... | https://mathoverflow.net/users/115764 | Confusion about locally cone-like spaces | Just to confirm your self-answer in the comments: That's right. In this case if you're looking for a neighborhood of $v$ in $X\_0=X^0=\{v,w\}$, then $U=\{v\}$ does it and the cone neighborhood $N$ in $\Sigma S^1$ is $\{v\}\times cS^1$.
| 1 | https://mathoverflow.net/users/6646 | 286843 | 126,630 |
https://mathoverflow.net/questions/286804 | 36 | This post is a sequel to: [Collaboration or acknowledgment?](https://mathoverflow.net/questions/191507/collaboration-or-acknowledgment)
The following has come to my attention. A senior mathematician (let us call him or her Alice) suggested a problem to a young mathematician (Bobby) who proceeded to solve it on her o... | https://mathoverflow.net/users/111456 | Dealing with unwanted co-authorship requests | Well, of course the young mathematician should simply discuss the
matter with the senior mathematician and perhaps the student until
they can come to an agreeable arrangement. My advice is that they
should all talk about it. Co-authorship is a matter upon which all authors must agree. What other answer could there be?
... | 35 | https://mathoverflow.net/users/1946 | 286844 | 126,631 |
https://mathoverflow.net/questions/286743 | 14 | Let $X$ be a smooth projective variety of Picard number one, and let $f:X\dashrightarrow X$ be a birational automorphism which is not an automorphism.
Must $f$ necessarily contract a divisor?
| https://mathoverflow.net/users/nan | Birational automorphisms of varieties of Picard number one | In contrast to an earlier answer on this question, I claim the answer here is **yes**.
In the accepted answer to this question:
[Pseudo-automorphisms on Fano varieties](https://mathoverflow.net/questions/179070/pseudo-automorphisms-on-fano-varieties?rq=1)
abx explains that for any smooth variety $X$ of Picard num... | 3 | https://mathoverflow.net/users/114758 | 286868 | 126,639 |
https://mathoverflow.net/questions/286865 | 11 | I'm looking at the proof of Higher Algebra Proposition 6.1.6.27, and in the very first sentence of the proof, Lurie states:
>
> The functor $(F\delta)\_{\Sigma\_n}$ is n-homogeneous by Proposition 6.1.5.4.
>
>
>
Checking back to 6.1.5.4, I see the statement
>
> Let $C$ be a small $\infty$-category which ... | https://mathoverflow.net/users/1353 | Where to find the correct result in Higher Algebra, incorrect reference | The correct reference is 6.1.4.14. (And the hypothesis of 6.1.6.27 should refer to countable limits and colimits, rather than finite limits and colimits.)
| 18 | https://mathoverflow.net/users/7721 | 286869 | 126,640 |
https://mathoverflow.net/questions/286612 | 5 | This question has been completely reformulated and a new property for the function $f\_q$ has been added due to a series of helpful comments by fedja.
Consider the integral from quantum field theory due to F.A. Smirnov (see this [MSE post](https://math.stackexchange.com/questions/2486031/prove-int-0-infty-big-gamma-... | https://mathoverflow.net/users/82588 | $q$-analog of an integral from quantum field theory? | The function
$$
f\_q(x,y,z)=\sum\_{cyc}e^z\frac{\theta\_q\left(e^{\frac{2 \pi i}{3}+x-z}\right) \theta\_q\left(e^{\frac{2 \pi i}{3}+y-z}\right)}{\theta\_q\left(e^{x-z}\right) \theta\_q\left(e^{y-z}\right)}
$$
satisfies all $4$ conditions and also has been confirmed numerically.
| 5 | https://mathoverflow.net/users/82588 | 286877 | 126,641 |
https://mathoverflow.net/questions/286791 | 2 | I've found myself looking at a structure $\mathbb{M}$ whose important properties are:
1. $\mathbb{M}$ is a discretely ordered additive monoid.
2. $\mathbb{M}$ has a least element, and this least element is the additive identity $0\in\mathbb{M}$.
3. All elements $m\in\mathbb{M}$ have unique additive decompositions onc... | https://mathoverflow.net/users/92164 | Name for this algebraic structure? | As Emil Jeřábek correctly suggests in the comments above, $\mathbb{M}$ is exactly the non-negative part of a discretely ordered group -- I humbly offer a proof to close the thread. The proof that the non-negative part of a discretely ordered group satisfies the above requirements is trivial.
For the opposite directio... | 3 | https://mathoverflow.net/users/92164 | 286885 | 126,643 |
https://mathoverflow.net/questions/286832 | 5 | Take $G$ a split reductive group (over a field of char 0) with Borel $B$, opposite Borel $\overline{B}$ and maximal split torus $T\subset B$. We write $X = G/\overline{B}$. Let $O \subset X$ be the big cell of $X$ for the Bruhat decomposition, it is a a dense open subset of $X$ which is isomorphic to $U$ the unipotent ... | https://mathoverflow.net/users/113062 | BGG resolution for characters of reductive groups | Modulo the messy bookkeeping sometimes encountered when passing to dual modules, what Kempf does in his paper is to exploit the geometric setting of Cousin complexes in order to find an independent approach to the BGG resolution. Though this is mostly limited to working over a sufficiently large field of characteristic... | 3 | https://mathoverflow.net/users/4231 | 286891 | 126,644 |
https://mathoverflow.net/questions/286872 | 22 | When Paul Gordan became a professor in 1875 he could show the binary form in any degree has some finite complete system of (general linear) invariants, but he could not actually give a complete system above degree 6. He discussed this limitation that year in *Uber das Formensystem binaerer Formen* (B.G. Tuebner, Leipzi... | https://mathoverflow.net/users/38783 | What is currently feasible in invariant theory for binary forms? | Let $F$ be a binary form of degree $d$, namely, a homogeneous polynomial of the form
$$
F(\mathbb{x})=\sum\_{i=0}^{d}\left(\begin{array}{c}d\\ i \end{array}\right)f\_i\ x\_1^{d-i}x\_2^i
$$
where $\mathbb{x}$ denotes the pair of variables $(x\_1,x\_2)$.
For $g=(g\_{ij})\_{1\le i,j\le 2}$ in $GL\_2$, define the correspon... | 29 | https://mathoverflow.net/users/7410 | 286892 | 126,645 |
https://mathoverflow.net/questions/286874 | 57 | It was an ambitious project of [Vladimir Voevodsky](https://en.wikipedia.org/wiki/Vladimir_Voevodsky)'s to provide new foundations for mathematics with [univalent foundations (UF)](https://en.wikipedia.org/wiki/Univalent_foundations) to eventually replace set theory (ST).
Part of what makes ST so appealing is its inc... | https://mathoverflow.net/users/8628 | In what respect are univalent foundations "better" than set theory? | I like your analogy with programming languages. If we think of ST as a low-level programming language and UF as a high-level one, then one advantage of UF is obvious: it is more convenient to write proofs (programs) in a high-level language. It is feasible to write proofs in UF, but it's virtually impossible to write d... | 47 | https://mathoverflow.net/users/62782 | 286895 | 126,646 |
https://mathoverflow.net/questions/286852 | 2 | I am considering the following question related to the randomness of Mobius function $\mu(n)$:
$\mu(n)$ is defined as :
$\mu(1)=1$,
$\mu(p\_1...p\_t)=(-1)^t$, $\forall t\in N^\*$ $p\_1,...,p\_t$ are different primes,
$\mu(n)=0$ if $\exists p$ is a prime, $p^2|n$.
My question is, given $k\in N^\*, (a\_1,...,a\... | https://mathoverflow.net/users/91939 | Is there some estimate numbers of the tuples come from Mobius function? | Studying the distribution of patterns of the Moebius function falls into an easy part, which deals with the distribution of zeroes, and a difficult part, which deals with the distribution of signs. Therefore it is more natural to separate these problems and ask for patterns of Liouville's $\lambda$-function. Here our k... | 2 | https://mathoverflow.net/users/37555 | 286903 | 126,651 |
https://mathoverflow.net/questions/286894 | 8 | Let $c\_1,c\_2,c\_3,c\_4,c\_5$ be a five chain of circles on a genus 2 surface (i.e $i(c\_k,c\_{k+1})=1$ and zero otherwise). Then $(T\_{c\_1} T\_{c\_2})^6 = (T\_{c\_4} T\_{c\_5})^6 = T\_c$ where $c$ is a separating curve that bounds a neighborhood of $c\_1$ and $c\_2$ (and also bounds a neighborhood of $c\_4$ and $c\_... | https://mathoverflow.net/users/101463 | Well definedness of square roots of separating Dehn Twists | They are different. In fact, they act differently on $H\_1(\Sigma\_2;\mathbb{Z})$. Let $V \subset H\_1(\Sigma\_2;\mathbb{Z})$ be the span of the homology classes of $c\_1$ and $c\_2$, and let $W \subset H\_1(\Sigma\_2;\mathbb{Z})$ be the span of the homology classes of $c\_4$ and $c\_5$. You can then calculate the $(T\... | 8 | https://mathoverflow.net/users/317 | 286905 | 126,653 |
https://mathoverflow.net/questions/286910 | 0 |
>
> Recall that a space $X$ is metaLindelof if every open cover of
> $X$ has a point-countable open refinement.
> A space $X$ is metacompact if every open cover of
> $X$ has a point-finite open refinement.
>
>
>
I have the following two questions, because the questons are similar, I put together here:
Is th... | https://mathoverflow.net/users/39873 | Is there a $\sigma$-metacompact space which is not metacompact? | In fact, there is a space which is the union of two paracompact spaces which is not metaLindelof. Let $X$ be a $\psi$-space, that is, a locally compact, pseudocompact space having a countable dense set of isolated points and such that the set of non-isolated points is uncountable, closed, and discrete. If $A$ is the se... | 3 | https://mathoverflow.net/users/89233 | 286914 | 126,657 |
https://mathoverflow.net/questions/286916 | 3 | Let $p$ be a prime. Let $K/\mathbb{Q}\_p$ be a finite extension and $\mathcal{O}=\mathcal{O}\_K$ its ring of integers.
Let $A\_K$ be an abelian variety and $\mathcal{A}\_\mathcal{O}$ denote its Neron model over $\mathcal{O}$. We don't assume that $\mathcal{A}\_\mathcal{O}$ has good or semistable reduction.
Let $\el... | https://mathoverflow.net/users/46108 | Finite subgroup scheme and Neron model of an abelian variety | In general the scheme $\mathcal A[l]\_{\mathcal O}$ is not finite because of the following lemma.
>
> Let $f:X\to Y$ be a separated quasi-finite flat morphism of noetherian schemes. Then it is finite iff the fibral rank is locally constant.
>
>
>
For a proof see the paper "Les schémas de modules des courbes el... | 8 | https://mathoverflow.net/users/115211 | 286917 | 126,659 |
https://mathoverflow.net/questions/283389 | 16 | **(Edited 10/17/17)**: With the hope of obtaining informed responses on the following intriguing remark of Marta Bunge on the status of Synthetic Differential Geometry, I have added a third question to the original two and expanded Bunge's quote to provide further context.
In her ["A Personal tribute to Bill Lawvere"... | https://mathoverflow.net/users/18939 | Query about SDG (Synthetic Differential Geometry) | In a paper by Marta Bunge and Eduardo Dubuc. "Local concepts in SDG and germ representability" (1987) certain axioms were laid down towards a synthetic theory of differential topology based on logical infinitesimal notions given by Jacques Penon in his 1985 Universite Paris VII thesis.
One of them was Postulate WAII... | 12 | https://mathoverflow.net/users/69712 | 286918 | 126,660 |
https://mathoverflow.net/questions/286876 | 3 | QUESTION EDITED: There was a mistake, the spectrum i had written before didn't even exist, so a big thanks to the people who made me notice that in the comments.
Let $X\_n$ be the spectrum such that $BP\_\*(X\_n) = \Sigma^{d\_n}BP\_\*/(v\_0^{p^{i\_0}},v\_1^{p^{i\_1}},\dots,v\_{n-1}^{p^{i\_{n-i}}})$. Recall that it's ... | https://mathoverflow.net/users/93775 | Studying the limit of a sequence of spectra knowing their BP-Homology | There are no spectra with the indicated $BP$-homology. The $BP$-homology of a spectrum is always a $BP\_\*BP$-comodule, and $BP\_\*/(v\_0^i,v\_1^j)$ only admits a comodule structure if $\eta\_R(v\_1)^j=v\_1^j\pmod{v\_0^i}$. Here $\eta\_R(v\_1)$ can be calculated from the relation
$$ \sum^F\_{i,j}t\_i\eta\_R(v\_j)^{p^i}... | 8 | https://mathoverflow.net/users/10366 | 286930 | 126,665 |
https://mathoverflow.net/questions/286882 | 3 | Let be $M$ a $n-$dimensional manifold. A distribution $D$ on $M$ is an assignment of subspace $D\_m \subset T\_mM$, for all $m\in M$.
A distribution $D$ on $M$ is said to be locally constant if for every $m\in M$ there is an open neighbourhood $U$ of $m$ such that $dim (D\_u)=k$ for all $u\in U$.
Is there any theor... | https://mathoverflow.net/users/110123 | Sufficient condition for a distribution to be locally constant | Here is a counterexample to such a theorem. Choose a point $x \in M$, and take $D\_x = \{0\}$ and $D\_m = T\_m M$ for $m \neq x$. Then $D$ is involutive (in the sense that for any vector fields $X, Y$ in $D$, the bracket $[X,Y]$ is also in $D$), but not locally constant. Check it locally:
$$[\sum\_i f\_i \partial\_i,\s... | 2 | https://mathoverflow.net/users/48261 | 286951 | 126,674 |
https://mathoverflow.net/questions/286963 | 4 | Suppose that $A$ is a reduced finitely generated algebra over a field and $\mathfrak{m}\subset A$ is a maximal ideal. Is it true that the localization $A\_{\mathfrak{m}}$ is analytically unramified, i.e. the completion
$$
\widehat{A\_{\mathfrak{m}}} = \lim\limits\_{\infty\leftarrow n}A\_{\mathfrak{m}}/(\mathfrak{m}A\_{... | https://mathoverflow.net/users/41487 | Is a localization of a reduced finitely generated algebra analytically unramified? | Let me expand my comment as an answer. There is a notion of excellent rings, for a precise definition look here <https://stacks.math.columbia.edu/tag/07QS> (and see Chapter 13 of Matsumura's book "Commutative Algebra" for a self-contained systematic development). We will need only one important feature of excellent rin... | 5 | https://mathoverflow.net/users/115211 | 286966 | 126,679 |
https://mathoverflow.net/questions/286734 | 10 | Let $k$ be an algebraically closed field of characteristic zero.
Let $A$ be the $\mathbf{Z}$-subalgebra of the Grothendieck ring of $k$-varieties $K\_0(\text{Var}\_k)$ generated by classes of semi-abelian varieties.
**Question 1.** What can we say about the ring homomorphism $A\to K\_0(\text{Var}\_k)$?
It is obvi... | https://mathoverflow.net/users/nan | $K_0$-equivalence of varieties | (Expanding my comments into an answer for more visibility.)
By [Larsen-Lunts](https://arxiv.org/abs/math/0110255v1) $K\_0(\operatorname{Var}\_k)/[\mathbb{A}^1]$ is the free abelian group on stable birational equivalence classes. It thus suffices to find a variety $X$ that is not stably birational to any variety in $\... | 6 | https://mathoverflow.net/users/51424 | 286968 | 126,680 |
https://mathoverflow.net/questions/286970 | 92 | I don't know if MO is the right place to ask such a question, but anyway it's my only hope to get an answer, and it's very important for me (not to say 'vital'); so let's try.
I'm at this time a Ph.D. student, and I plan to defend in the spring of 2018. I'm currently looking for a postdoc position for next year. I am... | https://mathoverflow.net/users/117681 | Coming out as transgender in the mathematical community | [Spectra](http://lgbtmath.org/) is an organization for LGBT mathematicians. I hope that you can find people to safely discuss your questions with there.
| 94 | https://mathoverflow.net/users/360 | 286973 | 126,682 |
https://mathoverflow.net/questions/286946 | 5 | Let $(X\_1,X\_2,\ldots)$ be a stationary, mixing sequence of real random variables. Then it holds (for example) for any event $A$ that is measurable in $\sigma(X\_1,X\_2,\ldots)$ and any $S \subseteq \mathbb{R}$ that
$$
\lim\_{i \to \infty} \big|\mathbb{P}[A,X\_i\in S] - \mathbb{P}[A] \cdot \mathbb{P}[X\_i \in S]\big|=... | https://mathoverflow.net/users/23661 | On a finitary version of mixing | Let $J\_n$ be a random variable independent of $X:=(X\_1,X\_2,\ldots)$ and uniformly distributed in the set $\{1,\ldots,n\}$. For each $i\in\{1,\ldots,n\}$, let
\begin{equation}
p(i):=P(A,X\_i\in S).
\end{equation}
We need to show that
\begin{equation}
p(J\_n)\to P(A)P(X\_1\in S)
\end{equation}
in probability unif... | 2 | https://mathoverflow.net/users/36721 | 286974 | 126,683 |
https://mathoverflow.net/questions/286085 | 10 | If there exists a Jónsson cardinal $\kappa$, then $x^\#$ exists for every $x\in V\_\kappa$ (in particular $V\neq L[x]$). It follows that if there is a proper class of Jónsson cardinals, then the sharp of every set should exist (this happens if for example if there is a proper class of Ramsey or measurable cardinals).
... | https://mathoverflow.net/users/78441 | Consistency of "the sharp of every set exists" | In one sense, closure under sharps is itself a standard point in the hierarchy of consistency strengths. Just like the exact consistency strength of "ZFC + measurable" is "ZFC + measurable", so is the case for closure under sharps. However, the value of the consistency strength hierarchy is in terms of the connections ... | 6 | https://mathoverflow.net/users/113213 | 286978 | 126,686 |
https://mathoverflow.net/questions/286983 | 5 | My question pertains to this paper by Terence Tao and Van Vu, <https://arxiv.org/abs/math/0703307>
Both my questions pertain to the argument presented in this paper in its section 6 (page 5). We are looking at a $n-$dimensional square random matrix $M\_n$ satisfying the conditions stated through Definitions 2.15, Def... | https://mathoverflow.net/users/89451 | A question about the paper "The Condition Number of a Randomly Perturbed Matrix" | 1. One does not need to have $n^{-B-3/2}/2$ to be equal to $0.1$, it is enough for it to be less than or equal to $0.1$, which is certainly the case for $n$ large enough.
2. Thanks for pointing out this typo (or more precisely, set of typos) in this paper. As you point out, the exponents here are adapted to the case of... | 9 | https://mathoverflow.net/users/766 | 286986 | 126,689 |
https://mathoverflow.net/questions/286226 | 6 | What is the branching rule for the subgroup $SU(p,q-1)\subset SU(p,q)$, i.e., the structure of the restriction of irreducible, finite-dimensional representations of $SU(p,q)$ to $SU(p,q-1)$? I would appreciate any reference and comments.
| https://mathoverflow.net/users/14181 | Branching rules for $SU(p,q)$ | As was indicated above your question is the equivalent to the branching rules for $sl\_{n-1} \to sl\_n$. This is a well-known branching rule and it is given by the following formula: If the representation of $sl\_n$ is given by the highest weight $(\lambda\_1 \geq \cdots \geq\lambda\_n)$ then it decomposes upon restric... | 6 | https://mathoverflow.net/users/37808 | 287001 | 126,695 |
https://mathoverflow.net/questions/287011 | 59 | For $n,m \geq 3$, define $ P\_n = \{ p : p$ is a prime such that $ p\leq n$ and $ p \nmid n \}$ .
For example :
$P\_3= \{ 2 \}$
$P\_4= \{ 3 \}$
$P\_5= \{ 2, 3 \}$,
$P\_6= \{ 5 \}$ and so on.
Claim: $P\_n \neq P\_m$ for $m\neq n$.
While working on prime numbers I formulated this problem and it has eluded me for a... | https://mathoverflow.net/users/117699 | A conjecture regarding prime numbers | This is true (for large $m$ and $n$) under ABC plus the assumption that there is a prime in $[x,x+x^{1/2-\delta}]$ for some positive $\delta$ (which is widely believed, but beyond RH). To see this, suppose $m <n$
and that they have the same radical $r$. Write $m=gM$ and $n=gN$ where $g$ is the gcd of $m$ and $n$, so t... | 42 | https://mathoverflow.net/users/38624 | 287017 | 126,698 |
https://mathoverflow.net/questions/286989 | 7 | There is a very simple formulation for the character of irreducible representations of $S\_n$ evaluated on an n-cycle, i.e. that it is 0 on all non-hook partitions, and $(-1)^m$ on hooks. Is there an analogous computation for irreducible characters of $B\_n$, the hyperoctahedral group, evaluated on signed 2n-cycles? Th... | https://mathoverflow.net/users/111128 | Evaluation of irreducible representations of the hyperoctahedral group at bipartition $(\lambda,\mu)=([n],\emptyset)$ | In general, if $(\lambda,\mu)$ is a bipartition of $n$, then
$$ \prod\_i(p\_{\lambda\_i}(x)+p\_{\lambda\_i}(y))\cdot\prod\_j
(p\_{\mu\_j}(x)-p\_{\mu\_j}(y)) = \sum\_{(\alpha,\beta)}
\chi^{\alpha,\beta}(\lambda,\mu)s\_\alpha(x)s\_\beta(y), $$
where $(\alpha,\beta)$ ranges over all bipartitions of $n$ and $\chi^{\alph... | 4 | https://mathoverflow.net/users/2807 | 287018 | 126,699 |
https://mathoverflow.net/questions/287020 | 13 | A [finite topological space](https://en.wikipedia.org/wiki/Finite_topological_space) is a finite family of finite sets that is closed under both union and intersection.
[Frankl's conjecture](https://en.wikipedia.org/wiki/Union-closed_sets_conjecture) states that for any finite union-closed family of finite sets, oth... | https://mathoverflow.net/users/7089 | Frankl's conjecture restricted to finite topological spaces | Consider the smallest nonempty set $S$ in our family $\mathcal F$ and pick any $s\in S$. Let $\mathcal F\_0$ be the subfamily of sets not containing $s$ (including $\varnothing$) and $\mathcal F\_1$ the subfamily of sets containing $s$.
If $s\not\in A\in\mathcal F$, then $S\cap A$ is a smaller element of $\mathcal F$... | 22 | https://mathoverflow.net/users/30186 | 287021 | 126,700 |
https://mathoverflow.net/questions/184286 | 16 | Every $f\colon\{-1,1\}^n\to \mathbb{R}$ can be repsenented as a multilinean polynomial of the form $$f(x\_1,x\_2,\ldots ,x\_n)=\sum \_{S\subseteq [n]} \hat{f}(S)\prod\_{i\in S} x\_i $$ The *degree* of the function is defined to be $\max \{|S|\,:\,\hat{f}(S)\neq0\}$.
Give $\{-1,1\}^n$ the uniform probability measure. ... | https://mathoverflow.net/users/38136 | What is the minimal $C_k$, such that every $f\colon \{-1,1\}^n\to \mathbb{R}$ of degree at most $k$ satisfies $\|f\|_2\le C_k\|f\|_1$ | When $f : \{-1,1\}^{n} \to \mathbb{C}$ is Walsh--Rademacher chaoes of degree $k$, i.e.,
$$
f(x) = \sum\_{1\leq j\_{1}<\ldots<j\_{k}\leq n} a\_{j\_{1}\ldots j\_{k}}x\_{j\_{1}}\cdots x\_{j\_{k}} \quad (\*)
$$
where $x = (x\_{1}, \ldots, x\_{n}) \in \{-1,1\}^{n}$ then one can improve by square root the bound in [Theorem ... | 8 | https://mathoverflow.net/users/50901 | 287030 | 126,703 |
https://mathoverflow.net/questions/286801 | 4 | It is known that every planar graph $G$ can be decomposed into at most 5 spanning star forests, which means that there exists at most five edge-disjoint spanning subgraphs of $G$ each of which is a forest with connected components being stars.
My question is as follows.
Does there exist a constant $k$ such that every... | https://mathoverflow.net/users/83519 | Decomposing planar graph into star forests with non-intersecting centers | There is such a constant, namely $k=5$.
The proof that there is a decomposition into five star forests in [1] is based on an acyclic $5$-colouring of the vertex set. A vertex $v$ cannot be the centre of a non-trivial star in the forest $F\_i$ unless it receives colour $i$ in the colouring. In particular, no vertex w... | 2 | https://mathoverflow.net/users/97426 | 287037 | 126,707 |
https://mathoverflow.net/questions/287012 | 3 | For $\frac{1}{4}<a<1$ consider the following function:
$$f(x)=\frac{|x|^{\frac{1}{2}}}{(x^2+1)^{a+ib}}$$
If $1>a>\frac{1}{2}$ then $f(x) \in L^2$ and the Fourier inversion theorem can be applied ($\mathcal{F^{-1}}$ is the inverse Fourier transform):
$$\mathcal{F^{-1}} \circ \mathcal{F} (f)=f$$
But if $\frac{1}{... | https://mathoverflow.net/users/38290 | Fourier transform inversion theorem for a function not in L1 or L2 | You can define the **distributionial Fourier transform** of a tempered distribution using all the abstract machinery established by Schwartz, and the thing you want to check is that it agrees with the **integral Fourier transform** defined by the usual integral.
For the first Fourier transform (from $x$ to $\xi$), yo... | 4 | https://mathoverflow.net/users/37103 | 287048 | 126,711 |
https://mathoverflow.net/questions/287043 | 9 | Consider the problem of finding the limit of the following diagram:
$$ \require{AMScd} \begin{CD}
& & & & E
\\ & & & & @VVV
\\ && C @>>> D
\\ & & @VVV
\\A @>>> B
\end{CD} $$
The abstract definition of the limit involves an adjunction related to collapsing the entire index category to a point. However, one could bre... | https://mathoverflow.net/users/nan | Calculating limits progressively | Let me try to turn my comments into an answer (I think it's also essentially what Vladimir was saying). Suppose you have some diagram $F: K \to \mathcal{C}$. To compute the limit of $F$ is the same as computing the right Kan extension $\epsilon\_\*F$ along the map $\epsilon: K \to \bullet$. The process you're describin... | 13 | https://mathoverflow.net/users/6936 | 287049 | 126,712 |
https://mathoverflow.net/questions/286992 | 4 | Let $\mathsf{T}\_1=(T\_1,\eta\_1,\mu\_1)$ and $\mathsf{T}\_2=(T\_2,\delta\_2,\mu\_2)$ be monads on a category $\mathcal{C}$. We say that an isomorphism $\delta\colon \mathsf{T}\_1\to \mathsf{T}\_2$ of monads is a natural isomorphism $\delta\colon T\_1\to T\_2$ satisfying $\mu\_2\circ \delta^2 = \delta \circ \mu\_1$ and... | https://mathoverflow.net/users/47345 | Isomorphisms of Kleisli categories | The more general result is that if $C$ is an object of a 2-category $K$ that admits Eilenberg-Moore objects, then the induced functor $\mathrm{EM} : \mathrm{Monads}(C) \to (K/C)^{\mathrm{op}}$ is fully faithful. This follows from Theorem 6 of Ross Street's *The formal theory of monads*, which says that this functor in ... | 7 | https://mathoverflow.net/users/49 | 287059 | 126,715 |
https://mathoverflow.net/questions/286886 | 5 | In Higher Algebra Lemma 6.1.6.3, most of the proof is pretty straightforward, but after thinking I understood it all correctly, I realized I had a gap in my understanding.
Suppose we have a homotopy cartesian square of $\infty$-categories (the Lemma has these as Kan complexes, but I think this is irrelevant to my que... | https://mathoverflow.net/users/1353 | Pointwise evaluation of the Beck-Chevalley map in $\infty$-categories | The Beck-Chevalley transformation $g^\*\_Y f\_\* \rightarrow f'\_\* g^\*\_X$ from a square of $\infty$-groupoids as above is an equivalence iff it's an equivalence when evaluated at every point of $Y'$, i.e. iff the transformation $p^\*g^\*\_Y f\_\* \rightarrow p^\*f'\_\* g^\*\_X$ is an equivalence for all maps $p : \*... | 2 | https://mathoverflow.net/users/1100 | 287060 | 126,716 |
https://mathoverflow.net/questions/287062 | 0 | A *hypergraph* is a pair $H=(V,E)$ where $V$ is a nonempty set, and $E\subseteq {\cal P}(V)\setminus\{\emptyset\}$ is a collection of non-empty subsets of $V$.
**Strong colorings.** If $\kappa$ is a cardinal and $H=(V,E)$ is a hypergraph, we call a map $c:V\to \kappa$ a strong coloring if for all $e\in E$ the restric... | https://mathoverflow.net/users/8628 | A weak version of the Erdös-Faber-Lovasz conjecture | Yes, and we do not even use the restrictions on mutual intersections of edges. For weak colorings, we may replace each edge with at least 2 vertices to an edge with exactly 2 vertices (possibly we get the same edge several times). It remains to properly color a graph with $n\geqslant 2$ edges with $n$ colors. This is d... | 4 | https://mathoverflow.net/users/4312 | 287063 | 126,718 |
https://mathoverflow.net/questions/287058 | 17 | There are already lots of questions on this subject like
[Is there an introduction to probability theory from a structuralist/categorical perspective?](https://mathoverflow.net/questions/20740/is-there-an-introduction-to-probability-theory-from-a-structuralist-categorical)
[Is there a combinatorial/topological tre... | https://mathoverflow.net/users/94970 | Good introduction to statistics from a algebraic point of view? | Lucien Le Cam developed an approach to statistics that largely disposes of measure-theoretic probability and replaced probability measures and random variables with certain Banach lattices. The approach can be found in Le Cam's book [Asymptotic Methods in Statistical Decision Theory](http://www.springer.com/de/book/978... | 12 | https://mathoverflow.net/users/35357 | 287064 | 126,719 |
https://mathoverflow.net/questions/232188 | 9 | I am interested in having an upper bound for the cardinality of
$\#\left\{n\leq x\,:\quad\omega(n)=k, \omega(n+2)=\ell\right\}$
for $k,\ell\geq 1$,
where $\omega(n)=\sum\_{p\vert n}1$ counts the number of (distinct) prime factors of the integer $n$.
I would actually like a sharp upper bound (up to a constant) ... | https://mathoverflow.net/users/88188 | Independence between the number of prime factors of $n$ and $n+2$ | The problem as been solved and [generalized](https://arxiv.org/pdf/1710.04877.pdf). Thread can be closed.
| 3 | https://mathoverflow.net/users/88188 | 287084 | 126,725 |
https://mathoverflow.net/questions/287065 | 2 | Consider the one dimensional Schrödinger hamiltonian $\mathcal{H}=-\frac{\hbar^2}{2} \frac{d^2}{dx^2} + V(x)$.
Suppose that $V:\mathbb{R} \rightarrow \mathbb{R}^+$ is a continuous and confining potential $\displaystyle \lim\_{\lvert x\rvert \to +\infty} V(x)=+\infty$
It is well known that $\mathcal{H}$ has pure dis... | https://mathoverflow.net/users/109757 | Pseudo-polynomial potentials for Schrödinger operators | The answer is yes. Suppose your $V$ equals to a polynomial $P$ when $|x|$ is large. Then there is a constant $c$ such that we have
$P-c<V<P+c$, which implies that $\lambda\_k^\prime-c<\lambda\_k<\lambda^\prime\_k+c$,
where $\lambda\_k^\prime$ are the eigenvalues of the potential $P$.
So $\lambda\_k$ and $\lambda\_k^\pr... | 1 | https://mathoverflow.net/users/25510 | 287093 | 126,729 |
https://mathoverflow.net/questions/287085 | 1 | Let $G$ be a (connected) graph with $n$ vertices. Is it true that the maximum cardinality of a minimal vertex cover of $G$ is $\geq \lfloor \frac{n}{2} \rfloor$? If so, can you point out any reference? If not, what is an easy counterexample?
Thanks!
| https://mathoverflow.net/users/103941 | Bound on the largest minimal vertex cover in a graph | Take a complete graph $K\_d$. For every its vertex $u$, take $d$ more vertices connected just to $u$. We get $d(d+1)$ vertices in total.
Every minimal vertex cover contains all but one vertices of $K\_d$. Either it contain all of them (and then it contains just $d$ vertices), or it does not contain some $u$ --- then ... | 3 | https://mathoverflow.net/users/17581 | 287094 | 126,730 |
https://mathoverflow.net/questions/287080 | 1 | Let $D$ denote the complex unit disk and $X \subset \mathbb{C}$ some subset. Let us consider a holomorphic motion $i \colon D \times X \rightarrow \mathbb{C}$ (denoted $i\_\lambda(z)$) meaning for each fixed $\lambda \in D$ the map $i\_\lambda(\cdot)$ is injective, for each fixed $z \in X$ the map $i\_{(\cdot)}(z)$ is ... | https://mathoverflow.net/users/117619 | Hölder continuity of holomorphic motions | The answer is given in @Misha's comment. This is an extended comment. You do not need the theory of quasiconformal (quasisymmetric) maps on arbitrary sets $X$ here, because there is a stronger form of $\lambda$-lemma: every holomorphic motion of any set extends to a holomorphic motion of the whole
Riemann sphere, with ... | 2 | https://mathoverflow.net/users/25510 | 287095 | 126,731 |
https://mathoverflow.net/questions/287054 | 6 | The Four Squares Theorem says that every natural number is the sum of four squares in $\mathbb Z$. What is known about coprime representations? Here we call a presentation $n=a^2+b^2+c^2+d^2$ coprime if the g.c.d. of the four numbers $a,b,c,d$ is 1. Does every natural number have a coprime presentation? If not, is ther... | https://mathoverflow.net/users/nan | Sums of four coprime squares | Let $R(n)$ denote the number of ways of writing $n$ as a sum of $4$ squares, and $r(n)$ the number of ways where gcd of $(a,b,c,d) =1$. Then grouping representations of $n$ as a sum of $4$ squares according to the gcd of the variables, clearly we have
$$
R(n) = \sum\_{k^2 | n} r(n/k^2),
$$
and so by Mobius inversion... | 14 | https://mathoverflow.net/users/38624 | 287098 | 126,734 |
https://mathoverflow.net/questions/286897 | 9 | A pseudo-Anosov foliation of a compact orientable surface $F$ is a one whose class in the space $\mathcal{PMF}(F)$ of projective measured foliations is preserved by some pseudo-Anosov homeomorphism of $F$. I saw it casually mentioned that pseudo-Anosov foliations are dense in $\mathcal{PMF}(F)$. What is a proper refere... | https://mathoverflow.net/users/23935 | Are pseudo-Anosov foliations dense? | The pseudo-Anosov foliations form a subset of $\mathcal{PMF}(F)$ which is invariant under the action of the mapping class group $MCG(F)$, because if $\Lambda\_+(\phi) \in \mathcal{PMF}(F)$ is the stable lamination of a pseudo-Anosov $\phi \in MCG(F)$ then $\psi(\Lambda\_+(\phi)) = \Lambda\_+(\psi\phi\psi^{-1})$ is the ... | 6 | https://mathoverflow.net/users/20787 | 287106 | 126,738 |
https://mathoverflow.net/questions/287109 | 26 | I know that $\cos(\pi/n)$ is a root of the Chebyshev polynomial $(T\_n + 1)$, in fact it is the largest root of that polynomial, but often that polynomial factors. For example, if $n = 2 k$ then $\cos(\pi/n)$ is the largest root of $T\_k$, which is a polynomial of lower degree, and if $n = 3$ then $\cos(\pi/n)$ is a ro... | https://mathoverflow.net/users/3319 | Minimal polynomial of cos(π/n) | [The minimal polynomial of $\cos(2\pi/n)$](https://www.jstor.org/stable/2324301?seq=1#page_scan_tab_contents) (by William Watkins and Joel Zeitlin, The American Mathematical Monthly
Vol. 100, No. 5 (May, 1993), pp. 471-474) has full clarity on this matter (just take their result for even $n$ to resolve your case).
| 55 | https://mathoverflow.net/users/1306 | 287113 | 126,740 |
https://mathoverflow.net/questions/287107 | 4 | Let's say we have two functions $h(s)$ and $g(s)$. We can easily simulate a stochastic integral, e.g.
$$t \mapsto \int\_0^t h(s) dB(s) \sim \mathcal{N}\bigg(0, \int\_0^t h(s)^2 ds \bigg). $$
What is the conditional distribution of stochastic integral of $g(s)$ with respect to $B(s)$ then?
$$ t \mapsto \int\_0^t g(s) dB... | https://mathoverflow.net/users/116749 | Conditional stochastic integration | (From the context of your post, it is apparent that $B$ is assumed to be a (say standard) Brownian motion.)
The joint normal distribution follows from the way the stochastic integral is defined. However, if you are already convinced that
\begin{equation}
I\_t(h):=\int\_0^t h(s) dB(s) \sim N\bigg(0, \int\_0^t h(s)^2 ds... | 4 | https://mathoverflow.net/users/36721 | 287116 | 126,742 |
https://mathoverflow.net/questions/285655 | 4 | Let $d=d\_1+d\_2$, $s\_1,s\_2>0$, $p>1$ and $(x\_1,x\_2)\in \mathbb{R}^{d\_1}\times \mathbb{R}^{d\_2}$, $(\xi\_1,\xi\_2)\in \mathbb{R}^{d\_1}\times \mathbb{R}^{d\_2}$. Define
$$
W^{s\_1,s\_2}\_{p}:=\left\{f: f=\mathcal{F}^{-1}\left(\frac{\mathcal{F}g(\xi)}{1+|\xi\_1|^{s\_1}+|\xi\_2|^{s\_2}}\right);\forall g\in L^p(\mat... | https://mathoverflow.net/users/69466 | Embedding theorem for anisotropic Sobolev spaces | There are several ways to proceed. Maybe the most elegant (if you are functional-analysis minded) is to use the fact that in $\mathbb{R}^d$ $(-\Delta)^{\alpha}$, for $0 < \alpha \leq 1$ is the generator of a Markovian semigroup satisfying Sobolev inequalities with ''dimensional constant'' given by $\alpha^{-1} \, d$. T... | 3 | https://mathoverflow.net/users/12604 | 287125 | 126,744 |
https://mathoverflow.net/questions/287115 | 11 | Consider the cohomology ring of the Grassmannian of k-planes in complex n-space. It has a standard presentation as a quotient of the ring of symmetric functions. In this presentation, the Schur functions are mapped to the Schubert classes, thus have a nice geometric interpretation.
One can generalise the Schur functi... | https://mathoverflow.net/users/425 | Is there a geometric interpretation of skew Schur functions? | This is discussed in Stanley's paper *Some combinatorial aspects of the Schubert calculus*. Corollary 3.7 says that under the natural isomorphism given by the Borel presentation of $H^\*(G/P)$ which sends an ordinary Schur function $s\_{\lambda}$ to the class of the Schubert variety $X\_{\lambda}$, a skew Schur functio... | 9 | https://mathoverflow.net/users/16002 | 287141 | 126,751 |
https://mathoverflow.net/questions/287120 | 4 | Let us say that a Hamel basis $H$ in an algebra $A$ is *closed under multiplication*, if $ab\in H$ whenever $a,b\in H$. It is an easy observation that if $A$ has such a basis then there it also has a character (a linear-multiplicative functional; see *Amer. Math. Monthly* **124** (2017), no. 7, 651–653.)
All characte... | https://mathoverflow.net/users/15129 | Bases closed under multiplication | Assume that a Banach algebra $B$ has such a basis $H$. Take any $h\in H$. Consider the element $x=ah+a^2h^2+a^3h^3+\dots$ where $a>0$ is chosen small enough to make the series converge in $B$. Assume $x=\sum\_{j=1}^n c\_jh\_j$ for some $h\_j\in H$. Write
$$
x=ah+a^2h^2+\dots+a^mh^m+a^mh^mx=ah+a^2h^2+\dots+a^mh^m+\sum\_... | 7 | https://mathoverflow.net/users/1131 | 287149 | 126,752 |
https://mathoverflow.net/questions/287142 | 2 | Let $X$ be an integer curve of (arithmetic) genus $g=0$. (the arithmetic genus $g$ is defined by $g:= 1 -\chi\_k(\mathcal{O}\_X)$ where $\mathcal{O}\_X$ is the structure sheaf of $X$ and $\chi\_k(\mathcal{O}\_X) := \sum \_{i \ge 0} (-1)^i dim\_k H^i(X, \mathcal{O}\_X)$ the Euler-Poincare characteristic of $\mathcal{O}\... | https://mathoverflow.net/users/108274 | Cartier Divisor generated by Global Sections | First of all, you definitely need to assume that your curve is proper to make sense of $\chi\_k(\mathcal O\_X)$. If it is not, then $H^i(X,\mathcal O\_X)$ is not finite dimensional over $k$, so $\chi\_X(\mathcal O\_X)$ is not well-defined. Actually, it is true that any proper curve is projective (Hartshorne, exercise I... | 6 | https://mathoverflow.net/users/115211 | 287157 | 126,754 |
https://mathoverflow.net/questions/287103 | 3 | Let $\mathcal{B}(F)$ the algebra of all bounded linear operators on a complex Hilbert space $F$. Let $M\in \mathcal{B}(F)^+$ (i.e. $M^\*=M$ and $\langle Mx\;, \;x\rangle\geq 0$ for all $x\in F$).
I want to show that $\mathcal{B}^M(F)\subseteq \mathcal{B}^{M^{1/2}}(F)$, where
$$\mathcal{B}^M(F)=\left\{S\in \mathcal{B}... | https://mathoverflow.net/users/116483 | Showing the following inclusion between two subalgebras of $\mathcal{B}(F)$ | The following seems like overkill to me; I'd like to see a solution with less machinery. So I just give a sketch.
* As $\newcommand{\im}{\operatorname{Im}} \im(M)^\perp = \ker(M)$ we may reduce to the case when $M$ is injective and has dense range, by compressing to $\im(M)$
* Notice we can work with $S^\*$ instead o... | 4 | https://mathoverflow.net/users/406 | 287162 | 126,756 |
https://mathoverflow.net/questions/287129 | 5 | The standard definition of computability, for a sequence $s\in\{0,1\}^\omega$, is that there is a Turing machine outputting $s[i]$ on input $i$.
I'm looking for strengthenings of this notion; for example, in the above definition it's not decidable whether there is a $1$ in $s$; or, given $i$, whether there is a $1$ i... | https://mathoverflow.net/users/10481 | Different notions of computable binary sequence | What a *computable sequence* is essentially follows from what computability is, and from what a sequence is.
Let us first agree that a sequence over $\mathbf{X}$ is a function $s : \mathbb{N} \to \mathbf{X}$. Then asking that whether or not a sequence over $\{0,1\}$ is constant be decidable amounts to
1. Solving t... | 1 | https://mathoverflow.net/users/15002 | 287173 | 126,760 |
https://mathoverflow.net/questions/287181 | 4 | Let $G$ be a locally compact, Hausdorff and $2^{nd}$-countable group and let $G\_{disc}$ be the same group with the discrete topology. We have a continuous (and bijective) homomorphism given by
$$
id:G\_{disc} \to G.
$$
Now, let $\mathcal{L}(G) \subset \mathcal{B}(L^2(G))$ and $\mathcal{L}(G\_{disc}) \subset \mathcal{... | https://mathoverflow.net/users/12604 | Does the inclusion of the discretized group into itself lift to the group von Neumann algebras? | I think this is only true if $G$ is discrete. As $\pi$ is assumed normal, it's pre-adjoint would give a map
$$ \pi\_\*: A(G) \rightarrow A(G\_{disc}) $$
between the associated Fourier algebras. As $\pi(\lambda\_{G\_{disc}}(g)) = \lambda\_G(g)$, the map $\pi\_\*$, considered as a map of function spaces, would be the for... | 5 | https://mathoverflow.net/users/406 | 287184 | 126,763 |
https://mathoverflow.net/questions/287175 | 2 | Let $\mathcal{B}(F)$ the algebra of all bounded linear operators on a complex Hilbert space $F$. Let $M\in \mathcal{B}(F)^+$ (i.e. $M^\*=M$ and $\langle Mx\;, \;x\rangle\geq 0$ for all $x\in F$.
>
> I want to show that $\mathcal{B}\_1(F)$ is not a subalgebra of $\mathcal{B}(F)$, where
> $$\mathcal{B}\_1(F)=\left\{... | https://mathoverflow.net/users/116483 | Why $\mathcal{B}_1(F)$ is not a subalgebra of $\mathcal{B}(F)$? | Let $M = {\rm diag}(0, 1, 0, \frac{1}{2!}, 0, \frac{1}{3!}, \dots) \in \mathcal B(F)^+$ and $S\in \mathcal B(F)$ be the backward unilateral shift,
$$
S = \left[\begin{matrix}0 & 1 \\ &0 & 1 \\ &&\ddots&\ddots\end{matrix}\right].
$$
It is easy to calculate that $MSM = 0$ and so $MSy = 0$ for all $y\in \overline{{\rm Im}... | 4 | https://mathoverflow.net/users/76593 | 287193 | 126,767 |
https://mathoverflow.net/questions/67181 | 5 | (I decided to repost this [from MathSE](https://math.stackexchange.com/questions/40240/lifting-local-compactness-in-covering-spaces), since the question seems to not be as easy as I had thought)
NB: In this question, local compactness is used in its weak form, i.e. in a locally compact space, every point has a compac... | https://mathoverflow.net/users/1058 | Lifting local compactness to a covering space | A suitable counterexample can be constructed as follows. Let $\mathbb I=\{x\in\mathbb R:0<x<1\}$ denote the open unit interval and let
$B=\{0\}\cup(\omega\times\mathbb I)\cup\{1\}$ be endowed with the topology $\tau\_B$ generated by the base consisting of the following sets:
$\bullet$ $\{n\}\times (a,b)$ for $0<a<b<... | 5 | https://mathoverflow.net/users/61536 | 287202 | 126,769 |
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