parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
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https://mathoverflow.net/questions/242723 | 5 | $\DeclareMathOperator{im}{im}$
I want to prove that Bott-Chern cohomology group $H^{p,q}\_{BC}=\frac{\ker\partial\cap \ker\bar{\partial}}{\im\partial\bar{\partial}}$ has finite dimension via Hodge decomposition like argument. References lead me to [this article](https://arxiv.org/abs/0709.3528) written by M. Schweitz... | https://mathoverflow.net/users/62635 | Hodge decomposition for Bott-Chern cohomology | $\DeclareMathOperator{im}{im}$
Based on @YangMills suggestion I examined Bruno Bigolin's paper titled "Gruppi di Aeppli" and as a result I am now able to show the following relation
>
> $$\ker\partial\cap\ker\bar\partial=\im\partial\bar\partial\oplus\_\perp\mathcal{H}.$$
>
>
>
Let $\psi\in\ker\partial\cap\ke... | 1 | https://mathoverflow.net/users/62635 | 242928 | 111,361 |
https://mathoverflow.net/questions/242648 | 3 | Let $\mu$ be a Gaussian measure on a separable Banach space $X$ and $q$ is the covariance operator of $\mu$. I am reading a proof for
$$\operatorname {supp} \mu = \bigcap\_{q(f, f) = 0} \ker f =: E$$
but there is a step I don't understand. So far I understand that the intersection is over an uncountable number of s... | https://mathoverflow.net/users/88505 | Representation of support of Gaussian measure by kernels of no-variance functionals | I think I might be to blame for this question. It looks very similar to something I once wrote, with the same gap. If so, sorry!
The result is true, but the approach described will not work. We have to choose the $f\_n$ with more care.
(Indeed, suppose $x$ is outside the linear span of your $x\_n$. Note that $E$ i... | 2 | https://mathoverflow.net/users/4832 | 242955 | 111,369 |
https://mathoverflow.net/questions/242965 | 3 | Let $X$ be a smooth projective surface over a field $k$. Is there a way to compute $H^1(X - \{x\},\mathcal{O}\_{X-\{x\}})$ in terms of similar invariants for $X$? Actually I'd like to remove even more points, finitely many points.
| https://mathoverflow.net/users/94276 | What's $H^*(X - \{x_1,\ldots,x_n\},\mathcal{O})$, when $X$ is a projective smooth surface? | The general reference for what follows is SGA 2, §1 to 4. Let $S\subset X$ be a finite subset, and $U:=X\smallsetminus S$. There is an exact sequence
$$H^1\_S(X,\mathcal{O}\_X)\rightarrow H^1(X,\mathcal{O}\_X)\rightarrow H^1(U,\mathcal{O}\_U)\rightarrow H^2\_S(X,\mathcal{O}\_X)\rightarrow H^2(X,\mathcal{O}\_X)$$
Now $... | 7 | https://mathoverflow.net/users/40297 | 242973 | 111,377 |
https://mathoverflow.net/questions/242911 | 5 | Steinberg's "Lectures on Chevalley Groups"
<https://math.depaul.edu/cdrupies/research/papers/chevalleygroups.pdf>
contain ``a complete list of isomorphisms" among the various finite simple Chevalley groups (Th. 37 on pp. 108--109). Unfortunately, the proofs are omitted. I am looking for a reference where the completen... | https://mathoverflow.net/users/9658 | Exceptional isomorphisms between finite simple Chevalley groups | I've tracked down, I think, the best references although I don't have access to them. A description of the history of this question is in Wilhelm Magnus' preface to the Dover edition of Dickson's *Linear groups*:
>
> In a later paper Dieudonné settled one of the fundamental questions
> which Dickson had left unans... | 5 | https://mathoverflow.net/users/801 | 242978 | 111,381 |
https://mathoverflow.net/questions/242920 | 10 | In many different places, I could find the notion on ''(poly)logarithm sheaves''. As is indicated in the name of it, I guess that it should have something to do with (poly)logarithm function: $\mathrm{Li}\_s(z)$. How are they related to each other? Any reference would be helpful, but it would be better if it requires l... | https://mathoverflow.net/users/44005 | Polylogarithm sheaves | I would suggest looking at Hain's article ([arXiv version here](http://arxiv.org/abs/alg-geom/9202022))
R. Hain. Classical polylogarithms. Motives Proceedings, vol II, Proc. Symposia Pure Math 55.2, 1994.
Very roughly, one writes out a multi-valued function on $\mathbb{P}^1\setminus\{0,1,\infty\}$ which takes valu... | 4 | https://mathoverflow.net/users/50846 | 242982 | 111,383 |
https://mathoverflow.net/questions/242677 | 9 | Does $\mathbb{CP}^2$ admit a Riemann surface lamination structure? Every paper or article I looked at, talk only about singular laminations on $\mathbb{CP}^2$. I was wondering why. If you know something about it or you can give some reference, it would be nice.
| https://mathoverflow.net/users/6822 | Does $\mathbb{CP}^2$ admit a Riemann surface lamination structure? | It is conjectured that $\mathbb {CP}^2$ contains no embedded compact laminated set (without singularities) apart the smooth algebraic curves.
This is a strong form of the "Minimal Exceptional" conjecture, stating that for a singular holomorphic foliation of $\mathbb{CP}^2$, every leaf accumulates in the singular set.... | 11 | https://mathoverflow.net/users/35428 | 242986 | 111,384 |
https://mathoverflow.net/questions/242945 | 15 | Littlewood established that $2e^{\gamma} \geq \limsup\_{t \to \infty} |\zeta(1+it)| / \log{\log{t}} \geq e^{\gamma}$, the lower bound unconditionally and the upper bound on RH. It now seems to be generally believed that the lower bound represents the truth, and even, in the most optimistic form, that quite possibly the... | https://mathoverflow.net/users/26522 | Does Littlewood's bound on $\zeta(1+it)$ extend to all the partial sums? | The short answer is yes, and this is treated explicitly for characters in the work of [Granville and Soundararajan](http://arxiv.org/pdf/math/9903196.pdf) (the paper appeared in J. Amer. Math. Soc.). Their Theorem 2 gives that on GRH for $x\le q$ and a primitive character $\chi \pmod q$ one has
$$
\Big| \sum\_{n\le x... | 11 | https://mathoverflow.net/users/38624 | 242987 | 111,385 |
https://mathoverflow.net/questions/242957 | 4 | Consider the graph with vertices $V=\mathbb Z$ and edges
$$E=\{(n,n+1):n\in\mathbb Z\}\cup\{(0,0)\},$$
that is,
the usual integer lattice with a self-edge at zero.
For some fixed parameters $a,b,n\in\mathbb N$,
I am interested in counting the number of walks from $a$ to $b$ in $n$ steps.
This is very easy in the case... | https://mathoverflow.net/users/76050 | Number of walks on integer lattice with self-edge at zero | There are two types of paths from $a$ to $b$: those that do not visit the origin ($0$) and those that do visit it. I start with analyzing the second kind of paths.
Paths that visit the origin
---------------------------
A path that visits the origin consists of three parts:
(i) a path from $a$ to $0$ that does not ... | 4 | https://mathoverflow.net/users/7076 | 242988 | 111,386 |
https://mathoverflow.net/questions/242919 | 10 | Let $\lambda=\text{unif}([0,1])$ be uniform distribution on $[0,1]$ and $B$ be any Borel set. Lebesgue's density theorem states that for $\lambda$-almost all $x\in[0,1]$ the limit
$$\lim\_{\epsilon\downarrow o}\frac{\lambda([x-{\epsilon},x+\epsilon]\cap B)}{2\epsilon}$$
exists and is either $0$ or $1$. Im interested in... | https://mathoverflow.net/users/94251 | Speed of convergence in Lebesgue's density theorem | The answer is negative. In fact, let us show that $a\_n$ as defined in the question may converge to $0$ however slowly. Indeed, let $(\epsilon\_j)$ is any sequence in $(0,1]$ converging to $0$ slowly and regularly enough, in the sense that
\begin{equation}
\epsilon\_{j-1}-\epsilon\_j\ge2^{2-j}\tag{1}
\end{equation}
e... | 5 | https://mathoverflow.net/users/36721 | 242990 | 111,387 |
https://mathoverflow.net/questions/242864 | 0 | I am a software developer with a rather simple problem. I don't really know how to express it in mathematical terms - I'll just try to write it down:
I have multiple different files... let's say 20 files. Each file can have a very different size. Some are very big, some are rather small. Each file is smaller than 3MB... | https://mathoverflow.net/users/94237 | Make multiple batches of maximum size, different sized objects | Your problem is called [bin packing](https://en.wikipedia.org/wiki/Bin_packing_problem), and there is a vast literature on exact and approximation algorithms (see the bibliography on the Wikipedia page).
| 2 | https://mathoverflow.net/users/12674 | 243017 | 111,394 |
https://mathoverflow.net/questions/234380 | 3 | Let $F\subseteq E$ be an algebraic field extension. Let $\alpha\in E$ be such that $\min(\alpha,F)$ has only one root in $E$ (which will be $\alpha$). Is it true that for any $p(x)\in F[x]$ we must have:
"$\min(p(\alpha),F)$ has only one root in $E$"
Another question: Does the above conjecture at least hold in char... | https://mathoverflow.net/users/32135 | If $\min(\alpha,F)$ has only one root in $E$, must $\min(p(\alpha),F)$ have only one root in $E$ | This is not true in general, even in characteristic $0$:
**Example.** Let $\alpha = \sqrt[3]{1+\sqrt{8}} \in \mathbb R$, and let $L = \mathbb Q(\alpha)$. The minimal polynomial of $\alpha$ over $\mathbb Q$ is
$$(x^3-1)^2 - 8 = x^6 - 2x^3 - 7.$$
If $\beta = \sqrt[3]{1-\sqrt{8}}$ and $\zeta\_3 = e^{2\pi i/3}$, then the... | 3 | https://mathoverflow.net/users/82179 | 243020 | 111,395 |
https://mathoverflow.net/questions/243000 | 3 | Let $P$ and $Q$ be two even, unimodular, positive definite quadratic forms of rank $n$. Let $r\_{k}(P)$ be the number of vectors of norm $k$, in symbols:
$$
r\_k(P)=\textrm{cardinality of }\{v\in \mathbb{Z}^n \; | \; P(v,v)=k\}
$$
It is well known that there exists a constant $c$, which is explicitly known and depe... | https://mathoverflow.net/users/48866 | Number of vectors of fixed norm | It is known at least since Hermite that, given integers $N$ and $D$ there are only finitely many equivalence classes of positive definite quadratic modules rank $n<N$ and discriminant $d<D$.
(His proof (as well as the modern proof using Minkowski's convex body theorem) doesn't even allude to theta series.)
| 1 | https://mathoverflow.net/users/39552 | 243025 | 111,396 |
https://mathoverflow.net/questions/243026 | 0 | In Nakahara's *Geometry, Topology and Physics* on page 375, he constructs a Lie-algebra-valued one-form $\omega$ on a principal bundle $P$ by "lifting" a Lie-algebra-valued one-form $\mathcal A\_i$ on an open covering $\{U\_i\}$ on the base manifold $M$. Given a $\mathfrak{g}$-valued one-form $\mathcal{A}\_i$ on $U\_i$... | https://mathoverflow.net/users/94315 | Exterior derivative on principal bundle | The expression $g^{-1} dg$ means the left invariant Maurer-Cartan 1-form on the Lie group $G$. It is only suggestive notation, but if $G$ is a subgroup of the general linear group $GL(n,\mathbb{R})$, then it has a precise meaning. The function $g : G \to \mathbb{R}^{n \times n}$ is the identity map, and then $dg : TG \... | 5 | https://mathoverflow.net/users/13268 | 243027 | 111,397 |
https://mathoverflow.net/questions/243030 | 4 | What are some papers or talks on the philosophy of mathematics which contains some statements about the unnecessary and unreasonable application of mathematics in other areas of science?
I found one paper as follows page 515, the last paragraph(before the discussion)
<http://www.sciencedirect.com/science/article/pi... | https://mathoverflow.net/users/36688 | Unreasonable application of mathematics to the other areas | One of the earliest contributions along this line is from Goethe, [Über Mathematik und deren Mißbrauch](https://books.google.nl/books?id=LHAHAAAAQAAJ&pg=PA167&lpg=PA167&dq=uber+mathematik+und+deren+missbrauch+goethe&source=bl&ots=4uE3j5NieH&sig=YVag1aSPp5D8lVRefPEi-BUQS8A&hl=nl&sa=X&redir_esc=y#v=onepage&q&f=false) (18... | 10 | https://mathoverflow.net/users/11260 | 243033 | 111,398 |
https://mathoverflow.net/questions/242411 | 5 | Is there a measure for matrix that is analogous to rank of the matrix, but it is continuous on matrix elements? Say, we could say the information in identity matrix $I\_n$ is $n$, and when the off-diagonal elements change from 0 to 1, the information contained in the matrix reduces continuously.
Example: considering... | https://mathoverflow.net/users/41445 | information measure for matrix that is analogous to rank | The notion of [stable rank](https://nickhar.wordpress.com/2012/02/29/lecture-15-low-rank-approximation-of-matrices/) is often used in the low-rank matrix approximation literature to offer a more tractable surrogate for rank. This does not necessarily satisfy all your requirements, but might be still suitable for your p... | 4 | https://mathoverflow.net/users/8430 | 243041 | 111,399 |
https://mathoverflow.net/questions/243042 | 1 | Consider the following situation: Let $\Omega =l^{\infty}(\mathbb{R})$ be the
space of all bounded sequences of real numbers. We will consider in $\Omega$ the metric:
$
d(x,y)=\sum\_{i\geq 1}\frac{|x\_i-y\_i|}{2^i}.
$
Set $\Omega\_k=[-k,k]^{\mathbb{N}},$ is clearly that $\bigcup \Omega\_k=\Omega$.
Denote by $B(\Om... | https://mathoverflow.net/users/nan | Could I affirm that $f$ is not identically 0? | No, $f$ could be identically zero. First let me describe a counterexample to a simpler situation. Let $\Omega = \mathbb{R}$ and let $\nu$ and each $\nu\_k$ be the measure whose restriction to $[n-1,n]$ is $\frac{1}{2^n}$ times Lebesgue measure, for $n = 1, 2, \ldots$. So $\nu\_k \to \nu$ because each $\nu\_k$ equals $\... | 1 | https://mathoverflow.net/users/23141 | 243046 | 111,400 |
https://mathoverflow.net/questions/243044 | 3 | Are the scanned images of Cambridge Mathematical Tripos papers from late 19th century available anywhere on Internet?
| https://mathoverflow.net/users/94325 | Cambridge Mathematical Tripos papers from late 19th century | Here is a [collection](https://archive.org/details/mathematicalprob00wolsrich), with solutions, from the period 1864-1878 (published by Joseph Wolstenholme). An earlier period, 1800-1820, was collected by [I.M.F. Wright.](https://books.google.nl/books?id=KBtRAAAAYAAJ&pg=PR6&lpg=PR6&dq=wright+tripos+problems+cambridge&s... | 4 | https://mathoverflow.net/users/11260 | 243050 | 111,403 |
https://mathoverflow.net/questions/243057 | 3 | Let $C$ be a complex curve of genus $g\ge 2$ and let $a\colon C\to J(C)$ be the Abel-Jacobi map. Is there a finite resolution of the ideal $\mathcal I\_{a(C)}$ whose terms are sums line bundles of the form $\mathcal O\_{J(C)}(m \Theta)$?
I think I remember seeing something of this type years ago, but I haven't been ... | https://mathoverflow.net/users/10610 | Resolution of the ideal of the Abel-Jacobi image of a curve? | This is not true, as soon as $g\geq 3$. Taking Chern classes this would imply that $c\_{g-1}(\mathcal{O}\_{a(C)})$ is an integral multiple of $\ \Theta ^{g-1}$ in $\ H^{2g-2}(JC,\mathbb{Z})$. But $c\_{g-1}(\mathcal{O}\_{a(C)})=(-1)^{g}(g-2)![a(C)]=(-1)^g\frac{\Theta ^{g-1}}{g-1}\ $, a contradiction.
| 6 | https://mathoverflow.net/users/40297 | 243060 | 111,405 |
https://mathoverflow.net/questions/242853 | 6 | I was trying to formulate intuitive descriptions of some large cardinals.
Roughly something equivalent to "A manifold is an object which looks like patches of $R^n$ glued together". Not perfectly rigorous, but hopefully conveys the basic picture.
Here are three descriptions I have in mind:
1) An inaccessible c... | https://mathoverflow.net/users/94232 | Intuitive descriptions of some large cardinals | (1) seems okay, but I'm afraid that most large cardinal properties beyond inaccessibility probably aren't going to admit such simple formulations.
I'm not sure I understand the precise meaning of (2), but in any case it doesn't seem right. For one thing, the existence of $0^\sharp$ is weaker than the existence of a m... | 7 | https://mathoverflow.net/users/1682 | 243066 | 111,406 |
https://mathoverflow.net/questions/242852 | 8 | Let $A = {\mathbb F}\_2[X]$, though the following can be adapted to $p \neq 2$ too. Order the elements of $A$ lexicographically. Equivalently, take a polynomial such as $P = X^4 + X + 1$, write its coefficients as binary digits $\mathbf{b}10011$ and find that it is the $19$th polynomial. Let $f$ be the resulting biject... | https://mathoverflow.net/users/3545 | Lexicographic distribution of irreducible polynomials | This is true.
By Gauss's theorem (the inclusion-exclusion formula for the number of irreducibles of a given degree), we may restrict to polynomials of a fixed degree $r$. A moment of reflection then shows that what is needed for lexicographical PNT is exactly the following: For every $k \in \mathbb{N}$, and every len... | 6 | https://mathoverflow.net/users/26522 | 243068 | 111,407 |
https://mathoverflow.net/questions/243045 | 7 | Let $T : V \to W$ be an isomorphism of vector spaces with bases $B\_V$ and $B\_W$, which may be of any cardinality.
>
> Does there exist a bijection $f : B\_V \to B\_W$ such that, for each
> $b\_V \in B\_V$, the coefficient of $f(b\_V)$ in $T(b\_V)$ is nonzero?
>
>
>
If $V,W$ are finite-dimensional, the answe... | https://mathoverflow.net/users/45505 | Bijection modeling isomorphism of infinite-dimensional vector spaces | Yes. Assume $B\_V, B\_W$ are infinite.
First (merely to simplify notation) reduce to $B\_V, B\_W$ countable: start with any $v \in B\_V$, include the finitely many $w \in B\_W$ that are involved in $T(v)$, include the finitely many $v \in B\_V$ that are involved in all $T^{-1}(w)$, include the finitely many $w \in B\... | 5 | https://mathoverflow.net/users/59248 | 243071 | 111,409 |
https://mathoverflow.net/questions/243072 | 1 | As summarized in the title, suppose there is an isomorphism between $G\_1 \times G\_2$ and $G\_3$, is it always true that $BG\_1 \times BG\_2$ is homotopy equivalent to $BG\_3$? If it is not always true could you give a counterexample and the condition for it to be true?
Similarly suppose there is an isomorphism betw... | https://mathoverflow.net/users/82645 | $G_1 \rtimes G_2 \cong G_3 \implies BG_1 \times BG_2 \simeq BG_3$? $G_1 \times G_2 \cong G_3 \implies BG_1 \times BG_2 \simeq BG_3$? | Let $p\_{G\_i}:EG\_i\rightarrow BG\_i$ be the universal fibration, $i=1,2$, then
$p\_{G\_1}\times p\_{G\_2}:EG\_1 \times EG\_2 \rightarrow BG\_1\times BG\_2$ is the universal fibration of $G\_1\times G\_2$ since it is a $G\_1\times G\_2$ principal bundle and $EG\_1\times EG\_2$ is contractible, thus $BG\_1\times BG\_2=... | 2 | https://mathoverflow.net/users/80891 | 243073 | 111,410 |
https://mathoverflow.net/questions/243076 | 3 | The well known nerve functor from small categories to simplicial sets has a left adjoint, namely the fundamental category functor. Does the double nerve functor $N^2:2Cat\rightarrow sSSet$ from 2-categories to bisimplicial sets have a similar left adjoint? I think I've seen a vague reference to it somewhere but nothing... | https://mathoverflow.net/users/84563 | Left adjoint to Double Nerve? | Yes, by the adjoint functor theorem, because it preserves all limits and both categories are locally presentable. It's also an instance of the general notion of [nerve and realization](https://ncatlab.org/nlab/show/nerve+and+realization) determined by a canonical cobisimplicial 2-category.
| 9 | https://mathoverflow.net/users/49 | 243077 | 111,411 |
https://mathoverflow.net/questions/242866 | 5 | Let $L / K$ be a solvable (or cyclic) Galois extension of totally real fields, and let $f$ be a Hilbert modular newform over $L$.
Suppose that, for every $\sigma \in Gal(L / K)$, the conjugate newform $f\_\sigma$ is twist-equivalent to $f$, i.e. there exists a Hecke character $\chi\_\sigma$ of $L$ such that $a\_{\sig... | https://mathoverflow.net/users/2481 | Hilbert modular forms twist-equivalent to their conjugates | Use Galois Representations. By Schur's Lemma, the projective representation extends to G\_K. By Tate's theorem, this projective representation lifts to a genuine representation of G\_K. The restriction of this representation to G\_L is a twist of the original representation. Hence, after twisting, the HMF is invariant ... | 3 | https://mathoverflow.net/users/94346 | 243079 | 111,412 |
https://mathoverflow.net/questions/243063 | 2 | Let $G$ be a reductive group defined over a field $F$. Let $\Sigma$ be the set of roots of $G$ with respect to a Borel subgroup $B=TU$ with torus $T$. Let $W=N\_G(T)/T$ be the Weyl group of $G$. For $\alpha\in \Sigma$, let $U\_\alpha$ be the root space of $\alpha$. Denote $x\_\alpha:F\rightarrow U\_\alpha$ the fixed is... | https://mathoverflow.net/users/13466 | If a Weyl element preserves a root, then it has a representative which preserves the root space? | This is true if $G$ is split. First of all $w$ is represented by some $\tilde w\in N\_G(T)\cap G(k)$ (see Borel-Tits, for example). Then $\text{Ad}\, \tilde w$ acts on $\mathfrak g\_\alpha$ by some scalar $c\in k^\*$. If there is $t\in T(k)$ with $\alpha(t)=c$ then $\dot w=\tilde wt^{-1}$ will do the trick. Now if $\al... | 6 | https://mathoverflow.net/users/89948 | 243082 | 111,414 |
https://mathoverflow.net/questions/243084 | 21 | One of the very common notations for syntactic substitution is $[\ /\ ]$.
However, there seems to be an inconsistency in the literature about its usage.
* Many write $[t/x]$ for "substitute $t$ for $x$" (Girard, Buss, ...).
* Others use $[x/t]$ for "replace $x$ with $t$" (van Dalen, Troelstra, Martin-Löf, ...).
I ... | https://mathoverflow.net/users/7507 | History of the notation for substitution | Some early examples of the form $[t/x]$ are due to Haskell Curry.
See:
* Haskell Curry & Robert Feys & William Craig, [Combinatory Logic. Volume I](https://books.google.it/books/about/Combinatory_Logic.html?id=avrcMwEACAAJ&redir_esc=y) (1958), page 54:
>
> Let $a$ and $b$ be obs and let $x$ be a variable; it is... | 20 | https://mathoverflow.net/users/42676 | 243086 | 111,416 |
https://mathoverflow.net/questions/243067 | 1 | Let X be a given matrix of dimension $p \times q$. Let $G$ be a $s \times p$ dimensional matrix of standard normal/Gaussian random variables.
* Are there cases where one can been able to quantify $P\_G [ \vert \vert \vert X \vert \vert - \vert \vert GX \vert \vert \vert > t ] $ ?
(choose any matrix norm for which ... | https://mathoverflow.net/users/38852 | Concentration of matrix norms under random projection. | That's a trivial simplification of the [Johnson-Lindenstrauss lemma](https://en.wikipedia.org/wiki/Johnson%E2%80%93Lindenstrauss_lemma) .
The matrix $X$ can be seen as a set of $q$ points in $p$ dimensions. Let us first prove the following result, than your question will follow directly with a union bound: (by the way... | 2 | https://mathoverflow.net/users/40231 | 243090 | 111,417 |
https://mathoverflow.net/questions/243075 | 1 | Is there an evaluation of this sum (possibly involving gamma functions)? $k$ and $n$ are natural numbers and $x$ is real with $0<x<1$.
$$ \sum\_{\substack{k=0\\n-k\text{ even}}}^n \frac{(-1)^{(n-k)/2}}{k+x} \frac{(n+k)!}{k!(\frac{n+k}{2})!(\frac{n-k}{2})!}$$
Any help is much appreciated.
EDIT: I know the result for a... | https://mathoverflow.net/users/94200 | Evaluation of sum of factorials | Plugging the sum into MAPLE gives
$$
g1:=(-1)\text{^}((n-k)/2)\*1/(k+x)\*(n+k)!/k!/((n+k)/2)!/((n-k)/2)!;
$$
Plug in $n=2m$ even:
$$
g2:=\text{subs}(k=n-2\*l,n=2\*m,g1);
$$
Then perform summation:
$$
Seven:=\text{SumTools[DefiniteSummation]}(g2,l=0..m);
$$
Result (modulo typos):
$$
Seven=\frac{(-1)^m4^m\Gamma(1+\frac x... | 2 | https://mathoverflow.net/users/89948 | 243091 | 111,418 |
https://mathoverflow.net/questions/234457 | 12 | Risking to be downvoted, here is a very lightweight question.
In various fields - say, algebraic geometry, nonstandard analysis, synthetic differential geometry - infinitely small quantities, i. e. those very-very close to zero, are represented by nilpotents, sometimes just square zero elements suffice to do quite a ... | https://mathoverflow.net/users/41291 | Multiplicative infinitesimals in q-analogs? | There is more than one question that is being asked here so I will leave aside the one about $q$-analogues for the simple reason that one can take any question, say $X$, in mathematics, and ask for its $q$-analog, $X\_q$, so things can get pretty monotonous.
As far as the multiplicative version of being "very small" ... | 2 | https://mathoverflow.net/users/28128 | 243095 | 111,419 |
https://mathoverflow.net/questions/242897 | 6 | *In what follows I'm going to use $V\_{\theta\_s}$ for the little adjoint representation af a Lie algebra i.e. the representation associated with the highest short rooth $\theta\_s$.*
Is easy to see that simple algebras of types $B\_n$, $C\_n$ and $F\_4$ can be found as subalgebras of respectively $D\_{n+1}$ with $n>... | https://mathoverflow.net/users/37771 | Involutions and Little Adjoint Representations of Simple Algebras | The involutive compact Lie algebras yield symmetric spaces and the representations in point are the corresponding isotropy representations.
The first symmetric space is the sphere $SO\_{2n+2}/SO\_{2n+1}=S^{2n+1}$, whose isotropy representation is the standard action of $SO\_{2n+1}$ on $\mathbb R^{2n+1}$. Its highest... | 4 | https://mathoverflow.net/users/15155 | 243117 | 111,426 |
https://mathoverflow.net/questions/243092 | 6 | A metrizable space $X$ will be called a *generalized Bernstein set* if every closed completely metrizable subspace $C$ of $X$ has cardinality $|C|<|X|$.
It is well-known that the real line contains a (generalized) Bernstein set of cardinality $\mathfrak c$.
Moreover, for every cardinal $\kappa$ with $\kappa^\omega... | https://mathoverflow.net/users/61536 | Bernstein sets of large cardinality | After thinking a night on this question and waking up, I realized that the answer is almost trivial: there are restrictions on possible cardinalities of generalized Bernstein set.
Any metrizable space $X$ of density $\kappa$ has cardinality $|X|\le\kappa^\omega$ and contains a discrete (and hence completely metrizab... | 2 | https://mathoverflow.net/users/61536 | 243124 | 111,429 |
https://mathoverflow.net/questions/241617 | 2 | Consider a 1-dimensional stochastic heat equation on $[0, 1]$, with boundary conditions of Neumann's type:
\begin{equation}\left\{
\begin{aligned}
&\partial\_t u(t, x) = \frac{1}{2}\partial\_x^2 u(t, x) - U(u(t)) + \dot W(dt, dx), \\
&\partial\_x u(t, 0) = \partial\_x u(t, 1) = 0, \\
&u(0, x) = v(x).
\end{aligned}\ri... | https://mathoverflow.net/users/44590 | The (infinite) invariant measure of an SPDE | I check it with the standard Garlerkin method and confirmed that it is right, in both cases (i) and (ii).
Discribe the proof briefly (under (ii)):
Take a CONS of $H$ as $h\_1 = 1$ and $h\_k(x) = \cos[(k-1)\pi x]$.
First notice that if $V(u) = V^\dagger(\langle u, h\_1 \rangle, \ldots, \langle u, h\_N \rangle)$... | 0 | https://mathoverflow.net/users/44590 | 243132 | 111,434 |
https://mathoverflow.net/questions/243126 | 8 | Very little is known about Euclid's life--much less than about other famous ancient Greek mathematicians, which is puzzling. It is also strange to me that Euclid didn't write about the Eratosthenes sieve.
Thus I'd like to ask mathematical historians and everybody else: is there any clear proof that Euclid and Eratost... | https://mathoverflow.net/users/8385 | Euclid vs Eratosthenes | [C.K. Raju](https://en.wikipedia.org/wiki/C._K._Raju) goes to some length to argue that Euclid did not exist at all, in [Good-Bye Euclid!](http://ckraju.net/papers/MathEducation1Euclid.pdf)
He starts from the established fact that, while Euclid was first mentioned by the 5th century philosopher [Proclus](https://en.w... | 5 | https://mathoverflow.net/users/11260 | 243133 | 111,435 |
https://mathoverflow.net/questions/243136 | 2 | Let $P(n,m)$ denote the set of all positive integer partitions of $n$ into parts that are pairwise distinct and bounded by $m$. Let $p(n,m) = |P(n,m)|$.
After some numerical experiments it appears
$$
p(n,m) + p(n+m,m) \geq p(n+k,m)
$$
for all $1 \leq k \leq m$.
(An even sharper inequality may hold, but the ab... | https://mathoverflow.net/users/94267 | An inequality on partitions into distinct bounded parts | At first, for any $x$ and $s\in \{1,2,\dots,m\}$ we have $p(x+s,m)+p(x-s,m)\geqslant p(x,m)$ by obvious injection (remove or add part equal to $s$). At second, numbers $p(x,m)$ when $m$ is fixed and $x$ varies increase upto $m(m+1)/4$ and decrease after that. This was discussed [here](https://mathoverflow.net/questions... | 2 | https://mathoverflow.net/users/4312 | 243141 | 111,440 |
https://mathoverflow.net/questions/242237 | 1 | I was computing some GIT quotients and came up with the following question: to compute $\mathrm{Proj}(\mathbb C [f\_1,f\_2,f\_3,f\_4,f\_5,f\_6]/I)$ where $f\_i$'s are homogeneous polynomials of same degree and $I$ is the ideal generated by $$\{f\_3f\_6-f\_4f\_5, f\_1f\_5-f\_3^2-f\_2f\_3, f\_1f\_6-f\_3f\_4-f\_2f\_4, f\_... | https://mathoverflow.net/users/93909 | Proj of some graded algebra | The OP clarified that the surface $S\subset \mathbb{P}^n$ satisfies all of the following properties: (a) $S$ is smooth, (b) $S$ is rational so that $h^1(S,\omega\_S)$ is zero, and (c) the Hilbert polynomial of $S$ equals $$ p(t) = 1+ d\frac{(t+1)t}{2},$$ for some integer $d$. Up to replacing $\mathbb{P}^n$ by the span ... | 4 | https://mathoverflow.net/users/13265 | 243154 | 111,441 |
https://mathoverflow.net/questions/243152 | 3 | Is there an ordered 4-tuple of rational numbers $(a,b,c,d)$ such that $(b,d)\ne(0,0)$ and $2a^2+3b^2+30c^2+45d^2=2$?
The former (deleted) question was just about cases $(a,b,c,d)\ne(1,0,0,0)$ but it was quite silly :( I apologize and I think now it makes sense. I guess there is a canonical proof of nonexistence or an... | https://mathoverflow.net/users/94407 | sum of four squares with some coefficients | Actually infinitely many, and a parametrization of all rational solutions is e.g.
$a:=\frac{3B^2+30C^2+45D^2-2E^2}{3B^2+30C^2+45D^2+2E^2}$
$b:=\frac{4BE}{3B^2+30C^2+45D^2+2E^2}$
$c:=\frac{4CE}{3B^2+30C^2+45D^2+2E^2}$
$d:=\frac{4DE}{3B^2+30C^2+45D^2+2E^2}$.
| 7 | https://mathoverflow.net/users/6101 | 243161 | 111,445 |
https://mathoverflow.net/questions/243171 | 1 | There is a beautiful [paper](http://arxiv.org/abs/1604.08657) on the arXiv by Andrew Suk containing an asymptotic result about the [Erdös-Szekeres convex polygon problem](https://en.wikipedia.org/wiki/Happy_ending_problem). I am struggling with one of the estimates he makes on page 4. He claims that for $n$ *large enou... | https://mathoverflow.net/users/3995 | estimating binomial coefficients | We have $$\binom{x}k=\frac{x(x-1)\dots (x-k+1)}{k!}\leqslant \frac{x^k}{k!}\leqslant x^k$$
for positive integers $x, k$. Applying this to $k=[2n^{3/4}]-2$, $x=n+k-2\leqslant 2n$ (for large $n$) we get $$\binom{n+k-2}{n-2}=\binom{n+k-2}{k}\leqslant (2n)^k=e^{k\log(2n)},$$
the rest follows from $4/5>3/4$.
| 4 | https://mathoverflow.net/users/4312 | 243173 | 111,447 |
https://mathoverflow.net/questions/243167 | 8 | A "cloven fibration" is a fibration for which we have an explicit choice of cartesian liftings; this is often phrased as, "We can pick a lifting without using the axiom of choice".
Firstly, I'm a bit perplexed by this language, since the intensional character of the formal proof of a statement ought not to factor in ... | https://mathoverflow.net/users/51336 | Constructively, are all fibrations cloven? | The Elephant defines a cloven fibration as a fibration equipped with a cleavage
(B1.3), where a cleavage is a particular map lifting arrows from the base
category. This doesn't require talking about (the axiom of) Choice, though
you could also describe it as a choice of liftings. However, it is an
instance of a common ... | 4 | https://mathoverflow.net/users/33143 | 243180 | 111,451 |
https://mathoverflow.net/questions/243008 | 4 | The setting is the same as in my last question [commutative diagram with $K\_{i+1}(A)\to K\_i(A\rtimes\_{\rho} \mathbb{R})$ (for $C^\*$-algebras)](https://mathoverflow.net/questions/241884/commutative-diagram-with-k-i1a-to-k-ia-rtimes-rho-mathbbr-for-c) :
Let $A$ be in the bootstrap category (=N in the other thread) ... | https://mathoverflow.net/users/nan | commutativity of a diagram in cohomology of $C^*$-algebras | Observe that the $KK$-class $\sigma \in KK\_1(A/J,J)$, which you mention in your edited paragraph only depends on the extension
$$
0 \to J \to A \to A/J \to 0
$$
and not on $B$. So we have $\delta\_1^n(x) = \sigma \otimes\_J x$ for $x\in KK\_n(J,B)$. If $\delta\_3 \colon K\_\*(A/J) \to K\_{\*+1}(J)$ denotes the bounda... | 2 | https://mathoverflow.net/users/3995 | 243192 | 111,456 |
https://mathoverflow.net/questions/243185 | 3 | I've seen a number of combinatorial interpretations for the coefficients of the compositional inverse (aka reversion) of a power series. Is there a known combinatorial interpretation for the coefficients of the *reciprocal* of a power series?
Specifically: I'm looking for a family of combinatorially defined sets $S\_... | https://mathoverflow.net/users/21690 | Combinatorial interpretation for coefficients of reciprocal of power series | Since $a\_0+a\_1x+a\_2x^2+\cdots=a\_0(1+(a\_1/a\_0)x+(a\_2/a\_0)x^2+\cdots)$,
we can assume $a\_0=1$. Then
$$ b\_n = \sum (-1)^k a\_{i\_1}\cdots a\_{i\_k}, $$
where the sum is over all $2^{n-1}$ compositions $(i\_1,\dots,i\_k)$ of
$n$. Thus we can take $S\_n$ to be the set of compositions of $n$, etc.
| 10 | https://mathoverflow.net/users/2807 | 243194 | 111,458 |
https://mathoverflow.net/questions/243183 | 5 | I am a little bit unsure about the mirror symmetry statement for elliptic curves; specifically, how the flipping of the Kähler and complex moduli works. Perhaps I should say at the outset, the reason I have been thinking about this, is that I am doing a computation involving a torus with parameter $\tau \in \mathbb{H}$... | https://mathoverflow.net/users/83496 | Confusion regarding statement of mirror symmetry for elliptic curves | This is related to the fundamental question about how to define the Kähler moduli space. The Kähler moduli space is often not as naive as one thinks. A solution for a genus 1 curve is given by Bridgeland in the last section of [this article](http://arxiv.org/abs/math/0212237), where the "extended" Kähler moduli is iden... | 4 | https://mathoverflow.net/users/21014 | 243199 | 111,459 |
https://mathoverflow.net/questions/243138 | 2 | In the book *Infinite abelian groups Vol. I* by L. Fuchs, on page 154, the notion of the generalized $p$-height of an element in an abelian group is defined, as follows:
Let $A$ be an abelian group and let $p$ be a prime number. First we define for every ordinal $\sigma$ a subgroup $p^\sigma A$ of $A$, recursively, ... | https://mathoverflow.net/users/42440 | Generalized height of elements in abelian groups | As suggested in my comment, define $h^\*\_p(a)$, as Fuchs does, to be the smallest ordinal $\sigma$ with $a\not\in p^{\sigma+1}A$ if there is such a $\sigma$, but if there is no such $\sigma$ then define $h^\*\_p(a)=\infty$, where $\infty$ is just a symbol that is defined to be greater than every ordinal.
If $\varphi... | 1 | https://mathoverflow.net/users/22989 | 243205 | 111,460 |
https://mathoverflow.net/questions/243198 | 8 | Let $H$ be a finite dimensional hilbert space. Let $L:H\otimes H\rightarrow H\otimes H$ be a unitary transformation. Then the equation
$$(L\otimes I)(I\otimes L)(L\otimes I)=(I\otimes L)(L\otimes I)(I\otimes L)$$
where $I:H\rightarrow H$ is the identity mapping is known as the Yang-Baxter equation.
We shall call a li... | https://mathoverflow.net/users/22277 | Are there any unitary matrices which satisfy the Yang-Baxter equation which are universal for quantum computation? | [Braiding Operators are Universal Quantum Gates](https://arxiv.org/abs/quant-ph/0401090), by Louis Kauffman and Samuel Lomonaco (2004)
>
> In this paper, we prove that certain solutions to the Yang-Baxter equation together with local unitary two-dimensional operators form a universal set of quantum gates. In partic... | 5 | https://mathoverflow.net/users/11260 | 243209 | 111,462 |
https://mathoverflow.net/questions/243207 | 5 | Let $a(n)$ be the number of lattice paths in ${\mathbb{Z}^2}$ of length $n$ which start at the origin $(0,0)$ and end up at $(n,0)$ and have only up-steps $U:(i,j) \to (i + 1,j + 1)$, down-steps $D:(i,j) \to (i + 1,j - 1)$ and horizontal steps $H:(i,0) \to (i + 1,0)$ on the $x-$axis.
Is there a direct combinatorial way... | https://mathoverflow.net/users/5585 | A follow up question to: Number of walks on integer lattice with self-edge at zero | Let me explain why $a(2n+1)=5^n$. I need a well-known identity $\sum\_{a+b=n}\binom{2a}{a}\binom{2b}{b}=4^n$, which has nice combinatorial proofs. Now we prove that the number of walks of length $2n+1$ from 0 to 0, in which every step is $+1$, $-1$ or staying at 0 (call this a loop) equals $5^n$. Induction in $n$, base... | 5 | https://mathoverflow.net/users/4312 | 243212 | 111,464 |
https://mathoverflow.net/questions/243215 | 4 | Let $\lambda, \mu$ be the Perron-Frobenius eigenvalues of two non-negative matrices $A,B$ respectively. I am interested in knowing whether there are any results available on the upper bound of $|\lambda-\mu|$ in terms of norms of $\|A-B\|$.
| https://mathoverflow.net/users/94452 | upper bound on the difference between two Perron-Frobenius eigenvalues | I guess you want a bound in terms of a norm of $A-B$, since one obviously has $|\rho(B) - \rho(A)| \leq ||A|| + ||B||$ for any operator norm $|| \cdot ||$.
If $A$ is irreducible, then its Perron eigenvector $x\_A$ (normalized so that $||x\_A||\_1 = 1$) has strictly positive entries, hence $K(A) = \max\_i (x\_A)\_i^{... | 4 | https://mathoverflow.net/users/21724 | 243217 | 111,465 |
https://mathoverflow.net/questions/243147 | 12 | I was wondering what an equivalent of the Collatz conjecture might be for finite fields. In a Collatz sequence a number is moved down within a set $\{2^k n : k \in \mathbb{Z}^\* \}$ for some odd $n$ or jumped to another such set via $n \mapsto 3n+1$. The analogy of the sets $\{2^k n \}$ in $F\_p$ are the cosets of the ... | https://mathoverflow.net/users/94375 | Collatz-like properties of finite fields | Here is a proof of the generalization of your Weak conjecture to the ring $\mathbf{Z}/m\mathbf{Z}$ where $m$ is any odd positive integer. First let me clarify what is being proved. Let $S$ be the subgroup of $(\mathbf{Z}/m\mathbf{Z})^\times$ which is generated by $2$, and consider a directed graph whose vertices are th... | 8 | https://mathoverflow.net/users/30412 | 243222 | 111,468 |
https://mathoverflow.net/questions/242382 | 3 | The following question came up when thinking about equidistribution of Satake parameters of elliptic curves. Let $G$ be a compact Lie group with Haar measure $\mathrm{d} x$. Recall that a sequence $\{x\_n\}$ of points in $G$ is equidistributed if for all $f\in C(G)$, we have equality:
$$
\lim\_{N\to\infty} \frac{1}{N} ... | https://mathoverflow.net/users/6856 | Criterion for convergence of sums for non-continuous functions | The paper "λ-equidistributed sequences of partitions and a theorem of the De Bruijn–Post type", by Chersi and Volčič, proves that if $(X,d,\lambda)$ is a separable metric space with probability measure whose support is $X$, then "$\frac{1}{N}\sum\_{n\leqslant N} f(x\_n)\to \lambda(f)$ for all $\lambda$-equidistributed ... | 2 | https://mathoverflow.net/users/6856 | 243254 | 111,479 |
https://mathoverflow.net/questions/243239 | 1 | Does anybody have a good reference that lists spectral sequences that may be used to compute Hom sets in derived categories (of coherent sheaves, say)?
| https://mathoverflow.net/users/94462 | Spectral sequences to compute Hom's in derived category | Here are two reference sheets for computing Hom sets in derived categories, which you may find useful:
1. [Perverse Sheaves Quick Reference Guide](https://www.math.lsu.edu/~pramod/tc/ps/psqr.pdf) and
2. [Derived Categories Cheat Sheet](https://www.math.lsu.edu/~pramod/tc/14s-7260/dercat.pdf).
Here are two more ref... | 4 | https://mathoverflow.net/users/58421 | 243264 | 111,481 |
https://mathoverflow.net/questions/242444 | 2 | Let $M$ be a Riemmanian manifold, $p\in M$ and $V\in T\_p(M)$.
Suppose $f^{-1}:U\_p \mapsto U$ is a diffeomorphism of a neighborhood of p to an open subset of $\mathbb{R}$ and define the sequence:
\begin{equation}
\{p\_i := G\_{p\_{i-1}}^{-1}(V)+ p\_{i-1} \}\_{n \in \mathbb{N}},
\end{equation}
where $p\_0=p$ and $G\... | https://mathoverflow.net/users/36886 | Convergence of Discrete Geodesic | I have some doubts about this discretization of the geodesic flow. I believe you should have another component of the map: one that transforms the velocity vector $V$. By the way, technically $V$ should be a momentum covector (belonging to the cotangent bundle).
If you want a discrete version of the geodesic flow, I... | 2 | https://mathoverflow.net/users/75853 | 243266 | 111,482 |
https://mathoverflow.net/questions/243232 | 5 | Let $A,B$ be two ternary quadratic forms with real coefficients, given by symmetric matrices
$$\displaystyle 2A = \begin{pmatrix} 2a\_{11} & a\_{12} & a\_{13} \\ a\_{12} & 2a\_{22} & a\_{23} \\ a\_{13} & a\_{23} & 2a\_{33} \end{pmatrix}, 2B = \begin{pmatrix} 2b\_{11} & b\_{12} & b\_{13} \\ b\_{12} & 2b\_{22} & b\_{23... | https://mathoverflow.net/users/10898 | Stabilizers of pairs of ternary quadratic forms | It's actually order 8 for no real zeroes and order 4 for two real zeroes,
not the other way around. (See the bottom of page 1038 of
[the paper](http://annals.math.princeton.edu/wp-content/uploads/annals-v162-n2-p10.pdf).)
The symmetry groups are taken modulo $\{ \pm 1 \}$, so we work
projectively in ${\rm PGL}\_2({\... | 4 | https://mathoverflow.net/users/14830 | 243267 | 111,483 |
https://mathoverflow.net/questions/243270 | 4 | Consider the inclusion of presheaves on $\mathbb{C}$ into families of sets indexed by $\mathbb{C}$-objects (which proceeds by forgetting the action on morphisms). Is there a left adjoint to this inclusion functor?
I thought that I had constructed something that seemed reasonable, but now I am doubtful whether it is i... | https://mathoverflow.net/users/51336 | Does the inclusion of presheaves into families of sets have a left adjoint? | Although I wouldn't call it an inclusion functor, the answer is yes and in fact the forgetful functor $Set^{C^{op}} \to Set/C\_0$ is monadic (as well as comonadic).
I think the most illuminating way to see this is to regard a presheaf as a set $F: X \to C\_0$ over $C\_0$, equipped with a $C$-action which is a map $C... | 7 | https://mathoverflow.net/users/2926 | 243273 | 111,485 |
https://mathoverflow.net/questions/243241 | 8 | The following question was asked by a colleague of mine. For any prime $p$ consider
$$ M\_p:=\min\_{z\_1,\dots,z\_p}\max\_{j,k}\left|z\_1^k+\dots+z\_j^k\right|,$$
where $z\_1,\dots,z\_p$ are the complex $p$-th roots of unity in any order, and $j,k\in\{1,\dots,p-1\}$ are arbitrary. Is $M\_p$ bounded?
| https://mathoverflow.net/users/11919 | Power sums of p-th roots of unity | This is closely related to the problems surrounding Turán's power sum method. For example see the chapter in Montgomery's Ten Lectures book, or this paper of [Gonek](http://projecteuclid.org/euclid.mmj/1029002459). Lemma 1 (attributed to Cassels) there shows that if $b\_j >0$, and $|z\_j|=1$ (for $j=1$, $\ldots$, $N$) ... | 8 | https://mathoverflow.net/users/38624 | 243276 | 111,486 |
https://mathoverflow.net/questions/243102 | 6 | This is a cross-post of my unanswered (more than a week) [question on Math.SE](https://math.stackexchange.com/questions/1827803/can-local-martingales-be-characterized-only-using-their-fv-process-and-bm). Since it covers topics from my graduate-level course on stochastic processes, I thought it might be appropriate to t... | https://mathoverflow.net/users/93694 | Can all local martingales be represented using only Brownian motion and finite variation processes? | First, a martingale is always only specified with respect to a filtration, and so is thus a local martingale. You do not specify any filtration in your problem, so I assume you mean the natural filtration of the local martingale (i.e., the smallest filtration w.r.t. which $X$ is a local martingale).
Second, your prov... | 9 | https://mathoverflow.net/users/20026 | 243277 | 111,487 |
https://mathoverflow.net/questions/243287 | 0 | I was trying to read a paper on Inverse Galois problem . I understands what the inverse Galois problem is. It asks if every finite group is the Galois group of some extension of the rationals.
The authors says that If we can show that every extension of $\mathbb{Q}$ the specialization of a Branched cover of $P^{1}$ w... | https://mathoverflow.net/users/92070 | Inverse Galois Problem ; Galois group of some branched cover of $P^{1}$ defined over $\mathbb{Q}$ | You could try [Malle-Matzat], "Inverse Galois Theory".
| 2 | https://mathoverflow.net/users/nan | 243289 | 111,490 |
https://mathoverflow.net/questions/243280 | 0 | Given $r$ numbers $a\_1,a\_2,...,a\_r$ and $n=qP$ where $P$ is the product of these $r$ numbers. $q$ is a natural number such that $q \geq 2$.
Also given is a matrix $A$ of the following form: $$A=\begin{pmatrix}x\_{11}&x\_{12}&...&x\_{1r}\\x\_{21}&x\_{22}&...&x\_{2r}\\:&:&:&:\\x\_{n1}&x\_{n2}&...&x\_{nr}\end{pmatrix... | https://mathoverflow.net/users/86494 | Application of the EGZ theorem | We may not care that elements are zeroes or ones. By EGZ applied to a first column (several times) we may partition rows to $n/a\_i$ blocks so that the sum in each block is divisible by $a\_i$. Now we consider only permutations for which these blocks are consecutive. Consider the second column, we have $n/a\_i$ numbers... | 1 | https://mathoverflow.net/users/4312 | 243290 | 111,491 |
https://mathoverflow.net/questions/243294 | 4 | Assume that $P\to M$ is a principal $G$-bundle where $G$ is some (compact) Matrix group. Let $\rho\colon G \to \operatorname{Gl}(\mathbb{R}^n)$ be the tautological representation and $\rho^\prime\colon G\to \operatorname{Gl}(V)$ some other representation.
Let
$$ E = P \times\_{\rho}\mathbb{R}^n, \qquad \text{and} \qq... | https://mathoverflow.net/users/93925 | Associated vector bundles and Characteristic Classes | The first Chern class of the dual $-L$ of a line bundle $L$ is the negative of the first Chern class of $L$. In general Chern classes of different associated vector bundles are unrelated, and when they are related the story is complicated. In your example, when you tensor line bundles, the first Chern class scales.
| 4 | https://mathoverflow.net/users/13268 | 243296 | 111,493 |
https://mathoverflow.net/questions/243258 | 6 | The multiplicative group $\Bbb Q^+$ can be viewed as a $\Bbb Z$-module. To see this, note that any rational can be decomposed into the form
$2^{n\_2} \cdot 3^{n\_3} \cdot 5^{n\_5} \cdot ...$
The tuple of coefficients $(n\_2, n\_3, n\_5, ...)$ is then an element in the module $\Bbb Z^{(\omega)}$, the set of integer ... | https://mathoverflow.net/users/24611 | Extending the topology on a set to the group/vector space it generates | The answer to your questions 1 and 3 is that there is a universal way to do it, but I don't know how this works for infinite linear combinations.
Unfortunately, I only know how to do it in the category of compactly generated topological spaces (or any cartesian closed category of spaces, really). Hence by a topologic... | 2 | https://mathoverflow.net/users/12547 | 243297 | 111,494 |
https://mathoverflow.net/questions/243295 | 7 | It is known that no nontrivial connected cover of $\operatorname{SL}(2,\mathbb R)$ admits a faithful finite dimensional linear representation (see, for example, page 143 in Fulton-Harris and Exercise 11.9 therein). I am looking for a reference with a proof of this fact and an information who observed it first.
**EDIT... | https://mathoverflow.net/users/23500 | Reference for nonlinearity of covers of $\operatorname{SL}(2,\mathbb R)$ | See two papers by Kubota: *Ein arithmetischer Satz über eine Matrizengruppe* (1966, MR0188194) and *Topological Covering of SL(2) Over a Local Field* (1967, MR0204422). Maybe these are the first references, although in isome sense it goes back (at least) to Weil's famous Acta paper *Sur certains groupes d'opérateurs un... | 10 | https://mathoverflow.net/users/6030 | 243304 | 111,496 |
https://mathoverflow.net/questions/243305 | 2 | **Question summary:**
If I have a two-sided bound, can I immediately get a one-sided bound with tighter constants?
**Question details:**
Let $\mathbf X = X\_1,...,X\_n$ be $n$ i.i.d. real-valued random variables where $X\_i \in [a,b]$ and $\mu = \mathbf E[X\_1]$. For $\delta\in(0,1)$, let $f(\mathbf X, \delta)$ b... | https://mathoverflow.net/users/84393 | Symmetry of concentration bounds on mean | The answer is negative. Indeed, for simplicity, let $a=-1$ and $b=1$.
In the case when $\delta=1/10$, $n=1$, and $X\_1$ is uniformly distributed on $[-1,1]$ (so that $\mu=0$), let $f(\mathbf X,\delta):=1-1/10$.
Then
$
\Pr\left (\left | \mu -\frac{1}{n} \sum\_{i=1}^n X\_i \right | \leq f(\mathbf X, \delta) \right )... | 3 | https://mathoverflow.net/users/36721 | 243307 | 111,498 |
https://mathoverflow.net/questions/243316 | 0 | Let $1<p<2$. Let $(f\_{n})\_{n}$ be a normalized weakly null sequence in $L\_{p}$ such that the sequence $(f\_{n})\_{n}$ contains no subsequence that is equivalent to the unit vector basis of $l\_{p}$.
Question: Does $(f\_{n})\_{n}$ admit a subsequence $(f\_{k\_{n}})\_{n}$ such that $$\|\sum\_{n=1}^{m}a\_{n}f\_{k\_{... | https://mathoverflow.net/users/41619 | Weakly null sequences in $L_{p}(1<p<2)$ | This question, as stated, has an easy negative answer: $L\_p$, $1<p<2$ contains a weakly null sequence equivalent to the unit vector basis of $\ell\_p$, and so does not satisfy the mentioned condition.
| 0 | https://mathoverflow.net/users/85406 | 243320 | 111,499 |
https://mathoverflow.net/questions/243319 | 0 | In my research of operator algebras and their connection with machine learning I of course use the well know result:
>
> For the map $ tr:M\_n \to M\_n $ denoting the transpose map of matrices (meaning that $ tr(A)=A^{tr} $ we know for the completely bounded norm (cb-norm) that $ ||tr||\_{cb}=n $
>
>
>
I keep... | https://mathoverflow.net/users/89375 | A nice proof that completely bounded (cb) norm of transpose map on $ M_n $ is n | I literally just googled "cb norm of transpose" and got a link to [this math.stackexchange question](https://math.stackexchange.com/questions/1320435/calculating-norms-for-the-transpose) which links to a proof.
Edit: the answer which contained the link disappeared, so I used Google magic again and found [Tomiyama's o... | 3 | https://mathoverflow.net/users/23141 | 243324 | 111,500 |
https://mathoverflow.net/questions/243291 | -2 | (Note: This question is related to my previous mathoverflow question, "Critical Points in $ZF$ without Choice".)
In the *Stanford Encyclopedia of Philosophy* entry "Non-Wellfounded Set Theory" (Section 2.2, "The Foundation Axiom"), one has the following statement (my comments regarding it are in brackets):
>
> Th... | https://mathoverflow.net/users/20597 | Critical points and the Foundation Axiom | As far as I understand your question, this is the answer:
Suppose $V$ is a model of $ZF$ minus Foundation (call this theory "$ZF^-$"). Then the following are equivalent:
* $V$ satisfies Foundation - that is, $V$ is in fact a model of all of $ZF$.
* "$V=\bigcup\_{\alpha\in ON} V\_\alpha$" - that is, for each $x\in V... | 4 | https://mathoverflow.net/users/8133 | 243333 | 111,502 |
https://mathoverflow.net/questions/243279 | 4 | It is known that for every abelian scheme $A$ over a ring $R$, there exists a subring $R\_0$ of $R$ that is of finite type over $\mathbb{Z}$ and an abelian scheme $A\_0$ over $R\_0$ such that $A$ is deduced from $A\_0$ by base change.
Using the relevant theorems in EGA, I see how one can get $A\_0$ over $R\_0$ satisf... | https://mathoverflow.net/users/nan | Elimination of noetherian hypothesis for abelian schemes | I am just writing my comments as an answer. As nfdc23 explains, there are stronger results that require weaker hypotheses, but let me assume that $R\_0$ is a finitely generated algebra contained in $R$, and let $A\_0$ be a proper, flat $R\_0$-scheme whose geometric fibers are reduced and whose base change $A$ to $R$ ha... | 3 | https://mathoverflow.net/users/13265 | 243336 | 111,505 |
https://mathoverflow.net/questions/243314 | 4 | In my research in operator theory, specifically in C\* algebras and enveloping, I came across this strange footnote in a text (locally published in non English where I study) which states the following:
>
> Suppose we have X a compact topological space, now suppose we have A, a sub-algebra of the algebra of continu... | https://mathoverflow.net/users/69446 | The C*-envelope of the algebra of continuous functions on a compact topological space is commutative | What they are aiming at is the following result:
Let $A \subset C(X)$ be a uniform algebra. Then there exists a unique compact set $F \subset X$, known as the Shilov boundary w.r.t. to $A$, such that every function in $A$ achieves its maximum modulus on $F$. Moreover,
$$
C\_e^\*(A) \cong C(F).
$$
See chapter 16 in Co... | 6 | https://mathoverflow.net/users/94503 | 243340 | 111,508 |
https://mathoverflow.net/questions/243338 | 3 | I'm reading *Clifford Algebra to Geometric Calculus* by Hestenes, and struggling with an early result about reversion inside of a grade-projection operator.
It is noted that $A\_r$ and $B\_s$ are homogeneous multivectors of grades $r$ and $s$. Just to be sure I understand the basic concepts, a homogeneous multivector... | https://mathoverflow.net/users/94484 | How does grade projection act on homogeneous multivectors in geometric algebra? | I think that this is not the same notation as you will find elsewhere. Usually, the Clifford algebra is taken to be $\mathbb Z/2\mathbb Z$-graded, rather than $\mathbb Z$-graded, precisely because an apparently homogeneous multivector of grade $n \ge 2$ can often be traded for the sum of two multivectors, of degrees $n... | 4 | https://mathoverflow.net/users/2383 | 243346 | 111,509 |
https://mathoverflow.net/questions/243350 | 3 | Let $p>2$ and $X$ a subspace of $L\_{p}$.
Then Kadec and Pelczynski proved that either $X$ is isomorphic to $l\_{2}$ or $X$ contains a subspace isomorphic to $l\_{p}$.
>
> **Question:** if $X$ is isomorphic to $l\_{2}$, does $X$ contain a subspace that is $(1+\epsilon)$-isomorphic to $l\_{2}$?
>
>
>
| https://mathoverflow.net/users/41619 | Subspaces of $L_{p}(2<p<\infty)$ | Yes. That follows, e.g., from the Krivine-Maurey theory of stable spaces even if it was known before their work. For $p<2$ you get from their theory, and more or less classical considerations, Aldous' theorem that every subspace of $L\_p$ contains for every $\epsilon > 0$ a subspace that is $1+\epsilon$-isomorphic to $... | 2 | https://mathoverflow.net/users/2554 | 243356 | 111,511 |
https://mathoverflow.net/questions/243361 | 11 | Is there any literature regarding the fastest known algorithm to compute the homology groups of a simplicial complex (on n vertices)? What about computing the fundamental group? The context is to tell whether a given simplical complex is contractible by showing that the fundamental group and all reduced homology groups... | https://mathoverflow.net/users/90324 | Computational complexity of computing simplicial homology | Homology groups can be computed with Smith normal form (see [this survey](http://ljk.imag.fr/membres/Jean-Guillaume.Dumas/Publications/DHSW.pdf)). As for deciding if a simplicial complex is contractible, that is difficult. It is undecidable to tell if a simplicial complex is contractible (see appendix A of [this paper]... | 14 | https://mathoverflow.net/users/51668 | 243362 | 111,512 |
https://mathoverflow.net/questions/243363 | 5 | I am working on a project which requires that I calculate homotopy limits of homotopy theories (i.e. $(\infty,1)$-categories). It may be relevant that the homotopy limits which interest me are in the shape of towers; that is, the indexing category looks like $\cdots\rightarrow\cdot\rightarrow\cdot$. Because I am intere... | https://mathoverflow.net/users/28033 | Methods for defining/calculating homotopy limits of quasicategories | When working with quasi-categories, it is often more convenient (and more compatible with existing machinery) not to work with actual strict diagrams of quasi-categories but rather with **coCartesian fibrations**. In your case this would be a coCartesian fibration of the form $\pi:\mathcal{C} \to N(I)$ where $I$ is you... | 6 | https://mathoverflow.net/users/51164 | 243369 | 111,513 |
https://mathoverflow.net/questions/243312 | 9 | I would like to know whether the following metatheorem on nonabelian $H^2$ has been ever stated and/or proved.
Let $k$ be a perfect field and $k^s$ its fixed separable closure.
Let $X^s$ be a *variety with additional structure* over $k^s$
(I don't want to specify what I mean by additional structure).
By a $k$-model o... | https://mathoverflow.net/users/4149 | Nonabelian $H^2$ and Galois descent | Let me elaborate more on the remark above. Let $k$ be a perfect field. Let $\mathrm{Field}\_k$ denote the category of finite extensions of $k$, i.e., the objects of $\mathrm{Field}\_k$ are fields $k'$ equipped with an embedding $k \to k'$ such that $k'$ is finite dimensional over $k$. The morphisms are the maps of fiel... | 9 | https://mathoverflow.net/users/51164 | 243372 | 111,514 |
https://mathoverflow.net/questions/243321 | 4 | Let $H$ be a CM field and $F$ be the maximal totally real subfield of $H$. Can we construct a Katz $p$-adic L-functions of Hecke characters without the ordinary condition (i.e every prime of $F$ above $p$ splits in $M$)?
| https://mathoverflow.net/users/46460 | Katz $p$-adic L function and ordinary condition | A p-adic L-function is expected to depend on (at least) two pieces of data: a family $V$ of representations of $G\_{\mathbf{Q}}$ over some base space $X$ (which should be a p-adic formal scheme or rigid space); and a family of subspaces $V^+$ of $V$ stable under $G\_{\mathbf{Q}\_p}$ (a "p-refinement" or "p-stabilisatio... | 6 | https://mathoverflow.net/users/2481 | 243375 | 111,515 |
https://mathoverflow.net/questions/243345 | 0 | Let us consider the polynomial ring $\Bbb C[x\_1,...,x\_s]$ and $\alpha(x\_i)= x\_i + \mu\_i$ where $\mu\_i \in \Bbb C$ are not all zero. Then $\alpha \in \mathrm{Aut}(\Bbb C[x\_1,...,x\_s])$.
>
> What are the fixed points of $\alpha^n-\mu\_j$ for a fixed $j$, i.e. what are the $a\_j \in \Bbb C[x\_1,...,x\_s]$ s.t... | https://mathoverflow.net/users/85472 | What are the fixed points of $\alpha^n-\mu_j$ for a fixed $j$? | Let $L$ be the line generated by $\underline{\mu} = (\mu\_1,\dots,\mu\_s)$ in $V = \mathbb{C}^s$, and consider a projection $p : V \rightarrow V$ onto $L$, with kernel $H$. Any polynomial function of the form
$$
a\_j(\underline{x}) = b(\underline{x} - p(\underline{x})) + \ell\_j( p (\underline{x})),
$$
where $b$ is a p... | 1 | https://mathoverflow.net/users/21724 | 243378 | 111,517 |
https://mathoverflow.net/questions/243373 | 1 | This relate to that paper:
<http://www.stat.purdue.edu/docs/research/tech-reports/1982/tr82-17.pdf>
Let $U\_1,...,Un$ be iid uniform on (0,1).
Set $L\_n=\max\_{i\leq n} U\_i$.
Also $S(n)= \inf\{i\leq n| U\_i = L\_n \}$ the time were the highest value is attained and
$Z(n)= \inf\{i\leq n| U\_i = L\_{S(n)-1} \}... | https://mathoverflow.net/users/92127 | Question on a random vector | For $1 \leq s' < s \leq n$ and $t,t' \in \mathbb{R}$, one has
$$
n(1-L\_n) > t, (S(n)-1)(1-\frac{L\_{S(n)-1}}{L\_n}) > t',S(n)=s, Z(n) = s'
$$
precisely when
$$
U\_1,\dots,U\_{s'-1} < U\_{s'},\\
U\_{s'+1},\dots,U\_{s-1} \leq U\_{s'}, \\
U\_{s+1},\dots,U\_{n} \leq U\_{s},\\
U\_{s'} < \left(1 - \frac{t'\_+}{s-1} \right)... | 1 | https://mathoverflow.net/users/21724 | 243382 | 111,518 |
https://mathoverflow.net/questions/243364 | 1 | Let $p$ be a prime, $\mathbb C\_p$ be the completion of a algebraic closure of $\mathbb Q\_p$ and $(u\_n)\_{n\in\mathbb N}$ be a sequence of $\mathbb C\_p$ converging towards $0$. Suppose that for all $n\in\mathbb N$, one has $1+u\_n\ne0$. Can $\prod\_{n\in\mathbb N}(1+u\_n)$ be zero?
I know that is trivially true in... | https://mathoverflow.net/users/33128 | null infinite product in the p-adic setting | If $n\_0$ is such that $|u\_n|<1$ for $n \geq n\_0$, then $|1+u\_n| = 1$ for $n \geq n\_0$, so the norm of the infinite product is the norm of the product of the first $n\_0$ terms and hence nonzero.
| 3 | https://mathoverflow.net/users/5743 | 243384 | 111,519 |
https://mathoverflow.net/questions/243383 | 6 | In his paper [Automorphic forms with singularities on Grassmannians](https://arxiv.org/abs/alg-geom/9609022), Borcherds poses Problem 16.5:
"Describe how the correspondence in this paper behaves under
the
action of Hecke operators."
Since the "correspondence in this paper" is the construction of special orthogonal... | https://mathoverflow.net/users/nan | Definition of Hecke operators on orthogonal modular forms | For any reductive group $G$ over a number field $F$, so in particular for the orthogonal group of a quadratic form, there exists a theory of Hecke operators. Let's say you have a congruence arithmetic group $\Gamma\subset G({\mathbb Q})$ and $\alpha\in G({\mathbb Q})$. Let's also say that a modular form is a $\Gamma$-l... | 5 | https://mathoverflow.net/users/nan | 243388 | 111,522 |
https://mathoverflow.net/questions/243389 | 6 | In the his book
*Linear algebraic groups*, by T.A. Springer,
there is a list of possible Tits-Indexes. For the $E\_7$ case, there is an index shown, such that vertex $1$ and $7$ are circled (Bourbaki notation). I just realized that yesterday by coincidence. However in Tits's original paper this index is not listed.... | https://mathoverflow.net/users/51251 | Wrong Tits-Index of E7 from Springer 's book | That is a typo. In index 14 on p. 321 the vertex 6 should be black. In Proposition 17.8.2, this index is correct. It corresponds to $E\_{7,3}^{28}$ in Tits' notation.
| 11 | https://mathoverflow.net/users/89948 | 243390 | 111,523 |
https://mathoverflow.net/questions/243377 | 6 | Consider a random walk on the real time, starting from $0$. But this time assume that we can decide, for each step $i$, a step size $t\_i>0$ to the left or the right with equal probabilities.
To formalize this, we have $(X\_n)\_{n\geq 0}$ such that $X\_0=0$, $Pr[X\_n=t\_n]=0.5$ and $Pr[X\_n=-t\_n]=0.5$ (hence still ... | https://mathoverflow.net/users/37612 | Random walk to stay in an interval forever | Yes. Indeed, if $s = \sum\_{i \geq 1} t\_i^2 <1$, then
$$
\mathbb{P}[ \ \ \forall n, \sum\_{i=1}^n X\_i \in [-1,1] \ \ ] \geq 1-s > 0.
$$
To see this, note that $M\_n = |\sum\_{i=1}^n X\_i|$ is a nonnegative submartingale, so that Doob's martingale inequality yields
$$
\mathbb{P}[ \max\_{1 \leq j \leq n} M\_j > 1 ] \l... | 10 | https://mathoverflow.net/users/21724 | 243391 | 111,524 |
https://mathoverflow.net/questions/243400 | 2 | Suppose $(X,\mathcal A,\mu,T)$ is a finite measure-preserving system. Then we define a new measure system $(X^{(K)},\mathcal A^{(K)},\mu^{(K)},T^{(K)})$ defined by $X^{(K)}=X\times \{1,2,...,K\}$ for any positive integer $K$, $\mu^{(K)}(A\times \{k\})=\dfrac{\mu(A)}{K}$ for $1\leq k\leq K$ and $A\in\mathcal A$, and def... | https://mathoverflow.net/users/66278 | Measurable isomorphism between two non-totally ergodic systems | Assume $(Y,\nu,S)$ is ergodic (but not totally ergodic), let $T = S^K$, where $K$ is the least positive integer $n$ for which $S^n$ is not ergodic, and let $Z$ be an ergodic component of $(Y,\nu,T)$. Since $S$ is ergodic, the ergodic decomposition of $(Y,\nu,T)$ is $Y = Z \cup S Z \cup \cdots \cup S^{K-1} Z$. So it is ... | 3 | https://mathoverflow.net/users/68305 | 243408 | 111,528 |
https://mathoverflow.net/questions/243402 | 6 | *This possibly a very basic descriptive set-theory question; if it is too basic for MO, feel free to migrate.*
Throughout we work in ZF+AD. My question is:
>
> If $A$ is an uncountable OD set of reals, need $A$ have an OD perfect subset?
>
>
>
Motivation: The *Solovay sequence* is given by:
* $\theta\_0$... | https://mathoverflow.net/users/8133 | Ordinal-definable witnesses to the perfect set property? | Let $A$ be the set of those reals that are *not* OD. Then $A$ is OD (since I've just defined it), and it's uncountable (since its complement, being a well-orderable set of reals, must be countable under AD). But any perfect OD set $P$ has an element that is OD and thus outside $A$, namely the first element of $P$
(in t... | 12 | https://mathoverflow.net/users/6794 | 243409 | 111,529 |
https://mathoverflow.net/questions/243401 | 1 | Is a (Cartier) divisor on a variety uniquely determined by its restriction to curves inside the variety? If so, how do we see this?
| https://mathoverflow.net/users/16356 | Divisor on variety determined by its restriction to curves | If you assume projective and smooth, this is not hard. Induct on dimension, $\dim X=1$ being the hypothesis. If $\dim X=n\geq 2$ and result proved for smaller dimensions, if $Y\in \mathcal{O}(mH)$, $m>>0$, $H$ a hyperplane section, then we may assume $Y$ is smooth and by induction, for the line bundle $L$ in question, ... | 6 | https://mathoverflow.net/users/9502 | 243411 | 111,530 |
https://mathoverflow.net/questions/243380 | 8 | Let $A/S$ be an abelian scheme such that the dual abelian scheme $A^{\vee}/S$ exists and let $\lambda : A \to A^{\vee}$ be a morphism of abelian schemes. Is the locus of points in $S$ where $\lambda$ is a polarization open?
EDIT: Recall that a polarisation on $A/S$ is a morphism of abelian schemes $\lambda :A \to A^t... | https://mathoverflow.net/users/nan | Is a polarization on an abelian scheme an open condition? | A nice way to understand this is to give a "better" characterization of polarizations that avoids any reliance on structures that only are available on geometric fibers. The claim is that a homomorphism $\lambda:A \rightarrow A^t$ is a polarization if and only if it satisfies the following conditions: (i) $\lambda$ is ... | 8 | https://mathoverflow.net/users/81332 | 243414 | 111,532 |
https://mathoverflow.net/questions/243403 | 14 | Let $X$ be the Fermat quartic $x^4+y^4+z^4+w^4=0$ in $\mathbb P^3$. It is known that $X$ contains infinitely many $(-2)$-curves, that is, smooth rational curves. (One way to obtain in infinitely many is to use the various elliptic fibrations on $X$, and use translations in the fibers.) Note however that these curves ar... | https://mathoverflow.net/users/94548 | Rational curves on the Fermat quartic surface | I am posting my comments as an answer. I am concerned that I misunderstand the OP, so let me state first the result. There exists a finite field extension $K/\mathbb{Q}$ such that for every closed immersion $\mathbb{P}^1\_\mathbb{C} \hookrightarrow X\otimes\_{\mathbb{Q}}\mathbb{C}$, there exists a closed immersion $\ma... | 13 | https://mathoverflow.net/users/13265 | 243415 | 111,533 |
https://mathoverflow.net/questions/243397 | 11 | In ["The mixing time of the giant component of a random graph"](http://arxiv.org/abs/math/0610459) by the aforementioned authors, in the last proof on page 19 it says something along the lines of
"It is well known and easy to verify that, since $m=O(n)$
, asymptotically almost surely the maximum degree occurring in $... | https://mathoverflow.net/users/60768 | Question on a paper by Benjamini/Kozma/Wormald about a “well known fact” | I have no idea where the figure $n^{0.02}$ comes from. I would usually say it's well known that the maximum degree is $O(\log n)$ (actually even this is an overestimate). It's for example found in an Erd\H{o}s-R\'enyi paper with a title about random matrices, I think; or surely in any of the 'Random Graphs' books.
In... | 10 | https://mathoverflow.net/users/36212 | 243416 | 111,534 |
https://mathoverflow.net/questions/243368 | 3 | Asking [this](https://mathoverflow.net/questions/241946/on-the-transitivity-of-the-action-of-the-unitary-group) question I have made a mistake joining my main question with two simple ones, so it hasn't received enough attention, however there was a partial answer, which was not elaborated, and now I am completely lost... | https://mathoverflow.net/users/53155 | A question on linear groups | Here's a complete proof of:
>
> Every subgroup of $\mathrm{GL}\_n(\mathbf{R})$ containing $\mathrm{SO}\_n(\mathbf{R})$ is either contained in the group of similarities $\mathbf{R}^\*\mathrm{O}\_n(\mathbf{R})$, or contains $\mathrm{SL}\_n(\mathbf{R})$.
>
>
>
(This is equivalent to the statement that for every $... | 6 | https://mathoverflow.net/users/14094 | 243433 | 111,537 |
https://mathoverflow.net/questions/243443 | 7 | Let $p(x)$ be a degree $n$ polynomial over $[-1, 1]$, and let $q(x) = p'(x) \sqrt{1-x^2}$. Is it true that
$$
\|q\|\_1 \leq O(n) \|p\|\_1
$$
where we define $\|f\|\_p := \left(\int\_{-1}^1 |f(x)|^pdx\right)^{1/p}$?
For reference, Bernstein's inequality shows that
$$
\|q\|\_\infty \leq n\|p\|\_\infty
$$
with equality ... | https://mathoverflow.net/users/94567 | L1 analog of Bernstein's inequality | Appendix A4 of the book
>
> P. Borwein, T. Erdelyi, Polynomials and Polynomial inequalities, Graduate Texts in Mathematics 161, Springer
>
>
>
should be a good source for your question. In particular, (A.4.22) gives
$$\|P'\|\_p\leq cn^2\|P\|\_p,$$
for every polynomial $P$ of degree $n$ and $0<p<\infty$. Appar... | 4 | https://mathoverflow.net/users/89429 | 243447 | 111,544 |
https://mathoverflow.net/questions/243452 | 3 | Let $G=SL\_n$ and let $P\_i$ be a maximal parabolic corresponding to a simple root say $\alpha\_i$. Let $W\_{P\_i}$ be the Weyl group of $P\_i$. Is there an efficient way to compute the longest coset representative $w^{max}$ for $wP\_i$ ? How can we go from $w^{min}$ to $w^{max}$ ? It seems there is a command to find t... | https://mathoverflow.net/users/94573 | Maximal Coset representative for the Weyl group of a Parabolic | The Weyl group is the symmetric group $S\_n$. The Weyl group of $P\_i$ is $S\_i\times S\_{n-i}$ acting on the right. Let
$$
w=(a\_1,\ldots,a\_i,a\_{i+1},\ldots,a\_n)\in S\_n
$$
Then $w^{min}$ (or $w^{max}$) is obtained by bringing the first $i$ and the last $n-i$ entries into an increasing (or decreasing, respectively)... | 7 | https://mathoverflow.net/users/89948 | 243453 | 111,546 |
https://mathoverflow.net/questions/243449 | 6 | By a *partial function* from $\omega$ to $\omega$ we understand a function $f:dom(f)\to\omega$ defined on an infinite subset of $\omega$.
A family $\mathfrak F$ of partial functions from $\omega$ to $\omega$ is called *domain almost disjoint* (briefly, *DAD*) if the family $(dom(f))\_{f\in\mathcal F}$ is almost disj... | https://mathoverflow.net/users/61536 | Is $\mathfrak b_a$ a new cardinal characteristic of the continuum? | After thinking some time I realized that $\mathfrak b\_a=\mathfrak b$, so this question has answer "No" and this "No" does not help to solve the o[riginal question](https://mathoverflow.net/questions/243365/cofinal-monotone-maps-from-omega-omega-to-kappa-kappa).
To show that $\mathfrak b\_a=\mathfrak b$, consider the... | 3 | https://mathoverflow.net/users/61536 | 243455 | 111,547 |
https://mathoverflow.net/questions/243432 | 2 | Let $1<p<\infty$. Johnson and Schechtman (Multiplication operators on $L(L\_{p})$ and $l\_{p}$-strictly singular operators, 2008, DOI: [10.4171/JEMS/141](http://dx.doi.org/10.4171/JEMS/141), [eudml](https://eudml.org/doc/277694), [arxiv](http://arxiv.org/abs/0708.0560)) observed that if $(x\_{n})\_{n}$ is a sequence in... | https://mathoverflow.net/users/41619 | Sequences in $L_{p}(1<p<\infty)$ that is equivalent to the unit vector basis of $l_{p}$ or $l_{2}$ | The answer is no. Let $2<p<\infty$ and in $\ell\_p \oplus\_p \ell\_2$ (which embeds isometrically into $L\_p$) consider $x\_n := e\_n \oplus \alpha \delta\_n$, where $(e\_n)$; respectively, $(\delta\_n)$, is the unit vector basis of $\ell\_p$; respectively, $\ell\_2$, and $0<\alpha <1$. One can show that the norm of an... | 1 | https://mathoverflow.net/users/2554 | 243457 | 111,549 |
https://mathoverflow.net/questions/243425 | 7 | We consider the following simple fact about matrices. Then we try to generalize it in the context of smooth manifolds;
Let $L$ be the collection of all $n \times n$ real matrices $A=(a\_{ij})$ with the following property:
$$\sum\_{i=1}^{n} a\_{ij}=0$$ for every fixed $j$.
Obviousely $L$ is a Lie algebra.([As I ha... | https://mathoverflow.net/users/36688 | Infinite dimensional version of a simple fact on certain singular matrices | For the first question, the answer is not necessarily.
**Very rough idea**: The rank-nullity theorem doesn't always hold on infinite dimensional spaces.
**Rough idea**: Let the operator $A$ be defined on $L^2(M)$ be a injective mapping such that its range does not include the constant function. More precisely, si... | 5 | https://mathoverflow.net/users/3948 | 243462 | 111,550 |
https://mathoverflow.net/questions/243202 | 5 | Let $A$ be an abelian variety and $\hat A$ be the dual abelian variety. If $P$ is the (normalized) Poincare line bundle, then Mukai defines $R\hat S:D(A)\to D (\hat A)$ via $R\hat S(?)=Rp\_{\hat A,\*}(Lp\_A^\*(?)\otimes P)$ and $RS:D(\hat A)\to D(A)$ via $R S(?)=Rp\_{ A,\*}(Lp\_{\hat A}^\*(?)\otimes P)$. He then shows ... | https://mathoverflow.net/users/19369 | Fourier Mukai transform for non-quasi coherent sheaves | I think it is clear that $R\hat{S}$ is not an equivalence on the categories of all $O$-modules. Indeed, if it were an equivalence, it would send product to product, and also preserve quasi-coherence. Let $M\_x$ be a sky-scraper sheaf at $x\in A$; its image under $R\hat{S}$ is an invertible $O$-module on $A^\vee$ (up to... | 3 | https://mathoverflow.net/users/2653 | 243464 | 111,551 |
https://mathoverflow.net/questions/243431 | 5 | Let $g \geq 1$ be a positive integer, and let $p$ be a prime. Consider the symplectic group $G := \operatorname{Sp}\_{2g}(\mathbb{F}\_p)$ of symplectic matrices with entries in $\mathbb{F}\_p$. Let $M \subset G$ be a maximal subgroup and let $S = \bigcup\_{h \in G} hMh^{-1}$.
Question: Is it true that $|S| = \alpha \... | https://mathoverflow.net/users/75970 | Bounding the union of conjugates of a maximal subgroup of the Symplectic group over a finite field | Let $\Omega$ be the set of cosets of $M$, and consider the natural action of $G$ on $\Omega$. The set $S^C$ (the complement of $S$) is the set of *derangements* in this action.
So the upper bound you seek is equivalent to a lower bound on the proportion of derangements in the action of $G$ on $\Omega$. The lower boun... | 5 | https://mathoverflow.net/users/801 | 243466 | 111,553 |
https://mathoverflow.net/questions/243293 | 2 | Consider the surface group $\Gamma=\langle a,b,c,d\mid [a,b][c,d]=1\rangle$: it is a Gromov hyperbolic group; its Gromov boundary $\partial\Gamma$ is homeomorphic to $S^1$ (the unit circle).
I would like to define a family of subsets of $\partial\Gamma$ as follows: fix $x,y\in\Gamma$. Then
$$
U(x,y):=\left\{\xi\in\part... | https://mathoverflow.net/users/80571 | Subsets of the boundary of a surface group | You definition is closely related to the notion of the "cone type" introduced by Jim Cannon. As Yves noted, $U(x,y)$ is compact. It is also connected.
1. Compactness part is immediate from the Arzela-Ascoli theorem: Take a sequence of rays $r\_i: [0,\infty)\to X$ (where $X$ is the Cayley graph; here it does not matt... | 3 | https://mathoverflow.net/users/21684 | 243468 | 111,554 |
https://mathoverflow.net/questions/243365 | 11 | Given a cardinal $\kappa$ consider the set $\kappa^\kappa$ of all functions from $\kappa$ to $\kappa$, endowed with the partial order $f\le g$ iff $f(\alpha)\le g(\alpha)$ for all $\alpha\in\kappa$.
A map $f:P\to Q$ between two partially ordered sets is called
* *monotone* if for any points $x\le y$ in $P$ we get ... | https://mathoverflow.net/users/61536 | Cofinal monotone maps from $\omega^\omega$ to $\kappa^\kappa$ | My former doctoral student Lubomyr Zdomskyy has resolved this problem, noticing that adding $\omega\_2$ Cohen reals to a model of GCH produces a model in which the cardinal $\omega\_1$ admits a monotone cofinal map $\omega^\omega\to(\omega\_1)^{\omega\_1}$.
Alternatively the same result can be derived from the exist... | 5 | https://mathoverflow.net/users/61536 | 243470 | 111,555 |
https://mathoverflow.net/questions/214883 | 16 | This is a very naive question. I have the impression that the area of "Spin geometry" is not an active research field. Sure Spin geometry is used in many different branches of mathematics and physics as a tool, but I don't see papers published on the development of Spin geometry by itself. Is this true? Loosely speakin... | https://mathoverflow.net/users/66688 | Open questions in "Spin geometry" | As @314159 has explained, Spin geometry on "spin manifolds" is a very active field of research, but the subject goes way beyond the domain spin manifolds. The key point to notice is that every pseudo-Riemannian spin manifold $(M,g)$ admits a bundle of irreducible Clifford modules over the bundle of Clifford algebras $C... | 8 | https://mathoverflow.net/users/94585 | 243471 | 111,556 |
https://mathoverflow.net/questions/243460 | 8 | Let $A$ be a unital algebra over a field $K$, $C^n(A)$ a space of all $n+1$ linear maps into scalar field $k$ (I'm interested in case $k=\mathbb{C}$) and
$$(bf)(a\_0,...,a\_{n+1})=\sum\_{i=0}^n(-1)^if(a\_0,...,a\_ia\_{i+1},...,a\_{n+1})+(-1)^{n+1}f(a\_{n+1}a\_0,a\_1,...,a\_n)$$
for $f \in C^n(A)$. One checks that $b^2... | https://mathoverflow.net/users/24078 | Isomorphism in cyclic cohomology vs isomorphism in Hochschild cohomology | This was something that used to puzzle me when I was first learning this stuff. Refreshing my memory just now, I think that the trick is to use the fact that the Connes–Tsygan sequence has some zero entries in low degrees. It starts
$$ 0 \to HC^0(A) \to HH^0(A) \to 0 \to HC^1(A) \to HH^1(A) \to HC^0(A) \to HC^2(A) \to ... | 9 | https://mathoverflow.net/users/763 | 243473 | 111,557 |
https://mathoverflow.net/questions/242548 | 22 | Let $X\_1,...,X\_n$ be independent uniform random variables in [0,1] and assume $c>1/2$. Is it true that $$\mathbb{P}\left[\sum\_{i=1}^n X\_i \leq n \cdot c\right]$$ is increasing with respect to $n$?
I know that the sum of uniform independent random follows Irwin–Hall distribution, but it seems hard to work with the... | https://mathoverflow.net/users/nan | On the sum of uniform independent random variables | The answer is yes. Let $S\_n:=\sum\_{i=1}^n X\_i$, $x\in\mathbb R$, $n=2,3,\dots$, and
\begin{equation}
G\_n(x):=P(S\_n/n\le x)=\frac1{n!}\,\sum\_j(-1)^j\binom nj (nx-j)\_+^n,
\end{equation}
where $u\_+:=0\vee u$; cf. **[[Irwin--Hall distribution](https://en.wikipedia.org/wiki/Irwin%E2%80%93Hall_distribution)]**.
T... | 5 | https://mathoverflow.net/users/36721 | 243483 | 111,561 |
https://mathoverflow.net/questions/242632 | 3 | The technique of quiver folding (please see [Folding by Automorphisms](http://www.math.lsa.umich.edu/~jrs/papers/folding.pdf)) can be used to prove statements about non-simply laced quivers (i.e. valued quivers) when they are already known in the simply-laced case.
I wonder whether we can use this technique to fold ... | https://mathoverflow.net/users/73892 | Quiver folding and maximal green sequences | This will certainly work fine in finite type. Folding $Q$ to $Q'$ corresponds to an inclusion of $W'$ into $W$, where the reflections of $W'$ are mapped to products of commuting reflections in $W$. $c'$-sortable elements of $W'$ give you a sublattice of the $c$-sortable elements of $W$. Thus, if you take a maximal gree... | 3 | https://mathoverflow.net/users/468 | 243493 | 111,567 |
https://mathoverflow.net/questions/243508 | 5 | Is there a natural example of a discrete subgroup $\Gamma\leq PSL\_2(\mathbf{R})$ such that
(1) $\Gamma$ has finite covolume
(2) $\mathfrak{h}/\Gamma$ is not compact ($\mathfrak{h}$ being the upper half-plane)
(3) $\Gamma$ is **not** commensurable to a conjugate of $PSL\_2(\mathbf{Z})$.
I cannot think of any su... | https://mathoverflow.net/users/11765 | Examples of discrete subgroups of $PSL_2(\mathbf{R})$ with finite covolume and which are not co-compact | I would guess that as soon as you have more than three cusps, the Teichmuller space associated to the surface is non trivial (a complex manifold with dimension > 0) whereas there are countably many (up to isometry) surfaces whose associated group is commensurable to $PSL\_2(\mathbf{Z})$.
So most discrete groups associa... | 10 | https://mathoverflow.net/users/6129 | 243511 | 111,572 |
https://mathoverflow.net/questions/207266 | 14 | I am looking for the integrability condition of the following system of pde:
$$\partial\_{[\nu}\Gamma^\kappa\_{\mu]\lambda}+\Gamma^\kappa\_{[\nu|\rho|}\Gamma^\rho\_{\mu]\lambda}=\frac{1}{2}R\_{\mu\nu\lambda}{}^{\kappa},\,\,\,\,\,\,\,\,\,(1)$$
given sufficiently smooth functions $R\_{\mu\nu\lambda}{}^{\kappa}$ on so... | https://mathoverflow.net/users/25516 | Does the Riemann-Christoffel curvature determine the connection? | Part of the difficulty in providing an answer to your question is the fact that the expression "the integrability condition" is a somewhat vague notion, and it's used in slightly different senses in different contexts.
The usual, somewhat imprecise, sense is that, for a given system of PDE, its 'integrability condit... | 5 | https://mathoverflow.net/users/13972 | 243517 | 111,575 |
https://mathoverflow.net/questions/243477 | 12 | $\newcommand{\talg}{\mathcal{T}(V)}$$\newcommand{\clalg}{\mathcal{Cl}\_q(V)}$$\newcommand{\qalg}{\mathcal{I}\_q(V)}$**Is there a way to embed Clifford algebras into the corresponding tensor algebra?**
There are simple and straightforward embeddings of the underlying vector space $V$ into its corresponding tensor alge... | https://mathoverflow.net/users/93694 | Is there a way to embed Clifford algebras into the corresponding tensor algebra? | $\newcommand{\qalg}{\mathcal{I}\_q(V)}$As K. Conrad points out, this question has actually been answered already on MathOverflow by user MTS: see these answers [here](https://mathoverflow.net/a/68411/93694) and [here](https://mathoverflow.net/a/60606/93694).
The essential idea is this: despite the fact that Clifford ... | 5 | https://mathoverflow.net/users/93694 | 243524 | 111,577 |
https://mathoverflow.net/questions/243505 | 0 | Assume that $H$ is an infinite dimensional Hilbert space.The space of all bounded operators on $H$ is denoted by $B(H)$.We consider the Lie algebra structure $[T,S]=TS-ST$ on $B(H)$.
>
> Is there a bounded linear operator $\phi: B(H)\to H$ such that$ \ker \phi$
> is a Lie subalgebra but $\phi$ is not in the form o... | https://mathoverflow.net/users/36688 | Bounded operators $T: B(H)\to H$ whose Kernel is a Lie algebra | The answer is yes :
Choose injective $A \in B(H)$ and $h \in H$ such that $h$ and $Ah$ are linear independent and define $\phi(T) = A T h$.
Then it is easy to see that there doesn't exist a $k \in H$ such that $A T h = T k$ for all $T \in B(H)$ .
| 1 | https://mathoverflow.net/users/17261 | 243531 | 111,580 |
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