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 |
|---|---|---|---|---|---|---|---|---|---|
https://mathoverflow.net/questions/206219 | 5 | Can someone give me a proof sketch for this:
Let $\mathscr{P}\_n$ be the set of all graphs which do not contain a path on $n$ vertices as a subgraph. Define the type of a graph inductively as: the type of the single vertex graph is $1$. The type of a graph $G$ is at most $n$ if there exists a vertex $v\in G$ such that ... | https://mathoverflow.net/users/43701 | A claim from "Graph minors - a survey" by Robertson and Seymour | This is trickier than it seems.
**Definition.** Let $\mathscr{R}\_n$ denote the set of all *connected* graphs $G$ for which there is a vertex $v\in G$ such that $G$ contains no path on $n$ vertices starting at $v$.
**Proposition 1.** For $n\geq 2$, we have $\mathscr{R}\_n\subseteq\mathscr{T}\_{n-1}$.
**Proof.**... | 8 | https://mathoverflow.net/users/11919 | 206228 | 98,681 |
https://mathoverflow.net/questions/206222 | 0 | Let $A \subseteq B$ and $A \subseteq C$ be commutative noetherian domains.
Assume that $A$ and $C$ are regular rings (=every localization at a maximal ideal is a regular local ring).
Assume that $B$ is a free $A$-module of finite rank $r$, so $B \cong \oplus\_{i=1}^{r} A$ as $A$-modules.
Is it possible to decide whethe... | https://mathoverflow.net/users/72288 | Regularity of a tensor product | Alright, I'm posting this as an answer.
As in the comments above, there are counterexamples to your general question.
As for the specific situation, unless you want to assume something more on $p$, $q$ and $w$ I guess it's still false: assume $p=x$ and $q=y$, so that $A=k[x,y]$, and take $B=k[x,y,w]/(w^2-x)$, $M=(x... | 3 | https://mathoverflow.net/users/5516 | 206236 | 98,682 |
https://mathoverflow.net/questions/205923 | 4 | This question is connected with my previous: [Heisenberg group: function without vertical derivative](https://mathoverflow.net/q/204537/15946).
Here I am trying to look from another side: what is a difference between Sobolev space and horizontal Sobolev space on Carnot group?
Outline some notations.
A **Carnot gro... | https://mathoverflow.net/users/15946 | Horizontal Sobolev space on Carnot group | If $V\_1 \ne \mathfrak{g}$, then $H W^{1, p} \ne W^{1, p}$.
The construction is based on the notion of dilation on the Carnot group $\mathbb{G}$. The dilation $\delta\_r$ is defined by for $X \in V\_i$ by
$$
\delta\_r \exp (t X) = \exp (i \log r X).
$$
This dilation is an automorphism on $\mathbb{G}$.
If $\mu$ is t... | 3 | https://mathoverflow.net/users/42047 | 206259 | 98,692 |
https://mathoverflow.net/questions/206221 | 0 | Let $A \subseteq B$ be commutative noetherian domains.
Of course, if $M$ is an $A$-module, then $M \otimes\_A B$ is a $B$-module.
I am curious to know if there exist additional conditions on $A$ and $B$, such that every $B$-module $N$ is necessarily of the form $M \otimes\_A B$ for some $A$-module $M$.
I do not mind ... | https://mathoverflow.net/users/72288 | When every module is a scalar extension? | An example: if $B$ is the henselization of a local ring $A$, then for any finite type $B$-module $N$ there exists a finite type $A$-module $M$ such that $N$ is a direct summand of $M\otimes\_AB$. You can find this result and some similar others in <http://arxiv.org/pdf/0707.4197v3.pdf>
I think there are more recent p... | 1 | https://mathoverflow.net/users/36672 | 206261 | 98,693 |
https://mathoverflow.net/questions/206016 | 6 | Let me first recall some pretty standard notations:
* $\text{cov}(\mathcal{M})$ is the covering number of the ideal $\mathcal{M}$ of all meager subsets of $\mathbb{R}$;
* $\mathfrak{b}$ is the bounding number;
* $\mathfrak{s}$ is the splitting number.
It is well-known that there are no relations in ZFC between thos... | https://mathoverflow.net/users/15860 | $\text{cov}(\mathcal{M})$ vs. $\mathfrak{b}$ vs. $\mathfrak{s}$ | In Diego Mejía's reference (mentioned by Habic)
<http://arxiv.org/abs/1305.4739>
there are models for triples $(\kappa,\lambda,\mu)$ for
1) $\mathfrak{s}=\kappa<\mathfrak{b}=\mathrm{cov}(\mathcal{M})=\lambda<\mathfrak{c}=\mu$,
2) $\mathfrak{s}=\mathfrak{b}=\kappa<\mathrm{cov}(\mathcal{M})=\lambda<\mathfrak{c}=... | 3 | https://mathoverflow.net/users/67193 | 206263 | 98,694 |
https://mathoverflow.net/questions/206270 | 27 | I'm guessing everyone is familiar with [Four Color Theorem](https://en.wikipedia.org/wiki/Four_color_theorem) which was proved by Appel and Haken using computers. A weaker version of this theorem is [Five Color Theorem](https://en.wikipedia.org/wiki/Five_color_theorem) which states that a planar graph is 5-colorable.
I... | https://mathoverflow.net/users/41938 | Algebraic proof of Five-Color Theorem using chromatic polynomials by Birkhoff and Lewis in 1946 | I tried to go through Birkhoff and Lewis many years ago but it is not easy because they use different variables and the style is so different to modern proofs.
In modern terms, the key idea is that if a graph contains a vertex $v$ of degree k, then you can get an expression of the form
$$P(G,\lambda) = (\lambda-k) P(... | 34 | https://mathoverflow.net/users/1492 | 206278 | 98,700 |
https://mathoverflow.net/questions/206256 | 3 | Let $X$ be an algebraic variety over an algebraically closed field $k$ of characteristic $0$.
A locally trivial $\mathbf{A}^n$-fibration is a morphism $\pi \colon Y \to X$ such that $\pi^{-1}(U)\cong U\times \mathbf{A}^n$, and that $\pi\colon U\times \mathbf{A}^n \to U$ is the projection onto the second factor, for eve... | https://mathoverflow.net/users/38052 | Chow groups of locally trivial affine fibrations | Not sure if your argument is correct, but the statement is Lemma 2.2 in Totaro's Group cohomology and algebraic cycles.
| 1 | https://mathoverflow.net/users/60618 | 206285 | 98,702 |
https://mathoverflow.net/questions/206239 | 6 | There are a number of variations on the Laplace transform that turn up all over math. Some examples:
* $\int\_{-\infty}^{\infty} f(t)e^{-st} dt$ - The Laplace transform
* $\sum\_{-\infty}^{\infty} f(t)z^{-t}$ - The Z-transform
* $\int\_{-\infty}^{\infty} f(t)e^{-i\omega t} dt$ - The Fourier transform
* $\sum\_{-\inft... | https://mathoverflow.net/users/24611 | Variations on the Mellin and Dirichlet transforms | I know that the discrete Mellin transform was defined by V.S.Ryko:
<http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=ivm&paperid=5138&option_lang=rus>
English reference: Soviet Mathematics (Izvestiya VUZ. Matematika), 1991, 35:8, 63–66
He also developed a very strong method with many page tables to sum seri... | 5 | https://mathoverflow.net/users/49208 | 206290 | 98,706 |
https://mathoverflow.net/questions/204759 | 1 | For an open subset $U$ of $\mathbb{C}^{2}$ containing $0$ and a holomorphic map $f:U\to \mathbb{C}^{2}$ which has a unique zero at the origin we associate a natural singular holomorphic foliation by complex curves, the foliation arising from $\dot z=f(z)$. In this case origin is called a singularity.
>
> What is an... | https://mathoverflow.net/users/36688 | Stability of singularity in singular holomorphic foliation | Such an example is impossible. We can always assume $W$ is a polydisc, and part of its boundary $\partial W$ is included in the $3$-space $T=\{(x,y) : |y|=r\}$. Take $p\in T\cap\bar W$. If the foliation were stable then the image of $t\mapsto z(t))$, for small $t$ and with $z(0)=p$, would be included in the adherence o... | 3 | https://mathoverflow.net/users/24309 | 206297 | 98,708 |
https://mathoverflow.net/questions/206282 | 7 | In their 1943 paper [On non-denumerable graphs](http://projecteuclid.org/download/pdf_1/euclid.bams/1183505209), Erdos and Kakutani suggest as likely the following proposition.
(EK\*) Suppose CH fails and $\lbrace M\_n : n \in \omega \rbrace$ is a countable family of sets of rationally independent real numbers; then ... | https://mathoverflow.net/users/57583 | Under $\neg CH$, have countable unions of rationally independent numbers inner measure zero? | Addendum: P. Komjath has pointed out that the following argument is originally due to Erdos and Kunen. See page 136 [here.](https://www.renyi.hu/~p_erdos/1978-40.pdf)
Claim 1: (Mycielski) Suppose $A$ is a compact subset of plane of positive area. Then there exist perfect sets $P, Q$ such that $Q$ has positive length ... | 8 | https://mathoverflow.net/users/2689 | 206306 | 98,710 |
https://mathoverflow.net/questions/206305 | 2 | In Bushnell and Henniart, The Local Langlands conjecture for GL(2), there is a proposition on p. 184 in which they prove the following:
Let $F$ be a non-archimedean local field, $\mathcal W\_F$ its Weil group, and $\tau$ an irreducible smooth representation of $\mathcal W\_F$. Then, if $\tau(\mathcal W\_F)$ is finite... | https://mathoverflow.net/users/56228 | Weil group of a local field, small notational problem | If $G$ is any profinite group, and $a\in\widehat{\mathbf{Z}}$, then for any sequence $(a\_n)$ of integers converging in $\widehat{\mathbf{Z}}$ to $a$, the sequence $(g^{a\_n})$ converges in $g$ to an element $g^a$ which is independent of the chosen sequence. Thus one can take $\widehat{\mathbf{Z}}$-exponents in a profi... | 3 | https://mathoverflow.net/users/4351 | 206313 | 98,712 |
https://mathoverflow.net/questions/206310 | 11 | Given a category $\mathcal{C}$ and a set (let's not bother with size issues here) $\mathcal{W} \subseteq \text{Mor}(\mathcal{C})$ we may form the category $\mathcal{C}[\mathcal{W}^{-1}]$ obtained by formally inverting all arrows belonging to $\mathcal{W}$. If we're lucky, then the localization functor $j: \mathcal{C} \... | https://mathoverflow.net/users/3824 | Reflective Localizations vs. categories of local objects | To avoid confusing myself, I will write $L : \mathcal{C} \to \mathcal{C} [\mathcal{W}^{-1}]$ for the localising functor and $R : \mathcal{C} [\mathcal{W}^{-1}] \to \mathcal{C}$ for its right adjoint. (Note that $R$ is *automatically* fully faithful – the hard part is existence!)
As you say, for any object $Y$ in $\m... | 9 | https://mathoverflow.net/users/11640 | 206318 | 98,714 |
https://mathoverflow.net/questions/206319 | 1 | Let $G$ be a semi-simple linear algebraic group over the complex numbers, e.g. the special linear group. Can you find an example of a finite sub-group $H$ of $G$ which does not normalize any maximal torus? or any non-trivial torus? In other words, given $H$, is it possible to find a torus $T$ in $G$ such that $H$ is co... | https://mathoverflow.net/users/48866 | Finite groups normalizing a torus | Yes, take the symmetry group of a platonic solid in $SO(3)$. It acts irreducibly on the adjoint representation (which is the standard representation), hence the only connected subgroups it normalizes are the trivial group and all of $SO(3)$. Neither is a torus.
| 6 | https://mathoverflow.net/users/18060 | 206322 | 98,715 |
https://mathoverflow.net/questions/206315 | 5 | Let $\mathbb{F}$ be a finite field.
Let $A\le \mbox{Mat}\_n(\mathbb{F})$ be a matrix algebra.
Is there a good bound on the number $k$ of random elements $a\_1,\dots,a\_k\in A$
that one needs to take such that, with high probability, the algebra generated by $a\_1,\dots,a\_k$ is $A$?
What if $A$ is generated by a *gr... | https://mathoverflow.net/users/2415 | How many random matrices does it take to generate a matrix algebra? | At least $n^2/4$. Divide $n \times n$ matrices into four $n/2 \times n/2$ blocks, and consider the subalgebra of matrices that are zero outside the upper right block. Then because the product of any two elements is zero, your elements generate as an algebra if and only if they generate as a vector space. So you need at... | 8 | https://mathoverflow.net/users/18060 | 206324 | 98,716 |
https://mathoverflow.net/questions/204437 | 27 | I have read and heard several times that it is “important” that unramified maps are not required to be locally of finite *presentation*, but only locally of finite *type*.
Apart from this issue with unramified I have always preferred “locally of finite *presentation*” as it seems to be conceptually and morally better... | https://mathoverflow.net/users/21815 | Why are unramified maps not required to be locally of finite presentation? | Various people have imposed finite presentation instead of finite type: most notably Grothendieck for unramified and Demazure–Gabriel for proper maps. Both these cases seem to have been motivated by the simple functorial characterization of locally of finite presentation. I would however argue that finite presentation,... | 34 | https://mathoverflow.net/users/40 | 206333 | 98,719 |
https://mathoverflow.net/questions/206337 | 1 | Given a matrix $M \in \mathbb{R}^{n \times m}$, I would like to find
$\min\_{x \in \mathbb{R}^m} \|Mx\|\_0$ such that $x \neq 0^m$,
where the $\ell\_0$ "norm" is measured by simply counting the number of nonzero entries in the vector. In my application, $n$ is typically $10^7$, $m$ is typically $10^4$, and $M$ is ... | https://mathoverflow.net/users/69974 | Given $M$, minimize $|Mx|_0$ | The key phrase to google is "sparsest vector".
| 1 | https://mathoverflow.net/users/5963 | 206339 | 98,721 |
https://mathoverflow.net/questions/204764 | 13 | For $r > 0$, define $f(n) = \lfloor {nr}\rfloor$ if $n$ is odd and $f(n) = \lfloor {n/r}\rfloor$ if $n$ is even. For which real numbers $r$ is the set $\{n,f(n), f(f(n)),\dots\}$ bounded for every nonnegative integer $n$?
So far, I have only computer-generated evidence. If $r = \sqrt 3$, the iterates reach $1$ for $n... | https://mathoverflow.net/users/61426 | Which irrationals yield bounded sets of iterates? | Let me write the iteration as $x(k+1, r) = f(x(k,r)) =\lfloor r x(k,r) \rfloor$ if $x(k,r)$ is odd, $\lfloor x(k,r)/r \rfloor$ if $x(k,r)$ is even,
where $x(0,r) = 1$ (for convenience). I claim there is an uncountable set of $r$ for which all $x(k,r)$ are odd, and in particular $x(k,r) \to \infty$ as $k \to \infty$. Th... | 4 | https://mathoverflow.net/users/13650 | 206344 | 98,724 |
https://mathoverflow.net/questions/206347 | 1 | In the paper
>
> GEOMETRY OF TRUNCATED SYMMETRIC PRODUCTS AND REAL
> ROOTS OF REAL POLYNOMIALS, JACOB MOSTOVOY, Bull. London Math. Soc. (1998) 30 (2):
> 159-165,
>
>
> **Theorem 2.** (b): $TP^n(\mathbb{R}P^1)$ is homeomorphic to $\mathbb{R}P^n$.
>
>
>
By page 161 line 2-line 3 of the above paper, the unorder... | https://mathoverflow.net/users/41075 | Unordered configuration space of $\mathbb{R}P^1$ | (1) and (2) are perfectly compatible, to the extent which (1) makes sense. $\mathbb{R}P^k$ is naturally a CW complex with one cell in each dimension, and $\mathbb{R}P^{k-2}$ is a subcomplex. Assuming that the inclusions are in the standard way, the difference $\mathbb{R}P^k \setminus \mathbb{R}P^{k-2}$ is not a union o... | 4 | https://mathoverflow.net/users/5010 | 206349 | 98,725 |
https://mathoverflow.net/questions/206352 | 32 | **QUESTION**
Numerical calculation with **gp** (first to the default 38-digit
precision, then tripled) supports the conjecture that
$$
\int\_0^\infty x \, [J\_0(x)]^5 \, dx =
\frac{\Gamma(1/15) \, \Gamma(2/15) \, \Gamma(4/15) \, \Gamma(8/15)}
{8\sqrt{5} \, \pi^4}
= \frac{2}
{\sqrt{5} \, \Gamma(7/15) \, \Gamma(11/15... | https://mathoverflow.net/users/14830 | $\int_0^\infty x \, [J_0(x)]^5 \, dx$: source and context, if any? | See *Densities of Short Uniform Random Walks* (with an appendix by Don Zagier) by
Jonathan M. Borwein, Armin Straub, James Wan, and Wadim Zudilin,
Canad. J. Math. Vol. 64 (5), 2012 pp. 961–990. <http://cms.math.ca/10.4153/CJM-2011-079-2>
Your integral is $p\_4(1)$ in Equation (2.1). They write it in terms of hypergeo... | 30 | https://mathoverflow.net/users/69507 | 206381 | 98,738 |
https://mathoverflow.net/questions/206382 | 3 | Hopefully this question does not double another. If so, don't bother to close this.
An extended topological quantum field theory is sometimes called, 'fully local".
Why is that? I can imagine that such a theory Has local structure, while an ordinary TQFT has not.
The question is vague, but a more precise answer, or... | https://mathoverflow.net/users/21965 | Why is an extended T(Q)FT called fully local? | I believe there's an explanation on the nlab page for extended topological quantum field theories (<http://ncatlab.org/nlab/show/extended+topological+quantum+field+theory>) and on page 13 of Lurie's cobordism paper. We can compute the value of an extended TQFT $Z$ on a manifold $X$ simply by computing it locally, and t... | 3 | https://mathoverflow.net/users/nan | 206386 | 98,740 |
https://mathoverflow.net/questions/142194 | 4 | *(This question was posted on MSE on 8. September (<https://math.stackexchange.com/questions/487729>) but got no answer so far; hence the posting on MO.)*
It is an open problem whether the number $\pi$ is disjunctive in base $10$, i.e., whether every finite sequence appears (at least once) in the base $10$ expansion ... | https://mathoverflow.net/users/11025 | $\pi$, disjunctive numbers, and finite sequences of given length | This is known since 2010 at least for $n\leq 11$ -- see [this entry](http://oeis.org/A036903) in the OEIS or F. Bellards's [page about digits of $\pi$](http://bellard.org/pi/pi2700e9/pidigits.html). In fact, every sequence of length $11$ occurs once in the first $2\ 512\ 258\ 603\ 207$ digits of $\pi$.
| 3 | https://mathoverflow.net/users/11025 | 206393 | 98,741 |
https://mathoverflow.net/questions/206401 | 2 | A couple of months ago, i saw a construction, that somehow looks like the construction of the tautological section of the pullback of a vector bundle to its total space, i am trying to piece it together, hope that somebody can help me:
Let $X$ be a projective scheme, say a surface over $\mathbb{C}$, let $S=Spec(R)$ b... | https://mathoverflow.net/users/70593 | Canonical (tautological) section of a family of sheaves | What you have is the fiber diagram.
$$
\begin{array}[c]{ccc}
X\times T & {\stackrel{\beta}{\rightarrow}} & X\times S \\
{\scriptstyle \delta} \downarrow & & \downarrow\scriptstyle{\pi}\\
T & {\stackrel{\alpha}{\rightarrow}} & S
\end{array}
$$
Now, there is a natural morphism
$\alpha^\* V = \alpha^\*\pi\_\*\mathcal... | 4 | https://mathoverflow.net/users/10941 | 206406 | 98,745 |
https://mathoverflow.net/questions/204143 | 8 | A(k,4,r) is the independence number of the Johnson graph J(k,r).
What is the best known asymptotic lower bound on A(k,4,floor(k/2)) ?
I only obtained $\frac{{k\choose\lfloor{k/2}\rfloor}}{\lfloor{k^2/4}\rfloor}$.
| https://mathoverflow.net/users/71090 | lower bound on A(k,4,floor(k/2)) | A better bound than that is known. Define a function $f:\{0,1\}^n\to \{0,1,\dots,n-1\}$ as follows:
$$ f(c\_0,c\_1,\dots,c\_{n-1})=\left(\sum\_{i=0}^{n-1} i\cdot c\_i\right)\bmod n.$$
Then define the following codes,
$$ C\_a=\{ c=(c\_0,c\_1,\dots,c\_{n-1})\in\{0,1\}^n ~|~ wt(c)=w, f(c\_0,c\_1,\dots,c\_{n-1})=a \},$$
wh... | 7 | https://mathoverflow.net/users/9044 | 206407 | 98,746 |
https://mathoverflow.net/questions/206364 | 6 | Let $A\_\ast$ and $F\_\ast$ be the functors $\textrm{Var}\_\mathbb C\to \textrm{Ab}$ of Chow groups and constructible functions, respectively, with respect to proper maps. Then the *Chern-Schwartz-MacPherson class* is the unique natural transformation $$c\_{\textrm{SM}}:F\_\ast\to A\_\ast$$ taking the value $c\_{\textr... | https://mathoverflow.net/users/30827 | What does the Chern-Schwartz-MacPherson class of a singular variety look like? | There are some special cases where the CSM classes of singular varieties can be computed easily.
For example there is Ehler's formula for $c\_{SM}(X)$ where $X$ is any complete toric variety. Let $\Sigma$ be the fan of $X$ with torus orbits $B\_\sigma$ for $\sigma \in \Sigma$, then the CSM class of $X$ is given by
... | 8 | https://mathoverflow.net/users/12402 | 206417 | 98,751 |
https://mathoverflow.net/questions/206415 | 8 | Define the function $[0,+\infty) \rightarrow R$:
$$ f = \cos (t) + \cos (\sqrt{2} t) + \cos (\sqrt{3} t) + \cos (\sqrt{5} t ) . $$
I want a number $t $ bigger than $10^7$ such that
$$ f(t) > 4 - 10^{-9} . $$
Can anyone give me such a number? Ultimately, I want an algorithm which works for arbitrary precision (... | https://mathoverflow.net/users/7063 | An algorithm for Poincare recurrence time | You want to find an $s$ such that $s, \sqrt{2} s, \sqrt{3} s, \sqrt{5} s$ are all close to integer. Your $t$ is then given by $2\pi s.$ The first question is a problem in simultaneous Diophantine approximation, an algorithm for which (using lattice reduction) is given by [W.Bosma](http://math.ucsd.edu/~kedlaya/ants10/b... | 11 | https://mathoverflow.net/users/11142 | 206419 | 98,753 |
https://mathoverflow.net/questions/206371 | 4 | Let $\textbf{Grph}$ be the category whose objects are graphs $G = (V,E)$ such that $V$ is a set and $E \subseteq \mathcal{P}\_2(V) := \{\{a,b\} \subseteq V: a\neq b\}$. We sometimes write $E(G)$ for $E$. The morphisms are maps $f:G\to H$ such that whenever $\{v,w\}\in E(G)$ then $\{f(v),f(w)\}\in E(H)$.
How can regul... | https://mathoverflow.net/users/8628 | Regular epimorphisms in the category of simple undirected graphs | I think the paper ["A canonical factorization for graph homomorphisms", Barry Fawcett, Can J. Math. 29 (4), 1977, 738-743](https://cms.math.ca/10.4153/CJM-1977-077-3), answers the question.
Theorem 3 states that in $\textbf{Grph}$, strict epimorphisms are the same as extremal epimorphisms, which are the same as "full... | 5 | https://mathoverflow.net/users/22989 | 206421 | 98,754 |
https://mathoverflow.net/questions/204405 | 3 | By the 2-shift map I mean the map $T:\{0,1\}^\mathbb{Z}\to \{0,1\}^\mathbb{Z}$ that shifts the sequence leftwise. By a root I mean an homeomorphism $\psi:\{0,1\}^\mathbb{Z}\to\{0,1\}^\mathbb{Z}$ that commutes with $T$ and such that $\psi^2=T$.
I read somewhere that it does not have one, but could not find a proof.
| https://mathoverflow.net/users/49898 | Does the 2-shift map have a root automorphism? | **(05/05/2015)** If $T$ is a permutation of any set $X$ with a (finite) odd number of 2-cycles, then $T$ is not a square in the group of permutations of $X$ (because if $T=U^2$ then $T$ commutes with $U$ --this assumption in the question is thus redundant--, hence $U$ preserves the union $J$ of 2-cycles of $T$ and acts... | 6 | https://mathoverflow.net/users/14094 | 206429 | 98,756 |
https://mathoverflow.net/questions/206423 | 2 | A have a question related to the boundary regularity of a solution of a Poisson equation on a bounded domain. But to make the question easier to pose I will state it on $ R\_+^2:=\{ x \in R^2:x\_2>0\}$ but now the statement may not be true, but keep in mind i am only concerned about local issues near the boundary.
S... | https://mathoverflow.net/users/66623 | elliptic boundary regularity, tangential regularity | Boundedness of $f$ isn't enough to get extra regularity for $\phi\_1$. Take for example $\phi = r^{\gamma}\sin(\theta)$, which solves an equation of the desired form with $f = (\gamma^2-1)\sin(\theta)$, which is discontinuous at the origin. Then
$$\phi\_1(1/{\sqrt{2}},1/{\sqrt{2}}) = \frac{1}{2}(\gamma-1),$$
so by hom... | 2 | https://mathoverflow.net/users/16659 | 206441 | 98,760 |
https://mathoverflow.net/questions/205861 | 5 | Let $(V,f)$ be a $3$-dimensional positive definite quadratic space over $\mathbf Q$.
Let $G(V)$ be a set of representatives of the isometry classes of maximal integral lattices on $V$.
To an element $L$ of $G(V)$, one associates the usual Theta series
$$\Theta\_L:=\sum\_{x\in L} q^{2f(x)} \in \mathbf Q[[q]]\ \ \... | https://mathoverflow.net/users/39552 | genus 2 Siegel theta series of 3-dimensional lattices | It is known, by Kitaoka's theory of characteristic sublattices, that if two lattices of rank $n$ with the same discriminant representing the same collection of lattices of rank $n - 1$, then the two lattices must be isometric. See, for example, Kitaoka's book "Arithmetic of quadratic forms", Chapter 6, Section 4.
| 5 | https://mathoverflow.net/users/29241 | 206443 | 98,761 |
https://mathoverflow.net/questions/206433 | 2 | I'm trying to solve the optimization problem
$\min\_x \frac{1}{n} \sum\_{i=1}^n f\_i(x)$
where $f\_i$ are (strongly) convex, smooth, lower semi-continuous, etc.
However, I am not able to do conventional stochastic gradient descent (SGD),
because the $f\_i$ are distributed in multiple machines.
I understand there ar... | https://mathoverflow.net/users/42202 | Rate of convergence for cyclic gradient descent | Methods of these type sometimes go under the name "[Kaczmarz method](http://en.wikipedia.org/wiki/Kaczmarz_method)". Kaczmarz method is a method for solving $Ax=b$ by iteratively (e.g. cyclic") projection onto the solutions of the equations given by the rows of the system. If the system is underdetermined and initializ... | 2 | https://mathoverflow.net/users/9652 | 206449 | 98,765 |
https://mathoverflow.net/questions/206450 | 4 | $f(x)=\sum\_{i=0}^\infty a\_ix^i$ with $a\_i\in\Bbb Q$. Let $S$ be finite set of places of $\Bbb Q$ such that:
**1.** $\forall p\notin S$, $|a\_i|\_p\leq1$ $\forall i\geq0$.
**2.** $\forall v\in S$, $f(x)$ extends to a meromorphic function on a disc $D\_v$ of radius $R\_v$ in $\Bbb C\_v$ and $\prod\_{v\in S}R\_v>1$... | https://mathoverflow.net/users/10035 | Is there a multivariate analog of Dwork's theorem? | *Update* (Nov, 2019). The multivariate rationality criterion is true, and it follows from the reference to Andre's paper below. However, in my suggested proof scheme there was a point I had overlooked at the time of answering this question. When fully worked out, the straightforward outline with ``$Qf - P$'' below lead... | 8 | https://mathoverflow.net/users/26522 | 206463 | 98,768 |
https://mathoverflow.net/questions/206469 | 1 | Is it true that any metric space consisting of $n$ points can be isometrically imbedded into $n-1$ dimensional Euclidean space? Hyperbolic space?
(For $n=3$ this is true.) If not, what are necessary/sufficient conditions?
The case $n=4$ is the first unknown to me case.
What happens with imbeddings into the unit sphe... | https://mathoverflow.net/users/16183 | Isometric imbedding of finite metric space into standards spaces | The condition in the Euclidean case is that the Gram matrix is positive semi-definite (put one vertex at the origin, then the entries of the matrix are $\|v\_i\|$ on the diagonal, and $\langle v\_i, v\_j\rangle$ elsewhere. These can be expressed in terms of lengths via the parallelogram law (see my arxiv preprint "some... | 3 | https://mathoverflow.net/users/11142 | 206479 | 98,773 |
https://mathoverflow.net/questions/206476 | 3 | Let $dg-Cat$ denote the category of (small) dg-categories and $Ho(dg-Cat)$ denote the localization of $dg-Cat$ at quasi-equivalence. Using the model structure on $dg-Cat$ we can describe the morphisms in $Ho(dg-Cat)$ as the homotopy classes of morphisms between cofibrant and fibrant objects.
In Toen's [Lectures on dg... | https://mathoverflow.net/users/24965 | A question about the morphisms in the homotopy category of dg-Cat | More generally one has the following statement: if $u : C \to D$ is a quasi-fully faithful functor of dg-categories, then the induced morphism of mapping spaces in the model category of dg-categories (with the Dwyer-Kan model structure)
$$ Map(E, C) \to Map(E, D) $$
induces an injection $\pi\_0$ and an isomorphism on ... | 5 | https://mathoverflow.net/users/2503 | 206480 | 98,774 |
https://mathoverflow.net/questions/206203 | 2 | While trying to prove the inequality $$c\_{p\_n-m}+c\_{m-n}>p\_n+2$$ I tried the bounds of $c\_n$ (denotes the $n$-th composite number) given in [this paper](http://www.emis.de/journals/JIPAM/images/027_01_JIPAM/027_01_www.pdf) to prove that the sum $c\_{p\_n-m}+c\_{m-n}$ satisfies the following properties, $$
\begin{a... | https://mathoverflow.net/users/nan | A question on the bounds of the $n$-th composite $c_n$ | I assume $p\_n \gt m \gt n$ in the inequality. I have a feeling that this will be as challenging as $\pi(x) + \pi(y) \gt \pi(x+y)$ to solve. The essence to me is that
composites are sparsest (primes are densest) near the origin, and that this sparsity
is measured by satisfaction of the inequality. However, looking at a... | 0 | https://mathoverflow.net/users/3206 | 206488 | 98,775 |
https://mathoverflow.net/questions/206481 | 9 | Let $S\subset\mathbb{R}$ be a convex set and $\mathbb{S}^{n}$ be the set of real symmetric matrices of order $n\times n$.
A matrix valued function $\Gamma: S \rightarrow \mathbb{S}^{n}$ is said to be convex if for all $x\_1,x\_2 \in S$ and for all $\lambda \in (0,1)$ one has
$$\Gamma\left(\lambda x\_{1}+\left(1-\lam... | https://mathoverflow.net/users/22389 | Convexity of the product of two exponential matrices | I think the $3\times 3$ Jordan block
$$
A=\begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0\end{pmatrix}
$$
is a counterexample.
For this $A$, we have
$$
\exp{At}=\begin{pmatrix} 1 & t & \frac{t^2}{2} \\ 0 & 1 & t \\ 0 & 0 & 1\end{pmatrix}
$$
so
$$
f(t) = \begin{pmatrix} 1+t^2+\frac{t^4}{4} & t+\frac{t^3}{2} & \... | 12 | https://mathoverflow.net/users/13360 | 206490 | 98,777 |
https://mathoverflow.net/questions/206396 | 1 | Is it possible to generalize [Degree of sum of algebraic numbers](https://mathoverflow.net/questions/26832/degree-of-sum-of-algebraic-numbers?rq=1) (especially Pete L. Clark's answer, based on Keith Conrad's answer)
in the following way:
Let $D$ be a (noetherian) UFD of zero characteristic which is not a field, and le... | https://mathoverflow.net/users/72288 | Degree of sum of integral elements over a UFD | When you don't work over a field your previous experience with field degrees can break down pretty badly because over a ring that is not a field (even over a ring as simple as $\mathbf Z$) a finite free module can be a submodule of another finite free module with the same rank and the two modules don't have to be the s... | 2 | https://mathoverflow.net/users/3272 | 206496 | 98,779 |
https://mathoverflow.net/questions/206459 | 5 | Let $L(s,\chi)$ be the $L$-function of a non-trivial Hecke character of a general number field $K$, so that $L(s,\chi)$ which has no pole or zero at $s=1$.
I am looking for a reference for upper bounds for $\alpha$ in the convexity estimate $$L(s,\chi)\ll\_{\epsilon,K,\chi}|\Im(s)|^{\alpha+\epsilon}, \qquad\frac{1}{2... | https://mathoverflow.net/users/9232 | Subconvexity bound for Hecke $L$-functions in the $s$-aspect | You have to be careful: the implied constant also depends on $\chi$ (not just on $\epsilon$), in particular it depends on the number field $K$. (I edited your post to reflect this.)
There is a better (subconvex) bound available, namely there is an absolute constant $\lambda<1/2$ such that
$$ L(\sigma+it,\chi)\ll\_{\e... | 6 | https://mathoverflow.net/users/11919 | 206500 | 98,781 |
https://mathoverflow.net/questions/206493 | 2 | It seems, there are two definitions of the Siegel upper half space.
1) One used by, say, Krantz in "Explorations in Harmonic Analysis; Page 252, or, by So Chin Chew and Mei Chi Shaw in "Partial Differential Equations in Several Variables": Page 171:
$\mathbb{H}\_{n}=\left\{ \left(z\_{1},z\_{2},\cdots,z\_{n}\right)... | https://mathoverflow.net/users/73657 | Connection between the two definitions of Siegel Upper Half Space | These are two different "classical domains". An on-line discussion of the four families of classical domains, related semi-simple real Lie groups, and some related matters is at <http://www.math.umn.edu/~garrett/m/lie/> in the note "classical groups, domains, cones".
| 3 | https://mathoverflow.net/users/15629 | 206502 | 98,783 |
https://mathoverflow.net/questions/206321 | 7 | Denote $I=(0, 1)$, and let $\mu$ be the Lebesgue measure on $I$. Does there exist a function $f$ on $I\times I$ viewed as an element of the space $L^\infty(\mu\times\mu)$ such that
$$
f^2=f
$$
(that is, $f$ takes values 0 and 1);
$$
f(x, y)+f(y, x)=1
$$
(that is, if $f(x, y)=0$ then $f(y, x)=1$, and vice versa);
and $... | https://mathoverflow.net/users/38629 | An indicator of a planar subset as an element of a tensor product | Well, I am not sure, but let me try to prove that the answer is negative.
The idea is to prove that any function $f(x,y)$ given by $\int g\_t(x)h\_t(y)dt$ is continuous with respect to appropriate admissible metric, where admissible means ``separable on the set of full measure.'' Namely, for any $x$ define $F\_x(t)=... | 2 | https://mathoverflow.net/users/4312 | 206522 | 98,791 |
https://mathoverflow.net/questions/206526 | 5 | I have a question on the definition of *Saturated Sets*, as particular subset of the set of strongly normalizing terms in lambda calculus.
Here is the definition: a set $S$ of strongly normalizing $\lambda$-terms is
said *saturated* if:
1) For all $x : var$ and for all sequences $a\_{1} \dots a\_{n}$ of strongly
n... | https://mathoverflow.net/users/73670 | Explanation of the definition of Saturated Sets in Lambda Calculus | Aha! I can give *some* answer to this one: saturated sets are a tool, designed specifically to allow the proof of strong normalization of [System F](http://en.wikipedia.org/wiki/System_F).
First notice that the strongly normalizing terms are a *particular instance* of saturated sets. However they are not the only one... | 7 | https://mathoverflow.net/users/36103 | 206529 | 98,793 |
https://mathoverflow.net/questions/204485 | 6 | I am trying to understand the proof of Theorem 4.6.2.1 in the book on algebraic stacks by Laumon and Moret-Bailly. The setting is this: $S$ is a Noetherian scheme, $f\colon X \rightarrow S$ is a projective $S$-scheme such that $f\_\*\mathscr{O}\_X = \mathscr{O}\_S$ universally (i.e. also after any base change on $S$), ... | https://mathoverflow.net/users/70964 | Algebraicity of the stack of coherent sheaves | As I understand it, you have $f: X\to S$ a projective morphism and $\mathcal{F}$ a sheaf on $X$ which is flat and relatively globally generated. Since your question is local we can assume $S$ is affine and $f\_\*\mathcal{F}$ is free.
You want to find an integer $N$ and a surjective morphism $$\alpha: \mathcal{O}\_X^... | 4 | https://mathoverflow.net/users/6950 | 206531 | 98,794 |
https://mathoverflow.net/questions/206536 | 6 | Does anyone know, where I can find the proof of necessary and sufficient conditions for differentiating under the integral sign in case of Henstock integral? Here are the theorems but not all the proofs:
>
> *Necessary and sufficient conditions for differentiating under the integral sign*
>
> Erik Talvila
>
... | https://mathoverflow.net/users/73676 | Henstock, Differentiation under the integral sign | Bartle's *A modern theory of integration* has it on [pp. 199-200](https://books.google.com/books?id=6-9royzoBL8C&pg=PA199).
| 2 | https://mathoverflow.net/users/19276 | 206538 | 98,796 |
https://mathoverflow.net/questions/206547 | 4 | Let $A$ be an associative algebra over a filed $k$.
**Q) What are the condition we can impose on $A$ such that there exists a $G$ such that $A=k[G]$, the group algebra generated by $G$?**
I am particularly interested in the following cases:
1) When $k=\mathbb{Q},\mathbb{R}$ or $\mathbb{C}$.
2) $A=M\_n(k),$ the... | https://mathoverflow.net/users/9485 | Relation between Associative algebra and group algebra | If $k$ has characteristic zero and $A$ is finite-dimensional, by Maschke's theorem it must be semisimple. If $k$ is in addition algebraically algebraically closed and $A \cong k[G]$, then $A \cong \prod\_i M\_{n\_i}(k)$ where $n\_i$ are the dimensions of the irreducible representations of $G$. So in this case the quest... | 10 | https://mathoverflow.net/users/290 | 206549 | 98,800 |
https://mathoverflow.net/questions/206546 | 0 | Let $(X,\*)$ be a pointed topological space.
Let $F(X,k)=\{(x\_1,\cdots,x\_k)\in X^k\mid \forall i\neq j: x\_i\neq x\_j, \}$.
Let $F(X,k)/S\_k$ be the $k$-th unordered configuration space.
Is there an inclusion $F(X,k)/S\_k\to F(X,k+1)/S\_{k+1}$ for each $k\geq 1$?
Note that $[x\_1,\cdots,x\_k]\mapsto [x\_1,\... | https://mathoverflow.net/users/41075 | unordered configuration space of pointed space | With your definitions, assuming you mean the configuration space of distinct points, and that the inclusions need to be compatible with the actual locations of the points in some way, there is not such a map for homotopy reasons.
I will write $C(X,k)$ for the unordered configuration space.
Let $X$ be the circle $S^... | 6 | https://mathoverflow.net/users/5010 | 206551 | 98,801 |
https://mathoverflow.net/questions/206550 | 9 | Let $K$ be a number field, $R$ the ring of integers of $K$,
${\mathbf{A}^f}$ the ring finite adeles of $K$, and ${\widehat{R}}\subset {\mathbf{A}^f}$ the ring of integral adeles.
Let $G$ be an affine group scheme of finite type over $R$ with smooth generic fiber $G\_K$.
Let
$$c(G)=G({\widehat{R}})\backslash G({\math... | https://mathoverflow.net/users/4149 | Group schemes, adeles, double cosets, and étale cohomology | Given an element of $c(G)$, we are looking for an etale $G$-torsor which is trivial when restricted to $\text{Spec}(K)$ and each $\text{Spec}(R\_\mathfrak{p})$, for $\mathfrak{p}$ a finite prime of $R$. Because this $G$-torsor will be trivial over $K$, it will in fact be trivial over some Zariski-open $U$ of $\text{Spe... | 7 | https://mathoverflow.net/users/6950 | 206553 | 98,802 |
https://mathoverflow.net/questions/201022 | 4 | Consider a morphism of commutative rings $h\colon R\rightarrow S$. This gives rise to a functor $h^\*\colon{\sf Mod}(R)\rightarrow{\sf Mod}(S)$, called scalar extension by means of $h$. This functor has a right adjoint, hence it commutes with inductive limits. We may ask now whether or not $h^\*$ commutes with *project... | https://mathoverflow.net/users/11025 | Extension of scalars and projective limits | *(This was answered in the comments, essentially by abx.)*
The scalar extension functor by means of $h\colon R\rightarrow S$ commutes with infinite products if and only if $S$, considered as an $R$-module by means of $h$, is of finite presentation.
A proof can be found in T.Y.Lam, *Lectures on modules and rings,* P... | 2 | https://mathoverflow.net/users/11025 | 206568 | 98,806 |
https://mathoverflow.net/questions/206570 | 3 | This question is pretty basic, so I apologize in advance if it is unsuitable for MO. If so, please do let me know and I will migrate it over to MSE.
Essentially, by work of Kanold, we know that the natural density of the perfect numbers $N$ for which
$$\sigma(N) = 2N$$
is $0$. Now my question is: Is it known that a s... | https://mathoverflow.net/users/10365 | On the natural density of almost perfect numbers | The natural density is $0$. This is due to the function $\sigma(n)/n$ possessing a continuous distribution, i.e. there exists a continuous function $\Phi:\mathbb{R}\to \mathbb{R}$ such that for all $a<b$ real numbers one has $$\lim\_{x\to \infty}\frac{1}{x}\#\Big\{1\leq n \leq x: a<\frac{\sigma(n)}{n} \leq b\Big\}=\Phi... | 11 | https://mathoverflow.net/users/9232 | 206571 | 98,807 |
https://mathoverflow.net/questions/206567 | 3 | I'm looking for a classification of $3$-forms over a real vector space of dimension $7$ as for the $3$-forms in dimension $6$. References on the latter case are R. Bryant *On the geometry of almost complex $6$-manifolds* [available here](http://arxiv.org/abs/math/0508428) and P. Baier's PhD thesis [available here](http... | https://mathoverflow.net/users/32185 | Classification of 3-forms in dimension 7 | The paper
Djoković, Dragomir Ž.(1983). Classification of trivectors of an eight-dimensional real vector space, Linear and Multilinear Algebra, 13: 1, 3 — 39
gives a systematic classification of 3-forms over a real vector space of dimension 8, including the classification of 3-forms over a real vector space of dimen... | 4 | https://mathoverflow.net/users/3380 | 206573 | 98,808 |
https://mathoverflow.net/questions/206587 | 3 | I have a problem I have been stuck with since several weeks now, and yet I believe it should be easy to specialists.
Let $k$ be an algebraically closed field, $m$ and $n$ two integers. Let $H\_1,\dots,H\_m$ be $m$ (smooth if that helps) hypersurfaces of degrees $d\_1, \dots,d\_m$ in the projective space $\mathbb P^n$... | https://mathoverflow.net/users/9317 | Upper bound on Betti numbers of an intersection of hypersurfaces (or quadrics) | No. When $m = 1$ and $d \ge 3$ the Betti numbers grow exponentially in $n$; see, for example, [this blog post](https://qchu.wordpress.com/2014/06/16/hypersurfaces-4-manifolds-and-characteristic-classes/) for the computation for smooth hypersurfaces (which is over $\mathbb{C}$, but then we can appeal to the Weil conject... | 4 | https://mathoverflow.net/users/290 | 206591 | 98,811 |
https://mathoverflow.net/questions/206588 | 1 | Inspired by the line bundle case, I have the following question:
Given an equivariant holomorphic vector bundle over complex projective space, is it true that tensoring it by line bundles often enough can change it from a bundle with no holomorphic sections to a bundle with holomorphic sections. Where could one find ... | https://mathoverflow.net/users/42100 | Tensoring by Line Bundles to Produce Holomorphic Sections | I am not sure what you mean by equivariant. What you want is certainly true, provided the line bundle is ample (positive), over any projective variety. This follows e.g. from asymptotic Riemann-Roch (which gives control of the Euler characteristic), together with the fact that tensoring with an ample line bundle kills ... | 2 | https://mathoverflow.net/users/22294 | 206594 | 98,813 |
https://mathoverflow.net/questions/206593 | 1 | I would like to prove uniform convergence of function series like :
$$\sum\limits\_{n=1}^{\infty} \chi(n) f(nx)$$ where $\chi$ is a primitive character and $f(x)$ a function decreasing to zero in zero and infinity.
Typically: $$F(x)= \sum\limits\_{n=1}^{\infty} \chi(n) n^2 x^2 e^{-nx}$$
On intervals $[\epsilon, \... | https://mathoverflow.net/users/38290 | Uniform convergence of infinite sum with Dirichlet characters | Your specific series is not uniformly convergent on any interval of the form $[0,\delta]$. If it were, then it would be uniformly Cauchy on that interval; in particular, there would exist $P$ such that for all $M\ge N>P$ we would have
$$
\bigg| \sum\_{n=N}^M \chi(n)n^2x^2e^{-nx} \bigg| < \frac12
$$
for all $x\in [0,\de... | 3 | https://mathoverflow.net/users/5091 | 206595 | 98,814 |
https://mathoverflow.net/questions/206602 | 1 | R. Carter prooved that in finite soluble groups $G$ Carter subgroups $C$ exist and that they are conjugated. Furthermore they are exactly the nilpotent projectors: For every normal subgroup $N$ of $G$ the factor group $CN/N$ is maximal nilpotent in $G/N$ .
My question is whether in non-finite soluble groups Carter s... | https://mathoverflow.net/users/57804 | Are Carter subgroups nilpotent projectors? | This is not true for infinite insoluble groups.
Let $K$ be the direct product of countably infinitely many copies $\langle a\_i,b\_i \rangle$ ($i \in {\mathbb N}$) of $S\_3 = \langle a,b \mid a^2=b^3=(ab)^2=1 \rangle$, and let $G$ be the semidirect product of $K$ with a group $\langle t \rangle$ of order $2$ that act... | 1 | https://mathoverflow.net/users/35840 | 206608 | 98,817 |
https://mathoverflow.net/questions/205865 | 8 | Based from Harminc and Sotak's result, [www.fq.math.ca/Scanned/36-3/harminc.pdf](http://www.fq.math.ca/Scanned/36-3/harminc.pdf)
We know that under certain condition, an arithmetic progression can contain an infinitely many palindromes.
My question will be, if I have a system of Arithmetic Progression such as
\be... | https://mathoverflow.net/users/68943 | A Question about Palindromic Numbers and System of Arithmetic Progression | Not in general. For instance, there does not exist a natural number $t$ such that $t$ and $1000t+27$ (say) are both palindromes. Indeed, if $1000t+27$ has the last three digits of $027$, hence has the first three digits of $720$ if it is a palindrome, hence $t$ has first three digits of $720$, hence $t$ has last three ... | 11 | https://mathoverflow.net/users/766 | 206623 | 98,822 |
https://mathoverflow.net/questions/206614 | 1 | Given a sequence of real numbers $c\_k\to-\infty$, is there always a $C^\infty$ subharmonic function $f$ on $\mathbb R^2$ and a sequence $z\_k\to\infty$ with $|z\_k|<k$ such that
$$\displaystyle\limsup\_{z\to\infty} \frac{f(z)}{\log |z|}<\infty\ \ \text{and}\ \ f(z\_k)<c\_k\ ?$$
I do not even know the answer for no... | https://mathoverflow.net/users/1573 | Oscillation of subharmonic functions of slow growth | Yes, of course. Take any convergent series with positive terms $a\_k$.
Consider the function $u(z)=\sum a\_k\log|1-z/z\_k|$. This is a subharmonic function,
$u(z\_k)=-\infty$, and satisfies $u(z)=O(\log|z|)$ because the Riesz mass is finite.
It is continuous (in the extended sense, $-\infty$ is allowed).
To make it s... | 3 | https://mathoverflow.net/users/25510 | 206625 | 98,823 |
https://mathoverflow.net/questions/206615 | 5 | I know that
$$
\min\_{\|x\|\_2=1=\|y\|\_2} \left(\sum\_{k=1}^nx\_ky\_k\right)^2-\sum\_{k=1}^nx\_k^2y\_k^2 \geq -1/2
$$
with equality whenever $|x\_k|=\frac{1}{\sqrt{2}}=|y\_k|$ for two coordinates.
I'm wondering if there's a sharper version that one can prove wherein $x$ is *fixed* and we only minimize over $y$, i.e... | https://mathoverflow.net/users/57784 | Minimum of squared sum minus sum of squares | To elaborate some on the last two sentences of Robert's answer, here's what I would view as the standard procedure to analyze the rank one perturbation $D^2-xx^t$. I'll proceed as in [this answer](https://mathoverflow.net/questions/193527/about-adding-a-negative-definite-rank-1-matrix-to-a-symmetric-matrix/193592#19359... | 5 | https://mathoverflow.net/users/48839 | 206626 | 98,824 |
https://mathoverflow.net/questions/206555 | 8 | I am learning about Quantum Homology which I have to use in my research, and I see that in many papers (For example in FOOO, "Spectral invariants with bulk, Quasimorphisms and Lagrangian Floer theory", <http://arxiv.org/abs/1105.5123>), the homology ring being computed is given in terms of a Superpotential function. I ... | https://mathoverflow.net/users/14105 | Learning Quantum (Co)Homology and Landau Ginzburg Superpotential | For compact toric manifolds, Floer cohomology of non-displaceable Lagrangians can be detected by their superpotentials. This is in some sense the $\mathfrak{m}\_0$ term in the $A\_\infty$ structure which represents the obstruction to defining Floer cohomology. This construction can be found in the paper of FOOO: <http:... | 7 | https://mathoverflow.net/users/43423 | 206642 | 98,830 |
https://mathoverflow.net/questions/203060 | 15 | The cake-cutting game is usually played between individuals. What if we try to play it between groups?
>
> A certain land has to be divided between two states. There are $n$ citizens in each state. Each citizen in each state has a subjective continuous value measure over the land. How can the land be divided such... | https://mathoverflow.net/users/34461 | Fair cake-cutting between groups | In ["Consensus-halving via theorems of Borsuk-Ulam and Tucker"](https://math.hmc.edu/~su/papers.dir/tucker.pdf), Su and Simmons describe a way to divide a cake to two parts such that each of $n-1$ people believe the parts have the same value. This can be done with $n-1$ cuts ($n$ pieces), which is optimal.
Once we ha... | 1 | https://mathoverflow.net/users/34461 | 206647 | 98,832 |
https://mathoverflow.net/questions/206622 | 4 | Let $E\to X$ is a a (smooth real) vector bundle with structure group some Lie group $G$. Suppose we have a (linear) connection $\nabla$ on $E$.
>
> Is it true that if $A$ is the connection 1-form of the connection
> $\nabla$ in local coordinates in some trivialisation, regarded as a 1-form with matrix
> coefficie... | https://mathoverflow.net/users/39229 | What is the space for the coefficients of the connection 1-form of a connection in a vector bundle? | This answer just extends the remark by Liviu Nicolaescu above. For a general connection on a vector bundle with structure group $G$, you cannot say anything about the connection coefficients. (As an extreme example look at the case of a trivial bundle which has structure group $\{e\}$ and still admits lots of non-trivi... | 5 | https://mathoverflow.net/users/64141 | 206654 | 98,834 |
https://mathoverflow.net/questions/49915 | 79 | This is a question posed by Adam Chalcraft. I am posting it here because I think it deserves wider circulation, and because maybe someone already knows the answer.
A *polyomino* is usually defined to be a finite set of unit squares, glued together edge-to-edge. Here I generalize it to mean a finite set of unit hyperc... | https://mathoverflow.net/users/3106 | Does every polyomino tile R^n for some n? | A positive answer to this question has just appeared in the arXiv:
Tiling with arbitrary tiles;
Vytautas Gruslys, Imre Leader, Ta Sheng Tan;
<http://arxiv.org/abs/1505.03697>
| 38 | https://mathoverflow.net/users/22608 | 206655 | 98,835 |
https://mathoverflow.net/questions/206657 | 5 | Good morning everybody, I am currently reading through the book of Banyaga "Structure of classical diffeomorphism groups" [link](ftp://softbank.iust.ac.ir/MathBooks/b/Banyaga%20-%20The%20Structure%20of%20Classical%20Diffeomorphism%20Groups.pdf/), and I am particularly interested in the question of factorizing volume-pr... | https://mathoverflow.net/users/57571 | Kernel of flux homomorphism (Calabi invariant) for volume-preserving maps on a compact manifold | If M is simply connected and closed (and oriented, but this is the case if it has a volume form), then $H^{1}(M)$ is zero and hence by Poincaré duality $H^{n-1}(M)$ is also zero. So the flux homomorphism is trivial and this answers your question.
Otherwise I think that the answer to your question is no. If you have ... | 4 | https://mathoverflow.net/users/61960 | 206658 | 98,837 |
https://mathoverflow.net/questions/206612 | 11 | Given a smooth map from $\phi: B \rightarrow M$ where $B$ is a Banach Space and $M$ is a finite dimensional smooth manifold (for example, the end point map for a control system), what is the strongest analogue of Sard's theorem which holds.
| https://mathoverflow.net/users/41654 | Sard's Theorem For Banach Spaces | The classical result is the celebrated Sard-Smale Theorem, see **[1](http://www.jstor.org/stable/2373250?origin=crossref&seq=1#page_scan_tab_contents)**.
>
> **Theorem (Smale).** Let $\phi \colon B \longrightarrow M$ be a $C^q$-Fredholm map between separable Banach manifolds, with $q > \max \{0, \, \textrm{index of... | 16 | https://mathoverflow.net/users/7460 | 206662 | 98,838 |
https://mathoverflow.net/questions/189337 | 3 | Let $M$ be a closed, triangulated manifold of dimension $m$ and $K(M)$ be its triangulation. Let $f\_i$ denote the number of $i$-simplices of $K(M)$. As proved by Klee the face numbers satisfy the following *Dehn-Sommerville* relations
$$ f\_k = \sum\_{i=k}^m (-1)^{i+m} \binom{i+1}{k+1} f\_i,$$
for $k = 0, 1, \dots, m$... | https://mathoverflow.net/users/7494 | Dehn-Sommerville relations for $\Delta$-complexes | I think the answer is yes and the following is a sketch of proof:
* The second barycentric subdivision of a $\Delta$-complex is a triangulation.
* The f-vector of a $\Delta$-complex and of its barycentric subdivision are related by a matrix that preserves the Dehn-Sommerville relations.
| 3 | https://mathoverflow.net/users/22608 | 206667 | 98,840 |
https://mathoverflow.net/questions/206405 | 17 | The following statement may be "well known but not well known enough", and my question is which reference would state it explicitly:
The construction of Jet bundles is a comonad on suitable bundles over a given base $X$. A differential operator $D \colon \Gamma\_X(E\_1) \to \Gamma\_X(E\_2)$ is equivalently a morphism... | https://mathoverflow.net/users/381 | Differential operators are coKleisli morphisms of the jet co-monad | It turns out that the statement in question appeared in full beauty in
* Michal Marvan, *[A note on the category of partial differential equations](http://ncatlab.org/nlab/show/jet%20comonad#Marvan86)*, in Differential geometry and its applications, Proceedings of the Conference in Brno August 24-30, 1986,
There i... | 13 | https://mathoverflow.net/users/381 | 206669 | 98,841 |
https://mathoverflow.net/questions/206672 | 0 | Assume $A$ and $B$ are infinite abelian groups, $B$ is a subgroup of $A$. Is it true that if every homomorphism from $B$ to $\mathbb Z$ can be extended to a homomorphism from $A$ to $\mathbb Z$, then for all $a\in B$ and all natural number $n$, if there is $c\in A$ such that $nc=a$ in $A$, then there is $x\in B$ such t... | https://mathoverflow.net/users/73380 | Subgroups with some closed property | No.
counterexample: $A={\mathbb Q}/{\mathbb Z}$.
$B=$ subgroup generated by any nonzero element of $A$. There are no nonzero homomorphisms from $B$ to ${\mathbb Z}$ since $B$ is torsion.
edit: torsion free counter example:
$A={\mathbb Q}$ and $B=\{\frac{p}{q} \in A \ | \ q=2^n \ \text{for some } n\in {\mathbb ... | 0 | https://mathoverflow.net/users/45347 | 206677 | 98,846 |
https://mathoverflow.net/questions/206618 | 23 | Let $G$ be a simply connected Lie group and $\mathcal O= G(v)=G/G\_v$ a $G$-orbit in some finite-dimensional $G$-module $V$. By the homotopy exact sequence, its fundamental group $\Gamma$ is the component group of the stabilizer $G\_v$:
$$
\Gamma= \pi\_1(\mathcal O) = \pi\_0(G\_v) = G\_v/G\_v^{\mathrm o}.
$$
>
> *C... | https://mathoverflow.net/users/19276 | How bad can $\pi_1$ of a linear group orbit be? | I think we only obtain central-by-finite-by-abelian finitely generated groups (I don't claim we get all of them; maybe we only get finite-by-abelian groups and this is what I expect).
(I use the convention that a group is P-by-Q if it lies in an exact sequence with kernel satisfying P and quotient satisfying Q.)
Th... | 11 | https://mathoverflow.net/users/14094 | 206678 | 98,847 |
https://mathoverflow.net/questions/206676 | 6 | For $\chi$ being an irreducible character of the symmetric group $S\_n$ and being $M$ a complex $n\times n$-matrix, I would like to show
$$
\sum\_{\sigma, \rho \in S\_n} \overline{\chi(\sigma)} \chi(\rho) \prod\_{j=1}^n M\_{\sigma\_j, \rho\_j} = \frac{ n! }{\chi(e)} \sum\_{\sigma \in S\_n} \chi(\sigma) \prod\_{j=1}... | https://mathoverflow.net/users/57506 | Sum identities with immanants | I hope this is not homework.
\begin{eqnarray\*}
\text{lhs}&=&\sum\_{\sigma,\rho} \overline{\chi(\sigma)}\chi(\rho)\left(\prod\_{i=1}^nM\_{\sigma(i),\rho(i)}\right)\\
&=&\sum\_{\sigma,\rho} \overline{\chi(\sigma)}\chi(\rho)\left(\prod\_{i=1}^nM\_{i,\rho\sigma^{-1}(i)}\right)\\
&=&\sum\_{\tau}\left(\sum\_{\sigma}\ov... | 3 | https://mathoverflow.net/users/8430 | 206688 | 98,851 |
https://mathoverflow.net/questions/206691 | 3 | Let $H\in M$ be a convex hypersurface, where $M$ is a complete Riemannian manifold and $H$ is an embedded (complete as a induced metric space) hyper surface without boundary and with positive definite second fundamental form. Is it true that $H$ bound a convex domain $D$ in $M$? i.e. any two point $x, y \in D$ can be c... | https://mathoverflow.net/users/3922 | Does convex hypersurface necessarily bound a convex domain? | Counterexample 1:
If $D$ is a ball of radius $\frac13$ on the torus $T=\mathbb R^n/\mathbb Z^n$, you can take two points on the opposite sides of the ball such that the minimal geodesic connecting them does not stay in $D$.
The boundary $\partial D$ has a strictly positive definite second fundamental form.
Counterexa... | 6 | https://mathoverflow.net/users/55893 | 206693 | 98,853 |
https://mathoverflow.net/questions/206601 | 5 | It is a classical fact (e.g. [here](http://ncatlab.org/nlab/show/topos+of+algebras+over+a+monad#ToposProperty)) that for an (accessible) right adjoint comonad on a (sheaf) topos, its Eilenberg-Moore category of coalgebras is itself a (sheaf) topos.
I suppose this remains true for $\infty$-toposes, for hypercomplete $... | https://mathoverflow.net/users/381 | higher Eilenberg-Moore-toposes of left exact derived comonads | If $E$ is an ∞-topos and $T: E \to E$ is an accessible left exact comonad, then indeed the ∞-category $E^T$ of $T$-coalgebras is an ∞-topos. Moreover, it is hypercomplete if $E$ is. I will first show:
>
> **Lemma.** Let $E'$ be presentable and let $U:E'\to E$ be a conservative functor that preserves colimits and pu... | 3 | https://mathoverflow.net/users/20233 | 206695 | 98,854 |
https://mathoverflow.net/questions/206641 | 3 | Atiyah and Singer proved that the nontrivial component of the set of skew-adjoint Fredholm operators $ \hat{\mathcal{F}\_{\*}}(\mathscr{H})$ is homotopic to the loop space of Fredholm operators $\Omega\mathcal{F}(\mathscr{H})$.
But in their paper and in other sources what they prove is homotopy to a so-called **relat... | https://mathoverflow.net/users/73712 | Loop space of Fredholm operators from a Relative loop space | On a complex Hilbert space $H$ we can easily identify the space of skew-adjoint operators with the space of self-adjoint operators. (Multiplication by $\sqrt{-1}$ will do the trick. The space $\newcommand{\FS}{\mathscr{FS}}$ $\FS$of Fredholm selfadjoint operators on $H$ has three components $\FS\_{\pm}$, $\FS\_\*$. The... | 0 | https://mathoverflow.net/users/20302 | 206697 | 98,855 |
https://mathoverflow.net/questions/206699 | 7 | Fix a first-order signature $\Sigma$. There is an equivalence relation $\sim$ on the class $\Sigma\mathrm{-Str}$ of all $\Sigma$-structures given by $M \sim N$ iff $M$ and $N$ are elementarily equivalent. Define a relation $\to$ on the class of all $\Sigma$-structures by $M\to N$ if there exists an elementary embedding... | https://mathoverflow.net/users/2362 | If two structures are elementarily equivalent, is there a zigzag of elementary embeddings between them? | The [Keisler–Shelah theorem](http://dx.doi.org/10.1007/BF02771574) implies that the following are equivalent:
* $M$ and $N$ are elementarily equivalent.
* For some set $X$ and some ultrafilter $U$ on $X$, $M^X / U$ and $N^X / U$ are isomorphic.
Thus, recalling Łoś's theorem, any two elementarily equivalent structur... | 9 | https://mathoverflow.net/users/11640 | 206700 | 98,857 |
https://mathoverflow.net/questions/206703 | 8 | I am looking for families of invariant integrals $\int\_G dg f(g)$ (where $dg$ is a Haar measure) over a semisimple Lie group that can be evaluated in closed form, together with references where I can find proofs.
I don't know yet what I'll need, but $f$ can be quite involved, so I want to study the techniques that ... | https://mathoverflow.net/users/56920 | explicit integrals over a Lie group | [Integration with respect to the Haar measure on unitary, orthogonal and symplectic group](http://arxiv.org/abs/math-ph/0402073) will tell you pretty much all we know for integrals of polynomial functions. (For a more recent paper, see [Elementary derivation of Weingarten functions of classical Lie groups](http://arxiv... | 7 | https://mathoverflow.net/users/11260 | 206713 | 98,861 |
https://mathoverflow.net/questions/206708 | 3 | What is the smallest known $c$ so that for any $n\geq 2$ there are at least $n/\log\_2{n}$ primes between $n$ and $cn$ (inclusive)?
The [prime number theorem](http://en.wikipedia.org/wiki/Prime_number_theorem) seems to give an asymptotic result so I am not sure how to get a particular constant from it.
I am interes... | https://mathoverflow.net/users/45564 | Smallest constant so that there are at least $n/\log_2{n}$ primes between $n$ and a constant multiple of $n$ | Assuming I made no mistake, $c=11/5$ is the smallest such value.
Let $c=11/5$. Using Theorem 1 of [Rosser-Schoenfeld](https://projecteuclid.org/euclid.ijm/1255631807), we see easily that
$$\pi(cn)-\pi(n-1)\geq(\log 2)\frac{n}{\log n},\qquad n\geq 74,$$
where $\log$ is the natural logarithm as usual in analytic number... | 11 | https://mathoverflow.net/users/11919 | 206719 | 98,864 |
https://mathoverflow.net/questions/206707 | 5 | I have seen two versions of a result called "Hecke Equidistribution" and I wanted to know if they were the same or different.
---
[**#1**](https://mathoverflow.net/questions/133410/hecke-equidistribution) Let $p = 4k+1 = (a+bi)(a-bi) = e^{i\theta}\sqrt{a^2 + b^2}$. Then $\theta$ equidistributed as $p \to \infty$.... | https://mathoverflow.net/users/1358 | Equidistribution of Hecke points and $p = (a+bi)(a-bi) = e^{i\theta}\sqrt{a^2 + b^2}$ | These are two very different results. Briefly, the first result depends on the nonvanishing of (certain) Hecke $L$-functions at the edge of the critical strip, while the second result depends on a nontrivial bound for Hecke eigenvalues of Maass forms on the modular surface. As you can see, both are related to objects n... | 5 | https://mathoverflow.net/users/11919 | 206722 | 98,865 |
https://mathoverflow.net/questions/206723 | 3 | Let $R$ be a local (Noetherian) integral domain of dimension greater than one. Can the integral closure (i.e. normalization) of $R$ have a maximal ideal of height one?
| https://mathoverflow.net/users/19045 | Can height one maximal ideals in the normalization contract to non-height one primes in the base? | Example 2 of Appendix A1 of Nagata's "Local Rings" with $m = 0$ is an example of such a ring.
| 2 | https://mathoverflow.net/users/60618 | 206725 | 98,866 |
https://mathoverflow.net/questions/206738 | 6 | For this question, a *structure* means a first-order structure in a computable language with domain $\omega$; a *copy* of a structure $\mathcal{A}$ is a structure $\mathcal{B}\cong\mathcal{A}$.
---
Given a structure $\mathcal{A}$, we can try to understand the computability-theoretic complexity of $\mathcal{A}$ in... | https://mathoverflow.net/users/8133 | Minimal degrees of structures |
>
> Question 1: Does every structure $\mathcal{A}$ have an $\mathcal{A}$-minimal degree?
>
>
>
No. [Diamondstone, Greenberg, and Turetsky](http://diamondstone.me/natural_degree_spectra.pdf) show that the set of *array noncomputable* degrees is a degree spectrum.
And Downey, Jockusch and Stob showed that a degr... | 5 | https://mathoverflow.net/users/4600 | 206741 | 98,874 |
https://mathoverflow.net/questions/206749 | 6 | Given a fusion category $\mathcal C$, the *Grothendieck Ring* $K\_0(\mathcal C)$ is the $\mathbb Z$-based ring whose basis elements correspond to isomorphism classes of simple objects and whose multiplication is given by
$$
X\times Y = \sum\_Z N\_{XY}^Z Z, N\_{XY}^Z=\vert \mathcal C(X\otimes Y,Z)\vert
$$
Two fusion... | https://mathoverflow.net/users/25642 | For what $G$ is $Rep(D(S_3))_{ad}$ Grothendieck equivalent to $Rep(G)$? | Despite my love of the finite group game, let me give an argument that doesn't use the classification of groups of order 18. The 1-dimensional objects correspond to representations of the abelianization. So your group must have abelianization $C\_2$, and so its commutator subgroup must be a group of size 9. Note that t... | 6 | https://mathoverflow.net/users/22 | 206756 | 98,880 |
https://mathoverflow.net/questions/206751 | 8 | While reading some papers about $\psi$DOs I found some spaces of vector valued functions which I am not familiar with. I am looking for references about the Schwartz space of functions with values in a Frechet space $E$ i.e. $\mathcal{S}(\mathbb{R},E)$.
For example when $E=L^{m}\_{Cl}(\Omega)$ is the space of $\psi$... | https://mathoverflow.net/users/40090 | Schwartz space of functions with values in a Frechet space | $\mathcal S(M,E) = \mathcal S(M)\bar\otimes E$ for the completed injective or projective tensor product, which agree since $\mathcal S(M)$ is a nuclear spaces. See H. Jarchow. Locally convex spaces. Teubner, Stuttgart, 1981. Fourier transform you can apply just to the left hand side of the tensor product.
Also, have ... | 6 | https://mathoverflow.net/users/26935 | 206757 | 98,881 |
https://mathoverflow.net/questions/198492 | 7 | Consider an element $e \in \pi\_n^s(S^0)$, in the stable homotopy groups of sphere. Let $C$ denote the spectrum which is cone of $e$, i.e. $C$ fits in the cofiber sequence
$$ S^n \to S^0 \to C.$$
$\textbf{Question:}$ I want to know under what condition $e$ viewed as a self map of $C^{\wedge k}$ (the $k$-fold smash ... | https://mathoverflow.net/users/19186 | When does a map in the stable homotopy group gets killed when smashed with cone of itself? | The answer to your third question is "yes".
A little more generally, if $Sq^{n+1}$ acts nontrivially in the mod $2$ cohomology of $C$, then $Sq^{(k+1)(n+1)}$ acts nontrivially in the mod $2$ cohomology of $C^{\wedge k+1}$, by the Cartan formula. If $e$ smashed with the identity on $C^{\wedge k}$ were null-homotopic, ... | 4 | https://mathoverflow.net/users/9684 | 206759 | 98,882 |
https://mathoverflow.net/questions/206755 | 5 | *(This was posted [previously](https://math.stackexchange.com/questions/1270902/) in MSE without getting any answers.)*
It is known that given *primitive* (co-prime) integer solutions to,
$$x\_1^4+x\_2^4+x\_3^4+x\_4^4 = z^4$$
then there is one $x\_i$ such that $z^4-x\_i^4$ is divisible by $d\_4=5^4$. Additionally... | https://mathoverflow.net/users/12905 | Congruence properties of $x_1^6+x_2^6+x_3^6+x_4^6+x_5^6 = z^6$? | To answer this question, one needs to understand why Theorem holds. In fact, it is a corollary of a stronger statement: if
$$x\_1^k+x\_2^k+x\_3^k+\dots+x\_k^k = z^k\qquad (1)$$
and $k+1$ is prime, then all $x\_j$, except possibly one, are multiples of $k+1$. (Letting $x\_i$ be this exceptional value, we trivially get t... | 3 | https://mathoverflow.net/users/7076 | 206763 | 98,884 |
https://mathoverflow.net/questions/203982 | 8 | While searching for a use for the "sum invariant" of indefinite binary quadratic forms of discriminant $D = n^2 + 4$ (see <https://cs.uwaterloo.ca/journals/JIS/VOL17/Smith/smith5.html>), I believe I have proved the following lower bound for the form class number $D$. Let $\tau(n)$ denote the number of positive divisors... | https://mathoverflow.net/users/5373 | Lower bound on class number of binary quadratic forms of discriminant of the form $n^2+4$ | see [Mollin's paper](https://projecteuclid.org/download/pdf_1/euclid.pja/1195512513) and the various references given there. You will probably also find relevant
material in his book "Quadratics".
| 3 | https://mathoverflow.net/users/3503 | 206768 | 98,885 |
https://mathoverflow.net/questions/206765 | 1 | Sorry for my ignorance in advance, this should be a very naive question and I would be happy for a reference.
Let $G$ be an arbitrary group (not necessary finite) acting on two (connected) manifolds $M$ and $N$. The action on $M$ is on the left and the action on $N$ is on the right. Suppose that the action
$M\times... | https://mathoverflow.net/users/65818 | Homology of manifold with action of group | No, this is in general quite far from being true (and also somewhat ill-posed, since there is not a natural candidate for what "=" means since there is no map).
For example, take $M = S^1$ and $N=S^2$, both with the antipodal $C\_2$-action (which you can view as either right or left action).
Then $M/C\_2 \simeq S^1... | 11 | https://mathoverflow.net/users/6574 | 206770 | 98,886 |
https://mathoverflow.net/questions/206790 | 4 | I have a construction on two vector bundles and I would like to give it a name and a symbol but I can't find anything.
For two vector bundles $A=\{(x,A\_x) : x \in X\}$ and $B=\{(y,B\_y):y\in Y\}$ there is a vector bundle on $X\times Y$:
$$
A \mathop{?} B = \{((x,y),\text{L}(A\_x,B\_y)) : (x,A\_x) \in A, (y,B\_y) \... | https://mathoverflow.net/users/73756 | Name for construction on two vector bundles | Some references, e.g., Berline–Getzler–Vergne, *Heat kernels and Dirac operators*, will define, for convenience, a modified tensor product $A \boxtimes B \to X \times Y$ by $$A \boxtimes B := \operatorname{proj}\_1^\ast A \otimes \operatorname{proj}\_2^\ast B,$$ where $\operatorname{proj}\_1 : X \times Y \to X$ and $\o... | 2 | https://mathoverflow.net/users/6999 | 206794 | 98,893 |
https://mathoverflow.net/questions/206513 | 3 | I found a theorem about multigraded Hilbert series stated as follows:
Let $R$ be a Noetherian multigraded algebra $R:=\bigoplus\_{j\in\mathbb{N}^m}{R\_j}$ over $R\_0=\mathbb{C}$. If $R$ is generated by $h$ homogeneous polynomials of multidegrees $\alpha\_1,\dots,\alpha\_h$, then its Hilbert series $HS\_R(\mathbf{t})$... | https://mathoverflow.net/users/41283 | I need to refind a reference on multigraded Hilbert series | As @Richard points out, this follows easily from proof with m = 1. In particular, you can do an induction on h and use the fact that length is additive on short exact sequences. However if you are looking for a specific reference, [this](http://www.math.lsa.umich.edu/~hderksen/Publications/denom2005.pdf) paper might be... | 1 | https://mathoverflow.net/users/13288 | 206802 | 98,895 |
https://mathoverflow.net/questions/204200 | 8 | For $n > 0$, let $P$ be an integral polytope, that is, the convex hull in $\mathbb{R}^n$ of points in $\mathbb{Z}^n$. Suppose that $\dim(P) = n$.
**Question:** Given $d > n + 2$ is it true that
$$ dP \cap \mathbb{Z}^n \stackrel{?}{=} (n+2)P \cap \mathbb{Z}^n + (d-(n+2))P \cap \mathbb{Z}^n \ ? $$
That is for any i... | https://mathoverflow.net/users/71056 | Integer decomposition of dilated integral polytopes | The answer is YES in the following stronger form:
**Lemma 1:** Let $d>n$ and let $p\in dP\cap \mathbb{Z}^n$. Then there is an $i\in\{1,\dots, n\}$ and points $p'\in iP\cap \mathbb{Z}^n$, $p\_1,\dots, p\_{d-i}\in P\cap \mathbb{Z}^n$ such that
$$
p = p' + p\_1 + \cdots + p\_{d-i}.
$$
For the proof I will (as is custo... | 7 | https://mathoverflow.net/users/22608 | 206806 | 98,896 |
https://mathoverflow.net/questions/206823 | 6 | It is well-known that the direct product of any family of abelian groups
is an elementary extension of the direct sum of the family
(see e.g. Lemma A.1.6 in the book `Model Theory' by W. Hodges,
where this is proven for modules).
Can this result be generalized to some other classes of groups?
More concrete questions:... | https://mathoverflow.net/users/68935 | Elementary equivalence of the direct product and direct sum of groups | Consider the sentence $P:\forall x\exists y:y^3=1\neq y,yx=xy$. Let $(F\_i)$ be an infinite family of groups, each of which possesses an element of order 3. Then $\bigoplus F\_i$ satisfies $P$.
Now assume that each $F\_i$ is non-abelian of order 6: in $F\_i$, elements of order 2 never commute to elements of order 3.... | 7 | https://mathoverflow.net/users/14094 | 206827 | 98,902 |
https://mathoverflow.net/questions/206820 | 4 | I want to know if there is a uniqueness (in any sense) theorem for the symplectic structure in a neighborhood of a symplectic surface in a four dimensional symplectic manifold. Or more generally for any configuartion of symplectic surfaces.
| https://mathoverflow.net/users/31475 | neighborhood of symplectic surfaces | Here is the version of the Darboux-Weinstein theorem that you want to use: Let $(M\_1,\omega\_1)$ and $(M\_2,\omega\_2)$ be symplectic manifolds of dimension $2n$ and let $\iota\_i:P\to M\_i$ be smooth embeddings with the property that there exists an isomorphism $\phi:\iota\_1^\*(TM\_1)\to \iota\_2^\*(TM\_2)$ of vecto... | 7 | https://mathoverflow.net/users/13972 | 206828 | 98,903 |
https://mathoverflow.net/questions/206645 | 19 | I've read in a number of places that, building on previous work of T. Nagell, W. Ljunggren proved in **1** that the Diophantine equation
$$\frac{x^{n}-1}{x-1} = y^{2}$$
doesn't admit solutions in integers $x>1, y>1, n>2$, except when $n=4, x=7$ and $n=5, x=3$.
Since neither I have been able to spot a copy of Ljun... | https://mathoverflow.net/users/1593 | On a result attributed to W. Ljunggren and T. Nagell | I went to my office today and scanned the Ljunggren's paper that OP asked for. I provide some bibliographical information first:
>
> Wilhelm Ljunggren, *Noen setninger om ubestemte likninger av formen $\frac{x^n - 1}{x-1}=y^q$*, Norsk. Mat. Tidsskrift, **25** (1943), 17 -- 20 ( = Collected Papers of W. Ljunggren e... | 26 | https://mathoverflow.net/users/26538 | 206830 | 98,904 |
https://mathoverflow.net/questions/206826 | 6 | This is a tweak of Henry Segerman's question
[Can an arbitrary collection of circles of total area 1/2 fit into a circle of area 1?](https://mathoverflow.net/questions/24184) , but restricted to the point of possibly having a
proof in the literature.
If one takes the question above and restricts it to ask for a pair... | https://mathoverflow.net/users/35626 | Pigeonholing Polygons: Can two rigid regions fit in twice the space needed? | Suppose you want to pigeonhole pack $aC$ and $bC$ into $C$, $a^2+b^2=1/2$, and $a\neq b$. Let $c=\tfrac12(a+(1-b))$, then $a<c<(1-b)$. Pick any direction $u\in S^1$, and let $w=w\_u(C)$ be the width of $C$. There are two parallel lines $L\_1$ and $L\_2$ perpendicular to $u$ that bound $C$ and are a distance $w$ apart. ... | 2 | https://mathoverflow.net/users/20186 | 206831 | 98,905 |
https://mathoverflow.net/questions/206798 | 3 | I give [here](http://www.mathcounterexamples.net/a-topological-vector-space-with-no-non-trivial-continuous-linear-form/) the classical example of the space $E = L^p([0,1])$ which has no open convex subsets apart from $\emptyset$ and $E$. Consequently, there is no non-trivial continuous linear form on $E$.
Apart from ... | https://mathoverflow.net/users/41060 | Examples of TVS with no non-trivial open convex subsets | $W^{p,\infty}(\mathbb R^n)$ for $0<p<1$, which is the space of all smooth functions such that each partial derivative is in $L^p$. Instead of $\mathbb R^n$ one can also take a Riemannian manifold of bounded geometry, where now each iterated covariant derivative has to be in $L^p$ (of sections of the appropriate tensor ... | 1 | https://mathoverflow.net/users/26935 | 206837 | 98,907 |
https://mathoverflow.net/questions/206753 | 4 | Let $R\_{n,k,b}$ be the number of $b$-ary strings of length $n$ that contain some run of length at least $k$ from some $(b-1)$-ary subalphabet. Let $N\_{n,k,b}=b^n-R\_{n,k,b}$ be the size of the complement. I am interested in values like, say, $N\_{21,8,4}$.
Can we compute $N\_{n,k,b}$ efficiently? Is there a reasona... | https://mathoverflow.net/users/4600 | Strings with no long runs from proper subalphabets | Here is a dynamic programming approach to compute $N\_{n,k,b}$ in time $O\_{k,b}(n)$. You can extract from it a recurrence for $N\_{n,k,b}$ in the $n$ variable.
Given a suitable sequence, for each of the $b$ letters, we can count the number of steps from the last occurrence of each letter to the end, which is a numbe... | 3 | https://mathoverflow.net/users/18060 | 206842 | 98,908 |
https://mathoverflow.net/questions/206813 | 5 | In [that](http://www.mathcounterexamples.net/a-topological-vector-space-with-no-non-trivial-continuous-linear-form/) post, I give an example of a TVS for which the topological dual is equal to $0$. But in the example, there is no open convex subset different from the empty set or the space itself.
Do you have an exa... | https://mathoverflow.net/users/41060 | TVS with null topological dual space | There can't be such an example.
Given an open convex subset $U$ not containing the origin of a (Hausdorff) TVS $E$, there is by the geometric version of [Hahn-Banach](https://books.google.com/books?id=9kXY742pABoC&pg=PA46) as given in Schaefer's book on topological vector spaces, a closed hyperplane $H$ disjoint from... | 8 | https://mathoverflow.net/users/73783 | 206856 | 98,911 |
https://mathoverflow.net/questions/206838 | 8 | I have no experience with category theory at all, but I recently stumbled upon the following construction. Since it is extremely elementary and seems rather natural, it *should* be known, but I have not been able to find it in the literature, probably because I was looking in the wrong places and/or for the wrong keywo... | https://mathoverflow.net/users/38566 | Vectorisation of a category | As Eric Wofsey observed in the comments, your construction is almost the what you get by freely adjoining coproducts to the linearization of your category. Indeed, your construction is the full subcategory of this on those objects with no repeated summands (a decently natural condition from representation theory); but ... | 12 | https://mathoverflow.net/users/78 | 206859 | 98,914 |
https://mathoverflow.net/questions/206854 | 6 | Suppose $\mathbb{P}$ is a regular suborder of the separative partial order $\mathbb{Q}$ (see below for definitions). Must there always exist some complete boolean algebra $\mathbb{B}$ such that:
1. $\mathbb{P}$ is a dense suborder of $\mathbb{B}$ (so $\mathbb{B}$ is the boolean completion of $\mathbb{P}$)
2. $\mathbb... | https://mathoverflow.net/users/26319 | completions of regular suborders | $\newcommand\P{\mathbb{P}}\newcommand\Q{\mathbb{Q}}
\newcommand\Q{\mathbb{Q}}\newcommand\B{\mathbb{B}}\newcommand\Z{\mathbb{Z}}
\newcommand\RO{\text{RO}}$Unless I am mistaken (and please correct me if I am, since these issues are sometimes confusing), I believe the answer is yes. Let $\B$
consist of the elements of $\R... | 5 | https://mathoverflow.net/users/1946 | 206871 | 98,917 |
https://mathoverflow.net/questions/206866 | 3 | If for some collection of open sets $\cup\_{i\in I} (A\_i \cup^\* -A\_i)=\mathbb S^d$, then is there an $x\in \mathbb S^d$ and $i\_1,\ldots i\_d\in I$ for which $x\in A\_{i\_1}\cap \ldots \cap A\_{i\_d}$?
Important that the question implicitly requires $A\_i\cap -A\_i=\emptyset$, as $\cup^\*$ denotes disjoint union.
... | https://mathoverflow.net/users/955 | If $\cup (A_i \cup -A_i)=\mathbb S^d$, then is there an $x$ in $d$ sets? | Unfortunately, I think your conjecture is wrong: Embedd $S^3 \subset \mathbb C^2$ in the standard way. Consider the three sectors
$$U\_1 = \{z \in \mathbb C \ | \ z = re^{i\varphi}, r > 0, \varphi \in (0, 2\pi / 3)\}$$
$$U\_2 = \{z \in \mathbb C \ | \ z = re^{i\varphi}, r > 0, \varphi \in (2\pi / 3, 4\pi/3)\}$$
$$U\_3 ... | 3 | https://mathoverflow.net/users/14233 | 206873 | 98,918 |
https://mathoverflow.net/questions/206869 | 3 | Let $f:X\to S$ be a finite type affine morphism of schemes where $S$ is an integral noetherian affine regular scheme whose function field is of characteristic zero.
Assume that all geometric fibers of $f$ are non-empty and that the generic fibre of $f$ is smooth.
Does have $f$ have a section up to replacing $S$ by ... | https://mathoverflow.net/users/73790 | Sections of morphisms up to fppf covering | The answer is negative; here is a counterexample. Let $S = \mathrm{Spec}(R)$ be the spectrum of a DVR whose fraction field of characteristic $0$ (to fulfill your requirements) and take $X := \eta \sqcup s$, where $\eta$ is the generic point and $s$ is the closed point of $S$. Then for any fppf cover $S' \rightarrow S$ ... | 3 | https://mathoverflow.net/users/5498 | 206875 | 98,920 |
https://mathoverflow.net/questions/206877 | 5 | I am looking at exercise 6.3.3 in Mcduff's and Salamon's book J-holomorphic curves and Symplectic topology, which basically gives an example of a moduli space whose actually dimension is greater than its virtual dimension. I am wondering how to compute the actual one.
The example is the following:
Take your symplect... | https://mathoverflow.net/users/73791 | actual dimension of concrete moduli space of holomorphic curves vs its virtual dimension | **Moral:** This is giving a glimpse at the troubles with *branched* multiply-covered $J$-holomorphic curves. The virtual-dimension is a Fredholm index, involving $\chi(C)$ of your curve. If $C$ is multiply-covered with $b$ branch points then the index scales as $b$. But the actual dimension scales as $2b$ (the branch p... | 2 | https://mathoverflow.net/users/12310 | 206880 | 98,922 |
https://mathoverflow.net/questions/206872 | 18 | For a topological space M, It is known from homotopy theory that the elements of the first cohomology $H^1(M;\mathbb{Z})$ are in 1-1 correspondence with homotopy classes of maps $[M,S^1]$
In my case of interest M is a smooth manifold. Take $\alpha$ and take smooth $f\colon M \to S^1$ representing $\alpha$ under the a... | https://mathoverflow.net/users/14105 | When can a class in $H^1(M;\mathbb{Z})$ be represented by a fiber bundle over $S^1$ | Let me assume throughout this answer that $M$ is closed, oriented, and connected. Here are some necessary conditions.
If you ask for a smooth fiber bundle, then a necessary condition is that the tangent bundle of $M$ has a trivial quotient of rank $1$, or equivalently a trivial subbundle of rank $1$. This is possibl... | 9 | https://mathoverflow.net/users/290 | 206886 | 98,924 |
https://mathoverflow.net/questions/205924 | 2 | It is well-known that any non-compact manifold admits a nowhere vanishing vector field. If we have a Riemannian metric we may pick such a vector field and normalize it so that at every point it has unit length. But what I additionally need is a control on the derivatives of the vector field and here it seems to me that... | https://mathoverflow.net/users/13356 | Nowhere vanishing, normalized vector field with bounded derivatives | I could find now a solution in the literature. In
S. Weinberger, *Fixed-point theories on noncompact manifolds*, J. Fixed Point Th. Appl. **6** (2009), 15-25
it is shown that a normed vector field with bounded derivatives exists if and only if the Euler class $e(M) \in H\_0^{\mathrm{uf}}(M)$ vanishes.
| 3 | https://mathoverflow.net/users/13356 | 206901 | 98,931 |
https://mathoverflow.net/questions/206900 | 7 | Let $\mathcal{M}\_2$ be the moduli space of genus two curves and $\mathcal{A}\_2$ the moduli space of principally polarized abelian surfaces. Then the Abel-Jacobi map gives an open embedding $\mathcal{M}\_2 \hookrightarrow \mathcal{A}\_2$. My question is, for which compactification $\overline{\mathcal{A}}\_2$ of $\math... | https://mathoverflow.net/users/21014 | Extending the Abel-Jacobi map over the DM-compactification $\overline{\mathcal{M}}_2$? | This map is usually called the Torelli map, not the Abel-Jacobi map. In any case, Mumford observed that a certain toroidal compactification of $\mathscr{A}\_g$ admits an extension of the Torelli map; the original reference is [this paper of Namikawa](http://link.springer.com/article/10.1007%2FBF01433145), I think. That... | 8 | https://mathoverflow.net/users/6950 | 206903 | 98,932 |
https://mathoverflow.net/questions/206898 | 3 | Let $\omega^\omega$ denote the collection of all functions $f:\omega\to\omega$. For $f,g\in\omega$ we define
* $f\leq g$ if $f(n)\leq g(n)$ for all $n\in\omega$;
* $f\leq^\* g$ if there is $N\in\omega$ such that $f(n)\leq g(n)$ for all $n\in \omega$ with $n\geq N$.
The cardinals ${\frak b}, {\frak d}$ are defined a... | https://mathoverflow.net/users/8628 | ${\frak b}$ and ${\frak d}$ defined with $\leq$ instead of $\leq^*$ | Using "$\le$" instead of "$\le^\*$," I claim $\mathfrak{b}'=\omega$ and $\mathfrak{d}'=\mathfrak{d}$.
**To see $\mathfrak{b}'=\omega$**, let $f\_n$ be the constant function $x\mapsto n$; then setting $S=\{f\_n: n\in\omega\}$, there is clearly no function which is $\ge$ every element of $S$.
**To see $\mathfrak{d}'=... | 7 | https://mathoverflow.net/users/8133 | 206904 | 98,933 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.