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/120808 | 14 | It is well known and more or less proven in Hartshorne's 'Algebraic Geometry' (p. 209) that for every noetherian scheme $X$ and every collection of abelian sheaves $\mathcal{F}\_i$ the canonical map
$$ \bigoplus\_i H^\*(X; \mathcal{F}\_i) \to H^\*(X; \bigoplus\_i \mathcal{F}\_i) $$
is an isomorphism.
My line of i... | https://mathoverflow.net/users/2039 | When does sheaf cohomology commute with arbitrary direct sums? | A Grothendieck topology is called *noetherian* if every object is quasi-compact (which is defined as usual). On such a topology, sheaf cohomology commutes with filtered colimits (and in particular with arbitrary direct sums). A proof can be found in Tamme's *Intoduction to Etale cohomology*, Theorem §3.3.11.1. For the ... | 12 | https://mathoverflow.net/users/2841 | 120847 | 68,005 |
https://mathoverflow.net/questions/120820 | 2 | The question is essentially in the title: while attempting to compute the colimit of a diagram of cell complexes, my colleague and I find ourselves stumped with the following graph theoretic problem:
>
> Given a directed (unweighted) graph $\Gamma$ and vertices $x,y$, does there exist a vertex $z$ along with direct... | https://mathoverflow.net/users/18263 | Finding a vertex equidistant from two given vertices in a digraph | Form the product graph $\Gamma \times \Gamma$ whose vertices are pairs $(u, u') \in \Gamma \times \Gamma$ and edges $(p, p') : (u, u') \to (v, v')$ are pairs of edges $p : u \to v$, $p' : u' \to v'$. Let $\Delta = \lbrace (u, u) \mid u \in V(\Gamma) \rbrace$ be the diagonal. Your question is equivalent to the question ... | 5 | https://mathoverflow.net/users/1176 | 120851 | 68,008 |
https://mathoverflow.net/questions/120843 | 4 | This question may be very trivial, I apologize if it is so.
I have subspace $V\subset \mathbb Z\_2^r$ with the property that for every choice of a subset $I$ of $k$ elements in $\{1,2,\dots r\}$, the projection of $V$ onto the corresponding $k$ coordinates is surjective.
QUESTION: Can one find a lower bound on $\d... | https://mathoverflow.net/users/10610 | Lower bound on the dimension of a subspace of $\mathbb Z_2^r$? | Let $\pi\_i:V\to\mathbb Z\_2$ be the projection to the $i$-th coordinate. The $\pi\_i$ are elements of the dual space $V^\star$, and by assumption, any $k$ of these $\pi\_i$ are linearly independent. Let $C$ be subspace of $\mathbb Z\_2^r$ consisting of those tuples $(a\_1,\dots,a\_r)$ such that $\sum a\_i\pi\_i=0$. So... | 7 | https://mathoverflow.net/users/18739 | 120856 | 68,009 |
https://mathoverflow.net/questions/97046 | 13 | In Exposé 195 of the Séminaire Bourbaki, Grothendieck states the following two theorems of non-flat descent.
>
> **Theorem 1.** Let $\Lambda$ be a noetherian ring and $C$ the category of $\Lambda$-algebras which are finite type artinian $\Lambda$-modules. Let $F:C\to (Set)$ be a covariant functor. Then $F$ is pro-r... | https://mathoverflow.net/users/17988 | Les deux théorèmes d'existence en théorie formelle des modules | You should look at A H M Levelt's notes:
[Sur la pro-representabilite de certains foncteurs en geometrie algebrique](http://www.math.ru.nl/~ahml/prorep.pdf)
where, I think, proofs of both Theorem 1 and Theorem 2 are written out. Also, he has written a paper "Foncteurs exacts à gauche" (Inventiones + errata) which d... | 5 | https://mathoverflow.net/users/40 | 120860 | 68,010 |
https://mathoverflow.net/questions/120798 | 8 | Suppose $V$ is a finite-dimensional vector space (over $\mathbb{C}$) and $\lambda$ is a partition of $n$ (not necessarily the dimension). Let $S^\lambda(V)=(V^{\otimes n})\_\lambda$ be the $\lambda$'th Schur functor applied to $V$ (a.k.a. the $\lambda$-component of the $S\_n$ representation $V^{\otimes n}$). Pick a bas... | https://mathoverflow.net/users/7108 | A basis for Schur functors | I think this exact question answered in Theorem 1 of §8.1 of [William Fulton's "Young Tableaux"](http://books.google.com/books/about/Young_Tableaux.html?id=cYA_RpBLJUkC). Lazy as I am, I am unsure whether I have ever read the proof, but the answer is the following (unless I got Fulton's notations wrong): Whenever $p$ i... | 8 | https://mathoverflow.net/users/2530 | 120863 | 68,012 |
https://mathoverflow.net/questions/120862 | 2 | I've been given the following BVP:
\begin{align\*}
-\Delta u = u- u^3,\: x\in \Omega
\end{align\*}\begin{align}
u = 0,\: x\in \partial \Omega
\end{align}
where $\Omega\subset \mathbb{R}^N$ is bounded.
I am supposed to show that $-1< u(x)< 1$ for all $x\in\mathbb{R}^N$.
I have experimented with sub/sup solutions, b... | https://mathoverflow.net/users/31181 | Boundedness of a given boundary value problem. | Multiply your equation times the function $v(x) = \max(u-1,0)$. After you integrate by parts the Laplacian you will get an equality with opposite signs in each side that gives you a contradiction unless $v \equiv 0$.
Then do the same thing with $\min(u+1,0)$.
There are other ways to do it, including using sub/super... | 1 | https://mathoverflow.net/users/26672 | 120864 | 68,013 |
https://mathoverflow.net/questions/120753 | 3 | Honestly, I have been looking for an a **finite Quasicommutative semigroup** by surfing the web, but I could't. May I ask here to give me an example for such this kind of semigroup. I tried to built one of them by using GAP, but it fails. Thank you so much. I am new to it.
| https://mathoverflow.net/users/13898 | Asking about a quasicommutative semigroup | The simple examples are Hamiltonian groups. Then you can construct Clifford semigroups from them.
| 4 | https://mathoverflow.net/users/18814 | 120866 | 68,014 |
https://mathoverflow.net/questions/120858 | 1 | Let $f:\mathbb R^n\to\mathbb R$ be a Morse function with uniform nondegenerate Hessian at critical points, i.e. for some $\delta>0$
$$
\forall x \in \{\nabla f=0\}\;\forall \xi\in\mathbb R^n: \quad |\langle \xi, \nabla^2 H(x)\xi \rangle | \geq \delta |\xi|^2 .
$$
*Edit:* Liviu Nicolaescu pointed out a condition to ensu... | https://mathoverflow.net/users/31180 | Integration by parts wrt. a Morse function on its basin of attraction | What troubles me is the noncompactness of $\newcommand{\bR}{\mathbb{R}}$ $\bR^n$. The boundary $\partial \Omega$ could be noncompact in a rather unpleasant way, difficult to control. For example, there might exists a sequence of critical points $p\_\nu$ of index $1$ going to $\infty$ as $\nu\to \infty$ such that there ... | 0 | https://mathoverflow.net/users/20302 | 120869 | 68,016 |
https://mathoverflow.net/questions/120836 | 2 | I am working on the class of the Fulton-Macpherson compactification of configuration space in the Grothendieck ring of varieties, over the field of complex numbers. As a first step, I am wondering about the case of 2 points, i.e. the motive of $X[2]/S\_2$ for a nonsingular variety $X$. For the case the two points coinc... | https://mathoverflow.net/users/30661 | Motive of unlabeled fulton-macpherson configuration space? | If we identify $T\_x^2 / T\_x$ with $\mathbb A^n$, then the action of $S\_2$ is multiplication by $-1$. So $S\_2$ acts trivially on the projectivization and you are just asking about the existence of a cell decomposition of projective space.
---
A useful reference might be the last parts of Ezra Getzler's old pr... | 0 | https://mathoverflow.net/users/1310 | 120876 | 68,019 |
https://mathoverflow.net/questions/120865 | 7 | **Question:** Let $X$ be a noetherian integral scheme. Is there a dense open subscheme $U\subset X$ such that $U$ is Jacobson?
I am happy to allow $X$ to be excellent and then the question of course immediately reduces to $X$ excellent and regular. Equivalently, we could also ask whether every noetherian/excellent sc... | https://mathoverflow.net/users/40 | Are noetherian schemes generically Jacobson? | Here is a counterexample. Let $k$ be a field, $R=k[x,y]$. Choose a set $\Sigma$ of closed points of $\mathbb{A}^2\_k=\mathrm{Spec}(R)$ such that:
(1) $\Sigma$ is Zariski-dense,
(2) for every $s\in\Sigma$ there is a curve $C$ containing $s$ such that $C\cap\Sigma$ is finite.
EDIT: (For instance, if $\mathrm{char... | 10 | https://mathoverflow.net/users/7666 | 120881 | 68,022 |
https://mathoverflow.net/questions/120870 | 0 | Is there an obvious way to write
$T\_{\mathbb{P}^2}\otimes T\_{\mathbb{P}^2}$ and $S^2(T\_{\mathbb{P}^2})$?
| https://mathoverflow.net/users/13803 | Symmetric power of tangent space | One can write explicit resolutions by line bundles ---
$$
0 \to O \to O(1)^6 \to O(2)^9 \to T \otimes T \to 0
$$
and
$$
0 \to O(1)^3 \to O(2)^6 \to S^2T \to 0
$$
respectively.
| 6 | https://mathoverflow.net/users/4428 | 120889 | 68,025 |
https://mathoverflow.net/questions/120879 | 3 | Apologies for the length question. Those acquainted with the analytics industry will know that the next big thing in the information technology world will be the Big Data revolution where huge volumes of data will be processed. Big Data revolution will imply huge requirement of storage space/memory hence it is critical... | https://mathoverflow.net/users/23388 | Mathematical techniques to reduce the amount of storage memory | This is one of the basic problems that led to the field of [information theory](http://en.wikipedia.org/wiki/Information_theory). It would take a while to explain all that is known about this, but the following will get you started.
Suppose we assign each book $b$ a binary string $\sigma\_b$ such that no two strings ... | 9 | https://mathoverflow.net/users/2000 | 120892 | 68,028 |
https://mathoverflow.net/questions/120897 | 2 | I asked this question in a similar form on math.se [here](https://math.stackexchange.com/questions/288297/towers-of-perfect-fields-of-mostly-order-pt), where it has been unanswered for a little over a week some work on it can be found there. The motivation for this question was another [question](https://math.stackexch... | https://mathoverflow.net/users/255 | How does an irreducible polynomial of prime power order split over an extension of prime power degree | Just for concreteness, here's an explicit counterexample along the lines of my comment. Let $L=\mathbb{Q}$ and let $f(t)=t^7 - 7t+3$ and $g(t)=t^7-14t^4+42t^2-21t-9$. Then $f(t)$ and $g(t)$ are irreducible in $L[t]$. Now let $K=\mathbb{Q}(c)$ where $g(c)=0$, so $[K:L]=7$. Then $f(t)$ factors in $K[t]$ into irreducibles... | 9 | https://mathoverflow.net/users/30412 | 120910 | 68,036 |
https://mathoverflow.net/questions/120893 | 2 | Let $B$ be a centrally symmetric convex body in $\mathbb R^n.$ The maximal function associated to $B$ is defined by
$$
Mf = \sup\_{r>0}(\chi\_{B})\_{r}\*|f|.
$$
Bourgain (<http://www.jstor.org/stable/info/2374532>) proved that this operator is bounded on $\mathbb R^n$ for all $p>3/2$ with constant depending only on $p.... | https://mathoverflow.net/users/31187 | Maximal function associated to convex bodies | In the case of the cube, it was shown by Bourgain ([very recently](http://arxiv.org/abs/1212.2661)) that the constants remain independent of $n$ for all $p>1$. The problem appears to be open for the case of more general centrally symmetric convex bodies. On page 3 of his recent preprint, Bourgain writes:
"While it is... | 2 | https://mathoverflow.net/users/630 | 120911 | 68,037 |
https://mathoverflow.net/questions/120872 | 11 | A relative category is a category $C$ with a subcategory $W$ containing all the objects of $C$.
Given a relative category $(C,W)$, $W$ is said to satisfy the ``2 implies 6'' property if, for any collection of three composable maps,
$$X\rightarrow Y\rightarrow Z\rightarrow A$$
the presence of the composites $X\ri... | https://mathoverflow.net/users/9581 | (Homotopy theory) When does the 2 of 3 property not imply 2 of 6? | I'm not sure that any examples naturally come up, of cases where you have the 2 out of 3 condition but not the 2 out of 6. Of course, if membership in W is defined by requiring certain functors to take a morphism to isomorphisms (as is so often the case in applications), then you always have 2 out of 6 (because a morph... | 11 | https://mathoverflow.net/users/6666 | 120917 | 68,039 |
https://mathoverflow.net/questions/120915 | 1 | Consider the transform $\widehat{b}=T^{-1}b$, where $T=\begin{bmatrix}b & Ab & A^2b & \dots & A^{n-1}b \end{bmatrix}$ has full rank. Is it possible to find an explicit expression for
\begin{equation}
\widehat{b}=\begin{bmatrix} \widehat{b}\_1 \\\ \vdots \\\ \widehat{b}\_n \end{bmatrix}
\end{equation}
in terms of the c... | https://mathoverflow.net/users/31193 | Linear algebra and Cayley Hamilton | Let $T^{(i)}=A^{i-1}b$ for $i=1,\cdots n$.
You can solve $\widehat{b}$ from the equation $T\widehat{b}=b$.
By Cramer's rule,
$$
\widehat{b\_1}=\frac{det[b, T^{(2)}, \cdots, T^{(n)}]} {det T}
$$
$$
\widehat{b\_2}=\frac{det[b, b, T^{(3)}, \cdots, T^{(n)}]} {det T}
$$
$\cdots$
$$
\widehat{b\_n}=\frac{det[b, T^{(2)}, ... | 1 | https://mathoverflow.net/users/21090 | 120929 | 68,045 |
https://mathoverflow.net/questions/120931 | 2 | Let $T\_1, \ldots, T\_n \in GL(n,\mathbb{F}\_p)$. Suppose for all $\vec{v} \in \mathbb{F}\_p^n$ we have $\det (T\_1 \vec{v}, T\_2 \vec{v}, \ldots, T\_n \vec{v}) = 0$. Now, let $k$ be a finite extension of $\mathbb{F}\_p$. Is it true that $\det(T\_1 \vec{v}, \ldots, T\_n \vec{v})=0$ for all $\vec{v} \in k^n$?
I know t... | https://mathoverflow.net/users/31197 | linear independence of orbits via a set of transformations in char p | Not in general: here is a counterexample. Take $p = 2$, and consider the matrices $T\_1 = [1,0,0; 0,0,0; 0,0,0], T\_2 = [0,0,0; 0,1,0; 0,1,0], T\_3 = [0,0,0; 1,0,0; 0,1,0]$. (I'm using semi-colons to separate rows; a bit of LaTeX trouble formatting the matrix ...)
Then if $v$ is the column vector $(x,y,z)$, we get th... | 5 | https://mathoverflow.net/users/2698 | 120935 | 68,048 |
https://mathoverflow.net/questions/120906 | 4 | This is a somewhat subjective question, about the past, present and especially future of algebraic number theory. I'm not at all in this area, but I'd be interested in an answer.
As we all know, algebraic number theory is one of the oldest and most developed areas in math. New and spectacular results come very often,... | https://mathoverflow.net/users/29333 | Algebraic number theory: building and simplifying | A major simplification in algebraic number theory occurred in the beginning of the 20th century when Hensel explicitly introduced his $\mathfrak{p}$-adic numbers. Compare the original cumbersome definition of the Hilbert symbol with the modern definition using local fields.
Another major simplification occurred in th... | 34 | https://mathoverflow.net/users/2821 | 120938 | 68,050 |
https://mathoverflow.net/questions/120928 | 1 | I am asking for a reference that contains a proof of Theorem 4, which is on page 315 of the following text:
>
> Hirsch, Morris W., and Stephen Smale.
> *Differential equations, dynamical systems, and linear algebra*. Vol. 60.
> Academic press, 1974.
>
>
>
Let $W$ be an open set in a vector space and $\mathca... | https://mathoverflow.net/users/1434 | Reference Request: Structural Stability of Gradient Fields | This came out of J. Palis' 1967 Thesis:
J. Palis "On Morse-Smale dynamical systems" Topology 8, 1969, 385--405.
But that dealt with dimension $\leq 3$. The result you mention seems to first appear as a corollary in
>
> J. Palis and S. Smale "Structural stability theorems" in Global analysis proceedings Symp. Pu... | 1 | https://mathoverflow.net/users/13923 | 120940 | 68,052 |
https://mathoverflow.net/questions/120909 | 21 | One can construct a category of probability spaces, but this [category has no products](https://mathoverflow.net/questions/49426/is-there-a-category-structure-one-can-place-on-measure-spaces-so-that-category-th). Now probability theory relies strongly on the ability to build independent products, the product measure. I... | https://mathoverflow.net/users/35357 | Can one view the Independent Product in Probability categorially? | Very nice question! ;) I wrote a short paper about this question about ten years ago, see <http://arxiv.org/abs/math/0206017> (My apologies for advertising my own work, but this is exactly the question I asked myself at that time).
The product of probability spaces is tensor product in the sense of category, as Marti... | 17 | https://mathoverflow.net/users/30364 | 120957 | 68,061 |
https://mathoverflow.net/questions/120965 | 3 | Hello,
Does anyone know if there is a result that relates a quantity such as an average degree to the fact the a (simple and connected graph) has no cut vertices?
e.g. if a graph has a Hamiltoninan cycle then it has not cut vertex.
By Ore's theorem, if deg(v) >= n/2 for each vertex v (n is the number of vertices ... | https://mathoverflow.net/users/31040 | Non-separable simple connected graphs. | I do not think you can do anything better than $\frac{n}{2}$. Consider the following construction: Take two complete graphs of equal size $\frac{n-1}{2}$. Add one vertex $v$ and connect this vertex to all vertices of the two complete graphs. Each vertex has degree at least $\frac{n-1}{2}$ and this graph has a cut verte... | 6 | https://mathoverflow.net/users/15684 | 120966 | 68,064 |
https://mathoverflow.net/questions/114779 | 6 | I'm in need of a condition that is analogous to the "finality" condition in the following lemma:
Lemma: A functor $F\colon A\to B$ is final if and only if for any functor $x\colon B\to Set$, the natural map $colim (xF)\to colim(x)$ is an isomorphism.
This lemma could be taken instead as a definition of *final funct... | https://mathoverflow.net/users/2811 | Local finality condition (for re-indexing parameterized colimits) | Since nobody has said so, I will mention that the notion you describe is a particular case of the known --- but perhaps obscure --- concept of Guitart exact square. One can read about it in [the nlab page](http://ncatlab.org/nlab/show/exact+square) and in [an article by Maltsiniotis](http://arxiv.org/abs/1101.4144). Ev... | 7 | https://mathoverflow.net/users/21095 | 120970 | 68,066 |
https://mathoverflow.net/questions/120946 | 1 | Hello,
I am trying to solve an equation of the form
$C\_1 f(k\_1 z) + C\_2 f(k\_2 z) + C\_3 f(k\_3 z) + C\_4 f(k\_4 z) = C\_5 z^2$
for $f(z)$. Everything is complex. The $C\_i$'s and $k\_i$'s depend on some other parameters $\{ \lambda\_i \}$ through a very complicated dependence.
Any ideas?
Thanks!
| https://mathoverflow.net/users/31201 | How to solve a linear algebraic complex equation in one function evaluated at different arguments? | Make the change of the independent variable setting $g(z)=f(e^z)$ then $g(z+a)=f(ke^z)$ where
$k=e^a$. Now you have a linear difference equation. Difference equations have been well studied.
A general method is some kind of Fourier transform, look for a solution in the form
of exponential, then take a sum of those.
See... | 1 | https://mathoverflow.net/users/25510 | 120973 | 68,068 |
https://mathoverflow.net/questions/120918 | 6 | According to Corollary 1.2(3) of the paper Silver: Noncommutative Localizations and Applications. J. of Alg. 7(1964), 44-67:
If $R$ is a (commutative) field and $\alpha: R \to S$ an epimorphism in the category of rings, then $\alpha$ is an isomorphism.
**Question:** Is $\alpha$ also an isomorphism if $R$ is not a ... | https://mathoverflow.net/users/18571 | Are epimorphisms from a division ring isomorphisms ? | I think the question can be answered affirmatively.
Since $0,R$ are the only ideals of $R$, $\alpha$ is injective (I assume $\alpha(1\_R)=1\_S$) and $R$ can be considered as subring and as left $R$-submodule of $S$. Let $\mu:S \otimes\_R S \to S$ be the multiplication. A splitting of $\mu$ is given by $j: S \to S \o... | 5 | https://mathoverflow.net/users/10194 | 120974 | 68,069 |
https://mathoverflow.net/questions/120976 | 2 | Suppose we have a family of objects $\Xi \to S$ over a base smooth projective scheme $S$. Take a closed point $p\in S$ and consider the tangent space to $S$ at $p$. Can one construct an "induced family" over this tangent space (or probably rather over its projectivized space) starting from $ \Xi$ ? what interpretation ... | https://mathoverflow.net/users/4096 | does there exist a family of objects over the tangent space to the base space of a family of objects? | Imagine that $S$ is an open subset of some fine moduli space which does not contain any rational curves. This does happen although I don't know any examples. Then any family over an affine space will have to be constant, so the answer would be no.
On the other hand, of course there is formal family over the completio... | 2 | https://mathoverflow.net/users/3847 | 120983 | 68,072 |
https://mathoverflow.net/questions/120912 | 1 | In "Motivic Homotopy Theory" on page 153 it is stated that there exists a canonical Hurewicz map relating the motivic stable homotopy category with the category $\operatorname{DM}\_\\_^{eff}(k)$. Unfortunatly, I was not able to find a definition of such a map and every attempt to define one myself ended in vain.
| https://mathoverflow.net/users/31192 | Definition of Hurewicz map relating $SH(k)$ with $DM_\_^{eff}(k)$ | There's a free-forgetful adjunction between $SH\_s(k)$ and $DM^{eff}(k)$, where $SH\_s(k)$ is the category of $S^1$-spectra (as opposed to $\mathbb{P}^1$-spectra). The right adjoint simply takes a sheaf of chain complexes with transfers in $DM^{eff}(k)$ to its underlying sheaf of spectra (i.e. view chain complexes as s... | 2 | https://mathoverflow.net/users/20233 | 120988 | 68,076 |
https://mathoverflow.net/questions/120984 | 28 | I've been getting interested in the (Bott--)Borel--Weil theorem lately. As a (mainly) geometer it is very interesting to see representation appearing (from nowhere as far as I can see) in the theory of complex geometry.
What I can't seem to find out, though, is if this fact has been used to prove any interesting res... | https://mathoverflow.net/users/1648 | Rep Theory Consequences of Bott--Weil--Borel | It's always a good idea to ask (as students typically do) why one is studying a particular subject or theorem. Here are some of my views, from the algebraic side of representation theory:
1) The original theorem here was proved by Borel and Weil, though never written up formally by them. Serre reported on it at the B... | 37 | https://mathoverflow.net/users/4231 | 120999 | 68,082 |
https://mathoverflow.net/questions/120354 | 2 | I am trying to find the maximum likelihood estimate of the parameters for the t-copula. Ideally I'd want to use a gradient-based method for optimization. However, I am having some difficulty in finding the partial derivative of the inverse of the standard, univariate t-CDF with v degrees of freedom with respect to v.
... | https://mathoverflow.net/users/31052 | Taking the partial derivative of the t-CDF with respect to the degrees of freedom | In case anyone comes across this same problem, I've found an answer. Consult the article and its web supplement "Derivatives and Fisher information of bivariate copulas" by Schepsmeier and Stober, forthcoming in Statistical Papers.
| 2 | https://mathoverflow.net/users/31052 | 121011 | 68,089 |
https://mathoverflow.net/questions/121005 | 2 | It is well known that for a proper smooth variety $X$ over an algebaically closed field $k$, the Picard functor $Pic\_{X/k}$ is representable by a smooth group scheme over $k$. My question is, when we retain everything except by replacing the base scheme $k$ by an artin ring, say the dual number $k[\epsilon]/\epsilon^2... | https://mathoverflow.net/users/5661 | The Picard Group over artin ring | In FGA, no.232, Thm 3.1, Grothendieck shows that if $f: X\to S$ is flat, projective and finitely presented, with reduced and irreducible geometric fibers, then $\operatorname{Pic}\_{X/S}$ is representable by a separated $S$-scheme, locally of finite presentation over $S$.
As for smoothness, unfortunately the "well-k... | 2 | https://mathoverflow.net/users/6950 | 121013 | 68,090 |
https://mathoverflow.net/questions/120943 | 2 | Let $f(x)$ be an irreducible polynomial in $\mathbb{Z}[x]$ or $\mathbb{F}\_{q}[x]$ with $deg(f(x)) \ge 2$ (assume constant coefficient is $1$).
Let $a \in \mathbb{Z}$ or $\mathbb{F}\_{q}$. Let $f(a)$ be composite if $a \in \mathbb{Z}$.
Let $char(\mathbb{F}\_{q}) \ne 2$.
Is it always possible to find polynomials $... | https://mathoverflow.net/users/10035 | Decomposing irreducible polynomials with a prescribed condition - Existence | The problem as stated is unsolvable in the case of $\mathbb{Z}$ coefficients with $f(a)$ a prime power. **Proof:** Let $a=0$, let $f(0) = p^n$ and let the coefficient of $x$ in $f$ not be divisible by $p$. Then $p(0)$, $q(0)$, $r(0)$ and $r(0)$ are all of the form $\pm p^k$, and the hypotheses forbid that $k=0$. So all... | 4 | https://mathoverflow.net/users/297 | 121030 | 68,096 |
https://mathoverflow.net/questions/121038 | 8 | Does there exist a compact hyperbolic 3-manifold $M$ that is not diffeomorphic to a geometrically finite hyperbolic manifold? If yes, can such $M$ have incompressible boundary?
I think the answer should be yes to both questions but I cannot find this in the literature.
**Remarks:** as usual, a *compact hyperbolic m... | https://mathoverflow.net/users/1573 | Hyperbolic 3-manifolds with no geometrically finite structure | [Edited several times] As the comments say, the answer to the first and hence to the second question is "no". Suppose that $M$ is the compact manifold and $N$ is its interior. Let $\rho$ be the given hyperbolic structure on $N$. If $M$ is without boundary then the volume of $\rho$ is finite and we are done.
Suppose ... | 8 | https://mathoverflow.net/users/1650 | 121058 | 68,109 |
https://mathoverflow.net/questions/121031 | 71 | The concept of relation in the history of mathematics, either consciously or not, has always been important: think of order relations or equivalence relations.
Why was there the necessity of singling out a particular kind of relations, namely the functional ones? I guess (but I don't have data about this) historicall... | https://mathoverflow.net/users/4721 | Why is Set, and not Rel, so ubiquitous in mathematics? | Regarding question 3, one can make an argument that actually the fundamental object is "Set together with Rel". The bijective-on-objects inclusion of Set into Rel is a categorical structure that can be expressed as an [F-category](http://ncatlab.org/nlab/show/F-category), a [proarrow equipment](http://ncatlab.org/nlab/... | 39 | https://mathoverflow.net/users/49 | 121068 | 68,113 |
https://mathoverflow.net/questions/121060 | 16 | The full braid group on $n$ strands $B\_n$ admits a surjective homomorphism $p\colon\thinspace B\_n\to \Sigma\_n$ onto the symmetric group on $n$ letters, which takes a braid to the induced permutation of its ends. The kernel $P\_n$ is well understood; it is the pure braid group on $n$ strands.
What about $p^{-1}(A\_... | https://mathoverflow.net/users/8103 | Does this subgroup of "even braids" have a name? | I don't know if these groups have been studied before, but I can say something about their cohomology rings, at least over $\mathbb{Q}$. Namely, we have $H^k(E\_n;\mathbb{Q}) = \mathbb{Q}$ if $k=0,1$ and $H^k(E\_n;\mathbb{Q}) = 0$ for $k \geq 2$. Of course, this is the same as the cohomology of the ordinary braid group... | 16 | https://mathoverflow.net/users/317 | 121073 | 68,118 |
https://mathoverflow.net/questions/121052 | 2 | A very famous [theorem of Poncelet](https://en.wikipedia.org/wiki/Poncelet%27s_closure_theorem) states that for an elliptic billiard all $n$-periodic trajectories are tangent to some ellipse. As far as I know, Poncelet proved this theorem while sitting in Russian jail so he didn't write it down. Could anybody give me a... | https://mathoverflow.net/users/21800 | Reference question: Poncelet theorem | Poncelet published his theorem ("Poncelet's porism) in 1822, after he returned to France following his captivity as war prisoner in Russia:
J.V. Poncelet, Traité des propriétés projectives des figures (Paris, 1822).
The book has been scanned and can be read [here](https://docnum.unistra.fr/digital/collection/coll7/... | 7 | https://mathoverflow.net/users/11260 | 121079 | 68,121 |
https://mathoverflow.net/questions/121083 | 6 | Is it known what is the centralizer of the complex conjugation in the absolute Galois group (i.e. the Galois group of the field of complex algebraic numbers over the rationals)? and, what would be a good reference for this question?
| https://mathoverflow.net/users/31241 | Centraliser of the complex conjugation in the absolute Galois group | If some element centralizes the complex conjugation, then it must preserve the real numbers as a set. Now, since any automorphism of the real numbers preserves the set of squares, it must preserve the order; and hence be continuous. Since $\mathbb Q$ is fixed, this implies that the real numbers are fixed pointwise. It ... | 14 | https://mathoverflow.net/users/8176 | 121085 | 68,123 |
https://mathoverflow.net/questions/121081 | 5 | In "introduction to algebraic geometry and algebraic groups" Gabriel and Demazure proved the following theorem(section 4 theorem 4.1.):
Let $\mathcal{LRS}\rightarrow{Sh}\_{Zar}$ be the functor that acts on objects as follows:$$X\mapsto{\mathcal{LRS}(Spec(-),X)}$$
Then it has a left adjoint.
$\mathcal{LRS}$ denotes... | https://mathoverflow.net/users/30916 | demazure's and gabriel's book, problem with the proof of a theorem | You probably mean Theorem 4.1 in Chapter I, Paragraph 1. With your formulation, it is in fact wrong. But note that Demazure-Gabriel have been careful as for the set-theoretic foundations: See page xiii for the general conventions. Instead of the full category of rings, they consider a full subcategory $\mathsf{M}$ of "... | 11 | https://mathoverflow.net/users/2841 | 121091 | 68,127 |
https://mathoverflow.net/questions/121066 | 2 | Numerical analysis of the first several hundred n suggests the following inequality:
$\varphi(3^n-2) \ge 2\cdot3^{n-1}$
| https://mathoverflow.net/users/31236 | Inequality with Euler's totient function | Take $n=382315009082231724951830011$. Then $3^n-2$ is divisible by the primes $5, 19, 23, 47, 71, 97, 149, 167, 173, 263, 359, 383, 389, 461, 479, 503, 557$. Furthermore, $\varphi(3^n-2)<2/3\cdot(3^n-2)<2\cdot 3^{n-1}$.
The following Sage code verifies this examples. I believe that this is close to a minimal countere... | 12 | https://mathoverflow.net/users/18739 | 121100 | 68,129 |
https://mathoverflow.net/questions/121113 | 12 | Let $G$ be a finitely generated discrete group with a finite symmetric generating set $S=S^{-1}\subset G$. For every group element $g$, define $\|g\|\_S$ to be the length with respect to $S$, i.e. the minimal length of any word in $S$ that represents $g$. For a natural number $n$, define $B\_n=${$ g\in G : \|g\|\_S\leq... | https://mathoverflow.net/users/29404 | Do finitely generated groups of polynomial growth satisfy a "uniform covering property?" | The answer is yes.
Moreover,
given symmetric generating set $S$,
any sets $B\_n$ has uniform covering property for all large $n$.
Indeed, let $C\_n$ be the minimal number of $g\cdot B\_{ n}$ needed to cover $B\_{2\cdot n}$.
Let $d$ be the word metric for $S$.
Then the sequence $(G,d/n)$ is precomact in the Gromov--... | 9 | https://mathoverflow.net/users/1441 | 121117 | 68,136 |
https://mathoverflow.net/questions/117648 | 7 | Hello,
Could you help me with a reference to elementary properties of Groebner bases in rings of formal power series over a field? I am especially interested in generic initial ideals.
Thank you in advance,
Serge
| https://mathoverflow.net/users/29992 | Groebner bases for power series rings (reference request) | To expand on Michael's comment, the Greuel, Pfister book Section 6.4 is about standard bases in formal power series rings. Quoting,
>
> The main result is that they can be computed, if the ideal is
> generated by polynomials. This is the basis for computations in local analytic
> geometry. The theory of standard ... | 2 | https://mathoverflow.net/users/5495 | 121121 | 68,137 |
https://mathoverflow.net/questions/113963 | 11 | While the common approach to algebraic groups is via representable functors, it seems that there is no such for differential algebraic groups (defined by differential polynomials). Neither the book by E. Kolchin, nor the texts by Ph. J. Cassidy contain anything like this — they work only with the groups of points over ... | https://mathoverflow.net/users/5018 | Differential/difference algebraic groups as "group schemes" | A functorial-schematic approach to differential/difference algebraic groups is surely possible. Why this is not to be found in the literature is probably due to historic reasons. The major bulk of results in differential and difference algebra was obtained in a time where algebraic geometry in the style of Weil's "Foun... | 10 | https://mathoverflow.net/users/21885 | 121124 | 68,138 |
https://mathoverflow.net/questions/121096 | 11 | I am currently studying the paper "CONTINUITY OF SOLUTIONS OF PARABOLIC AND ELLIPTIC EQUATIONS" by John Nash (cf. American Journal of Mathematics, Vol. 80, 1958, <https://doi.org/2372841>). The author there establishes some a priori bounds on solutions of parabolic and elliptic equations.
I would be interested in the f... | https://mathoverflow.net/users/29973 | Nash's paper on parabolic equations | De Giorgi solved [Hilbert's 19th problem](https://en.wikipedia.org/wiki/Hilbert%27s_nineteenth_problem).
Nash independently and almost simultaneously obtained the parabolic version of the same result. Nash's result implies that all quasilinear parabolic equations, under some very reasonable assumptions, have smooth s... | 12 | https://mathoverflow.net/users/26672 | 121125 | 68,139 |
https://mathoverflow.net/questions/121136 | 7 | Let $G=GL\_k(\mathbb C)$ be the complex linear group. Then the infinite Grassmannian is a model for the classifying space $BG$.
We can write the infinte Grassmannian as a colimit of the finite Grassmannians $Gr(k,n)$,
which are honest algebraic varieties, that have the following nice properties:
1) They admit an aff... | https://mathoverflow.net/users/2837 | Nice algebraic approximations of classifying spaces | There is good hope for other groups in nice infinite families. $BGl\_k$ is the classifying space of $k$-dimensional vector bundles. $Gr(k,n)$ is the classifying space of $k$-dimensional vector subbundles of an $n$-dimensional trivial vector bundle.
Similarly, $BSP\_{2k}$ is the classifying space of $2k$-dimensional s... | 2 | https://mathoverflow.net/users/18060 | 121142 | 68,145 |
https://mathoverflow.net/questions/121137 | 5 | Let $(X,L)$ be a polarized abelian variety over $k=\overline{k}$, and let $K(L)$ be the kernel of the isogeny $X\to X^\vee$ that sends $x$ to $t\_x^\*L\otimes L^{-1}$. The theta group $\mathscr{G}(L)$ of $(X,L)$ is a central extension
$$0\to k^\times\to\mathscr{G}(L)\to K(L)\to0.$$
Now, $\mathscr{G}(L)$ acts on $H^0(X,... | https://mathoverflow.net/users/14143 | Theta group representation | I think the answer is no due to the following counterexample (there might be a mistake somewhere though):
Take an elliptic curve $X$ with $L = O\_X(3P\_0)$ (characteristic $\neq 2, 3$), then $K$ will be the $3$-torsion subgroup and $K\_1$ will be generated by a $3$-torsion point $P$. Then in the embedding by $|L|$ of... | 5 | https://mathoverflow.net/users/3847 | 121143 | 68,146 |
https://mathoverflow.net/questions/121141 | 10 | I've heard it conjectured that a finitely presentable group $G$ is hyperbolic if it satisfies the following two conditions.
1. $G$ contains no subgroup isomorphic to a Baumslag-Solitar group $BS(n,m)$ (including $BS(1,1) \cong \mathbb{Z}^2$).
2. $G$ is rationally of finite type in the sense that all the groups $H\_k(... | https://mathoverflow.net/users/31252 | Nonhyperbolic groups that contain no free abelian groups or Baumslag-Solitar groups | Noel Brady's finitely presented non-hyperbolic group embeds in a hyperbolic group (and hence satisfies (1)), but has infinitely generated third integral homology.
I'm guessing, but don't remember with certainty, that the rational third homology is infinitely generated.
Noel Brady, Branched coverings of cubical co... | 7 | https://mathoverflow.net/users/1335 | 121149 | 68,148 |
https://mathoverflow.net/questions/121159 | 0 | Hi,
How to find, or at least express, the equilibrium of a zero-sum game with an $n\*n$ payoff matrix (each player has $n$ strategies) and the payoff of the entry $(i,j)$ is $u(i,j)$. $u$ a random function of the strategies $i$ and $j$.
What about the case where $n \rightarrow \infty$.
Any reference or code is we... | https://mathoverflow.net/users/29611 | Equilibrium of random zero-sum game, | Assuming the $u(i,j)$ are iid with a continuous distribution, the probability that $(i,j)$ is
a saddle point, i.e. that $u(i,j)$ is the greatest entry in its column and the least in its row, is
$((n-1)!)^2/(2n-1)! \approx 2^{1-n^2} \sqrt{\pi/n}$ as $n \to \infty$. Thus the probability
that the game has a saddle point g... | 1 | https://mathoverflow.net/users/13650 | 121163 | 68,155 |
https://mathoverflow.net/questions/121178 | 21 | Is it true that given a finitely presented group $G$, either all primes
or only finitely many of them occur as orders of elements of $G$?
| https://mathoverflow.net/users/28104 | Primes occurring as orders of elements of a finitely presented group | No. The set of primes can be whatever you want (added: within reason! As Benjamin Steinberg points out, it can in fact be any *recursively enumerable* set of primes).
First, note that for infinitely presented groups, the torsion can be whatever you like: the torsion in the group
$\*\_i \mathbb{Z}/p\_i$
is precise... | 25 | https://mathoverflow.net/users/1463 | 121189 | 68,164 |
https://mathoverflow.net/questions/121190 | 0 | If I have a sequence of random variables $X\_1, X\_2, \ldots, X\_n$ (possibly infinite) such that all pairwise cdf's are factorized:
$$F(X\_i, X\_j) = F\_i(X\_i) F\_j(X\_j)$$
for all pairs $(X\_i, X\_j)$, does it mean that the joint cdf is also factorized? That is:
$$F(X\_1, \ldots, X\_n) = \prod\_{i=1}^{n} F\_i(... | https://mathoverflow.net/users/757 | Are all variables in a set of random variables independent if all pairs are independent? | The simplest of the many standard counterexamples is when $(X\_1,X\_2,X\_3)$ takes the values $(1,1,1)$, $(1,0,0)$, $(0,1,0)$ and $(0,0,1)$ all equiprobably.
| 5 | https://mathoverflow.net/users/10503 | 121192 | 68,166 |
https://mathoverflow.net/questions/121193 | 0 | It is known that the universal Teichmuller space $T(1)=\{quasisymmetric \ homeomorphisms \ of \ S^1 \}/ SL (2, \mathbb R)$ is a group. My question is, under what conditions does the Teichmuller space $T(G)$ of a Fuchsian goup $G$ which is finitely generated and of the first kind a group. Or, basically, e.g., (under wha... | https://mathoverflow.net/users/30776 | when is the Teichmuller space a group? | Since you do not say what group operation you have in mind, your question is rather difficult to answer. But what you seem to be proposing in your "Or, basically..." sentence does not work.
For $f : \mathbb{H} \to \mathbb{H}$ to be compatible with $G$ means that the Fuchsian groups $G$ and $f G f^{-1}$ are conjugate ... | 1 | https://mathoverflow.net/users/20787 | 121198 | 68,170 |
https://mathoverflow.net/questions/121188 | 1 | I would like to know how are encoded the real-analytic functions on the interval by the computers. When I think in a real-analytic function I just think in a composition of the ''typical'' analytic functions of every day, in all of this cases it is possible to find an ''exact''(because there is an explicit formula for ... | https://mathoverflow.net/users/31217 | How are real-analytic functions encoded in computer algebra? | The function $\sum \frac{x^n}{n^n}$ has been [discusssed](https://mathoverflow.net/questions/109160). Other choices for the denominator might also give convergent functions defined by power series whose behavior is complicated. Maybe that is relevant.
| -1 | https://mathoverflow.net/users/8008 | 121209 | 68,176 |
https://mathoverflow.net/questions/121200 | 1 | Let $K$ be a field and $K\{X\}$ be the free non-associative algebra, freely generated by the countably infinite set $X$. We consider elements of $K\{X\}$ as (non-associative) polynomials in the variables of $X$.
Given $\mathfrak{F}\subseteq K\{X\}$, one can consider the variety of algebras defined by $\mathfrak{F}$. ... | https://mathoverflow.net/users/20105 | Is a variety of algebras a set? | Summarizing the above, the answer is no.
| 2 | https://mathoverflow.net/users/15934 | 121210 | 68,177 |
https://mathoverflow.net/questions/121187 | 13 | I'm looking for a reference for the following result.
Theorem. Let $K$ be a complete, non-archimedean field, and let $X/K$ be a projective scheme, with analytification $X^\mathrm{an}$. Then the analytification functor from coherent $\mathcal{O}\_X$-modules to coherent ${\mathcal{O}}\_{X^\mathrm{an}}$-modules is an eq... | https://mathoverflow.net/users/13647 | Reference for rigid analytic GAGA | I am quite surprised by the attribution to Kiehl that you saw. Anyway, I think the result is due to Ursula Köpf (not only over a field $K$ but actually over an affinoid space): "Über eigentliche Familien algebraischer Varietäten über affinoiden Räumen", Schriftenreihe Univ. Münster, 2 Serie, Heft 7 (1974).
Brian Conr... | 12 | https://mathoverflow.net/users/4069 | 121215 | 68,181 |
https://mathoverflow.net/questions/121202 | 12 | It is an easy consequence of the Serre open image theorem that for the torsion point count on elliptic curves, the following possibilities arise.
1. If $E/\bar{\mathbb{Q}}$ is an elliptic curve without CM, then the number of torsion points $x \in E$ with $[\mathbb{Q}(x):\mathbb{Q}] \leq d$ is $\asymp d^{3/2}$ as $d \... | https://mathoverflow.net/users/26522 | The torsion point count in higher dimension | As far as I know, we expect that the image of the Galois group in $GL\_{2g}(\mathbb A\_\mathbb Q)$ is open in $G(\mathbb A\_\mathbb Q)$ for $G$ the monodromy group of the Galois representation (which is expected to be independent of $l$. Then the asymptotic for this function clearly depends only on the connected compon... | 9 | https://mathoverflow.net/users/18060 | 121218 | 68,182 |
https://mathoverflow.net/questions/121168 | 15 | How much has been the group of diffeomorphisms of a manifold " been studied.
I got this information from wiki.
" Quite a lot is known about the group of diffeomorphisms of the circle. Its Lie algebra is (more or less) the Witt algebra, which has a central extension called the Virasoro algebra, used in string theory and... | https://mathoverflow.net/users/30081 | group of diffeomorphisms of a manifold | One can approach the study of diffeomorphism groups from many perspectives: topology, geometry, differential equations, and dynamics. I'll mention a few results that I'm aware of, giving links to literature surveys on different topics.
There is a short exact sequence $$Diff\_0(M)\to Diff(M)\to MCG(M),$$
where $Diff\_... | 16 | https://mathoverflow.net/users/1345 | 121224 | 68,185 |
https://mathoverflow.net/questions/121182 | 1 | Does the network of $X$ equal to the network of $C\_p(X)$?
$C\_p(X)$ denotes the set of all real-valued continuous functions on $X$ endowed with the topology of pointwise convergence.
Thanks!
| https://mathoverflow.net/users/18465 | Does the network of $X$ equal to the network of $C_p(X)$? | The following is Theorem I.1.3 in *Topological Function Spaces* by A.V. Arkhangelski:
>
> For any space $X$, $nw(X)=nw(C\_p(X))$.
>
>
>
Where $nw(X)$ denotes the network weight of $X$ (i.e. the minimal cardinality of a network in $X$).
| 4 | https://mathoverflow.net/users/17836 | 121229 | 68,187 |
https://mathoverflow.net/questions/121152 | 8 | Hamkins showed that his infinite time Turing machine has the power to decide some $\Delta\_2^1$ sets. I wonder if some modifications of the machine could be made to reach level $\Sigma\_1^2$ sets, or, if no modifications on sight, if the power of his machine plus infinitely iterated super jumps from super oracles (thos... | https://mathoverflow.net/users/27974 | Can a Hamkins infinite time Turing Machine with infinite Super Turing jumps (from higher type oracles) get the power to decide $\Sigma_1^2$ sets? | One should think of the class $\Delta^1\_2$ as truly enormous, closed under powerful set-theoretical constructions. It
may help to keep in mind that the minimal transitive model of ZFC, if it exists, is contained inside $\Delta^1\_2$, and so
one cannot jump out of $\Delta^1\_2$ with a computational operation that is a... | 17 | https://mathoverflow.net/users/1946 | 121230 | 68,188 |
https://mathoverflow.net/questions/121247 | 10 | I want to start studying differential geometry but I can't seem to find a proper starting path. Whenever I try to search for differential geometry books/articles I get a huge list. I know that it is a broad topic, but I want some advice for you regarding the books and articles. I want to learn differential geometry and... | https://mathoverflow.net/users/24541 | Differential geometry study materials | I would recommend Lee's book "Introduction to Smooth Manifolds." It's a long book but is comprehensive, has complete proofs, and has lots of exercises.
| 18 | https://mathoverflow.net/users/22781 | 121248 | 68,195 |
https://mathoverflow.net/questions/121103 | 9 | Terminology:
Cohesive sets: $A\subset \omega$, for each recursively enumerable set $W\_e$, either $A\cap W\_e$ is finite or $A\cap(\omega\setminus W\_e)$ is finite.
Non-high degrees: Degree $a$ such that $a'\not\geq 0''$.
I'm wondering if it is possible to construct a cohesive set using some non-high 1-generic de... | https://mathoverflow.net/users/23835 | Cohesive sets with degree below some non-high 1-generic degrees? | The answer to this question is in Jockusch and Stephan's 1993 paper 'A cohesive set which is not high.'
<http://www.comp.nus.edu.sg/~fstephan/coh.ps>
Take $A$ and $B$ such that $A \le\_T B$, $A$ is Cohesive and $B' \not\ge\_T 0''$. This implies that $A' \not \ge\_T 0''$ and so by the paper mentioned above $A$ compute... | 7 | https://mathoverflow.net/users/8309 | 121262 | 68,202 |
https://mathoverflow.net/questions/121268 | 4 | Suppose f is a computable function from a recursively enumerable set U to the natural numbers and that L,K are r.e. subsets of U. Is f(L-K) a difference of r.e. subsets? The motivation comes from
[Primes occurring as orders of elements of a finitely presented group](https://mathoverflow.net/questions/121178/primes-oc... | https://mathoverflow.net/users/15934 | Computable images of differences of r.e. sets | Is f injective? If so, yes. If not, no. In the second case, you could achieve any $\Sigma^0\_2$ set.
| 3 | https://mathoverflow.net/users/32178 | 121274 | 68,208 |
https://mathoverflow.net/questions/121028 | 4 | Let $M$ be a $2^n \times 2^n$ matrix over real number field, where the rows and columns are indexed by subsets of $[n] := \{1,2,\ldots,n\}$, and defined as follows,
$
M\_{A, B} = 1
$
if $A \subseteq B$;
$M\_{A, B} = -1$
if $B \subsetneq A$;
$M\_{A, B}$ can take arbitrary value over $\mathbb{R}$.
In words, $M$ is a ma... | https://mathoverflow.net/users/26659 | Rank of a matrix with missing entries | Amazingly (to me) the rank can be as low as $n$. Simply define $M\_{A,B}=1$ when $|A| \le |B|$ and $M\_{A,B}=-1$ when $|A| \gt |B|.$ This is consistent with the previous requirements and makes the $A$ row depend only on $|A|.$
| 3 | https://mathoverflow.net/users/8008 | 121275 | 68,209 |
https://mathoverflow.net/questions/121253 | 23 | A category is a **skeleton** if, roughly speaking, no two distinct objects within the category are isomorphic. To every category is associated a skeleton, and two categories are categorically "equivalent" if and only if their skeletons are isomorphic. A fuller definition can be found [here](http://en.wikipedia.org/wiki... | https://mathoverflow.net/users/24611 | Skeleton category of the category of skeleton categories? | I'm not entirely sure what you're looking for in an answer, but maybe I'll flesh out my comment.
It looks like what you're describing is equivalent to the homotopy category associated to the model structure on Cat where the weak equivalences are equivalences of categories. (I can say "the" because there is only one s... | 9 | https://mathoverflow.net/users/6936 | 121279 | 68,210 |
https://mathoverflow.net/questions/121273 | 3 | For the Lie group $SO(n,1)$ I believe the maximal nilpotent subgroups are conjugate to either a diagonal group times a compact group or a unipotent group times a compact group. In either case the compact group will commute with the other group. Is this true and if so how do I prove it?
| https://mathoverflow.net/users/31071 | Maximal nilpotent subgroups of SO(n,1) | Let's look first at maximal solvable subgroups, i.e. Borel subgroups. If $G=KAN$ is an Iwasawa decomposition of $G$, Borel subgroups are conjugate to $MAN$, where $M$ is the centralizer of $A$ in $K$. In the case of $SO(n,1)$, we have $K\simeq SO(n),A\simeq\mathbb{R}$(this is the maximal diagonalizable subgroup), $N\si... | 2 | https://mathoverflow.net/users/14497 | 121283 | 68,213 |
https://mathoverflow.net/questions/114399 | 10 | I am looking for a list/reference which explores the consequences of Legendre's conjecture, which states that one can always find a prime number between $n^2$ and $(n+1)^2$.
| https://mathoverflow.net/users/21884 | Consequences of Legendre's conjecture | In the absence of much other contributions yet this still being open, I promote my slightly expanded comments to an answer:
The Legendre conjecture, while of historical relevance, nowadays does not seem to play too much of a role in research, it is thus unlikely to have many things that are specifically consequences ... | 7 | https://mathoverflow.net/users/nan | 121316 | 68,222 |
https://mathoverflow.net/questions/121306 | 35 | *I originally [posted this question on StackExchange](https://math.stackexchange.com/questions/295358/how-are-infinite-dimensional-manifolds-most-commonly-treated), where it was suggested I post here. It was also suggested I read about Hilbert manifolds and Fréchet manifolds. Nevertheless, I am still looking for an ans... | https://mathoverflow.net/users/19956 | How are infinite-dimensional manifolds most commonly treated? | Banach manifolds have found many uses such as, gauge theory (Donaldson theory, Seiberg-Witten theory, Floer theory), symplectic topology (Gromov-Witten theory), to name few I am more familiar with.
One great advantage of Banach manifolds over Frechet manifolds is the implicit function theorem which in the Banach cont... | 23 | https://mathoverflow.net/users/20302 | 121318 | 68,223 |
https://mathoverflow.net/questions/121123 | 2 | Suppose one has an inclusion $\iota : D \hookrightarrow S$ where $D$ is a finite distributive lattice and $S$ is a finite join-semilattice.
If $\iota$ preserves all meets and joins one can show that $|J(D)| \leq |J(S)|$ i.e. $D$ has no more join-irreducibles than $S$.
Does this also hold if $\iota$ only preserves f... | https://mathoverflow.net/users/5152 | Distributive lattice embedding into a finite lattice. | Well, I proved that it is true with a friend yesterday, so I'll include it here.
Recall that we have an inclusion $\iota : D \hookrightarrow S$ where $S$ is a finite join-semilattice and $D$ is a finite distributive lattice. More explicitly: $S$ has all finite joins, so it is a complete lattice with a top and bottom ... | 1 | https://mathoverflow.net/users/5152 | 121325 | 68,227 |
https://mathoverflow.net/questions/121270 | 6 | Let A be a Borel set in R^n. Must then A + B(0,1) be Borel?
Here B(0,1) is the *closed* ball centered at 0 of radius 1.
I know that Erdos and Stone gave an example of a compact set (it is Cantor) and a G\_\delta set, whose Minkowski's sum is not Borel. But can we have an example with one of them being a closed ball?
... | https://mathoverflow.net/users/31283 | Must the Minkowski sum of a Borel set and a *closed* ball be Borel? | Idea from <http://arxiv.org/abs/1211.0430> (Example 2.4).
Take a Borel set $A' \subset [0,1]^2$ with the property that its projection to the first coordinate is not Borel. Now put this set on a cylinder in $\mathbb{R}^3$ and call it $A$.
The set $$(A + B(0,1)) \cap (\mathbb{R}\times\{0\}^2)$$ (the Minkowski sum inte... | 11 | https://mathoverflow.net/users/11716 | 121331 | 68,231 |
https://mathoverflow.net/questions/121228 | 4 | Given the group ring $\mathbb{Z}[G]$ of a finite group $G$ over $\mathbb{Z}$, is there a way to generalize the notion of the "frobenius algebra" in some cases? One can show that every group ring $\mathbb{Q}[G]$ is a frobenius algebra and thus the projektive and injective modules correspond. This seems to be wrong in ge... | https://mathoverflow.net/users/31271 | Injective Modules over Group Rings | I claim that if $R$ is a left Noetherian ring and $M$ a finitely generated left $R$-module, then $M$ is injective as an $R$-module iff it is injective in the category of finitely generated $R$-modules.
Suppose $M$ is injective in the category of f.g. $R$-modules. Then for any left ideal $I$ of $R$ and any left $R$-mo... | 5 | https://mathoverflow.net/users/19045 | 121348 | 68,236 |
https://mathoverflow.net/questions/121340 | 23 | In order to realise the K-groups of a ring as the homotopy groups of some space associated to that ring, Quillen proposed the following (roughly-sketched) construction:
Recall that $K\_1(R) = GL(R)/E(R)$, so we're, at least, looking for a space $X$ with $\pi\_1(X) = K\_1(R)$. The classifying space of $K\_1(R)$ is obv... | https://mathoverflow.net/users/19313 | Plus construction considerations. | Here are some thoughts, gathered from reading many texts about algebraic K-theory. Let me start with some historical remarks, then try to give a more revisionist motivation of the plus construction.
First of all, it's true as you say that the already-divined definitions of the lower K-groups made it seem like the hig... | 29 | https://mathoverflow.net/users/3931 | 121351 | 68,238 |
https://mathoverflow.net/questions/121288 | 6 | Well, the title almost says it all. I would like to list as many examples as possible of moduli functors, for which a coarse moduli space does not exist (and maybe explain why). So, examples such as $[\mathbb{A}^1/G\_m]$ are not what I mean. I would really like to see moduli functors that come from some classifying pro... | https://mathoverflow.net/users/4096 | examples of moduli functors for which coarse moduli space does not exists | $\overline{\mathfrak{M}}\_{0,0}$, the stack of pre-stable genus zero curves. This parameterizes genus zero curves having at worst nodal singularities. There are an infinite number of isomorphism classes of such curves, but they all occur as a specialization of a trivial family (just do repeated blowups of the central f... | 8 | https://mathoverflow.net/users/9617 | 121357 | 68,239 |
https://mathoverflow.net/questions/121315 | 4 | Let $X\subset \mathbb CP^n$ be a smooth submanifold whose normal bundle is
$$\bigoplus\_{i=1}^{codim X}O(k\_i).$$
Is there some general enough additional condition of $X$ that implies that $X$ is a complete intersection?
For example, would $dimX\ge 2$ suffice (to exclude things like $X=\mathbb CP^1$)?
| https://mathoverflow.net/users/13441 | Projective submanifolds of $\mathbb CP^n$ whose normals bundles are sums of linear. | I enter my comment as an answer. The paper *On the normal bundle of submanifolds of $\mathbb{P}^n$* by Lucian Badescu contains some answers to the question. Here are the links:
Published version: <http://www.ams.org/journals/proc/2008-136-05/S0002-9939-08-09255-1/>
On arXiv version: <http://arxiv.org/pdf/math/07014... | 4 | https://mathoverflow.net/users/16046 | 121362 | 68,243 |
https://mathoverflow.net/questions/121364 | 6 | Originally posted here: <https://math.stackexchange.com/questions/276167/fixed-point-theorem-on-graphs>
--
I have a graph $G=(V,E)$ where to each vertex $v$ I have associated a value, $\hat{v}$ (ie I have a "network" in the terminology here <http://snap.stanford.edu/snap/index.html> ).
Let $\phi : \hat{V} \righta... | https://mathoverflow.net/users/6360 | Fixed point theorem on graphs? | If I am not mistaken, this problem was discussed in the article
Zbl 0194.13702
Lyubich, Yu.I.; Tabachnikov, M.I. Subharmonic functions on a directed graph.
Siberian Math. J. 10, 432-442 (1969).
(unfortunately in Russian; I don't know whether there is an English translation)
| 7 | https://mathoverflow.net/users/18814 | 121367 | 68,246 |
https://mathoverflow.net/questions/121345 | 5 | For a holomorphic vector bundle $E$ over a complex manifold $M$, we denote its space of smooth sections by $\Gamma^{\infty}(E)$, and its space of holomorphic sections by $\Gamma^{hol}(E)$. Now I've been looking at the line bundles $L\_k$ over the complex projective spaces ${\bf C} P^N$, and I have managed to show that ... | https://mathoverflow.net/users/1648 | When are the Smooth Sections of a Bundle Generated as a Module (over Smooth Functions) by the Holomorohic Sections | Swan has proved that taking global section gives an anti-equivalence between finitely generate projective $\Gamma^{\infty}(M)$-modules and $C^{\infty}$ vector bundles on $M$; this correspondence is functorial in $M$. Hence a set of section of a $C^{\infty}$ vector bundle $E$ on $M$ generated $\Gamma^{\infty}(E)$ if and... | 5 | https://mathoverflow.net/users/4790 | 121378 | 68,251 |
https://mathoverflow.net/questions/121129 | 4 | Henry Sherwin's Mathematical Tables achieved some popularity. The first edition was published ~1706 and the fifth and last in 1771. Some editions were more erroneous than others and the error rate was not monotonically decreasing. The 1873 Royal Society Report says:
"Sherwin's tables are of historical interest as for... | https://mathoverflow.net/users/4111 | Biographical Information Concerning Henry Sherwin | Next to nothing is known about the life of Henry Sherwin. He is mentioned by some scholars, for instance in the Sudelbücher of G. C. Lichtenberg, and his Tables were not only in public libraries but also in private possession of scholars like Gauss. But his person is not touched. Encyclopedia Britannica, British Nation... | 8 | https://mathoverflow.net/users/nan | 121382 | 68,254 |
https://mathoverflow.net/questions/121376 | 5 | We know that the Riemann tensor is antisymmetric with respect to the first two vectors (the vectors that we parallel transport the third vector around the parallelogram made by their integral curves).
But what is the geometric interpretation?
Why do we have to expect that by changing the order of parallel transportat... | https://mathoverflow.net/users/25609 | Interpretation of Riemann tensor antisymmetry | This is a re-flavouring of Alexander's answer but in a language I prefer.
Take two vectors $v,w \in T\_p N$, and consider the `rectangle' $exp(xv+yw)$ where $0 \leq x \leq a$ and $0 \leq y \leq b$. The holonomy around the boundary of this rectangle is an orthogonal transformation of $T\_p N$, and it looks approximat... | 13 | https://mathoverflow.net/users/1465 | 121392 | 68,258 |
https://mathoverflow.net/questions/111022 | 10 | At a conference not too long ago I gave a talk on (left) Bousfield localization and was asked an interesting question afterwards. The question was whether I knew any examples of model categories which were not left proper, but which did admit Bousfield localization. There are some trivial examples, but I'm interested n... | https://mathoverflow.net/users/11540 | Is there a notion of a “model category which admits left Bousfield localization?” | One relevant reference, though not answering any of the original questions, is a paper by George Raptis [On the cofibrant generation of model categories](http://arxiv.org/abs/0907.2726), where it is shown that under Vopenka's principle, every cofibrantly generated model category is Quillen equivalent to a combinatorial... | 8 | https://mathoverflow.net/users/30641 | 121403 | 68,266 |
https://mathoverflow.net/questions/121389 | 3 | Suppose that you have vectors $a\_{1},...,a\_{m}$ in $\mathbb{R}^n$. Can you take $n$ among them, let $v\_{1},...,v\_{n}$ such that $a\_{1},...,a\_{m}$ belong to the convex hull of $(cn)v\_{1},-(cn)v\_{1},...,(cn)v\_{n},-(cn)v\_{n}$ for some constant $c$?
My idea was to use gram-schmidt process where at step $i$ I ch... | https://mathoverflow.net/users/nan | Find a convex hull that contains given points? | Take $v\_1,v\_2,\dots,v\_n$ which span a parallelepiped of maximal volume.
If
$$a\_i=x\_1\cdot v\_1+\dots+x\_n\cdot v\_n$$
then $|x\_k|\le 1$, otherwise exchanging $v\_k$ to $a\_i$ will increase the volume.
Hence $a\_i$ belongs to the convex hull of $\{\pm n\cdot v\_{i}\}$; i.e. $c=1$.
---
*Below is the original... | 4 | https://mathoverflow.net/users/1441 | 121407 | 68,269 |
https://mathoverflow.net/questions/121393 | 12 | Consider tri-linear forms, $\{A\_{ijk}\}$ where $i=1,..,n\_1$, $j=1,..,n\_2$, $k=1,..n\_3$, over a field of zero characteristic, up to the equivalence $A\to (U\_1,U\_2,U\_3)(A)$, by three matrices.
What is known about the classification? (Complete) invariants?
Unlike the case of bi-linear forms, in general the classifi... | https://mathoverflow.net/users/2900 | basics of classification of trilinear forms (when is it non-discrete) | I'll phrase my answer for an algebraically closed field.
As soon as all of the $n\_i$ are at least 3 you get moduli. So the interesting cases are $(2,2,n)$ and $(2,3,n)$ which I claim have finitely many orbits.
Given a tensor of type $(2,2,n)$, you can always write it as $\sum e\_i \otimes f\_j \otimes h\_{i,j}$ wh... | 5 | https://mathoverflow.net/users/321 | 121415 | 68,274 |
https://mathoverflow.net/questions/120232 | 8 | Added Background: The pair correlation of the zeros of the Riemann zeta function is influenced by the the derivative of the logarithmic derivative $(\zeta^\prime(s)/\zeta(s))^\prime$; see for example the answers to
[this question](https://mathoverflow.net/questions/83027/what-is-ricardo-perez-marcos-ene-product-does-i... | https://mathoverflow.net/users/6756 | Pair correlation for the Riemann zeros and $(\zeta^\prime(s)/\zeta(s))^\prime$ | I believe the comments of joro and Carlo Benakker right under your question is to the point. Since the zeroes close to $s$ will be the ones that contributes in the sum, the zeroes close to s must be computed first and in order to do that several values of the zeta-function must certainly be calculated and the time for ... | 5 | https://mathoverflow.net/users/10811 | 121418 | 68,276 |
https://mathoverflow.net/questions/121406 | 22 | [I have updated the question after initial comments in the hope of clarifying it.]
I do quite a bit of reasoning, typically about topology and metric spaces, in "non-standard" foundations, such as inside of a particular topos, in type theory, or a predicative constructive setting. These typically do not have anything... | https://mathoverflow.net/users/1176 | Where in ordinary math do we need unbounded separation and replacement? | I asked the same question about the replacement axiom not long ago at the $n$-Category Café, and the [answer](http://golem.ph.utexas.edu/category/2012/12/rethinking_set_theory.html#c042853) I got back from Mike Shulman is that it's used for example in the transfinite construction of free algebras, which really refers t... | 20 | https://mathoverflow.net/users/2926 | 121421 | 68,279 |
https://mathoverflow.net/questions/121411 | 13 | Let $X\sim B(n,p)$ denote a binomial random variable. Is there any approximation available for the quantity $E(\sqrt{X})$? Clearly Jensen's inequality holds, but rudimentary tooling around with Maple hasn't turned up anything more substantial.
| https://mathoverflow.net/users/31323 | Expectation of square root of binomial r.v. | $\newcommand{\E}{\mathbf{E}}$
$\renewcommand{\P}{\mathbf{P}}$
$\DeclareMathOperator{\var}{Var}$
If we use Taylor expansion (as Anthony suggested) for $\sqrt{x}$ around 1, we get:
$$\sqrt{x}\approx 1 + \frac{x-1}{2} - \frac{(x-1)^2}{8} .$$
We can use this to get an approximation of
$$\E(\sqrt{X})\approx 1-\frac{\var(X... | 19 | https://mathoverflow.net/users/1061 | 121424 | 68,282 |
https://mathoverflow.net/questions/121408 | 3 | Let $\operatorname{Klein}$ denote the category of principal homogeneous bundles. An object in this category is a tuple $\mathbf Q = (Q, P; G, H; q, a, \tilde a)$, where:
* $G$ is a Lie group, and $H$ is a closed subgroup;
* $P$ is a [$G$-torsor](http://math.ucr.edu/home/baez/torsors.html) and the manifold $Q$ is diff... | https://mathoverflow.net/users/238 | What are the symmetries of a principal homogeneous bundle? | No, in general $G=G(\mathbf{Q})$ can be strictly smaller that ${\rm Aut}(\mathbf{Q})$.
Let $G$ be a Lie group and $H\subset G$ be a Lie subgroup. Set $P=G$, $\ Q=G/H$, and define the maps in the obvious way. We get a principal homogeneous bundle $\mathbf{Q}=(Q,P,\dots)$ in your sense.
Now take $G=SL(n,\mathbb{C})$,... | 5 | https://mathoverflow.net/users/4149 | 121425 | 68,283 |
https://mathoverflow.net/questions/121402 | 4 | I am not very sure if this is a proper question, but I'm trying to investigate what the area of math can offer in researching the differential equation in polar coordinates:
$r'^2+r^2=(kt)^2$, $r(t=0)=0$, k- Const or in other notation:
$r'(\theta)^2+r(\theta)^2=\theta^2$, $r(\theta=0)=0$
| https://mathoverflow.net/users/10903 | What is symmetry group of non-linear equation? | As for asking about whether the symmetries of this equation would help you solve it, here are a few remarks that you may (or may not) find useful:
I assume that you want to consider what are usually called the 'point symmetries', i.e., the transformations of the $r\theta$-plane that carry (graphs of) solutions of the... | 18 | https://mathoverflow.net/users/13972 | 121429 | 68,286 |
https://mathoverflow.net/questions/121379 | 37 | In [this post](http://golem.ph.utexas.edu/category/2012/04/principal_bundles.html) on the *n*-Category Café, Urs Schreiber says that, "The theory of [G-principal bundles](http://ncatlab.org/nlab/show/principal+bundle) makes sense in any [$(\infty,1)$-topos](http://ncatlab.org/nlab/show/%28infinity%2C1%29-topos)." I fol... | https://mathoverflow.net/users/238 | What is an $(\infty,1)$-topos, and why is this a good setting for doing differential geometry? | Derived versions of differential topology are becoming prominent tools in symplectic geometry. Whether or not you think of them via topoi is not crucial (I certainly can't), and perhaps the terminology turns off more people than it draws, but these ideas are being put to serious use by very serious no-nonsense mathemat... | 33 | https://mathoverflow.net/users/582 | 121436 | 68,288 |
https://mathoverflow.net/questions/121445 | 4 | I am interested in a more specific reference or explanation of "the categorical view" explained in the article [http://ncatlab.org/nlab/show/theory#CategoricalView.](http://ncatlab.org/nlab/show/theory#CategoricalView) In particular, I am interested in trying to prove full completeness for a geometric model of multipli... | https://mathoverflow.net/users/20343 | Completeness of a Theory from the Categorical Viewpoint | "I interpret that as being that for a particular model of a theory, I want to define a functor which can be identified with that theory"
This sentence seems wrong. The theory itself isn't any functor, but rather the category CT. Or, if you allow models in arbitrary categories with the right sort of structure (the te... | 7 | https://mathoverflow.net/users/4177 | 121447 | 68,293 |
https://mathoverflow.net/questions/44907 | 4 | What is an example of a non-paracompact topological vector space?
I'm aware of [this question](https://mathoverflow.net/questions/3241/when-is-a-locally-convex-topological-vector-space-normal-or-paracompact), but I don't care if my tvs is locally convex. In fact the wilder the better. The only criterion is that it sh... | https://mathoverflow.net/users/4177 | An example of a non-paracompact tvs (over the reals, say) | Henno Brandsma provided this answer in a comment to anon's answer, which I think is the easiest and best, hence I'm adding it as a community wiki answer.
Consider an uncountable set $\mathfrak{n}$ and the space $\mathbb{R}^\mathfrak{n}$ in the product topology. This is [not normal](http://www.ams.org/journals/bull/19... | 3 | https://mathoverflow.net/users/4177 | 121450 | 68,296 |
https://mathoverflow.net/questions/121297 | 9 | While trying to prove some properties of a subset of a prime spectrum I arrived at the following question: Let $R$ be a commutative ring and let $P \in \mathrm{Spec}(R)$. We can consider $\mathrm{Spec}(R\_P)$ as a subset of $\mathrm{Spec}(R)$. My question is: What kind of subset (subspace) is this? I guess it is not tr... | https://mathoverflow.net/users/8070 | What kind of subset is Spec(R_P) in Spec(R)? | As already said, in general the image is not open and even not constructible. But it is pro-constructible (that is, locally an intersection of locally constructible subsets, see EGA IV.1.9.4, and 1.9.5(ix) because $X:=\mathrm{Spec}(R)$ is affine hence quasi-compact).
Denote this image by $S$. This is a subset of $X$... | 6 | https://mathoverflow.net/users/3485 | 121452 | 68,298 |
https://mathoverflow.net/questions/102864 | 6 | One of the ways to define the Morley rank of a definable set is with respect to a model, say $M$, i.e. a set has rank $\alpha+1$ if there are infinitely many definable subsets with parameters in $M$ of rank $\alpha$ (and a similar clause for limit ordinals). One then shows that once $M$ is $\aleph\_0$-saturated then co... | https://mathoverflow.net/users/2234 | computing Morley rank using parameters from an arbitrary model | The distinction between Morley rank as defined by arbitrary formulas and by definable families of formulas is essential. $\aleph\_1$- categoricity in particular implies the rank can defined by
definable families. Since my 1973 [?] article in the transactions AMS or Shelah's book or
say Pillay's geometric model theory ... | 5 | https://mathoverflow.net/users/31339 | 121454 | 68,299 |
https://mathoverflow.net/questions/121434 | 4 | Hi,
In the Arthur-Selberg trace formula for $G = GL(2)/\mathbf Q$ (as seen for example in
Gelbart's "Lectures on the Trace Formula"), the spectral side includes terms
like:
$$
\int\_{-\infty}^\infty tr (\rho(\mu,it)(f))dt
$$
where $f$ is the test function, $\mu$ is a Hecke character, and $\rho(\mu,s)$
is the induced ... | https://mathoverflow.net/users/36285 | vanishing of spectral term in Arthur-Selberg trace formula for GL(2)? | *Sorry, my original answer was adressing vanishing of the contribution of the continuous spectrum. You are asking for something different.*
If the $\infty$ component is a pseudo coefficient of the discrete series representations, the contribution you ask for vanish by the definition of a pseudo coefficient and the fa... | 2 | https://mathoverflow.net/users/10400 | 121463 | 68,302 |
https://mathoverflow.net/questions/121461 | 0 | I am reading the draft of "[Equations of Riemann Surfaces of Genus 4, 5 and 6 wih Large Automorphism groups](http://www.math.uga.edu/~davids/ivrg/SwinarskiEquations.pdf)" and the author starts using the notation $H^0(C, mK)$ on page 3, without explaining it. As the author is studying the action of the automorphism grou... | https://mathoverflow.net/users/6776 | Reference for notation $H^0(C, mK)$ | I can't access the draft you've linked but (almost always) this means the vector space of global sections of the $m^{th}$ power of the canonical bundle. Note for a Riemann surface the canonical bundle is just the cotangent bundle, the dual of the tangent bundle. For the general theory of algebraic curves, see e.g. Geom... | 3 | https://mathoverflow.net/users/22294 | 121465 | 68,303 |
https://mathoverflow.net/questions/121462 | 4 | Hello,
I am a graduate student in the field of discrete-time dynamics. I am wondering about applications of this field outside of mathematics. More precisely, I would like to know if there are "real life" situations where dynamical notions provide a significant insight, or even better, a power of prediction.
For e... | https://mathoverflow.net/users/19189 | Applications of discrete-time dynamics | The most natural way in which discrete time dynamics appears in physical systems is stroboscopically. The stroboscopic description of a periodically driven Hamiltonian system leads to the standard map, which exhibits a chaotic phase space. These systems have been realized with [microwaves](http://arxiv.org/abs/1302.128... | 4 | https://mathoverflow.net/users/11260 | 121471 | 68,307 |
https://mathoverflow.net/questions/121484 | 11 | Can anyone provide an example of a set S which is definable in ZFC and provable in ZFC to be
denumerably infinite, while at the same time, no set definable in ZFC can be proved in ZFC to
be an element of S? Such examples are easy to find if S is allowed to be uncountable-for instance
S could be the set of all non-measu... | https://mathoverflow.net/users/4423 | A question about definable non-empty sets containing no definable elements. | Here are a few partial answers.
First, I claim that if ZFC is consistent, then for every
ZFC-definable nonempty set $S$, whether countable or uncountable, it is consistent with ZFC that $S$
contains a definable element. Further, there will be a model of ZFC in which every element of $S$ is definable.
Suppose that ... | 8 | https://mathoverflow.net/users/1946 | 121494 | 68,320 |
https://mathoverflow.net/questions/121493 | 2 | Regarding the internalization of mathematics to a particular category as in the nLab article: [Internal Logic](http://ncatlab.org/nlab/show/internal+logic), there is a peculiar table mentioned in the section on [Categorical Semantics](http://ncatlab.org/nlab/show/internal+logic#CategoricalSemantics) in which there is a... | https://mathoverflow.net/users/20343 | Adjoint of Pushout as Modal Operators in Internal Logic | Probably you should get a hold of a textbook on categorical logic for some of these questions. Of the references given in the [Wikipedia article](http://en.wikipedia.org/wiki/Categorical_logic), the one by Lambek & Scott and the two with Lawvere as an author might be a good start. A study of these would probably help m... | 7 | https://mathoverflow.net/users/2926 | 121503 | 68,324 |
https://mathoverflow.net/questions/121366 | 2 | I am just a beginner in $D$-module, so this could be a stupid question, but I can't find an easy reference for it. I would like to define the notion of $D$-affine morphism. The most obvious way would be to say that a smooth morphism between smooth quasi-projective schemes:
$$ f : X \rightarrow Y$$
is *$D$-affine* if th... | https://mathoverflow.net/users/31278 | D-affine morphisms and composition | The situation as I see it is as follows:
The first definition you give is the natural one. It implies that D-modules on the $X$ are given by sheaves of modules for the sheaf of algebras $f\_\ast D\_X$ on $Y$. ~~However, I am curious if there are actually any interesting examples of such a morphism. Projective spaces... | 2 | https://mathoverflow.net/users/7762 | 121510 | 68,326 |
https://mathoverflow.net/questions/121504 | 5 | In nearly all (if not all) projective geometry texts I have bumped into the following theorem:
"Principle of duality: If in a theorem in $\mathfrak{P}$ one switches the word point for the word line and the corresponding incidence relations once again one obtains a theorem of $\mathfrak{P}$."
So far so good. Then I ... | https://mathoverflow.net/users/9187 | On duality on finite projective planes | I'd expect that, in the duality principle that you quoted from "most (if not all) projective geometry texts", the symbol $\mathfrak P$ refers to the theory of projective planes, not to an arbitrary particular projective plane. One reason for this expectations is that theories, not planes, are the sort of entity that ca... | 7 | https://mathoverflow.net/users/6794 | 121518 | 68,330 |
https://mathoverflow.net/questions/121516 | 1 | What efficient algorithms exist for the solving $x^N = a$ in GF(q)?
What are their complexities?
| https://mathoverflow.net/users/31356 | Find root in finite field | Pasting the title of your question into Google gives references like
<http://www.math.leidenuniv.nl/~astolk/monday/notes/stolk-roots.pdf> and
<http://www.ma.utexas.edu/users/voloch/Preprints/roots.pdf>
-- this should answer your question.
| 2 | https://mathoverflow.net/users/28104 | 121523 | 68,332 |
https://mathoverflow.net/questions/121520 | 1 | I'm trying to understand the definition of a differential form on $M$ in the context of Fréchet spaces or, more generally, locally convex spaces. The standard procedure defines a k-form as a map $\alpha: M \rightarrow L^k\_{\text{alt}}(TM)$ such that the coordinate representation $U \times E^k \rightarrow \mathbb{R}$ i... | https://mathoverflow.net/users/17047 | Cotangent bundle in the category of locally convex spaces | If evaluation $E\times E'\to \mathbb R$ is jointly continuous for any pair of lcs topologies which are compatible with duality, then $E$ has to be normable:
Namely, if $ev(U\times U')\subset (-1,1)$ for open $U$ and $U'$, then $U$ is contained in the polar of an open set, thus is bounded. So $E$ is normable. This exten... | 2 | https://mathoverflow.net/users/26935 | 121524 | 68,333 |
https://mathoverflow.net/questions/121227 | 3 | An easy question that I have never been able to answer.
Suppose we have the CA on $\{ 0,1,2 \}^{\mathbb{N}}$ with local rule given by $f(x,y)=A\_{x,y}$ and $A$ the $3\times 3$ matrix $A=(0,1,2,0,1,2,1,2,0).$ For example $(0,1,2,0,0,0,1,2,0,\ldots)\mapsto (1,2,1,0,0,1,2,1,\ldots),$ It is like a shift if the coordinate b... | https://mathoverflow.net/users/31217 | Invariant measures for Cellular automata | This problem is easy, but there is a related problem that is not, this is to find a fully supported invariant measure for our CA that is at the same time invariant for the action of the shift different from the Haar measure. Is this possible?
To solve the problem here we can do the following. Call our CA by $F.$ $F$... | 1 | https://mathoverflow.net/users/31217 | 121525 | 68,334 |
https://mathoverflow.net/questions/121527 | 4 | The Abel-Jacobi map from an algebraic curve $C$ to its Jacobian $J(C)$ is given analytically by $$p\to \left( \ldots, \int^{p}\_{p\_0} \omega\_i,\ldots\right),$$
where $p\_0$ is some point on $C$ and $\omega\_i$ form a basis of $H^0(C,K)$. Why is it an algebraic morphism?
| https://mathoverflow.net/users/10626 | Why is the Abel-Jacobi map an algebraic morphism? | Maybe there are easier ways to see it, but Chow's theorem/GAGA certainly gives you the result, since you have an analytic morphism of projective analytic varieties.
| 6 | https://mathoverflow.net/users/2698 | 121529 | 68,335 |
https://mathoverflow.net/questions/121526 | 1 | It appears that this is not true if H is of infinite dimension. My question is therefore the following: does anyone have a counter-example? Is there a caracterisation for the points of the boundary that are (non trivial) projections?
Thanks in advance.
| https://mathoverflow.net/users/31358 | Are all points x of the boundary of a convex set C of a Hilbert space H projections onto C of a point different than x? | I'm a little concerned that my original incorrect answer was accepted and the only feedback on the edit was a comment that it didn't make sense ... here's a slightly simpler counterexample.
Let $H = l^2$ and let $C$ be the set of sequences $(a\_n)$ satisfying $|a\_n| \leq \frac{1}{n}$ for all $n$. $C$ is clearly clos... | 4 | https://mathoverflow.net/users/23141 | 121532 | 68,338 |
https://mathoverflow.net/questions/121374 | 5 | It is interesting FACT that given $l,m,n\geq 2$, there is (are) a *finite group* with elements $a,b$ such that $o(a)=l, o(b)=m$, and $o(ab)=n$ (see [link](https://mathoverflow.net/questions/118035/order-of-elements) for a nice example by Derek Holt / B. Sury).
Although, the group
>
> $G\_{l,m,n}=\langle a,b\col... | https://mathoverflow.net/users/6761 | Finite Quotients of Free Groups | These groups are called von Dyck groups, see e.g. here. Von Dyck group $D(l,m,n)$ is finite if and only if it is of *spherical type*:
$$
\chi=-1+ l^{-1} + m^{-1} + n^{-1}>0.
$$
A side note: Von Dyck groups are fundamental groups of 2-dimensional oriented orbifolds. The number $\chi$ above is the orbifold Euler charac... | 7 | https://mathoverflow.net/users/21684 | 121534 | 68,340 |
https://mathoverflow.net/questions/121238 | 1 | Let $\xi\_{tn}(\theta),t=1,\dots,n$ be a real-valued martingale difference array indexed by a parameter $\theta \in \Theta \subset R$, where the set $\Theta$ is compact. Now, for all fixed $\theta \in \Theta$, the law of large numbers,
$\sum\_{t=1}^n \xi\_{tn} (\theta) \rightarrow 0$ in probability,
is assumed to... | https://mathoverflow.net/users/31275 | Uniform law of large numbers for martingale difference | Even if $\xi$'s are independent (a specific case of your martingale-difference stitation), uniform LLN sometimes holds and sometimes it does not. This type of questions has been studied in Machine learning and, specifically, in Vapnik-Chervonenkis theory. The Glivenko-Cantelli theorem (see wikipedia) describes one situ... | 1 | https://mathoverflow.net/users/2968 | 121535 | 68,341 |
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