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/123337 | 6 | Categorical models for linear logic with $\otimes$, $1$, $\&$, $\top$, $\oplus$, $0$, and $\multimap$ are typically symmetric monoidal closed categories (for modeling $\otimes$, $1$, and $\multimap$) with products (for modeling $\&$ and $\top$) and coproducts (for modeling $\oplus$ and $0$). Is it harmful to additional... | https://mathoverflow.net/users/25527 | Are exponentials in categorical models of linear logic harmful? | No, it does not make the structure collapse. For example consider the category of small categories $\mathbf{Cat}$ --- it is clearly complete and cocomplete cartesian closed category, and moreover it has another closed monoidal structure induced by the "funny tensor" and linear exponents $\mathbb{C} \multimap \mathbb{D}... | 11 | https://mathoverflow.net/users/13480 | 123341 | 69,207 |
https://mathoverflow.net/questions/123355 | 3 | The $k$-core of a finite graph is defined as follows. Delete all vertices of degree $< k$ and repeat until there are no such vertices left. If there is a nonempty subgraph remaining, necessarily of minimum degree $\ge k$, we call this graph the $k$-core. If the deletion process results in removing all vertices, we say ... | https://mathoverflow.net/users/4558 | Maximum number of edges $f(n,k)$ in a graph on $n$ vertices with no $k$-core? |
>
> $$f(n,k)= (n-k)(k-1)+\frac{k(k-1)}{2}, \; \mathrm{for} \; n \geq k, $$
> $$f(n,k) = \frac{n(n-1)}{2}, \; \mathrm{for} \; n \leq k. $$
>
>
>
**Proof:** Every graph with no $k$-core on $n$ vertices contains a vertex of degree $\leq k-1$ which one can delete, obtaining a graph with no $k$-core. Thus $$f(n,k) \... | 7 | https://mathoverflow.net/users/8733 | 123367 | 69,217 |
https://mathoverflow.net/questions/123356 | 3 | Let $\alpha:\mathbb{R}^n\to\mathbb{R}^n$, $n\geq 2$, be a $\mathbb{Q}$-linear bijection with the following properties:
1) $\alpha$ sends straight affine $\mathbb{R}$-lines to straight affine $\mathbb{R}$-lines
2) If $r$ and $s$ are parallel straight affine $\mathbb{R}$-lines, then so are $\alpha(r)$ and $\alpha(s)$... | https://mathoverflow.net/users/4721 | A little question on certain parallel-lines-preserving maps | The [fundamental theorem of projective geometry](http://www.wikipedia.org/wiki/Collineation) proven by von Staudt around 1850 (at least, this is when he published the proof) states:
Let $n\ge 2$, $F$ be a field and $f: FP^n\to FP^n$ is a *collineation*, i.e., a bijection preserving collinearity of points. Then $f$ is... | 8 | https://mathoverflow.net/users/21684 | 123373 | 69,219 |
https://mathoverflow.net/questions/123358 | 6 | Given a topological space $X$ instead of forming the fundamental groupoid $\pi(X)$ which is the category whose objects are the points and morphisms the homotopy classes of paths one can also form the fundamental 2-groupoid which is the bicategory with objects = points, 1-morphisms = paths and 2-morphisms = homotopy cla... | https://mathoverflow.net/users/3824 | When is this braiding not a symmetry? | If $X=\Omega Y$, that braided monoidal category (indeed groupoid) classifies the homotopy type of $P\_3Y$, the $3$-type of $Y$. Such $3$-type is completely determined by the map $\eta^\*\colon \pi\_2(Y)\rightarrow \pi\_3(Y)$ defined by precomposition with the Hopf map $\eta\colon S^3\rightarrow S^2$. This map is quadra... | 5 | https://mathoverflow.net/users/12166 | 123380 | 69,222 |
https://mathoverflow.net/questions/123382 | 5 | Does there exist a torsion-free group $(G,.)$ with the following property?
There exists finite, non-empty subsets $S\_1,S\_2\subset G$ such that for all $s \in S\_1.S\_2$ there are at least two solutions to the equation $s=s\_1.s\_2$ where $s\_1\in S\_1$ and $s\_2\in S\_2$.
-
I imagine that such a group $(G,.)$ d... | https://mathoverflow.net/users/30022 | finite subsets of torsion-free groups | These are the "unique product groups". Not every torsion-free group satisfies this property. See Promislow, S. David, A simple example of a torsion-free, nonunique product group.
Bull. London Math. Soc. 20 (1988), no. 4, 302–304.
| 9 | https://mathoverflow.net/users/nan | 123385 | 69,225 |
https://mathoverflow.net/questions/123372 | 14 | Let $\Gamma$ be a discrete subgroup of a connected finite dimensional Lie group $G$. Let $K$ be a maximal compact subgroup of $G$ and denote $X=G/K$. It is well-known that $\Gamma$ acts properly on $X$ via multiplication on the left.
Now suppose that a discrete group $\tilde{\Gamma}$ contains $\Gamma$ as a finite ind... | https://mathoverflow.net/users/31670 | discrete subgroups of Lie groups and actions on homogeneous spaces | Here is a simple counter-example. Let $p : S\_3 \to S\_2$ be a degree 2 covering map from the closed genus 3 surface to the closed genus 2 surface, inducing an index 2 injection $p\_\* : \pi\_1(S\_3) \to \pi\_1(S\_2)$. Pulling back any hyperbolic structure from $S\_2$ to $S\_3$ via $p$ defines an embedding of Teichmull... | 9 | https://mathoverflow.net/users/20787 | 123387 | 69,226 |
https://mathoverflow.net/questions/123333 | 1 | Is there a difference or is it just terminology?
| https://mathoverflow.net/users/31865 | "Non-oriented" vs "undirected" graph | Many people consider that an "oriented graph" is what you get from a simple undirected graph when you assign a direction to each edge. The difference between that and "directed graph" is that a directed graph can have cycles of length 2 even if it is qualified as "simple", whereas an "oriented graph" cannot. On the oth... | 5 | https://mathoverflow.net/users/9025 | 123389 | 69,227 |
https://mathoverflow.net/questions/123392 | 7 | Let $G$ be a finite group. Is it possible that there are "many" one-dimensional representations, and "very few" high-dimensional irreducible representations?
Originally, I thought it's impossible for the following reason. If $\chi\_1, \chi\_2$ are two one-dimensional representations such that $\chi\_2 \chi\_1^{-1} \n... | https://mathoverflow.net/users/26659 | Can we have many 1-dimensional rep, and very few high dimensional reps in a finite group? | Let $p$ be an odd prime and $G$ the semi-direct product of the additive group
$ {\mathbb F}\_p= {\mathbb Z}/p{\mathbb Z}$ and the group $({\mathbb Z}/p{\mathbb Z})^\*$ of the units in the finite field of $p$ elements, acting by multiplication on ${\mathbb F}\_p$. The group $G$ has $p-1$ characters (one dimensional repr... | 32 | https://mathoverflow.net/users/23291 | 123394 | 69,230 |
https://mathoverflow.net/questions/123399 | 1 | Let $G$ be a finite $p$-group, $p$ odd, and let $A$ be a maximal elementary abelian normal subgroup of $G$. Assume that $x \in G$ centralizes $A$ and $x^p$ is a central element. Is it true that $x \in AZ(G)$?.
| https://mathoverflow.net/users/31883 | Centralizers of abelian normal subgroups of p-groups | I don't think so. I can define a group $G = \langle v,w,x,y,z \rangle$ of order $p^5$, with $A = \langle x,y,z \rangle$ elementary abelian, $w^p=z$ and $w \in C\_G(A)$ (so my $w$ is your $x$), $v^p=1$, $[v,w]=z$, $[v,x]=y$, $[v,y]=[v,z]=1$. Then $Z(G) = \langle y,z \rangle < A$, so $w \not\in AZ(G)$.
(Note that $G$ i... | 2 | https://mathoverflow.net/users/35840 | 123403 | 69,234 |
https://mathoverflow.net/questions/123397 | 1 | Hello
As I checked the atlas of finite simple groups if $p$ is a prime then according to the notations of the atlas of finite simple groups, $2.L\_2(p)$ has an irreducible character of degree $(p-1)/2$ or $(p+1)/2$.
Why this character exists and is it true for each prime number $p>3$?
Thanks so much for your help.
| https://mathoverflow.net/users/31045 | The irreducible character of $2.L_2(p)$ where p is a prime | OK, so if I am not mistaken the non-split extension of $L\_2(p)$ should simply be $SL\_2(p)$. You are now asking whether $SL\_2(p)$ has a character of degree $(p-1)/2$ or $(p+1)2$ when $p \neq 2$. Indeed this is true. The generic character table of $SL\_2(q)$ is given in Table 5.4 of Bonnafe's book "Representations of ... | 3 | https://mathoverflow.net/users/22846 | 123404 | 69,235 |
https://mathoverflow.net/questions/123402 | 9 | In a nutshell my question is "Are there any easy, educational wall-crossing formulae?".
Recently I often hear the word "(Kontsevich-Soibelman's etc) wall-crossing formula" in algebraic geometry talks. I wonder what they are like but am not really ready to read those relevant papers with high technology. I understand ... | https://mathoverflow.net/users/31553 | Example of wall-crossing formulae? | I can tell you the gist of a "wall crossing formula". Typically you have a *space of parameters* $\newcommand{\eS}{\mathscr{S}}$ $\Lambda$, a *configuration space* $\newcommand{\eC}{\mathscr{C}}$ $\eC$, and a *parametrized moduli space* $\newcommand{\eM}{\mathscr{M}}$ $\eM$ which is a subvariety $\eM\subset \eC\times \... | 23 | https://mathoverflow.net/users/20302 | 123409 | 69,237 |
https://mathoverflow.net/questions/123388 | 8 | This is a sequel to this MO question:
[The multiplicative order of 2 modulo primes](https://mathoverflow.net/questions/60441/the-multiplicative-order-of-2-modulo-primes)
As shown in Charles Matthews' paper linked to there, it is not hard to show that for each $\delta > 0$ there is a $c = c(\delta) > 0$ such that t... | https://mathoverflow.net/users/26522 | The critical exponent in the multiplicative order of 2 modulo primes | The Generalized Riemann Hypothesis (for Dedekind zeta functions of certain Kummerian fields) would show that the "critical exponent" for the multiplicative order problem is $1$. In fact, GRH implies that if $g$ is any function tending to infinity, then (asymptotically) 100% of primes $p$ satisfy $n\_p > p/g(p)$. This f... | 5 | https://mathoverflow.net/users/31889 | 123416 | 69,240 |
https://mathoverflow.net/questions/123413 | 2 | I am trying to understand the relationship between the simplicial path space and loop space with the path space of a topological space, and the loop space of a topological space.
I have understood that the simplicial path space of a simplicial object $A$ is homotopy equivalent to the constant simplicial object $A\_0$... | https://mathoverflow.net/users/31887 | Simplicial path and loop spaces | I'll refer to my ancient book "Simplicial objects in algebraic topology". It is best to restrict to Kan complexes $K$ with a single vertex. In 23.3 and 23.4, it is shown that the path projection $PK \to K$ is a particularly nice kind of simplicial bundle provided that its
fiber $L(K)$ is a simplicial group, which usual... | 4 | https://mathoverflow.net/users/14447 | 123419 | 69,242 |
https://mathoverflow.net/questions/121167 | 4 | Hello,
I have recently read about the construction of the descent map in Kasparov KK theory, which, for a group $G$ and two $G$-equivariant $C^\*$ algebra $A$ and $B$ send $KK\_i^G(A,B)$ to $KK\_i(A \rtimes\_{red} G, B \rtimes\_{red} G)$.
I havn't checked all the detail, but it seems to me that the same constructio... | https://mathoverflow.net/users/22131 | On the descent homomorphsim of Kasparov equivariant KK theory | This descent map sometimes appears in the literature, see e.g. my contribution to:
Mislin, Guido & Valette, Alain (2003), Proper Group Actions and the Baum-Connes Conjecture, Basel: Birkäuser, ISBN 0-8176-0408-1.
One reason to consider it, is that the full crossed product has better functoriality properties than th... | 1 | https://mathoverflow.net/users/14497 | 123434 | 69,249 |
https://mathoverflow.net/questions/123411 | 4 | Suppose $f$ is a bounded continuous function on $[0,\infty)$ such that $\int\_0^\infty f(t) \exp(-xt) \: dt \rightarrow 0$ as $x \rightarrow 0^+$. Does it follow that $\int\_0^\infty f(t) \exp(-xt^2) \: dt \rightarrow 0$ as $x \rightarrow 0^+$? Is the reverse implication true?
I suspect that the answer is "no" in bot... | https://mathoverflow.net/users/3621 | Using a quadratic kernel instead of a linear kernel in the Laplace transform | There exists a Tauberian theorem of the following form. Suppose that $a\in L^1\_{loc}(\mathbb{R}\_{>0})$ and for any $x>0$ the integral
$$ G(x):=\int\_0^\infty e^{-x^2t^2} a(t) dt $$
exists and satisfies
$$ \sup\_{x>0}|G(x)|<\infty, $$
and
$$ \lim\_{x\searrow 0} G(x) = A\in\mathbb{R}. $$
The function $G(x... | 4 | https://mathoverflow.net/users/20302 | 123436 | 69,251 |
https://mathoverflow.net/questions/17464 | 15 | Consider a function $f: \{0,1\}^n \rightarrow \{1..R\}$. This function can be interpreted as a coloring $Color(v)$ of vertices in a unit n-dimensional hypercube in $R$ colors.
We say there is a directed edge $(v1, v2)$ in the hypercube if $v1$ and $v2$ differ in only one coordinate in n-dimensional space and this co... | https://mathoverflow.net/users/2641 | Making a non-monotone function monotone | A combinatorial proof was found last year by Chakrabarty and Seshadhri: <http://eccc.hpi-web.de/report/2012/030/download>. In Theorem 3 they show that indeed $E(f) = \Omega(M(f))$.
| 3 | https://mathoverflow.net/users/2641 | 123437 | 69,252 |
https://mathoverflow.net/questions/123424 | 1 | If $M$ is a finitely generated module over a local ring $(R, \mathfrak{m})$, we can detect whether $M$ has a nonzero free direct summand as follows: Consider the natural map
$$\phi\_M\colon \mathrm{Hom}\_R(M,\mathfrak{m}) \longrightarrow \mathrm{Hom}\_R(M,R)$$
induced by the inclusion of $\mathfrak m$ into $R$. Then $M... | https://mathoverflow.net/users/460 | Detecting and counting free direct summands | If $F$ is a free direct summand of $M$ of maximal rank, then $M=F\oplus N$, where $N$ has no free direct summand. So $\mathrm{coker} \;\phi\_M\cong\mathrm{coker} \;\phi\_F\oplus\mathrm{coker} \;\phi\_N$, and, as you've pointed out, the dimension of $\mathrm{coker} \;\phi\_F$ is the rank of $F$ and $\mathrm{coker} \;\ph... | 3 | https://mathoverflow.net/users/22989 | 123441 | 69,253 |
https://mathoverflow.net/questions/123365 | 2 | Consider an undirected connected bipartite graph (with cycles) $G = (V\_1,V\_2,E)$, where $V\_1,V\_2$ are the two node sets and $E$ is the set of edges connecting nodes in $V\_1$ to those in $V\_2$. We assume that $|V\_1|=v\_1$, $|V\_2|=v\_2$ and $|E|=e$.
Then $(e-v\_1-v\_2+1)$ edges need to be removed to make $G$ a... | https://mathoverflow.net/users/26701 | Removing cycles from an undirected connected bipartite graph in a special manner | [edited]
It seems this question is NP-Complete.
The general idea:
In a graph which is a 3-regular graph minus an edge,
a spanning tree that minimizes $\max x\_i$ is (more or less) an Hamiltonian Path.
Some more work is needed in order to make it an Hamiltonian Cycle;
finding an Hamiltonian Cycle in a 3-regular bi... | 1 | https://mathoverflow.net/users/27663 | 123446 | 69,255 |
https://mathoverflow.net/questions/116633 | 25 | The description below comes from
* József Beck. **Combinatorial games. Tic-tac-toe theory**, Encyclopedia of Mathematics and its Applications, **114**. Cambridge University Press, Cambridge, 2008, [MR2402857 (2009g:91038)](http://www.ams.org/mathscinet-getitem?mr=2402857).
Given a finite set $S$ of points in the p... | https://mathoverflow.net/users/6085 | Sane bound on number of moves for Maker-Breaker game on $\mathbb R^2$ for $\{0,1,2,3,4\}$ | Eleven moves suffice.
François Brunault commented that the maker can get two moves to start on some hexagonal lattice (a lattice generated by unit vectors with an angle of $60$ degrees). In fact, by the fourth move, the maker can get three moves to start in a hexagonal lattice, and can choose these to be the vertices... | 19 | https://mathoverflow.net/users/2954 | 123449 | 69,257 |
https://mathoverflow.net/questions/123444 | 15 | Is every finitely generated projective $\mathbf{Z}[x]$-module free?
| https://mathoverflow.net/users/29980 | Is every projective $\mathbf{Z}[x]$-module free? | While it's certainly true (per Fernando's comment) that this is a special case of the Quillen-Suslin theorem, it was certainly known long before Quillen and Suslin came along.
There's a paper of Murthy from the mid-1960s which shows that every projective $R[x]$-module is extended whenever $R$ is a regular ring of di... | 28 | https://mathoverflow.net/users/10503 | 123450 | 69,258 |
https://mathoverflow.net/questions/123239 | 9 | Let $Q$ be a recursively presented group. Is it possible to embed $Q$ into a finitely presented group $G$ such that the image of $Q$ is malnormal in $G$?
Note that a subgroup $H$ of $G$ is malnormal if $H^g\cap H\neq 1$ implies that $g\in H$. That is, $H$ intersects each of its proper conjugates trivially.
This see... | https://mathoverflow.net/users/6503 | A malnormal embedding theorem? | As I wrote in Remark 5.23 cited by Ian Agol, the embedding from that paper should answer the first question. It certainly preserves all centralizers, that is basically proved in the paper. That is if $G$ is a finitely generated recursively presented group, $Q>G$ is the group constructed in that paper, then for every $x... | 6 | https://mathoverflow.net/users/nan | 123452 | 69,259 |
https://mathoverflow.net/questions/123317 | 15 | In [1] Grothendieck posits the following:
**Conjecture**. Let $S$ be a reduced connected scheme, locally of finite type over Spec($\mathbf{Z}$) or a field $k$, $A$ and $B$ two abelian schemes over $S$, $l$ a prime number, $u\_l: T\_l(A) \rightarrow T\_l(B)$ a homomorphism, and suppose that there exists a point $s\in ... | https://mathoverflow.net/users/2604 | Status of Grothendieck's conjecture on homomorphisms of abelian schemes | This is an expansion of my comments. Grothendieck's conjecture is now a theorem, except perhaps when $S$ is in characteristic $p$ and $\ell=p$. The explanation that follows is presumably what Grothendieck had in mind in his comment regarding the Tate conjecture.
First, we make some reductions: We can certainly assume... | 11 | https://mathoverflow.net/users/7868 | 123454 | 69,261 |
https://mathoverflow.net/questions/123451 | 4 | In my work I've run into the following situation. In a model category, I have two reflexive coequalizers $A\_i \stackrel{\to}{\to} B\_i \to C\_i$ and a map of diagrams which is levelwise a weak equivalence (i.e. $A\_1\to A\_2$ and $B\_1\to B\_2$ are weak equivalences). I need to conclude $C\_1\to C\_2$ is a weak equiva... | https://mathoverflow.net/users/11540 | When do reflexive coequalizers preserve weak equivalences? | **Disclaimer:** I am *not* an expert on model categories.
$\newcommand{\MM}{\mathcal{M}}
\newcommand{\pair}{\mathsf{P}}
\newcommand{\dom}{\operatorname{dom}}
\newcommand{\codom}{\operatorname{codom}}
\newcommand{\colim}{\operatorname\*{colim}}
\newcommand{\hocolim}{\operatorname\*{hocolim}}
\newcommand{\id}{\mathrm{id}... | 9 | https://mathoverflow.net/users/21095 | 123466 | 69,265 |
https://mathoverflow.net/questions/123433 | 4 | Let $G$ be a finite $p$-group. Since we can embed $Z\_2(G)/Z(G)$ in $Hom(G,Z(G))$, we have $d\_2 \leq d(G)d(Z(G))$; where $d\_2(G)=d(Z\_2(G)/Z(G))$ and $d(G)$ denotes the minimal number of generators of $G$. The question is, does the equality $d\_2 = d(G)d(Z(G))$ imply that $Z(G)$ is cyclic?
| https://mathoverflow.net/users/31883 | Generators of p-groups | OK, here is an example of a group $Q$ of class 3 and order $p^{17}$, which will work for $p \ge 5$. We have $d(Q)=5$, $d(Z(Q)) = 2$, $d(Z\_2(Q))/Z(Q)) = 10$, with $Z(Q) = \langle Q.16, Q.17 \rangle$ and $Z\_2(Q) = \langle Q.6,\ldots,Q.17 \rangle$. All generators have order $p$ - in fact $Q$ has exponent $p$. All pairs ... | 6 | https://mathoverflow.net/users/35840 | 123470 | 69,267 |
https://mathoverflow.net/questions/123472 | 12 | There do exist manifolds which do not admit any smooth structure at all. But the only examples I've heard of are all compact.
>
> Are there any non-compact, non-smoothable manifolds?
>
>
>
| https://mathoverflow.net/users/13356 | Are there non-compact, non-smoothable manifolds? | The Cairns-Hirsch theorem says that a PL manifold $M$ is smoothable if and only if $M\times \mathbb{R}$ is smoothable, so you can take $M$ to be any one of the known compact, PL examples such as Kervaire's manifold and then $M\times\mathbb{R}^n$ is non-smoothable for $n \geq 1$.
| 30 | https://mathoverflow.net/users/428 | 123476 | 69,268 |
https://mathoverflow.net/questions/123462 | 3 | Let $\mathfrak{g}$ be a simple Lie algebra; let $R = \mathfrak{g}^{\*, reg}$ denote the regular locus in the dual Lie algebra. Consider the vector bundle $\mathfrak{z}$ over $R$, whose fiber over a point $\xi \in R$ is the stabilizer of $\xi$ in the dual to $\mathfrak{g}$. Using the isomorphism $R/G = \mathfrak{t}/W$, ... | https://mathoverflow.net/users/2623 | A fact about $t/W$ and the centralizer bundle on $\mathfrak{g}^{\text{reg}}$ | I think this can be understood in terms of Hamiltonian reduction.
The group $G$ acts on $\mathfrak g^{\ast,reg}$ inducing an action on $T^\ast \mathfrak g^{\ast,reg}$. The associated moment map $\mu: T^\ast \mathfrak g^{\ast,reg} \to \mathfrak g^\ast$ is:
$\mu(\xi,x) = coad(x)(\xi) - \xi$.
A basic result of Hami... | 4 | https://mathoverflow.net/users/7762 | 123489 | 69,273 |
https://mathoverflow.net/questions/108419 | 8 | I am given a matrix of the following form:
$$M = \begin{pmatrix}
a\_0 & -1 & \cdots & \cdots & -1 \newline
-1 & a\_1 & \ddots & & \vdots \newline
\vdots & \ddots & \ddots & \ddots & \vdots \newline
\vdots & & \ddots & \ddots & -1 \newline
-1 & \cdots & \cdots & -1 & a\_n
\end{pmatrix}$$
with $a\_i \in \mathbb{Z}... | https://mathoverflow.net/users/13488 | Smith normal form of a Matrix with -1 outside the diagonal | According to <http://www-math.mit.edu/~rstan/transparencies/snf.pdf>,
the Smith Normal form for a nonsingular matrix $M$ is
$$
\textrm{Diag}(e\_1,\ldots, e\_n),
$$
where $e\_1\cdots e\_i = \gcd \big( i\times i\textrm{ minors of }M \big)$.
There are infinitely many $a\_1,\ldots, a\_n$ such that $M$ is nonsingular, ... | 6 | https://mathoverflow.net/users/21090 | 123498 | 69,277 |
https://mathoverflow.net/questions/123460 | 3 | Why are there only finitely many cubic fields of a given discriminant? Is this true for higher dimension too?
What other invariants are needed to classify cubic fields? number of real and complex embeddings, Galois closure?...
| https://mathoverflow.net/users/25762 | Cubic Fields Up to Isomorphism | Your first two questions have already received good answers, so let me offer a few references about "classifying cubic fields", and point to some of my colleagues' interesting work in the subject.
What information determines the field? This question turns out to be quite subtle. In general, even the Dedekind zeta fu... | 5 | https://mathoverflow.net/users/1050 | 123507 | 69,280 |
https://mathoverflow.net/questions/123486 | 7 | For the purposes of teaching my elementary course in algebraic geometry I am looking for a reference (or notes) that contains a complete proof of a higher-dimensional weak Bezout theorem. I only want to learn about a proof that is based on the approach when dimensions and degrees of projective varieties are introduced ... | https://mathoverflow.net/users/13441 | Higher dimensional Bezout via Hilbert polynomials: a reference | This is to answer the last question on why the $F\_i$ form a regular sequence.
In my first attempt I was trying to do this without using the unmixedness of an ideal $I$ generated by $\mathrm{height } I$ number of elements, but it seems that I cannot. (Hat tip to Hailong).
So here is the statement that one needs:
... | 9 | https://mathoverflow.net/users/10076 | 123509 | 69,282 |
https://mathoverflow.net/questions/123522 | 5 | As far as I know, toric Fano manifolds are classified only up to dimension 4. In dimension one the projective line is the only example. In dimension two we have five examples: $\mathbb P ^2$, $\mathbb P^1 \times \mathbb P^1$ and $\mathbb P^2$ blown-up at 1,2 or 3 points in general position.
Toric Fano 3-folds were cl... | https://mathoverflow.net/users/35428 | Toric Fano manifolds with Picard number 1 | I don't think the Fano condition plays an essential role here; Any smooth toric variety with Picard number 1 is a projective space. This can be verified using usual toric geometry machinery: The condition for a toric variety $X$ to be of dimension $n$ and Picard number 1 means that the fan $\Delta$ is generated by $n+1... | 5 | https://mathoverflow.net/users/3996 | 123525 | 69,289 |
https://mathoverflow.net/questions/123537 | 9 | Let $N \geq 5$ be a prime, and $H$ a subgroup of $GL\_2(\mathbb{F}\_N)$. As shown in Chapter IV of [DeRap], there is a curve $X\_H(N)$, defined over $K\_N := \mathbb{Q}(\zeta\_N)^{\det H}$, which is a coarse moduli space of elliptic curves with **level-$H$ structure**; that is, given any $L$ an extension of $K\_N$, the... | https://mathoverflow.net/users/5744 | Which level structures on elliptic curves are twist-invariant? | The criterion is that H contains -I.
| 8 | https://mathoverflow.net/users/431 | 123544 | 69,296 |
https://mathoverflow.net/questions/123535 | 2 | Hello,
Let $X$ be a smooth variety in char. 0. Let us call a $D$-module on $X$ constant, if it is isomorphic to a finite direct sum of the $D$-modules $O$ (the sheaf of regular functions with the usual $D$-module structure).
Then a subquotient of a constant $D$-module should be constant. But how to show it?
The reaso... | https://mathoverflow.net/users/2095 | A submodule of a constant D-module is constant | If your base field $k$ is algebraically closed (or if $X$ has a rational point), then this follows directly from the fact that the category of $\mathcal{O}\_X$-coherent flat connections is neutral Tannakian, i.e. equivalent to the category of finite dimensional $k$-representations of some affine $k$-group scheme $G$. F... | 8 | https://mathoverflow.net/users/259 | 123550 | 69,299 |
https://mathoverflow.net/questions/123551 | 1 | Let $\mathcal{F}$ be a filed, $\mathcal{G}=GL(n,\mathcal{F})$, all n×n invertible matrices
on $\mathcal{F}$, What is $Aut(\mathcal{G})$ ?
| https://mathoverflow.net/users/22161 | the automorphism group of the general linear group GL(n,F) | See Chapt.IV in the book
Dieudonné, J. A.
La géométrie des groupes classiques. Springer-Verlag (1971).
| 7 | https://mathoverflow.net/users/18814 | 123553 | 69,301 |
https://mathoverflow.net/questions/123543 | 1 | I come across an optimization problem of the following form.
$$\max {\frac{1+v}{1-u}} \qquad \text{s.t.} \qquad ux^2+vy^2-xy \ge 0, \quad \forall x,y\in\mathbb{R}$$
I do not know much of optimization. What I have done is that $ux^2+vy^2\ge 2\sqrt{uv}xy\ge xy$, so I let $uv=\frac{1}{4}$ and get the seemingly correct... | https://mathoverflow.net/users/31927 | What kind of optimization problem is this? | Let us simplify your constraints. Namely, set $x=r\sin\phi$, $y=r\cos\phi$. Then it simplifies to $u\sin^2\phi + v\cos^2\phi \geq \sin\phi \cos\phi$, for all $0\leq \phi\leq 2\pi.$ Then divide both sides by $\cos^2\phi$, and set $z:=\frac{\sin\phi}{\cos\phi}$. You'll get quadratic inequality $f(z):=uz^2-z+v\geq 0$, for... | 2 | https://mathoverflow.net/users/11100 | 123560 | 69,305 |
https://mathoverflow.net/questions/123559 | 1 | Let $R$ be a commutative noetherian ring, and let $M$ be an $R$-module. How I can show that if any localization $M\_p$ at a prime ideal $p$ of the ring $R$ is injective over $R\_p$, then $M$ is injective?
| https://mathoverflow.net/users/31803 | injectivity is a local property over noetherian rings | Let $A \to B$ be an injection of $R$-modules. We want to show that $\mathrm{Hom}(B,M) \to \mathrm{Hom}(A,M)$ is surjective. It suffices to check this locally (since localization is an exact functor and the cokernel of the map is zero iff it's zero locally). So let $p$ be a prime, and consider the localized map $\mathrm... | 6 | https://mathoverflow.net/users/27153 | 123570 | 69,308 |
https://mathoverflow.net/questions/121468 | 5 | By inspecting the accepted answer to this question
[Are epimorphisms from a division ring isomorphisms ?](https://mathoverflow.net/questions/120918/are-epimorphisms-from-a-division-ring-isomorphisms)
one obtains the following necessary condition for epimorphisms:
>
> Let $R \le S$ be rings with identity $1\n... | https://mathoverflow.net/users/18571 | Epimorphisms and free submodules | A simple counterexample can be constructed using free objects:
For a set $X$ let $\mathbb{Z}\langle X\rangle$ be the free ring on $X$ and let $\mathbb{Z}F(X)$ be the group ring of the free group $F(X)$ on $X$. The inclusion $X \hookrightarrow F(X)$ induces a ring homomorphism $i: \mathbb{Z}\langle X\rangle \to \math... | 10 | https://mathoverflow.net/users/10194 | 123579 | 69,311 |
https://mathoverflow.net/questions/123569 | 1 | I have a finite set of points, and plot the graph log(N) vs. log(e). I see a polygonal chain (the final slope, starting at some size of e, is zero, of course). If the set represents some physical property of a complex system, is there any interpretation for that kind of series of slopes? Any literature around?
And, gen... | https://mathoverflow.net/users/31856 | Fractal dimension of 1D set, what if logN vs log(e) is a polygonal chain? | The canonical reference for this material (at least with the people I hang out with) is Ken Falconer's Fractal Geometry Mathematical Foundations and Applications.
For most sets that are fully self-similar with infinite levels of geometry such as the Cantor set the function $\log(N)/\log(e)$ will be a similar polygon... | 2 | https://mathoverflow.net/users/11332 | 123587 | 69,315 |
https://mathoverflow.net/questions/123513 | 30 | Absolute neighborhood retracts (ANRs) are topological spaces $X$ which, whenever $i\colon X\to Y$ is an embedding into a normal topological space $Y$, there exists a neighborhood $U$ of $i(X)$ in $Y$ and a retraction of $U$ onto $i(X)$.
They were invented by Borsuk in 1932 (*Über eine Klasse von lokal zusammenhängenden... | https://mathoverflow.net/users/10696 | The role of ANR in modern topology | Another reason you might not see the word ANR these days is that compact finite-dimensional spaces are ANRs if and only if they are locally contractible. Thus, "finite-dimensional and local contractible" can replace ANR in the statement of a theorem (and might help the result appeal to a wider audience).
In comparis... | 34 | https://mathoverflow.net/users/31951 | 123593 | 69,317 |
https://mathoverflow.net/questions/123005 | 5 | Hi!
There is a risk that this question might be utterly trivial, and if it is, my sincerest apologies, but I haven't been able to find anything in the literature.
Let $f:X \rightarrow Y$ be an étale morphism of schemes. This gives rise to a geometric morphism of topoi $f^\ast:Sh(Y) \rightleftarrows Sh(X) : f\_\ast... | https://mathoverflow.net/users/7607 | Essential geometric morphisms on the étale site. | The answer is always. Let $\mathrm{Et}(X)$ and $\mathrm{Et}(Y)$ denote the étale sites. There is a functor $f\_! : \mathrm{Et}(X) \rightarrow \mathrm{Et}(Y)$ sending an étale $X$-scheme $p : U \rightarrow X$ to $f \circ p : U \rightarrow Y$. This functor is cocontinuous (SGA4.III.2.1) and continuous (SGA4.III.1.1). By ... | 9 | https://mathoverflow.net/users/32 | 123598 | 69,319 |
https://mathoverflow.net/questions/123605 | 3 |
>
> Do there exist nonzero integers $a,b,c$ for which the equation $$aX + bY = cZ$$ has infinitely many solutions with $X,Y,Z$ distinct prime powers?
>
>
>
For example, if there are infinitely many Sophie Germain primes $p$ (so that $2p+1$ is also prime), then we can take $a = 2$, $b = c = 1$. Triples of the fo... | https://mathoverflow.net/users/13410 | Catalan-type equations for prime powers | It is known by results of Green and Tao that there are infinitely many 3-term sequences of primes in arithmetic progression. So taking $a = b = 1$ and $c = 2$, you have infinitely many solutions in primes.
| 5 | https://mathoverflow.net/users/21146 | 123607 | 69,324 |
https://mathoverflow.net/questions/123608 | 13 | Hi everyone,
This is a question I have been asking from long, but none of my colleagues could ever answer me:
It is a well-known fact that the axiom of choice (AC) allows one to prove the existence of some set with some property $P$, though we cannot *exhibit* such a set: for instance, one knows that there exists a... | https://mathoverflow.net/users/31954 | Non-constructive existence proofs without AC? | If ZF is consistent, then the answer is no.
ZF proves that there is an $x$ such that $P(x)$: either $x$ is an $L$-generic Cohen real or there is no $L$-generic such real. (Here, by $L$ I mean the constructible universe of Gödel.)
In a universe with $L$-generic Cohen reals, $P$ holds only of them, but in a universe... | 12 | https://mathoverflow.net/users/1946 | 123612 | 69,326 |
https://mathoverflow.net/questions/123610 | 4 | Is there a recently published book on the Classical Moment Problems and related theory?
I have seen a couple of old books by Tamarkin and a few other books by Russian authors. Want to know what else can be a good reference.
| https://mathoverflow.net/users/31955 | Book on the Moment Problem | The classic reference and still one of the best places to learn about this stuff is Akhiezer's book *The classical moment problem and some related problems in analysis*, (English translation 1965). This isn't a recently published book, but it is in my view the gold standard.
A much more recent book *Unbounded selfad... | 7 | https://mathoverflow.net/users/20302 | 123616 | 69,328 |
https://mathoverflow.net/questions/123575 | 5 | This is probably trivial but has been bothering me all day.
Suppose $f:\Sigma\_g\to \mathbb{S}^2$ is a $g+1$ fold branched conformal map with $\Sigma\_g$ a connected genus $g$ surface and $f$ having $4$ branch points (all of order $g+1$).
Is it true that $f:\Sigma\_g^\* \to (\mathbb{S}^2)^\*$ (the cover obtained by... | https://mathoverflow.net/users/26801 | Branched Regular Cover over 4-times punctured sphere | The answer is *no*:
The monodromy group of the cover, as a permutation group on the fiber $f^{-1}(b)$ of a non-branch point $b$, is generated by four elements $s\_1,s\_2,s\_3,s\_4$ with $s\_1s\_2s\_3s\_4=1$. By Riemann's existence theorem, the question is equivalent to the following, with $n=g+1$:
>
> Let $s\_1,s... | 6 | https://mathoverflow.net/users/18739 | 123621 | 69,331 |
https://mathoverflow.net/questions/92527 | 26 | Let $a$, $b$ be two coprime natural numbers. Let $A \subseteq \{0,1,\ldots, a-1\}$ and $B \subseteq \{0,1,\ldots,b-1\}$ be two nonempty sets, which we think of as sets of residues mod $a$ and $b$ respectively.
I would like to know if anyone has ever seen (or knows a proof for) the following result: that any interval ... | https://mathoverflow.net/users/22485 | long enough interval of integers to solve a simultaneous congruence | [**Edited** *again mostly to spell out why the $m\_j \bmod a$ are distinct*.]
Yes, the desired result is true for all $k$. The following proof
is elementary but possibly more algebraic than expected (apparently
some kind of variant of the "polynomial method" in combinatorics, though
with no need for anything as advan... | 14 | https://mathoverflow.net/users/14830 | 123636 | 69,339 |
https://mathoverflow.net/questions/123633 | 15 | I apologize in advance if this is somewhat elementary, but:
>
> Let $(M,g)$ be a compact Riemannian manifold. Is there a "characterization" of which symmetric bilinear tensors $B\in Sym^2(M)$ are Hessians of functions $f\colon M\to \mathbb R$?
>
>
> In other words, for which $B\in Sym^2(M)$ there exists $f\colon ... | https://mathoverflow.net/users/15743 | Characterizing Hessians among symmetric bilinear tensors | There *are* local conditions, but they typically involve the curvature tensor of the underlying metric. For example, if the metric is flat, so that one can choose orthonormal coordinates $x\_i$ in which the covariant derivatives are the ordinary derivatives, then the condition that a quadratic form $H = h\_{ij}\ dx\_id... | 23 | https://mathoverflow.net/users/13972 | 123644 | 69,343 |
https://mathoverflow.net/questions/123632 | 8 | I've been reading about the formal structure of gauge theories and am a little confused by the notation. Could someone clarify this for me?
Suppose that $A$ is a Lie algebra valued 1-form corresponding to the gauge potential of a $U(1)$ gauge theory. Then several sources define the field strength tensor by
$$F=dA+A... | https://mathoverflow.net/users/22337 | Wedge Product of Lie Algebra Valued One-Form | There's something about the notation you should know before you get confused when trying to do non-abelian gauge theory. The second term in the field strength should involve a combination of the wedge product of forms and the Lie bracket: the field strength (in the case of an arbitrary gauge group $G$ with Lie algebra ... | 18 | https://mathoverflow.net/users/21375 | 123648 | 69,345 |
https://mathoverflow.net/questions/123500 | 4 | A few days ago I was asked by the director of the [Center for Undergraduate Research and Scholarship](http://www.aug.edu/curs/) at Georgia Regents University (formerly known as MCG and Augusta State) to contribute an article for the new **Faculty Handbook: Mentoring Undergraduates in Research and Scholarship** which is... | https://mathoverflow.net/users/7442 | Faculty Handbook: Mentoring Undergraduates in Research and Scholarship | I'd recommend asking for input from professors who have a track record of supervising successful undergraduate research, such as [Joseph Gallian](http://www.d.umn.edu/~jgallian/tanglewood.html) or the people who run the [Ross Program](http://www.math.osu.edu/ross/).
Frank Morgan also has some [good advice](http://sit... | 8 | https://mathoverflow.net/users/3106 | 123652 | 69,346 |
https://mathoverflow.net/questions/123647 | 2 | Sometimes definite integral is defined using antiderivatives:
$$\int\_{a}^b{f(t)dt}=F(b)-F(a)$$
where $F$ is any continuous function such that:
$$(\forall t\in[a,b]\setminus C)(F'(t)\text{ exists and }F'(t)=f(t))$$
where $C$ is a countable set.
Then (if I have written the definition correctly) it can be proved, the i... | https://mathoverflow.net/users/31968 | Defining definite integral using indefinite integral | To add to Gerald Edgar’s answer:
1) Kurzweil–Henstock integral satisfies $\int\_a^bf(x)\,dx=F(b)-F(a)$ even under the weaker assumptions that $F$ is continuous and $F'(x)=f(x)$ for all but countably many $x\in[a,b]$, hence it fully subsumes the integral you want to define. It is, however, strictly more general: if $A... | 1 | https://mathoverflow.net/users/12705 | 123659 | 69,350 |
https://mathoverflow.net/questions/123654 | 2 | Is there any character formula for demazure modules in arbitary kac moody settings which does not use demazure operators?
| https://mathoverflow.net/users/21155 | character formula for demazure modules | Ryom-Hansen, Steen(DK-CPNH)
Littelmann's refined Demazure character formula revisited. (English summary)
Sém. Lothar. Combin. 49 (2002/04), Art. B49d, 10 pp.
The review:
"The author provides a purely combinatorial proof the Demazure character formula, a generalisation of Weyl's character formula. This is done usin... | 2 | https://mathoverflow.net/users/4366 | 123664 | 69,353 |
https://mathoverflow.net/questions/123663 | 2 | I've been reading a proof concerning S-arithmetic subgroups of algebraic groups and I'm having trouble determining why the following step should be true. First, the setup:
Let $G$ be a connected absolutely simple adjoint algebraic group defined over a nonarchimedean local field $F$. Let $K \subset F$ be a global fiel... | https://mathoverflow.net/users/31495 | S-arithmetic subgroup question | This is strong approximation. $\Gamma$ is an $irreducible$ lattice in the product group $G\_S$. If there is more than one place $v$ in $S$ other than $v\_0$ where $G(K\_v)$ is not compact, then the projection of $\Gamma$ to $G(K\_{v\_0})$ is dense, and therefore cannot be discrete. Margulis' book itself may contain thi... | 4 | https://mathoverflow.net/users/23291 | 123682 | 69,360 |
https://mathoverflow.net/questions/123677 | 13 | I need to evaluate some (one-variable) integrals that neither SAGE nor Mathematica can do symbolically. As far as I can tell, I have two options:
(a) Use GSL (via SAGE), Maxima or Mathematica to do numerical integration. This is really a non-option, since, if I understand correctly, the "error bound" they give is not... | https://mathoverflow.net/users/398 | Rigorous numerical integration | [Interval arithmetic](http://en.wikipedia.org/wiki/Interval_arithmetic) methods will permit rigorous bounds. You might try [INTLAB](http://www.ti3.tu-harburg.de/rump/intlab/). There are various books on rigorous numerics, e.g., [Warwick Tucker's *Validated Numerics*](http://books.google.com/books/about/Validated_Numeri... | 9 | https://mathoverflow.net/users/1847 | 123683 | 69,361 |
https://mathoverflow.net/questions/123594 | 9 | This is a basic question concerning the local Langlands correspondence for $GL\_2(\mathbb{Q}\_p)$, in particular the compatibility with tensor products . I apologize if this is too basic but I hope someone with more experience can answer it quickly.
Let $G=GL\_2(\mathbb{Q}\_p)$, $B \subset G$ the subgroup of upper tr... | https://mathoverflow.net/users/31952 | On local Rankin-Selberg L-functions for Steinberg representations | I apologize for answering my own question, but I think that the answer might be of some interest to others.
Since I was getting no answers I asked Professor Jacquet directly today. He explained that the expression for $L(St\_G \times St\_G,s)$ in the book cited above is not correct. The correct $L$-function appears i... | 9 | https://mathoverflow.net/users/31952 | 123685 | 69,362 |
https://mathoverflow.net/questions/123655 | 4 | The higher dimensional Lehmer problem asserts that if $\alpha\_1,\ldots,\alpha\_r$ are multiplicatively independent non-zero algebraic numbers generating an extension of $\mathbb{Q}$ of degree $d$, then $h(\alpha\_1) \cdots h(\alpha\_r) \geq c(r)/d$. This arose in the work of Amoroso and David, who have shown this up t... | https://mathoverflow.net/users/26522 | The elliptic Lehmer problem for several independent algebraic points | ADDENDUM: I looked at Masser's 1989 article, and a quick back-of-the-envelope calculation seems to give the result, at least for two points. Thus if $P$ and $Q$ are independent points generating a field of degree $d$, then
$$ \hat h(P)\hat h (Q) \ge \frac{C(E)}{d^3(\log d)^2}. $$
The idea is to apply the main theorem i... | 3 | https://mathoverflow.net/users/11926 | 123688 | 69,365 |
https://mathoverflow.net/questions/123670 | 1 | This will be a simple problem on paper, but the brute force method is not really suitable for a computer, so I'm after a tricky algorithm that works in practice too: if $n$ positive half-integers $p\_i$ are given, one can form $2^n$ linear combinations with coefficients $\pm 1$:
$$P = e\_1 p\_1 + \ldots + e\_n p\_n$$... | https://mathoverflow.net/users/4526 | All possible linear combinations of positive half-integers with coefficients +/- 1 | Indeed it is the Subset Sum Problem (or closely related to it). This problem asks essentially whether $\mu(P)>0$.
It is only weakly NP-complete, and there is a pseudopolynomial dynamic-programming
algorithm for it. You can find it in any textbook on algorithm design.
More precisely, if the sum of the $p\_i$'s is $B$, y... | 3 | https://mathoverflow.net/users/30800 | 123689 | 69,366 |
https://mathoverflow.net/questions/123681 | 13 | For reasons which the margin of this page is too small to hold, I have been reading parts of a recent paper by O. Selim
*On submeasures on Boolean algebras*, arXiv [1212.6822v3](http://arxiv.org/abs/1212.6822v3)
and in Section 7 the following technical lemma is given (Lemma 7.5 in the paper)
**Lemma (paraphrased)... | https://mathoverflow.net/users/763 | Invertibility of a certain matrix indexed by the Hamming cube | The argument of Kim and Roush looks as follows after translating out the semigroup theory (and is essentially using a Mobius inversion idea).
Let $T\colon \mathbb Z^S\to \mathbb Z^S$ be the group homomorphism corresponding to left multiplication by $A$. We show that in appropriate bases for the domain and codomain th... | 8 | https://mathoverflow.net/users/15934 | 123699 | 69,370 |
https://mathoverflow.net/questions/123690 | 7 | In the study of representation theory of $S\_n$, we know that the irreducible characters of $\chi\_\lambda$ of $S\_n$ are indexed by partitions $\lambda \vdash n$. There are several methods in determining the dimension of each $S^\lambda$, $f^\lambda$. One of the method is by considering
$ f^\lambda = \frac{n!}{\prod... | https://mathoverflow.net/users/31978 | Dimension of Specht Modules $S^\lambda$ | The opposite is true. It is a result of D. Craven, settling a conjecture of A. Moreto, that given any $k$, for all large enough $n$, there are at least $k$ distinct irreducible representations of $S\_n$ all of the same dimension.
<http://arxiv.org/pdf/0709.0897.pdf>
| 15 | https://mathoverflow.net/users/36466 | 123701 | 69,371 |
https://mathoverflow.net/questions/123700 | 1 | Let $ f: \mathbb{P}^n \dashrightarrow (\mathbb{P}^1)^n , [x\_0:\dots :x\_n] \mapsto ([x\_0,x\_1],[x\_0,x\_2], \dots ,[x\_0,x\_n])$ a birational map.
Is $f$ a Crepant birational map?
| https://mathoverflow.net/users/31524 | Crepant Birational Map | Kontsevich proved in '95 that the hodge polynomial is independent of crepant resolution over a field of characteristic zero. He created the field of motivic integration to prove this fact. Here are the respective hodge polynomials of the two spaces in question where I repress the fact that we are working over the field... | 2 | https://mathoverflow.net/users/5031 | 123711 | 69,376 |
https://mathoverflow.net/questions/123706 | 1 | I heard this name that it can construct GL(2) automorphic forms or L-functions from GL(1)?
I did not find it anywhere.
Or does it have another name which we are familiar with?
| https://mathoverflow.net/users/2666 | Maass-Hecke construction | Automorphic Induction. Look at Bumps book on Automorphic reps in the first chapter.
| 1 | https://mathoverflow.net/users/10400 | 123723 | 69,378 |
https://mathoverflow.net/questions/123684 | 2 | The subject says it all. I would like to know if Proposition 3.1 in
Arthur-Clozel's book on the trace formula holds for local fields of positive
characteristic.
Thanks!
EDIT: Here is Prop 3.1 of Arthur-Clozel: (Notation will be explained after the statement)
### Proposition 3.1
Assume $\phi\in C\_c^\infty(GL\_n... | https://mathoverflow.net/users/36285 | Arthur-Clozel Prop 3.1 for Function Fields? | This can be found in Laumon Cohomology of Drinfeld modules fourth chapter.
| 3 | https://mathoverflow.net/users/10400 | 123724 | 69,379 |
https://mathoverflow.net/questions/123714 | 4 | Let $X$ be a Banach space; $K\subset X$ nonempty, closed and convex; and $f:K\to \mathbb R$ lower semicontinuous, convex functional. Let also $f$ be coercive, i.e., $f(x)\to +\infty$ as $\|x\|\to +\infty$.
Now, it is well-known that:
>
> If $X$ is reflexive, then $f$ has a minimum.
>
>
>
The proof goes esse... | https://mathoverflow.net/users/26039 | existence of a minimum for a convex functional on a non-reflexive space | An equivalent formulation of your question is to prove that
$f:X^\prime \to \mathbb{R}$ be written as the supremum of affine functions, i.e.
$f(x^\prime)=\sup\_{i \in \mathbb{N}} \langle x^\prime,x\_i\rangle+c\_i$.
This is equivalent to weak star lower semicontinuity (cf Ambrosio, Fusco, Pallara Functions of Boun... | 3 | https://mathoverflow.net/users/28090 | 123725 | 69,380 |
https://mathoverflow.net/questions/123728 | 2 | Let $V$, $W$ be varieties (affine or projective) over an algebraically closed field $K$.
Let $p \in V$ and $q \in W$. Is there a description of the local ring of $V\times W$ at $(p,q)$ in terms of the local ring of $V$ at $p$ and the local ring of $W$ at $q$?
| https://mathoverflow.net/users/31532 | Local ring of product of varieties | Let $k$ be a field. The fiber product of two $k$-schemes $X,Y$ (or even locally ringed spaces, see [here](http://arxiv.org/pdf/1103.2139v1.pdf)) has as points triples $(x,y,\mathfrak{p})$, where $\mathfrak{p}$ is a prime ideal in $\kappa(x) \otimes\_k \kappa(y)$, or equivalenty a prime ideal $\mathfrak{q}$ in $\mathcal... | 2 | https://mathoverflow.net/users/2841 | 123733 | 69,384 |
https://mathoverflow.net/questions/123715 | 2 | Let $ f: \mathbb{P}^n \dashrightarrow (\mathbb{P}^1)^n , [x\_0:\dots :x\_n] \mapsto ([x\_0,x\_1],[x\_0,x\_2], \dots ,[x\_0,x\_n])$ a birational map.
In particular, if $X$ is the blow-up of $\mathbb{P}^n$ at $r+n-1$ points and $X'$ is the blow-up of $(\mathbb{P}^1)^n$ at $r$ points.
Consider $\phi : X \dashrightarrow ... | https://mathoverflow.net/users/31524 | Crepant Birational Map on the Blow-up | Suppose for the moment that $n=2$. Then the birational map $f\colon \mathbb{P}^n\dashrightarrow(\mathbb{P}^1)^n$ described in the question, which is $f\colon [x:y:z] \dashrightarrow([x:y],[x:z])$ has two base-points, $[0:0:1]$, $[0:1:0]$ and the inverse is $f^{-1}\colon ([a:b],[c:d]) \dashrightarrow[ac:bc:ad]$ and has ... | 3 | https://mathoverflow.net/users/23758 | 123740 | 69,385 |
https://mathoverflow.net/questions/123734 | 3 | I am interested in the relationship between the Hecke eigenvalue at $p$ and at $p^k$ for $k \geq 2$ in the unramified and ramified situation for modular/Maass forms.
More precisely, I know from a representation theoretic perspective that the Hecke eigenvalue at $p$ determines uniquely the factor of the automorphic re... | https://mathoverflow.net/users/10400 | Hecke eigenvalue at p and at p^k | For each $k$ there is such a polynomial and can be determined recursively from the usual relation $\lambda(m)\lambda(n)=\sum\_{d\mid (m,n)}\chi(d)\lambda(mn/d^2)$, where $\chi$ is the nebentypus (for weight zero Maass newforms). Am I missing something?
Actually, for unramified primes $P\_k(x)=U\_k(x/2)$, where $U\_k$... | 6 | https://mathoverflow.net/users/11919 | 123741 | 69,386 |
https://mathoverflow.net/questions/123736 | 5 | Suppose I have a smooth projective variety $X$ and a semi-orthogonal decomposition of its bounded derived category of coherent sheaves $D^b(X)$. Then I can apply right or left mutations to the full asmissible triangulated subcats that make up the semi-orth decomposition. Algorithmically it is clear to me what happens, ... | https://mathoverflow.net/users/4096 | what are mutations of sheaves all about? | I wouldn't say that in general there is some special geometry behind mutations. This just explains that whenever you have one sod, you have many of them.
However, in some cases there is some meaning. For example, if $D(X) = < A, {}^\perp A >$ then the left mutation functor $L\_A:{}^\perp A \to A^\perp$ is isomorphic... | 6 | https://mathoverflow.net/users/4428 | 123742 | 69,387 |
https://mathoverflow.net/questions/123747 | 1 | It is well-known that log terminal singularities are rational, but log canonical
singularities are not. On the other hand, rational singularities are not necessarily
$\mathbb Q$-Gorenstein, so there are rational singularities that are not even log
canonical (and hence not log terminal). On the third hand (if someone ha... | https://mathoverflow.net/users/31847 | Rational, but not log canonical singularity | Actually, if you look at my [answer](https://mathoverflow.net/questions/123286/rational-and-log-canonical-singularity-that-is-not-log-terminal/123287#123287) to your previous [question](https://mathoverflow.net/questions/123286/rational-and-log-canonical-singularity-that-is-not-log-terminal), the same computation gives... | 3 | https://mathoverflow.net/users/10076 | 123748 | 69,388 |
https://mathoverflow.net/questions/123744 | 8 | I've been tasked with proofreading an Engineering/Mathematics thesis paper. I was always told that numbers under 10 should be spelled out (one, two, three, ...) but I was wondering if this rule holds in math and science fields as well. This paper understandably uses a lot of numbers in it and switching back and forth b... | https://mathoverflow.net/users/31992 | Formal writing: numbers under 10 | Generally speaking you should try to distinguish between an English number and a mathematical number. As in "Over the next five chapters we will prove that 5 is a prime number." Never spell out a number which is a subject of study, like the "5" in that sentence.
However ordinals like in "Section 4" and "Lemma 2" are ... | 26 | https://mathoverflow.net/users/9025 | 123749 | 69,389 |
https://mathoverflow.net/questions/123713 | 8 | I'm quite sure I don't understand very well the links between proof theoretical ordinals of theories, the axioms of transfinite induction and the objects a theory can prove to exist.
For instance I'm considering *Peano Axioms* ($\mathbf{PA}$), of proof theoretic ordinal $\epsilon\_0$, *Primitive Recursive Arithmetic*... | https://mathoverflow.net/users/31984 | ERA, PRA, PA, transfinite induction and equivalences | This entirely depends on what exactly you mean by $TI$, as there are several options (I actually do not understand what the $\{\alpha\in\epsilon\_0\}$ part of the notation is supposed to mean either, but I will assume it just means transfinite induction up to $\epsilon\_0$):
1. $TI\{\alpha\in\epsilon\_0\}$ is the sch... | 9 | https://mathoverflow.net/users/12705 | 123750 | 69,390 |
https://mathoverflow.net/questions/123739 | 16 | Of the mathematical objects that I am familiar with, it is normally the case that the product of 2 objects is an object of the same type and that an equivalence relation on an object induces a quotient object of the same type.
I think I have some understanding as to why the product of 2 fields is not a field, because... | https://mathoverflow.net/users/29763 | Is there a conceptual reason why topological spaces have quotient structures while metric spaces don't? | You *can* define a (pseudo)metric on a quotient of a metric space. Let $X$ be a metric space with metric $d$ and an equivalence relation $\sim$. Say that a chain between two points $x,y\in X$ is a sequence of points $x=a\_0\sim b\_0$, $a\_1\sim b\_1$, $\ldots$ $a\_n\sim b\_n=y$, and define the length of such a chain to... | 32 | https://mathoverflow.net/users/75 | 123753 | 69,391 |
https://mathoverflow.net/questions/123721 | 9 | The symmetric group $S\_n$ acts on $[n]:=\{1,\ldots,n\}$, thereby inducing an action on the set $$\wp\_k(n)=\{\: A\subseteq[n] \::\: \#A=k \:\}$$ of subsets of cardinality $k$, simply by $$(g,A)\mapsto g(A)=\{\: g(a) \::\: a\in A \:\}.$$
This finally yields an action on $V\_k:=\mathbb C[\wp\_k(n)]$.
By Lemma 4 in [this... | https://mathoverflow.net/users/25237 | Permutation character of the symmetric group on subsets of certain size | A classical construction of the Specht modules of $S\_n$ says that $\chi\_{(n-k,k)}$ is present in the character $\pi\_k$ of the action $S\_n$ on $V\_k$, with multiplicity 1. Indeed $M^\lambda:=V\_k$ is the permutation module arising along the way of constructing the Specht module $S^\lambda$ for the partition $\lambda... | 6 | https://mathoverflow.net/users/11100 | 123754 | 69,392 |
https://mathoverflow.net/questions/123104 | 7 | given an isogeny between two abelian varieties $\varphi: A\rightarrow B$ (everything definied over a number field $K$), we can factor $\varphi$ through a multiplication-by-$n$-endomorphism on $A$ and a `remaining' isogeny $\psi:A\rightarrow B$.
>
> What do we know or expect about an upper bound $N$ on the degree of... | https://mathoverflow.net/users/12668 | Results and conjectures on bounds on degrees of isogenies | Mein lieber Stefan,
I think that these sorts of questions are wide-open. As far as I know, even if you fix the number field, and vary over AVs up to some fixed dimension, then it is *not* clear that there exists a bound on the possible degrees of your $\psi$ (which I'm guessing is a *cyclic* isogeny). You in any case... | 7 | https://mathoverflow.net/users/5744 | 123755 | 69,393 |
https://mathoverflow.net/questions/123743 | 2 | Let $R$ be a local ring. By an open idempotent I mean an $R$-module $F$ equipped with a homomorphism $e : F \to R$ such that $e \otimes F = F \otimes e$ is an isomorphism $F \otimes F \cong F$ (this notion in a monoidal category is due to Drinfeld, Boyarchenko). For example, $0 \to R$ and $R \to R$ are open idempotents... | https://mathoverflow.net/users/2841 | Open idempotents in modules over a local ring | The answer is no in general. It is yes if $R$ is $\mathfrak{m}$-adically separated (e.g. noetherian), where $\mathfrak{m}$ is the maximal ideal.
For a counterexample, assume $R$ is a valuation ring (not a field) and $\mathfrak{m}=\mathfrak{m}^2$. Take $F=\mathfrak{m}$, $e=$ the inclusion in $R$. Now $e\otimes \mathf... | 6 | https://mathoverflow.net/users/7666 | 123757 | 69,395 |
https://mathoverflow.net/questions/123731 | 9 | If M is a sufficiently nice model category and D is a small category then there are two natural model structures we can impose on the functor category $Fun(D,M)$ where the weak equivalences are the levelwise weak equivalences.
* The **projective** model structure, which is characterized by the fibrations being the l... | https://mathoverflow.net/users/184 | When is the projective model structure cartesian? When is the internal hom invariant? | I got interested in a similar issue last summer, namely: "When does passage to the diagram category preserve the pushout product axiom?" I ended up finding a paper on arXiv by Sinan Yalin called "[Classifying Spaces and module spaces of algebras over a prop](http://arxiv.org/abs/1207.2964)" which gives conditions on $M... | 6 | https://mathoverflow.net/users/11540 | 123780 | 69,403 |
https://mathoverflow.net/questions/123774 | 2 | Hi.
For a Dirichlet character $\chi$ mod $N$, the (weak) multiplicity one theorem states the following:
Let $f, g \in S\_k^{\text{new}}(\Gamma\_0(N), \chi)$ be such that for every $m \in \mathbb{N}$ s.t. $(m,N)=1$, we have
$$T(m)f = \lambda\_m f,~~ T(m)g = \lambda\_m g$$
i.e. $f,g$ are simultaneous eigenforms with ... | https://mathoverflow.net/users/20431 | Counterexamples for (weak) multiplicity one | Very concretely: if $f\in S\_k^{\text{new}}(\Gamma\_0(N), \chi)$, and $d>1$ is an integer, then $f(z)$ and $f(dz)$ are elements of $S\_k(\Gamma\_0(dN), \chi)$ with the same Hecke eigenvalues at $m$ coprime with $dN$, yet they are not multiples of each other. Of course this is the same what Marc was saying, but in more ... | 5 | https://mathoverflow.net/users/11919 | 123781 | 69,404 |
https://mathoverflow.net/questions/123765 | 8 | Given a $2n\times 2n$ real, symmetric, Hamiltonian matrix $W$ (anticommutes with the symplectic metric), is there an orthogonal, symplectic matrix $R$ such that $R^\top WR$ is block-diagonal?
Being symmetric, for sure one can diagonalize $W$ by an orthogonal matrix. However, we only want block-diagonalization. This ... | https://mathoverflow.net/users/21919 | Symplectic block-diagonalization of a real symmetric Hamiltonian matrix | Yes, you can always do this. The right way to understand this is to look at the complex symmetric matrix $X = x + i y$. You want to act on this by a complex matrix $C = c + is$ where $C$ is unitary, by the rule, $X\mapsto C^{T}XC$, and you want to know whether you can arrange that $C^{T}XC$ is a real, diagonal matrix. ... | 6 | https://mathoverflow.net/users/13972 | 123784 | 69,406 |
https://mathoverflow.net/questions/123746 | 2 | I am trying to show the equality of two complex space germs related to double points of singular map germs $f:(\mathbb C^n,0)\to (\mathbb C^p,0)$. This two spaces are given by two (possibly non reduced) ideals, let's say $I,J$, in $\mathcal O\_{2n+s}$. I can show $J\subseteq I$ and, if we call $K$ the ideal wich define... | https://mathoverflow.net/users/31991 | Product and quotient of ideals | This is a partial answer. An ideal $K$ is a cancellation ideal if for any ideals $I$ and $J$, $IK=JK$ implies that $I=J$. Your question (1) says that $K$ is a cancellation ideal.
It is known that a principal ideal $K=(a)$ in a commutative ring $R$ is a cancellation ideal if and only if $a$ is not a zero divisor.
Als... | 2 | https://mathoverflow.net/users/30062 | 123785 | 69,407 |
https://mathoverflow.net/questions/123763 | 4 | I am currently trying to understand intermediate extensions of perverse sheaves, specifically the proof of Gabber's purity theorem, which states that the intermediate extension of a pure perverse sheaf is pure.
As part of the proof, in §5 of "Faisceuax pervers" by Bernstein, Beilinson & Deligne, I've come across the ... | https://mathoverflow.net/users/13647 | Pullbacks of intermediate/middle extensions and Gabber's purity theorem | Note : [BBD]=Astérisque 100 by Beilinson-Bernstein-Deligne
The answer to the first question is basically Deligne's "generic base change", by which I mean theorem 1.9 of SGA 4 1/2 [Th. finitude].
I suppose that you are in part (b) of the second proof of corollary 5.3.2 in [BBD]. So you're assuming that X is affine, ... | 9 | https://mathoverflow.net/users/31960 | 123787 | 69,408 |
https://mathoverflow.net/questions/123752 | 5 | Let $\Gamma$ be a discrete cocompact subgroup of the euclidean motion group
$$
G={\mathbb R}^d\rtimes O(d).
$$
Let $\phi:G\to O(d)$ the projection homomorphism.
Is it true that $\phi(\Gamma)$ is finite?
| https://mathoverflow.net/users/nan | Lattice in motion group | The answer is yes. This is a theorem of Bieberbach (see Corollary (8.26) of the book "Discrete Subgroups of Lie Groups" by M.S.Raghunathan (Springer Ergebnisse Tract).
| 5 | https://mathoverflow.net/users/23291 | 123789 | 69,409 |
https://mathoverflow.net/questions/123772 | 0 | Let G be a Fuchsian Schottky group defined by a **possibly infinite** set of disjoint halfplanes {C\_i}\_i. Let F be the fundamental domain obtained by intersecting the complements of the C\_i's. If H i a subgroup of G then:
Can a fundamental domain for H be found in terms of the domain F?
Is H of schottky type?
Do the... | https://mathoverflow.net/users/31997 | Fundamental domain for subgroup of fuchsian Schottky group. | Here is a concrete construction of a fundamental domain for $H$ that works regardless of normality and finite generation. The translates of the fundamental domain $F$ by the action of $G$ form a tiling of $\mathbb{H}^2$. Let $T$ denote the dual tree of this tiling, with one vertex for each translate of $F$, and with tw... | 1 | https://mathoverflow.net/users/20787 | 123790 | 69,410 |
https://mathoverflow.net/questions/123771 | 10 | I am trying to get a sense of how often the commutator subgroup $[G,G]$ of a (Gromov) hyperbolic group $G$ is infinitely generated.
**Remarks:**
1. $[G,G]$ is infinitely generated if is $G$ is free noncyclic, and of course $[G,G]$ is finitely generated when $G$ has finite abelianization.
2. There are examples when... | https://mathoverflow.net/users/1573 | Hyperbolic groups with infinitely generated commutator subgroups | No, consider a hyperbolic 3-manifold $M$ with $b\_1(M)=2$, and all faces of the Thurston norm fibered. Then there are only finitely many surjections to $\mathbb{Z}$ with infinitely generated kernel, corresponding to the (projective classes of the) vertices of the Thurston norm ball.
For an explicit example, consider... | 11 | https://mathoverflow.net/users/1345 | 123795 | 69,412 |
https://mathoverflow.net/questions/123762 | 3 | It looks very easy but I must admit I am struggling with this problem.
Okay, let $M$ be a von Neumann algebra acting on a Hilbert space $H$ and let $K$ be another Hilbert space. Suppose $h\colon M\to B(K)$ is a \*-homomorphism which is continuous in the strong operator topology. Is the image $N=h(M)$ a von Neumann al... | https://mathoverflow.net/users/31994 | Is the image of a vNA under a SOT-continuous morphism still a vNA? | If you consider $M / \mathrm{ker}h$ instead of $M$ ($\mathrm{ker}h$ is SOT-closed, so you end up with a von Neumann algebra), you may assume that $h$ is injective, hence isometric, since it as a $\ast$-homomorphism between $C^{\ast}$-algebras. We now wish to show that its image is SOT-closed. This is probabliy not very... | 4 | https://mathoverflow.net/users/24953 | 123799 | 69,415 |
https://mathoverflow.net/questions/123779 | 2 | I have been trying to learn about snappy's method for encoding once-punctured torus bundles (<http://www.math.uic.edu/t3m/SnapPy/manifold.html#snappy.Manifold>). As you can see from the link, they are imported via the Manifold() function.
I have looked through documentation on the snappy website and much of the twist... | https://mathoverflow.net/users/27453 | Once punctured torus bundles in snappy/twister | I believe that your question is answered by section (3e) at [this](http://www.geom.uiuc.edu/software/snappea/) page at the Geometry Center. Note that b+ and bo are equivalent as are b- and bn (o and n stand for orientation preserving and reversing, respectively). You can check a few examples using the is\_isometric met... | 3 | https://mathoverflow.net/users/1650 | 123815 | 69,424 |
https://mathoverflow.net/questions/123813 | 3 | Let $E$ be an elliptic curve over $\mathbb{Q}$. It is known from Gross and Zagier that if $\textrm{rank}\_{\textrm{an}}(E) \leq 1$, then
$$\textrm{rank}(E) \geq \textrm{rank}\_{\textrm{an}}(E).$$
Instead, suppose I know that $\textrm{rank}(E) \leq 1$, are there any (unconditional) inequalities that relate rank and... | https://mathoverflow.net/users/31105 | Inequality relating rank and analytic rank | This is not really the right question. I think it's easier to just list the two things that are known unconditionally and then you can figure out if this answers your question.
1) If analytic rank is zero then algebraic rank is zero.
2) If analytic rank is 1 then algebraic rank is 1.
Proof: see quid's comment.
... | 6 | https://mathoverflow.net/users/30035 | 123820 | 69,425 |
https://mathoverflow.net/questions/123406 | 12 | **Description**
I my work, I met fixed-point functional equations of a very specific form. Since I am not expert in the domain of functional equations, I have not pushed the study of these very far. Here is a description of the objects.
Let $X := \lbrace x\_1, \dots, x\_k \rbrace$ be an alphabet of mutually commuti... | https://mathoverflow.net/users/15551 | Series defined by a fixed-point functional equation | You can get recurrence relations for the coefficients by considering, for each monomial, which lower degree monomials will contribute to its coefficient.
For example, suppose we want to find the coefficients of $S$ without iteration. Iteration here is particularly bad: the degree of $S\_{k+1}$ will be $3^k$, so it w... | 7 | https://mathoverflow.net/users/18086 | 123827 | 69,428 |
https://mathoverflow.net/questions/123586 | 2 | I am curious about weak solutions of the parabolic problem
$ u\_t - \Delta u + a(x,t) \cdot \nabla u = f(x,t)$ in $ \Omega \times (0,\infty)$ with $ u=0$ on $ \partial \Omega$. Here $a(x,t)$ is some divergence free vector field with minimal smoothness properties. We assume that $ u(x,0)$ is smooth.
My question is: ... | https://mathoverflow.net/users/29444 | Regularity of parabolic equation; divergence free drift | The $L^\infty$ a priori estimate for u does not depend on $a$ at all provided that the initial data is either bounded (you would apply maximum principle) or $L^1$ (you would apply Nash bound on the heat kernel).
The only subtlety (if any) would be at the definition of weak solution. If $a$ is too rough, weak solution... | 6 | https://mathoverflow.net/users/26672 | 123829 | 69,429 |
https://mathoverflow.net/questions/123826 | 6 | ### The bare question:
Let $\mathcal{C}$ be an $\infty$-topos, and let $\tau\_{\leq 0}\mathcal{C}$ be the subcategory of 0-truncated objects (which is the nerve of an ordinary Grothendieck topos: see HTT 6.4.1.3).
>
> Does the inclusion $\tau\_{\leq 0}\mathcal{C} \hookrightarrow \mathcal{C}$ preserve filtered (or... | https://mathoverflow.net/users/1797 | FIltered colimits of truncated objects in $\infty$-topoi | I believe the answer is YES and, more generally, that $\tau\_{\leq n}\mathcal{C}\subset\mathcal{C}$ preserves filtered colimits for any $\infty$-topos $\mathcal{C}$. For the $\infty$-topos of $\infty$-groupoids this is well-known. This implies the result in any presheaf $\infty$-topos since colimits and truncations are... | 10 | https://mathoverflow.net/users/20233 | 123831 | 69,430 |
https://mathoverflow.net/questions/123845 | 4 | How is the algebraic topologist James R. Munkres' last name "Munkres" pronounced? Is it "Munkrees" or "Munkers" or something else entirely? There is some disagreement among my acquaintances.
Apologies for the non-mathematical question. But I think that knowing the correct pronunciation is important, given Munkres has... | https://mathoverflow.net/users/31084 | How is Munkres pronounced? | Something like Munkrehs -- the e on the end is pronounced, but as an eh rather than ee I believe.
| 8 | https://mathoverflow.net/users/17969 | 123847 | 69,436 |
https://mathoverflow.net/questions/123842 | 2 | This is a cross-posting of a MSE [question](https://math.stackexchange.com/questions/311376/orthogonality-between-three-vectors-whose-coordinates-are-nondecreasing) (which did not receive any feedback there so far).
Say that a vector $x=(x\_1,x\_2, \ldots ,x\_n)\in {\mathbb R}^n$ is **nondecreasing** if $x\_1 \leq x\... | https://mathoverflow.net/users/10341 | Orthogonality between vectors whose components increase | Consider the nondecreasing vector $v=(1,1,\ldots,1)$. It's not too hard to show that two nondecreasing nonzero vectors orthogonal to $v$ must have a positive inner product. Here is a proof.
Let $a=(a\_1,\ldots, a\_n)$ and $b=(b\_1,\ldots, b\_n)$ be orthogonal to $v$. Suppose that $a\_j$ is the last negative entry of ... | 4 | https://mathoverflow.net/users/3075 | 123850 | 69,438 |
https://mathoverflow.net/questions/123836 | 1 | Let a finite group $G$ acts on an orientable manifold $X$ freely. Denote $\pi:X\rightarrow Y=X/G$ be the quotient map. This covering map defines two maps between cohomology groups $\pi^\*=H^\ast(\pi):H^\*(Y,\mathbb{Z})\rightarrow H^\*(X,\mathbb{Z})$ and $\pi\_!:H^\*(X,\mathbb{Z})\rightarrow H^\*(Y,\mathbb{Z})$. The lat... | https://mathoverflow.net/users/32008 | A question on composites of pushforward and pullback | The first identity $\pi\_! \circ \pi^\* = \vert G \vert \cdot \mathrm{Id}$ holds, and follows from knowing that $\pi\_!$ is a $H^\*(Y)$-module map via $\pi^\*$, so
$$\pi\_!( \pi^\*(x)) = \pi\_!(1)\cdot x$$
and $\pi\_!(1) = \vert G \vert$ as may be seen from the $G$-cover over a point.
The second proposed identity $\p... | 3 | https://mathoverflow.net/users/318 | 123852 | 69,439 |
https://mathoverflow.net/questions/123860 | 2 | Any other definition for algebraic number than the root of algebraic equation?
| https://mathoverflow.net/users/14024 | Any other definition for algebraic number than the root of algebraic equation? | In model theory, an object is *algebraic* in a structure $M$ if it satisfies a property that only finitely many other objects in $M$ exhibit, where by "property" here we mean one that is expressible in the first-order language of the structure. This is a weakening of definability, since $a$ is *definable* in $M$ if it ... | 12 | https://mathoverflow.net/users/1946 | 123861 | 69,443 |
https://mathoverflow.net/questions/123864 | 4 | In "Many homotopy categories are homotopy categories" (M. Cole / Topology and its Applications 153 (2006) 1084–1099), Cole generalizes the construction of the model category of Ström to any bicomplete category which is enriched, tensored and cotensored over compactly generated spaces and which satisfies an additional c... | https://mathoverflow.net/users/24563 | About the Cole-Ström model category structure with a locally presentable category | Actually, there is an error in the Cole paper you're talking about, as recently discovered by Richard Williamson and very recently corrected by Tobi Barthel and Emily Riehl in ["On the Construction of Functorial Factorizations for model categories"](http://arxiv.org/abs/1204.5427). The introduction to this paper is ver... | 4 | https://mathoverflow.net/users/11540 | 123869 | 69,446 |
https://mathoverflow.net/questions/123866 | 4 | What are the known sufficient conditions,analagous to the planar curvature condition, in terms of functions of theta and phi, on the support function h(theta,phi) of a surface in 3D which imply it is the boundary of a convex body?
[The origin of this question is an elementary approach to exploring 3D (convex) sets of... | https://mathoverflow.net/users/24669 | Support Functions Of 3D Convex Bodies In Spherical Polar Coordinates | For me, the simplest way to figure out whether a function, which is often defined as a function of the unit sphere, is a support function or not is to extend it to all of $\mathbb{R}^n$ as a function homogeneous of degree $1$. Then the function is a support function if and only if it is convex.
You can figure out whe... | 4 | https://mathoverflow.net/users/613 | 123872 | 69,448 |
https://mathoverflow.net/questions/123814 | 1 | Let con(ZFC) be a sentence in ZFC asserting that ZFC has an omega-model M. Let $A\_{M}$ be an wff over M. Let S be the theory ZFC+con(ZFC). Is the reflection for S: $Bew\_{S}(A\_{M}) \implies A\_{M}$ is satisfied?
I asking also for an explanation of the paradox in the link
<http://cs.nyu.edu/pipermail/fom/2007-Octob... | https://mathoverflow.net/users/29570 | Reflection principles | I suppose the "paradox" you're asking about is the passage marked with >> at the link you gave, but with "$\omega$-model" in place of "model" and with "has an $\omega$-model" in place of "is consistent". But then there is no longer any justification for the statement (on lines 9 & 10) that there's a proof in ZFC of the... | 4 | https://mathoverflow.net/users/6794 | 123877 | 69,451 |
https://mathoverflow.net/questions/123879 | 16 | Given complete, locally convex Hausdorff vector spaces $E$ and $F$, let
$$ E \otimes\_i F, \qquad E \otimes\_\pi F$$ denote the (completed) inductive and projective tensor products respectively. The inductive (resp. projective) tensor product has the universal property for separately (resp. jointly) continuous bilinear... | https://mathoverflow.net/users/22470 | Inductive tensor product and smooth functions | If you mean by LF-space a *strict* inductive limit of Frechet spaces (as it was done by Dieudonne and Schwartz) I think the answer is yes.
Here is what I believe could be made a proof: Since the inductive tensor product
respects inductive limits you have $C^\infty(M) \otimes\_i F = \lim C^\infty(M) \otimes\_i F\_n$ i... | 14 | https://mathoverflow.net/users/21051 | 123885 | 69,454 |
https://mathoverflow.net/questions/123759 | 8 | Let $T^n$ be the $n$-dimensional torus and $g$ be a Riemannian metric on $T^n$. Let $\tilde g$ be the induced metric on the universal covering; using suitable coordinates, $\tilde g$ is therefore a $\mathbb{Z}^n$-periodic metric on $\mathbb{R}^n$ (I shall conflate the lattice $\mathbb{Z}^n$ with the fundamental group o... | https://mathoverflow.net/users/4961 | Is displacement controled by stable norm? | If you allow an additive term $C(n)diam(g)$ rather than $2diam(g)$, then yes, the statement is true. In the paper D.Burago, "Periodic metrics", Adv. Soviet Math. 9, (1992), 205-210, he proves that for every periodic metric on $\mathbb R^n$ there is a constant $C$ such that
$$
| d(x,y)-\|x-y\|\_{st} | \le C
$$
for all ... | 8 | https://mathoverflow.net/users/4354 | 123892 | 69,459 |
https://mathoverflow.net/questions/123875 | 1 | Up to what order do the moments of the [Kolmogorov distribution](http://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Smirnov_test#Kolmogorov_distribution) exist? References would be appreciated.
| https://mathoverflow.net/users/32017 | Moments of the Kolmogorov distribution | The Kolmogorov distribution is defined by the distribution of the random variable $K:=\sup\_{0\leqslant t\leqslant 1}|B(t)|$, where $B(t)$ is the Brownian Bridge.
The problem of existence of moments for $K$ is actually the same as the study of moments of $K':=\sup\_{0\leqslant t\leqslant 1}|W(t)|$, where $W(t)$ is a... | 2 | https://mathoverflow.net/users/17118 | 123895 | 69,460 |
https://mathoverflow.net/questions/123828 | 0 | Let $X$ is a linearly lindelof subspace of $Z$ and $b$ is not $\omega$-separated from $X$, i.e., for any closed $G\_\delta$ set $P$ of $Z$ which contains $b$, $P\cap X \not=\emptyset$. If $\tau < \aleph\_\omega$, how to show that $b$ is not $\tau$-separated from $X$, i.e., for any closed $G\_\tau$ set $P$ of $Z$ which ... | https://mathoverflow.net/users/18465 | A question on linearly lindelof space | I think you need some separation axioms (see my comment above). Here is a proof when $Z$ is regular. I hope I did not overlook something.
Suppose that there is some closed $G\_\tau$ $P\ni b$ disjoint from $X$, so $P=\cap\_{\alpha<\tau}U\_\alpha$ for $U\_\alpha$ open in $Z$. By regularity of $Z$, one can choose induct... | 1 | https://mathoverflow.net/users/29491 | 123898 | 69,463 |
https://mathoverflow.net/questions/123893 | 8 | This is a very naive question, but I'm trying to understand the difference between the various categories when it comes to embedding surfaces in 4-dimensional manifolds. The situation I'd really like to understand is when we have a surface neatly and properly embedded in the 4-ball $D^4$, so that its boundary lies in $... | https://mathoverflow.net/users/22424 | Differences between various categories of surface embeddings in 4-space | Regarding (1), yes there's a unique (up to smooth isotopy) smoothing of a PL locally-flat embedding. You can similarly ensure that if your original surfaces are close in some PL compact-open sense, that their smoothings can be made to be close in the smooth C^1 compact-open sense.
Regarding (2), yes, there are knots... | 4 | https://mathoverflow.net/users/1465 | 123899 | 69,464 |
https://mathoverflow.net/questions/123884 | 5 | It is [well known](https://en.wikipedia.org/w/index.php?title=Vietoris%E2%80%93Rips_complex&oldid=491420022#Relation_to_.C4.8Cech_complex) that the nerve (or Čech complex) of a covering consisting of metric balls with a common fixed radius is nicely approximated by the Vietoris-Rips complex. Being a flag complex on its... | https://mathoverflow.net/users/8415 | Are there "geometrically nice" sets from which to construct coverings that admit "Vietoris-Rips like" approximations to the nerve? | Since the question is rather broad, the answer got unwieldy. Sorry, I hope some of this is relevant to the sort of questions you are asking.
**Complexity of Approximating the Čech Complex**
The computational tractability of the Vietoris-Rips complex is an engaging - but largely false - contemporary myth. While it i... | 5 | https://mathoverflow.net/users/18263 | 123908 | 69,468 |
https://mathoverflow.net/questions/106624 | 3 | For given $n,\ell\in\mathbb N\_0$, I am interested in studying the following recursion relation for some $\mu\in\mathbb R$:
$$\sqrt{-1} \tfrac{j(\ell-j+1)(n-j+1)(n+j+1)}{2(2j-1)(2j+1)} a\_{j-1} - \tfrac {j(j+1)}2 a\_j
- \sqrt{-1} \tfrac{(j+1)(\ell+j+2)}2 a\_{j+1} = \mu a\_j,$$
for $j\ge0$, assuming $a\_{-1}=0$.
The ... | https://mathoverflow.net/users/26295 | Three term recurrence relation. | Racah are involved
| -2 | https://mathoverflow.net/users/26295 | 123919 | 69,472 |
https://mathoverflow.net/questions/123894 | 20 | The isometry group of a metric space is a topological group (with the compact open topology). The isometry group of a Riemann Manifold is a Liegroup. (Thm. of Steenrod-Myers)
So, is every topological group isomorphic (in the category of topological groups) to the isometry group of a metric space?
And what about the... | https://mathoverflow.net/users/32022 | Is every topological (resp. Lie-) group the isometrygroup of a metric space (resp. Riemannian manifold)? | This seems to be completely answered in the topological category by Thm 1.4 of <http://arxiv.org/pdf/1202.3368v3.pdf>.
**Edit**
If X is a topological space which is not Dieudonne complete (meaning its topology cannot be given by a complete uniformity), then it seems theorem 7.24 of <http://books.google.com/books?i... | 8 | https://mathoverflow.net/users/15934 | 123939 | 69,483 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.