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https://mathoverflow.net/questions/58963 | 4 | Let $f:X\to Y$ be a universal homeomorphism of regular (excellent finite-dimensional) schemes, $Z\subset Y$ be a regular subscheme. Is $f^{-1}(Z)$ necessarily regular?
| https://mathoverflow.net/users/2191 | Is the pre-image of a regular subscheme with respect to a universal homeomorphism of regular schemes regular? | Well, $f^{-1}Z$ could easily be non-reduced (for example, take the relative Frobenius morphism $\mathbb A^1\_k \to \mathbb A^1\_k$, defined by the embedding $k[y] = k[x^p] \subseteq k[x]$, where $k$ is a field of characteristic $p > 0$, and let $Z \subseteq \mathbb A^1$ be defined by $y = 0$), so I would guess that the... | 8 | https://mathoverflow.net/users/4790 | 58975 | 36,657 |
https://mathoverflow.net/questions/58961 | 2 | Let $\alpha$ be an irrational number, $n\geq 1$ and
$ X\_n=\lbrace (x,y) \in {\mathbb Z}^2 | |y| \leq n, \ x+y\alpha >0 \rbrace$
Now let $(x\_n,y\_n)$ minimize the quantity $x+y\alpha$ on $X\_n$. This
pair is unique because $\alpha$ is irrational.
Now the problem is to deduce all the "location" of $\alpha$ from
th... | https://mathoverflow.net/users/2389 | Best rational approximation in a special sense | The dots $\{ x+ y \alpha \colon |y|\leq n \}$ move continuously with $\alpha$, so we should consider when $x\_n+y\_n \alpha$ can *stop* being the smallest positive dot. First, note that $y\_n = 0$ is impossible. We need $x\_n+y\_n \alpha > 0$, so that $\alpha > - x\_n/y\_n$ if $y > 0$ and $\alpha < -x\_n /y\_n$ if $y\_... | 3 | https://mathoverflow.net/users/935 | 58977 | 36,658 |
https://mathoverflow.net/questions/58839 | 12 | Consider *injective* homolomorphic functions $f:\mathbb D\to \mathbb C$ on the unit disk $|z|\leq 1$, normalized by the conditions $f(0)=0$ and $f'(0)=1$.
Thus for $|z|\leq 1$ we have $ f(z)=\sum\_{k=0}^{\infty} a\_k z^k $ with $a\_0=0$ and $a\_1=1$.
Ludwig Bieberbach conjectured in 1916 and Louis de Branges proved ... | https://mathoverflow.net/users/450 | Does de Branges's theorem extend to several variables? | Such a result would have to be quite different in several variables, because the holomorphic automorphism group of $\mathbb{C}^n$ is very big when $n \geq 2$. For injectivity, we need to look at equidimensional mappings $F$ from the domain (whatever it may be), and into $\mathbb{C}^n$ say. For simplicity, choose $n = 2... | 14 | https://mathoverflow.net/users/13218 | 58981 | 36,659 |
https://mathoverflow.net/questions/58979 | 1 | One can prove Elimination of Quantifiers of ACVF finding an extension of any partial embedding of a model $K$ into a $|K|^+$ Saturated one using the language $\mathcal{L} = ( 0,1,+,\*, U, \mid )$. In this Language $U$ is the unary predicate standing for the Valuation Ring of the model, and $\mid $ is a binary relation ... | https://mathoverflow.net/users/13782 | Completeness of Algebraically Closed Valued Fields(ACVF) Theory | Just like with algebraically closed fields, for completeness you need to specify the characteristic of the field and the characteristic of the residue field.
There is a general trick if you have a theory $T$ with quantifier elimination and a structure $A$ that is embedded in every model of $T$, then $T$ is complete.... | 7 | https://mathoverflow.net/users/5849 | 58985 | 36,662 |
https://mathoverflow.net/questions/58972 | 0 | Let $G=(V,E)$ be an undirected graph and $p \colon E \mapsto (0,1]$ defines weights of its edges.
Let's fix two connected vertices $v\_1, v\_2 \in V$.
Random graph $G'=(V,E')$ is obtained from $G$ by removing each edge $e \in E$ with probability $1-p(e)$.
What is the probability that connectivity between $v\_1$ a... | https://mathoverflow.net/users/13778 | Probability of preserving connectivity between pair of vertices in weighted graph | Let L be the set of all simple paths in G from $v\_1$ to $v\_2$. By inclusion-exclusion, the probability that $v\_1$ and $v\_2$ are connected is
$\sum\_{A \subseteq L} (-1)^{|A|-1} P(\cup A \subseteq E')$
where for any set S of edges,
$P(S \subseteq E') = \prod\_{e \in S} p(e)$.
Although explicit computation won... | 1 | https://mathoverflow.net/users/13650 | 58989 | 36,663 |
https://mathoverflow.net/questions/58994 | 9 | The Parker loop has order $2^{13}$, and reducing it modulo its 'center' yields -- not just a $12$-dimensional $\mathbb{Z} /(2)$-vector space, but one which can be naturally identified with the subspace of $(\mathbb{Z} /(2))^{24}$ which is the extended $24$-bit Golay code.
Analogous to the extended $24$-bit Golay code... | https://mathoverflow.net/users/12610 | Parker-like loop of order 2187? | Parker's loop can be viewed as a special case of Griess's construction of code loops for doubly even binary codes ("Code loops", J. Algebra 100 (1986), 224-234, <http://deepblue.lib.umich.edu/bitstream/2027.42/26195/1/0000274.pdf>). Over $F\_p$ with $p$ odd, things are more subtle, but there are some constructions.
F... | 7 | https://mathoverflow.net/users/4720 | 59011 | 36,675 |
https://mathoverflow.net/questions/59007 | 4 | I'm wondering if analytic number theorists can prove results which have the following flavor:
So let $N$ be a large positive integer.
Q: Can you always find a prime number $p$ in the interval $(N, 3N/2)$ for which
there exists an odd prime $q$ which divides $p-N$ with multiplicity exactly one?
If such a result ca... | https://mathoverflow.net/users/11765 | Multiplicity one prime in the factorisation of p-N | The number of primes in $[N,3N/2]$ grows as $\frac{N}{\log N}$, while the number of [powerful numbers](http://en.wikipedia.org/wiki/Powerful_number) in $[1,N/2]$ grows as $\sqrt{N}$, so pretty quickly you will find primes $p\in [N,3N/2]$ so that $p-N$ is not powerful, i.e. has a prime divisor which has multiplicity 1.
... | 6 | https://mathoverflow.net/users/2384 | 59019 | 36,679 |
https://mathoverflow.net/questions/58980 | 1 | Hello,
Given positive integers $k$ and $n$. Are there upper bounds on coefficients $A$ and $B$ such that they depends only on $k$ (eg., $2 k^k$) and for all non-negative integer sequences $(a\_i)\_{1}^n, (b\_i)\_{1}^n$ and non-negative increasing real sequence $(p\_i)\_{1}^n$, the following inequality holds?
$$
\su... | https://mathoverflow.net/users/12981 | Coefficient bounds of an inequality | This is true.
I prefer to denote $q\_i=p\_i^{-1}$, $\alpha\_i=a\_ip\_i$, $\beta\_i=b\_ip\_i$, $A\_i=\sum\_{j=1}^i\alpha\_j$, $B\_i=\sum\_{j=1}^i\beta\_j$. Now we have to check that
$$
\sum\_i q\_i\beta\_iA\_i^k\le C\sum\_i q\_i\alpha\_iA\_i^k+C\sum\_i q\_i\beta\_iB\_i^k
$$
This is linear in $q\_i$, so we just need t... | 7 | https://mathoverflow.net/users/1131 | 59028 | 36,683 |
https://mathoverflow.net/questions/59003 | 6 | While looking at a preprint I've just bumped into a question about the longest element $w\_0$ of a Weyl group $W$ (say irreducible of a Lie type $A$ - $G$ and of rank $n>1$, to simplify). Suppose this element is written as a not necessarily reduced product of the form $w s\_n w'$, where both $w$ and $w'$ lie in the pro... | https://mathoverflow.net/users/4231 | Occurrences of a simple reflection in the longest element of a Weyl group? | First, I'll show that you can assume that $ws\_nw'$ is a reduced product.
To see this, first note that if we multiply $s\_n$ by a reduced expression for $w'$, the result will still be reduced, and this can then be extended to a reduced expression for the longest element by multiplying some reduced expression $s\_{i\_... | 4 | https://mathoverflow.net/users/7434 | 59037 | 36,689 |
https://mathoverflow.net/questions/58986 | 2 | Let $f:X\longrightarrow S$ be a flat projective morphism of regular integral noetherian schemes such that that the generic fibre $X\_\eta\longrightarrow K(S)$ is a smooth projective connected curve over $K(S)$.
*Assumptions.* For simplicity, let us assume that $S$ is one-dimensional, i.e., $S$ is a connected Dedekind... | https://mathoverflow.net/users/4481 | Sections of morphisms of schemes up to a finite morphism | Consider the Stein factorization $X\to S\_1\to S$ of $f$ (thus $S\_1=\mathrm{Spec}(f\_\*\mathcal{O}\_X)$). Then $S\_1\to S$ is finite, and since $X$ is normal, I think $S\_1$ must be the normalization of $S$ in $K(X)$. Then $f$ factors through $S'$ if and only if $S\_1\to S$ does, which in turn is equivalent to Karl's ... | 3 | https://mathoverflow.net/users/7666 | 59043 | 36,694 |
https://mathoverflow.net/questions/39366 | 7 | Let $\mathfrak g$ be a Lie algebra (if it matters, right now I only care about finite-dimensional Lie algebras in characteristic $0$, although I'm never opposed to hearing about more general cases). Recall that it determines a differential graded algebra ("the complex that computes Lie algebra cohomology"), with $k$th ... | https://mathoverflow.net/users/78 | Is a quasi-iso in Lie algebra cohomology necessarily an iso? | Here are some comments to your answer that I hope will be helpful (it's still sufficiently confusing that I might make some mistakes). Given an $L\_{\infty}$ algebra, we can define the "Koszul dual" Chevalley $\it{chains}$ $C\_\*(L)$(for several reasons it's more natural to think of the Chevalley chains rather than coc... | 5 | https://mathoverflow.net/users/6986 | 59047 | 36,697 |
https://mathoverflow.net/questions/59054 | 10 | I just started to read Shimura - Automorphic forms and number theory (Lecture notes in mathematics, 54). On page 20 or so, he mentions that every projective variety which is an algebraic group, is necessary abelian.
Why?
| https://mathoverflow.net/users/10400 | Why can projective varieties just have abelian group operations? | I borrow this proof from [Birkenhake-Lange, Complex Abelian Varieties, Lemma 1.1.1].
Let $X$ be a projective variety having a group structure. I assume that we are working over $\mathbb{C}$.
Consider the commutator map $\Phi(x,y)=xyx^{-1}y^{-1}$, and let $U$ be a coordinate neighborhood of $1 \in X$. By the continu... | 9 | https://mathoverflow.net/users/7460 | 59057 | 36,702 |
https://mathoverflow.net/questions/59053 | 2 | This is related to [this MO question](https://mathoverflow.net/questions/59026/meaning-of-cauchy-integral-theorem-the-cohomology-viewpoint), I don't know if it's really "research-level". As in that question, let $U$ be a domain of the complex plane $\mathbb{C}$, i.e. an open connected subset. Let
$$ \kappa\_a:=\frac{1}... | https://mathoverflow.net/users/4721 | Homology of a region of the plane | Integrating the form $\kappa\_a$ computes the winding number of $\gamma$ about $a$. This is a special case of a linking pairing.
Alexander Duality is the statement that if $X\subset S^n $ is compact and is a deformation retract of some open $U$ and $Y=S^n-X$ then $H\_i(Y)$ is isomorphic to $H^{n-i-1}(X)$. The pairing... | 4 | https://mathoverflow.net/users/4304 | 59060 | 36,704 |
https://mathoverflow.net/questions/59065 | 0 | (Related [question](https://mathoverflow.net/questions/58320/intersections-of-conjugates-of-lie-subgroups))
Let $I$ be the group of orientation preserving symmetries of a regular [icosahedron](http://en.wikipedia.org/wiki/Icosahedral_symmetry). This is a $60$ element subgroup of $SO(3)$, isomorphic with the alternati... | https://mathoverflow.net/users/8103 | Intersections of conjugates of the icosahedral group in SO(3) | Rotate a quarter turn around the axis passing through the midpoints of two antipodal edges.
That gives a different copy of the original icosahedron. A half turn preserves both
icosahedra. So the statement is wrong.
| 5 | https://mathoverflow.net/users/4794 | 59068 | 36,709 |
https://mathoverflow.net/questions/59022 | 16 | In this question, all representations are finite-dimensional representations over $\mathbb{C}$.
Fix some $n \geq 3$. Assume that $V$ is a representation of $\text{SL}(n,\mathbb{Z})$. Also, assume that $W$ is a subrepresentation of $V$ and set $V' = V/W$, so we have a short exact sequence
$$0 \longrightarrow W \long... | https://mathoverflow.net/users/13790 | Question about the representation theory of SL(n,Z) | Consider the surjective map of $SL(n,\Bbb Z)$-modules $Hom\_{\Bbb C}(V',V)\to Hom\_{\Bbb C}(V',V')$.
Tim tells us that the identity map from $V'$ to $V'$ lifts to an $f:V'\to V$ which is invariant under a finite index subgroup $\Gamma $ of $SL(n,\Bbb Z)$. Then by averageing one can make it invariant under $SL(n,\Bbb Z)... | 10 | https://mathoverflow.net/users/4794 | 59072 | 36,710 |
https://mathoverflow.net/questions/58834 | 27 | Let $f\colon X \to S$ be a smooth proper morphism of schemes. If $S$ is of characteristic zero (i.e., $S$ is a $\mathbb Q$-scheme), then Deligne has shown:
1. $R^af\_\*\Omega^b\_{X/S}$ is locally free for all $a,b \geq 0$.
2. The Hodge-De Rham spectral sequence
$E^{ab}\_1 = R^af\_\*\Omega^b(X/S) \Rightarrow H\_{\rm D... | https://mathoverflow.net/users/13302 | Degeneration of the Hodge spectral sequence | [I misunderstood Torsten Ekedahl's earlier comment. I'm reverting the lemma
to its original form which was a bit stronger.]
Since the question seemed to resonate with me, I've been thinking about this on and off (but mostly off) for a couple of days now. Here's what I've come up with.
What seems to make the example... | 11 | https://mathoverflow.net/users/4144 | 59074 | 36,711 |
https://mathoverflow.net/questions/59077 | 7 | I recently came across some polynomials with some remarkable properties.
A polynomial $P(u,v) \in \mathbb{R}[u,v]$ in 2 variables is remarkable if
the set of solutions to the system $P(u,v)=P(v,u)=0$ is a finite number of points in $\mathbb{C}^2$ such that each point is of the form $(x, \overline{x}).$
Here are some... | https://mathoverflow.net/users/1056 | Bivariate polynomials with special properties | Let $V\_1$ be the defining 3-dimensional representation of SU(3) with character $\chi\_1$. Likewise, let $V\_2$ be the conjugate representation with character $\chi\_2 = \overline{\chi\_1}$. Then every irreducible representation of SU(3) has a character which is a $\mathbb{Z}-$polynomial in $\chi\_1$ and $\chi\_2$.
W... | 9 | https://mathoverflow.net/users/12301 | 59083 | 36,713 |
https://mathoverflow.net/questions/59079 | 2 | Let $(X,d)$ be a complete metric linear space whose balls are convex. Let $Y\subseteq X$ be a bounded, closed and convex subset that verifies the following property: for all $y\_0\in Y$, the distance function $y\rightarrow d(y\_0,y)$ attains its sup in $Y$.
Does $Y$ have extreme points?
I was trying to adapt the clas... | https://mathoverflow.net/users/13809 | Existence of extreme points | There is a counter-example.
Note that in any normed space, its unit ball satisfies your supremum-attaining property. Indeed, for any $x\_0\in X$ the supremum of $d(x\_0,\cdot)$ on the ball is attained at the point $-x\_0/\|x\_0\|$ if $x\_0\ne 0$, and at any point of the sphere if $x\_0=0$.
It remains to construct a... | 10 | https://mathoverflow.net/users/4354 | 59084 | 36,714 |
https://mathoverflow.net/questions/59082 | 4 | Let $m\_1,\ldots, m\_n$ be pairwise coprime natural numbers $\geq 1$. We consider the product $$G(m\_1,\ldots,m\_n) := \prod\_{i=1}^{n} \mathbb{Z} / m\_i \mathbb{Z}.$$ We define $M(n)$ as the set $n$-tuple of natural numbers $\geq 1$ with the property that the entrys are pairwise coprime. We define $l : M(n) \rightarro... | https://mathoverflow.net/users/13763 | A question concerning products of finite cyclic groups | We have $l(m\_1,\cdots,m\_n)$ is ~~at least~~ equal to the maximum length of a sequence of consecutive integers each of which is divisible by some $m\_i$. Let's suppose for a moment that $m\_i$ is the $i$th prime. In "On the problem of Jacobsthal", Iwaniec proves that this number is $\ll (n\log n)^2$, while your bound ... | 2 | https://mathoverflow.net/users/2384 | 59099 | 36,723 |
https://mathoverflow.net/questions/59076 | 1 | This question is probably trivial for those with decent training in math. Unfortunately, I have very limited training, and I got it over forty years ago and have forgotten a lot. I have tried getting this problem answered on other sites, but those sites seem to be designed to help students in high school or early in co... | https://mathoverflow.net/users/13807 | Maximizing a Definite Integral Subject to Constraints | I think this isn't too bad, and your guess in below point (d) is justified. I doubt it depends on the detailed formulae.
Suppose we concentrate on *g - f*, which is constrained to have integral 0? And by a change of origin to have a graph that is non-positive on the negative real axis, and non-negative on the positi... | 1 | https://mathoverflow.net/users/6153 | 59107 | 36,728 |
https://mathoverflow.net/questions/59095 | 7 | Inspired by other questions i have two questions about modules over division rings: given a division ring $D$ with center $Z(D)=K$. One has the notion of dimension for left modules (vector spaces) $V$ over $D$, which is well behaved like in linear algebra.
But does the following also hold: Given a left vector space $... | https://mathoverflow.net/users/3233 | Bimodules over division rings | The answer to both questions is positive if $D$ is finite-dimensional over its center $K$, and negative, in general, otherwise.
Q1. Suppose $V$ is a $D$-bimodule over $K$ (i.e., a $D\otimes\_K D^{op}$-module), while $W$ is a left $D$-module of dimension $n$, as in your question. Then $W$ is isomorphic to a direct sum... | 12 | https://mathoverflow.net/users/2106 | 59112 | 36,733 |
https://mathoverflow.net/questions/59104 | 9 | Let $K$ be a local field, for example the $p$-adic numbers. In Neukirch's book "Algebraic number theory", there is the statement: if $K$ contains the $n$-th roots of unity and if the characteristic of $K$ does not divide $n$, and we set $L=K(\sqrt[n]{K^{\times}})$, then one has $N\_{L/K}(L^{\times})=K^{\times n}$.
My... | https://mathoverflow.net/users/13726 | local class field theory (Norm map) | Let $L/K$ be a finite abelian extension of local fields. Although, there is no generic form for the image of the norm map, $N^{L}\_K$, in practice one can follow the following procedure to determine its image.
Choose a uniformizer $\pi\_L$ in $\mathcal{O}\_L.$ Then $L^{\times}$ is equal to the group generated by $\p... | 11 | https://mathoverflow.net/users/13816 | 59127 | 36,744 |
https://mathoverflow.net/questions/59117 | 7 | Let $\Gamma=(G,E)$ be a connected undirected graph, with no loops or multiple edges. $G$ is finite or countably infinite. For each edge $e=\{x,y\}\in E$, we assign a positive, symmetric edge weight $c\_e := c\_{\{x,y\}} = c\_{xy} = c\_{yx}$. I would like to know for which graphs $\Gamma$ it is possible to choose $(c\_e... | https://mathoverflow.net/users/5153 | Assigning positive edge weights to a graph so that the weight incident to each vertex is 1. | Here is a solution along the lines of JBL's answer.
First a couple of definitions:
>
> A [disjoint cycle cover](http://en.wikipedia.org/wiki/Vertex_cycle_cover) of a graph is a collection of cycles of our graph which are disjoint subgraphs, and contain all the vertices of our graph. A special case of a disjoint c... | 8 | https://mathoverflow.net/users/2384 | 59131 | 36,748 |
https://mathoverflow.net/questions/59149 | 11 | Why can the elements of the dual space of $\ell^\infty(\mathbb N)$ be represented as sums of elements of $\ell^1(\mathbb N)$ and Null$(c\_0)$?
EDIT: As [confirmed in the comments](https://mathoverflow.net/questions/59149/dual-space-of-ell-infty/59168#comment148669_59168), the OP intended to ask about this sentence
"... | https://mathoverflow.net/users/13799 | Dual space of $\ell^\infty$ | Obviously, the OP intended to ask about this sentence
"$f\in\ell\_\infty^\*$ is the sum of an element of $\ell\_1$ and an element null on $c\_0$"
from the paper D. H. Fremlin and M. Talagrand: A Gaussian Measure on $l^\infty$ <http://jstor.org/stable/2243023>
(Which is different claim from what was in the question.)
... | 26 | https://mathoverflow.net/users/8250 | 59168 | 36,770 |
https://mathoverflow.net/questions/59155 | 20 | The 200-th anniversary of the birth of Galois will be on October 25th, 2011. For Abel's bicentennial birth year in 2002, Norway had a big conference and initiated the Abel prize. A cursory web search doesn't reveal any major (or minor) conference to mark the bicentennial for Galois. Is there something being planned tha... | https://mathoverflow.net/users/3272 | Galois Bicentennial? | There is a conference at the IHP in Paris on October 24-28 of this year in honour of this anniversary. [Here](http://www.galois.ihp.fr) is the web-site, which doesn't have much information on it yet.
UPDATE: There is now quite a bit of information posted.
| 19 | https://mathoverflow.net/users/2874 | 59171 | 36,773 |
https://mathoverflow.net/questions/59167 | 0 | What is the profinite completion of the group $S^1$?
where $S^1= \{ z\in\mathbb{C}: |z|=1 \}$ is a compact and abelian group.
| https://mathoverflow.net/users/13643 | Profinite completion | It is trivial, as $S^1$ has no non-trivial (normal) finite index subgroups.
Recall that the profinite completion is the inverse limit of the $G/N$ where $N$ is a normal subgroup of finite index. So, if the only normal finite index subgroup is the full group, the profinit completion is trivial. And, for your group th... | 12 | https://mathoverflow.net/users/nan | 59175 | 36,776 |
https://mathoverflow.net/questions/59123 | 1 | i just wanted to know some necessary and sufficient conditions
| https://mathoverflow.net/users/13814 | condition for the existence of lines on degree four hypersurface in P^3.. | Here's how to obtain such a condition in principle. Doing it in practice will require a computer algebra system, and may need a lot of computer power:
The generic line in $\mathbb{P}^3$ can be parameterized as $(x:y:ax+by:cx+dy)$ for some $(a,b,c,d)$. More specifically, this is the dense Schubert cell in $G(2,4)$. Le... | 3 | https://mathoverflow.net/users/297 | 59179 | 36,778 |
https://mathoverflow.net/questions/59178 | 11 | Up to $10^6$:
$\sigma(8n+1) \mod 4 = OEIS A001935(n) \mod 4$
[A001935 Number of partitions with no even part repeated](https://oeis.org/A001935)
Is this true in general?
It would mean relation between restricted partitions of $n$ and divisors of $8n+1$.
Another one up to $10^6$ is:
$\sigma(4n+1) \mod 4 = ... | https://mathoverflow.net/users/12481 | Up to $10^6$: $\sigma(8n+1) \mod 4 = OEIS A001935(n) \mod 4$ (Number of partitions with no even part repeated ) | Let's call A001936(n) by $a(n)$. Here is a sketch of why $$a(n)\equiv \sigma(4n+1)\pmod{4}$$
Firs note that the generating function of $a(n)$ is
$$A(x)=\sum\_{n\geq 0}a(k)x^n=\prod\_{k\geq 1}\left(\frac{1-x^{4k}}{1-x^k}\right)^2$$ for $\sigma(2n+1)$ the generating function is $$B(x)=\sum\_{k\geq 0}\sigma(2k+1)x^k=\pro... | 23 | https://mathoverflow.net/users/2384 | 59185 | 36,782 |
https://mathoverflow.net/questions/59166 | 6 | EDIT: In this question I forgot to put one of the assumptions, and the question was easier than it should be. Here is the [revised question](https://mathoverflow.net/questions/59313/example-of-a-non-normal-infinite-index-subgroup-of-a-non-amenable-group-with-cert). Please vote to close this question as it is no longer ... | https://mathoverflow.net/users/2631 | Example of an infinite index subgroup of a non-amenable group whose normalizer is of non-zero finite index, and such that the Schreier graph is of subexponential growth | The answer to the question is yes. Here are two examples, the first one answers the question as it is posted and in the second one I assume that $N(H)$ is the normal closure of $H$ in $G$. The second example seems to be more interesting, in the sense that one have to do more computation on Schreier graph, compering to ... | 9 | https://mathoverflow.net/users/8699 | 59194 | 36,787 |
https://mathoverflow.net/questions/59183 | 2 | I was wondering if any vector bundles on a manifold other than the tangent bundle give topological invariants. I guess stiefel Whitney classes also come from the inverse bundle - but other than that.
| https://mathoverflow.net/users/13827 | important vector bundles | The top exterior power of the tangent bundle determines orientablility (the most basic of topological invariants, after dimension) in the sense that $\wedge^nTM$ is a trivial line bundle iff $M^n$ is orientable.
The complexified tangent bundle $TM\otimes\_{\mathbf{R}} \mathbf{C}$ is used to define the [Pontryagin cla... | 5 | https://mathoverflow.net/users/8103 | 59198 | 36,788 |
https://mathoverflow.net/questions/59213 | 48 | Here is a very natural question:
Q: Is it always possible to generate a finite simple group with only $2$ elements?
In all the examples that I can think of the answer is yes.
If the answer is positive, how does one prove it? Is it possible to prove it without using the
classification of finite simple groups?
| https://mathoverflow.net/users/11765 | Generating finite simple groups with $2$ elements | The answer to your question is yes. Moreover, if you pick two random elements from a finite simple group, then they generate the whole group with probability which tends to 1 as the size of the group grows. There are even stronger results in this direction, but I am not an expert in the subject so you will have to look... | 46 | https://mathoverflow.net/users/5034 | 59216 | 36,797 |
https://mathoverflow.net/questions/59208 | 4 | This is probably a very simple question, but I can't seem to see the answer. Given a (Zariski) closed subset $Z$ in $\mathbb P^n$ of codimension $d$, we can always find a linear subspace $L \cong \mathbb P^{d-1}$ such that $L \cap Z = \varnothing $. I'm wondering if a similar statement can be said about Grassmannians.
... | https://mathoverflow.net/users/11661 | Subvarieties of the Grassmannian of lines | Indeed much more is true.
Suppose $X$ is an irreducible algebraic manifold admitting a transitive action
of a linear algebraic group $G$. If $Y$ and $Z$ are irreducible subvarieties of $X$ then
for a general $g \in G$ the intersection of $gY$ (the translate of $Y$ by $g$) and $Z$
is empty or equidimensional of dimen... | 10 | https://mathoverflow.net/users/605 | 59217 | 36,798 |
https://mathoverflow.net/questions/59101 | 9 | The "classical Beurling density" of a subset of the natural numbers is $d(A)=lim\_{n\rightarrow\infty}\frac{|A\cap[1,n]|}{n}$, when it exists. It defines a finitely additive probability measure on the natural numbers which is invariant with respect to the sum. Here is my question: does there exist a "nice formula" to d... | https://mathoverflow.net/users/13809 | A density on the natural numbers invariant with respect to the multiplication | The natural thing to do here is to replace the intervals $[1,n]$ (and $n = |[1,n]|$) in the definition of $d(A)$ with a sequence $F\_n$ of subsets of $\mathbb{N}$ which is multiplicatively asymptotically invariant (or, in other words, a Folner sequence for the semigroup $(\mathbb{N},\cdot)$). For an exploration of this... | 7 | https://mathoverflow.net/users/7392 | 59224 | 36,802 |
https://mathoverflow.net/questions/59182 | 4 | Hi,
this is again a question from me which did not get any answer at math.stackexchange (Link: <https://math.stackexchange.com/questions/27366/>)
This question is about how well one can choose a partition of unity.
So suppose $(M,g)$ is an open Riemannian manifold. I would like to have the following statement to ... | https://mathoverflow.net/users/13356 | Existence of a partition of unity with uniformly bounded derivatives. | As Deane Yang points out, you want bounds from below on the injectivity radius and from above on the curvature. You may want to consider a manifold of *bounded geometry*, which means that in addition all derivatives of the curvature tensor are bounded. This last condition is equivalent to having an atlas of coordinate ... | 4 | https://mathoverflow.net/users/13840 | 59226 | 36,804 |
https://mathoverflow.net/questions/59157 | 10 | I'm looking for references for the following facts concerning the ultrafilter lemma (~ "there exist non-principal ultrafilters"):
1. The ultrafilter lemma is independent of ZF.
2. ZF + the ultrafilter lemma does not imply the Axiom of Choice.
I would prefer an overview article / book that links to the original pape... | https://mathoverflow.net/users/8494 | Reference Request: Independence of the ultrafilter lemma from ZF | In any of the formulations mentioned (so far) in the comments, the ultrafilter lemma is independent of ZF but weaker than AC. That the strongest form (all filters can be extended to ultrafilters) doesn't imply AC is a theorem of J.D. Halpern and A. Lévy ["The Boolean prime ideal theorem does not imply the axiom of choi... | 22 | https://mathoverflow.net/users/6794 | 59227 | 36,805 |
https://mathoverflow.net/questions/59070 | 2 | I want to explicitly write down equations for a variety in
$$\mathbb{P}(T\_{\mathbb{P}^2}\oplus T\_{\mathbb{P}^2})$$
What would be the natural way to do this? I know how the equations should look like over each point of $\mathbb{P}^2$ but it doesn't seem trivial to me how to "translate" them into global equations.
| https://mathoverflow.net/users/13803 | Projectivization of the tangent bundle | A twisted cubic on $P^3$ can be realized as the degeneration locus of a map $A\otimes O(-1) \to B\otimes O$, where $A$ and $B$ are vector spaces of dimension 2 and 3. In a relative situation you should replace vector spaces by vector bundles on the base. So, choose a pair of a vector bundles $A$ and $B$ on $P^2$ of ran... | 2 | https://mathoverflow.net/users/4428 | 59237 | 36,810 |
https://mathoverflow.net/questions/59229 | 4 | Hi,
I would like to know if there is a formula for the relative canonical bundle when performing a blow-up along an "authorized" subvariety. More precisely:
Let $X$ be a projective, irreducible, normal variety and let $Y \subset X$ be a smooth subvariety such that $X$ is normally flat along $Y$. Let $p : Z = Bl\_{Y... | https://mathoverflow.net/users/13841 | normal flatness and canonical bundle | Franz, to complement Karl's answer let me remark that actually you cannot expect a formula without further conditions. I will use your notation.
If $X$ is singular, $K\_X$ may not be *Cartier* or even $\mathbb Q$-Cartier (i.e., that some non-zero integer multiple would be Cartier). Without this it is hard to make sen... | 5 | https://mathoverflow.net/users/10076 | 59242 | 36,814 |
https://mathoverflow.net/questions/59238 | 2 | Assume F is null bordant. Does it imply that the total space of fiber bundle
$F\hookrightarrow E\to M$
is null bordant?
in particular what if $F$ is sphere?
| https://mathoverflow.net/users/3922 | Is the total space of Fiber bundle bordant to 0 if the fiber is null bordant? | There exist oriented surface bundles $E \to B$ on closed surfaces such that $E$ has nonzero signature (first found by Atiyah and Hirzebruch). Hence $E$ is not (oriented) nullbordant, even though base and fibre are nullbordant.
Textbook reference: Morita, Geometry of characteristic classes.
A lot of material on char... | 5 | https://mathoverflow.net/users/9928 | 59246 | 36,815 |
https://mathoverflow.net/questions/59211 | 10 | For the usual $4$-dimensional Minkowski space $M$, the standard Dirac operator is given by
$$
D: C^{\infty}(M) \to C^{\infty}(M), ~~~~~ f \mapsto \sum\_{i=1}^4 \gamma\_i\frac{\partial f}{\partial x\_i},
$$
where the $\gamma\_i$ are the usual [gamma matrises](http://en.wikipedia.org/wiki/Gamma_matrices). As we all know,... | https://mathoverflow.net/users/1095 | Dirac's Original Operator and the Hodge--Dirac Operator | As Sebastian and A.J. Tolland point out, the problem is that the two operators act on different bundles. However, the exterior algebra is canonically isomorphic to the Clifford algebra as vector spaces. Using this identification, you find two operators $D$ and $d+d^{\ast}$. Both operators have first order and they have... | 12 | https://mathoverflow.net/users/9928 | 59250 | 36,817 |
https://mathoverflow.net/questions/59243 | 4 | S. Lang gives a statement on page x of his 'Algebra':
>
> Most of our diagrams are composed of triangles and squares as above, and to verify that a diagram consisting of triangles and squares is commutative it suffices to verify that each triangle and square in it is commutative.
>
>
>
If we want to prove this... | https://mathoverflow.net/users/13842 | Diagrams consisting of triangles and squares | Think of a diagram as a planar directed graph with an equivalence relation on directed paths which respects concatenations of paths. A commutative diagram is one where this equivalence relation identifies all paths with common sources and sinks.
The remark above says that to check that a diagram is commutative it is ... | 4 | https://mathoverflow.net/users/2384 | 59251 | 36,818 |
https://mathoverflow.net/questions/59244 | 11 | I can't quite figure this problem yet. There is an ant at one vertex of a cube. The ant goes from one vertex to another by choosing one of the neighboring vertices uniformly at random. What is the average minimum time it takes to visit all vertices?
| https://mathoverflow.net/users/13822 | What is the cover time of a random walk on a cube? | An additional reference:
Chapter 12 in *Problems and Snapshots from the World of Probability* by
Blom, Holst, and Sandell is devoted to an elementary exposition of such
cover problems.
A related problem:
The solution to Problem 6556 in the *American Mathematical Monthly*
(Vol. 96, No. 9, Nov. 1989, pages 847... | 7 | https://mathoverflow.net/users/nan | 59261 | 36,824 |
https://mathoverflow.net/questions/59161 | 2 | I am completely stuck in the following linear algebra problem.
Consider a finite group $H$ and $N\times N$-matrices $M\_{g,h}$ with entries in $\mathbb{Z}$ for all $g,h\in H$. Assume $\sum\_{h\in H} M\_{g,h}=D$ for all $g\in H$, where $D$ is some $N\times N$-matrix with $\det(D)\neq 0$. Furthermore $M\_{g,h}=M\_{1,g^... | https://mathoverflow.net/users/43085 | Nonsingularity of certain block matrices | Even if $H$ is Abelian (see [MO question](https://mathoverflow.net/questions/44954)), the answer is **No**. Here is a counter-example where $H=\mathbb Z\_2$ and $N=2$. We have $M=\begin{pmatrix} A & B \\\\ B & A \end{pmatrix}$, where $D=A+B$. The assumption is $\det(A+B)\ne0$. One verifies easily that $\det M=0$ if and... | 3 | https://mathoverflow.net/users/8799 | 59266 | 36,826 |
https://mathoverflow.net/questions/59257 | 4 | I am reading about Lyapunov functions for Markov processes, and I am having trouble thinking of examples to keep in mind as I read. If $X\_t$ is a continuous-time Markov process with generator $L$, a Lyapunov function is supposed to be a function $V$, in the domain of $L$, with $V \ge 1$ such that $LV \le -aV + b 1\_C$... | https://mathoverflow.net/users/4832 | Examples of Lyapunov functions for Markov processes | This is called a "drift condition" in the applied probability literature -- this is used quite often when dealing with MCMC simulations, for example.
In continuous time, what about the good old Ornstein-Uhlenbeck process $dz = -zdt + \sqrt{2}dW$ and generator $L \phi(x) = -x \phi'(x) + \phi^{''}(x)$: the Lyapunov fun... | 4 | https://mathoverflow.net/users/1590 | 59274 | 36,833 |
https://mathoverflow.net/questions/59252 | 3 | Is geometric realization $|\cdot|:\mathbf{Top}^{\mathbf{\Delta}^{\textrm{op}}}\rightarrow \mathbf{Top}$ a left Quillen functor? If so, under what model structure on $\mathbf{Top}^{\mathbf{\Delta}^{\textrm{op}}}$? I would guess the Reedy model structure.
A reference would be ideal.
Thanks
| https://mathoverflow.net/users/11300 | geometric realization on $\mathbf{sTop}$ | The results on homotopy invariance of the geometric realization of simplicial spaces go back at least to May's *The Geometry of Iterated Loop Spaces*, although he doesn't explicitly mention model categories. It is indeed true that the geometric realization of simplicial spaces is a left Quillen functor with respect to ... | 3 | https://mathoverflow.net/users/12547 | 59285 | 36,834 |
https://mathoverflow.net/questions/59262 | 15 | There are various open problems in the subject of logical number theory concerning the possibility of proving this or that well-known standard results over this or that weak theory of arithmetic, usually weakened by restricting the quantifier complexity of the formulas for which one has an induction axiom. It particula... | https://mathoverflow.net/users/10909 | Induction, the infinitude of the primes, and workaday number theory | First, let me discuss the precise open question and why it is interesting to logicians. Then I will discuss potential ramifications outside of pure logic.
The main open question is whether the theory IΔ0 can prove the existence of arbitrarily large primes. The theory IΔ0 is the theory over the language of basic arith... | 22 | https://mathoverflow.net/users/2000 | 59286 | 36,835 |
https://mathoverflow.net/questions/59230 | 2 | Given an $N$-dimensional Riemannian manifold $M$, with associated Hodge $\ast$-mapping $\ast$, we have the chain complex
$$
\Omega^{0} {\buildrel {\text d}^\ast \over \longleftarrow} \Omega^{N} {\buildrel {\text d}^\ast \over \longleftarrow}\cdots \Omega^{N}
$$
For a Kahler manifold $M$ of complex dimension $N$, with... | https://mathoverflow.net/users/1648 | When does the Anti-Holomorphic Chain Complex Exist for Non-Kahler Manifolds? | Hodge $\*$ and $\overline{\partial}^\*$ are defined and have the named properties on any complex manifold with respect to any Hermitian metric. No Kahlerness, nor compactness, is necessary. Note that Voisin's book presents them in the chapter before she defines the Kahler condition.
---
John McCarthy points out ... | 8 | https://mathoverflow.net/users/297 | 59292 | 36,838 |
https://mathoverflow.net/questions/59291 | 38 | For a poset $P$ there exists an embedding $y$ into a complete and cocomplet poset $\hat{P}$ of downward closed subsets of $P$. It is easy to verify that the embedding preserves all existing limits and no non-trivial colimits --- i.e. colimits are freely generated. $\hat{P}$ may be equally described as the poset of all ... | https://mathoverflow.net/users/13480 | Completion of a category | Yes, it's a general construction which is related to so-called Isbell conjugation.
Let $C$ be a small category. It is well-known that the free colimit cocompletion is given by the Yoneda embedding into presheaves on $C$, $y: C \to Set^{C^{op}}$. The presheaf category is also complete. Dually, the free limit-completi... | 44 | https://mathoverflow.net/users/2926 | 59295 | 36,840 |
https://mathoverflow.net/questions/58000 | 13 | Let us consider polynomial contact structures on $\mathbb RP^3$, i.e. contact structures on $\mathbb R^3$ defined by a form $w=Pdx+Qdy+Rdz,\ P,Q,R\in \mathbb R[x,y,z]\ $ in an affine part and then extended to $\mathbb RP^3$, and $ w \wedge dw \ne 0$ everywhere.
One can find all such forms $w$ that $deg P, deg Q, deg ... | https://mathoverflow.net/users/4298 | Polynomial contact structures on $RP^3$ | **Polynomial distributions on $\mathbb P^n$.** The following works for any field $k$.
The polynomial $1$-forms defined on $\mathbb A^{n+1}$ which induce distributions on $\mathbb P^n$ are those invariant by homotheties and annihilated by the Euler vector field $R = \sum\_{i=0}^n x\_i \partial\_i$. Explictly these can ... | 13 | https://mathoverflow.net/users/605 | 59303 | 36,845 |
https://mathoverflow.net/questions/59302 | 23 | Let $f:\mathbb R\to\mathbb R$ be a smooth, orientation preserving diffeomorphism of the real line.
Is it the case that there always exist another diffeomorphism $g:\mathbb R\to\mathbb R$
such that $g\circ g = f$?
*Note:* it is relatively easy to show that a continuous $g$ exists, but I am not managing to find a sm... | https://mathoverflow.net/users/5690 | square root of diffeomorphism of R: does it always exist? | The answer is no, assuming that you seek an orientation preserving square root.
(I see unknown's answer appeared while I'm typing. I don't quite understand it at the moment but the construction looks different.)
Consider a diffeomorphism $f$ such that $f(0)=0$, $f(1)=1$, $f(t)>t$ for all $t\in(0,1)$, $f'(0)=a>1$, $... | 19 | https://mathoverflow.net/users/4354 | 59311 | 36,848 |
https://mathoverflow.net/questions/59322 | 0 | Let $(S,\cdot)$ be a semigroup and $W\subseteq S$ be a subset. Let me call $W$ "tile" if the following property is satisfied: there exist $s\_1,...s\_k\in S$ such that the sets $s\_i\cdot W$ are pairwise disjoint and cover $S$. For instance, tiles for the semigroup $(\mathbb N,+)$ are given by the multiples of some fix... | https://mathoverflow.net/users/13809 | How can we pave the multiplicative semigroup $(\mathbb N,\cdot)$? | Prime factorization provides an isomorphism between the semigroup $(\mathbb N,\cdot)$ and an infinite direct sum of copies of $(\mathbb N,+)$.
So you can reduce your problem to the case that you already know how to solve.
*Warning:* my first $\mathbb N$ does not contain the element zero, whereas my second $\mathbb N$... | 5 | https://mathoverflow.net/users/5690 | 59323 | 36,853 |
https://mathoverflow.net/questions/59329 | 6 | I've spent quite a bit of time studying the Mathieu Groups, and I own the ATLAS.
My question is about M12. It is based on the ternary Golay Code, and is the automorphism
group of a Steiner S(5,6,12) system. Now, all of these Steiner systems are isomorphic
up to labelling. The order of M12 is 95040, which is 132 x 720... | https://mathoverflow.net/users/10350 | M12 simple sporadic group | You probably want
[The golay codes and the Mathieu groups](http://books.google.com/books?id=upYwZ6cQumoC&pg=PA299) by John Conway
| 7 | https://mathoverflow.net/users/51 | 59335 | 36,860 |
https://mathoverflow.net/questions/57946 | 3 | This question arises from my discussion with a Master student. It concerns with the following situation: let $\phi: R \to S$ be a homomorphism between Noetherian commutative rings. Suppose the $R$-module structure of $S$ has a presentation matrix with all entries in the Jacobson radical of $R$ (so $S$, as $R$-module, i... | https://mathoverflow.net/users/2083 | Freeness of modules along ring homomorphisms | Let me answer and accept this in CW so that it will not be bumped periodically as not answered by the software. It was hoped that the case of $\phi$ surjective can be generalized, but as Laurent pointed out in the comments, one can not hope to get any reasonable statement.
| 1 | https://mathoverflow.net/users/2083 | 59336 | 36,861 |
https://mathoverflow.net/questions/59270 | 13 | More generally, are there any remarkable properties enjoyed by the $\infty$-category of stable $\infty$-categories?
| https://mathoverflow.net/users/13848 | Is the $\infty$-category of stable $\infty$-categories stable? | No. It's pointed by the zero category, but then taking loops of a stable category C (in the sense of the pullback of 0 --> C along itself) always gives the zero category, so loops is definitely not an equivalence.
One important structural feature of the category of stable categories along these lines is that it has s... | 15 | https://mathoverflow.net/users/3931 | 59342 | 36,863 |
https://mathoverflow.net/questions/59309 | 4 | There sohould be a list of K-theory and K-homology groups for the the standard spaces, like circle, spheres, (non-commutative) tori, but despite I've googled for it, I have found nothing satisfying. Maybe someone can give a reference?
| https://mathoverflow.net/users/4807 | Lists of K-homology Groups | For many basic examples, the usual tools of (co)homology theory work just fine (and very similarly) in K-theory and K-homology.
Here's an example. How does one compute, say, the De Rham cohomology of $S^1$? There are lots of ways, but one way is to use the Mayer-Vietoris sequence - the same thing works in K-theory (a... | 5 | https://mathoverflow.net/users/4362 | 59343 | 36,864 |
https://mathoverflow.net/questions/59339 | 6 | Recall the following version of Szemerédi's Theorem: let $r\_k(N)$ be the largest cardinality of a subset of $[N]:=\{1,\ldots, N\}$ which does not contain an arithmetic progression of length $k$. Then, for any $k\ge 3$, $r\_k(N)/N\to 0$ as $N\to\infty$.
It follows that the same is true for any finite pattern. i.e. i... | https://mathoverflow.net/users/11009 | Bounds on the size of sets not containing a given finite pattern | Your question is quite broad, I won't be able to answer everything.
Let me start by answering your most concrete question regarding the pattern
$ \{1,2,m\} $. I prefer to use $\{0,1,k\}$, which is equivalent.
Recall that the existence of a three term arithmetic progression can be recast as asking for a soution to $... | 4 | https://mathoverflow.net/users/nan | 59345 | 36,866 |
https://mathoverflow.net/questions/59316 | 11 | After over half a year's study on operator algebra (especially on von Neumann algebra) by doing exercises in Fundamentals of the theory of operator algebras 1, 2 --Kadison, I was told that the current research focus is on the Ⅱ1 factor, and certain background on group theory is necessary, such as studying the free prod... | https://mathoverflow.net/users/9305 | Group theory required for further study in von Neumann algebra | As for a book on group theory that may be useful or interesting to read for further study of $II\_{1}$ factors, I think that de la Harpe's book *Topics in Geometric Group Theory* is good for this. The reason I say this is that geometric group theory is concerned with the "large scale" structure of groups, and concerns ... | 15 | https://mathoverflow.net/users/6269 | 59346 | 36,867 |
https://mathoverflow.net/questions/59247 | 6 | I am interested in computing confidence intervals for the mean of a random variable $X$ given $\require{cancel}\xcancel{N \text{ i.i.d. samples}}$ an i.i.d. sample of $N$ copies of $X$, where $N$ is $\operatorname{Binomial}(n, p)$. Any time I read about confidence intervals for the mean it is assumed that the ~~number ... | https://mathoverflow.net/users/13843 | Confidence intervals when the number of samples is random | Let $\hat{\mu}=\frac{\sum\_i X\_i}{N}$ and $\mu = \mathbf{E}(X)$
$$\mathcal{P}(\mu \in [x,y]\,|\, a) = \sum\_i \binom ni p^i(1-p)^{n-i}\mathcal{P}(\mu \in [x,y]\,|\,a,N=i)$$
If $pN$ is relatively large, and $\hbox{var}(X) = \sigma^2$ you can represent the true mean as taken from a mixture of gaussian with mean $a$ ... | 2 | https://mathoverflow.net/users/8737 | 59350 | 36,870 |
https://mathoverflow.net/questions/59290 | 8 | I am currently reading Shalika's article "Representation of the two by two unimodular group over local fields" and various other related articles, which deal with the classification of complex representation of reductive groups over local rings. It is cumbersome, that some authors consider only local rings with residue... | https://mathoverflow.net/users/10400 | Why is p=2 special, if we want to classify cplx. representation of GL2(Zp)? | Shalika's paper that you mention deals not with $\mathrm{GL}\_2(\mathfrak{o})$, but with $\mathrm{SL}\_2(\mathfrak{o})$. The representation theory of $\mathrm{GL}\_2(\mathfrak{o})$ is uniform in the residue characteristic $p$, in the sense that for any $r,n\in\mathbb{N}$, there exist polynomials $f\_{r,n}(X)\in\mathbb{... | 12 | https://mathoverflow.net/users/2381 | 59352 | 36,872 |
https://mathoverflow.net/questions/59314 | 6 | For projective $N$-space $CP^{N}$, there is a canonical Kähler metric called the Fubini-Study metric. Do there exist other Kähler metrics for $CP^N$. If so, is there any classification of such metrics?
More generally, how does this work for the Grassmanians, or even flag manifolds?
| https://mathoverflow.net/users/2612 | Kähler metrics for projective space that are not the Fubini-Study metric | Every complex manifold that admits one Kahler metric $w$ admits a lot of them, indeed $w+i\partial \bar \partial f$ is a Kahler metric if the second derivatives of $f$ are not too large. This is why, asking if one can classify Kahler metrics on $\mathbb CP^n$ is more-less equivalent to ask if one can classify functions... | 13 | https://mathoverflow.net/users/943 | 59358 | 36,876 |
https://mathoverflow.net/questions/59359 | 1 | It is easy to find a choice function on all finite subsets of $\mathbb R$, but without using the axiom of choice, not on all subsets. Is there an "explicit" choice function on the countable subsets of the real line? Can it be used to create paradoxical objects like, say, Vitali sets in the same way that a choice functi... | https://mathoverflow.net/users/4903 | Choice function on the countable subsets of the reals | The standard construction of a [Vitali set](http://en.wikipedia.org/wiki/Vitali_set) only involves making choices from countable subsets of $\mathbb{R}$ (specifically, from sets of the form $(r+\mathbb{Q}) \cap [0,1]$). It is well known that ZF (assuming consistent) does not prove the existence of a non-measurable subs... | 9 | https://mathoverflow.net/users/11771 | 59362 | 36,879 |
https://mathoverflow.net/questions/59357 | 21 | Given two morphisms between chain complexes $f\_\bullet,g\_\bullet\colon\,C\_\bullet\longrightarrow D\_\bullet$, a chain homotopy between them is a sequence of maps $\psi\_n\colon\,C\_n\longrightarrow D\_{n+1}$ such that $f\_n-g\_n= \partial\_D \psi\_n+\psi\_{n-1}\partial\_C$. I can motivate this definition only when t... | https://mathoverflow.net/users/2051 | Why chain homotopy when there is no topology in the background? | Here's one way to look at it: There is a chain complex $Hom(C,D)$ in which the $n$th chain group is the product over $k$ of $Hom(C\_k,D\_{n+k})$ and the boundary is given by $\partial (f(c))=(\partial f)(c)+(-1)^{|f|}f(\partial c)$. A chain map is a $0$-cycle, and two of them are chain homotopic if they differ by a bou... | 29 | https://mathoverflow.net/users/6666 | 59363 | 36,880 |
https://mathoverflow.net/questions/59305 | 2 | Fix a finite extension $F$ of $\mathbb{Q}\_p$. Consider its ring of integers $\mathfrak{o}$ with maximal ideal $\mathfrak{p}$. Set $R\_n = \mathfrak{o}/\mathfrak{p}^n$. Let $\mathrm{B}$ be the upper triangular matrices. Consider a character $\mu : \mathrm{B}(R\_n)\rightarrow \mathbb{C}^\times$. When is the induced repr... | https://mathoverflow.net/users/10400 | Parabolic induction for GL(2,Z/pn) | A sufficient criterion for irreducibility is given, for example, in Theorem 4.6 in Hill: Semisimple and cuspidal characters of $\mathrm{GL}\_n(\mathcal{O})$. Hill's result is more general, and holds for certain representations of $\mathrm{GL}\_n(\mathcal{O})$, for $n\geq 2$. For $\mathrm{GL}\_2(\mathcal{O})$ it says th... | 4 | https://mathoverflow.net/users/2381 | 59364 | 36,881 |
https://mathoverflow.net/questions/59232 | 10 | Let $(M , \omega)$ be a symplectic manifold, $J$ a compatible almost complex structure. We call *pseudo holomorphic strip* a solution $u : \mathbb R \times I \to M$ of the equation $\partial\_s u + J \partial\_t u = 0.$ Given a pair of Lagrangian submanifolds $L\_0, L\_1$, such a strip is said to be bounded by the pair... | https://mathoverflow.net/users/12386 | Index theorem interpretation of the spectral flow for a pseudo holomorphic curve | In some sense this really goes back to pre-index theory days
to Vekua and was one of the motivations for the index theorem (for a
reference to Vekua see Gromov's psuedo-holomorphic curves paper and I think
there is long discussion in Booss and Bleecker). Vekua proved the following.
Take a map from
$f:S^1 \to \mathbb{C... | 9 | https://mathoverflow.net/users/12605 | 59367 | 36,884 |
https://mathoverflow.net/questions/59370 | 6 | This question may be trivial to people with the right background, but I do not see the answer.
>
> Let $\Bbbk$ be an algebraically closed field. Can any one-dimensional group variety (over $\Bbbk$) act transitively on $\mathbb{P}^1\_{\Bbbk}$?
>
>
>
Here is my reasoning so far:
-Every $g \in G$ fixes a point... | https://mathoverflow.net/users/5094 | Is it true that no one-dimensional group variety acts transitively on $\mathbb{P}^1$? | The case of a 1-dimensional connected affine group (which in particular is solvable) is settled most easily by invoking the Borel Fixed Point Theorem: a connected solvable (affine) group acting as an algebraic group on an irreducible projective variety has a fixed point. If the given "group variety" is an abelian varie... | 9 | https://mathoverflow.net/users/4231 | 59378 | 36,889 |
https://mathoverflow.net/questions/59381 | 2 | Recently, I devised a new convex hull algorithm. Is there any forum where I can submit my work?
| https://mathoverflow.net/users/10429 | Where to submit a new convex hull algorithm? | Computational geometry has several journals (Discrete and Computational Geometry, Computational Geometry: Theory and Applications, International Journal of Computational Geometry and Applications, and Journal of Computational Geometry) and conferences (ACM Symposium on Computational Geometry, European Workshop on Compu... | 13 | https://mathoverflow.net/users/440 | 59383 | 36,893 |
https://mathoverflow.net/questions/59385 | 1 | In the paper by Freed et al. "Topological Quantum Field Theories From Compact Lie Groups" they say
...the stack of G-bundles with connections is $\star // G = BG$...
My question is what's the notation $\star // G$? Is it the same as the symplectic quotient;
i.e., take a contractible space $\star$ and quotient out ... | https://mathoverflow.net/users/13132 | question about notation | I can't say much about being related to a symplectic quotient but it is a very important example in the theory of stacks. The short answer, without going into too many details, is that $[S/G]$ is category whose objects are principal homogeneous $G$-bundles with a $G$-equivariant morphism to S. The morphisms are pullbac... | 5 | https://mathoverflow.net/users/13863 | 59392 | 36,899 |
https://mathoverflow.net/questions/59390 | 22 | Under what circumstances is a quasi-isomorphism between two complexes necessarily a homotopy equivalence? For instance, this is true for chain complexes over a field (which are all homotopy equivalent to their homology). It's also true in an $\mathcal{A}\_\infty$ setting.
Is it true for chain complexes of free Abelia... | https://mathoverflow.net/users/5010 | When is a quasi-isomorphism necessarily a homotopy equivalence? | If your complexes are bounded, this is always true for any ring more generally replacing free modules with projectives. The statement is that $\mathrm{D}^b(A\text{-}mod)$ is equivalent to $\mathrm{Ho}(Proj\text{-}A)$ and you can find it in Weibel Chapter 10.4. If your complexes are unbounded things are more tricky. The... | 22 | https://mathoverflow.net/users/6986 | 59396 | 36,900 |
https://mathoverflow.net/questions/22203 | 1 | I have a follow-up question to this one:
[unbiased estimate of the variance of a weighted mean](https://mathoverflow.net/questions/11803/unbiased-estimate-of-the-variance-of-a-weighted-mean)
Specifically, how do I generalise the result given here (and on Wikipedia) for the unbiased sample estimate of the variance o... | https://mathoverflow.net/users/5546 | Unbiased estimate of the variance of an *unnormalised* weighted mean | Hi,
Rather long after your question, but it can be done directly in the same way Matus did it, or you can simply use the following:
Matus assumed weights $W\_i$ which sum to $1$. Suppose you have weights Ui, and write $V\_1 = \sum U\_i$, and $V\_2 = \sum U\_i^2$, consistent with the Wikipedia entry for weighted sampl... | 2 | https://mathoverflow.net/users/13547 | 59403 | 36,904 |
https://mathoverflow.net/questions/59405 | 3 | What are the units of $\mathbb{Z}/4\mathbb{Z}[x]$? Anything of the form $\pm 1 + 2 x p(x)$ for $p(x) \in \mathbb{Z}\_4[x]$ works, and is in fact its own inverse. It's easy to see that any unit must have constant coefficient $\pm 1$ and, if the unit is not constant, it must have highest non-zero coefficient $2$, which l... | https://mathoverflow.net/users/13834 | What are the units of $\mathbb{Z}/4\mathbb{Z}[x]$? | Yes, these are the only units. Consider the obvious map ${\bf Z}\_4[x] \to {\bf F}\_2[x]$; if $f(x)\in ({\bf Z}\_4[x])^\*$ then $\psi(f(x))\in ({\bf F}\_2[x])^\*$ and so $\psi(f(x)) = \pm 1$. Thus $f(x) \pm 1 \in \ker(\psi)$ and so $f(x) = \pm 1 + 2g(x)$ for some $g(x)\in {\bf Z}\_4[x]$.
More generally this works fo... | 5 | https://mathoverflow.net/users/6701 | 59407 | 36,906 |
https://mathoverflow.net/questions/59414 | 6 | Suppose $V$ is a vector bundle with structure group $SO(3)$, and suppose that it can be lifted to a $\text{Spin}(3) = SU(2)$ bundle (i.e. $w\_2(V) = 0$). Let us call the lifted bundle $E$. Then it is stated on page 42 in *The Geometry of Four-Manifolds* by Donaldson and Kronheimer that we have the relation $p\_1(V) = -... | https://mathoverflow.net/users/11031 | Characteristic classes of lifted bundles | Let me rephrase your question so that I understand it: let $P \to X$ be an $SU(2)$-principal bundle. Then we get a $2$-dimensional complex vector bundle $E \to X$ by $P \times\_{SU(2)} C^2$ (with the defining representation). And we get a $3$-dimensional real vector bundle $V:= P \times\_{SU(2)} R^3$ (with the adjoint ... | 2 | https://mathoverflow.net/users/9928 | 59416 | 36,909 |
https://mathoverflow.net/questions/59263 | 2 | For any commutative ring $A$, the set of idempotents of $A$ will be denoted as $E(A)$. This set has a (canonical) ring structure. With addition defined by:
$$e+'f=e(1−f)+f(1−e)$$
where $+$ and $−$ are operation in the ring itself. The multiplication operation is the same as the ring itself.
Suppose now I have a commu... | https://mathoverflow.net/users/1245 | ring of idempotents of the integral extension of a ring | Every element $e$ of $E(B)$ is already integral since it satisfies $e^2 - e = 0$. (This was effectively in a comment above, but perhaps it's better placed as an answer.)
| 1 | https://mathoverflow.net/users/2926 | 59419 | 36,911 |
https://mathoverflow.net/questions/59422 | -1 | Let $\phi\_{n}(x)$ be the $n$-th cyclotomic polynomial. What are the restrictions to $n$ (if any) to have $\phi\_{n}(x)$ divides $\phi\_{2n}(x)$ (where division is in $\mathbb{Z}[x]$)?Or is it true that $\frac{\phi\_{2n}(x)}{\phi\_{n}(x)}\in\mathbb{Z}[x]$ for all integers $n$?
| https://mathoverflow.net/users/13875 | Cyclotomic Polynomials | When is a primitive \*n\*th root of unity also a primitive 2\*n\*th root of unity? Please note that the answer is never, and this can also be seen by unique factorisation.
| 4 | https://mathoverflow.net/users/6153 | 59426 | 36,915 |
https://mathoverflow.net/questions/58843 | 4 | Let $\pi:Y\longrightarrow X$ be a finite morphism of smooth projective geometrically connected curves over a number field $K$. Let $h:\mathcal{X}\longrightarrow \textrm{Spec} \ O\_K$ be a regular integral flat projective $O\_K$-scheme with generic fibre $X$. Here $O\_K$ is the ring of integers of $K$.
I would like t... | https://mathoverflow.net/users/4333 | Extending finite morphisms of curves to finite morphisms of arithmetic surfaces | The vertical ramification can't be seen from the branch locus $D$. For example, consider
$$ Y : y^2 = f(x), \quad f(x)\in O\_K[x]$$
(say with $f(x)$ monic and separable in all residue fields of $O\_K$) and
$$ Y' : y^2 = tf(x), \quad t\in O\_K.$$
Then $Y\to \mathbb P^1\_K$ extends in an obvious way to $\mathcal Y\to \ma... | 3 | https://mathoverflow.net/users/3485 | 59431 | 36,919 |
https://mathoverflow.net/questions/59393 | 0 | As we know, the inversion formula of Fourier transformation holds pointwise for Schwartz class.
We also have a general result concerning the inversion of Fourier transformation on locally compact abelian groups, which says that if $f$ belongs to the intersection of the $L^1$-algebra and the Fourier-Stieltjes algebra o... | https://mathoverflow.net/users/13244 | Inversion of Fourier Transformation | The analogues of Schwartz functions on general locally compact abelian groups are called [Schwartz-Bruhat functions](http://en.wikipedia.org/wiki/Schwartz-Bruhat_function), and are mapped to Schwartz-Bruhat functions under Fourier transforms. Their dual spaces are spaces of tempered distributions on such groups which a... | 7 | https://mathoverflow.net/users/51 | 59435 | 36,921 |
https://mathoverflow.net/questions/59398 | 5 | Let $R$ be a regular local ring, $I$ a prime ideal and $J$ an $I$-primary ideal in $R$. Is it true that if $R/I$ is CM then also $R/J$ is CM?
This question is in some way the inverse of [this one](https://mathoverflow.net/questions/47428/cm-for-radical-ideal).
| https://mathoverflow.net/users/5998 | CM for primary ideal | A useful way to think about this issue is to consider $J=I^{(n)}$, the $n$-*symbolic power* of $I$, which by definition is the $I$-primary component of $I^n$.
When $R$ is a polynomial rings over $\mathbb C$, this is the ideal consisting of functions vanishing to order at least $n$ on $X = \text{Spec}(R/I)$.
It is ... | 6 | https://mathoverflow.net/users/2083 | 59444 | 36,924 |
https://mathoverflow.net/questions/59368 | 1 | **Question.** Let $k$ be a field of characteristic $0$. Let $L$ be a $k$-vector space. Consider the subspace $S$ of $L\otimes L\otimes L\otimes L$ spanned by all tensors of the form
$\left[a,\left\lbrace b,\left[c,d\right]\right\rbrace \right]$ for $a,b,c,d\in L$,
where $\left[u,v\right]$ denotes $u\otimes v-v\otim... | https://mathoverflow.net/users/2530 | Invariant space of lifted Chevalley automorphisms of the tensor algebra | Starting with the tensor $[e\_1,\{e\_2,[e\_3,e\_4]\}]$, I calculated $[e\_{\sigma(1)},\{e\_{\sigma(2)},[e\_{\sigma(3)},e\_{\sigma(4)}]\}]$ for $\sigma$ running over a set of representatives of the cosets of (34) (since interchanging the last two inputs only changes the sign of the original tensor). Encoding the 12 resu... | 6 | https://mathoverflow.net/users/12301 | 59446 | 36,926 |
https://mathoverflow.net/questions/59442 | 6 | Let $x$ and $y$ be two permutations of $\mathbb{Z}^2$ defined as follows. The permutation $x$ sends $(n,0)$ to $(n+1,0)$ and fixes all else while $y$ sends $(0,n)$ to $(0,n+1)$ and fixes all else. Is the group generated by $x$ and $y$ amenable?
I do know that the group does not contain a copy of the free group on two... | https://mathoverflow.net/users/13878 | Is the group generated by two almost disjoint infinite cycles amenable? | The derived subgroup of your group consists of permutations with finite support. Indeed, suppose that $w$ is a commutator word in $a$ and $b$ so the total exponent of $a$ (of $b$) is 0. Take a point $(m,n)$ where $m$ or $n$ are very large (comparing to $|w|$). Then $w(a,b)$ fixes that point. Therefore your group is an ... | 8 | https://mathoverflow.net/users/nan | 59447 | 36,927 |
https://mathoverflow.net/questions/59296 | 14 | Consider the following $n\times n$ random matrix $V\_{n}$ where the $(p,q)$ entry is given by
$$
V\_{n}(p,q):= \frac{1}{\sqrt{n}}\exp(2\pi i(p-1) x\_{q})
$$
where $x\_{1},x\_{2},\ldots,x\_{n}$ are iid random variables with uniform distribution on $[0,1]$.
It is not difficult to prove that $V\_{n}^{\*}V\_{n}$ has the ... | https://mathoverflow.net/users/13825 | A Question on Random Matrices | It is actually more like $e^{-\sqrt n}$. Let's look at the norm of the inverse matrix. The entries are $\pm\prod\_{i:i\ne j}\frac 1{z\_j-z\_i}\sigma\_m(z\_1,\dots,z\_{j-1},z\_{j+1},\dots,z\_n)$ where $z\_k=e^{2\pi i x\_k}$ is a random point on the unit circle and $\sigma\_m$ is the $m$-th symmetric sum. Since $\log |Z-... | 11 | https://mathoverflow.net/users/1131 | 59453 | 36,929 |
https://mathoverflow.net/questions/59452 | 4 | For a Kahler manifold $M$ we have two well-known Laplacians: the de Rham Laplacian $\Delta\_{\text{d}} = ($d$ + $d$^\ast)^2$, and the Dolbeault Laplacian $\Delta\_{\overline{\partial}} = (\overline{\partial} + \overline{\partial}^\ast)^2$. Now on smooth functions, these two operators are related by the well-known formu... | https://mathoverflow.net/users/1648 | Relation between the de Rham and Hodge Laplacians on the Exterior Algebra | If $(X,\omega)$ is Kähler, then it is always true that
$$
\Delta'=\Delta''=\frac 12\Delta,
$$
where these three Laplacians are with respect, in order, to $\partial$, $\bar\partial$ and $d$. This is valid when they act on any space of complex-valued differential forms.
More generally, you can look to differential form... | 12 | https://mathoverflow.net/users/9871 | 59455 | 36,930 |
https://mathoverflow.net/questions/57980 | 22 | Richard Laver proved that there is a unique binary operation $\*$ on $\{1,\ldots,2^n\}$ which satisfies $$a\*1 \equiv a+1 \mod 2^n$$
$$a\* (b\* c) = (a\* b) \* (a \* c).$$
This is the $n$th Laver table $(A\_n,\*)$.
There is an algorithm for computing $a \* b$ in $A\_n$, but in general (and especially for small values... | https://mathoverflow.net/users/10774 | What is the largest Laver table which has been computed? | I've been in contact with Patrick Dehornoy and Ales Drapal and both thought that $A\_{28}$ is likely the current record for a Laver table computation.
| 8 | https://mathoverflow.net/users/10774 | 59467 | 36,937 |
https://mathoverflow.net/questions/59454 | 6 | Let $X$ be a complex projective manifold of complex dimension $n$ and $A\to X$ an ample line bundle. Let $L\to X$ be a line bundle such that
$$
c\_1(L)^k\cdot c\_1(A)^{n-k}>0,\quad k=1,\dots,n.
$$
Is it true that then $L$ is big?
The answer is yes if $n\le 2$: for $n=1$ there is nothing to prove, and for $n=2$ the po... | https://mathoverflow.net/users/9871 | A line bundle not big but with good intersection numbers | Here is a very straightforward contre-example. Let $X=\mathbb CP^2\times \mathbb CP^1$ blown up in one point. Denote by $E$ the exceptional divisor, and denote by $\pi$ the projection of $X$ to $\mathbb CP^2$, and take the following bundle:
$$L\_n=\pi^\*(O(n))\otimes O(E),$$
where $n$ satisfies two conditions: $$c\_1... | 6 | https://mathoverflow.net/users/943 | 59468 | 36,938 |
https://mathoverflow.net/questions/59470 | 5 | If $f:X \rightarrow S$ is locally of finite type, there is unique largest open subset $U$ in $X$ such that $f|U$ is etale.
Suppose $f$ is finite and $U$ is nonempty. Is it true that $f|U$ is finite etale?
Thanks in advance.
| https://mathoverflow.net/users/1363 | Restricting finite morphism to finite etale morphim | No, because $U\to S$ finite implies that $U\to X$ is finite (at least when $X\to S$ is separated), so $U$ would be closed in $X$.
If you want an example, take a non-trivial morphism from a projective smooth curve $X$ to the projective line over $\mathbb C$.
You might ask whether $U\to f(U)$ (if the latter is open... | 12 | https://mathoverflow.net/users/3485 | 59483 | 36,948 |
https://mathoverflow.net/questions/59503 | 6 | The category of presheaves $Pre(C)$ on a small category $C$ is the category of functors $C^{op}\to Sets$. Since the category of sets is co-complete and every presheaf is a colimit of representable ones (i.e. presheaves of the form $Hom(-,c)$ for an object $c$ of $C$) $Pre(C)$ may be interpreted as the co-completion of ... | https://mathoverflow.net/users/2625 | Question on the interpretation of a presheaf category as a co-completion | You should think of $Pre(C)$ as a *formal* (i.e., free) cocompletion of $C$. Even if $C$ is cocomplete, by passing to $Pre(C)$ there will be "new" colimits that were not in $C$ (are not in the essential image of the Yoneda embedding). A simple example of this is the empty presheaf (which returns the empty set when eval... | 17 | https://mathoverflow.net/users/2926 | 59507 | 36,956 |
https://mathoverflow.net/questions/59498 | 10 | I recently saw a conjecture that a modular form is a modular form for a congruence subgroup of $SL\_2(Z)$ if and only if it has bounded denominators. Are both directions conjectures, or is one already known to be true?
| https://mathoverflow.net/users/13886 | Bounded denominators for modular forms | As Ramsey pointed at, it is known that forms on congruence subgroups have bounded denominators. The other direction is still a conjecture, though some partial progress has been made. You might try looking at some of the papers by Ling Long ( <http://orion.math.iastate.edu:80/linglong/> ). She has been interested in thi... | 11 | https://mathoverflow.net/users/nan | 59518 | 36,960 |
https://mathoverflow.net/questions/59511 | 10 | I've been learning about the Circle Method (at the level of the book
"An Invitation to Modern Number Theory," by Miller and Takloo-Bighash). The arguments in the book show how the
Circle method can be applied successfully to the ternary Goldbach problem, but fails for the binary problem (since the minor arc
contribu... | https://mathoverflow.net/users/8955 | The Circle Method and the binary Goldbach Problem | I would highly recommend Nathanson's book.
Also, you might want to look at the following papers for some fairly recent applications of the circle method:
Vu, Van (2000). "On a refinement of Waring's problem". Duke Mathematical Journal 105 (1).
Wooley, Trevor (2003). "On Vu's thin basis theorem in Waring's problem... | 10 | https://mathoverflow.net/users/10898 | 59519 | 36,961 |
https://mathoverflow.net/questions/59488 | 5 | Is there any (conjectural) characterization of $\overline{\bf{Q}}$-points
on Shimura varieties?
The question of course does not always make sense
for ${\bf{Q}}$-points: a theorem of Shimura shows that a quaternionic Shimura curve has
no ${\bf{R}}$-points, and a theorem of Mazur shows that the modular curve
$Y\_0(N)$... | https://mathoverflow.net/users/6121 | Characterization of algebraic points on Shimura varieties? | If you haven't, you should first think about these questions just for modular curves, which are the simplest Shimura varieties. Then there are only finitely many $N$ for which the modular curve of level $N$ has genus $< 2$. Once the genus is at least $2$, there are only finitely many points over any fixed number field ... | 10 | https://mathoverflow.net/users/2874 | 59528 | 36,965 |
https://mathoverflow.net/questions/59531 | 2 | Is it possible for a group (non-simple and non-abelian) that solvability of all of its proper subgroups leads the whole group to be solvable?
| https://mathoverflow.net/users/13898 | About solvable groups | No. $SL(2,5)$ is a non-simple non-solvable group with the property that all its proper subgroups are solvable.
| 15 | https://mathoverflow.net/users/12858 | 59533 | 36,968 |
https://mathoverflow.net/questions/59527 | 15 | Let $\Omega$ be a connected open set in the complex plane. What is the closure of the polynomials in $\mathcal{H}(\Omega)$ the set of holomorphic functions on $\Omega$? The topology is the usual compact convergence topology. Take, for instance, an annulus such as $D(r,R)$, the set of all complex $z$ such that $r<|z|< R... | https://mathoverflow.net/users/13700 | What holomorphic functions are limits of polynomials? | Let $\Sigma\supset\Omega$ be the union of $\Omega$ and all bounded components of ${\mathbb C}\setminus \Omega$. The algebra you get is the algebra of all holomorphic functions on $\Sigma$.
First, every $f\in{\cal H}(\Sigma)$ is a locally uniform limit of polynomials as a consequence of Runge's Theorem, see Corollary ... | 21 | https://mathoverflow.net/users/nan | 59536 | 36,971 |
https://mathoverflow.net/questions/59541 | 3 | Poinsot’s construction describes the motion of a freely rotating rigid body in terms of an ellipsoid rolling on a plane. (<http://www.phys.ttu.edu/~huang24/Teaching/Phys5306/CH5C.pdf>), and the path of the point of contact is called the herpolhode, and gives the direction of the angular velocity. It seems to me that th... | https://mathoverflow.net/users/13900 | Herpolhode equation | I believe there is quite a large classical literature on the herpolhode. For example
<http://www.archive.org/stream/cu31924005727965#page/n477/mode/2up>
and following pages. I actually came across the term first, I think, in Greenhill's book on the application of elliptic functions - seems to come up via the inters... | 2 | https://mathoverflow.net/users/6153 | 59543 | 36,976 |
https://mathoverflow.net/questions/59537 | 0 | As J.S.Rose noted in his book "A Course on Group Theory" : There is a section of GLn(F) which is isomorphic to PSLn(F), n≥1, F is a field"?. I ask that "What can this section be?"
| https://mathoverflow.net/users/13898 | Groups GLn(F) and PSLn(F) | No clue about the book but there is no non-trivial homomorphism from PSL(2,5) to GL(2,5). Use MAGMA or GAP to check it if it is not clear to you.
| -1 | https://mathoverflow.net/users/5301 | 59553 | 36,981 |
https://mathoverflow.net/questions/59479 | 11 |
>
> **Question.** Is every $(p, \, p)$ closed form ($p\geq1$) in a contractible open set of $\mathbb{C}^n$ $\partial \bar{\partial}$ exact?
>
>
>
We know that every $d$-exact $(p, \,p)$-form on a compact Kahler manifold is $\partial \bar{\partial}$ exact (by the Hodge theorem), but unfortunately, that can't be ... | https://mathoverflow.net/users/3709 | $\partial \bar{\partial}$ lemma for contractible domains | Okay, here is a counter-example. Let $X$ be the following open subset of $\mathbb{C}^2$:
$$X:= \{ (z\_1, z\_2) : |z\_1| < 2,\ |z\_2| < 1 \} \cup \{(z\_1, z\_2): |z\_1| < 1,\ |z\_2| < 2 \}$$
This is the standard example of a contractible space for which $H^1(X, \mathcal{O})$ is nonzero. We know that $H^1(X, \mathcal{O... | 18 | https://mathoverflow.net/users/297 | 59554 | 36,982 |
https://mathoverflow.net/questions/59319 | 10 | This question is about the classical Adams spectral sequence. Squaring operations are defined on its $E\_2$ term. I'd like to know how to compute some of the non-trivial operations, such as $Sq^2 ( c\_0 ) = h\_0 e\_0$. I feel like this ought to be doable in the May spectral sequence, but I don't know the details.
I'm... | https://mathoverflow.net/users/13856 | Computing squaring operations in the Adams spectral sequence | I think for $k>0$ currently no-one knows an efficient **algorithmic** way to compute the $Sq^k$, e.g., from a minimal resolution. (The $Sq^0$ is easy since it is induced by the "Frobenius" map on $A\_\ast$ and one just needs to compute a chain map).
For the May spectral sequence you might want to check out *Nakamura... | 4 | https://mathoverflow.net/users/8824 | 59564 | 36,988 |
https://mathoverflow.net/questions/59560 | 0 | Let $(X,\mathfrak{M},\mu)$ be a positive finite measure space, then
define $\rho(A,B)=\int\_X |\chi\_A-\chi\_B|d\mu$.
Is $(\mathfrak{M},\rho)$ a complete metric space(modulo sets of measure 0)?
I am trying very hard to look for any references, but I cannot find any.
So, if $(\mathfrak{M},\rho)$ is a complete metr... | https://mathoverflow.net/users/13905 | Is this metric space complete? | To complement Michael Renardy's response, a reference is: "1.12.6. Theorem" in Bogachev's Measure Theory, Volume 1. The proof goes as follows. Take any Cauchy sequence in the metric, then pass to a subsequence in which the mutual distances converge to zero very fast, then the original sequence converges in the metric t... | 2 | https://mathoverflow.net/users/11919 | 59565 | 36,989 |
https://mathoverflow.net/questions/59520 | 57 | Less tongue in cheek, is it known what the relative consistency is for theorems proved with an automatic theorem prover? Of course this depends somewhat on what assumptions one makes with respect to predicativity and so on. But for concreteness take one of the popular packages with its standard installation.
Perhaps ... | https://mathoverflow.net/users/4177 | How true are theorems proved by Coq? | For systems like Coq that are based on type theory, this question is trickier to answer than you might expect.
First of all, what does it take to "know" the consistency strength of some system? Classically, the most thoroughly studied logical systems are based on first-order logic, using either the language of elemen... | 45 | https://mathoverflow.net/users/3106 | 59577 | 37,001 |
https://mathoverflow.net/questions/59490 | 4 | First lets fix a prime $p$ (I really care about $p=2$ but would be happy to know about other primes as well). When localized at a prime the spectrum $ku$ (Complex connective K-theory) splits as a wedge of suspensions of a spectrum $l$. In their paper on the Cooperation algebra of the Adams Summand, Baker-Richter state ... | https://mathoverflow.net/users/3901 | Complex orientation of the Adams Summand | Yes, the complex orientation can be factored through these truncations of BP. Either classical methods (the Baas-Sullivan theory of manifolds with singularity - see Baas' "On bordism theory of manifolds with singularities") or more modern methods (see e.g. Strickland's "Products on MU-modules") produce truncated Brown-... | 6 | https://mathoverflow.net/users/360 | 59585 | 37,005 |
https://mathoverflow.net/questions/59583 | 3 | Hello all, I'm trying to find a good resource for a discussion on the relation between say, the p-norm of a vector (from a finite dimensional vector space) and its Euclidean norm. In my search on the internet and in various books, I only encounter basic, standard inequalities such as the Cauchy-Schwarz and Holder's ine... | https://mathoverflow.net/users/5534 | Inequalities and bounds for relating p-norms (Reference request) | It is no better than what you get from Holder. Take the case where one of the vectors is zero.
| 2 | https://mathoverflow.net/users/2554 | 59587 | 37,007 |
https://mathoverflow.net/questions/59515 | 8 | Suppose $G\to GL(V)$ is a linear representation of an irreducible algebraic group over a field $k$.
>
> Suppose $C\subseteq V$ is a $G$-invariant closed cone that spans $V$, and that the stabilizer of any point of $C$ is linearly reductive. Must $V$ be a direct sum of 1-dimensional representations?
>
>
>
[Edit... | https://mathoverflow.net/users/1 | If a representation has enough reductive stabilizers, is it a direct sum of characters? | The answer is yes. We know G is reductive, take B=TU a Borel.
Decompose V into a direct sum of weight spaces for T: $V=\oplus V\_\lambda$. I claim that C contains a non-zero wieght vector. Proof: First lets consider T=Gm. If v=Σivi in C with t in Gm acting on vi by multiplication by ti, consider $\lim\_{t\to \infty}t... | 6 | https://mathoverflow.net/users/425 | 59590 | 37,008 |
https://mathoverflow.net/questions/59106 | 26 | I am interested in different contexts in which Gödel's incompleteness theorems arise. Besides traditional Gödelian proof via arithmetization and formalization of liar paradox it may also be obtained from undecidability results of Turing and Church. In the context of complexity theory, it is not hard to see that Gödel's... | https://mathoverflow.net/users/6307 | Proofs of Gödel's theorem | Apart from usual proofs with diagonalization, have a look at model-theoretic proofs (Kotlarski's proof, Kreisel's left-branch proof, etc..), then there are some other proofs that formalize paradoxes (Kikuchi's, Boolos's, etc... there are about a dozen, most of them mentioned in Kotlarski's book).
If you don't want fu... | 21 | https://mathoverflow.net/users/11751 | 59591 | 37,009 |
https://mathoverflow.net/questions/59596 | 0 | Hi,
I have a finite field Fp with p = 11 mod(12) and I am trying to get the third nontrivial root of unity in Fp^2 = Fp^2[x]/(x^2+1). So, i need x where x^3=1.
Somehow I came into a source saying that it would be:
(p-1)/2 + (3^((p+1)/4)) mod(p))\*i where i^2=-1.
But it seems not to be correct:
Any idea?
| https://mathoverflow.net/users/13911 | nontrivial cube root of unity | OK, this is about imitating the formula for a complex cube root of unity. Write p as 12k - 1. The real issue is only why 3 to the power 3k should act as square root of 3 in this field. Square it and apply Fermat's little theorem to see why. (There is a missing factor 2 in the formula you gave.)
| 0 | https://mathoverflow.net/users/6153 | 59602 | 37,016 |
https://mathoverflow.net/questions/59593 | 18 | The *Pontryagin square* (at the prime 2) is a certain cohomology operation
$$
\mathfrak P\_2: H^q(X;\Bbb Z\_2) \to H^{2q}(X;\Bbb Z\_4)
$$
which has the property that its reduction mod 2 coincides with $x\mapsto x^2$. Furthermore,
If $x\in H^q(X;\Bbb Z\_2)$ is the reduction of an integral class $y$, then $\mathfrak P\_2... | https://mathoverflow.net/users/8032 | Geometric interpretation of the Pontryagin square | This is just a guess, based on your comment about the case when $Q$ and $M$ are both oriented.
Claim: If $M$ is oriented and $Q$ is unoriented (and perhaps nonorientable), then the self-intersection of $Q$ is well defined in $\mathbb Z$ if $q$ is odd and well-defined in $\mathbb Z\_4$ is $q$ is even.
Proof: Let $Q'... | 11 | https://mathoverflow.net/users/284 | 59603 | 37,017 |
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