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/64931 | 30 | Given a finite-dimensional semisimple complex Lie algebra $\mathfrak{g}$, the Bernstein-Gelfand-Gelfand category $\mathcal O$ is the full subcategory of $\mathfrak g$-modules satisfying some finiteness conditions. It contains all finite-dimensional modules as well as all highest-weight modules, it's Noetherian and Arti... | https://mathoverflow.net/users/2177 | Why the BGG category O? | I'd add to what Ben says the observation that people have found a number of different module categories valuable for different purposes within this same Lie algebra context. (And some Lie theory people don't really find category $\mathcal{O}$ to be all that important in their own work.) Even two people close to the ori... | 21 | https://mathoverflow.net/users/4231 | 64933 | 40,112 |
https://mathoverflow.net/questions/64927 | 7 | Let $(g\_k)$ be a sequence of independent standard gaussians variables on a fixed probability space $\Omega$. Let $(\epsilon\_k)$ be a sequence of independent rademacher variables.
What is the best constant in the following inequality:
$$
\vert\vert\sum\_{k}\epsilon\_k \otimes x\_k \vert\vert\_{L^2(\Omega,E)}
\leq
K... | https://mathoverflow.net/users/5210 | Best constant in comparison between Rademacher and gaussian averages? | $\sqrt{\pi\over 2}$, the reciprocal of the $L\_1$ norm of a standard gaussian, is the best constant. Let $x\_k$ be the kth unit basis vector in $\ell\_1$ and let the sum go from $1$ to $N$. The square of the left hand side is $N^2$ and the square of the right hand side is $N+N\sqrt{2\over \pi}(N-1)\sqrt{2\over \pi}$ (m... | 10 | https://mathoverflow.net/users/2554 | 64944 | 40,118 |
https://mathoverflow.net/questions/64940 | 1 | Consider the sequence
$$
a(n) = \prod\_{u^n=1,u \neq 1}( (1+u)^n+1)
$$
Some terms are:
$$
1,1,0,9,121,2704,118336, 4092529,0,97734390625, \ldots
$$
Alonso del Arte asks:
Question: What are the multiples of $3$ such that
$$
a(3k) =0
$$
I tried some factorization of cyclotomic polynomials without success.
May... | https://mathoverflow.net/users/11016 | Zeros of a sequence related to roots of unity | Suppose that $n=3m$, where $m$ is odd, and $u=e^{2\pi i/3}$. Then
$$ (1+u)^n+1 = ((1+u)^3)^m+1 = (-1)^m+1=0, $$
so $a(n)=0$.
| 10 | https://mathoverflow.net/users/2807 | 64946 | 40,120 |
https://mathoverflow.net/questions/64921 | 3 | Hi
I'm interested in packing the 3 space as dense as possible using equally sized tori whose major radius is much bigger than their minor radius in.
Do you have any idea how to attack this problem?
I'm fairly new to this topic and I haven't found many papers for non-convex objects.
(I think that the torus is someho... | https://mathoverflow.net/users/44243 | Torus in $\mathcal{R}^3$ | This is a remark about [Villarceau circles](http://en.wikipedia.org/wiki/Villarceau_circles)
The torus completely decrivbed by two radii, say $(R,r)$.
($R>r$ and your torus is $r$-nbhd of a circle of radius $R$).
Let us denote by $d(R,r)$ the density with which the space can be packed by your torii.
Then
$$\limsup\_... | 5 | https://mathoverflow.net/users/1441 | 64961 | 40,131 |
https://mathoverflow.net/questions/63132 | 13 | Can anyone help me out with proofs/counterexamples? I'm working on an operator-valued multiplicative ergodic theorem and need what may(?) be a well-known fact. This fact (if true) would help me get rid of an annoying asymmetry in the conclusion of a theorem.
I'm assuming that $T$ is an invertible ergodic transformati... | https://mathoverflow.net/users/11054 | Non-integrable ergodic theory | $\newcommand{\R}{\mathbb R}$
$\newcommand{\P}{\mathbf P}$
$\newcommand{\Z}{\mathbb Z}$
I found this question very interesting and gave it much thought this week. I believe I have a proof now. I think it would be interesting to see what generalizations one can get from this argument. Below I use probabilistic notation... | 10 | https://mathoverflow.net/users/1061 | 64963 | 40,133 |
https://mathoverflow.net/questions/64955 | 11 | The infinity axiom can be formulated by defining a function $S$ as
$$S(N) = \{0\} \cup \{n+1\\ |\\ n \in N\}$$
(FWIW, I'm assuming the von Neumann ordinals.) The axiom is then
$$\exists I . I = S(I)$$
which gives us our first infinite set. Then $\omega$ is the intersection of all the subsets of $I$ that are als... | https://mathoverflow.net/users/10828 | Is there a least-fixed-point formulation of inaccessible cardinals? | Every inaccessible cardinal is a fixed point of the operation $P$
that assigns to every set $X$ of ordinals the set $P(X)=\{2^{|\alpha|}:\alpha\in X\}\cup\bigcup X$.
On the other hand, every (nonempty) fixed point of $P$ is a strong limit, the least nonempty
fixed point being $\omega$.
Now here is the problem: If you... | 16 | https://mathoverflow.net/users/7743 | 64964 | 40,134 |
https://mathoverflow.net/questions/64982 | 46 | I'm a grad student getting close to submitting my first journal article (which will be single-authored). My understanding is that it's standard practice for authors to transfer the copyright of their paper to the journal in which it is published. I want my article to be published in a journal, but I don't want to trans... | https://mathoverflow.net/users/15105 | Publishing journals articles without transferring copyright. | Most journals in math allow you to publish a version of the paper which was previously posted to the arxiv.org. They ask you often to take the copyright for the published version which just slightly differs from the arxiv version. So there is not much difference between having it public or having a slightly different v... | 23 | https://mathoverflow.net/users/35833 | 64983 | 40,144 |
https://mathoverflow.net/questions/64977 | 3 | Suppose I had a complete bipartite graph with edges each given some numerical "cost" value. Is there a way to select a subset of those edges such that each vertex on each side of the graph is mapped to each vertex on the other (one to one) and the total "costs" is maximized (or minimized)?
Has anyone ever formulated ... | https://mathoverflow.net/users/15103 | "Bipartite" Travelling Salesman Problem? | You are looking for the max-flow-min-cut theorem: <http://en.wikipedia.org/wiki/Max-flow_min-cut_theorem>
| 2 | https://mathoverflow.net/users/1056 | 64990 | 40,148 |
https://mathoverflow.net/questions/64877 | 5 | The [Polygonal number theorem](http://en.wikipedia.org/wiki/Fermat_polygonal_number_theorem) states that every positive integer is the sum of at most $n$ $n$-gonal numbers.
I do think that expecting a "Proof without words" of this theorem is asking for something unreasonable, but is there any way to "visualize" or t... | https://mathoverflow.net/users/7144 | Is there a geometrical interpretation to Fermat's polygonal number theorem? | Let me make some comments on the polygonal number theorem along the theme that things are less thrilling for $k \gt 4$.
1. Triangular and square numbers are pleasing to represent as dot patterns, after that $k$-gonal numbers are less attractive.
2. Around 1796 Gauss proved that every integer is a sum of 3 triangular ... | 6 | https://mathoverflow.net/users/8008 | 64999 | 40,153 |
https://mathoverflow.net/questions/64973 | 3 | Let $d$ be an integer $\geq 2$, and let $\Omega = \lbrace 0,1 \rbrace^d$, $A \subseteq \lbrace 0,1 \rbrace^2 $ and $i,j$ integers with $1 \leq i < j \leq d$. If we select an element $(x\_1,x\_2, \ldots, x\_d)$ in $\Omega$ randomly, the probability $p(\Omega,i,j,A)$ that $(x\_i,x\_j)\in A$ is exactly $\frac{|A|}{4}$.
... | https://mathoverflow.net/users/2389 | Sufficiently random sample | I like the question. You are actually asking, what is the smallest finite probability space (in size of $\Omega'$) on which one can have $d$ distinct pair-wisely independent (but not necessarily independent) events of probability $\frac{1}{2}$. Just to explain the reformulation: once the $i$-th event is defined to be
$... | 3 | https://mathoverflow.net/users/14763 | 65004 | 40,156 |
https://mathoverflow.net/questions/65002 | 6 | Whenever possible, I like to present Cantor's diagonal proof of the uncountability of the reals to my undergraduates. For simplicity, I usually restrict to showing that the subset
$$
A = \{x \in [0,1) \mid \text{ the decimal representation of $x$ uses only 0's and 1's} \}
$$
is already uncountable. I was thinking recen... | https://mathoverflow.net/users/8871 | A Bijection Between the Reals and Infinite Binary Strings | Just for the fun, we can use continued fractions to map the sequences of positive integers injectively to [0,1] the sequence may end with $\infty$ meaning that we get a finite fraction (a rational number). Now, the mapping 01010111100110110... to 001010111100110110... to 2,1,1,1,1,4,2,2,1,2,... is a clear bijection (ad... | 7 | https://mathoverflow.net/users/1131 | 65014 | 40,163 |
https://mathoverflow.net/questions/65018 | 2 | Recall that a morphism $f:X \to B$ is called **shrinkable** is there exists a section $s:B \to X$ together with a homotopy $$H:I \times X \to X$$ from $sf$ to $id\_X$ over $B,$ i.e. for all $t$, the map $$H\_t:X \to X$$ is a map in $Top/B$ from $f$ to $f$.
Call a morphism $f:X \to B$ **parashrinkable** if for every m... | https://mathoverflow.net/users/4528 | Shrinkable maps and universal weak equivalences | Suppose that $f:X\to B$ is such that for every $n$, for every map $D^n\to B$, the resulting map $X\times\_BD^n\to D^n$ is a weak equivalence. Then $f$ is a weak equivalence.
Proof: It suffices to show that for every point $b\in B$ the homotopy fiber of $f$ w.r.t. $b$ is such for every $n\ge 0$ every map of $S^{n-1}$ ... | 3 | https://mathoverflow.net/users/6666 | 65022 | 40,166 |
https://mathoverflow.net/questions/65045 | 8 | I'm looking for an example of a finitely presented and finitely generated amenable group, that has a subgroup which is not finitely generated.
The question is easy for finitely generated amenable group and an example is the lamp-lighter group $C\_2\wr \mathbb{Z}$.
An Abelian and finitely generated group has no suc... | https://mathoverflow.net/users/15125 | Example of an amenable finitely generated and presented group with a non-finitely generated subgroup | I don't know much about amenable groups I am afraid, but according to the Wikipedia article, all solvable groups are amenable. So we can take the Baumslag-Solitar group
$B(1,n) = \langle x,y \mid y^{-1}xy = x^n \rangle.$
If we let $N$ be the normal closure of the subgroup generated by $x$, then $N$ is abelian with ... | 13 | https://mathoverflow.net/users/35840 | 65053 | 40,182 |
https://mathoverflow.net/questions/65041 | 4 | **Question.** Let $p$ be a prime. Let $q$ be a power of $p$. Let $P^0$, $P^1$, $P^2$, ... be elements of some associative $\mathbb F\_q$-algebra $A$. (Here, $P^i$ does not mean $P$ to the $i$-th power; instead, $i$ is an upper index.) Asume that the power series $\sum\limits\_{a,k} \left(tu\right)^a P^a P^k$ and $\sum\... | https://mathoverflow.net/users/2530 | Adem-Wu relations from Bullett-Macdonald identities | Larry Smith is not really using complex integration. Instead, he is using the residue map, which can be defined algebraically by the rule
$$ \text{res}\left(\sum\_{k=-N}^\infty a\_k z^k dz\right)=a\_{-1} $$
and the fundamental transformation property that
$$ \text{res}\left(\sum\_{k=-N}^\infty a\_k \;f(z)^k\;f'(z) dz... | 7 | https://mathoverflow.net/users/10366 | 65056 | 40,184 |
https://mathoverflow.net/questions/64000 | 2 | Let $\Phi $ denote the standard normal CDF, and $\phi$ the standard normal PDF. Fix $\alpha > 0$.
Let
$$ Z\left( r\right) =r\int\_{0}^{\infty } e^{-(r+\alpha )t} \mathbb{E} \left[ \Phi
\left( \frac{e^{-\alpha t}(\theta -\omega )}{\sqrt{\frac{1%
}{2}(1-e^{-2\alpha t})}}\right) \frac{1-\Phi (\theta )}{\phi(\theta)} \;... | https://mathoverflow.net/users/7967 | Limit of an integral involving the normal CDF | Let $U$, $X$ and $Y$ denote independent random variables with $U$ uniform on $(0,1)$ and $X$ and $Y$ standard normal. For every positive $s$, introduce
$$
K(s)=\mathbb{E}\left[U^{s}\Phi\left(W(X-Y)\right)\frac{1-\Phi(X)}{\phi(X)}\right],\qquad
W=\frac{U}{\sqrt{\frac12(1-U^2)}}.
$$
Some simple computations show that for... | 3 | https://mathoverflow.net/users/4661 | 65062 | 40,186 |
https://mathoverflow.net/questions/65047 | 4 | I have come across the following surface: let $X$ be the double covering of $\mathbb{P}\_\mathbb{Z}^2$ defined by the equation
$$y^2=x\_0^6+x\_1^6+x\_2^6$$
where $y$ is a variable of degree 3.
There is an obvious action of $\mu\_6 \times \mu\_6$.
Less obvious, I have made some short calculations that show a great l... | https://mathoverflow.net/users/2024 | How is this surface related to the square of that CM elliptic curve? | This is probably more complicated then necessary, but anyway:
Let $C\_1$ be the smooth genus 2 curve with affine equation $y^2=x^6+1$ and $C\_2$ be the genus 10 curve with affine equation $w^6=z^6+1$. There is a natural $\mu\_6$-action on $C\_1\times C\_2$ such that $X$ is birational to the quotient $(C\_1\times C\_2... | 10 | https://mathoverflow.net/users/8621 | 65067 | 40,189 |
https://mathoverflow.net/questions/65068 | 1 | I wonder why one requires that the base manifold of a Lie groupoid is second-countable?
| https://mathoverflow.net/users/15128 | Why is the base manifold of a Lie groupoid required to be second-countable? | **Answer #1:**
There is *no real reason* for imposing that the base manifold of a groupoid be second countable.
**Answer #2:**
You lose some desirable properties if you don't impose second countability:
For example, without it,
the homotopy type of the geometric realisation of the nerve
will no longer be an inv... | 4 | https://mathoverflow.net/users/5690 | 65076 | 40,192 |
https://mathoverflow.net/questions/65019 | 11 | Wireless networks are typically modeled as random geometric graphs. The number of nodes $N$ in the network is drawn from a Poisson distribution with intensity $\lambda$
$$P(N = n) = \frac{\lambda^n e^{-n}}{n!}.$$
Once this number has been chosen, the $N$ nodes are placed uniformly at random over a circular (or sq... | https://mathoverflow.net/users/13822 | (almost) statistical independence of nodes degrees in a graph | This is a partial answer concerning the validity of the second approximation. I'll consider the case $k=1$ for simplicity: all the difficulties and tricks will be clear from it already.
First of all, when people claim that some probability estimate agrees well with simulations, they always mean the difference metric.... | 4 | https://mathoverflow.net/users/1131 | 65079 | 40,195 |
https://mathoverflow.net/questions/65065 | 2 | Fix a compact Hausdorff space $K$ and think about $C(K)$ as a C\*-algebra acting on a Hilbert space $H$. Suppose that $C(K)$ is closed in $\mathcal{B}(H)$ in:
* $\sigma$-strong
* $\sigma$-strong\*
topology. Must $K$ be extremelly disconnected?
| https://mathoverflow.net/users/15129 | When C(K) is closed in sigma strong topology? | Recall that if two compact spaces $K\_1$, $K\_2$ are such that $C(K\_1)\cong C(K\_2)$, then $K\_1\cong K\_2$.
The space $K$ is called the spectrum of the abelian *C*\*-algebra $C(K)$.
Since $C(K)$ is closed in the σ-strong topology, it is a von Neumann algebra (that condition is equivalent to being closed in the σ-st... | 5 | https://mathoverflow.net/users/5690 | 65080 | 40,196 |
https://mathoverflow.net/questions/65040 | 3 | I am wondering if there are "elementary" ways to check for $n$-connectedness of simplicial complexes in terms of simple conditions on vertices.
For example
Let $X$ be the curve complex on a surface $S$ (the exact type of curve complex does not matter). That is, the vertices of $X$ correspond to curves on $S$. A $p$... | https://mathoverflow.net/users/14243 | High connectivity arguments for simplicial complexes | I don't know whether this is useful in practice, but the straightforward generalization of your 0-connectedness criterion to a 1-connectedness criterion goes as follows.
Any 1-cycle of the form $[(a\_0, a\_1), (a\_1, a\_2), \ldots, (a\_k, a\_0)]$ can be transformed into the trivial (empty) 1-cycle by a sequence of th... | 3 | https://mathoverflow.net/users/284 | 65082 | 40,198 |
https://mathoverflow.net/questions/65070 | 4 | I know about geometrical method of construction of discrete subgroups of $SL(2,\mathbb{R})$ using Lobachevsky plane (e.g. B.A. Dubrovin, A.T. Fomenko, S.P. Novikov, Modern Geometry --- Methods and Applications, Springer) via fundamental polygon. Such construction has many applications and some relevant themes were alre... | https://mathoverflow.net/users/14175 | A construction of generators of discrete subgroups of SL(2,R) | For a pleasant introduction that includes many beautiful pictures, I suggest the book "Indra's Pearls" by Mumford, Series, and Wright. They also give examples of explicit computation of the relevant matrices. [Here's](http://books.google.co.uk/books?id=kC5kdUirHHoC&printsec=frontcover&dq=Indra%2527s+pearls&hl=en&ei=JUr... | 4 | https://mathoverflow.net/users/1650 | 65085 | 40,200 |
https://mathoverflow.net/questions/65086 | 25 | Motivation: $T\_{\mathbb P^2}$ isn't an extension of line bundles
-----------------------------------------------------------------
Here's a trick to show that the tangent bundle $T$ of $\mathbb P^2$ is not an extension of line bundles. If it were, we would have a short exact sequence
$$\def\O{\mathcal O}
0\to \O(a)... | https://mathoverflow.net/users/1 | If the total Chern class of a vector bundle factors, does it have a sub-bundle? | The answer for projective spaces is negative. I think the simplest example are 2-bundles on $\mathbb{P}^3(\mathbb{C})$. In that case the Schwarzenberger condition is that $c\_1c\_2$ should be even. Atiyah and Rees have proved that for any pair $(c\_1,c\_2)$ satisfying this there are holomorphic vector bundles $\xi$ wit... | 19 | https://mathoverflow.net/users/2349 | 65088 | 40,202 |
https://mathoverflow.net/questions/65071 | 4 | Hi,
Let $V$ be an integral domain with an ideal $m\subset V$ and put $K = S^{-1}V$ where
$S = {1}\cup m$ (a multiplicatively closed subset). Is it true that the category of almost
$(V,m)$-modules is equivalent to the category of $K$-modules via $-\otimes\_V K$? By the category of almost $(V,m)$-modules I mean the cat... | https://mathoverflow.net/users/36285 | Faltings' category of almost modules | At least in Faltings's setting, $m$ is the maximal ideal of a non-Noetherian valuation domain $V$. If we let $S$ be the non-zero elements of $m$, then the localization of $V$ at $S$ will
be the field of fractions $K$ of $V$. The category of $K$-modules (i.e. $K$-vector spaces)
*is* obtained as a Serre quotient of $V$-m... | 7 | https://mathoverflow.net/users/2874 | 65089 | 40,203 |
https://mathoverflow.net/questions/65099 | 5 | I seem to recall once hearing a result to the effect that $\emptyset^{(\omega)}$ was the double jump of some other degree, but could not be the triple jump of any degree. However I'm unable to find the exact result. Does anyone know what I might be thinking of (or what is actually known about jump inversion on $\emptys... | https://mathoverflow.net/users/8991 | Jump Inversion of Arithmetic | This doesn't exactly answer your question but...
If $A$ is any upper bound for the arithmetic degrees then $0^{(\omega)}$ is recursive in $A^{\prime\prime}$. Enderton and Putnam proved that there
upper bounds with $A^{\prime\prime}=0^{(\omega)} $
Dave
| 8 | https://mathoverflow.net/users/5849 | 65102 | 40,209 |
https://mathoverflow.net/questions/65109 | 4 | Let $F$ be a field. Let $p$ and $q$ be monic members $F[x]$. Let $I = \{p\cdot r : r\in F[x]\}$ and $J = \{q\cdot r : r\in F[x]\}$. I know that if $F[x]/I$ is isomorphic to $F[x]/J$ then ($\operatorname{deg}(p) = \operatorname{deg}(q)$ and (p is reducible if and only if q is reducible)). For cases where we do have $\op... | https://mathoverflow.net/users/nan | checking if F[x]/I is isomorphic to F[x]/J | If $F$ is finite, $p$ and $q$ are irreducible, and have the same dagree $d$, then $F[x]/I$ and $F[x]/J$ are isomorphic $F$-algebras, since there is (up to isomorphism) only one extension field of a finite field of any given degree (see for example Birkhoff & Mac Lane, A survey of Modern Algebra, Section 15.6).
Things... | 6 | https://mathoverflow.net/users/9672 | 65111 | 40,215 |
https://mathoverflow.net/questions/65115 | 7 | Let $L$ be a finite-dimensional nilpotent subalgebra of the Lie algebra $W\_n$ of all vector fields in $n$ variables (I am interested both in polynomial and formal vector fields). Does there exist a bound in terms of $n$ on the index of nilpotency of $L$?
For $n=1$ the answer is trivially "yes": every finite dimensio... | https://mathoverflow.net/users/1223 | Nilpotent Lie algebras of vector fields | I am afraid not: the subalgebra of $W\_2$ spanned by $\partial\_x, \partial\_y, x\partial\_y, \ldots, x^m\partial\_y$ has the nilpotency index $m+1$.
| 6 | https://mathoverflow.net/users/1306 | 65125 | 40,223 |
https://mathoverflow.net/questions/65059 | 24 | Surely yes, and in more generality, but can it be proved?
It seems that most, if not all, statements about quadratic forms representing primes fall back on algebraic number theory (i.e. splitting of primes in $\mathbb{Q}(\sqrt{7})$) for their proofs, and so are incompatible with the condition that $0 < y < x/10$.
S... | https://mathoverflow.net/users/1050 | Does the quadratic form $x^2 - 7y^2$ represent infinitely many primes, with the restriction that $0 < y < x/10$? | The units of $k=\mathbf{Q}(\sqrt{7})$ have the form $\pm (8+3 \sqrt{7})^n$ with $n \in \mathbf{Z}$. If $\pi = x+y\sqrt{7}$ is a prime element of $k$, then $\lambda(\pi):= \log |x+y\sqrt{7}|$ is well-defined in $\mathbf{R}/\alpha \mathbf{Z}$ where $\alpha = \log(8+3\sqrt{7})$. Note that $\lambda$ factors as $\lambda = f... | 16 | https://mathoverflow.net/users/6506 | 65140 | 40,232 |
https://mathoverflow.net/questions/65138 | 2 | Let $A$ be a ring, $\mathfrak{a}\subset A$ an ideal. Then is the $\mathfrak{a}$-adic topology on $A$ necessarily a metric space? I can see that it is true when $A$ is a DVR, but is it true in general?
| https://mathoverflow.net/users/5292 | Metrizability of $\mathfrak{a}$-adic topology | As explained in the other answers, in general there is a problem of seperatedness. If $\bigcap\_{n=1}^{\infty} \mathfrak{a}^n \ne 0$ then the topology is not Hausdorff.
On the positive side however, you have for example Krull's intersection theorem which says that if $A$ is a noetherian integral domain, then for any ... | 3 | https://mathoverflow.net/users/3759 | 65144 | 40,234 |
https://mathoverflow.net/questions/65143 | 1 | The following definition is given as the Fourier transform of a Borel probability measure $\mu$ on $E$, a Banach Space (Real):
$\hat{\mu}: E^\*\rightarrow \mathbb{C}$ defined by
$\hat{\mu}(x^\*):=\int\_E \; \exp(-i(x,x^\*))d\mu(x),$
where $(\cdot,\cdot)$ is the duality pairing.
One defines the Fourier transfo... | https://mathoverflow.net/users/2048 | Fourier Transform of measure on Banach Space (a question about Pontryagin Duality) | Yes. Let $\phi:(E,+)\rightarrow S^1$ be a continuous homomorphism. For each $x\in E$, the map $\mathbb R\rightarrow S^1; t\mapsto \phi(tx)$ is continuous and a homomorphism, so there is some $\mu(x)\in\mathbb R$ with $$\phi(tx) = \exp(i t \mu(x)) \qquad (x\in E,t\in\mathbb R).$$ Then, for $x,y\in E$ and $\lambda\in\mat... | 4 | https://mathoverflow.net/users/406 | 65145 | 40,235 |
https://mathoverflow.net/questions/65149 | 11 | Let $f\_\alpha$ be a family of continuous positive functions $\mathbb R\to \mathbb R$
where the index $\alpha$ runs in a compact metric space
and the map $\alpha\to f\_\alpha$ is continuous
with respect to compact-open topology on the target. Suppose there is a uniform
upper bound on the integrals of $f\_\alpha$'s o... | https://mathoverflow.net/users/1573 | Is the supremum of continuous functions integrable? | No. Let the compact metric space be $[0,1]$ with the standard topology and define
$$
f\_\alpha(x) =\alpha\max(1-\alpha \vert x\vert,0)
$$
for $\alpha\in[0,1]$. This satisfies the properties asked for, with the upper bound $\int f\_\alpha(x)\\,dx\le1$ (and, equality for $\alpha\not=0$).
But,
$$
\sup\\_\alpha f\_\alpha(x... | 11 | https://mathoverflow.net/users/1004 | 65150 | 40,237 |
https://mathoverflow.net/questions/65156 | 4 | On $\mathbf R^d$ the Ornstein-Uhlenbeck operator is defined as ($\partial\_i = \frac{\partial}{\partial x\_i}$).
$$L = \frac12 \sum\_i \partial\_i^\* \partial\_i$$
where $\partial\_i^\* = -\partial\_i + 2 x\_i$.
Now we can form the semigroup $e^{-t L}$ and the corresponding maximal function
$$M^\* f(x) = \sup\... | https://mathoverflow.net/users/5295 | Maximal function related to the Ornstein-Uhlenbeck operator. | There is a considerable literature about such operators. See, for example,
P. Sj\"ogren, Operators associated with the Hermite semigroup - a survey. J. Fourier Anal. Appl. 3, Spec. Iss., 813-823 (1997).
S. Pérez and F. Soria, Operators associated with the Ornstein-Uhlenbeck semigroup.
J. Lond. Math. Soc., 61, No.3... | 4 | https://mathoverflow.net/users/12205 | 65163 | 40,242 |
https://mathoverflow.net/questions/65130 | 6 | Does there exist a prime 3-manifold such that its mapping class group has an abelian representation in which the 2$\pi$ rotation is represented by -1?
In detail:
Let $M$ be a closed orientable prime 3-manifold.
Let $D\_F(M,p)$ be the group of diffeomorphisms of $M$ that fix a point $p$ of $M$ and a frame there. Defin... | https://mathoverflow.net/users/15143 | Do abelian spinorial prime three manifolds exist? | Yes, there exists such a manifold $M$. This follows if there exists aspherical $M$ with $Diff(M)\simeq 0$ (contractible) and $H^2(M;\mathbb{Z}/2\mathbb{Z})=0$. I claim there exists such manifolds. Let's see why such $M$ suffice.
There are two fibrations:
$$ D\_F(M,p) \to Diff(M,p) \to GL(3,\mathbb{R})$$
and
$$ Diff(... | 8 | https://mathoverflow.net/users/1345 | 65167 | 40,245 |
https://mathoverflow.net/questions/65152 | 0 | Are there standard references on infinite products of rational functions and their convergence properties? I'd appreciate information on finite products too!
The original motivation for this is the (finite) product $f(n)=\prod\_{i=1}^{n-1}(1-\frac{i}{i^2+n})$ that I had to bound some time ago. Applying some calculus... | https://mathoverflow.net/users/1939 | Functions defined as infinite products | According to Maple, the finite product $f(n) = \frac{\Gamma(n - (1 + \sqrt{1-4n})/2) \Gamma(n-(1-\sqrt{1-4n})/2) \Gamma(1-\sqrt{-n}) \Gamma(1+\sqrt{-n})}
{\Gamma(n-\sqrt{-n}) \Gamma(n+\sqrt{-n}) \Gamma((1-\sqrt{1-4n})/2) \Gamma((1+\sqrt{1-4n})/2)}$.
The infinite product is 0, because $\sum\_i i/(i^2+n)$ diverges.
| 4 | https://mathoverflow.net/users/13650 | 65177 | 40,254 |
https://mathoverflow.net/questions/65157 | 4 | Hello,
given 2 different finite $p$-groups $G$ and $H$, $|G|=|H|=p^n$.
It has been shown by Ian Leary that the mod-p cohomology rings do not determine the groups $G$ and $H$. In fact he gave an example of such groups.
Are there more known examples of pairs of non-isomorphic groups that give rise to the same cohomolog... | https://mathoverflow.net/users/3102 | Examples of p-groups exhibiting isomorphic mod-p cohomology rings. | Let $\Gamma$ be the kernel of the reduction map $GL\_n(\mathbb{Z}/p^3\mathbb{Z}) \to GL\_n(\mathbb{Z}/p\mathbb{Z})$. $\Gamma$ is a non-abelian group of order $p^{2n^2}$ and $p$-rank $n^2$. Its mod-$p$ cohomology is
$$ H^\ast(\Gamma;\mathbb{F}\_p) = \mathbb{F}\_p[x\_1,...,x\_{n^2}] \otimes \Lambda(y\_1,...,y\_{n^2}).$$
... | 6 | https://mathoverflow.net/users/10194 | 65189 | 40,262 |
https://mathoverflow.net/questions/65183 | 13 | Let $p$ be an odd prime. I am interested in how many quadratic residues $a$ sre there such that $a+1$ is also a quadratic residue modulo $p$. I am sure that this number is
$$
\frac{p-6+\text{mod}(p,4)}{4},
$$
but I have neither proof nor reference. It is a particular case of the question in the title: if $a$ and $b$ ar... | https://mathoverflow.net/users/1168 | When is the sum of two quadratic residues modulo a prime again a quadratic residue? | Here's a copy-paste of something I wrote up a while ago:
Lemma: Let $q$ be odd, and let $Q$ be the set of quadratic residues (including $0$) in $\mathbb F\_q$. Then the number of elements $s\_q(c)$ in $\{x^2+c|x \in \mathbb{F}\_q\} \cap Q$ is given by
\begin{array}{|c|c|c|}
\hline
& c \in Q & c \notin Q \\
\hline
-1... | 16 | https://mathoverflow.net/users/262 | 65193 | 40,264 |
https://mathoverflow.net/questions/64653 | 7 | Consider the two (inequivalent) $\mathbb{Z}$-representations $\phi,\psi$ of the symmetric group $S=S\_3$ given by
$(1,2)^\phi=\left(\begin{array}{rr}0 &-1\\\ -1 & 0\end{array}\right), \qquad
(1,2,3)^\phi=\left(\begin{array}{rr}0 &1\\\ -1 & -1\end{array}\right);$
$(1,2)^\psi=\left(\begin{array}{rr}0 &1\\\ 1 & 0\end{... | https://mathoverflow.net/users/12961 | Embedding $S_3$ into $Aut(F_2)$ | As Tom Goodwillie noted in his comment, $GL(2,\Bbb Z)$ can be identified with $Out(F\_2)$, so the question can be rephrased in terms of lifting subgroups of $Out(F\_2)$ to $Aut(F\_2)$. There is a Realization Theorem for finite subgroups of $Aut(F\_n)$ and $Out(F\_n)$ which says that such a subgroup can always be realiz... | 8 | https://mathoverflow.net/users/23571 | 65198 | 40,267 |
https://mathoverflow.net/questions/63496 | 6 | We can sample a discrete value from the multinomial distribution.
We can also sample the parameters of the multinomial distribution from its conjugate prior the dirichlet distribution.
Since the dirichlet distribution is part of the exponential family, it too must have a conjugate prior distribution in the exponent... | https://mathoverflow.net/users/942 | What can be said about an infinite linear chain of conjugate prior distributions? | Let's say that you have a distribution $F$ in the [exponential family](http://en.wikipedia.org/wiki/Exponential_family) with density
\begin{align}
\newcommand{\mbx}{\mathbf x}
\newcommand{\btheta}{\boldsymbol{\theta}}
f(\mbx \mid \btheta) &= \exp\bigl(\eta(\btheta) \cdot T(\mbx) - g(\btheta) + h(\mbx)\bigr)
\end{align}... | 5 | https://mathoverflow.net/users/634 | 65203 | 40,270 |
https://mathoverflow.net/questions/65194 | 27 | Let me first define the majorization order (or dominance order) on partitions as $\lambda \succeq \mu$ iff $$\sum \_{i=1}^{k}\lambda\_i \geq \sum\_{i=1}^{k}\mu\_i$$ for all $1\le k\le l-1$ and $$\lambda\_1+\cdots+\lambda\_l=\mu\_1+\cdots+\mu\_l.$$
While playing around with some variation on [Muirhead's inequality](ht... | https://mathoverflow.net/users/2384 | Majorization and Schur Polynomials | This question is listed as a conjecture (conjecture 7.4 in the section "Open questions") in a recent [paper](http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WDY-526CFXB-3&_user=10&_coverDate=02%2F17%2F2011&_rdoc=1&_fmt=high&_orig=gateway&_origin=gateway&_sort=d&_docanchor=&view=c&_acct=C000050221&_version=1&... | 16 | https://mathoverflow.net/users/1306 | 65217 | 40,278 |
https://mathoverflow.net/questions/65215 | 1 | In an article of Sten Kaijser ("A note on dual Banach spaces") I find the assertion that $E = L^2\_{\text{loc}}({\mathbb R})$ modulo constants is a reflexive space.
Question 1: which is the 'natural' locally convex top. vector space structure of this space?
(maybe the proj. limit of $L^2(K\_n)$-spaces modulo consta... | https://mathoverflow.net/users/15166 | "L^2_loc mod constants" as a reflexive space | I'd say the natural structure on $E$ is one of a Frechet space, given by the family of seminorms $\|f\|\_{L^2(K\_n)}$. Each local space generated by each seminorm is reflexive (Hilbert) so the resulting topology is reflexive. Then if you take the quotient by the closed subspace of constant functions you get again a ref... | 2 | https://mathoverflow.net/users/7294 | 65218 | 40,279 |
https://mathoverflow.net/questions/65200 | 2 | Given 4 random points in 2D, how do I compute the area of the quadrilateral formed by the points?
I'm aware of formulae giving the area when I know the sides a,b,c,d and the diagonals p & q.
But how do I decide algorithmically which of the 6 connecting lines between the 4 points are sides and which are diagonals?
| https://mathoverflow.net/users/15161 | Quadrilateral from 4 random points | Here's a literal answer to your question: Sort the points, so that $a$ is has the leftmost $x$-coordinate, $b-a$ has the smallest argument (choosing the branch where arguments take values in $(-\pi,\pi]$), and $c-a$ has the largest argument. Take half the ($z$-coordinate of the) cross-product of $b-a$ with $c-a$ to get... | 3 | https://mathoverflow.net/users/121 | 65219 | 40,280 |
https://mathoverflow.net/questions/65220 | 3 | Hi. I have a feeling this is kinda a known thing, but I don't know it, so hopefully you can help me out. I'm a lowly software developer.
For a given n and k, is there a short-cut way of calculating (or approximating) the value of this expression?
$\sum\_{n}^{k}\frac{1}{n}$
It's a matter of computational efficien... | https://mathoverflow.net/users/15168 | sigma 1/n up to k | If you mean that the sum starts at 1 these are exactly the Harmonic numbers, see <http://en.wikipedia.org/wiki/Harmonic_number>
There are well-known asymptotic expansions (see the above mentioned page for a start, and below).
For actual computation, there is a nice blogpost on computing them by Fredrik Johansson: <... | 3 | https://mathoverflow.net/users/nan | 65221 | 40,281 |
https://mathoverflow.net/questions/65180 | 15 | I know very little about the fancy equivariant stable homotopy category, so I apologize if this question is silly for one reason or another, but:
I think that stable homotopy, in the non-equivariant case, corresponds to the homology theory associated to the sphere spectrum. And indeed, in the equivariant context we h... | https://mathoverflow.net/users/6936 | Why are equivariant homotopy groups not RO(G)-graded? | If $G$ acts linearly and isometrically on $V$ then the unit disk $D(V)$ and its boundary $S(V)$ are an appealing kind of equivariant disk and sphere. You can try taking the pairs $(D(V),S(V))$ as the building blocks for equivariant complexes, but you won't get all the equivariant homotopy types that you want, not even ... | 19 | https://mathoverflow.net/users/6666 | 65222 | 40,282 |
https://mathoverflow.net/questions/65211 | 8 | This might be the most stupid question I am ever posting here: I am asking for a proof or a counterexample to a problem [I proposed on MathLinks long ago](http://www.artofproblemsolving.com/Forum/viewtopic.php?f=42&t=137128).
Let $G$ be a bipartite graph, i. e., a graph such that the set of its vertices is the union ... | https://mathoverflow.net/users/2530 | Condition on a bipartite graph to have an $m$-factor | Take the general Ore-Ryser theorem:
Let $G$ be bipartite graph with vertex set $V=X\coprod Y$ ($X$, $Y$ parts) and $f:V\rightarrow \{0,1,2,\dots\}$ be a function such that $f\left(x\right)\leq\left(\text{degree of }x\right)$ for every $x\in X$. Then there exists a subgraph of $G$ (obtained from $G$ by removing some e... | 8 | https://mathoverflow.net/users/4312 | 65224 | 40,283 |
https://mathoverflow.net/questions/65191 | 2 | Say I have a set $X=x\_1,x\_2,\ldots,x\_n$ of random variables, and would like to find a size $k\leq|X|$ subset that contains as much information as possible. This is complicated because the variables may contain redundant information.
It seems, intuitively, that "information" behaves like the size or volume of a se... | https://mathoverflow.net/users/15159 | Is there some generalization of the "Maximum Coverage Problem" for information in random variables? | The prototypical way of measuring the "information content" of a set of random variables is by evaluating the [Shannon entropy](http://en.wikipedia.org/wiki/Entropy_%28information_theory%29) of their joint distribution. It is known that entropy is a [submodular set function](http://en.wikipedia.org/wiki/Submodular) and... | 4 | https://mathoverflow.net/users/8049 | 65225 | 40,284 |
https://mathoverflow.net/questions/65226 | 18 | I originally thought of asking this question at the Mathematics Stackexchange, but then I decided that I'd have a better chance of a good discussion here.
In the Wikipedia page on Ramanujan ([current revision](https://en.wikipedia.org/w/index.php?title=Srinivasa_Ramanujan&oldid=1131559732#Selected_publications_on_wor... | https://mathoverflow.net/users/7144 | The Ramanujan Problems | There is a survey [article](https://www.google.com/books?hl=en&lr=&id=TT1T8A94xNcC&oi=fnd&pg=PA215&dq=related:DJmDgKBpMq0J:scholar.google.com/&ots=ujqGzfahmA&sig=FpslhlMUpn6dFwItc5FINa7XYE0#v=onepage&q&f=false) by Berndt, Choi, and Kang devoted to the set of 58 Ramanujan's problems. They indicate that the questions had... | 14 | https://mathoverflow.net/users/5371 | 65237 | 40,288 |
https://mathoverflow.net/questions/65112 | 7 | Godel's theorem tells us that any sufficiently powerful consistent formal theory of the integers is incomplete; but what about formal theories of the real numbers? More precisely, what about theories of the real numbers that are categorical (i.e. have only one model)? One such theory is given by the ordered field axiom... | https://mathoverflow.net/users/3621 | incompleteness in real analysis | Let me address the updated version of your question.
There is a philosophical current running through parts of [descriptive set theory](http://en.wikipedia.org/wiki/Descriptive_set_theory), and this includes anything that might be described as classical real analysis, to the effect that the realm of Borel mathematic... | 7 | https://mathoverflow.net/users/1946 | 65240 | 40,290 |
https://mathoverflow.net/questions/65236 | 8 | Let $G$ be algebraic group acting on a smooth algebraic variety $M$. Assume the action is proper and free. Is the orbit space $M/G$ an algebraic variety? If so, could someone point to a reference? If not, could someone point to a counterexample? Thanks.
| https://mathoverflow.net/users/14993 | Quotient of smooth algebraic variety by proper free action of algebraic group | The answer is **no**.
A counterexample is given in the paper by James K. Deveney and David R. Finston
[A Proper $G\_a$ Action on $\mathbb{C}^5$ Which is Not Locally Trivial](http://www.jstor.org/stable/2160782),
Proc. Amer. math. Soc. **123** (1995).
Quoting from the introduction:
>
> An assertion which would... | 11 | https://mathoverflow.net/users/7460 | 65243 | 40,292 |
https://mathoverflow.net/questions/65239 | 0 | I hope this question isn't trivial. Let $L$ be a lattice in $\mathbb{C}$ generated by two complex numbers $w\_1,w\_2$ which are linearly independent over $\mathbb{R}$. Let $\gamma\in\mathbb{C}$ be a root of unity such that $\gamma\notin\mathbb{R}$ and $\gamma L=L$, and let $l\in L\backslash\{0\}$ be an element of short... | https://mathoverflow.net/users/14143 | Lattice generators | Since $\gamma L=L$, we have $\gamma w\_1=aw\_1+bw\_2$, $\gamma w\_2=cw\_1+d w\_2$ with integer $a,b,c,d$, which immediately implies that $\gamma$ is a root of a polynomial of degree $2$ with integer coefficients, so the root of unity in question can be of degree $3,4,6$ only (otherwise the cyclotomic polynomial is irre... | 2 | https://mathoverflow.net/users/1306 | 65245 | 40,294 |
https://mathoverflow.net/questions/39914 | 13 | Given an abelian category C, we can form the Yoneda extensions $YExt^i(X,Y)$ to the equivalent classes of $i$-extensions of X by Y.
Given any abelian category C, we can always formulate the derived category D(C), and define $Ext^i\_C(X,Y)$ to be $Hom\_{D(C)}(X,Y[i])$.
Now we can naturally associate a Yoneda $i$-exte... | https://mathoverflow.net/users/3848 | group of Yoneda extensions and the EXT groups defined via derived category | It is true in general and Verdier did the computation in his thesis Des Catégories Dérivées des Catégories Abéliennes Chap III Sect 3 ... the thesis is on line!
| 11 | https://mathoverflow.net/users/15175 | 65251 | 40,297 |
https://mathoverflow.net/questions/65023 | 14 | Let $P$ be a subset of $\mathbb N^d$ (or of some normal pointed affine semigroup), and suppose that $f:=\sum\_{p\in P}\ t^p\in\mathbb Z[[t\_1,\ldots,t\_d]]$ is a rational function. What can be said about the structure of $P$? In particular, must $P$ be a finite (disjoint) union of finitely generated modules over affine... | https://mathoverflow.net/users/30 | Which sets of lattice points have rational generating functions? | I feel like there has to be an easier proof of this, but I just posted [a note](http://www.math.lsa.umich.edu/~speyer/PowSerNote.pdf) on my webpage proving the following Theorem. The key is a [paper of Sam Payne's](http://arxiv.org/abs/0706.2438).
Let $f(t\_1, \ldots, t\_n)/g(t\_1, \ldots, t\_n) = \sum a(d\_1, \ldots... | 12 | https://mathoverflow.net/users/297 | 65252 | 40,298 |
https://mathoverflow.net/questions/65254 | 0 | Suppose we have a set $S$ of real numbers,such that
$ n \neq \Pi\_{m \in S-{n}} m^{r\_m}$....... $\forall n\in S$ where $r\_{m}$ is a rational number....
The set of prime number is an example of such a ser...now my question is that...Is such a set necessarily countable?
| https://mathoverflow.net/users/15176 | a question about subset of R | Since we are talking about rational exponents, I assume it makes sense to restrict to positive real numbers. If I got it right the question is whether a set $S$ where no element is a product of some others to rational powers is necessarily countable.
One can show non-constructively that the answer is no, that is, th... | 3 | https://mathoverflow.net/users/14302 | 65257 | 40,300 |
https://mathoverflow.net/questions/64007 | 7 | Let $U$ be some (unbounded) universe of elements, and let $\mathcal{S}$ be a collection of subsets of size $c$ each, such that any two elements from $\mathcal{S}$ have a non-empty intersection. Let $C \in \mathcal{S}$ be a special and unknown element of this collection. Now for any non-empty subset $V \subseteq U$ we d... | https://mathoverflow.net/users/11259 | Low rate c-uniform pairwise intersecting set systems | The projective plane example can be tweaked to give $f(c)\leqslant (2c)/(c^2-2c+4)\approx 2/c\ $ for general $c$. Let $p$ be the largest prime less than $c$ and take the projective plane of order $p$. Now let $A\_1,\ldots,A\_{p^2+p+1}$ be pairwise disjoint sets of size $c-p-1$ that are also disjoint to the points of th... | 2 | https://mathoverflow.net/users/12674 | 65269 | 40,306 |
https://mathoverflow.net/questions/65267 | 12 | In the most naive form my question would be as follows: If $f:X\to \mathrm{Spec}\;A$ is a flat morphism of schemes is it true that $H^0(X,\mathcal{O}\_X)$ is a flat $A$-module?
In general the answer is no: It is a theorem of Chase that a ring $A$ is coherent (meaning that every finitely generated ideal of $A$ is fini... | https://mathoverflow.net/users/2308 | Global sections of flat scheme also flat? | For another counterexample, let $A$ be $k[u,v,w]/uw-v^2$ and let $X$ be the complement of the origin in the plane $\text{Spec} \ k[x,y]$. Let $f$ be the restriction of the morphism of affine schemes corresponding to the map of rings $k[u,v,w]/uw-v^2 \to k[x,y]$ by $u\mapsto x^2$, $v\mapsto xy$, $w \mapsto y^2$. The mor... | 17 | https://mathoverflow.net/users/13265 | 65270 | 40,307 |
https://mathoverflow.net/questions/65253 | 1 | Given $S=\mathbb{P}^2$ and a locally free $O\_S$-module $E$ of rank r and an integer $l\geq 1$. Then it is known that the scheme $Quot(E,l)$ is irreducible, due to Ellingsrud and Lehn. Here $Quot(E,l)$ parametrizes zero dimensional quotients $E\rightarrow T$ of length $l$.
Are there any generalizations of this scheme... | https://mathoverflow.net/users/3233 | Generalized Quot-schemes | This can be thought of as a response to your comment to Sasha's answer.
Until you try fixing length, there is no issue. Indeed, consider the scheme
$$Quot(E)=\coprod Quot(E,k)$$ of all quotients of $E$. Then, as Sasha says, $Quot\_R(E)$ is obviously a closed subscheme.
You can now consider $Quot\_R(E,l)$ as subse... | 3 | https://mathoverflow.net/users/2653 | 65273 | 40,308 |
https://mathoverflow.net/questions/65282 | 27 | In 1936 Gödel announced a theorem to the effect that proofs of certain theorems $T\_1,T\_2,\ldots$ become dramatically shorter when one passes from a formal system, such as Peano arithmetic PA, to a stronger one, such as a system in which Con(PA) is provable. More precisely, given any computable function $f$, we can fi... | https://mathoverflow.net/users/1587 | Are any natural examples of Gödel speed-up known? | Friedman has given many examples of such speedups. One well known one is his finite version of [Kruskal's tree theorem](http://en.wikipedia.org/wiki/Kruskal%27s_tree_theorem). In particular he gave examples of reasonably natural statements that have very short proofs in 2nd order arithmetic, and can be proved in Peano ... | 16 | https://mathoverflow.net/users/51 | 65291 | 40,317 |
https://mathoverflow.net/questions/65284 | 13 | Having promised a longtime collaborator that I would clear my plate to finish up some joint work of ours, I am swallowing my pride and tossing up the following technical point of function field arithmetic to the experts:
Let $k\_0 = \mathbb{F}\_2(t)$, and let $k(P)$ be a finite separable extension of $k\_0$. Let $S$ ... | https://mathoverflow.net/users/1149 | Help wanted with Chebotarev condition in characteristic 2 | By my comments to the question, we can assume the infinite place is not in $S$ (because its splitting in the indicated quadratic extensions is automatic). Let the elements of $S$ be $\pi\_1,\dots,\pi\_r$, where the $\pi\_i$'s are distinct (monic) irreducibles in ${\mathbf F}\_2[t]$. For each $\pi\_i$, the Artin-Schreie... | 16 | https://mathoverflow.net/users/3272 | 65296 | 40,319 |
https://mathoverflow.net/questions/65295 | 1 | I am having trouble understanding one of the results in the following paper
<http://arxiv.org/PS_cache/math/pdf/0104/0104175v1.pdf>
In proposition 3.1, the author says
Let $(R,\frak{m})$ be a regular local ring, $p,q$ prime ideals s.t. $rad(p+q)= \frak{m}$ and $dim(R/p)+dim(R/q)=dim(R)$. If $R/p$ is regular then ... | https://mathoverflow.net/users/15191 | Symbolic powers in regular local rings | The short answer to what you're missing is that you ignored the dimension criterion. $R/\mathfrak{m}$ is zero-dimensional, so the only prime ideal $\mathfrak{p}$ to which the the theorem would apply is $\mathfrak{p} = (0)$.
| 3 | https://mathoverflow.net/users/nan | 65297 | 40,320 |
https://mathoverflow.net/questions/65301 | 2 | Let $S$ be a surface of general type (the canonical divisor $K\_{S}$ is big and nef).
Then I have read that $\chi(\mathcal{O}\_{S}) \geq 1$.
Why?
If this requires vanishing theorems, then assume that they hold.
| https://mathoverflow.net/users/6254 | Cohomology of structure sheaf on surfaces of general type with $p_g=0$ | This result can be found in [Beauville, Complex Algeberaic Surfaces, Chapter X], see in particular Theorem X.4.
For the reader's convenience, let me give a short account of the proof.
Since $K\_S$ is big and nef, we have $K\_S^2 >0$. By Noether formula $$\chi(\mathcal{O}\_S) = \frac{1}{12}(K\_S^2+c\_2(S))$$ it is t... | 4 | https://mathoverflow.net/users/7460 | 65304 | 40,323 |
https://mathoverflow.net/questions/65292 | 13 | According to references, Gromov-Witten invariants were first defined for symplectic manifolds and later for projective varieties algebraically, and they coincide on the overlap. Because I thought Gromov-Witten invariants are important invariants for symplectic manifolds, I looked up papers about the computation. But re... | https://mathoverflow.net/users/11846 | Computation of Gromov-Witten invariants for symplectic manifolds | There are some very good reasons why the majority of calculations are done for algebraic manifolds. Maybe the most naive reason is as follows: *it is harder to solve PDEs than to draw lines through two points in a space*. Somehow everyone knows that for two points in $\mathbb CP^n$ there is exactly one line that passes... | 17 | https://mathoverflow.net/users/943 | 65307 | 40,325 |
https://mathoverflow.net/questions/65306 | 5 | Suppose $\kappa$ is an inaccessible cardinal, and let $P$ be the $< \aleph\_1-$support product of $Add(\alpha^{++}, 1)$ for singular cardinals $\alpha < \kappa.$
1- Does this forcing preserve cardinals?
2-(A weaker question) Does $\kappa$ remain inaccessible in the generic extension?
| https://mathoverflow.net/users/11115 | $< \aleph_1-$support Product of Cohen forcings | Your forcing notion will collapse all uncountable cardinals
below $\kappa$ to $\aleph\_1$. To see this, fix any
uncountable $\gamma\lt\kappa$. Consider the first
$\aleph\_1$ many cardinals after $\gamma$ at which forcing
occurs. For the $\xi^{th}$ such cardinal $\alpha$, we are adding a
subset to $\alpha^{++}$. Conside... | 11 | https://mathoverflow.net/users/1946 | 65314 | 40,329 |
https://mathoverflow.net/questions/65299 | 2 | Let $S$ be a projective smooth surface over $\mathbb{C}$ and let $E$ be a rank-$r$ vector bundle on $S$. Having fixed an Hilbert polynomial $P$, which is the expected dimension of the Quot-scheme $Quot\_S(E,P)$?
| https://mathoverflow.net/users/33841 | Expected dimension of Quot-schemes on algebraic surface | Let $F$ be a quotient of $E$ with Hilbert polinomial $P$ and $[F]$ the corresponding point of $Quot\_S(E,P)$. Let $K = Ker(E \to F)$. Then the tangent space to $Quot\_S(E,P)$ at $[F]$ is $Hom(K,F)$ and the obstruction space is $Ext^1(E,F)$. So, the expected dimension is
$$
\chi(K,F) = \sum (-1)^i \dim Ext^i(K,F),
$$
w... | 1 | https://mathoverflow.net/users/4428 | 65319 | 40,332 |
https://mathoverflow.net/questions/65327 | 6 | Let $M$ be a connected compact manifold without boundary, $\pi:\widetilde{M}\to M$ be the universal covering map. A fundamental domain of $(\pi,\widetilde{M}, M)$ is a compact subset $D\subset \widetilde{M}$ such that
1. the union of $\gamma D$ over all $\gamma\in \pi\_1(M)$ covers $\widetilde{M}$,
2. the colle... | https://mathoverflow.net/users/11028 | fundamental domain of universal covering | The first part of 3 follows from 1. The second part of 3 follows from 2.
There is always a contractible (in particular simply connected) fundamental domain:
Triangulate $M$. Take the union of all the codimension zero open simplices, together with just enough codimension one open simplices to make it connected. This w... | 5 | https://mathoverflow.net/users/6666 | 65330 | 40,338 |
https://mathoverflow.net/questions/65303 | 3 | Let $k$ be a field, $A$ be an integral domain, $B \subset A$, and $A, B$ are both finitely generated $k$ algebra. Let $p \subset B$ be a prime ideal. Suppose there exists prime ideals $q \subset A$, such that $q \cap B=p$, and $q$ is the minimal such ideal in the sense of inclusion. Then, is it true $ dim A\_q \leq dim... | https://mathoverflow.net/users/15139 | Comparation of dimensions of rings | Dear Li, first of all I think that when you write "... such that $q \cap B=p$, and $q$ is *the* minimal such ideal in the sense of inclusion", you mean "... and $q$ is *a* minimal ideal...".
The answer to your question is given, I think, by the following more general result
**Theorem** Let $\phi: B\to A$ be a morp... | 5 | https://mathoverflow.net/users/450 | 65333 | 40,339 |
https://mathoverflow.net/questions/65335 | 3 | Has anyone worked out a uniform way of constructing the Hopf algebra of a Chevalley group out of the root system (or, more precisely out of the root datum for reductive groups).
By "uniform", I mean a method that works for any type, not case by case.
If yes, can anyone point out a reference in the literature? Ideal... | https://mathoverflow.net/users/4763 | Hopf algebra of Chevalley group from the root system | A no-nonsense construction, over $Z$, following work of Kostant and Chevalley, is given in Lusztig's paper "Twelve bridges from a reductive group to its Langlands dual". The heart of the construction of the Hopf algebra is in Section 5.
This is easy enough to find online, and according to Lusztig's webpage, it can a... | 4 | https://mathoverflow.net/users/3545 | 65345 | 40,347 |
https://mathoverflow.net/questions/65274 | 10 | Let $Symm$ be the vector space with basis $(b\_\lambda)$ given by the
set of all partitions $\lambda$ (of all natural numbers), thought
of as Young diagrams. Let $e\_i$ be
the *degree $i$ Pieri operator* that takes
$b\_\lambda \mapsto \sum b\_{\lambda'}$ where $\lambda'$ has $i$ more
boxes than $\lambda$, no two in th... | https://mathoverflow.net/users/391 | Heisenberg algebra from Pieri operators and their transposes? | I believe that you mean for your basis $b\_\lambda$ to correspond to the basis of Schur functions in the ring of symmetric functions, in which case your $e\_i$ operator corresponds to multiplication with respect to the complete symmetric function $h\_i = b\_{(i)} = b\_i$ and your $e\_i^T$ corresponds to the adjoint of ... | 5 | https://mathoverflow.net/users/15200 | 65356 | 40,353 |
https://mathoverflow.net/questions/55447 | 15 | Let $\gamma\_0$ and $\gamma\_1$ be two simple closed curves in a closed surface $S$.
>
> What is the maximum Euler characteristic of a compact properly embedded surface $\Sigma \subset S\times [0,1]$ such that $\partial \Sigma = \gamma\_0 \times \{0\} \cup \gamma\_1 \times \{1\}$?
>
>
>
Of course, in order fo... | https://mathoverflow.net/users/6205 | Maximal euler characteristic of surfaces bounding two fixed curves | Here is a graph $\mathcal{G}(S,x)$ associated to a surface and homology class (similar to the 1-skeleton of the curve complex): For a fixed class in $x\in H\_1(S;\mathbb{Z}\_2)$, consider isotopy classes of embedded multicurves representing the homology class $x$ (one may assume no components are parallel and there are... | 7 | https://mathoverflow.net/users/1345 | 65358 | 40,355 |
https://mathoverflow.net/questions/65332 | 3 | First of all, this is no useful way to *decompose* a matrix -
you need to know the eigenvalues beforehand. But it popped up
naturally during my knot theory dabblings.
Assume that you know the characteristic equation
$$\prod\_{i=1}^n (S - e\_i I) = 0$$
with $S$ being an $n \times n$ matrix, $I$ the $n \times n$ ide... | https://mathoverflow.net/users/11504 | Matrix decomposition the other way | This is a fairly standard result in the theory of diagonalizable linear operators, sometimes known as the spectral decomposition theorem for diagonalizable operators. Indeed, a linear operator over some field is diagonalizable if and only if it has a decomposition of this form. You can read about this in Hoffman & Kunz... | 5 | https://mathoverflow.net/users/7392 | 65362 | 40,357 |
https://mathoverflow.net/questions/65265 | 1 | This problem arose when solving a continuous Markov chain exercise from a book I'm studying. Given a set of positive $q\_i$, with $i \in \mathbb{Z} $, and non-negative $\lambda$ and $\mu$ that add to 1 the solution amounts to solving
$p\_{0,i}^'(t) = \lambda q\_{i-1} p\_{0,i-1}(t) - q\_i p\_{0,i}(t) + \mu q\_{i+1} p... | https://mathoverflow.net/users/15181 | Solution to difference differential equation with constant coefficients | Thanks to Michael Renardy for the analogy with the heat equation. Let's assume $q\_i = q$, i.e. independent on $i$. The difference-differential equation can then be solved by using generating function technique. Let $G(z) = \sum\_{i \in \mathbb{Z}} z^i p\_{0,i}(t)$. Then the equation and initial condition imply that $G... | 1 | https://mathoverflow.net/users/15181 | 65367 | 40,359 |
https://mathoverflow.net/questions/65363 | 4 | I believe that it is "well known" that the following two statistics on Dyck paths have symmetric joint distribution:
1. number of returns to the axis $RET(D)$
2. height of the first peak (or length of the last descent) $HFP(D)$
That is: $\sum\_{D} x^{RET(D)}y^{HFP(D)} = \sum\_{D} x^{HFP(D)}y^{RET(D)}$
However, I ... | https://mathoverflow.net/users/3032 | Equidistribution of returns and height of first peak of Dyck paths | You can use the article ["A bijection on Dyck paths and its consequences"](https://doi.org/10.1016/S0012-365X(97)00097-6 "Discrete Math. 179, No. 1-3, 253-256 (1998); corrigendum ibid. 187, No. 1-3, 297 (1998). zbMATH review at https://zbmath.org/0890.05005") by E. Deutsch. The author has several papers on enumerative ... | 1 | https://mathoverflow.net/users/2384 | 65368 | 40,360 |
https://mathoverflow.net/questions/65369 | 22 | I had a discussion with one of my teachers the other day, which boiled to the following question:
>
> Assume ZF. Let $A,B$ be sets such that there exist $f\colon A\to B$ which is injective and $g\colon A\to B$ which is surjective.
>
>
> Is there $h\colon A\to B$ which is bijective?
>
>
>
Of course it is enou... | https://mathoverflow.net/users/7206 | Half Cantor-Bernstein Without Choice | If $A = \{0\}\times 2^{\omega}$ and $B = (\{0\}\times 2^{\omega})\cup (\{1\}\times \omega\_1)$ and there is no injection from $\omega\_1$ to $2^{\omega}$,
then there does not exist such a bijection.
Define $f : (\{0\}\times 2^{\omega}) \to ((\{0\}\times 2^{\omega})\cup (\{1\}\times \omega\_1))$ by $f(x) =... | 15 | https://mathoverflow.net/users/nan | 65373 | 40,362 |
https://mathoverflow.net/questions/64889 | 6 | Let $G$ be a $k$-transitive subgroup of the symmetric group $Sym(n)$, $k\geq 2$, $n$ large. (Make $k$ larger if you think it necessary to make the question below non-trivial/interesting.)
Write $C(g)$ for the centraliser of an element and $|C(g)|$ for its number of elements.
What can you say about $\min\_{g\in G} |... | https://mathoverflow.net/users/398 | Centralisers in the symmetric group | (Written before clarification at end of question was added). Here is a result which seems to be of a somewhat negative nature in the context of your problem, and your suggested line of attack, I think. I will denote the number of conjugacy classes of $G$ by $k(G)$. The group $ G = {\rm SL}(2,2^{n})$ (n>1) is a triply t... | 2 | https://mathoverflow.net/users/14450 | 65385 | 40,368 |
https://mathoverflow.net/questions/65376 | 6 | Let $k$ be a finite field. We define $N(d,g)$ to be the number of plane curves $f(x,y)$ defined over $k$ of degree $d$ with (geometric) genus $g$. If $D(d) := (d-1)(d-2)/2$ (the maximum possible genus), I would expect that as $d$ goes to infinity that the proportion of curves of degree $d$ with genus $D(d)$ would go to... | https://mathoverflow.net/users/2784 | How do the number of plane curves over a finite field of a fixed genus increase with the degree? | Fix $g$, the genus, and $q$, the order of $k$.
$N(d,g)$ should be $\approx C q^{3d}$, where $C$ is some constant dependent on $q$ and $g$. (Note that my $C$ has absorbed the $q^{-4}$ in Felipe's answer.) There are some nonrigorous details here.
There are finitely many isomorphism classes of pair $(X, L)$ where $X$ is... | 6 | https://mathoverflow.net/users/297 | 65389 | 40,371 |
https://mathoverflow.net/questions/65390 | 5 | Quick question: It's known that
$$\limsup\frac{n}{\varphi(n)\log\log n}=e^\gamma$$
but are there known C and N such that
$$\varphi(n)>\frac{Cn}{e^\gamma\log\log n}$$
for all $n>N$?
Failing that, what are good effective bounds on $\varphi$? The square root bound isn't good enough for me.
| https://mathoverflow.net/users/6043 | Effective bounds on Euler's totient | Yes. Look at
<http://en.wikipedia.org/wiki/Euler>'s\_totient\_function#Inequalities:
$$\varphi(n)>\frac{n}{e^\gamma\log\log n + \frac{3}{\log\log n}}$$
for $n>2$.
| 7 | https://mathoverflow.net/users/11142 | 65391 | 40,372 |
https://mathoverflow.net/questions/65387 | 0 | Let $\{X\_{\alpha} \}\_{\alpha \in A}$ be a collection of Banach spaces. It is easy to show that $ P = \{(x\_{\alpha}) : {\rm sup}\_{\alpha} \|x\_{\alpha} \| < \infty \} $ with $\| (x\_{\alpha} ) \| = {\rm sup}\_{\alpha} \| x\_{\alpha} \|$ is a banach space.
If the indexing set $A$ is finite, then it is easy to show... | https://mathoverflow.net/users/4002 | products in the category of banach spaces | [Promoted from comments, as requested]
The category of Banach spaces and bounded linear maps does not have arbitrary infinite products. Here is a simple example, using the notation of the question: take each $X\_\alpha$ and $Y$ to be a copy of the ground field, fix an unbounded function $\lambda:A\to{\mathbb R}$, and... | 3 | https://mathoverflow.net/users/763 | 65393 | 40,373 |
https://mathoverflow.net/questions/65396 | 7 | I am interested in bounding the following Salie-type ("twisted Kloosterman") sum
$$
S(a,b,\beta) = \sum\_{x \in \mathbb{Z}/{p^{\beta}}\mathbb{Z}} \left( \frac{x}{p^{\beta}} \right) \chi(ax + bx^{-1}).
$$
Here, $\left( \frac{\cdot}{q} \right)$ denotes the Jacobi symbol, $\chi(x) = \exp(2 \pi i x /p^{\beta})$, $p$ is... | https://mathoverflow.net/users/15212 | Salie-type sum bound | There is a general "elementary" formula for Salié sums for arbitrary modulus, involving roots of quadratic equations, and from which the bound is immediate. A quick derivation is in Sarnak's "Some applications of modular forms" but it can be found in many places.
| 2 | https://mathoverflow.net/users/20038 | 65398 | 40,376 |
https://mathoverflow.net/questions/65377 | 2 | The integral Chow groups are infinite dimensional. Can we say something about their dimension, for example, how many elements are required to generate them? Or their vector space dimension after tensoring with $\mathbb{Q}$. What is known in this direction? I recall hearing some statements whose proof uses the theory of... | https://mathoverflow.net/users/11395 | Dimension of Chow groups | Chow groups are not in general countable. For example, $CH^1(X) = Pic(X)$ is uncountable for a smooth projective curve $X$ of positive genus over any uncountable algebraically closed field.
The theory of Chow varieties allows one to prove the countability of Chow groups modulo algebraic equivalence (this follows easi... | 5 | https://mathoverflow.net/users/519 | 65399 | 40,377 |
https://mathoverflow.net/questions/65392 | 1 | Hello,
I was wondering if anyone is aware of an elementary proof of the claim in the title, assuming the existence of nilpotent injectors in soluble groups. By elementary I mean a proof that does not involve any recourse to Fitting classes etc.
Thanks in advance.
| https://mathoverflow.net/users/nan | Conjugacy of nilpotent injectors in soluble groups | I take a nilpotent injector of a finite solvable group $G$ to be a nilpotent subgroup $M$ of $G$
such that $M \cap N$ is a maximal nilpotent normal subgroup of $N$ whenever $N$ is subnormal in $G$.
Assuming existence of $M$ , I think uniqueness up to conjugacy follows inductively. Notice that
$M \cap H$ is a nilpotent ... | 3 | https://mathoverflow.net/users/14450 | 65409 | 40,385 |
https://mathoverflow.net/questions/65411 | 2 | Let $F$ be a number field and $E/F$ a Galois extension. Suppose we have a representation $\rho\_E : Gal(\overline{F}/E) \rightarrow GL\_n(\overline{Q}\_p)$. My question is : what are sufficiant conditions so that $\rho\_E$ can be extended to a representation $ Gal(\overline{F}/F) \rightarrow GL\_n(\overline{Q}\_p)$ ?
... | https://mathoverflow.net/users/10427 | How can we extend Galois representations ? | Pick a $\sigma \in \Gamma\_F$ such that $\sigma |\_E$ generates $\mathrm{Gal}(E/F)$.
There is an $A \in \mathrm{GL}\_n$ such that for all $x \in \Gamma\_E$, $\rho(\sigma x \sigma^{-1}) = A \rho(x) A^{-1}$, and if $q$ denotes the order of $\mathrm{Gal}(E/F)$, we get that $\rho(\sigma^q x \sigma^{-q}) = A^q \rho(x) A^{-q... | 6 | https://mathoverflow.net/users/5735 | 65416 | 40,387 |
https://mathoverflow.net/questions/65424 | 46 | Say $A$ and $B$ are symmetric, positive definite matrices. I've proved that
$$\det(A+B) \ge \det(A) + \det(B)$$
in the case that $A$ and $B$ are two dimensional. Is this true in general for $n$-dimensional matrices? Is the following even true?
$$\det(A+B) \ge \det(A)$$
This would also be enough. Thanks.
| https://mathoverflow.net/users/15221 | Determinant of sum of positive definite matrices | The inequality
$$\det(A+B)\geq \det A +\det B$$
is implied by the Minkowski determinant theorem
$$(\det(A+B))^{1/n}\geq (\det A)^{1/n}+(\det B)^{1/n}$$
which holds true for any non-negative $n\times n$ Hermitian matrices $A$ and $B$. The latter inequality is equivalent to the fact that the function $A\mapsto(\det A )^... | 65 | https://mathoverflow.net/users/5371 | 65430 | 40,392 |
https://mathoverflow.net/questions/65429 | 7 | What are the good books, online lecture notes or starting material on exponentials sums with applications in number theory for a beginner, apart from N. M. Korobov's book? The book or notes should cover methods of Weyl, van der Corput and Vinogradov, with some details.
| https://mathoverflow.net/users/14466 | Exponential sums for beginner. | Shparlinski has a nice set of lecture notes, aimed at beginners, with a view towards applications: <http://www2.ims.nus.edu.sg/Programs/coding/files/ishpar.ps>
| 8 | https://mathoverflow.net/users/2290 | 65440 | 40,398 |
https://mathoverflow.net/questions/65415 | 13 | Hi. This may be a very general question.
Are there any examples of problems in combinatorics which were open, but which found a solution when stated in algebraic terms?
If yes, could somebody mention some of these? I'm new to this and don't know many examples yet.
I know about the "Magic Squares", which refers t... | https://mathoverflow.net/users/15054 | Succesful applications of algebra in combinatorics | Stanley's proof of the Upper Bound Conjecture relied on a connection with free resolutions of graded algebras. This has led to the very active area of Stanley--Reisner theory, where combinatorial properties of simplicial complexes are related to algebraic properties of certain graded algebras.
For references, there'... | 5 | https://mathoverflow.net/users/4 | 65442 | 40,400 |
https://mathoverflow.net/questions/65418 | 4 | On the Moduli space $\overline{M}\_{g}$ of genus $g$ stable curves the *Hodge class* $\lambda$ induces a birational morphism $f$ on a projective variety contracting the boundary, that is the exceptional locus of $f$ coincides with the boundary of the moduli space.
**Is there a line bundle $L$ on the moduli space of p... | https://mathoverflow.net/users/14514 | Birational Contractions on Moduli of pointed Curves | A detailed description of the nef cones of the examples you mention is given in the thesis of William Rulla (The birational geometry of $\overline{M}\_3$ and $\overline{M}\_{2,1}$, The University of Texas at Austin, 2001.) From this, it follows that there are no such line bundles for the two examples that you mention. ... | 4 | https://mathoverflow.net/users/519 | 65445 | 40,403 |
https://mathoverflow.net/questions/65355 | 17 | In "Tilting Theory and Cluster Combinatorics" Buan, Marsh, Reineke, Reiten, and Todorov constructed cluster categories for mutation finite cluster algebras (without coefficients), and Amiot gives a construction of a cluster category given a quiver with potential whose Jacobian algebra is finite dimensional (in particul... | https://mathoverflow.net/users/5323 | Which cluster algebras have been categorified? | There are several different questions in here and what follows are only partial answers to some of these, mostly consisting of pointers to pieces of the literature.
* "In what other instances have cluster
categories been constructed?"
For surveys on cluster categories, I would recommend:
Cluster algebras, quiver ... | 15 | https://mathoverflow.net/users/13215 | 65447 | 40,404 |
https://mathoverflow.net/questions/65455 | 1 | Let $H \leq G$. Let $Z\_G$ denote the center $[G,G]$ the commutator subgroup. Assume $[G,G] \leq Z\_G$ (i.e. nilpotent of class 2). Then $G/Z\_G$ is abelian since $Z\_G$ contains the commutator subgroup. There is a theorem that states that $H$ must also be nilpotent of class at most 2, hence $[H,H] \leq Z\_H$ so $H/Z\_... | https://mathoverflow.net/users/15228 | Quotient of subgroups by center. | The answer to (1) is "yes." As Carnahan notes, you have $H\cap Z\_G\subseteq Z\_H$. Since $H/(Z\_G\cap H)\cong HZ\_G/Z\_G$ is isomorphic to a subgroup of $G/Z\_G$, then, $H/Z\_H$ is a quotient of $H/(Z\_G\cap H)$, hence isomorphic to a quotient of a subgroup of $G/Z\_G$, so $\mathrm{rank}(H/Z\_H) \leq \mathrm{rank}(G/Z... | 5 | https://mathoverflow.net/users/3959 | 65457 | 40,407 |
https://mathoverflow.net/questions/65170 | 3 | Let $G$ be a Banach-Lie group with Lie algebra $\mathfrak g$
and $\mathfrak h$ a closed subalgebra. Using the exponential
map and the Baker-Campbell-Hausdorff-formula one constructs a local
Lie group $\mathcal L\_H$ corresponding to $\mathfrak h$. Then the subgroup
$H = \langle \mathcal L\_H\rangle$
of $G$ naturally i... | https://mathoverflow.net/users/15155 | Integral subgroup theorem for Banach-Lie groups | I have been told by Karl-H. Neeb of the following neat counterexample due to K. H. Hofmann and S. Morris. Consider the Banach-Lie group $\mathfrak g=\ell^1(\mathbf R,\mathbf R)\times\mathbf R$ with the addition and take the closed discrete subgroup $D$ generated by the
elements $(n,\sum\_{x\in\mathbf R}n(x)x)$ where $... | 1 | https://mathoverflow.net/users/15155 | 65462 | 40,409 |
https://mathoverflow.net/questions/65451 | 8 | Suppose $T$ is a free finite rank $\mathbb{Z}\_p$-module with a continuous action of $\operatorname{Gal}(\overline{K} / K)$, where $K$ is a number field. There is a definition of local Tamagawa numbers $\operatorname{Tam}(T / K\_v)$ for each prime $v$ of $K$, going back to Fontaine and Perrin-Riou (or to Bloch and Kato... | https://mathoverflow.net/users/2481 | Do Tamagawa numbers of Galois representations stabilise in the cyclotomic tower? | As stated, the answer to your question is certainly no.
For instance, an elliptic curve $E/\mathbb Q$ with split multiplicative ordinary reduction at $p$ will have unbounded Tamagawa number at $p$ in the cyclotomic extension of $\mathbb Q$. To see this, you can check that the Tamagawa number is the order of $H^{2}({... | 2 | https://mathoverflow.net/users/2284 | 65469 | 40,412 |
https://mathoverflow.net/questions/65474 | 7 | Let $G$ be a compact connected Lie group, $T$ maximal torus, identified with $\mathbb{R}^n/\mathbb{Z}^n$, $X^\*(T)$
the set of characters of $T$, naturally identified with $\mathbb{Z}^n$. Let next $\Phi$ denote the set of roots, corresponding to $T$,
$$
\Lambda=\{v\in \mathbb{R}^n: \forall \alpha\in \Phi,\ 2(v,\alp... | https://mathoverflow.net/users/4312 | why are all characters of the maximal torus in a Lie group weights? | "Easiest" depends on how you set things up: everything really hinges on how you want to identify $X^\ast(T)$ with $\mathbb Z^n$. It's probably cleanest if you don't work explicitly with $\mathbb Z^n$, but instead state everything in terms of Lie algebras and their duals. I personally like the setup given in Knapp, *Lie... | 5 | https://mathoverflow.net/users/430 | 65479 | 40,417 |
https://mathoverflow.net/questions/65472 | 2 | As a lot of people know, graph partitioning is NP-Complete. In graph partitioning, you try to create k balanced (within some pre-specified epsilon) disjoint subsets of (possibly weighted) vertices such that the edgecut is minimized. (See <http://en.wikipedia.org/wiki/Graph_partitioning>).
But what about the simpler ... | https://mathoverflow.net/users/15230 | Official name and complexity of k-way balanced set partitioning? What is the best heuristic? | The problem is NP-complete, because it contains the problems Partition and 3-Partition (problems 41 and 46 in <http://www.csc.liv.ac.uk/~ped/teachadmin/COMP202/annotated_np.html>). If your instances are not extremely huge, I would give an integer programming formulation a try. The heuristics build into modern solvers w... | 1 | https://mathoverflow.net/users/12674 | 65485 | 40,422 |
https://mathoverflow.net/questions/65461 | 14 | I am having some trouble finding and/or understanding a general definition of subvarieties intersecting transversally. Assume that $Z\_1,\ldots,Z\_k$ are closed, irreducible subvarieties of a nonsingular algebraic variety $Y$.
Intuitively, I would say that these subvarieties **intersect transversally** if all variet... | https://mathoverflow.net/users/9947 | Transversal Intersection of Varieties | This naive answer is strictly intended to provoke a knowledgeable answer from someone. Transversal intersection at p should mean that all subvarieties are smooth at p, all contain p, and the codimension of the intersection of the tangent spaces equals the sum of the codimensions of the individual tangent spaces. ????
... | 5 | https://mathoverflow.net/users/9449 | 65496 | 40,427 |
https://mathoverflow.net/questions/65432 | 17 | One of the foundational results about mapping class groups of surfaces is that they are generated by Dehn twists. A mapping class is a connected component in a space of diffeomorphisms, so another way of stating this is "any diffeomorphism of a surface is generated by diffeomorphisms each of which is supported in an an... | https://mathoverflow.net/users/2051 | Elegant proof that mapping class groups are generated by Dehn twists? | I'm pretty sure there doesn't exist a "slicker" proof of this fact in the literature. The proof you describe exists in many forms starting with Dehn and Lickorish -- as I said in a comment, the particular arrangement of it you gave (making use of the complex of curves) is basically due to Ivanov. The only fundamentally... | 12 | https://mathoverflow.net/users/317 | 65500 | 40,429 |
https://mathoverflow.net/questions/65499 | 12 | Does anyone know how news of Gödel's incompleteness theorem spread? Did it do so little by little, or was it shouted in dramatic headlines throughout mathematical literature? If anyone can point me to the history of how people reacted to Gödel's result, I would be grateful.
I am asking this question because I have re... | https://mathoverflow.net/users/14510 | How quickly did Gödel's Incompleteness Theorem become known and heeded throughout mathematics | I believe the answer to the title of the question is "Quickly". The Wikipedia article on "Gödel's incompleteness theorems" has a nice discussion of the developments in the 1930's. The title of Gödel's paper on the incompleteness theorem is "Über formal unentscheidbare Sätze... I", and apparently he never needed to writ... | 8 | https://mathoverflow.net/users/14302 | 65512 | 40,438 |
https://mathoverflow.net/questions/65516 | 3 | For a finite field $\mathbb{F}$ of char $p$ $\geq 2$, let $f$ be a normalised eigenform of weight $k\geq 2$ that is ordinary at $p$ and $\overline{\rho}\_f$ be the mod $p$ Gal representation attached to it. Let $G\_p$ and $I\_p$ be the decomposition and inertia subgroups. We know that the restriction of $\overline{\rho... | https://mathoverflow.net/users/5310 | Why does $H^1(G_p/I_p,\mathbb{F}(\delta\epsilon^{-1}))$ vanish? | A variant of David Loeffler's answer...
If more generally $V$ is a $G\_{\mathbb Q\_{p}}$-representation with coefficients in a field, then the dimension of $H^{1}(G\_{p}/I\_{p},V)$ (EDIT : or rather $H^{1}(G\_{p}/I\_{p},V^{I\_{p}})$) is equal to the dimension of $H^{0}(G\_{p},V)$ because the former is the cokernel of... | 3 | https://mathoverflow.net/users/2284 | 65525 | 40,447 |
https://mathoverflow.net/questions/65477 | 7 | Let $G$ be a countable discrete residually finite group.
Is there a way to characterise the actions of $G$ that are orbit-equivalent to profinite ones?
Ozawa and Popa introduced the concept of weakly compact actions. Weakly compact actions are stable under orbit equivalence and profinite actions are weakly compact.... | https://mathoverflow.net/users/15125 | Actions orbit equivalent to profinite ones | If a group has a free profinite action then in particular it must be residually finite. On the other hand, any measure preserving action of an amenable group is weakly compact. Thus, any free measure preserving action of infinite amenable group $G$ which is not residually finite cannot be orbit equivalent to a profinit... | 9 | https://mathoverflow.net/users/6460 | 65538 | 40,456 |
https://mathoverflow.net/questions/65546 | 5 | I think this graph has a name: the vertices are bit strings of length $n$, and $(x\_1, \ldots , x\_n)$ is connected to $(x\_2, \ldots, x\_n, 0)$, $(x\_2, \ldots, x\_n, 1)$, $(0,x\_1, \ldots , x\_{n-1})$ and $(1, x\_1, \ldots , x\_{n-1})$. I'm wondering (a) what the name is and (b) where I can read more about this graph... | https://mathoverflow.net/users/15253 | What is the name of this graph? | They are called De Bruijn graphs (De Bruijn graphs are generally considered directed, and can be defined over any set of symbols, not just $\{0,1\}$).
| 9 | https://mathoverflow.net/users/14302 | 65549 | 40,462 |
https://mathoverflow.net/questions/65463 | 12 | A fractal set has a Hausdorff dimension.
In some cases, we may generate a fractal by iterating $f,$
and let the fractal be the set of starting points $x$ such
that $|f^{\circ n}(x)|$ is bounded as $n$ grows. (The julia set and the sierpinski triangle are such sets, if one allows $f$ to be a Hutchinson operator).
We... | https://mathoverflow.net/users/1056 | Hausdorff dimension for invariant measure? | There are a wide variety of notions of dimension of a measure. Your basic intuition is completely correct: for a dynamical system, the dimension of a natural invariant measure provides more relevant information than the dimension of the invariant set, since the system may spend more time in certain parts of the space.
... | 17 | https://mathoverflow.net/users/11009 | 65553 | 40,465 |
https://mathoverflow.net/questions/65556 | 8 | How to construct a CW complex $X$ with prescribed homotopy groups $\pi\_i(X)$ and prescribed actions of $\pi\_1(X)$ on the $\pi\_i(X)$'s?
| https://mathoverflow.net/users/9401 | Construct a CW complex with prescribed homotopy groups and actions of $\pi_1$. | One method is as follows.
1. Construct $Y\_i = K(\pi\_i, i)$ for $i \geq 2$ as a based space with an action of the group $\pi\_1$. You can do this manually (by attaching free orbits of cells along the group action) or canonically (using a functorial construction of Eilenberg-Mac Lane spaces, such as one obtained from... | 14 | https://mathoverflow.net/users/360 | 65565 | 40,475 |
https://mathoverflow.net/questions/65540 | 2 | Let $X$ be a smooth complete complex (algebraic) 3-fold, $D$ an effective divisor on $X$, and $C$ a smooth integral curve contained in the support of $D$. Let $X'$ be the blowup of $X$ along $C$, and denote by $D'$ the strict transform of $D$, $E$ the exceptional divisor, and $C'=D' \cap E$ the (set-theoretic) intersec... | https://mathoverflow.net/users/11661 | Blowing up a curve on a threefold | Let $X=\mathbb P^3$, $D$ a quadric cone and $C$ a line on $D$ that goes through the vertex of $D$. Then $D'$ is isomorphic to $D$ blown up at its vertex, in particular it is smooth, $E\simeq C\times \mathbb P^1$ and $C'$ is isomorphic to the union of one member of each of the rulings on $E$, that is $C\cup \mathbb P^1$... | 6 | https://mathoverflow.net/users/10076 | 65572 | 40,481 |
https://mathoverflow.net/questions/65541 | 2 | Is there a version of Gordan's Theorem in which $Ax=0$ has been replaced by $Ax=b$? That is, I want a condition (possibly including conditions on $A$) for when $Ax=b$ has a solution $x \neq 0$, for $A \in \mathbb{R}^{m \times n}$ and $b \in \mathbb{R}\_+^m \setminus \{0\}$
Gordan's Theorem says that for all $A \in \m... | https://mathoverflow.net/users/4974 | Gordan's Theorem with $Ax=b$ | All these results follow from the duality theory of linear programming.
Let $e$ be the vector in $R^n$ with all entries 1.
$Ax = b$, $x \ge 0$, $x \ne 0$ is solvable iff the problem P:
maximize $e^T x$ subject to $Ax = b$, $x \ge 0$
is unbounded. This implies that the dual problem D:
minimize $b^T y$ subje... | 4 | https://mathoverflow.net/users/13650 | 65575 | 40,484 |
https://mathoverflow.net/questions/65567 | 5 | No, it's not the Quadratic, algebraic sort that we're talking about here....Instead, consider this:
Imagine a 6 x 6 grid of 36 equal minisquares, or tiles. A "shape" consists of interconnected adjacent tiles on this grid. [Adjacent tiles must share an edge, not merely a vertex.]
**Prove the number of discrete combi... | https://mathoverflow.net/users/15255 | Completing the Square | You can find about 30 examples at [Mike Reid's Rectifiable polyomino page.](https://www.cflmath.com/%7Ereid/Polyomino/rectifiable_data.html) It would appear that for the 6x6 square there are two types of 9 cell tilers:
* Take the [center edge of a 3X6](https://www.cflmath.com/%7Ereid/Polyomino/9omino12_rect.html) rec... | 2 | https://mathoverflow.net/users/8008 | 65600 | 40,499 |
https://mathoverflow.net/questions/62834 | 9 | What are some concrete examples of theorems which can be deduced from the Arthur trace formula, which do not follow from the simple trace of Kazdhan and Flicker?
(So I do not mean weaker in the sense that the Arthur trace formula implies the simple trace formula.)
| https://mathoverflow.net/users/10400 | Why is the simple trace formula a weaker tool than the Arthur trace formula? | I must have missed this question a month ago, but hopefully you're still interested.
I haven't read that particular paper, but simple trace formulas have restricted test functions, which for one can only deal with representations which are supercuspidal at some place. So any theorem you prove, say about transfer of r... | 13 | https://mathoverflow.net/users/6518 | 65608 | 40,505 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.