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https://mathoverflow.net/questions/102954 | 8 | I'll begin this question with the finite-dimensional case, as a
warmup.
Let me say a continuous path $\omega : [0,1] \to \mathbb{R}^d$ is
**hyperplanar** if there exists a nonzero $x \in \mathbb{R}^d$ such
that $\omega(t) \cdot x = 0$ for all $t \in [0,1]$.
It is not hard to show:
>
> **Proposition.** Let $B\_t... | https://mathoverflow.net/users/4832 | Does infinite-dimensional Brownian motion live in hyperplanes? | As suggested in my comment, here's a simple fact which applies to any probability measure $\mu$ on (the Borel σ-algebra of) a second countable topological space $X$. There is a unique minimal closed subset $S$ of $X$ with $\mu(S)=1$ -- the support of $\mu$ -- and, if $X\_1,X\_2,\ldots$ is an IID sequence of random vari... | 8 | https://mathoverflow.net/users/1004 | 102963 | 59,707 |
https://mathoverflow.net/questions/102989 | 2 | Hi all! Why are skein modules 1-dimensional on closed 3-manifolds? The result seems clear on closed manifolds with vanishing first Betti number (e.g. $S^3$), but I don't see how to prove it for, say, $\mathbb{T}^3$. Can anyone point me to an old reference (I'm rather new to this field)?
| https://mathoverflow.net/users/25305 | Skein Modules on closed 3-manifolds | The skein module of a closed 3-manifold is not 1-dimensional. That of the 3-torus is infinite-dimensional, a basis is given by multicurves on the torus as [proved by Przytycki](http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=PC&pg7=ALLF&pg8=ET&review_fo... | 5 | https://mathoverflow.net/users/6205 | 102991 | 59,718 |
https://mathoverflow.net/questions/102979 | 7 | Dear Mathoverflow,
I would like to know if the nomenclature of mathematics has a name for Radon-Nikodym derivatives that are bounded away from zero and infinity almost everywhere. As in for equivalent measures $\mu, \nu$, there exists constants $c,C$ such that
$$ 0 < c \leq \frac{d\nu}{d\mu}(x) \leq C < \infty$$
fo... | https://mathoverflow.net/users/8769 | A name for Radon-Nikodym derivatives that are bound away from zero and infinity | Good question. I am not aware of any standard terminology, although this condition is quite natural and appears pretty often. I would rather call such measures **uniformly equivalent**. As for "correlated" - as you say, it would create wrong connotations.
| 5 | https://mathoverflow.net/users/8588 | 102994 | 59,720 |
https://mathoverflow.net/questions/102828 | 5 | Some questions about algebraic groups.
Let $G$ be an affine algebraic group over algebraically closed field $k$.
Questions: Let $H$ be a closed subgroup of $G$, then (as I learnt from some paper) the quotient $\pi\colon G\to G/H$ is faithfully flat, why? reference? When it is locally trivial, especially when $G$ is ... | https://mathoverflow.net/users/16762 | Why $G\to G/H$ is faithfully flat? | If you are really looking for a reference, here is one: Groupes algébriques, Demazure-Gabriel, Chapter III, §3, Proposition 2.5 p. 328. Note that the fact that the base is a field is not important. It might be any scheme. The important assumptions are
1. the subgroup $H$ has to be flat over the base (which of course... | 3 | https://mathoverflow.net/users/4763 | 102997 | 59,721 |
https://mathoverflow.net/questions/103001 | 0 | Everything over F\_2. Let us define Hamming norm of polynom |p(x)| = number of non-zero monoms.
Respectivly for a pair of polynoms |[p ; g]| = |p| +|g|.
Consider linear map $F\_2[x] \to F\_2[x] \oplus F\_2[x] $ given by
$p(x) \mapsto [ p(x)(x^2+1) ; p(x) (x^2+x+1)] $.
**Question**
Is it true that minimal Hamming ... | https://mathoverflow.net/users/10446 | Multiplication by polynomials x^2+1 ; x^2+x+1. Does minimal Hamming norm of image equal to 5 ? | Yes. Without loss of generality, we can assume $p(x)$ is not divisible by $x$. Then $p(x)(x^2+1)$ and $p(x)(x^2+x+1)$ both have constant term $1$, so the combined Hamming weight is at least $4$ from the constant and leading terms. The only way it could be $4$ is if they were $x^n+1$ and $x^m+1$ for some $n$ and $m$. Ho... | 6 | https://mathoverflow.net/users/4720 | 103002 | 59,723 |
https://mathoverflow.net/questions/103014 | 1 | Is there a standard definition for a lacunary sequence?
Suppose $0 < a\_1 < a\_2 < \cdots.$
I've read two papers using the term recently. One requires
$$
\liminf\_n\frac{a\_{n+1}}{a\_n}>1
$$
while the other only requires
$$
\lim\_na\_{n+1}-a\_n=+\infty.
$$
The two differ, of course: $a\_1=1,\ a\_{n+1}=a\_n+\sqrt{... | https://mathoverflow.net/users/6043 | Lacunary sequence | I believe historically the 'lacunary' terminology derives from Hadamard (around [Ostrowski--Hadarmard's Gap Theorem](http://mathworld.wolfram.com/Ostrowski-HadamardGapTheorem.html)), and there are other results like this, where one needs the condition on the ratio (as opposed to the difference).
[Indeed in some pape... | 2 | https://mathoverflow.net/users/nan | 103019 | 59,733 |
https://mathoverflow.net/questions/103015 | 3 | Suppose $\mathcal{M}$, $\mathcal{N}$, and $\mathcal{P}$ are Riemannian manifolds (compact and of dimension 2, if it matters). It seems well-known that if $\phi:\mathcal{M}\rightarrow\mathcal{N}$ is (weakly) conformal and $f:\mathcal{N}\rightarrow\mathcal{P}$ is harmonic, then $f\circ\phi:\mathcal{M}\rightarrow\mathcal{... | https://mathoverflow.net/users/25311 | Harmonic/conformal map composition between manifolds in either order? | Because it is not true. For example, suppose that $f: D^n \to D^n$ (the unit disk in $R^n$ with Euclidean metric) is harmonic -- i.e. the components are bounded harmonic functions in the ordinary sense satisfying $f\_1^2 + \ldots + f\_n^2 < 1$. Let $\phi: D^n \to D^n$ be the identity map, where the domain has the Eucli... | 10 | https://mathoverflow.net/users/17969 | 103022 | 59,734 |
https://mathoverflow.net/questions/103018 | 8 | My problem is to characterise (or find useful information on) the cone $C$ of $N\times N$ matrices $M$ ($N\geq 1$) such that $$V^t M V\geq 0$$ for every vector $V $ with non-negative entries. Is this cone of matrices familiar to anyone?
Remark 1: $C$ clearly contains the convex cone $S$ of semi-definite positive matr... | https://mathoverflow.net/users/16934 | Characterising semi-definite positiveness on vectors with non-negative entries | The cone $C$ is called the cone of copositive matrices and its dual $C^\*$ is called the cone of completely positive matrices. Here are some references.
The paper most relevant to your question is probably "On Non-Negative Forms In Real Variables Some Or All Of Which Are Non-Negative," in which P. H. Diananda shows t... | 10 | https://mathoverflow.net/users/5963 | 103029 | 59,738 |
https://mathoverflow.net/questions/102974 | 3 | Hello all,
I have happened upon the following sum:
$ 1^2 + \Big(1 \times \frac{1}{3} + \frac{1}{3} \times 1 \Big)^2 + \Big(1 \times \frac{1}{5} + \frac{1}{3} \times \frac{1}{3} + \frac{1}{5} \times 1 \Big)^2 + \Big(1 \times \frac{1}{7} + \frac{1}{3} \times \frac{1}{5} + \frac{1}{5} \times \frac{1}{3} + \frac{1}{7} ... | https://mathoverflow.net/users/13358 | Question on a Basel-like sum | Interesting problem. Here is my version.
$$\begin{align}
f(u) &= u+ \frac{u^3}{3}+\frac{u^5}{5}+\dots = \frac{1}{2}\log\frac{1+u}{1-u}
\cr
f(u)^2 &= u^2 + \left(1\cdot\frac{1}{3}+\frac{1}{3}\cdot 1\right)u^4 +
\left(1\cdot \frac{1}{5}+\frac{1}{3}\cdot\frac{1}{3}+\frac{1}{5}\cdot 1\right)u^6+\dots
\end{align}$$
out of t... | 4 | https://mathoverflow.net/users/454 | 103039 | 59,747 |
https://mathoverflow.net/questions/103020 | 5 | Let me recall the definition which seems the most standard of Fricke polynomials.
Let $G$ be the free group with two generators $u,v$. It is not very hard to prove that there exists a unique application $t : G \rightarrow \mathbb{Z}[X,Y,Z]$
such that
(1) $t(u)=X,t(v)=Y,t(uv)=Z$,
(2) $t$ is the character (i.e. the... | https://mathoverflow.net/users/9317 | Which polynomials are Fricke polynomials ? | I do not think that there is a complete answer to this question. However, one can give some necessary conditions (which show that any answer must be complicated).
One can show that a triple $(x,y,z) \in \mathbb C^3$ comes from traces of matrices in $SU(2)$ (i.e. there exist $u,v \in SU(2)$ such that $(x,y,z)=(t(u),t(... | 2 | https://mathoverflow.net/users/8176 | 103040 | 59,748 |
https://mathoverflow.net/questions/103044 | 2 | Is it possible for two different $n$-element sets, each of which consists of $n$ unique positive integers (they can appear in both sets, though) to have the same sum when the squares of their elements are added?
Edit: For obvious reasons, I'm not considering the case $n=1$.
| https://mathoverflow.net/users/25314 | Question on Sums of Squares | Yes. One way to see this is that there are more $n$-element subsets with terms up to $N$ than there are possible sums of squares, giving an answer by the pigeonhole principle.
A more beautiful answer was given by Prouhet in the 1850's, who exhibited for each $n$
an explicitly-defined pair of sets $A$ and $B$ of size ... | 13 | https://mathoverflow.net/users/11054 | 103045 | 59,751 |
https://mathoverflow.net/questions/103023 | 3 | Given a quadratic differential $q$ on a surface of genus $g$, we say that $q\in \mathcal Q(k\_1,\ldots,k\_n)$ if $q$ has $n$ distinct zeroes of order $k\_1,\ldots,k\_n$ respectively. The set $\mathcal Q(k\_1,\ldots,k\_n)$ is called the stratum of $q$.
A polygon $P$ with angles equal to rational multiples of $\pi$ giv... | https://mathoverflow.net/users/13832 | Strata of quadratic differentials from rational billiards | Dear Alex Becker,
You can find the answer to your question in the excellent survey "Rational billiards and flat structures" of H. Masur and S. Tabachnikov: for a free online version see here (<http://math.uchicago.edu/~masur/handbook.dvi>).
As you can check in this survey, it is not very hard to deduce the singul... | 3 | https://mathoverflow.net/users/1568 | 103046 | 59,752 |
https://mathoverflow.net/questions/103025 | 5 | Let $X$ be an affine algebraic variety over an algebraically closed field $k$ of characteristic zero. Let $G$ be a reductive algebraic group acting on $X$. In this setting, there exists a categorical quotient variety $X / G$. Are there nice conditions (involving $X$ and/or $G$) that imply that $\mathrm{dim}(X / G) = \m... | https://mathoverflow.net/users/9960 | When does dimension behave nicely for quotients of affine algebraic varieties by the action of a group? | A sufficient condition is that there should exist a closed orbit of maximal dimension (i.e. of dimension $\dim(G)$). Indeed, when it is the case, the stable locus is nonempty, and the image of the stable locus in $X/G$ is an open subset of $X/G$ of dimension $\dim(X)-\dim(G)$. This happens for instance if $G$ acts prop... | 7 | https://mathoverflow.net/users/2868 | 103048 | 59,753 |
https://mathoverflow.net/questions/102999 | 9 | When dealing with some lifting problems, I came across the following problem, which probably has a well-known answer, but anyway:
Suppose I have a (locally contractible) connected topological group $G$, such that $\pi\_1(G) \cong \mathbb{Z}$. Let $\tilde{G} \to G$ be its universal covering group. Let $P \to X$ be a p... | https://mathoverflow.net/users/3995 | line bundles and universal covers | I don't think there is a canonical construction of the associated line bundle, this will be only defined up to homotopy, in some sense.
Here is a slightly more geometric construction. Let $G'$ be the group obtained by substituting $\mathbb Z \subseteq \widetilde G$ with $\mathbb R$; that is, $G' = (\widetilde G \time... | 6 | https://mathoverflow.net/users/4790 | 103050 | 59,754 |
https://mathoverflow.net/questions/103049 | 12 | Given a knot in the 3-sphere in [Bridge Position](http://books.google.ca/books?id=s4eGEecSgHYC&lpg=PA114&ots=GH9_c5YQUN&dq=bridge%2520number%2520rolfsen&pg=PA114#v=onepage&q=bridge%2520number%2520rolfsen&f=false) you can find a presentation for the fundamental group of the complement (a Wirthinger presentation) contain... | https://mathoverflow.net/users/1465 | Knot theory question: bridge number vs. min generators of fundamental group of complement | The (p,q) torus knot has a presentation with two generators, namely $\langle x,y \mid x^p = y^q\rangle$, but if $p,q>2$ then it's non-alternating and so it must have bridge index greater than 2.
| 13 | https://mathoverflow.net/users/428 | 103051 | 59,755 |
https://mathoverflow.net/questions/103055 | 8 | I've found a few articles that write the ring of formal Laurent series in $t$ as $R((1/t))$, but what's the underlying meaning of $\cdot ((\cdot))$?
A mathematician of my acquaintance swears that $R((t))$, not $R((1/t))$, should be used to denote the ring of formal Laurent series in $t$. We can't decide who's right w... | https://mathoverflow.net/users/3621 | notation for formal Laurent series | I agree with the mathematician of your acquaintance -- well, okay, I am the mathematician of your acquaintance.
Here are some references for the notation $K((x))$ for the field of formal Laurent series $\sum\_{n \geq n\_0} a\_n x^n$ over $K$:
>
> The [wikipedia article on formal power series](http://en.wikipedia.... | 7 | https://mathoverflow.net/users/1149 | 103058 | 59,757 |
https://mathoverflow.net/questions/103054 | 14 | Is there a characterization of modules $N$ for which the functor $N\otimes-$ reflects exact sequences?
| https://mathoverflow.net/users/23011 | When tensor reflects exact sequences? | As Alex Becker pointed out, $N$ is faithfully flat iff $N\otimes\_R -$ preserves and reflects exact sequences. Since $N$ is flat iff $N\otimes\_R -$ preserves exact sequences, you might think $N$ is faithful iff $N\otimes\_R -$ reflects exact sequences. This is wrong, as we'll show below. These faithfully flat modules ... | 8 | https://mathoverflow.net/users/11540 | 103059 | 59,758 |
https://mathoverflow.net/questions/103035 | 6 | I know Major Macmahon conjectured the formula $$ \prod\_{m=1}^\infty \frac{1}{(1-q^m)^m}=1 + \sum\_{n=1}^\infty PL(n)q^n$$
but who was the first to prove it?
| https://mathoverflow.net/users/12337 | Who proved that the plane partition generating function is valid? | The answer is MacMahon himself, who proved this in his book *Combinatory Analysis* as a corollary of a more general theorem about plane partitions. See Sections IX and X.
There is some additional historical information in the Notes to Chapter 7 of Richard Stanley's book *Enumerative Combinatorics*, volume 2.
| 8 | https://mathoverflow.net/users/3106 | 103061 | 59,760 |
https://mathoverflow.net/questions/103057 | 30 | This is a question in two parts.
Say that $\mathbf{On}$ is the proper class of all ordinal numbers in ZFC. We can define a binary operator over $\mathbf{On}$ which corresponds to the **commutative** version of ordinal addition; this has been called "Hessenberg addition" and "natural addition" before. It's also the op... | https://mathoverflow.net/users/24611 | Are Conway's omnific integers the Grothendieck group of the ordinals under commutative addition? | There is an obvious extension of Cantor normal form to the Grothendieck group of the ordinals. Then the standard argument that $\sqrt{x}$ does not lie in the ring $\mathbb Z[x]$ applies to $\sqrt{\omega}$. Specifically, $\sqrt{\omega}$ must have a Cantor normal form $a + b \omega + $ higher-order terms, which squares t... | 19 | https://mathoverflow.net/users/18060 | 103064 | 59,762 |
https://mathoverflow.net/questions/103062 | 3 | Our question arises from wondering about the systems of natural numbers in models ZFC + Con(ZFC) and ZFC + $\neg$Con(ZFC). In thinking of the systems of natural numbers of these models, we came to the following remark and questions. If we take the ultrapower of a model of Peano arithmetic we get a model of Peano arithm... | https://mathoverflow.net/users/nan | Models of the natural numbers in ultrapowers in the universe. | An ultrapower of a model $M$ of ZFC will have non-standard natural numbers if and only if either $M$ has non-standard natural numbers or the ultrafilter used in forming the ultrapower is countably incomplete. The natural numbers of the ultrapower of $M$ with respect to an ultrafilter $U$ are isomorphic to the ultrapowe... | 6 | https://mathoverflow.net/users/6794 | 103066 | 59,763 |
https://mathoverflow.net/questions/103041 | 6 | Who was the first to develop the asymptotic formulae for the distinct parts version of $p(n)?$
| https://mathoverflow.net/users/12337 | Who discovered the asymptotic formula for the number of partitions of n into distinct parts? | According to Dickson, History of the Theory of Numbers, Volume 2, page 162, "G. H. Hardy and S.Ramanujan proved that the logarithm of the number $p(n)$ of partitions of $n$ is asymptotic to $\pi\sqrt{2n/3}$, and the logarithm of the number of partitions of $n$ into distinct positive integers is asymptotic to $\pi\sqrt{... | 7 | https://mathoverflow.net/users/3684 | 103070 | 59,766 |
https://mathoverflow.net/questions/103056 | 17 | When is an acyclic chain complex contractible?
I know an acyclic chain complex of free modules over a PID (or field) are always contractible, but what about over a more complicated ring, like a graded algebra over Z/p (for instance the mod p Steenrod Algebra)?
EDIT: I want to assume the chain complex is bounded b... | https://mathoverflow.net/users/24021 | When is an acylic chain complex contractible | There is a useful characterization in Brown: Cohomology of Groups, Prop. 0.3:
>
> A chain complex $C$ over any ring is contractible iff it is acyclic and each short exact sequence $0 \to \ker(d\_n) \to C\_n \to \operatorname{im}(d\_n) \to 0$ splits.
>
>
>
This immediately explains the OP's PID example: If $C... | 19 | https://mathoverflow.net/users/10194 | 103071 | 59,767 |
https://mathoverflow.net/questions/102580 | 3 | Say you have a finite-dimensional vector space $V$ with an $\ell^p$ norm on it. In general, the norm induced on a subspace $V\_s$ of doesn't have to be another $\ell^p$ norm, so the unit sphere in $V\_s$ using this induced can be some strange shape.
Given $V\_s$ and a norm $||·||$ induced this way on it, how can one ... | https://mathoverflow.net/users/24611 | How do you compute the dual norm of an induced norm on a subspace of a finite-dimensional $\ell^p$-normed vector space? | An exact solution can be found here using the Hahn-Banach Theorem: <http://math.unl.edu/~s-bbockel1/928/node25.htm>
Using this, you can show that $V^∗\_S$ is isometrically isomorphic to $V^∗/S$°, where $S°$ is the subspace in V∗ for which $s(t)=0$ for $s$ in $S°$ and $t$ in $S$. – Mike Battaglia Jul 21 at 4:06
| 2 | https://mathoverflow.net/users/24611 | 103075 | 59,770 |
https://mathoverflow.net/questions/38891 | 19 | I've been trying a learn a little more about group schemes by working through a set of exercises on Brian Conrad's website. Exercise 8.3 of <http://math.stanford.edu/~conrad/papers/gpschemehw1.pdf> reads:
>
> If $k$ is a perfect field and $G$ is a locally finite type $k$-group prove that $G\_{red}$ is a closed $k$-... | https://mathoverflow.net/users/3384 | Is there a connected $k$-group scheme $G$ such that $G_{red}$ is not a subgroup? | The new edition of SGA3 by Philippe Gille et Patrick Polo provides a connected example due to Raynaud. It is in SGA3 Exposé VIA Exemples 1.3.2 (2) and you may find it at <http://www.math.jussieu.fr/~polo/SGA3/Exp6A-23mai11.pdf>.
This example is $2$-dimensional, but it is easy to modify it to get a $1$-dimensional exa... | 12 | https://mathoverflow.net/users/2868 | 103080 | 59,775 |
https://mathoverflow.net/questions/102990 | 3 | Let $X$ be a smooth complex surface. Assume that $2$-dimensional cohomology class $[A] \in H^{2}(X, \mathbb{C})$ belongs to $H^2(X,\mathbb{Z}) \cap H^{1,1}(X)$. Then the class $[A]$ can be represented by a complex-algebraic submanifold of $X$. Does this follow from Lefschetz theorem on $(1,1)$-classes?
Thanks
| https://mathoverflow.net/users/25306 | Algebraic cycles | Here is a more detailed answer which somehow sums up the comments and remarks above.
To begin with, what I say is valid more generally on every smooth projective manifold: you don't need to restrict your attention to surfaces! On the other hand, you must require projectivity (which you don't do in your question) otherw... | 3 | https://mathoverflow.net/users/9871 | 103084 | 59,778 |
https://mathoverflow.net/questions/103085 | 12 | Are applications of group theory known to exist in numerical analysis?
One particular aspect I am curious about is whether matrix groups have been successfully used to derive algorithms.
Also, are there aspects of numerical analysis that would have been difficult to conceive or derive WITHOUT the help of group theory?
... | https://mathoverflow.net/users/25326 | Applications of group theory in numerical analysis? | [Alain Connes](http://en.wikipedia.org/wiki/Alain_Connes)& K mentions some "Butcher group" ( <http://arxiv.org/abs/hep-th/9904044> ):
>
> ... We emphasize the unifying role which the Butcher group, discovered in the study of **numerical integration of ordinary differential equations**, plays in QFT.
>
>
>
See ... | 8 | https://mathoverflow.net/users/10446 | 103091 | 59,781 |
https://mathoverflow.net/questions/102751 | 17 | In connection with [this MO post](https://mathoverflow.net/questions/102725/a-mixing-property-of-linear-map-over-finite-fields), here is a question somewhat implicitly contained in a [joint paper of S. Kopparty, S. Saraf, M. Sudan, and myself](http://arxiv.org/pdf/1003.3736.pdf).
Let ${\mathbb F}$ be a finite field, ... | https://mathoverflow.net/users/9924 | A mixing property for finite fields of characteristic $2$ | The use of Birch/Swinnerton-Dyer in my previous and David Speyer's answer is vaste overkill!
Actually in David's example one can compute exactly the value set sizes. I do only the relevant case $m=1$. (As the size is exact, there is no need for the asymptotic consideration $m\to\infty$, only $r\to\infty$ matters.)
*T... | 10 | https://mathoverflow.net/users/18739 | 103095 | 59,784 |
https://mathoverflow.net/questions/103092 | 5 | Does anybody know of any identities or combinatorial interpretations for alternating sums of alternate Stirling numbers?
I am particularly interested in expressions of the form:
$$\pm\sum\_{k}(-1)^k|s(n,2k)|=\mp|s(n,2)|\pm|s(n,4)|\mp|s(n,6)|\pm\ldots $$
or:
$$\pm\sum\_{k}(-1)^k|s(n,2k+1)|=\pm|s(n,1)|\mp|s(n,3)|... | https://mathoverflow.net/users/4078 | Alternating sums of alternate Stirling numbers | The generating function here is
$$\sum\_{n\geq 0}s(n,k) x^n y^k=\sum\_{n\geq 0} \frac{x^n}{n!}y(y-1)\cdots (y-n+1)=e^{y\log(1+x)}.$$
If we put $y=i$ the coefficient of $x^n$ becomes $A\_n+iB\_n$ where $A\_n$ and $B\_n$ are your sequences. It is pretty clear from here that the exponential generating function for $A\_n$ ... | 13 | https://mathoverflow.net/users/2384 | 103102 | 59,788 |
https://mathoverflow.net/questions/103074 | 3 | Are there any estimates for the eigenvalues of the Laplace operator for $\Gamma \backslash SL(2, \mathbb{C})/SU(2)$ known beyond the main term? Here, $\Gamma$ should be congruence subgroup in $SL(2,o)$ for $o$ the ring of integers in an imaginary quadratic field.
I have checked Mathscinet and couldn't find anything e... | https://mathoverflow.net/users/10400 | Weyl law for SL(2,C) | For general compact manifolds (of dimension $n$), the error term (on the number of eigenvalues less than $T^2$, counted with multiplicity) is $O(T^{n-1})$. So for $\Gamma$ co-compact, the error term is $O(T^2)$.
For merely co-finite $\Gamma$, it should be possible to bound the error term, but I am unaware of any resu... | 4 | https://mathoverflow.net/users/6753 | 103106 | 59,792 |
https://mathoverflow.net/questions/103003 | 3 | The tangent numbers $(T\_{2n+1})=(1,2,16,272,7936,...)$ (cf. OEIS: A000182) satisfy many recurrences. I would be interested to find references for the following which I think must be very old:
$T\_3 -2T\_1=0$, $T\_5 -8T\_3 =0,$ $T\_7 -18T\_5 +8T\_3 =0,...$ or more generally
$${T\_{2n + 1}} = \sum\limits\_{j \ge 1} {}... | https://mathoverflow.net/users/5585 | Reference request for an identity for tangent numbers | The following is too long for a comment, so let me type it as an answer though it does not literally answer your question.
Using the standard formula
$$
T\\_{2k-1}=(-1)^{k-1}2^{2k}(2^{2k}-1)\frac{B\_{2k}}{2k},
$$
your formula can be rewritten as
$$
(2^{2n+2}-1)B\_{2n+2}=\sum\_{j\ge1}(-1)^{n-j+1}\binom{n+1}{2j}(2^{... | 2 | https://mathoverflow.net/users/1306 | 103109 | 59,793 |
https://mathoverflow.net/questions/103116 | 7 | Let $a,b$ be sets, we write $a\leq^\ast b$ if either $a=\varnothing$ or there exists a surjection $f\colon b\to a$. With the axiom of choice this is a linear ordering equivalent to the usual ordering of cardinals $a\leq b$ if there is an injection from $a$ into $b$).
Without the axiom of choice this order need not be... | https://mathoverflow.net/users/7206 | $\Theta$ and the Hartogs of $2^\mathbb R$ | Yes, it is consistent with AD. In $L(\mathbb{R})$, if AD holds then $\Theta$ injects into $\mathcal{P}(\mathbb{R})$, or equivalently, into $2^{\mathbb{R}}$. The OD sets of reals are Wadge-cofinal and $\Theta$ is regular, so we can define an injection $F : \Theta \to \mathcal{P}(\mathbb{R})$ by induction. Let $F(\alpha)... | 4 | https://mathoverflow.net/users/1682 | 103118 | 59,794 |
https://mathoverflow.net/questions/101834 | 10 | Is the Shapiro inequality for $n=23$ an open problem? The reason why I am asking is I have two contradictory pieces of information from two different articles.
The first article titled "The validity of Shapiro’s cyclic inequality" published by B.A. Troesch in the journal "Mathematics of Computation" in 1989 claims th... | https://mathoverflow.net/users/nan | Shapiro inequality for $n=23$ | There is no incongruity between the two papers that have been cited in the question.
According to [A.M. Fink](http://books.google.de/books?id=xtqZ7_BfKJAC&lpg=PA247&ots=TJZeH60U0O&dq=shapiro%2520inequality&pg=PA247), what is missing is completely analytical proof of the case $n=23$. So when Bushell wrote that $n=23$... | 7 | https://mathoverflow.net/users/8430 | 103123 | 59,796 |
https://mathoverflow.net/questions/103111 | 7 | ~~Strauch & Tóth [1]~~ Georges Grekos [3][4] showed that for any choice of upper and lower density, there is some subset of $\mathbb{N}$ with the chosen densities, provided the lower is no more than the upper and both are in [0, 1].
Mišík [2] extended this to show that for any choice of upper density, lower density, ... | https://mathoverflow.net/users/6043 | Prescribed values for the uniform density | OK. I think you can do this pretty easily by hand.
First you need a way to generate sequences with very uniform density. The Sturmian sequences are perfect for this. A [Sturmian sequence](http://en.wikipedia.org/wiki/Sturmian_word) with parameter $\alpha$ has the property that sub-blocks of length $N$ have density con... | 5 | https://mathoverflow.net/users/11054 | 103127 | 59,799 |
https://mathoverflow.net/questions/103128 | 10 | Wikipedia credits Bourbaki with coining it, but doesn't provide a source. Does anyone happen to know the motivation for using this term?
| https://mathoverflow.net/users/25342 | What is the origin of the term magma? | The second definition in <http://www.larousse.com/en/dictionnaires/francais/magma/48543> would seem to answer your question and agrees with the wiki page.
Update: here is what the link said to avoid broken links as YCor pointed out.
>
> magma
>
> nom masculin
> (latin magma, résidu, du grec magma)
>
>
> ... | 6 | https://mathoverflow.net/users/15934 | 103131 | 59,800 |
https://mathoverflow.net/questions/103132 | 24 | I'm not very experienced in this topic, but I read a short description of the Yang-Mills existence and mass gap problem, and as long as I understood it has mainly physical consequences and implications. Therefore, I was just curious what it has to do with mathematics? And what are the mathematical and general consequen... | https://mathoverflow.net/users/24541 | What does Yang-Mills and mass gap problem has to do with mathematics? | There is a long, long list of mathematical subjects that were either pioneered or significantly inspired by results in quantum field theory. However, while physicists may trust the manipulations they do in QFT, and the results of those manipulations have been spectacularly successful, for almost every interesting quant... | 22 | https://mathoverflow.net/users/947 | 103139 | 59,803 |
https://mathoverflow.net/questions/102463 | 3 | Is it known whether or not every sheaf of ideals of the etale structure sheaf of a Noetherian scheme is generated by finitely many of its sections? Of course it is trivially true for some widely used special cases. But is it known one way or the other, in this generality?
| https://mathoverflow.net/users/38783 | Ideals of etale structure sheaves | I must apologize for posting a false answer. in writing up a proof i discovered a gap which grew to a counterexample.
In fact not every sheaf of ideals of an etale structure sheaf is finitely generated. I have added a counterexample to the end of my ArXiv paper on cohomology in second order arithmetic arXiv:1207.0276... | 3 | https://mathoverflow.net/users/38783 | 103142 | 59,805 |
https://mathoverflow.net/questions/103150 | 3 | Fix some positive integers $N$ and $d\_k$, $k=1,2,\dots$ with $N=\sum\_{k=1}^\infty d\_k$.
Suppose you have a graph $G$ taken randomly uniformly among the set of all (unoriented) graphs with $N$ vertices, $d\_k$ of which have degree $k$ (i.e., they are connected by an edge to $k$ other vertices).
Now suppose that y... | https://mathoverflow.net/users/4129 | The degrees in a random subgraph | It doesn't matter that $G$ was chosen randomly. The choice of $G$ might matter if you asked for something more complicated about the distribution than the expected value.
The probability that a vertex $v$ is included is $M/N$.
Let the degree of $v$ be $h \ge k$ in $G$. The chance that precisely $k$ of its neighbors... | 3 | https://mathoverflow.net/users/2954 | 103151 | 59,810 |
https://mathoverflow.net/questions/103129 | 21 | Jonathan Sondow elegantly proves the irrationality of *e* in his aptly titled *A Geometric Proof that e Is Irrational and a New Measure of Its Irrationality* (The American Mathematical Monthly, Vol. 113, No. 7 (Aug. - Sep., 2006), pp. 637, <http://www.jstor.org/stable/27642006>).
In his argument, he constructs a sequ... | https://mathoverflow.net/users/22971 | Irrationality proof technique: no factorial in the denominator | The same proof technique, for modified versions of the **Fact**, proves that some values of some hypergeometric functions are irrational. For example, the Bessel functions of the first kind have the following power series:
$$J\_n(x) = \sum\_{i=0}^\infty \frac{(-1)^i}{i! (i+n)!} \bigg(\frac x 2\bigg)^{2i+n} $$
For a... | 12 | https://mathoverflow.net/users/2954 | 103161 | 59,815 |
https://mathoverflow.net/questions/103130 | 1 | Hello everyone,
I'm trying to look at Mumford's Paper, [The Pathologies of Modular Surfaces](http://www.dam.brown.edu/people/mumford/Papers/DigitizedAlgGeomPapers--ForNon-CommercialUse/61b--Path1.pdf).
On page 341, section II he says a certain surface can be constructed as the join of 3 graphs $E\_0 \rightarrow \ma... | https://mathoverflow.net/users/25343 | Question about Terminology in Mumford | The **join** of two varieties $X,Y\subseteq \mathbb{P}^n$ is
$$ J(X,Y) = \overline{\bigcup\_{\substack{x\in X,~y\in Y\\x\ne y}} \ell(x,y)}$$
where $\ell(x,y)$ denotes the projective line through $x$ and $y$. The join of $k$ varieties $X\_1,\ldots,X\_k\subseteq \mathbb{P}^n$ is defined to be the closure of the union of... | 1 | https://mathoverflow.net/users/9947 | 103167 | 59,819 |
https://mathoverflow.net/questions/103160 | 0 | Recently, I met an equation about the integration by parts and surface integrals. It says:$$
\int\_{|\xi|\geq\epsilon}D\_i\Gamma(\xi){\partial\over\partial{\xi\_i}}f(\xi)d\xi=-\int\_{|\xi|=\epsilon}D\_i\Gamma(\xi)f(\xi){\xi\_i\over{|\xi|}}dS$$
here $\Gamma$ satisfy $\triangle\Gamma(\xi)=0$ and $f\in C\_0^\infty(R^n)$. ... | https://mathoverflow.net/users/25094 | the relationship between integration by parts and surface integrals | This follows immediately from Green's formula, which says for $X$ vector field,
$\Omega$ open set,
$n$ the unit exterior normal to the boundary $\partial \Omega$
$$
\int\_{\Omega}div X\ dx=\int\_{\partial \Omega}X\cdot n d\sigma.
$$
Apply this to the vector field $X=u\nabla v$ and you get
$$
\int\_{\Omega}(\nabla u\cdo... | 2 | https://mathoverflow.net/users/21907 | 103170 | 59,821 |
https://mathoverflow.net/questions/102964 | 14 | I have a sequence $\{X\_n\}$ of random variables supported on the real line, as well as a normally distributed random variable $X$ (whose mean and variance are known but irrelevant). I know that the moments of the $X\_n$ converge to the corresponding moment of $X$, that is, for every $k\ge1$,
$$
\lim\_{n\to\infty} \mu\... | https://mathoverflow.net/users/5091 | Convergence of moments implies convergence to normal distribution | It is theorem 30.2 in Billingsley's *Probability and Measure* (I own a second Polish edition, so numbering may differ a little).
It's quite easy to prove it, once you estabilish Prokhorov's theorem; namely use boundedness of some moments to conclude that your sequence of distributions is tight and then it suffices to... | 13 | https://mathoverflow.net/users/24953 | 103174 | 59,824 |
https://mathoverflow.net/questions/103154 | 2 | The total space $T$ of an embedded into $\mathbb{R}^n$ pure $n$-dimensional simplicial complex (in other words, the union of finitely many $n$-dimensional compact convex polytopes) sometimes admits an "almost partition" into $n$-simplices (i.e. $T$ equals the union of these simplices, and the interiors of these simplic... | https://mathoverflow.net/users/11100 | coarser than triangulations "almost partitions" into simplices | The term "triangulation" tends to be ambiguous, as they appear both in a geometric and topological context. In this case, what you want is called a *dissection*, at least in the discrete geometry literature (see e.g. [here](http://www.math.ucla.edu/~pak/book.htm) and [there](http://www.springer.com/mathematics/geometry... | 4 | https://mathoverflow.net/users/4040 | 103177 | 59,825 |
https://mathoverflow.net/questions/103147 | 10 | I'm studying a particular kind of curve evolution on Riemannian manifolds. It would help me
to know the answer to the following kinda weird question:
Does there exist a closed Riemannian manifold $M$ and a pair of distinct closed geodesics
$\gamma, \alpha$ in $M$ satisfying the follow properties?
(1) $\gamma$ and ... | https://mathoverflow.net/users/23743 | A strange question about closed geodesics on a closed manifold | Is it a homework problem?
Define the distance between curves as
$$d(\gamma,\gamma')=\inf\_h\sup\_x|\gamma'(x)-\gamma\circ h(x)|,$$
where $h:\mathbb S^1\to\mathbb S^1$ is reparametrization.
Fix small $\delta$ so that if $d(\gamma,\gamma')\le \delta$
and $\mathop{\rm length}\gamma=\mathop{\rm length}\gamma'$
then $\... | 6 | https://mathoverflow.net/users/1441 | 103180 | 59,826 |
https://mathoverflow.net/questions/103175 | 2 | According to ["EQUIVALENCES TO THE RIEMANN HYPOTHESIS](http://aimath.org/pl/rhequivalences) p.4
Let $g(n)$ be the maximal order of a permutation of n objects
RH Equivalence 3.3. The Riemann Hypothesis is equivalent to
$\log{g(n)} < Li^{-1} (n)$ for n large enough.
$Li$ is strictly increasing for $n>1$ and $\log{g... | https://mathoverflow.net/users/12481 | What are the fallacies that this RH inequality may fail at most finitely often? | I suspect the mistake is in relying on Sage or Maple in the last step. Instead use the asymptotic expansion of li(x) ( <http://en.wikipedia.org/wiki/Logarithmic_integral_function> )
$$\operatorname{li}(x)=\frac x {\log x}+\frac{x}{\log^2 x}+O \left(\frac x {\log^3 x} \right)$$
to obtain an asymptotic expression of $G(n... | 7 | https://mathoverflow.net/users/10811 | 103181 | 59,827 |
https://mathoverflow.net/questions/103179 | 2 | Suppose I have a family of $n$ linearly-independent elements $v\_i$ of the Hilbert space $\mathbb{C}^m$, which are not necessarily orthogonal. Can I always find a partial isometry $f: \mathbb{C} ^m \to \mathbb{C} ^n$ such that the vectors $f(v\_i)$ are orthogonal and nonzero?
If not, under what conditions on the fami... | https://mathoverflow.net/users/799 | Partial isometries making families of linearly independent vectors orthogonal | The answer is no in general: if $n=m$ a necessary and sufficient condition is that the $v\_i$'s are already orthogonal.
The answer is yes if $n \leq m/2$ (proof: if $m \geq 2n$ without loss of generatlity we can write $\mathbb C^m \simeq \mathbb C^n \oplus \mathbb C^n \oplus \mathbb C^{m-2n}$ and assume that the firs... | 5 | https://mathoverflow.net/users/10265 | 103185 | 59,830 |
https://mathoverflow.net/questions/102122 | 4 | I define a *$n$-labeling* of a directed acyclic graph $G = (V, E)$ as a function $f$ from $V$ to the power set of {1, ..., $n$} such that for any $x, y \in V$, $x \neq y$, we have $f(y) \subset f(x)$ iff $x \rightarrow^+ y$ (i.e. there is a path of length >0 from $x$ to $y$ in $G$). Clearly, any DAG $G$ admits a $|V|$-... | https://mathoverflow.net/users/16035 | Minimal labeling of a directed acyclic graph | Joel David Hamkins is right, the question is equivalent to finding the smallest $n$ for which a given partial order embeds into the power set lattice $\langle P(\lbrace 1, ..., n\rbrace), \subset\rangle$. It turns out that this question is well-known: the quantity $n$ is called the 2-dimension (because $n$ is the minim... | 0 | https://mathoverflow.net/users/16035 | 103190 | 59,832 |
https://mathoverflow.net/questions/84227 | 7 | Let $(S,\mathcal{S})$ and $(T,\mathcal{T})$ be measurable spaces. A *transition probability* from $S$ to $T$ is a function $\pi:S\times\mathcal{T}\to [0,1]$ such that $\pi(s,\cdot)$ is a probability measure for all $s\in S$ and $\pi(\cdot,B)$ is measurable for all $B\in\mathcal{T}$.
Now let $(\Omega,\Sigma,\mu)$ be a... | https://mathoverflow.net/users/35357 | Random Functions and Transition Probabilities | I've found a rather simple solution. No additional assumptions are necessary. There is a natural method of composing transition probabilities that gives rise to a new transition probability (see for example (2) [here](http://arxiv.org/pdf/1205.1488.pdf)).
We can identify probability measures with transition probabil... | 2 | https://mathoverflow.net/users/35357 | 103191 | 59,833 |
https://mathoverflow.net/questions/102730 | 5 | The differential of the exponential map on a symmetric space can be expanded
(abusing some notation) as
$d{\rm Exp}\_X=\sum\_{n=0}^{\infty}\frac{({\rm ad}X)^{2n}}{(2n+1)!}.$
This is an old (1958) result of Helgason.
*Question* (EDITED):
Is there any generalization to reductive homogeneous spaces?
[EDIT 2017] ... | https://mathoverflow.net/users/9833 | The differential of the exponential map: reductive homogeneous space | I think the formula will be essentially the same.
The formula you wrote is valid in general for the exponential map of analytic manifolds equipped with an analytic affine connection. It is stated and proved in [this paper](http://www.mscand.dk/article.php?id=1601) by Helgason (see pages 6-7 of the linked .pdf): *Som... | 4 | https://mathoverflow.net/users/7031 | 103192 | 59,834 |
https://mathoverflow.net/questions/103202 | 4 | Let $A$ and $B$ be integrally closed, commutative Noetherian integral domains, and let $f: A \to B$ be a finite étale injective homomorphism. Let $d$ be the degree of $f$ (i.e. the rank of $B$ as an $A$-module).
If $p$ is a height 1 prime ideal of $A$, and $q$ is a prime ideal of $B$ lying above $p$, then $B/qB$ is a... | https://mathoverflow.net/users/12706 | Splitting of primes in extension of integral domains | $A$ is a Krull ring, so the localization $A\_p$ is a DVR, thus it is a Dedekind domain, so the statement is true for $A\_p$. Since $B\_p$ is etale over $A\_p$ and the splitting of $p$ in $B\_p$ is the same as the splitting of $p$ in $B$, this statement is also true for $A$.
| 4 | https://mathoverflow.net/users/18060 | 103207 | 59,841 |
https://mathoverflow.net/questions/102966 | 3 | Let $T^\*$ denote upper triangular matrices (of the appropriate size) with positive diagonal entries and $\mathrm{UT}$ upper triangular matrices with all diagonal entries equal to 1.
>
> Does every (abstract group) embedding $\varphi:\mathrm{UT}(n,\mathbb{R})\to\mathrm{UT}(m,\mathbb{R})$ extend to $\bar{\varphi}:T^... | https://mathoverflow.net/users/22599 | Does every embedding of one unipotent group (over R) in another extend to an embedding of the respective upper triangular matrix groups? | Here's a counterexample ($m=n=3$) which is a continuous homomorphism and actually probably also works for your second question with $\mathbf{Q}$.
In short: most automorphisms of $UT(3)$ do not extend to $T^\*(3)$.
Since I deal with continuous automorphisms, it boils down to a Lie algebra problem. An automorphism of... | 3 | https://mathoverflow.net/users/14094 | 103208 | 59,842 |
https://mathoverflow.net/questions/42146 | 1 | I'm looking for a copy of
"J Schmid, On the degree complexity of Hilbert's 17th problem and the real nullstellensatz. Habilitationsschrift, Universitat Dortmund, 1998."
Articles referring to this work mention the effective bounds on the complexity, but don't have the explicit expression.
PS a question in general: ... | https://mathoverflow.net/users/nan | does anyone have a copy of schmid's effective work on hilbert 17th? | Here is a copy:
<http://math.usask.ca/fvk/schmid.ps>
| 3 | https://mathoverflow.net/users/15506 | 103209 | 59,843 |
https://mathoverflow.net/questions/103115 | 11 | Context: I want to compare the sample probability distributions (PDFs) of two datasets (generated from a dynamical system). These datasets depend on a set of parameters, and I want a concise way to evaluate the distance between the two PDFs over several different parameter regimes, ideally by a single number. For a fix... | https://mathoverflow.net/users/25339 | Distance metric between two sample distributions (histograms) | In the first place the answer depends on the nature of your data (e.g., numerical continuous, numerical discrete, nominal etc.). In each of these cases the empirical measures on the range of your data have to be compared by using the corresponding specific methods. I presume that your values are reals (since you evoke ... | 10 | https://mathoverflow.net/users/8588 | 103210 | 59,844 |
https://mathoverflow.net/questions/95650 | 3 | Let $(E,\mathscr E)$ be a measurable space and $P,\tilde P$ be two stochastic kernels on that space. I wonder how the induced measures $\mathsf P\_x$ and $\tilde{\mathsf P}\_x$ differ on the space of finite trajectories $(\Omega\_n,\mathscr F\_n)$ and on the space of infinite trajectories $(\Omega,\mathscr F)$. I've al... | https://mathoverflow.net/users/11768 | Difference in probability distributions from two different kernels | It's much easier to understand and answer this question by using the (equivalent) language of families of transition probabilities corresponding to stochastic kernels. So, let $(\pi\_x)$ (resp., $(\tilde\pi\_x)$ be the family of transition probabilities corresponding to the kernel $P$ (resp., $\tilde P$). Then $$\|P-\t... | 2 | https://mathoverflow.net/users/8588 | 103216 | 59,846 |
https://mathoverflow.net/questions/103205 | 4 | This is only my second question on mathoverflow, so my apologies if this would be more appropriate at a physics site. My question concerns a modification to the Einstein-Hilbert action. The standard action is given (in the absence of matter and with cosmological constant $\Lambda=0$) by
$$ \mathcal{S\_{EH}}(g\_{\mu\n... | https://mathoverflow.net/users/25075 | Work on an Einstein-Hilbert type action but with the *absolute value* of scalar curvature? | In the vacuum case this is not greatly different from the Einstein-Hilbert action.
Let $(M,g)$ be a classical solution to the variational problem as you posed. Suppose $p\in M$ is such that $R(p) \neq 0$, then by continuity in a small neighborhood of $p$, the scalar curvature $R$ is signed, and hence locally in that... | 4 | https://mathoverflow.net/users/3948 | 103220 | 59,848 |
https://mathoverflow.net/questions/103228 | 1 | There is a natural fraction linear transform of $SL(2,\mathbb{R})$ on $\mathbb{C}P^1$ given by:
$$
\begin{pmatrix} a & b \\
c & d \end{pmatrix} \cdot[z,w]=[az+bw,cz+dw].
$$
Let $\mathbb{Z}/2=\{ 1,s \}$ be the group with 2 elements.
My question is: is there an action of $\mathbb{Z}/2$ on $\mathbb{C}P^1$ such that i... | https://mathoverflow.net/users/24965 | Find an action of $\mathbb{Z}/2$ on $\mathbb{C}P^1$ which is compatible with the fraction linear transform of $SL(2,\mathbb{R})$ | Such $s$ does not exist since it would have to fix fixed points of all parabolic elements of $SL(2,R)$ and, hence, a circle. Note that your compatibility notion is usually called commutation.
| 4 | https://mathoverflow.net/users/21684 | 103230 | 59,852 |
https://mathoverflow.net/questions/103182 | 11 | An acyclic category (also called loopfree category or scwol (small category without loops)) is a small category where only identity morphisms have inverses, and any morphism from an object to itself is the identity.
Every poset P can be regarded as an acyclic category by identifying the set of objects with the elemen... | https://mathoverflow.net/users/20356 | Acyclic categories related to structures in algebraic topology | One thing that has come up a little bit in my work and I believe also e.g. in work of Dmitry Feichtner-Kozlov is such an extension of the widely used fact (popularized by Gian-Carlo Rota) that the Moebius function $\mu\_P(x,y) $ of a poset $P$ may be interpreted as the reduced Euler characteristic of a simplicial compl... | 8 | https://mathoverflow.net/users/23408 | 103247 | 59,860 |
https://mathoverflow.net/questions/103258 | 9 | There is a popular (and I think helpful) example of etale covers, namely covers of Riemann surfaces with ramification points removed. Is there a similarly accessible example to motivate Nisnevich covers?
| https://mathoverflow.net/users/38783 | Is there a typical example of Nisnevich covers? | Well, a representative example is where you take some arbitrary etale cover Y of X which splits over a closed subvariety Z of X, then form the Nisnevich cover of X consisting of the open complement X - Z together with the open subscheme Y' of Y where you remove all but one of the copies of Z lying above Z. For instance... | 13 | https://mathoverflow.net/users/3931 | 103260 | 59,869 |
https://mathoverflow.net/questions/103141 | 7 | I'm familiar with the tensor product of modules, but I've also come across functor tensor product (in emily riehls paper on homotopy limits), what are they, and how are they (if they are) related to traditional tensor products? (Emily shows that they can be defined as a particular coend, but that doesn't really provide... | https://mathoverflow.net/users/22002 | What is the functor tensor product? | It is easy to be explicit. Not in full generality, given a (small) closed symmetric monoidal category $\mathcal C$ with coequalizers, a covariant functor $M\colon \mathcal C\to \mathcal C$ and a contravariant functor $N\colon \mathcal C \to \mathcal C$, the tensor product
$N\otimes\_{\mathcal C} M$ is the coequalizer ... | 17 | https://mathoverflow.net/users/14447 | 103264 | 59,872 |
https://mathoverflow.net/questions/103257 | 6 | The Nisnevich topology on $Sch$ is a Grothendieck topology strictly finer than the Zariski topology, and the etale topology is strictly finer than the Nisnevich topology.
Colin McLarty asked me for an example of a Nisnevich cover which is not a Zariski cover. The standard example I have seen in several places is rath... | https://mathoverflow.net/users/4177 | A Nisnevich cover which is not Zariski | My standard example is an $n$-gon of $\mathbb P^1$'s covering the nodal cubic. For some reason, I especially like the case $n=2$.
Here's the affine version, which always takes me a bit to work out. It's two parabolas joined at two points covering the nodal cubic:
$$
\def\spec{\mathrm{Spec\,}}
\spec k[s,t]/(t^2-(s^2-... | 6 | https://mathoverflow.net/users/1 | 103272 | 59,877 |
https://mathoverflow.net/questions/103294 | 0 | When solving non-linear equations via Newton's method, load increments are often used to improve convergence. In mechanics for example, if the final load in 90N, one could choose 3 load steps of 30N each. At each load step several Newton iteration are used until convergence, and the final converged result is used as th... | https://mathoverflow.net/users/16941 | solving non linear equations | To answer completely, one should know what is the application that you have in mind and if there is any specific convergence result for the Newton method.
But, in general, I think that you are right: solving an inexact model to full precision is not needed. If you are solving an inexact problem $P'$ whose solution $x... | 1 | https://mathoverflow.net/users/1898 | 103296 | 59,886 |
https://mathoverflow.net/questions/103298 | 0 | I'm reading the papaer "On the Reduction of a Matrix to Diagonal Form" of Epstein and Flanders (Amer. Math. Monthly 62, (1955). 168–171.
Let $S$ denote the trace function.
The authors stated that a well-known result in the theory of algebras of matrices is:
A matrix algebra $\mathbb U$ over a field $\mathbb F$ o... | https://mathoverflow.net/users/24864 | Matrix Algebras | Any book on Ring Theory covering Artin-Wedderburn's Theorem :-))
To be fair, you need some dexterity in using it:
If $U$ is not semisimple, then it has nonzero nilpotent ideal $I$. All elements $x\in I$ will have zero trace since $x^n=0$. Hence $S(Iy)=0$ for any $y$.
In the opposite direction, let $I$ be the kern... | 2 | https://mathoverflow.net/users/5301 | 103304 | 59,888 |
https://mathoverflow.net/questions/103279 | 9 | What are the known relations between isoperimetric and Poincaré inequalities on manifolds?
For example, for manifolds with a lower bound on Ricci curvature, the Cheeger-Buser inequality relates the isoperimetric constant to the first eigenvalue of the Laplacian. Does this imply a Poincaré inequality for such manifold... | https://mathoverflow.net/users/39082 | Isoperimetry and Poincaré Inequality | Everything here is for closed Riemannian manifolds.
If you have a lower bound on Cheeger's isoperimetric constant $h(M)$, then Cheeger's inequality
$\lambda\_1(M)\geq \frac{h(M)^2}{4}$ gives you a lower bound on the first eigenvalue of the Laplacian.
Thanks to the variational characterization of $\lambda\_1(M)$, this i... | 14 | https://mathoverflow.net/users/13168 | 103305 | 59,889 |
https://mathoverflow.net/questions/103301 | 5 | Is there a finite index **torsion-free** subgroup $G$ of $GL\_n(\mathbb Z)$, where $n\ge 3$,
such that the coinvariants group $\mathbb Z^n\_G$ is finite?
Here $G$ acts on $\mathbb Z^n$ in the standard way, and $\mathbb Z^n\_G$ by definition is the quotient of $\mathbb Z^n$ by the subgroup generated by the set $\{gz-... | https://mathoverflow.net/users/1573 | Subgroups of $GL_n(\mathbb Z)$ with finite coinvariants | For any finite index subgroup $G$ there is a nonzero integer $m$ so that $G$ contains the elementary matrices $e\_{ij}(m)$ that have ones on the diagonal, $m$ at the $(i,j)$ entry and zeroes elsewhere. So the span of $\{gz-z: g\in G, z\in\mathbb Z^n\}$ contains all multiples of $m$
and ${\mathbb Z}^n\_G$ is finite. Now... | 7 | https://mathoverflow.net/users/4794 | 103309 | 59,890 |
https://mathoverflow.net/questions/103308 | 9 | I'm looking for a reference that gives an overview of the most important properties of Arakelov intersection theory (on arithmetic varieties of arbitrary dimension) and that describes basic properties of the Arakelov Chow ring. There's a similar MO question asking about
[survey articles on (classical) intersection theo... | https://mathoverflow.net/users/11926 | Overview of Arakelov intersection theory and the Arakelov Chow ring | A good reference in my humble opinion is Bost's paper in Bourbaki:
Théorie de l'intersection et théorème de Riemann-Roch arithmétiques
Séminaire BOURBAKI. Novembre 1990. 43ème année, 1990-91, n° 731
Another reference would be Soule's book on Arakelov geometry "Lectures on Arakelov geometry" written with Abramovic... | 8 | https://mathoverflow.net/users/4333 | 103312 | 59,891 |
https://mathoverflow.net/questions/103313 | 5 | Dear all.
I'm a theoretical physicist trying to understand the structure equations and their geometrical significance, this for their gravitational applications.
I know the relation between the Lie algebras and the Maurer-Cartan structure equations, but as far as I know, they are used in manifolds which are not nec... | https://mathoverflow.net/users/25356 | Maurer-Cartan structure equation derivation | What you are asking for is an introduction to the theory of $G$-structures, for which the (Maurer-Cartan) structure equations are a basic tool.
There are many sources for this material, starting, of course, with the fundamental works of Élie Cartan on the subject, though many find his expositors, who use more modern... | 15 | https://mathoverflow.net/users/13972 | 103317 | 59,893 |
https://mathoverflow.net/questions/103288 | 5 | This possibly easy question is related to [this one.](https://mathoverflow.net/questions/102933/reconstructing-a-word) Let $s\_1,...,s\_n$ be a sequence of natural numbers (some of them may be equal to 0). Consider the following sequence of multisets of 2-vectors (each vector is counted with its multiplicity) of natu... | https://mathoverflow.net/users/nan | A combinatorial question | Unless I made mistake a counterexample is formed by the binary $m$-sequence of length 7 $s=(1,0,0,1,0,1,1)$ and its reversal $\tilde{s}=(1,1,0,1,0,0,1)$ that is not a cyclic shift of $s$. Both lead to the sequence of generating functions $1+2a+2b+2ab$, $2a+b+a^2+2ab+a^2b$, $a+a^2+2ab+a^3+2a^2b$, $a^2+ab+2a^3+2a^2b+a^3b... | 5 | https://mathoverflow.net/users/15503 | 103318 | 59,894 |
https://mathoverflow.net/questions/103287 | 4 | My question is:
Can we judge a manifold that can admit a (p,q) metric?
I only know the case that the existen of a lorentz metric is equivalent to Euler Character is zero
| https://mathoverflow.net/users/25054 | A Existence Problem of (p,q) metric | The criterion for existence of a $(p,q)$ metric is (assuming $p+q=dim X$) that the tangent bundle splits as a direct sum of two subbundles of dimensions $p$ and $q$.
EDIT : I doubt there is an easy algebraic topology criterion in general, as characteristic classes [EDIT after Lennart Meier's comment: other than Eule... | 7 | https://mathoverflow.net/users/6451 | 103329 | 59,900 |
https://mathoverflow.net/questions/103316 | 3 | Dear all,
When dealing with General Relativity one uses the Levi-Civita connection with is torsion-free. Thus the commutator of the covariant derivatives yields
$[\nabla\_\mu,\nabla\_\nu]V^\rho = R\_{\mu\nu}{}^\rho{}\_\lambda V^\lambda.$
Usually, it is said that curvature is the responsible of the change of the d... | https://mathoverflow.net/users/25356 | Interpretation of Curvature and Torsion | [Here](http://www.lightandmatter.com/html_books/genrel/ch05/ch05.html#Section5.8) is my attempt to present the intuition behind torsion in an accessible way. [Here](https://mathoverflow.net/questions/20493/what-is-torsion-in-differential-geometry-intuitively) is a similar, previous thread on MathOverflow.
In your que... | 2 | https://mathoverflow.net/users/nan | 103330 | 59,901 |
https://mathoverflow.net/questions/103322 | 1 | I need a bibliographical reference for this fact: let $\mathcal{M}$ be a model category such that all objects are cofibrant; then the class of weak equivalences is the class of maps f such that $\mathcal{M}(f,T)/\simeq$ is a bijection for any fibrant object $T$ where $\simeq$ is the homotopy relation. I would prefer a ... | https://mathoverflow.net/users/24563 | Bibliographical reference needed (characterizing the weak equivalences of a model category) | This is Theorem 7.8.6 on page 133 of Hirschhorn. The first direction (that any weak equivalence $f$ gives a bijection $\mathcal{M}(f,T)/\sim$, for $T$ fibrant) is Corollary 7.7.4(1). So the proof of the theorem is really just the proof of the other implication.
| 2 | https://mathoverflow.net/users/11540 | 103346 | 59,910 |
https://mathoverflow.net/questions/33681 | 9 | It is well known that there is an isomorphism of $SL\_2=SL(V)$ representations
$$
Sym^n(Sym^m(V))\simeq Sym^m(Sym^n(V))
$$
called Hermite reciprocity (discovered in 1854).
My question is: Is there anything like this isomorphism
for $U\_q(sl\_2)$, at least for generic $q$?
| https://mathoverflow.net/users/7410 | Is there a quantum Hermite reciprocity? | There is in fact a reasonable way to define quantum analogues of symmetric and exterior powers of a finite-dimensional representation of $U\_q(\mathfrak{g})$. Let $V$ be such a representation, and let $\hat{R} : V \otimes V \to V \otimes V$ be the braiding of $V$ coming from the universal R-matrix.
It is a fact (see,... | 7 | https://mathoverflow.net/users/703 | 103366 | 59,920 |
https://mathoverflow.net/questions/103361 | 6 | Let $A=\lim\_{r \rightarrow +\infty} \frac{Vol(B(o,r))}{\omega\_{n} r^{n}}$ for any Riemannian manifold $(\mathbb{M}^{n},g)$ with nonnegative Ricci curvature. Here $\omega\_{n}$ is the volume of unit ball in $\mathbb{R}^n$. We call $A$ the cone angle at infinity or asymptotic volume ratio, and the manifold is cone-like... | https://mathoverflow.net/users/12904 | cone angle at infinity for product of cones | I claim that $A=A\_1A\_2$.
Indeed, the distance in the product is given by the standard Pythagorean formula, therefore
$$
B(o,r) = \bigcup\_{x\in B\_1(o\_1,r)} B\_2(o\_2,\sqrt{r^2-|o\_1x|^2})
$$
where $B$, $B\_1$ and $B\_2$ denote metric balls in $M$, $M\_1$ and $M\_2$, $o=(o\_1,o\_2)$ and $|o\_1x|$ denotes the Riem... | 6 | https://mathoverflow.net/users/4354 | 103369 | 59,922 |
https://mathoverflow.net/questions/103362 | 8 | For a partition $\mu$ of $n$, let $S^{\mu}$ be the associated Specht module, defined over $\mathbb{Z}$. For any field $k$, we can tensor $S^{\mu}$ with $k$ to get a representation $S^{\mu}\_k$ of the symmetric group $S\_n$ over $k$. Finally, let $A\_n \cong \mathbb{Z}$ be the sign representation of $S\_n$ over $\mathbb... | https://mathoverflow.net/users/25383 | Dual of a Specht module | Yes, this works over $\mathbb Z$, and the pairing can be explicitly realised with polytabloids. See Section 4 of my paper "On the structure of Specht modules", J. London Math. Soc. 67 (2003) 85–102. (In retrospect, I wish I'd given this paper a more helpful title.)
Briefly: start by defining a map
$[,]: M^\mu\times... | 11 | https://mathoverflow.net/users/6771 | 103371 | 59,924 |
https://mathoverflow.net/questions/103338 | 4 | In the paper
P.J. Webb: Bounding the ranks of ZG-lattices by their restrictions to elementary abelian groups. J. Pure Appl. Algebra 23 (3) (1982), 311-318.
the author writes in the introduction without giving any reference or explanation ($G$ denotes a finite group):
>
> If $M$ is any $\mathbb{Z}G$-lattice, ... | https://mathoverflow.net/users/24759 | Uniqueness of the rank of the core of a lattice | Even more is true: Two possible cores of $M$ lie in the same genus, i.e. they become isomorphic in the localization at any prime of $\mathbb Z$. ~~However, I couldn't find an explicit proof of this anywhere in literature~~ **Edit:** It's Proposition 5.4 in Gruenberg: "Relation Modules of Finite Groups" (though unfortun... | 4 | https://mathoverflow.net/users/17498 | 103378 | 59,929 |
https://mathoverflow.net/questions/103365 | 11 | Dear everyone,
I was unable to obtain the equivalence between the two statements of the Hodge conjecture. I searched for some previous questions that others asked here, to check whether someone has asked the same thing previously. But I didn't find any such instance, that is why I am asking.
We know that Hodge conj... | https://mathoverflow.net/users/24713 | Equivalence between statements of Hodge conjecture | The rotation number in question has to do with the behaviour of a differential form $\omega$ on the manifold $X$ under rotation $e^{i\theta}$ of tangent vectors : a complex $k$-form $\omega$ has rotation number $p-q$ iff $$\omega(e^{i\theta}v\_1,\dots,e^{i\theta}v\_k)=e^{i(p-q)\theta}\omega(v\_1,\dots,v\_k)$$ (with $p+... | 14 | https://mathoverflow.net/users/6451 | 103385 | 59,931 |
https://mathoverflow.net/questions/103386 | 4 | Let $G$ be a simple group of order $p^{2}-1$ where $p$ is Mersenne prime.
I am looking for a contradiction when $p\mid |Aut(G)|$. Does anyone
knows how we can get a contradiction? Thanks.
| https://mathoverflow.net/users/25390 | Automorphism group of a simple group | Yes, if you mean contradiction to the simplicity of $G.$ Let $P$ be a Sylow $p$-subgroup of ${\rm Aut}(G).$ Let $Q$ be a $P$-invariant Sylow $q$-subgroup of $G$ where $q$ is an odd prime divisor of $p-1$ (there is one such as the number of Sylow $q$-subroups of $G$ is certainly not divisible by $p,$ while $P$ permutes ... | 6 | https://mathoverflow.net/users/14450 | 103390 | 59,933 |
https://mathoverflow.net/questions/103395 | 8 | Assume $\Sigma\_1$ and $\Sigma\_2$ are two embedded compact surfaces
(say orientable) in an orientable 3-manifold $M$. Assume $\Sigma\_1$ and $\Sigma\_2$
are homotopic in $M$. Then are they isotopic?
| https://mathoverflow.net/users/25393 | Isotopy in 3-manifolds | No, generally they're not.
For example, there's only one homotopy class $S^2 \to \mathbb R^3$ but there's two isotopy classes of embeddings (given via how the embedding orients the compact 3-manifold it bounds).
edit: I think if your 3-manifold is irreducible and if your maps $S^2 \to M$ are not null homotopic th... | 8 | https://mathoverflow.net/users/1465 | 103398 | 59,935 |
https://mathoverflow.net/questions/103402 | 12 | The Bass-Papp Theorem asserts that a commutative ring $R$ is Noetherian iff every direct sum of injective $R$-modules is injective. Thus every non-Noetherian ring carries a counterexample.
If
$$
I\_1 ⊊I\_2⊊\cdots⊊I\_n⊊\cdots
$$
is an infinite properly ascending chain of ideals of $R$, then for all $n$ let $E\_n=E(R/... | https://mathoverflow.net/users/25394 | The direct sum of injective modules need not be injective | The standard proof that I am aware of is actually explicit in this regard.
As you note, assume that $I\_1\subsetneq I\_2\subsetneq\cdots$ is an infinite ascending chain of ideals of $R$, let $E(R/I\_n)$ be the injective envelope of $R/I\_n$ for each $n$, and let $$E=\bigoplus\_{n=1}^{\infty}E(R/I\_n)$$
be the their d... | 17 | https://mathoverflow.net/users/3959 | 103405 | 59,939 |
https://mathoverflow.net/questions/103394 | 1 | Assume that we have a differential operator such as $-\frac{\partial}{\partial x^2} + id$ on $\mathbb{R}^1$
We also then argue that if a fundamental solution has compact support, then it is supported on the origin.
My follow up question is how can one then show assuming that the fundamental solution is compactly supp... | https://mathoverflow.net/users/17532 | Fundamental Solutions with compact support (distributions) | For a constant coefficient partial differential operator P(D), the fundamental solution of P can never belong to $\epsilon'(\mathbb{R}^{n})$,i.e.have compact support.
In fact,assume we have $P(D)u=f$,where u is a distribution,then u have compact support $\Leftrightarrow$ $\frac{f}{P(\xi)}$ is analytic(The result can ... | 2 | https://mathoverflow.net/users/23078 | 103417 | 59,946 |
https://mathoverflow.net/questions/103376 | 4 | I can't solve following exercise in a note about prime numbers. I need this for study about large gaps of consecutive prime numbers.
Prove that f $0<1-\delta<1$ then
$$\sum\_{p\le y}\frac{1}{p^{1-\delta}}\le\frac{y^\delta}{\log(y^\delta)}+\log(1/\delta)+O\left(\frac{y^\delta}{\delta(\log y)^2}+1\right)$$.
In the ... | https://mathoverflow.net/users/25386 | An estimate of the sum related to primes | I only get
$$\sum\_{p\le y}\frac{1}{p^{1-\delta}}\le \frac{y^{\delta}}{\log(y^\delta)}
+e^2\log(1/\delta)+O\Bigl(\frac{y^\delta}{\delta^2(\log y)^2}+1\Bigr).$$
The first sum is
$$\sum\_{p\le e^{2/\delta}}\frac{1}{p^{1-\delta}}=
\sum\_{p\le e^{2/\delta}}\frac{p^\delta}{p}\le e^2
\sum\_{p\le e^{2/\delta}}\frac{1}{p}... | 2 | https://mathoverflow.net/users/7402 | 103437 | 59,954 |
https://mathoverflow.net/questions/103423 | 2 | Let $\mathcal{D} \approx \mathbb{P}^{\delta\_d}$ be the space of homogeneous degree $d$
polynomials in three variables (up to scaling), where $\delta\_d = \frac{d(d+3)}{2}$.
Define $\mathcal{A}$ to be space of degree $d$ curves with a strict node at
the point $[1,0,0]$, ie
$$ \mathcal{A} := \{ f \in \mathcal{D}: ... | https://mathoverflow.net/users/4463 | Does a generic curve inside the space of curves with a node at a specific point have only finitely many nodes? | Ritwik: I believe we have discussed this before. Consider the blowing up at $[1,0,0]$,
$$
\nu:X\to \mathbb{P}^2
$$
with exceptional divisor $E$. Now for the family $\overline{\mathcal{A}}$ of plane curves $C$ of degree $d$ having a singularity at $[1,0,0]$, consider the family of transform curves $\nu^\*C - 2E$. By con... | 2 | https://mathoverflow.net/users/13265 | 103447 | 59,959 |
https://mathoverflow.net/questions/103448 | 6 | What is known about ZF without powerset but with an axiom "every set
has a set of all its countable subsets"?
This seems stronger than positing that the set of natural numbers has
a powerset, though I do not know a proof that it is.
More generally, for any definable cardinal $\alpha$, what about the
axiom "every se... | https://mathoverflow.net/users/38783 | What is known about size-restricted power set axioms? | Isn't $H(\mathfrak c^+)$, the collection of sets whose transitive closures have cardinality at most $\mathfrak c=2^{\aleph\_0}$, a model of your theory (but not the power set axiom)? The point is that a set of cardinality $\mathfrak c$ has only $\mathfrak c$ countable subsets.
EDIT: Colin actually asked whether this ... | 7 | https://mathoverflow.net/users/6794 | 103454 | 59,961 |
https://mathoverflow.net/questions/103445 | 3 | Let $X$ be a compact Kahler surface which is a ball quotient. Can such $X$ contain a torus $T$ such that the fudamental class of $T$ is non-trivial? I expect this is false as $\pi\_{1}(X)$ is a hyperbolic group, thus $\mathbb{Z} \times \mathbb{Z}$ can not occur as a subgroup of $\pi\_{1}(X)$ (here I consider the subgro... | https://mathoverflow.net/users/25402 | Question on Ball Quotients | Fundamental group of $X$ is torsion-free, so the image of $\eta: \pi\_1(T^2)\to \pi\_1(X)$ is either trivial or infinite cyclic. In any case, you can realize $\eta$ by a composition of maps
$$
T^2\to S^1\to X.
$$
The first map will kill the fundamental class of the torus, since $H\_2(S^1)=0$.
| 7 | https://mathoverflow.net/users/21684 | 103456 | 59,963 |
https://mathoverflow.net/questions/103426 | 6 | As we know, continuous spectrum and residual spectrum are two cases in the spectrum of an operator, which only appear in infinite dimension.
If $T$ is a operator from Banach space $X$ to $X$, $aI-T$ is injective, and $R(aI-T)$ is not $X$. If $R(aI-T)$ is dense in $X$, then $a$ belongs to the continuous specturm, it ... | https://mathoverflow.net/users/9946 | Why do we distinguish the continuous spectrum and the residual spectrum? | I'm afraid that this is more or less a reformulation of what you asked, but: a bounded operator $T$ on a Banach space $X$ is invertible if and only if it is bounded below (i.e. there is some constant $C>0$ such that $||Tx|| \geq C||x||$ for all $x\in X$) and has dense range. Bounded below implies injective, and it also... | 3 | https://mathoverflow.net/users/703 | 103460 | 59,964 |
https://mathoverflow.net/questions/103339 | 2 | A well-known theorem in topology says that for a smooth manifold $M$ of dimension $n$ the map $f: M \rightarrow point$ satisfies
$$f^! \mathbf R = \mathbf R[n]$$
Here $\mathbf R$ is the constant sheaf.
Here is my question: is there any kind of converse statement to this? (I.e. if $f$ is such that the above equation h... | https://mathoverflow.net/users/18116 | smooth manifold vs. exceptional inverse image | Let $X$ be a homology sphere which is not homotopic to a sphere. (For example, the Poincare $3$-sphere.) Denote by $M$ the suspension of $X$. Then I believe that $f^!\mathbb{R}=\mathbb{R}[4]$, yet $M$ is not even a topological manifold. (Homologically one cannot distinguish $M$ from a $4$-manifold.)
| 5 | https://mathoverflow.net/users/20302 | 103466 | 59,967 |
https://mathoverflow.net/questions/103414 | 2 | Parallel translation of a vector along a geodesic in a surface is characterized by the following three properties:
1. The vector being transported moves continuously.
2. It has constant norm.
3. It maintains a constant angle with the geodesic.
In [V.I. Arnold's book on mechanics](http://books.google.com.uy/books?id... | https://mathoverflow.net/users/7631 | Parallel translation on surfaces | The easiest way I see is to define parallel translation the usual way.
(This angle property as well as linearity is evident.)
Then show that angle property alone plus approximation give the same translation.
| 2 | https://mathoverflow.net/users/1441 | 103471 | 59,970 |
https://mathoverflow.net/questions/103475 | 1 | Hi,
Consider the zeta function of a definable set over a finite field. More precisely, let $\varphi$ be a formula in the language of fields and let $X$ be a definable subset of $F^n$ given by $X=\{(a\_1,\dots,a\_n) \in F^n| F\models \varphi(a\_1,\dots,a\_n) \}$ where $F$ is a finite field of characteristic $p$, then... | https://mathoverflow.net/users/nan | Riemann hypothesis for zeta function of definable sets over finite fields | No.
The zeta function of the set $\{x|x \neq 0\}$ over $\mathbb F\_p$ is:
$e^{\sum\_{i=1}^\infty \frac{p^i-1}{i} t^i} = \frac{1-t}{1-pt}=\frac{1-p^{-s}}{1-p^{1-s}}$
It has a zero where $s=0$.
More generally, every affine algebraic curve minus a point has a zeta function that is zero at $s=0$.
| 5 | https://mathoverflow.net/users/18060 | 103477 | 59,974 |
https://mathoverflow.net/questions/103453 | 3 | Hi, I need an estimation or an exact closed form expression for the following integral
$\int\_{0}^{2\pi} K\_N^4(s) ds $
where $K\_N(s)= \frac{1}{N2\pi} (\frac{sin(Ns/2)}{sin(s/2)})^2$, the Fejer kernel.
I don't know how to obtain an estimation better than
$\int\_{0}^{2\pi} K\_N^4(s) ds < N^4$
Does anyone kno... | https://mathoverflow.net/users/19133 | Does anybody know an estimation of L4 norm of fejer kernel ? | $\displaystyle \int\_0^{2\pi} K\_N^4(s)\ ds = \frac{c\_{N-1}}{8 \pi^3 N^4}$ where $c\_n$ is the coefficient of $z^{4n}$ in $(1 + z + \ldots + z^n)^8$.
$c\_n$ appears to have the closed form
$$ c\_n = \frac{\left( 315+1284 n + 2734 n^2+3300{n}^{3}+2335{n}^{4}+906{n}^{5}
+151{n}^{6} \right) \left( n+1 \right)}{315}
$... | 8 | https://mathoverflow.net/users/13650 | 103483 | 59,977 |
https://mathoverflow.net/questions/103265 | 13 | Let $F$ be a finite field. Let $F[X]$ and $F[[X]]$ denote the ring of polynomials and power series over $F$, respectively. I'm trying to show a statement like the following:
Fix a $d > 0$. Let $g\in F[[X]]$. If there exists a set $C\subseteq F[X]$ of polynomials (with no constant term) of degree at most $k$ such that... | https://mathoverflow.net/users/5534 | Can formal power series become polynomial often, when composed with polynomials? | I claim that, if $g(x) \in k[[x]]$ is not a polynomial, then $g \circ c$ is a polynomial for at most $|F|^{k/2}$ polynomial $c$ of degree $\leq k$. We will always use the letter $c$ to represent a polynomial with $c(x)=0$.
**Case 1:** $g$ is transcendental over the field $k(x)$. In this case, I claim that $g \circ c$... | 6 | https://mathoverflow.net/users/297 | 103484 | 59,978 |
https://mathoverflow.net/questions/103451 | 2 | This is by any means elementary, but since I have asked this question on Stark Exchange but received no satisfactory answers I decide to post it here.
It is well known that a symmetric matrix over field $\Bbb F$ is congruent to a diagonal matrix, i.e., there exists some A s.t. $A^TUA=D$ with $U$ symmetric and $D$ dia... | https://mathoverflow.net/users/22727 | On certain decomposition of unitary symmetric matrices | Here is a way to answer the second question. I am assuming complex
matrices and using '$X^{\*}=\overline{X}^{\mathrm{T}}$. The first question (already answered by Suvrit ) I discuss at the end.
Assume '$U^{\*}U=I$ and $U^{\mathrm{T}}=-U$. If $\mathbf{v}$ is nonzero and $U\mathbf{v}=\lambda\mathbf{v}$ then $|\lambda|=... | 2 | https://mathoverflow.net/users/6133 | 103488 | 59,981 |
https://mathoverflow.net/questions/102737 | 30 | Let $\Sigma\_n\subset G$ be a set of generators of the symmetric group $S\_n$. It is a well-known conjecture that the diameter of the Cayley graph $\Gamma(S\_n,\Sigma\_n)$ is at most $n^C$ for some absolute constant $C$. (The diameter of the Cayley graph is just the maximum of $\ell(g)$ for $g\in S\_n$, where $\ell(g)$... | https://mathoverflow.net/users/398 | Diameter of symmetric group | What follows is an incomplete answer. I am in the middle of Russian woods, so would rather have somebody else trace all the refs, etc. but looking at the bounty expiration date decided that it's worth stating what is known.
The answer is NO to all, but that's a conjecture not a theorem. I have seen this conjecture st... | 11 | https://mathoverflow.net/users/4040 | 103490 | 59,983 |
https://mathoverflow.net/questions/103492 | 13 | The wikipedia article(s) as well as the nlab article(s) about Grothendieck topologies and Grothendieck pretopologies are careful to differentiate the two very emphatically and to point out that distinct pretopologies can give rise to the same topology on a category. My (almost certainly trivial) questions are the follo... | https://mathoverflow.net/users/25415 | Grothendieck Topologies versus Pretopologies | A) Let $\tau$ be a Grothendieck pretopology on any category with fiber products. Define a new Grothendieck pretopology $\tau'$ where a cover $\{U\_i \to X\}$ is a $\tau'$ cover if and only if there exists a refinement $\{V\_{ij} \to X\}$ (i.e. there exists for each $ij$ an $X$-morphism $V\_{ij} \to U\_i$) such that $\{... | 12 | https://mathoverflow.net/users/12914 | 103499 | 59,988 |
https://mathoverflow.net/questions/103486 | 3 | Define $\tau: \mathbf{Ord} \to \mathbf{Ord}$ such that $\tau(\alpha)$ is the order type of the minimal set $S$ of ordinals such that $\alpha \in S$ and $S$ is closed under ordinal exponentiation.
We have the following:
$\tau(0)=2, \tau(1)=1$ and for all $\alpha \in [2, \omega), \tau(\alpha)=\omega$.
*Question*: Wha... | https://mathoverflow.net/users/9550 | Order type of the minimal set closed under ordinal exponentiation | Take $\alpha$ to be any ordinal greater than or equal to $\omega$. The set of ordinals $S(\alpha)$ obtained generated from $\alpha$ using ordinal exponentiation are the ordinals of the form $\alpha^{\alpha^{E(\alpha)}}$ where $E(\alpha)$ is an exponential polynomial over the base $\alpha$. By "exponential polynomial ov... | 5 | https://mathoverflow.net/users/23338 | 103503 | 59,991 |
https://mathoverflow.net/questions/102559 | 16 | The tensor rank of a three dimensional array $M[i,j,k], i,j,k\in [1,\ldots,n]$ is the minimal number of vectors $x\_i,y\_i,z\_i$, such that $M=\sum\_{i=1}^d x\_i\otimes y\_i\otimes z\_i$.
From dimension argument it easily follows that there exists a tensor of tensor rank at least $\frac{1}{3}n^2$. One can also easily ... | https://mathoverflow.net/users/4246 | What is the largest tensor rank of $n \times n \times n$ tensor? | The lower bound can be improved slightly to $n^3/(3n-2)$ by noting that in $x\otimes y\otimes z$ one can assume $|y|=|z|=1$. See also Chapter 20, "Typical Tensorial Rank", in the book *Algebraic Complexity Theory*, by Peter Bürgisser, Michael Clausen, and Mohammad Amin Shokrollahi.
An upper bound of $n^2−n−1$ is show... | 13 | https://mathoverflow.net/users/408 | 103506 | 59,994 |
https://mathoverflow.net/questions/93877 | 1 | I have been looking at hyperelliptic curves over an algebraically closed field $k$ of characteristic two, with a view towards finding the basis for the vector space of holomorphic differentials. To do this I have viewed the curves as the function field $k(x,y)$, originally restricting to those defined by
$$
y^2 - y = f... | https://mathoverflow.net/users/16082 | Hyperelliptic curves over characteristic two fields | If anyone is interested, a thorough treatment is given by the book referenced in the comments here: <https://math.stackexchange.com/questions/137495/artin-schreier-extensions-over-characteristic-two-fields>
| 1 | https://mathoverflow.net/users/16082 | 103511 | 59,996 |
https://mathoverflow.net/questions/103450 | 3 | Let $S$ be a finite commutative semigroup with identity. Under what conditions (on the semigroup $S$) it is possible to find a ring $R$ such that the multiplicative structure of $R - \{0\}$ is isomorphic to $S$?
| https://mathoverflow.net/users/24864 | Embedding Semigroups in Rings | $S$ must be a cyclic group of order $p^n-1$ for some prime $p$ and natural $n$. Indeed, since $R\setminus \{0\}$ is a semigroup under multiplication, $R$ does not have zero divisors. Hence $R$ is a division ring. Since $S$ is finite, $R$ is a finite division ring, hence, by Wedderburn, a finite field. Therefore $S$ mus... | 6 | https://mathoverflow.net/users/nan | 103520 | 60,002 |
https://mathoverflow.net/questions/103047 | 16 | My question is assume that we know that the degree of some irreducible variety is small does it possible to conclude that there exists polynomial of small degree vanishing on this variety.
Let us make the question more concrete:
Let $V\subset A^{2n}$ be an irreducible algebraic variety of dimension $n$ and degree $... | https://mathoverflow.net/users/4246 | Minimal degree of polynomial vanishing on the variety of small degree. | The minimal degree of a polynomial that vanishes on $V$ is the minimal $m$ such that $h\_I(m)\ne 0$, where $h\_I$ denotes the Hilbert function of the variety $V$. Hence, you are interested in upper (and lower?) bounds on the Hilbert function of certain varieties. There are two papers I know that deal with such bounds:
... | 4 | https://mathoverflow.net/users/9947 | 103526 | 60,006 |
https://mathoverflow.net/questions/103538 | 2 | When studying the Lorentz group $O(1,3)$, one can decompose it into four parts... physicist usually called these
* Proper-orthochronuos $\mathscr{L}^{\uparrow}\_+$,
* Proper-asynchronous $\mathscr{L}^{\downarrow}\_+$,
* Improper-orthochronuos $\mathscr{L}^{\uparrow}\_-$,
* Imroper-asynchronous $\mathscr{L}^{\downarro... | https://mathoverflow.net/users/25356 | Decomposition of Lorentz-like groups | Provided that $n,p>0$, $O(n,p)$ has four connected components as well. There are many ways to see this. $O(n,p)$ is a matrix subgroup of the general linear group of $\mathbb{R}^{n+p}$:
$$ O(n,p) = \lbrace a \in \operatorname{GL}(n+p,\mathbb{R}) \mid a^T \eta a = \eta\rbrace$$
where
$$\eta = \begin{pmatrix} -I\_n & 0 \c... | 6 | https://mathoverflow.net/users/394 | 103546 | 60,014 |
https://mathoverflow.net/questions/103545 | 9 | It is well-known that, given a normalized eigenform $f=\sum a\_n q^n$, its coefficients $a\_n$ generate a number field $K\_f$.
In their 1995 [paper](http://www.math.mcgill.ca/darmon/pub/Articles/Expository/05.DDT/paper.pdf) "Fermat's Last Theorem", Darmon, Diamond, and Taylor remark that, at the time of writing, ver... | https://mathoverflow.net/users/10547 | Number Fields Arising from Newforms | For (1), see Ribet's wonderful article *Galois representations attached to eigenforms with Nebentypus* (<http://dx.doi.org/10.1007/BFb0063943>). It's proposition 3.2.
| 7 | https://mathoverflow.net/users/1021 | 103554 | 60,019 |
https://mathoverflow.net/questions/103364 | 6 | Let $f=(f\_0,\ldots,f\_{n-1})$ be a vector in $V\_n=\{\pm 1\}^n$. Let
$F=(F\_0,\ldots,F\_{n-1})$
be its (discrete) Fourier transform defined by
$$
F\_k=\sum\_{x=0}^{n-1} f\_x \omega\_n^{x k}
$$
where $\omega\_n=\exp(2 \pi i/n)$. Let
$$\theta\_n=\min \left( \max\_{0\leq k\leq n-1}|F\_k|: f \in V\_n \right).$$
Is an... | https://mathoverflow.net/users/17773 | Growth rate of the infinity norm of Discrete Fourier Transform of +1,-1 vectors | Here is my proposed answer based on what I have found out so far: Since the set $V\_n$ is a subset of the sphere $S\_{n-1}(\sqrt{n})$ in $R^n$ the minimum $\theta\_n$ is lower bounded by the minimum, cal it $\alpha\_n$, on this sphere. A vector $f=(\sqrt{n},0,\ldots,0)$ and all its cyclic shifts achieves equal magnitud... | 1 | https://mathoverflow.net/users/17773 | 103555 | 60,020 |
https://mathoverflow.net/questions/103556 | 2 | Is there a unified description (or a set of axioms) of the zeta function of an algebraic curve over a finite field $\mathbb{F}\_q$ and the Riemann zeta function?
| https://mathoverflow.net/users/nan | A unified description of zeta functions of a curve over $\mathbb{F}_q$ and Riemann $\zeta$ function | Let $D$ be a ring which is infinite, not a field, and in which $D/I$ is finite for any nonzero ideal $I$ (what Pete Clark calls a [abstract number ring](http://alpha.math.uga.edu/~pete/aant.pdf)). Then we can define $N(I) = |D/I|$ and write down a zeta function
$$\zeta\_D(s) = \sum\_{I \neq 0} \frac{1}{N(I)^s}.$$
Thi... | 7 | https://mathoverflow.net/users/290 | 103558 | 60,021 |
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