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https://mathoverflow.net/questions/106999 | 5 | Background
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I'm working to simplify the Lagrangian formalism of classical field theory for the situation of a vector bundle with a bundle metric and a metric connection. Particularly, I want to specify the Euler-Lagrange equations and the Noether theorem for this case.
---
Consider a vector bundle $(E... | https://mathoverflow.net/users/26429 | 1-jet bundle on vector bundle with metric connection |
>
> Is there a canonical way to identify an Element... ?
>
>
>
Yes: an element $j\in J^1E$ is the same as subspace $R\subset T\_{\phi}E$ of dimension $\dim(M)$ transversal to $VE$. Since your metric connection gives a splitting $T\_\phi E=V\_\phi E\oplus H\_\phi E$ and since $V\_{\phi}E\cong E\_{\pi(\phi)}$ and ... | 8 | https://mathoverflow.net/users/745 | 107033 | 61,785 |
https://mathoverflow.net/questions/107012 | 3 | Wigner's theorem states that every symmetry of complex Hilbert space is either unitary or anti-unitary, up to multiplication by a unit scalar function. Here $f:\mathcal{H} \rightarrow \mathcal{H}$ is called a symmetry if it is a bijection and $\left| \langle f(\mathbf{v}),f(\mathbf{w}) \rangle \right| = \left| \langle ... | https://mathoverflow.net/users/6133 | Can one derive Wigner’s theorem in the complex case from the real case? | I think the answer to the question is "no", but from the follow-up comment I gather the underlying question is "what proof of Wigner's theorem can one give to physics students". Here's what I would do, inspired by <http://arxiv.org/abs/0808.0779>
I would focus on the case of an $N=2$ dimensional Hilbert space of a sp... | 2 | https://mathoverflow.net/users/11260 | 107037 | 61,787 |
https://mathoverflow.net/questions/107043 | 25 | The extension of the 2-adic valuation to the reals used in the usual proof clearly uses AC. But is this really necessary? After all, given an equidissection in $n$ triangles, it is finite, so it should be possible to construct a valuation for only the algebraic numbers, and the coordinates of the summits (with a finite... | https://mathoverflow.net/users/17164 | Is Monsky's theorem dependent on the axiom of choice? | No choice is needed. If, in a choiceless universe, there were a counterexample, then that counterexample amounts to finitely many real numbers (the coordinates of the relevant points). It would still be a counterexample in the sub-universe of sets constructible (in Gödel's sense) from those finitely many reals. But tha... | 37 | https://mathoverflow.net/users/6794 | 107044 | 61,789 |
https://mathoverflow.net/questions/106955 | 1 | Given an $\infty$-category $\mathcal{C}$ with whatever descriptors you wish, how do higher internal homs work? Specifically, given two $n$-morphisms, $f$ and $g$, is it possible (I don't see why not a priori) to have the collection of $n+1$-morphisms between $f$ and $g$ itself be an $n$-morphism? Or, is it more sensibl... | https://mathoverflow.net/users/11546 | Internal Homs in Infinity Categories | I'm sure someone might give a better answer, but here are my two cents. I'd like to start out by pointing out that your question is difficult already for n=0:
First, it would help to know what one means by an enriched oo-category. Other than special, incredibly natural cases like Spaces, Chain complexes, and Spectra,... | 2 | https://mathoverflow.net/users/3593 | 107050 | 61,792 |
https://mathoverflow.net/questions/107046 | 3 | Let $G$ be an infinite, countable, finitely generated group. Let $H$ be a finite index subgroup of $G$. Let $S$ be a finite, symmetric set of generators of $G$, and let $d(\cdot,\cdot)$ be the word length metric on $G$ induced by $S$. Then $d$ is also a metric on $H$. The question is the following: is $d$ in the same b... | https://mathoverflow.net/users/23661 | Lowering metrics to finite index subgroups | Even more is true. A metric space is called D-separated if the distance between any distinct points is at least D.
Claim. Let $X, X'$ be $1$-separated metric spaces. Let $f: X\to X'$ be a bijective quasi-isometry. Then $f$ is bilipschitz.
Proof. Inequality $d(f(x), f(y))\le L d(x,y)+A$ implies that $d(f(x), f(y))\... | 6 | https://mathoverflow.net/users/21684 | 107051 | 61,793 |
https://mathoverflow.net/questions/107052 | 2 | Fix an algebraic closure, $\overline{\mathbb{Q}}$ for the rationals and consider the set, $B\_p$, of all places of $\overline{\mathbb{Q}}$ over a fixed (possibly infinite) prime, $p$, of $\mathbb{Q}$. Let $G\_\mathbb{Q}=\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$.
1: Let $G\_\mathbb{Q}$ act on $B\_p$ in the natural... | https://mathoverflow.net/users/4701 | Place stabilizers for the absolute Galois Group | If $p$ is a rational prime, then choosing a prime $v$ of $\overline{\mathbf{Q}}$ lying over $p$ amounts to choosing an embedding $i:\overline{\mathbf{Q}}\hookrightarrow\overline{\mathbf{Q}}\_p$. This gives rise to a map $\varphi:G\_{\mathbf{Q}\_p}\rightarrow G\_{\mathbf{Q}}$defined as follows: given $s$ in the source, ... | 6 | https://mathoverflow.net/users/4351 | 107057 | 61,796 |
https://mathoverflow.net/questions/107056 | 4 | I'm reading Illusie's book "complexe de cotangent et Deformations I". And I'm puzzled on the definition of cotangent complex.
I formulate my question as follows:
Suppose $C$ and $D$ are abelian categories, and $F:C \to D$ is a functor. I would like to consider its left-derived functor $LF$. There are two ways:
(1... | https://mathoverflow.net/users/26460 | resolution by simplicial objects versus resolution by chain complex | The point in Illusie's (and earlier Quillen's) work is that you are not really working in an abelian category! Look at Quillen's paper, his homology is the homology of commutative algebras (and the abelian objects there are not that interesting as they have trivial multiplication). The homology is the derived functor o... | 8 | https://mathoverflow.net/users/3502 | 107078 | 61,802 |
https://mathoverflow.net/questions/107062 | 13 | Dear MO,
**Question 1.** Do you know of an example of a Fano variety which is **not** Frobenius split?
**Background**
(1) A variety $X$ in characteristic $p$ is called *Frobenius split* if there is a "$p$-th root" map $\sigma: \mathcal{O}\_X \to \mathcal{O}\_X$, that is, an additive map satisfying $\sigma(f^p g) ... | https://mathoverflow.net/users/3847 | Frobenius splitting of Fano varieties | For question 1, the following example works, and you'll see how to construct many more (in higher characteristic).
Set $k$ to be a perfect field of characteristic $2$.
$$X = \text{Proj } k[x,y,u,v]/\langle x^3+y^3+u^3+v^3 \rangle \subseteq \mathbb{P}^3\_k.$$
---
###Fano check:###
I'm certain you already know ... | 18 | https://mathoverflow.net/users/3521 | 107085 | 61,807 |
https://mathoverflow.net/questions/107041 | 1 | I have asked a similar question involving some finance background some time ago here [math.stackexchange](https://math.stackexchange.com/questions/137011/futures-pricing-and-futures-price-process-under-the-real-world-measure), however no really good answer came up. I was able to find a solution at least for a special c... | https://mathoverflow.net/users/15411 | Solving an Ornstein-Uhlenbeck-like SDE $y(t,T)=H_t + \mathbb{E}[\int_t^T y(s-,T)dX_s|\mathcal{F}_t]$ | In the level of generality you have stated, I have no ideas. But in some cases this can be reformulated as a type of backward SDE, as studied in section 2.2 of <http://arxiv.org/pdf/0801.3505.pdf> in the case that $X$ and $H$ are continuous. It is no simple problem, but a couple of special cases seem approachable.
Th... | 4 | https://mathoverflow.net/users/26459 | 107097 | 61,810 |
https://mathoverflow.net/questions/107095 | 9 | I have been reading Deligne-Milne's Tannakian Categories and got to the point at the end of part 3 where they discuss what went wrong with Saavedra's definition. Motivated by their counterexample, they ask the following question:
Let $C$ be a $k$-linear rigid abelian tensor category over a field $k$. Let us further a... | https://mathoverflow.net/users/5301 | Saavedra's Definition of Tannakian Category | The answer is yes; in fact, this is mentioned in a footnote to the definition of Tannakian category in Milne's [corrected version of the Deligne-Milne paper,](http://www.jmilne.org/math/xnotes/tc.html) on page 32. This is proved in Deligne's *Catégories tannakiennes*, Theorem 1.12.
| 11 | https://mathoverflow.net/users/396 | 107103 | 61,811 |
https://mathoverflow.net/questions/107096 | 18 | The following question was asked by me on Mathematics.SE. Unfortunately, no one answered it so I thought I might give it a try one level higher. Below the line you can find the slightly edited question, the original one can be found [here](https://math.stackexchange.com/questions/155565/can-all-convex-polytopes-be-real... | https://mathoverflow.net/users/24381 | Can all convex polytopes be realized with vertices on surface of convex body? | Yes, there is such a body. Actually there is one very close to the standard unit ball and containing disjoint representatives of each combinatorial type (but these representatives are very small).
Indeed, every combinatorial type of a $d$-polytope has a realization which looks as follows: there is a "large" $(d-1)$-d... | 24 | https://mathoverflow.net/users/4354 | 107113 | 61,815 |
https://mathoverflow.net/questions/106784 | 7 | Suppose I have a convex set $C\subset\mathbb{R}^n$ such that $0\in C$ and every Cauchy sequence in $C$ converges in $C$, but $C$ need not be bounded. (Actually I want unbounded $C$). Consider the set
\begin{equation}\mathfrak{L}=\lbrace T:\mathbb{R}^n\rightarrow \mathbb{R}^n, \text{ $T$ is linear}, T(C)\subseteq C\rbr... | https://mathoverflow.net/users/651 | Duality between extremal points and extremal maps | I'm not sure what counts as a "relation" here.
It is tempting to think that an extreme point of $\mathfrak L$ would map at least some extreme points of $C$ to extreme points of $C$, but this is not true.
For example, in ${\mathbb R}^2$ let $C$ be the convex hull of $(-1,-2)$, $(-1,2)$, $(2,-1)$ and $(2,1)$.
Then one ... | 2 | https://mathoverflow.net/users/13650 | 107118 | 61,817 |
https://mathoverflow.net/questions/107116 | 8 | The question is in the title, but let me elaborate a little.
Background
----------
[Penrose Tilings](http://en.wikipedia.org/wiki/Penrose_tiling) are really pretty and satisfy some remarkable properties. For instance, I believe the following is true: even though a Penrose tiling is aperiodic by definition, given a ... | https://mathoverflow.net/users/18263 | What are Penrose Tilings, and how do they relate to Quasicrystals? | As explained [here](http://en.wikipedia.org/wiki/Aperiodic_tiling), there is an infinite number of distinct tilings that can be constructed using the three sets of tiles introduced by Roger Penrose (rhomb, kite-dart, boat-star). The distinction between tiles and tilings is often not made, and one informally speaks of t... | 3 | https://mathoverflow.net/users/11260 | 107119 | 61,818 |
https://mathoverflow.net/questions/107109 | 3 | let $A=\{1,2,3...,N\}$ and $B\_1,B\_2,B\_3\dots,B\_n$ be a series of subsets of $A$ which satisfied that $|B\_i|=m$,
$|B\_i\cap B\_j|\le k$. what is the maximum of $n$? ($k< m< N$)
it can be easily showed that
$n\le [C(N−1,k)/C(m−1,k)]/[N/m]$ (by counting twice, $[X]$ is integer part of $x$)
I wonder is there any... | https://mathoverflow.net/users/26487 | set and subset series combinatorics | Yes, this is an active research area. For a quick survey, see the introduction to [this paper of Furedi and Sudakov.](http://www.math.uiuc.edu/~z-furedi/PUBS/furedi_sudakov_kwise.pdf) (they also prove many interesting extensions).
| 1 | https://mathoverflow.net/users/11142 | 107120 | 61,819 |
https://mathoverflow.net/questions/107082 | 4 | Suppose $G$ is a finitely generated subgroup of $GL\_n(Z)$, $n\ge 3$. I suspect that there is no decision procedure for deciding whether or not such $G$ is finitely presented. How can this proved?
| https://mathoverflow.net/users/25762 | Decision Problem for finitely generated subgroups | Various forms of this question were discussed on MO several times, see e.g. [here](https://mathoverflow.net/questions/47961/undecidability-in-matrix-groups/90840#90840). The key is that $SL(4,Z)$ contains
$H=F\_2\times F\_2$, direct product of rank 2 free groups. Then for every finitely-presented group $Q$ (i.e., a gr... | 5 | https://mathoverflow.net/users/21684 | 107124 | 61,821 |
https://mathoverflow.net/questions/107126 | 12 | Smoothly embed a genus g surface in $\mathbb{R}^3$, and pick a normal vector pointing "out" of the surface at each point. Then on each tangent plane, I have a map which rotates the tangent plane 90 degrees in the direction given by the right hand rule. This gives an almost complex structure on the manifold, and every a... | https://mathoverflow.net/users/1106 | Does every smoothly embedded surface $\mathbb{R}^3$ inherit a natural complex structure, and if so, which one? | Question 1: Looks good to me.
Question 2: is a duplicate of [this question](https://mathoverflow.net/questions/53999/conformal-embedding-of-riemann-surfaces-into-3-space) (the answer is: every conformal structure can be so realized).
Question 3/4. There are algorithms to compute the conformal structure given a surf... | 11 | https://mathoverflow.net/users/11142 | 107127 | 61,823 |
https://mathoverflow.net/questions/107005 | 1 | Consider Hilbert spaces $V$,$H$; a closed quadratic form $a$ with domain $V$; and its associated operator $A$ on $H$. (If necessary, the form can be assumed to be coercive.) For the sake of simplicity, assume the embedding of $V$ into $H$ to be compact (even trace class, if necessary), so that $A$ has purely point (rea... | https://mathoverflow.net/users/26039 | Weyl asymptotics vs. form perturbations | Let me elaborate a little bit on my idea. Denote by $\lambda\_j$ the eigenvalues of the operator $A$. Then Weyl-asymptotics means that
$$
N(E) = \\#(j: \lambda\_j < E) < C \cdot E^\alpha.
$$
Consider now the operator $(A)^{-1}$ with eigenvalues $\lambda^{-1}$ ... and consider its Schatten $p$ norm
$$
\|A^{-1} \|\_p =... | 1 | https://mathoverflow.net/users/3983 | 107130 | 61,826 |
https://mathoverflow.net/questions/106740 | 5 | My question is regarding the definition of $\mathcal{D}$-modules of level $m$ given in [this](http://arxiv.org/abs/0811.1168) paper. As an example, let $X=\mathbb{A}^1$ over $S=\text{Spec }\overline{\mathbb{F}\_p}$; I was told that a $\mathcal{D}$-module of level $m$ is a module over $\mathbb{k} \langle x, \partial\_x,... | https://mathoverflow.net/users/2623 | $\mathcal{D}$-modules of level m | I find it helpful to first work through the definition of multiplication on $\mathcal{D}^{(m)}$ when $m = \infty$, in which case it reduces to the "classical" ring of differential operators in the sense of Grothendieck; read sections 16.7 and 16.8 of [EGA 4, Quatrième partie](http://www.numdam.org/numdam-bin/feuilleter... | 10 | https://mathoverflow.net/users/6827 | 107131 | 61,827 |
https://mathoverflow.net/questions/107034 | 4 | are there any results on the number of graphs on n vertices that has chromatic number=k ?
I mean how many graphs on n vertices has x(G)=2,x(G)=3,......,x(G)=n-1 ?
**update**:
infact equivalently, is there a way to count unique k-partite graphs?
after that to get x(G)=k, just subtract #k-partite-#(k-1)-partite.
I've... | https://mathoverflow.net/users/26449 | Counting graphs on n vertices by chromatic number | The formula for chromatic number 2 can't quite be right, as it gives $3/2$ when $n=2$. [Number](http://oeis.org/A000683) of 2-colored labeled graphs on $n$ nodes is tabulated at the Online Encyclopedia of Integer Sequences. The listing there agrees with the formula here for $n=3$ and $n=4$ but gives 360, not 375, for $... | 2 | https://mathoverflow.net/users/3684 | 107132 | 61,828 |
https://mathoverflow.net/questions/107069 | 4 | Take two graphs (of bounded valency) or manifolds (f bounded geometry) $G$ and $G'$. Assume there is a quasi-isometry $f:G \to G'$, and assume the Poisson or Martin boundary of $G$ is known, what may one say about the Poisson or Martin boundary of $G'$ (or how does $f$ transforms Poisson or Martin boundaries)?
EDIT:... | https://mathoverflow.net/users/18974 | How does a quasi-isometry affect Poisson or Martin boundaries? | 1. Terry Lyons in "Instability of the Liouville property for quasi-isometric Riemannian manifolds and reversible Markov chains", Journal of Diffe Geometry, 1987, constructs examples of graphs (and Riemannian manifolds) $G, G'$ which are quasi-isometric, but one admits no (nonconstant) positive harmonic functions while ... | 7 | https://mathoverflow.net/users/21684 | 107136 | 61,830 |
https://mathoverflow.net/questions/107107 | 20 | Dusan Pokorny and Jan Rataj have just posted a paper (<http://arxiv.org/abs/1209.2305>) in which they prove the identity
$$
\det (A-B) = \frac 1{d!} \sum\_{k=0}^d (-1)^k \binom dk \det((d-k)A + kB)
$$
where $A,B$ are $d \times d$ matrices (this is the corrected version of the formula, in response to Paseman's initial i... | https://mathoverflow.net/users/26486 | a determinantal identity | See the comments of Elkies and Grinberg:
*Looks like an application of the finite-difference formula to
the polynomial $P(k) := \det(dA - k(A-B))$ which has degree $d$
with leading coefficient $(-1)^d \det(A-B)$.* (Noam Elkies)
*More generally: if $A$ is a commutative ring, and $P\in A\left[X\right]$ is a polynom... | 8 | https://mathoverflow.net/users/26486 | 107140 | 61,831 |
https://mathoverflow.net/questions/106478 | 8 | I'm reading a paper by V.S. Vladimirov and I.V. Volovich and they make a particular claim which is supposed to be discussed in:
N.M. Krylov, Dokl. Akad. Nauk SSSR, **60**, 687 (1947).
I have not been able to reproduce the claimed result, so, it would be nice to see what precisely is done in the reference above.
... | https://mathoverflow.net/users/24854 | Trying to find Russian paper from 1947 | Does this sound like it?
Крылов Н.М. О кватернионах Роана Гамильтона и понятии моногенности// ДАН СССР.- 1947.- т.55, № 9. с.799-800.
In English: N.M.Krylov, On Rowan Hamilton quaternions and the notion of monogeny, Doklady Akademii Nauk SSSR (Proceedings of the Academy of Sciences of the USSR), 1947, vol. 55, no 9... | 12 | https://mathoverflow.net/users/1131 | 107142 | 61,833 |
https://mathoverflow.net/questions/105272 | 45 | I've been collecting some of the many unpublished manuscripts of Bill Thurston over the years. His recent passing inspired me to ask the following. I've seen a number of references (for instance, in his wikipedia biography [here](http://en.wikipedia.org/wiki/William_Thurston)) to a senior thesis that he wrote when he w... | https://mathoverflow.net/users/317 | Thurston's senior thesis at New College | I've emailed you a pdf copy of "A Constructive Foundation for Topology" by Bill Thurston (June 14, 1967; New College Senior Thesis; submitted to Roger Renne).
In consideration of the comments above, I'm hesitant to post the file in a publicly accessible location. Perhaps the mathoverflow community can figure out whet... | 18 | https://mathoverflow.net/users/22971 | 107146 | 61,835 |
https://mathoverflow.net/questions/107143 | 5 | Consider the category $Space(G)$ of $G$-spaces. For $H\subset G$, there is a forgetful functor from $Space(G)$ to $Space(H)$. Also, for an object $X$ in $Space(G)$ and another object $X'$ in $Space(G')$, $X\times X'$ is in $Space(G\times G')$.
Consider also the category of $Mod(G)$ of modules over $H\_G(pt)$. For $H\... | https://mathoverflow.net/users/5420 | Terminology for a notion of "categories parameterized by another (symmetric monoidal) category" | Zhen is right that you can think of it as a lax monoidal pseudofunctor. In [this paper](http://www.tac.mta.ca/tac/volumes/20/18/20-18abs.html), I called an equivalent structure a "monoidal fibration".
| 7 | https://mathoverflow.net/users/49 | 107147 | 61,836 |
https://mathoverflow.net/questions/99432 | 1 | Hello,
I am looking for matrix representation of Tits group $^2F\_4(2)'$ of size 17 971 200. Atlas of finite groups offer several matrix representations but they are not embedded in U(27). They are in GL(27, C).
I am looking really for embedding of $^2F\_4(2)'$ in $E\_6$ compact Lie group. I tried to guess the emb... | https://mathoverflow.net/users/nan | Matrix representation of 2F4(2)' in unitary U(27) | I am happy to inform that Robert Wilson have sent me the complex matrices generating the 2F4(2)' group inside the E6 compact Lie group. See the paper
<http://arxiv.org/pdf/1208.4221.pdf>
for details.
Best regards,
Marek
| 1 | https://mathoverflow.net/users/nan | 107156 | 61,840 |
https://mathoverflow.net/questions/107133 | 5 | Let $X=Proj(A)$ be a projective scheme, one can the moduli space of coherent sheaves on $X$ with fixed Hilbert polynomial and stability. Since coherent sheaves on $X$ are all obtained as the sheafification $\widetilde{M}$ of a graded $A$-module $M$, it is reasonable to ask whether we can construct the moduli space of s... | https://mathoverflow.net/users/50973 | Moduli space of modules over non-commutative rings | I think the paper "Abstract Hilbert Schemes" by Artin and Zhang might be useful. This may be one of the other papers Peter had in mind in his answer.
The basic idea as far as I understand it seems to be the following; things work nicely if your ring is strongly noetherian! This condition means that whenever you tenso... | 4 | https://mathoverflow.net/users/21075 | 107157 | 61,841 |
https://mathoverflow.net/questions/107153 | 6 | Do any of you happen to know the history of the standard prime factorization proof of $\sqrt 2$ is irrational? I know this theorem was known to Aristotle, and that the Fundamental Theorem of Arithmetic, on which the proof rests, is found already in Euclid, but I've not been able to track down the origin of this particu... | https://mathoverflow.net/users/17457 | First known proof of $\sqrt 2$ is irrational with prime factorization? | "The Discovery of Incommensurability" by Kurt von Fritz [ <http://www.jstor.org/stable/1969021> ] indicates that the early Greek mathematicians did not explicitly use the Fundamental Theorem to prove the irrationality of √2. The proof known to Aristotle ("the diagonal of the square is incommensurate with the side, beca... | 9 | https://mathoverflow.net/users/11260 | 107158 | 61,842 |
https://mathoverflow.net/questions/107075 | 1 | Found a possible reduction from recognizing [lexicographic product of graphs](https://en.wikipedia.org/wiki/Lexicographic_product_of_graphs) to 2SAT
(since 2SAT is polynomial, the algorithm is polynomial).
Can't prove completeness of the algorithm and since it is related to the complexity
of graph isomorphism, almost... | https://mathoverflow.net/users/12481 | Counterexamples for this algorithm for recognizing lexicographic product of graphs? | Gerhard's suggestion:
>
> permute k of the rows and columns to get the adjacency matrix of a graph isomorphic to T. What does (\*) do?
>
>
>
Made the algorithm fail and this qualifies as a valid counterexample to me.
| 2 | https://mathoverflow.net/users/12481 | 107165 | 61,843 |
https://mathoverflow.net/questions/107179 | 14 | Suppose that for all $\alpha<\kappa$ we have that $A\_\alpha\subseteq\kappa$. We define the diagonal intersection to be $$\bigtriangleup\_{\alpha<\kappa}A\_\alpha = \left\lbrace\xi<\kappa\ \middle|\ \xi\in\bigcap\_{i<\xi}A\_i\right\rbrace$$
One of the most surprising theorems in basic set theory, I think, is that if ... | https://mathoverflow.net/users/7206 | Intuition behind the diagonal intersection | The diagonal intersection corresponds to the infimum in the boolean algebra $\mathbb{B}\_I:=P(\kappa)/I$ (where $I$ is the nonstationary ideal, or more generally where $I$ is any normal ideal on $\kappa$). More precisely: if $Z \subset P(\kappa)/I$ and $|Z| = \kappa$, then $Z$ has an infimum in $\mathbb{B}\_I$, and thi... | 16 | https://mathoverflow.net/users/26319 | 107181 | 61,853 |
https://mathoverflow.net/questions/107168 | 23 | I asked this question at [Maths Stack Exchange](https://math.stackexchange.com/q/194749/39599), but I haven't received any replies yet (I'm not sure how long I should wait before it is acceptable to ask here, assuming there is such a period of time).
---
The Wikipedia page for [Weitzenböck identities](http://en.w... | https://mathoverflow.net/users/21564 | Weitzenböck Identities | Here is roughly the philosophy of the Weitzenbock technique. (Most of what follows is taken from Berline-Getzler-Vergne book.)
Suppose that $E\_0,E\_1\to M$ are vector bundles on an oriented Riemann manifolds $M$ equipped with hermitian metrics. Denote by $C^\infty(E\_i)$ the space of smooth sections of $E\_i$.
A ... | 22 | https://mathoverflow.net/users/20302 | 107189 | 61,858 |
https://mathoverflow.net/questions/107159 | 7 | I have a minor result which I'm sure has come up somewhere before but I can't seem to find it.
Consider a confluent hypergeometric function of the form
$$\newcommand{\ff}{{}\_1F\_1}
\ff(b+k;b;z)\textrm{, for }k\in\mathbb{N}.$$
Numerical tests suggest that this is always a polynomial of degree $k$ multiplied by an ex... | https://mathoverflow.net/users/20729 | Pochhammer symbol of a differential, and hypergeometric polynomials | Formally using the inverse Mellin transform for x>0:
$$e^x f(x\tfrac{d}{dx})e^{-x}=e^x f(x\tfrac{d}{dx}) \frac{1}{2\pi i} \int\_{\sigma - i \infty}^{\sigma + i \infty} \frac{\pi}{\sin(\pi s)} \frac{x^{-s}}{(-s)!} ds$$
$$=e^x \frac{1}{2\pi i} \int\_{\sigma - i \infty}^{\sigma + i \infty} \frac{\pi}{\sin(\pi s)} f(-s... | 7 | https://mathoverflow.net/users/12178 | 107191 | 61,860 |
https://mathoverflow.net/questions/107195 | 6 | It is not difficult to show (even without Weyl criterion) that the sequence $\sqrt{n}$, $n=1,2,\ldots$ is equidistributed mod 1. However, I need a reference to this result. Can you help me? Thanks.
| https://mathoverflow.net/users/21700 | References for the result that $\sqrt{n}$ is equidistributed mod 1 | Fejer's theorem: If $w(t)$ is a function with continuous first and second derivatives whose signs are eventually constant, and if $t \cdot w'(t)$ goes to infinity for $t$ to infinity, and $w(t)/t$ goes to zero, then $(w(n): n=1,2,3,...)$ is uniformly distributed.
In particular, for any fixed $b>0$ and $\alpha$ betwe... | 16 | https://mathoverflow.net/users/14915 | 107196 | 61,863 |
https://mathoverflow.net/questions/107175 | 5 | In Shipley's paper <http://arxiv.org/abs/math/0209215> she proves a Quillen equivalence between the category of $H\mathbb Z$-modules and dg $\mathbb Z$-modules. So, to a chain complex $C$, she assigns a spectrum $HC$, but is it true in some sense that $\pi\_i HC \cong H\_i C$? I am aware there are some subtleties invol... | https://mathoverflow.net/users/10201 | Homotopy groups and homology groups for the $H\mathbb Z$ module-dg module correspondence. | Shipley's result (of which an alternative version can be found in EKMM, IV.2) proves that there is a string of Quillen equivalences between the derived category of chain complexes over $\mathbb{Z}$ and the category of $H\mathbb{Z}$-modules. This, in particular, gives rise to a chain of natural equivalences between thei... | 7 | https://mathoverflow.net/users/360 | 107197 | 61,864 |
https://mathoverflow.net/questions/107192 | 7 | Let $G$ be a finitely generated subgroup of a product of two finite rank free groups $F\_m \times F\_n$. If there is a Lipschitz retraction $F\_m \times F\_n \to G$ with respect to word metrics, then $G$ is undistorted in $F\_m \times F\_n$, and the Dehn function of $G$ has a quadratic upper bound.
Suppose now that w... | https://mathoverflow.net/users/20787 | Dehn function for undistorted subgroups of a product of free groups | The Bieri-Stallings subgroup of $F\_n\times F\_n$ is undistorted and finitely generated, but not finitely-presented, so in some sense it has an infinite Dehn function. It's the kernel of the map $F\_n\times F\_n\to \mathbb{Z}$ which sends each generator to 1, and it's generated by elements of the form $g\_ih\_j^{-1}$ w... | 10 | https://mathoverflow.net/users/25051 | 107200 | 61,867 |
https://mathoverflow.net/questions/107163 | 7 | For pedagogical purposes I am looking for a function $\mathbb{N}\to\mathbb{N}$ that is defined everywhere but has most of its values unknown. Although such a function cannot be simple by definition, I nevertheless hope that there is such a function that is **simple to explain**.
I have listed a few examples myself wh... | https://mathoverflow.net/users/26503 | A function that is defined everywhere but has unknown values | How about something based on a generalized twin prime conjecture? That is, $f(n)=1$ if there are infinitely many pairs of primes that differ by $2n$ (and $f(n)=0$ if not).
**Added 9/15/12:** It might be worth modifying the example to say $f(n)=1$ if and only if there are infinitely many pairs of *consecutive* primes ... | 8 | https://mathoverflow.net/users/15837 | 107204 | 61,868 |
https://mathoverflow.net/questions/107182 | 4 | Can one show that any abelian $p$-group (not necessarily finite) is the center of a $p$-group and of index $p$?
| https://mathoverflow.net/users/25762 | Center of p-groups | **Edit.** In fact, any nontrivial abelian $p$-group $A$ can be realized as the center of a $p$-group with index $p^n$ except in the case $n=1$ (if $A$ is trivial, then it cannot be the center of a nontrivial $p$-group). As has been noted, if $N\subseteq Z(G)$ and $G/N$ is cyclic, then $G$ is abelian, so no group can ha... | 13 | https://mathoverflow.net/users/3959 | 107207 | 61,870 |
https://mathoverflow.net/questions/107149 | 1 | Given a function F, how to find polynom which is best/good approximate with respect supreremum-norm, i.e. minimize over P\_{approx} sup|F-P\_{approx}| ?
I am intersted in polynoms in two variables of small degree, but the comments on the general situation (1variable, Nvariables) are also welcome. "Sup" is taken over ... | https://mathoverflow.net/users/10446 | Approximation by polynom 1) with respect to supremum-norm 2) I need F_{approx} > F_{exact} | The framework of sum ofsquares / semidefinite programming (SOS / SDP) allows one to compute arbitrarily good approximations to such problems. The general idea is as follows.
First, we write the problem as an optimization problem whose decision variables are the polynomial's coefficients and some other auxiliary varia... | 2 | https://mathoverflow.net/users/5963 | 107211 | 61,872 |
https://mathoverflow.net/questions/107215 | 7 | Let $A$ and $B$ be objects in a topos $\mathcal{E}$ with natural numbers object $\mathbb{N}$. If there is a monomorphism $m: \mathbb{N}\rightarrow A\times B$, is it necessarily the case that there is either a monomorphism from $\mathbb{N}$ to $A$ or a monomorphism from $\mathbb{N}$ to $B$?
This is clearly true when $... | https://mathoverflow.net/users/26517 | Monomorphisms from natural numbers objects into products. | Let us call an object $A$ *infinite* if there is a monomorphism $\mathbb{N} \to A$. Your question then asks whether $A$ or $B$ must be infinite in order for $A \times B$ to be infinite.
From now on we argue in the internal language of a topos. I am going to show that the [Lesser Limited Principle of Omniscience](http... | 10 | https://mathoverflow.net/users/1176 | 107222 | 61,877 |
https://mathoverflow.net/questions/107210 | 15 | The question is simple:
>
> What fraction of matrices in $G\_n = \text{GL}\_n(\mathbb{Z})$ have at least one unit entry (i.e., either $\lbrace\pm 1 \rbrace$)?
>
>
>
I'm not sure what the correct measure on $G\_n$ would be, so here is a suggestion: for each natural number $m \geq 1$, define $G\_n(m)$ to be the ... | https://mathoverflow.net/users/18263 | What fraction of n x n invertible integer matrices contain at least one unit? | Let $G\_p$ denote the subgroup $\mathrm{GL}\_n(\mathbf{F}\_p)$ consisting
of matrices with determinant $\pm 1$. Then
$G\_p$ is exactly the image of $\mathrm{GL}\_n(\mathbf{Z})$ under reduction mod $p$.
Any natural method of counting matrices of "height at most $T$" should have the following property: if one restricts... | 10 | https://mathoverflow.net/users/26524 | 107227 | 61,882 |
https://mathoverflow.net/questions/107231 | 10 | I'm wondering if there is a characterization of schemes over a a field $k$ whose dualizing complex is a perfect complex in terms of singularities. E.g. on a proper Cohen-Macauley scheme over a field, the dualizing complex is a sheaf, when is this sheaf have a finite locally free resolution? (Or this never happens unles... | https://mathoverflow.net/users/1657 | Characterization of schemes whose dualizing complex is perfect | As Hailong said in his comment this only happens in the Gorenstein case; here is a sketch of an argument.
Suppose $X$ is a quasi-compact quasi-separated scheme with a dualising complex $D$ and let us assume $D$ is perfect. Recall that $\mathsf{D}^\mathrm{perf}(X)$, the category of perfect complexes over $X$, is a rig... | 14 | https://mathoverflow.net/users/310 | 107240 | 61,888 |
https://mathoverflow.net/questions/107237 | 3 | I have a semisimple algebra $R$ over a field $k$ that looks like a group algebra $k[G]$ except that it's deformed slightly. That is, instead of a basis $e\_1 ..., e\_n$ closed under multiplication, I have a set of $1$-dimensional subspaces $v\_1,...,v\_n$, such that their direct sum is $R$ (that is, a choice of one gen... | https://mathoverflow.net/users/18060 | What is this deformed group algebra named? | Yes, Dima is right, these are twisted group algebras. They are discussed in [this MathOverflow Question](https://mathoverflow.net/questions/100098/whether-such-an-algebra-has-to-be-the-group-algebra/100101#100101)
| 3 | https://mathoverflow.net/users/5301 | 107246 | 61,891 |
https://mathoverflow.net/questions/107230 | 6 | The category of [convergence spaces](http://ncatlab.org/nlab/show/convergence+space) generalise topological spaces and form a quasi-topos, as topoi are allegedly nicer is there a nicer kind of topological-like space, the category of which forms a topos?
| https://mathoverflow.net/users/22002 | Is there a category of topological-like spaces that forms a topos? | Ronnie has already given the answer which immediately popped into my head when I saw the question. But I should sound a warning that the objects of the topological topos aren't exactly topological spaces, and indeed it seems likely that the only full subcategories of $Top$ that are toposes are fairly uninteresting for ... | 12 | https://mathoverflow.net/users/2926 | 107249 | 61,893 |
https://mathoverflow.net/questions/107256 | 1 | For any $k$ sums $T\_k = 1/|G|\sum\_{g\in G} g^k$ belong to the center of the group algebra, for finite group G.
For $k=2$ they "are" (up to details and interpretation) [Frobenius-Schur indicators](http://en.wikipedia.org/wiki/Frobenius-Schur_indicator).
For $k>2$ similar things called "higher" FS indicators.
**Ques... | https://mathoverflow.net/users/10446 | sum_g g^k, Frobenius-Schur indicators, S_n-invariants in freeAss(x_i), center of the group algebra | For question 1, they are integers, and this is related to Adams operations. The result basically follows from Newton's identities. However, there is no general bound on the size of the integers which may occur when $k >2.$
Later edit: In general, the $T\_{k}$'s can fail very badly to generate the center of the group ... | 4 | https://mathoverflow.net/users/14450 | 107263 | 61,901 |
https://mathoverflow.net/questions/102624 | 7 | I recently became interested in Maass cusp forms and heared people mentioning a "multiplicity one conjecture". As far as I understood it, it says that the dimension of the space of Maass cusp form for fixed eigenvalue should be at most one.
Since Maass cusp forms always are defined for a Fuchsian lattice, I wonder
... | https://mathoverflow.net/users/25218 | Multiplicity one conjecture | I rethought your question and have discovered a partial answer for 1) and 2). I add this as a disjoint answer, since my other answer adresses a totally different (negative) issue.
In Sarnak's article <http://web.math.princeton.edu/sarnak/baltimore.pdf>, he recalls one famous conjecture (Conjecture I, due to himself) ... | 3 | https://mathoverflow.net/users/10400 | 107264 | 61,902 |
https://mathoverflow.net/questions/107268 | 7 | Let $W$ be a cyclic word of length $n$ in a 2-letter alphabet $\{0,1\}$. It is clear that it has at most $n^2$ different subwords (because the number of subwords of length $i$ is at most $n$ for each $i$) and that the actual number of subwords is less than $n^2$ (because the number of subwords of length $1$ is not $n$,... | https://mathoverflow.net/users/nan | A word with maximal number of subwords | If you take a [de Bruijn sequence](http://en.wikipedia.org/wiki/De_bruijn_sequence) of length $2^k$, then you have every length $k$
sequence precisely once. This implies that the number of subwords is maximal, since each subword of length $\geq k$ is determined uniquely by its prefix, and each subword of length $ < k$... | 10 | https://mathoverflow.net/users/1345 | 107270 | 61,903 |
https://mathoverflow.net/questions/107287 | 10 | It follows from the modularity theorem for elliptic curves over $\mathbb{Q}$ that there are finitely many elliptic curves of a given conductor $N$. Moreover, one can algorithmically enumerate them. [**Edit:** As Emerton comments below, without further argument, this is only true for elliptic curves up to isogeny!]
>... | https://mathoverflow.net/users/683 | Finiteness of elliptic curves of a given conductor | As Noam says, this was well-known long before Wiles. The proof that he sketches can be used to prove Shafarevich's theorem for elliptic curves, i.e., given a finite set of primes $S$, there are only finitely many elliptic curves over $\mathbb{Q}$ with good reduction outside of $S$. So if you have a particular $N$ in mi... | 13 | https://mathoverflow.net/users/11926 | 107289 | 61,911 |
https://mathoverflow.net/questions/107291 | 4 | I have heard that this set is the disjoint union of two conics in $Gr(2,4)$, but I do not have an original reference. Does anyone either have such a reference, or know a way of seeing this?
| https://mathoverflow.net/users/26545 | How do I find the set of all lines lying on a general quadric in $\mathbb{CP}^3$? | The quadric $Q$ is isomorphic to $\mathbb{P}^1 \times \mathbb{P}^1$ embedded with the linear system $\mathcal{O}\_Q(1,1)$. The *rulings* of the quadric are the two families of lines $\mathbb{P}^1 \times \{a\}$ and $\{b\} \times \mathbb{P}^1$, and it is not difficult to see that these are the only lines in $Q$.
It is... | 10 | https://mathoverflow.net/users/7460 | 107293 | 61,913 |
https://mathoverflow.net/questions/107265 | 2 | I want to show that there is some $\gamma(n)=o(n^{-1})$ and some $C(n) \to -\infty$ such that for $\gamma \leq \theta \leq \pi$ we have
$\sum\_{k=1}^n -1+\cos(k\theta) \leq C(n)$.
If we rewrite this using the Dirichlet kernel, what I want is that: if $\gamma \leq \theta \leq \pi$ then
$-n+\frac{D\_n(\theta)}{2}-\... | https://mathoverflow.net/users/17086 | Bound on trigonometric sum | To elaborate on my comment: First, $\cos k \theta - 1 = -2 \sin^2 \frac 12 k \theta,$
so the inequality can be rephrased as saying that
$\lim\_{n\rightarrow \infty} \sum\_{k=1}^n \sin^2 k \theta/2 = \infty.$
For $\theta=O(1)$ you already know how to do this. For $\theta \ll 1/n,$ as per @Matt's suggestion one can ... | 3 | https://mathoverflow.net/users/11142 | 107299 | 61,918 |
https://mathoverflow.net/questions/107312 | 2 | What is known about S\_n group invariants in a free associative (noncommutative) algebra k < x\_1, ...x\_n >) ? (S\_n natural acts by permutations on generators).
What is Poincare series ? Is it finitely generated, is it free ? Are the generators as algebra/vector space known ?
The same question for the commutativ... | https://mathoverflow.net/users/10446 | S_n invariants in a free associative algebra ("noncommutative symmetric polynoms") | The paper <http://arxiv.org/pdf/math/0502082.pdf> shows the invariants are a free associative algebra and give an explicit basis. Hence it is not commutative. This is proved first in M. C. Wolf, Symmetric functions of noncommutative elements, Duke Math. J. 2 (1936), 626–637 without an explicit basis.
| 2 | https://mathoverflow.net/users/15934 | 107313 | 61,923 |
https://mathoverflow.net/questions/107308 | 4 | Good afternoon,
I'm just curious about this question, because I see that there are a lot of papers which study the value distribution of an entire curve $f\colon \mathbb{C}\to X,$ with X a complex manifold by using Nevanlinna theory. But I don't know the motivation of this research.
**My question :** Why do we nee... | https://mathoverflow.net/users/11376 | why do we need to study entire curves? | A good reference is S. Lang, Introduction to complex hyperbolic geometry.
Shortly, I can give the following reasons. Listed in historical order.
1. There are famous Picard theorems about entire curves whose target is of dimension 1.
They are 130 years old and they have a lot of connections with other areas of mathema... | 11 | https://mathoverflow.net/users/25510 | 107315 | 61,924 |
https://mathoverflow.net/questions/107301 | 9 | Let $E \subset \mathbb{P}\_\mathbb{C}^2$ be an elliptic curve. If $E$ has complex multiplication (by anything) then the theory of complex multiplication in particular tells us that if $\sigma \in \textrm{Aut}(\mathbb{C})$ then $E^\sigma$ will be isogenous to $E$. Suppose $\textrm{End}(E) \cong \mathbb{Z}$ do we still h... | https://mathoverflow.net/users/25854 | Isogeny classes of elliptic curves | We say that an elliptic curve $E$ over a number field $K$ is *an elliptic $\mathbf{Q}$-curve* if it is is isogenous to its Galois conjugates $E^\sigma$. These were first studied by Benedict Gross, but were later studied by Elkies, Ellenberg, Ribet, and many others. It's possible to show that $\mathbf{Q}$ curves are mod... | 16 | https://mathoverflow.net/users/3384 | 107319 | 61,926 |
https://mathoverflow.net/questions/107250 | 8 | In his classical 1956 paper
[Fully reducible subgroups of algebraic groups](http://www.jstor.org/stable/2372490?origin=crossref)
Mostow proves the following theorem:
>
> **Theorem 7.1.**
> Let $G$ be an algebraic group over a field $K$ of characteristic 0,
> $\mathfrak{N}$ the set of nilpotent elements in the r... | https://mathoverflow.net/users/4149 | Mostow's theorem on algebraic groups | I prefer the argument as I wrote it in the comments, since the brevity there conveys the structure of the proof most clearly. But since the OP requests it, below is a very detailed version (which might obscure the simplicity of the main idea due to its length).
Before explaining the "modern" cohomological proof more ... | 12 | https://mathoverflow.net/users/26145 | 107342 | 61,935 |
https://mathoverflow.net/questions/107338 | 5 | Suppose that $C,D$ are symmetric monoidal categories and $F:C\rightarrow D$ is a nonsymmetric monoidal functor.
1. Does it imply that there exists isomorphic functor $G$ which is a symmetric monoidal functor?
2. If not, then maybe additional assumption that $F$ is an equivalence should be imposed?
In other words w... | https://mathoverflow.net/users/26560 | symmetry in monoidal categories | The answer to both questions is no, since a monoidal category can have several non-equivalent symmetric structures. For instance, the category of $\mathbb{Z} / 2\mathbb{Z}$-graded vector spaces can be given the "usual" symmetry,
$$c(x \otimes y) = y \otimes x,$$
or the "super" symmetry,
$$c(x \otimes y) = (-1)^{\lvert... | 10 | https://mathoverflow.net/users/396 | 107343 | 61,936 |
https://mathoverflow.net/questions/107334 | 6 | Let
$$
f(x)=\sin(x)\sqrt{1+\cos^{2}(x)+\cos^{4}(x)}.
$$
In my study of almost commuting unitary matrices, $U$ and $V$, I
have need for a bound like
$$
\left\Vert \tilde{f}(V)U-U\tilde{f}(V)\right\Vert \leq C\left\Vert VU-UV\right\Vert ,
$$
where $\tilde{f}(e^{\pi ix})=f(x)$ so my functional calculus makes
sense. (Matr... | https://mathoverflow.net/users/6133 | A fourier series related to spin Chern numbers almost commuting matrices | Let $\|\cdot \|\_F $ denote the $\ell^1$ norm of the Fourier series of a function on $[0,2\pi]$. This is a Banach algebra norm on functions whose Fourier series converge absolutely.
Note that $$1 + \cos^2(x) + \cos^4(x) = \frac{15}{8} + \frac{1}{2} e^{2ix} + \frac{1}{2} e^{-2ix} + \frac{1}{16} e^{4ix} + \frac{1}{16} e... | 5 | https://mathoverflow.net/users/13650 | 107348 | 61,938 |
https://mathoverflow.net/questions/107347 | 9 | 1.Is there any classification of symmetric structures on monoidal category $Vect$(with respect to usual tensor product)?
2.Can one organize them in suitable category?
I hope that this is nice and instructive exercise, which is appropriate for mathoverflow.
| https://mathoverflow.net/users/26560 | Symmetric structures on Vect | There's just the one (up to symmetric monoidal equivalence).
For this answer, I'll need a lemma: if $C$, $D$ are $Vect$-enriched categories and $F, G$ are additive $Vect$-enriched functors $C \to D$, then any natural transformation $F \to G$ is $Vect$-enriched natural. This is a consequence of the fact that the unde... | 17 | https://mathoverflow.net/users/2926 | 107351 | 61,941 |
https://mathoverflow.net/questions/107324 | 2 | Let $A$ be an abelian variety over an algebraically closed field $K$, and $B$ an abelian subvariety. We know (for instance, from Milne course on AV) that there is an abelian "cofactor" $B'\subset A$ such that the addition $B\times B'\to A$ is an isogeny, which is the same as saying that $B\cap B'$ is a finite algebraic... | https://mathoverflow.net/users/25887 | Can we control the size of the intersection of two abelian subfactors of an abelian variety ? | Here is a (slightly more detailed) variant of what was pointed out by grp.
Let $E$ and $F$ be non-isogenous elliptic curves over $K$. Let $n$ be a positive integer. (If $p=char(K)>0$ and $p$ divides $n$ we assume additionally that both $E$ and $F$ are ordinary elliptic curves.) Then there are order $n$ cyclic subgrou... | 4 | https://mathoverflow.net/users/9658 | 107353 | 61,943 |
https://mathoverflow.net/questions/107327 | 28 | Given the *Dedekind eta function* $\eta(\tau)$. Define $m = (p-1)/2$ and a $24$th root of unity $\zeta = e^{2\pi i/24}$.
1. Let *p* be a prime of form $p = 12v+5$. Then for $n = 2,4,8,14$:
$$\sum\_{k=0}^{p-1} \Big(\zeta^{m k}\, \eta\big(\tfrac{\tau+m k}{p}\big)\Big)^n = -\big(\sqrt{p}\;\eta(p\tau)\big)^n\tag1$$
2. L... | https://mathoverflow.net/users/12905 | A 14th and 26th-power Dedekind eta function identity? | This question asks in effect to show that $\eta^n$ is a $\pm p^{n/2}$
eigenfunction for the Hecke operator $T\_p$. The claim holds because
each of these $\eta^n$ happens to be a CM form of weight $n/2$,
and $p$ is inert in the CM field ${\bf Q}(i)$ or ${\bf Q}(\sqrt{-3})$.
In plainer language, the sum over $k$ takes th... | 32 | https://mathoverflow.net/users/14830 | 107356 | 61,944 |
https://mathoverflow.net/questions/107355 | 2 | I am looking for a solver that allows me to solve an optimization problem of the form
$$\begin{array}{ll} \text{minimize} & x\_1 x\_2 \cdots x\_n\\ \text{subject to} & \color{gray}{\text{(some linear constraints)}}\end{array}$$
I've used Gurobi before. However, I couldn't find the way to include products in the obj... | https://mathoverflow.net/users/26568 | Minimizing product subject to linear constraints | This is a hard problem (maximizing the product is a bit better one, as sometimes one can take $\log$ of the objective function, and it becomes concave...). Your best shot might be to use the sum of squares approach for polynomial optimization, as implemented e.g. in [YALMIP](http://users.isy.liu.se/johanl/yalmip/pmwiki... | 1 | https://mathoverflow.net/users/11100 | 107358 | 61,945 |
https://mathoverflow.net/questions/107340 | 6 | Let $\rm{SL}\_n$ be the special linear group and let $\rm{Sym}\_n$ be the set of all symmetric matrices of size n. $\rm{SL}\_n$ acts on $(\rm{Sym}\_n)^m$ by $g(A\_1, \ldots , A\_m)=(gA\_1 g^{\rm T}, \ldots , g A\_m g^{\rm T})$. Clearly, in the case of $m=1$ the ring of invariants is generated by $\det(A)$. But what are... | https://mathoverflow.net/users/24296 | Invariants of group action: SL_n acts simultaneously on m symmetric matrices | If you are willing to replace $SL$ by $GL$ (so without the determinant) then this is a special case of a much more general result.
Apply the theory in the following paper to the quiver with one vertex and $m$ edges.
MR0958897 (90e:16048)
Le Bruyn, Lieven; Procesi, Claudio
Semisimple representations of quivers.
... | 3 | https://mathoverflow.net/users/3992 | 107363 | 61,948 |
https://mathoverflow.net/questions/107364 | 1 | In my understanding, random variable is a measurable function from a probability space to a measurable space. Suppose $X$ is a random variable from $(A, \sigma\_{A},P\_A)$ to $(B,\sigma\_{B})$. And $Y$ is a random variable from $(B, \sigma\_{B}, P\_B)$ to $(C, \sigma\_{C})$.
Then $Y(X)$ is a random variable from $A$ ... | https://mathoverflow.net/users/19283 | The Probability distribution of Random variable of Random variable | $Y(X)$ doesn't mean anything. You can't define the composition of random variables. What you can do is compose a random variable $X$ by a measurable function $f$ (provided the $\sigma$ algebras are the same) : $f(X)$.
So in your example, there are two different objects, measurable functions and random variables :
-... | 3 | https://mathoverflow.net/users/26018 | 107366 | 61,949 |
https://mathoverflow.net/questions/107369 | 8 | I ran into the following situation and it turned out to be more subtle than it looked.
I have a complete Riemannian manifold $M$ and my objective is to construct a sequence of functions $f:M \to [0,1]$ which have compact support, $L^2$ norm bounded away from zero, but all third derivatives converge uniformly to zero.... | https://mathoverflow.net/users/7631 | Almost constant bump function | Here is a counter-example.
Take a sequence of round 2-dimensional spheres $M\_n$ of radii $r\_n=n^{-1/2}$, $n=1,2,\dots$. Join them together into a long connected sum, namely connect each sphere to the next one by a tiny handle. Choose the points on the spheres where handles are attached carefully: on $M\_n$, the two... | 13 | https://mathoverflow.net/users/4354 | 107380 | 61,955 |
https://mathoverflow.net/questions/107329 | 3 | I've been going through a paper by Warshall where he shows the presence of dead-ends of unbounded depth$^1$ (for any generating set) in the (discrete) Heisenberg group. This turns out to be the case for the lamplighter group [Cleary and Taback] (and perhaps for many amenable non-virtually-abelian groups?), and the lamp... | https://mathoverflow.net/users/18974 | How does a group strictly grows? | For Q2, the answer is yes. It suffices to enumerate the vertices of the Cayley graph by natural numbers so that each vertex (except the first one) is adjacent to at least one with a smaller number and at least one with a larger number.
This is possible for any locally finite connected graph $\Gamma=(V,E)$ which enjoy... | 2 | https://mathoverflow.net/users/4354 | 107387 | 61,959 |
https://mathoverflow.net/questions/107302 | 10 | Given *any* manifold $M$, we can get a *symplectic* manifold by taking the **cotangent bundle** $T^\ast M$ with symplectic form $\omega=\sum dp\_i\wedge dq\_i$. Given *any* manifold $M$, we can get a *contact* manifold by taking the **projectivization of the cotangent bundle** $\mathbb{P}^\ast M=(T^\ast M-\lbrace0\text... | https://mathoverflow.net/users/12310 | 'Contactization' and Symplectization | The "pre-quantization" construction of a contact manifold out of symplectic manifold predates prequantization by a couple of decades:
see Boothby, W. M.; Wang, H. C.
On contact manifolds.
Ann. of Math. (2) 68 1958 721–734.
The analogue of the theorem for symplectic orbifolds is due to Thomas: Thomas, C. B.
Almost reg... | 6 | https://mathoverflow.net/users/25355 | 107388 | 61,960 |
https://mathoverflow.net/questions/107385 | 3 | In *"Graph theory as I have known it"*, [p.12, Knights Errant](https://books.google.com/books?id=uYW2tttqQ74C&pg=PA12), the late Tutte mentions as an aside the chess question *"Does either Black or White have a certain win from the initial position, given perfect play by both sides"*.
Is there any literature on that ... | https://mathoverflow.net/users/24669 | A chess question of W.T. Tutte | The question "does either Black or White have a certain win from the initial position, given perfect play by both sides" was first addressed by Wilhelm Steinitz in his 1896 "Theory of Perfect Play" (Chapter 6 of [Modern Chess Instructor](https://www.archive.org/details/modernchessinstr00steirich)). He concluded that "b... | 5 | https://mathoverflow.net/users/11260 | 107399 | 61,963 |
https://mathoverflow.net/questions/107409 | 1 | How many cusps does the congruence subgroup $\Gamma(N)$ have?
Thanks;
| https://mathoverflow.net/users/24766 | The number of Cusps | This can be found in most books on modular forms, there's a lot of detailed information in Chapter 3 of Diamond-Shurman. It's
$$ \frac 1 2 N^2 \prod\_{p \mid N}\left(1 - \frac 1 {p^2}\right)$$
if $N \geq 3$ and $3$ if $N=2$. The factor $N^2 \prod\_{p \mid N}\left(1 - \frac 1 {p^2}\right)$ arises as the number of elemen... | 4 | https://mathoverflow.net/users/1310 | 107412 | 61,965 |
https://mathoverflow.net/questions/107414 | 7 | Does there exist a matroid that is representable over $\mathbb{R}$ but not over $\mathbb{Q}$?
In particular, can one give a positive answer using a nonrational polytope, i.e., a combinatorial polytope that cannot be realized as the convex hull of rational vertices? (Such things do exist; see, e.g., p.94 of Grünbaum's... | https://mathoverflow.net/users/8604 | Matroid representable over $\mathbb{R}$ but not over $\mathbb{Q}$? | Jeremy, on the very same page 94 you will find a "point and line configuration" called *Perles configuration* which when viewed as set ov vectors in $\Bbb R^3$ is a matroid that is realizable over $\Bbb Q[\sqrt{5}]$ but not over $\Bbb Q$. In [my book](http://www.math.ucla.edu/~pak/book.htm) I even prove it (Ex 12.3) - ... | 11 | https://mathoverflow.net/users/4040 | 107416 | 61,968 |
https://mathoverflow.net/questions/107389 | 7 | Let $G$ be a compact connected semisimple Lie group, $\mathfrak{g}$ be its complexified Lie algebra and $\mathfrak{g}^\*$ its complex dual space. We can form the symmetric algebra $S(\mathfrak{g}^ \* ) $ and its $G$-invariant subalgebra $S(\mathfrak{g}^ \* ) ^G$. For a closed connected subgroup $H< G$ we have the same ... | https://mathoverflow.net/users/24965 | About the map $S(\mathfrak{g}^ * )^G\rightarrow S(\mathfrak{h}^ * )^H$ for $H < G$ | The symmetric algebra can be viewed as polynomial functions on $\mathfrak g$. The kernel of the map consists of $G$-invariant polynomial functions that vanish on $\mathfrak h$. Thus the kernel is trivial when the $G$-orbit of $\mathfrak h$ is Zariski-dense in $\mathfrak g$, for instance, when the $G$-orbit of $H$ is Za... | 6 | https://mathoverflow.net/users/18060 | 107418 | 61,969 |
https://mathoverflow.net/questions/107402 | 5 | It is possible for nonisomorphic fields to have the same discriminant, and even to be arithmetically equivalent, meaning with the same Dedekind zeta function. Previous questions which discussed this are "Number fields with same discriminant and regulator?" and "Are there two non-isomorphic number fields with the same d... | https://mathoverflow.net/users/26327 | Nonisomorphic number fields with the same Galois group and discriminant | My [answer](https://mathoverflow.net/questions/3448/are-there-two-non-isomorphic-number-fields-with-the-same-degree-class-number-and/29467#29467) to one of those previous questions provides an example of two Galois cubic fields of the same discriminant. The cyclic group of order three is simple.
**Added:** Just to gi... | 7 | https://mathoverflow.net/users/1021 | 107419 | 61,970 |
https://mathoverflow.net/questions/70724 | 13 | In potential theory, the $\textit{logarithmic energy}$ of a Radon measure $\mu$ acting on $\mathbb{C}$ is defined by
$$I(\mu)=\iint\log\frac{1}{|x-y|}\mu(dx)\mu(dy).$$ Of course it is not well defined for all measures and may takes values in $[-\infty,+\infty]$. To avoid this annoying fact, one typically restrict to me... | https://mathoverflow.net/users/15517 | What do we actually know about logarithmic energy ? | Concerning the first question: We have $I(\mu-\nu)>0$ whenever $I(\mu-\nu)$ is defined, finite or not, and $\mu$,$\nu$ are different signed Radon measures with equal total masses (or rather charges). This, with any finite dimensional Hilbert space in place of the plane, is Example 3.3 in
<http://www.ams.org/journals/t... | 2 | https://mathoverflow.net/users/26591 | 107423 | 61,973 |
https://mathoverflow.net/questions/107430 | 2 | Let $ f:X\to Y $ be a map in the pointed category of topological spaces $ Top\_\* $. And let $ U:Top\_\*\to Top $ be the "forgetful" functor (which "forgets" the basepoint). We can look at the reduced mapping cylinder $ M\_f $ and at the unreduced mapping cylinder $ M\_{U(f)} $ in the category $ Top $.
Until yesterday,... | https://mathoverflow.net/users/26595 | Does the reduced Mapping cylinder have the same homotopy type of unreduced Mapping cylinder? | I don't trust anything with degenerate basepoints, but I do think you are right about the homotopy type via the parenthetical "(or only using an explicit homotopy)''. However, I was working in compactly generated spaces, denoted $\mathcal{U}$, including the weak Hausdorff property. I don't think you can prove that $Mf$... | 5 | https://mathoverflow.net/users/14447 | 107432 | 61,977 |
https://mathoverflow.net/questions/107417 | 1 | A complex number $z$ is an integer if and only if $\sin(\pi z)=0$.
It follows that a complex number $z$ is an integer if and only $\sin^2(\pi z) = 0$. So for a **real** analytic function $f$ and any **real** number $x$, $x$ and $f(x)$ are both integers if and only if $\sin^2(\pi x) + \sin^2(\pi f(x)) = 0$.
Are ther... | https://mathoverflow.net/users/20411 | Analytical predicate for integers over complex numbers | [Reposted as an answer, as requested. -T.]
While this can be done for any given $f$ (as indicated by Angelo's answer), it can't be done in a way which is stable with respect to perturbations of $f$ (which rules out most "analytic" answers, such as the one in the real case). This is because zeroes of a complex analyti... | 4 | https://mathoverflow.net/users/766 | 107433 | 61,978 |
https://mathoverflow.net/questions/107434 | 8 | Suppose A is a symmetric positive semidefinite matrix. Is there a way to upper bound the largest eigenvalue using properties of its row sums or column sums?
For instance, the [Perron–Frobenius theorem](https://en.wikipedia.org/wiki/Perron%E2%80%93Frobenius_theorem) states that, for positive matrices, the largest eige... | https://mathoverflow.net/users/25102 | Upper bounds on eigenvalues of PSD matrix? | Yes, it is true that the largest eigenvalue is bounded by the largest absolute row sum or column sum. You can check [Gershgorin circle theorem](http://en.wikipedia.org/wiki/Gershgorin_circle_theorem).
Actually, all the eigenvalues lie in the union of all Gershorin circles.
| 20 | https://mathoverflow.net/users/26600 | 107438 | 61,980 |
https://mathoverflow.net/questions/107448 | 3 | Is the Birch and Swinnerton-Dyer conjecture known in positive characteristic?
| https://mathoverflow.net/users/nan | Birch and Swinnerton-Dyer conjecture in positive characteristic | Edit: This answer addresses an earlier version of the question, where the OP asked whether or not BSD made sense for elliptic curves over finite fields. It also however answers the current question.
The Birch and Swinnerton-dyer conjecture for an elliptic curve over a number field relates the rank of the Mordell-Weil... | 5 | https://mathoverflow.net/users/5101 | 107449 | 61,982 |
https://mathoverflow.net/questions/107458 | 12 | Let $M^n$ be a smooth manifold whose universal cover is homeomorphic $\mathbb{S}^n$, are there examples where $M^n$ is not homeomorphic to a space form ?
The answer may vary if you replace homeomorphic by diffeomorphic, and I'm also interested in this question under this restriction.
I came to this question while ... | https://mathoverflow.net/users/8887 | Manifold whose universal covering is a sphere but which is not a space form? | There are lots of fake lens spaces, and fake spherical space forms (search on these keywords). In particular, a construction of fake lens spaces is in chapter 12 of Milnor's ["Whitehead torsion"](http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183527946). Here are details:
start... | 12 | https://mathoverflow.net/users/1573 | 107461 | 61,986 |
https://mathoverflow.net/questions/107451 | 0 | Being new to $C$\* algebra, I'm trying to understand basic properties of *-homomorphisms of such algebras. Let $P$ be a set of commuting projections on Hilbert space ${\cal H}$.
Let ${\cal P}$ be the $C$*-algebra spanned by $P$ and the identity $I$.
Let $f$ be any \*-homomorphism from ${\cal P}$ to the complex numbers.... | https://mathoverflow.net/users/26607 | Structure of Homomorphisms of commutative C^* algebra | This depends on the structure of your set of projections. For example, if they are pairwise orthogonal, then $f$ can send at most one of them to 1, because the sum of any two is again a projection. But if one of their ranges is non-trivial and is included in the ranges of the others, then $f$ can map them all to 1. In ... | 0 | https://mathoverflow.net/users/6794 | 107463 | 61,988 |
https://mathoverflow.net/questions/107383 | 2 | As we know that context-free language is in P,any result or conjecture of computaional complexity of formal languange with rational generating function?And more,any result or conjecture of computaional complexity of formal languange with algebraic generating function?
| https://mathoverflow.net/users/14024 | Any result or conjecture of computaional complexity of formal languange with rational generating function? | To clarify Qiaochu's comment and make it explicit, I claim there are uncountably many languages with a rational generating function (namely, $\dfrac{1}{1-x}$). Only countably many of these can even be recursively enumerable. Namely, let $w\in \lbrace 0,1\rbrace^{\omega}$ be a right infinite word. Let $L(w)$ be the lang... | 3 | https://mathoverflow.net/users/15934 | 107464 | 61,989 |
https://mathoverflow.net/questions/107443 | 1 | Suppose I have a bag of `N` balls, each labeled with a distinct integer. The experiment starts when I draw a ball from the bag, record its label, and then return the ball back to the bag. I repeat this step until I draw a ball that I have already drawn before. Let `X` be the number of balls I drew in the experiment. Le... | https://mathoverflow.net/users/26604 | Probability of first collision with replacement | $P(X)=(X-1)N^{1-X}\frac{(N-1)!}{(N+1-X)!}$, for $X=2,3,\ldots N+1$.
| 1 | https://mathoverflow.net/users/11260 | 107471 | 61,993 |
https://mathoverflow.net/questions/107441 | 11 | I asked this on MSE yesterday ( <https://math.stackexchange.com/q/197873/39378> ) but no one has answered it yet. I hope it's not too soon to post it here.
Here are a few ways to formalize the question, so you can pick your favorite and answer it. Assume whatever large cardinals you like.
(1) Is it consistent with ... | https://mathoverflow.net/users/1682 | Does every nonempty definable finite set have a definable member? | I believe the answers to these questions are all positive. This kind of problem was discussed by Groszek and Laver in *Finite groups of OD-conjugates* [Period. Math. Hungar. 18 (1987), 87-97, [MR0895774](http://www.ams.org/mathscinet-getitem?mr=895774)]. Answering a question of Mycielski, they show that there can be tw... | 10 | https://mathoverflow.net/users/2000 | 107474 | 61,994 |
https://mathoverflow.net/questions/107481 | 9 | Is there an explicit description of the maximal tamely ramified extension of $\mathbf Q\_p$?
| https://mathoverflow.net/users/18116 | Maximal tamely ramified extension of $\mathbf Q_p$ | Yes, there is. The maximal unramified extension is obtained by adding all roots of unity of order prime to $p$. The maximal tame extension is obtained by adding on top of that all $n$-th roots of $p$, for $n$ prime to $p$; so its Galois group is isomorphic to $\prod\_{\ell \ne p} \mathbf{Z}\_\ell$, and conjugation by $... | 14 | https://mathoverflow.net/users/2481 | 107484 | 61,997 |
https://mathoverflow.net/questions/107483 | 2 | So $X$ and $Y$ are Hermitian matrices (or just symmetric real) of size $n$ by $n$ and suppose $Y\succeq X$, namely $Y-X$ is positive-semidefinite. Now write the eigenvalues of $Y$ as $\alpha\_1\leq\ldots\leq \alpha\_n$, and the eigenvalues of $X$ as $\beta\_1\leq\ldots\leq \beta\_n$. Is is necessarily true that $\alpha... | https://mathoverflow.net/users/7599 | if Y-X is positive semi-definite, are the eigenvalues of Y bigger? | This is true and well known. By the minimax principle, $\alpha\_k$ is the minimum over all $k$-dimensional subspaces of the norm of the quadratic form $v\mapsto(v,Yv)$ restricted to the subspace. And similarly for $\beta\_k$ and $(v,Xv)$. Since $(v,Yv)\ge(v,Xv)$ for every vector $v$, the same inequality holds for the n... | 9 | https://mathoverflow.net/users/4354 | 107494 | 62,002 |
https://mathoverflow.net/questions/107493 | 2 | Is there a way to sample a planar map uniformly at random? I am aware of the Cori-Vauquelin-Schaeffer bijection that can be used to sample and study uniformly random quadrangulations. There are other results in the literature that allow for sampling other classes of planar maps, but I haven't seen any results for the e... | https://mathoverflow.net/users/7717 | Uniformly random planar map | Something you may potentially be interested in :
<http://www.lix.polytechnique.fr/~fusy/Articles/FusyAofa.pdf>
Nathann
| 1 | https://mathoverflow.net/users/1715 | 107495 | 62,003 |
https://mathoverflow.net/questions/107478 | 4 | Suppose $M= \bigoplus\_{n\in \mathbb Z} M\_n$ is a finitely generated graded module over a Noetherian graded commutative ring $A=\bigoplus\_{n\in \mathbb Z}A\_n$.
If $A$ is positively graded ($A\_n=0$ if $n<0$), then each $M\_n$ is finitely generated as $A\_0$-module (Atiyah, McDonald: Introduction to commutative al... | https://mathoverflow.net/users/18571 | Are homogeneous components of f.g graded modules f.g ? | The answer is yes. Instead of showing that $M\_n$ is finitely generated we may show it has the
property that any ascending sequence of $A\_0$-submodules stabilizes. If $N$ is an $A\_0$-submodule of $M\_n$, consider $M\_n\cap AN$. It is a sum of the $M\_n\cap NA\_i$. One sees it is
$N$ itself. So $N$ can be recovered f... | 8 | https://mathoverflow.net/users/4794 | 107501 | 62,007 |
https://mathoverflow.net/questions/107491 | 4 | This is somewhat related to a question that I asked on Math.SE but, sadly, received no response. I apologize ahead of time if this is not appropriate for MO. Feel free to vote to close if this is the case.
I really have two (distinct) questions. The first is regarding a paper by Greenlees and May and the second is mo... | https://mathoverflow.net/users/1146 | Mackey(also Green and Tambara) functors and Greenlees-May | I thank you for the careful reading and apologize for the concision. This is a downwards induction on the size of subgroups. Using the explicit description of the $RV$ given top of page 239 and the
conventions recalled at the bottom of page 240, we arrived at the description of the relevant $RV$ given just below Prop. ... | 13 | https://mathoverflow.net/users/14447 | 107506 | 62,010 |
https://mathoverflow.net/questions/107508 | 8 | Every open set in the complex plane homeomorphic to an annulus is biholomorphic to exactly one annulus whose inner radius is 1 and whose out radius is $r>1$. Each value of $r$ gives a different complex manifold. This number r is called the "modulus" of the annulus. You could say that the set of real number $(0,\infty)$... | https://mathoverflow.net/users/1106 | Is there an explicit formula for the modulus of an annulus given a parameterization of the inner and outer boundries? | I think that there is no formula. The best one can do is to estimate. Here is a simpler problem of the same sort: suppose you have a parametrization of the boundary of a simply connected region, and
suppose that 0 is inside. Consider the Riemann mapping f of this region sending 0 to 0.
The problem is to find |f'(0)|. T... | 5 | https://mathoverflow.net/users/25510 | 107511 | 62,012 |
https://mathoverflow.net/questions/71124 | 3 | let $\sum\_0^n l\_i x^i$ and $\sum\_0^n 2^i x^i$ be generating function of L a given language and the closure over alphabet $\Sigma= \{0,1 \}$ when $n\to\infty$.
let$$D=\frac{\sum\_0^n l\_i }{\sum\_0^n 2^i }$$,$$d=\frac{ l\_i x^i}{ 2^i x^i}=\frac{ l\_i }{ 2^i }$$.
Obviously,$0 \leq D,d \leq 1$.when(under what condit... | https://mathoverflow.net/users/14024 | density of formal language？ | I suggest looking at Jean Berstel, Sur la densité asymptotique de langages formels. (French) Automata, languages and programming (Proc. Sympos., Rocquencourt, 1972), pp. 345–358. North-Holland, Amsterdam, 1973.
| 1 | https://mathoverflow.net/users/15934 | 107525 | 62,020 |
https://mathoverflow.net/questions/107540 | 2 | Let $X$ be a non-empty set. Consider $\mathcal{P}(X)$, the power-set of $X$. We say that $a,b \in \mathcal{P}(X)$ form an edge if and only if their symmetric difference is a singleton, i.e. $\textrm{card}((a\setminus b) \cup (b\setminus a)) = 1$.
It is clear that for a finite set $X$ the resulting graph has chromatic... | https://mathoverflow.net/users/8628 | Chromatic number of the power set | Yes, for $X$ infinite the resulting graph also has chromatic number 2. To see this, just use the fact that a graph is bipartite if and only if it does not contain an odd cycle (this remains true for infinite graphs). An odd cycle in your graph quickly leads to a contradiction due to parity reasons.
| 5 | https://mathoverflow.net/users/2233 | 107543 | 62,028 |
https://mathoverflow.net/questions/106815 | 6 | Consider the iterated function system $T\_{1}(x)=(\beta x,\tau y)$, $T\_{2}(x,y)=(\beta x+(1-\beta),\tau y+ (1-\tau))$ for $\beta\in(1/2,1)$ and $\tau\in (0,1/2)$ with self affine set $\Lambda\_{\beta,\tau}$.
---
It is known that for almost all $\beta$ and all $\tau$ $\dim\_{H}\Lambda\_{\beta,\tau}=\dim b=1-\log(... | https://mathoverflow.net/users/23542 | Measures of full Hausdorff dimension for self-affine sets | If $\beta^{-1}$ is Pisot, then **there is** an ergodic measure of maximal dimension. This is a special case of the rather difficult Theorem 2.15 in the paper [Dimension Theory of iterated function systems](http://arxiv.org/abs/1002.2036) by De-Jun Feng and Huyi Hu. Very roughly speaking, Feng and Hu adapt Ledrappier-Yo... | 4 | https://mathoverflow.net/users/11009 | 107546 | 62,030 |
https://mathoverflow.net/questions/107519 | 0 | Can some one tell me the reference or any idea how to take the Discrete Sobolev space work defined for a scalar valued map to the space of maps which are vector valued.Let's say
$f:\Omega \rightarrow R^n $ where $\Omega \subset R^d$. A Sobolev space of vector valued maps are defined. Though it looks pretty straigh... | https://mathoverflow.net/users/26265 | Discrete Sobolev space of $R^n$ valued maps | well, there are objects called bochner-lebesgue spaces which are nothing but lebesgue spaces of vector-valued functions. if the target space is finite dimensional, the theory is absolutely trivial, in the general case it is much more involved but still very well understood (see for instance [this book](http://books.goo... | 1 | https://mathoverflow.net/users/26039 | 107549 | 62,032 |
https://mathoverflow.net/questions/107541 | 2 | Let $\cal X$ be a DM stack of finite type over a field (if necessary, I will assume that $k=\mathbb{C}$ and $\cal X$ is a scheme, or even a variety) and $G$ be a finite group. Then we have a quotient stack $[\mathcal{X}/G]$ and in fact we have a natural morphism $\phi:\mathcal{X}\to [\mathcal{X}/G]$.
Is $\phi$ always... | https://mathoverflow.net/users/622 | Are quotients of stacks flat? | The map $\phi$ is in fact étale. By definition a map $B \to [\mathcal X/G]$ (say $B$ is a scheme) is a $G$-torsor $E \to B$ and a $G$-equivariant map $E \to \mathcal X$. The base change of $\phi$ is exactly $E \to B$. Since the property of being étale is stable under base change and local, it suffices to show that $E\t... | 8 | https://mathoverflow.net/users/1310 | 107552 | 62,034 |
https://mathoverflow.net/questions/107538 | 3 | Suppose we're working in the first-order language of the real numbers, and we write
$$\forall x (x > 0 \;\rightarrow\; 1/x > 0)$$
We want this to be true, however I feel like it doesn't quite work, because by the principle of universal instantiation, it follows that we can set $x=0$ in order to obtain
$$0 > 0 \;\... | https://mathoverflow.net/users/26080 | Dealing with undefined expressions in predicate logic | I would definitely not introduce any third truth value or other concept of "UNDEF". Here is how I might deal with the issue if it came up in a logic class. I would ask everyone to think of their favorite real number but not tell anyone else. That number is their own definition of 1/0. Now a sentence like $\forall x (x>... | 1 | https://mathoverflow.net/users/14302 | 107553 | 62,035 |
https://mathoverflow.net/questions/107556 | 1 | OK, I have to ask a dumb question again: Where do Lie groups enter in the
construction of the Reshitikhin-Turaev invariant? The parts of the proof I
understand are that 6j symbols take care of themselves. So why not define
6j symbols axiomatically (Biedenharn-Elliott plus symmetry plus function
value at a 0 argument s... | https://mathoverflow.net/users/11504 | Generalizing the Reshitikhin-Turaev construction possible? | Indeed any ribbon tensor category gives knot invariants. There's no reason why the ribbon tensor category has to come from a Lie group. However, we just don't know many other examples of ribbon tensor categories.
It's certainly plausible that you can get knot invariants from quantum versions of other points on the Vo... | 8 | https://mathoverflow.net/users/22 | 107558 | 62,038 |
https://mathoverflow.net/questions/107566 | 4 | Here is my setting: Let $E\in\mathcal{M}\_k(\Gamma\_0(N))$ be an Eisenstein series (of trivial Nebentypus) that is a normalized eigenform for all the Hecke operators at level $\Gamma\_0(N)$. Assume that the Fourier expansion of $E$ lies in $\mathcal{O}\_K[[q]]$ where $\mathcal{O}\_K$ is the integer ring of some number ... | https://mathoverflow.net/users/14967 | Fourier expansion of Eisenstein series at various cusps | Let $f$ be any modular form of weight $k$ whose coefficients at the cusp $\infty$ lie in $\mathcal{O}\_K$. Then (using GAGA and the $q$-expansion principle - see for example Katz's article on $p$-adic forms) $f$ gives rise to a section of the sheaf $\omega^k$ on $X\_0(N)$ (or $X\_1(N)$ + fixed under the diamond operato... | 2 | https://mathoverflow.net/users/12107 | 107581 | 62,049 |
https://mathoverflow.net/questions/107577 | 2 | Assume you have an almost connected simple Lie group G with trivial center. (In particular excluding non-algebraic examples such as the universal cover of SL\_2(R).)
Such a group should automatically be an algebraic group over the reals resp. the complex numbers.
Is this true and why?
Can we in addition conclud... | https://mathoverflow.net/users/12824 | Are certain simple Lie groups linear algebraic groups? | The answer is yes for complex Lie groups, and follows from the classification. (Root data are in fact defined over $\mathbb{Z}$: a complex semisimple group has not only an underlying algebraic, but even an arithmetic structure). For a more direct explanation, see Theorem 6.3 in the book "Lie Groups and Lie Algebras III... | 5 | https://mathoverflow.net/users/26522 | 107583 | 62,050 |
https://mathoverflow.net/questions/107563 | 1 | Dear all,
I have recently been breaking my head over this question. The idea is that a certain variable $Y$ is normally distributed with a parameter $X$ in both mean and variance.
$Y|X \sim N(\mu X,X^2)$
This parameter $X$ is assumed to be normally distributed as well with parameters $\alpha$ and $\beta$.
$X\... | https://mathoverflow.net/users/26640 | $Y|X \sim N(\mu X,X^2)$ and $X\sim N(\alpha, \beta)$. How is $Y$ distributed? | The characteristic function is
$$\eqalign{ E\left[e^{itY}\right] &= E\left[ E\left[ e^{itY}|X \right]\right] \cr
&= E \left[ \exp(it\mu X - t^2 X^2/2 \right] \cr
&= \frac{\exp \left(\left(-(\alpha^2+\beta \mu^2) t^2 + 2 i \mu \alpha t\right)/\left(2 t^2 \beta + 2\right)
\right)}{\sqrt {{t}^{2}
\beta+1}}\cr}$$
It is ... | 1 | https://mathoverflow.net/users/13650 | 107589 | 62,053 |
https://mathoverflow.net/questions/107551 | 5 | I [asked this question](https://math.stackexchange.com/questions/194666/power-means-inequality) at Stack Exchange but received no answer. The origins of the question are unclear, as I came across it rummaging through old notebooks from highschool, in one of which it was stated without any reference or proof. Let $x, y,... | https://mathoverflow.net/users/3773 | An inequality involving sums of powers | Let $s\_k = x^k + y^k + z^k + t^k$. First we check that the denominator is nonnegative. By Holder, we know $s\_4^{2/3}s\_1^{1/3} \ge s\_3$, rearranging that and using $s\_1 = 1$ we see that the denominator is indeed at least $0$.
Now we multiply out and rearrange, to see that the given inequality is equivalent to:
... | 8 | https://mathoverflow.net/users/2363 | 107592 | 62,054 |
https://mathoverflow.net/questions/107588 | 9 | This is very much the same post as I posted at [math.stackexchange](https://math.stackexchange.com/questions/195203/some-intuition-behind-o-minimal-systems).
I am following the definitions of an o-minimal system in "Tame Topology and O-minimal Structures" by Lou van den Dries.
It is immediate from the definition t... | https://mathoverflow.net/users/2044 | Intuition behind o-minimal structures. | Let $B$ be a slight rotation of the graph of the sine function and lets say that $B \in S\_2$. Then one should be able to show from the definitions that the set $A$ of reals where $B$ has a "local maximum" is in $S\_1$. But the set $A$ is not a finite union of points and intervals.
**Edit 1:** Here are some details a... | 8 | https://mathoverflow.net/users/17836 | 107596 | 62,057 |
https://mathoverflow.net/questions/97769 | 7 | Let $P(x)$ be a real polynomial of degree at most $d$. Assume $|P(x)| \leq 1$ for $|x| \leq 1$. I would like a bound saying that each coefficient of $P(x)$ is at most $C^d$ in magnitude, for some absolute constant $C$.
This is surely a well-known, basic fact in approximation theory and I'm looking for a proper refere... | https://mathoverflow.net/users/658 | Approximation theory reference for a bounded polynomial having bounded coefficients | Dear Ryan, I hope the following references will be useful for you:
V.A. Markov has solved your posed problem back in 1892, see pages 80-81 in
<http://www.math.technion.ac.il/hat/fpapers/vmar.pdf>
Compare also the book
I.P. Natanson: Constructive Function Theory, Vol. I. Uniform Approximation, F. Ungar Publishin... | 5 | https://mathoverflow.net/users/26649 | 107601 | 62,059 |
https://mathoverflow.net/questions/107516 | 0 | Hi there -
I'm Manny, a soon to be MSC thesist. I'm looking for a subject to write my thesis about - and recently I was caught by functional differential equations. Is there any neat reference for funtional equations in settings more general than the real line?
Maybe something that reads "Topological Functional Equ... | https://mathoverflow.net/users/26611 | References for functional equations in more general settings than the reals | Try the book "Functional identities" by Bresar, Chebotar, and Martindale. It deals with functional equations in the realm of associative algebras, Lie algebras and Jordan algebras. As I mentioned in one of my previous postings, the area has its own 2010 MSC code, 16R60.
| 1 | https://mathoverflow.net/users/6339 | 107611 | 62,066 |
https://mathoverflow.net/questions/107586 | 0 | Let $f:\mathbb{R}^n\rightarrow\mathbb{R}$ be a $C^2$ convex function which is strictly
positive. If $x\_n$ is a sequence of points such that $f(x\_n)\rightarrow 0$, show that (or
give a counterexample) the gradient $\nabla f(x\_n)$ also tends to zero.
| https://mathoverflow.net/users/26645 | Asymptotic behavior of convex functions | A counterexample is
$$f=\sqrt{y^2+e^{-x}}.$$
You can verify by computing the second derivatives that this is convex.
As a sequence $x\_n$ you can take $(n,1/n)$. Then $f(x\_n)\to 0$ but the derivative with respect to $y$ tends to 1. Thus the gradient does not tend to 0.
| 1 | https://mathoverflow.net/users/25510 | 107625 | 62,073 |
https://mathoverflow.net/questions/107574 | 2 | If $X$ is a smooth rigid analytic space over a $p$-adic field $K$ (of characteristic zero), then every coherent $\mathcal{O}\_X$-module with integrable connection is locally free. In his paper ["Finiteness theorems for the cohomology of an overconvergent isocrystal on a curve"](http://www.numdam.org/numdam-bin/fitem?id... | https://mathoverflow.net/users/13647 | Modules with connection over $p$-adic laurent series rings | There are two possible arguments, I think. First, using $A\_I$ is a PID you can factor $f$ into prime factors, and work with each of them. We need to prove that if $g$ is prime and $f=g^e$ for some $e\geq 1$ is such that $f'\in f$, then $f$ is a constant. Now, the first option is that you redo the theory of resultants ... | 1 | https://mathoverflow.net/users/18238 | 107632 | 62,077 |
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