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https://mathoverflow.net/questions/100794 | 18 | Let $K$ be a bounded symmetric ($-K=K$) open convex body in $\mathbb R^n$. The critical determinant $d(K)$ of $K$ is the least possible volume $|\operatorname{det}(a\_1\dots a\_n)|$
of the fundamental parallelepiped of a lattice $\Lambda=\{\sum\_{j=1}^n m\_j a\_j: m\_j\in\mathbb Z\}$ such that $K\cap \Lambda=\{0\}$. Cl... | https://mathoverflow.net/users/1131 | Is the ball reducible in some high dimension? | According to [a talk that I found on the web](http://www.math.rwth-aachen.de/~nebe/talks/lat2op.pdf), it is a theorem of Voronoi that every indecomposable root lattice is extreme. Also the $E\_8$ lattice is the union of two copies of the $D\_8$ lattice with the same sphere radius. And, of course, the rotational symmetr... | 7 | https://mathoverflow.net/users/1450 | 101015 | 58,661 |
https://mathoverflow.net/questions/101018 | 10 | In homotopy theory there are lots of nice constructions that seem designed to have some effect on the homotopy of a space, i.e. completing, localizing, and taking various homotopy (co)limits. It seems like some of these (for instance Quillen's + construction) have moderately visual interpretations by attaching cells an... | https://mathoverflow.net/users/11546 | Geometric Interpretations of Homotopy Theoretical Constructions | Though it is often satisfying and useful to have an appropriate way to visualize a construction, in fact the real innovation in homotopy theory that lead to the constructions you mention, among many others, was the realization that you need not limit your attention to geometric, visualizable operations and spaces but c... | 17 | https://mathoverflow.net/users/288 | 101024 | 58,666 |
https://mathoverflow.net/questions/100581 | 9 | **Q1:** My first question is about defining the category $\text{IndCoh}(S)$ for a $DG$ scheme $S$. So in page $18$ of [this paper](http://www.math.harvard.edu/~gaitsgde/GL/IndCohtext.pdf), they are defined as being the ind-completion of the category $\text{Coh}(S)$. Here $\text{Coh}(S) \subset \text{QCoh}(S)$ is define... | https://mathoverflow.net/users/2623 | Defining ind-coherent sheaves and their singular support | (Hopefully t3uji, tony pantev or Greg Stevenson will chime in with a more authoritative answer, but in the meanwhile..)
The notion of singular support of a coherent sheaf is an analog of the notion of singular support of a constructible sheaf or D-module. Let's quickly recall the latter: given a sheaf we can measure ... | 10 | https://mathoverflow.net/users/582 | 101025 | 58,667 |
https://mathoverflow.net/questions/101031 | 35 | Is there any resource that might help non-experts gains some understanding of why
the [Kervaire invariant](http://en.wikipedia.org/wiki/Kervaire_invariant) problem remains open now only in dimension $126$? ($126 =2^7-2=2^{j+1}-2$;
whether $\theta\_j=\theta\_6$ exists in the "$128$-stem"), i.e., why the celebrated Hill-... | https://mathoverflow.net/users/6094 | Kervaire invariant: Why dimension 126 especially difficult? | I'll give a shot at an answer. The relevant dimensions are of the form $2^j-2$. For
$j\leq 4$, it is easy and classical that we can construct manifolds of Kervaire invariant one. The
problem was ``reduced'' from differential topology to pure stable homotopy
theory by Browder in 1969. Direct calculational methods in
ho... | 51 | https://mathoverflow.net/users/14447 | 101033 | 58,670 |
https://mathoverflow.net/questions/44580 | 8 | I am a student of Saint Petersburg State Polytechnical University, chair of Theoretical Mechanics.
While looking into stability of ideal crystal lattices (2D and 3D) by means of molecular dynamics I have encountered a challenging analytical problem. To draw the stability regions I need to find the conditions on coeffic... | https://mathoverflow.net/users/10339 | Conditions on coefficients of a homogeneous polynomial of the third degree in three variables over R which allow it to be positive on a positive octant | Ms. Podolskaya,
Your 2nd question is related to [matrix copositivity](http://en.wikipedia.org/wiki/Copositive_matrix), I believe. Take a look at the 5th chapter of [Parrilo's doctoral dissertation](http://resolver.caltech.edu/CaltechETD%3Aetd-05062004-055516).
A quadratic form in $\mathbb{R}[x\_1,x\_2,x\_3]$ is of ... | 1 | https://mathoverflow.net/users/2741 | 101038 | 58,672 |
https://mathoverflow.net/questions/101045 | 3 | My question is about a small detail on page 132 of the above-mentioned book.
Let $R'$ be a faithfully flat $R$ algebra and $M'$ a $R'$-module. Let $\varphi: p\_1^\* M' \cong p\_2^\* M'$ be a covering datum, where $p\_1$ and $p\_2$ are projections onto the first and second factor from $R'' = R'\otimes\_{R} R'$ to $R'$... | https://mathoverflow.net/users/9035 | A small detail in Neron Models (Bosch-Lütkebohmert-Raynaud) on descent theory | Let $R,S$ be rings, $M$ a $R$-module and $N$ a $S$-module. Let $R \to S$ be a ring homomorphism. Then a $R$-module homomorphism $M \to N|\_R$ is said to be cocartesian over $R \to S$, when the corresponding $S$-module homomorphism $M \otimes\_R S \cong N$ is an isomorphism. If $M$ and $N$ are algebras, you recover the ... | 4 | https://mathoverflow.net/users/2841 | 101046 | 58,675 |
https://mathoverflow.net/questions/101041 | 1 | In what probability does cospectra of adjacent matrix of Cayley graph imply isomorphism of the corresponding group?
Further more,In what probability does cospectra of adjacent matrix imply isomorphism of the corresponding graphs
| https://mathoverflow.net/users/14024 | In what probability does cospectra of Cayley graph imply isomorphism of the corresponding group | For your second question, if by *probability* you mean $$\lim\_{n \to \infty} \frac{|S\_n|}{|G\_n|},$$ where $S\_n$ is the set of all possible spectra of simple $n$-vertex graphs, and $G\_n$ is the set of isomorphism classes of simple $n$-vertex graphs, then it is conjectured that the above probability is 1. That is, a... | 2 | https://mathoverflow.net/users/2233 | 101049 | 58,678 |
https://mathoverflow.net/questions/101008 | 0 | Suppose that we have a sequence of functions $u\_j$ that are in $L^{\infty}(0,1)$. Then the sequence of maps $N\_j(s) := \|u\_j(s)\|^2$ are also in $L^{\infty}(0,1)$. Hence they give rise to distributions and therefore has a distributional derivative. What is the explicit formula for $DN\_j$? Is it related to the class... | https://mathoverflow.net/users/24644 | Calculating a distributional derivative | First, I do not understand why do you need a sequence of functions when the question involves an individual function. Suppose that $u$ is real valued. Then the product of the distributions $u$ and $u'$ may not even be defined. (This is the case when $u$ is the Heaviside function.) However, if the distributional derivat... | 3 | https://mathoverflow.net/users/20302 | 101050 | 58,679 |
https://mathoverflow.net/questions/101023 | 8 | Let $k$ be a field and $n$ a nonnegative integer. For any matrix $U\in\mathrm{M}\_n\left(k\right)$, let $\mathrm{ad} U$ denote the map $\mathrm{M}\_n\left(k\right)\to \mathrm{M}\_n\left(k\right),\ V\mapsto UV-VU$. Thus, $\mathrm{ad} U$ is an element of the $k$-algebra $\mathrm{End}\_k\left(\mathrm{M}\_n\left(k\right)\r... | https://mathoverflow.net/users/2530 | ad (A^n) is a polynomial in ad A ? | I don't think it's true even when $A$ is diagonalisable.
Suppose that $A$ is a diagonal matrix, with $(i,i)$th entry $\lambda\_i$; thus $A = \rm{diag}$$(\lambda\_i)$. Then we can write $\rm{ad}(A) = \rm{diag}$$(\lambda\_i - \lambda\_j)$, so
$P(\rm{ad}$$(A)) = \rm{diag}$$ (P (\lambda\_i - \lambda\_j) )$
for any p... | 7 | https://mathoverflow.net/users/6827 | 101055 | 58,682 |
https://mathoverflow.net/questions/101011 | 8 | Let $E$ be a spectrum acted upon a finite group $G$. Is there a general way of computing the homology of the homotopy fixed point spectrum $E^{hG}$ in terms of that of $E$? (I'm aware that there is a spectral sequence for computing $\pi\_\* E^{hG}$ in terms of $\pi\_\* E$, but smashing with some other spectrum probably... | https://mathoverflow.net/users/344 | Homology of homotopy fixed point spectra | Consider $G$ of order $2$ acting trivially on the sphere spectrum $S^0$. In this key example, smashing with $H\mathbb Z/2$ drastically fails to commute with $^{hG}$: If it did commute, then the mod $2$ homology of the homotopy fixed point spectrum would be the homotopy of $(H\mathbb Z/2)^{hG}$, so $\mathbb Z/2$ in nonp... | 10 | https://mathoverflow.net/users/6666 | 101059 | 58,685 |
https://mathoverflow.net/questions/101063 | 0 | one can show that the relation between first Chern class and second Chern class of $CP^n$ is
$\frac{2(n+1)}{n} c\_2 (M)=c\_1 (M)^2$
here $c\_1 (M)^2=c\_1 (M)∧c\_1 (M)$.
So is there any recurrence formula for *i*-th Chern class of $CP^n$ ?
| https://mathoverflow.net/users/nan | recurrence formula for *i*-th Chern class of $CP^n$ | There is no need for a recurrence formula. If $h\in \mathrm H^2(\mathbb C\mathbb P^n, \mathbb Z)$ is the Poincaré dual of a hyperplane section, then $h^i$ generates $\mathrm H^{2i}(\mathbb C\mathbb P^n, \mathbb Z)$ for all $i = 0, \dots, n$. Then it follows from the Euler sequence that
$$
\mathrm c\_i(\mathbb C\mathbb ... | 4 | https://mathoverflow.net/users/4790 | 101064 | 58,688 |
https://mathoverflow.net/questions/101043 | 59 | I am interested in seeing if and how Morse Theory can "do everything". Some core things are handle decomposition, Bott periodicity, and Euler characteristic. But what do the normal (co)homology operations look like from Morse Homology?
**Poincare duality** $H\_\*(M)\cong H^{n-\ast}(M)$ is the symmetry $f\to -f$, i.e.... | https://mathoverflow.net/users/12310 | Operations via Morse Theory | Massey products are discussed in Section 1.3 of
>
> Fukaya, Kenji. Morse homotopy,
> $A\_{\infty}$-category, and Floer
> homologies. Proceedings of GARC
> Workshop on Geometry and Topology '93
> (Seoul, 1993), 1--102,
>
>
>
available [here](http://www.math.kyoto-u.ac.jp/~fukaya/mfikki.pdf) (pdf). The Mass... | 22 | https://mathoverflow.net/users/424 | 101072 | 58,694 |
https://mathoverflow.net/questions/101067 | 9 | Let $G$ be a reductive group over a local non-archimedean field $F$.
Can every irreducible supercuspidal representation of $G(F)$ be realized as the induction from an open subgroup, which is compact modulo the center?
| https://mathoverflow.net/users/10400 | Are all irreducible supercuspidal representation induced from compact-mod-center subgroups? | It is known for GL(N) and SL(N) (Bushnell and Kutzko), for classical groups when the residue characteristic is not $2$ and when no quaternionic algebra is involved (Stevens), for GL(N) of a division algebra (Stevens and Sécherre), for a general reductive group when the residue characteristic of $F$ is large enough (Kim... | 11 | https://mathoverflow.net/users/4767 | 101089 | 58,709 |
https://mathoverflow.net/questions/101105 | 0 | We know by some facts from Kobayashi, if the Kahler manifold $M$ has positive first Chern class, i.e., $c\_1 (M)>0$ then $M$ is simply connected. So if $c\_1 (M)<0$ under which assumption on $M$ , we have $π\_1(M)={e}$.
| https://mathoverflow.net/users/nan | sign of the First chern class fundamental group of Kahler Manifolds | This is really more of an extended comment, since I'm not sure how to give a definite answer.
When $\dim M=1$, $c\_1(M)>0$ if and only if $M= \mathbb{C}\mathbb{P}^1$ if and only if $M$ is simply connected, by the uniformization theorem. In the dimension $2$, things are more complicated. Certainly simply connected with ... | 2 | https://mathoverflow.net/users/4144 | 101107 | 58,718 |
https://mathoverflow.net/questions/101097 | 3 | The question of [existence of a universal inverse semigroup](https://math.stackexchange.com/q/104893/12490) of an arbitrary semigroup has been answered before (this is a construction similar to the Grothendieck group). Let's refer to the universal inverse semigroup of a semigroup $S$ as $G\_I[S]$ (for this question). I... | https://mathoverflow.net/users/20781 | Is the universal inverse semigroup of a commutative semigroup an embedding? | B. Schein described all semigroups embeddable into inverse semigroups in Schein, Boris M., Subsemigroups of inverse semigroups, Le Matematiche LI (1996), Supplemento, 205–227 (in fact the paper was written in the 50s). From that paper it easily follows that not every commutative semigroup embeds into an inverse semigro... | 6 | https://mathoverflow.net/users/nan | 101114 | 58,724 |
https://mathoverflow.net/questions/101120 | 5 | <http://www.springerlink.com/content/04x54gr171v556m4/fulltext.pdf>
On page 149 (DeRa-7), in the middle of the page, I can translate the middle paragraph that starts "3. La surface de Riemann ..." as follows:
3.The Riemann surface $X/\Gamma$ is not compact. Geometrically, this fact is reflected as follows: If $E\_\... | https://mathoverflow.net/users/15242 | Some help in digesting a paragraph in the introduction of Deligne/Rapoport's "Les Schemas de Modules de Courbes Elliptique" | 1. "Il arrive que..." means "sometimes". So the paragraph says that sometimes the minimal model of $E\_\eta$ over $\mathbf{C}[[t]]$ has bad reduction, which is true.
2. You're starting with an elliptic curve over $\mathbf{C}((t))$, not over $\mathbf{C}$. There's no reason that $E\_\eta$ admits a level $n$ structure ove... | 13 | https://mathoverflow.net/users/88 | 101124 | 58,732 |
https://mathoverflow.net/questions/101130 | 2 | I have a question that might be answered with a pointer to some references or with some discussion. I did some searching, to no avail, but I realized that I might not have the vocabulary to form a successful search.
Suppose we have a random (or pseudorandom) vector $r\in\mathbb{R}^{n}$ with $\left\|r\right\|=1$, e.g.... | https://mathoverflow.net/users/17416 | On Random Vectors and Eigenvectors of Symmetric Matrices | As pointed out in the comments, the matrix has nothing to do with the question, and you are simply trying to compute the distribution of $x \cdot v$ as $x$ varies over the unit sphere, and $v$ is a fixed unit vector. By rotational invariance, you might as well assume that $v = (1, 0, \dots, 0),$ at which point the ques... | 3 | https://mathoverflow.net/users/11142 | 101150 | 58,742 |
https://mathoverflow.net/questions/101129 | 10 | [W. V. D. Hodge](http://en.wikipedia.org/wiki/W._V._D._Hodge) is famous for his Hodge conjecture, one of the Millennium prize problems. Hodge might have had some rough heuristics or ideas that led him to the formulation of the conjecture.
I am looking for the history and background behind the formulation of Hodge Co... | https://mathoverflow.net/users/24713 | Heuristics for the Hodge Conjecture | The best answer I can imagine for a question like this is to quote the man himself:
"The second result of Lefschetz tells us that a necessary and sufficient condition
that a 2-cycle $\Gamma\_2$ in $V\_2$ be algebraic... This result has many geometrical applications...
It is clearly a matter of great importance to exten... | 22 | https://mathoverflow.net/users/4144 | 101151 | 58,743 |
https://mathoverflow.net/questions/101160 | 7 | In Peano Arithmetic, the induction axiom states that there is no proper subset of the natural numbers that contains 0 and is closed under the successor function. This is intended to rule out the possibility of extra natural numbers beyond the familiar ones. It doesn't accomplish that goal; there remains the possibility... | https://mathoverflow.net/users/24547 | Axiom to exclude nonstandard natural numbers | As long as you axiomatize set theory in first-order logic, the answer to your question is no. The axioms would be consistent with each finite subset of the following set of sentences involving a new constant symbol $c$: "$c$ is a natural number" and "$c\neq n$" for each (standard name of a) natural number $n$. By compa... | 22 | https://mathoverflow.net/users/6794 | 101165 | 58,751 |
https://mathoverflow.net/questions/99900 | 5 | I'm investigating the eigenvalue ratios
$$
\frac{\lambda\_1}{\sum\_{j=2}^N\lambda\_j}
\quad\mbox{and}\quad
\frac{\sum\_{j=1}^N\lambda\_j}{\sum\_{j=2}^N\lambda\_j}
$$
of the NxN matrix $B=AA^T$. $\lambda\_1$ denotes the largest eigenvalue. The ratios can be thought of as a measure of "rank-1-ness" of $B$.
I haven'... | https://mathoverflow.net/users/24536 | Has the largest-to-rest eigenvalue ratio of real symmetric matrices been researched before? | I am not sure if you are allowed to change your objective function, but a natural alternative for measuring "rank-1-ness" is
$$
\frac{\lambda\_1^2}{\lambda\_1^2+\lambda\_2^2+\dots+\lambda\_n^2}.
$$
This ratio is easy to compute: the denominator is the squared Frobenius norm of the matrix (i.e., sum of squares of all e... | 3 | https://mathoverflow.net/users/1898 | 101166 | 58,752 |
https://mathoverflow.net/questions/101163 | 1 | Evidently Legendre showed that, for positive primes, if $p \equiv 3 \pmod 8$ there is an integral solution to $x^2 - p y^2 = -2.$ Next, if $q \equiv 7 \pmod 8$ there is an integral solution to $x^2 -q y^2 = 2.$
What I would like, and seems to be true, is $x^2 - 2 p y^2 = -2$ for $p \equiv 3 \pmod 8,$ and $x^2 - 2 q y... | https://mathoverflow.net/users/3324 | probably Lagrange or Legendre, Pell variant | According to Dickson (History of numbers Vol. 2, Ch. XII, p.376), Göpel (Jour. für Math. 45, 1853, 1-14) proved your conjectures "by use of continued fractions".
Actually Jour. für Math. stands for Crelle's journal, and Göpel's paper (which is his 1835 doctoral dissertation) is available online [here](http://www.digi... | 2 | https://mathoverflow.net/users/11919 | 101181 | 58,762 |
https://mathoverflow.net/questions/101176 | 19 | Many introductory texts on algebraic geometry set up some sort of algebra-geometry dictionary in which radical ideals correspond to varieties, and so on. I am wondering if there is a geometric way to think about a subalgebra in a polynomial ring? For instance for invariant rings one may develop some intuition, but what... | https://mathoverflow.net/users/5495 | What is the geometric object corresponding to a subalgebra in a polynomial ring | Because geometry and algebra are connected by the contavariant $\textrm{Spec}$ functor, a subalgebra corresponds to the image of a dominant morphism. So subalgebras of $k[x\_1,..,x\_n]$ are affine schemes with dominant maps from $\mathbb A^n\_k$.
Sometimes, like in your example, this is in addition a surjective morph... | 23 | https://mathoverflow.net/users/18060 | 101184 | 58,763 |
https://mathoverflow.net/questions/101147 | 2 | Suppose we have a set of regular functions defined on a product of metric spaces, for instance the Banach space of the smooth functions from $\mathbb R^n$ to $\mathbb C$. We know, thanks to the Taylor series, that such a function can 'almost' be written as a linear combination of products $f\_1 \cdot \ldots \cdot f\_n$... | https://mathoverflow.net/users/20228 | How (and when) to factor a function defined on a product of metric spaces? | Is this the statement you want: any locally constant, compactly supported function from ${\bf Q}\_p^n$ to ${\bf C}$ is uniformly approximated by linear combinations of products $f\_1\cdots f\_n$ where each $f\_i$ is a locally constant, compactly supported function from ${\bf Q}\_p$ to ${\bf C}$? Yes, this follows from ... | 3 | https://mathoverflow.net/users/23141 | 101188 | 58,767 |
https://mathoverflow.net/questions/101157 | 3 | Let $G$ be an algebraic group defined over a char 0
local field $k$. Following Borel and Tits (73) we define
the group $G^+(k)$ or $G^+$ by the subgroup of $G(k)$
generated by the unipotent elements of $G(k)$.
Suppose $G$ is generated by a finite set of unipotent
$k$-subgroups, say $U\_1,\cdots, U\_n$. Is it true t... | https://mathoverflow.net/users/11056 | The group G^+ of algebraic groups over local fields | I haven't really checked the details, but I guess an argument works as follows. First, there exist $(k\_i)$ such that the product map $\prod U\_{k\_i}\to G$ is onto. I think this implies that its differential is surjective on a Zariski open set. Therefore it is onto at some point in $\prod U\_{k\_i}(K)$; it follows (by... | 2 | https://mathoverflow.net/users/14094 | 101190 | 58,769 |
https://mathoverflow.net/questions/101158 | 4 | I would like to compute the homology of certain low dimensional CW complexes and I am hoping to take advantage of software that handles simplicial sets as input. Thus, I would like to convert a CW complex into a homologically equivalent simplicial set.
In the case where the CW complex is regular, everything is easy. ... | https://mathoverflow.net/users/18263 | Constructing a simplicial set homology-equivalent to a given CW complex | For a 2-dimensional complex, I think the following works, but I may not have checked carefully enough. Given a finite 2-dimensional CW complex, I'll describe a finite simplicial set with the same homology. (Note: this procedure uses the full attaching maps, not just their degrees.)
First, the 1 skeleton is a graph, m... | 2 | https://mathoverflow.net/users/4042 | 101202 | 58,776 |
https://mathoverflow.net/questions/101204 | 1 | Let $\mathcal{C}^0([a,b],\mathbb{R})$ be the space of all continuous functions $f:[a,b]\rightarrow\mathbb{R}$ and $\mathcal{C}^\infty([a,b],\mathbb{R})$ the subspace of all smooth functions. Define $f\leq g:\Leftrightarrow(\forall x: f(x)\leq g(x))$ and $f\ll g:\Leftrightarrow(\forall x: f(x)< g(x))$. Is the following ... | https://mathoverflow.net/users/11317 | Order density of smooth functions among continuous functions? | Yes, since smooth functions are dense in $C^0([a,b])$ with the supremum norm. Since $[a,b]$ is compact, $\inf (h-f)$ is acheived somewhere in $[a,b]$ so must be positive. Thus any smooth function $g$ sufficiently near $\frac{f+h}{2}$ will suffice.
Indeed $(C^r([a,b],\mathbb R),\leq)$ is not a lattice for $r\in\mathbb... | 1 | https://mathoverflow.net/users/13832 | 101205 | 58,778 |
https://mathoverflow.net/questions/101159 | 5 | [Srinivasa Ramanujan](http://en.wikipedia.org/wiki/Srinivasa_Ramanujan) in his [first letter](http://books.google.co.in/books?id=Of5G0r6DQiEC&lpg=PP1&pg=PA21#v=onepage&q&f=false) to [G.H. Hardy](http://en.wikipedia.org/wiki/G.H._Hardy) stated many results for which he didn't give proofs. Among them the result taken fro... | https://mathoverflow.net/users/1483 | Request for the proof of a result from Ramanujan's letter to Hardy. | Thanks "@Charles Matthews". I did email Prof. Berndt (after seeing your comment) and he suggested me to look at this paper:
* *[Some definite integrals connected with Gauss's sums](http://books.google.co.in/books?id=oSioAM4wORMC&lpg=PA357&dq=Ramanujan%20collected%20papers&pg=PA59#v=onepage&q=Ramanujan%20collected%20p... | 10 | https://mathoverflow.net/users/1483 | 101213 | 58,782 |
https://mathoverflow.net/questions/100692 | 2 | Let $X$ and $Y$ be locally convex, Hausdorff topological vector spaces and let $[a,b] \subset \mathbb{R}$. Let $f: [a,b] \to \hom(X,Y)$ be continuous, where $\hom(X,Y)$ is the space of continuous linear maps from $X$ to $Y$ with the topology of uniform convergence on bounded subsets of $X$. Let $H \subset \hom(X,Y)$ be... | https://mathoverflow.net/users/22470 | Equicontinuity of continuous families of maps between topological vector spaces | Assume that there is a sequence $T\_n\in hom(X,Y)$ which converges to $0$ uniformly on all
bounded subsets of $X$ but is not equicontinuous (such a sequence should exist if $X$ fails to be $c\_0$-*quasibarrelled*, see chapter 8.2 of the book *Barrelled Locally Convex Spaces* of J. Bonet and P. Perez Carreras). I believ... | 1 | https://mathoverflow.net/users/21051 | 101221 | 58,785 |
https://mathoverflow.net/questions/101098 | 27 | I have often seen a relationship being alluded to between these two theories but I am unable to find any literature which proves/derives/explains this relationship.
I guess in the condensed matter physics literature this is the "same" thing which is referred to when they say that one has propagating chiral bosons on... | https://mathoverflow.net/users/2678 | The Chern-Simons/Wess-Zumino-Witten correspondence | Quantum field theories are understood/formalized at various levels of detail (e.g. action functional only, space of states/partition function only, full functorial QFT, full extended QFT). Accordingly there are such different levels at which people will say "It is well-known that...".
For the general [holographic pri... | 12 | https://mathoverflow.net/users/381 | 101223 | 58,787 |
https://mathoverflow.net/questions/101217 | 6 | Let $M$ denote a complex manifold of dimension $n$ and let $X\subset M$ denote an analytic hypersurface. Then it is a standard fact from several complex variables that around a given point $p\in X$ there are open subsets $V\subset X, W\subset \mathbb{C}^{n-1}$ and and a finite-sheeted covering $\pi: V\rightarrow W$ bra... | https://mathoverflow.net/users/24525 | Fundamental group of an analytic hypersurface | It should be said that van Kampen's paper "On the connection between the fundamental groups of some
related spaces". *Amer. J. Math.* 55 (1933) 261--267, gives a formula for the case of a union of two spaces with non-connected intersection, and this was needed for his work on algebraic curves: "On the Fundamental Grou... | 3 | https://mathoverflow.net/users/19949 | 101227 | 58,788 |
https://mathoverflow.net/questions/101228 | 2 | I computed the CW structure on U(n) using morse theory.I want to verify my answer.So I was wondering if someone here can supply the answer as I can't find any source by googling.Also I want to know if there is any reference for this.
| https://mathoverflow.net/users/21488 | CW structure on Unitary Group | A simply-connected compact Lie group $G$ has the same rational homotopy type (and rational cohomology ring) of as a product of odd-dimensional spheres $S^{2m\_1+1}\times\cdots\times S^{2m\_r+1}$ where the $m\_i$ are
invariants called *exponents* and $r$ is the rank of $G$. (The exponents are related to
many algebraic... | 2 | https://mathoverflow.net/users/15155 | 101232 | 58,790 |
https://mathoverflow.net/questions/101230 | 8 | Given two randomly chosen positive rational integers, the probability that the two numbers are coprime is $\frac{6}{\pi^2}$. This is also the probability that a positive integer is squarefree. Are there generalizations of these results for Gaussian integers? Or more generally for the ring of integers in an algebraic nu... | https://mathoverflow.net/users/24864 | Probability in the Primes | There are generalizations, see [this mathworld article](http://mathworld.wolfram.com/RelativelyPrime.html) for some results and references. A detailed exposition for arbitrary number fields is given [in this paper by G. Collins and J. Johnson](https://dl.dropbox.com/u/5188175/collinsjohnson.pdf)
| 11 | https://mathoverflow.net/users/11142 | 101233 | 58,791 |
https://mathoverflow.net/questions/101237 | 1 | It is known that Einstein-scalar Lichnerowicz equation
$\Delta\_gu-\frac{4(n-1)}{n-2}\Big(R\_g-|\nabla\psi|\_g^2\Big)u-\frac{4(n-1)}{n-2}\Big(Bu^{\frac{n+2}{n-2}}-Au^{-\frac{3n-2}{n-2}}\Big)=0.$
where $ R\_g $ is scalar curvature and $\psi$ is scalar field.
It stems from the of Einstein constraint equations in g... | https://mathoverflow.net/users/11859 | Geometric interpretation for Einstein-scalar Lichnerowicz equation | The constraint equations are simply the expression for relationships that the metric and second fundamental form of a hypersurface inside a (Lorentzian) Einstein manifold must satisfy, according to the Gauss and Codazzi equations. So they are highly geometric (at least in the case of the vacuum Einstein constraints -- ... | 2 | https://mathoverflow.net/users/17969 | 101243 | 58,793 |
https://mathoverflow.net/questions/101240 | 4 | Assume I have polytope in R^k given by N (k<< N) linear inequalities (A\_i x < b\_i).
I guess complexity of its volume calculate is higher than linear in "N", am I right ?
(Is the complexity known ? )
Example: k =120, N=2^24, so probably the only method for practical calculation is Monte-Carlo, am I right ?
Actual... | https://mathoverflow.net/users/10446 | Complexity of convex polytope volume calculation ? (Volume of Voronoi cell) (Error probability) | See [this very nice paper of Bringmann and Fried.](http://www.mpi-inf.mpg.de/~tfried/paper/CGTA1.pdf)
| 3 | https://mathoverflow.net/users/11142 | 101245 | 58,795 |
https://mathoverflow.net/questions/101246 | 16 | Consider $m$ random 0-1 vectors of length $n$. Let $L$ be the lattice spanned by them. What is the value of $m$ (as a function of $n$) for which it is true with positive probability that $L=Z^n$? More generally, let $V(L)$ be the rank of $Z^n/L$ (The volume of $L$). What is the behavior of $V(L)$ as a function of $n$ a... | https://mathoverflow.net/users/1532 | The latice spanned by $m$ random 0-1 vectors of length $n$ | I believe (but haven't fully checked) that you can get an upper bound of $m=cn \log^2 n$ using the second moment method. I'm including a sketched argument below.
I will assume WLOG that $m$ is even. I will also (for now) make a parity assumption: I will assume that, modulo $2$, the sum of all $m$ vectors is equal to... | 6 | https://mathoverflow.net/users/405 | 101261 | 58,800 |
https://mathoverflow.net/questions/101249 | 4 | Let $E$ be a finite semigroup. According to N. Bourbaki (Algèbre I p. 121 exerc. 14 c), if $M$ and $M'$ are minimal right ideals in $E$, then they are isomorphic. I spent some time browsing through Clifford and Preston's monography "the algebraic theory of semigroups", also reading a few articles by Rees, Clifford and ... | https://mathoverflow.net/users/24869 | Minimal right ideals in finite semigroup | This is in Clifford-Preston. First every minimal left (right) ideal is inside the minimal two-sided ideal $I$ (which is unique), see Exercise 13 on page 84. Second, the ideal $I$ is a simple semigroup (obvious, but is also in C-P). Third, by Sushkevich's theorem (Appendix A), all maximal subgroups of $I$ are isomorphic... | 2 | https://mathoverflow.net/users/nan | 101268 | 58,803 |
https://mathoverflow.net/questions/101256 | 1 | I am currently in the midst of a summer research project and have run across an interesting summation: $F(n) = \sum\limits\_{i=1}^{\lfloor\frac{n}{2}\rfloor}(n - 2i + 1)P\_2(i)$.
And here are some necessary definitions:
* $P\_2(i)$ is the number of primitive words over the binary alphabet, and I have from one of my... | https://mathoverflow.net/users/24871 | Finding a Big Theta Bound for a Summation Involving the Möbius Inversion Formula | If a word is not primitive, it is a power of a word of length at most half as long. There aren't many of those compared with all words of full length. So, $P\_2(i)/2^i \to 1$ as $i \to \infty$ and you only need that the limit infinum is not $0$.
| 1 | https://mathoverflow.net/users/2954 | 101272 | 58,805 |
https://mathoverflow.net/questions/101276 | 3 | Hello everyone,
I am currently studying set theory on my own on the book *Set Theory, an Introduction to Large Cardinals* by Frank R. Drake and I have a couple of serious doubts.
Drake first introduces the usual (not formal) definition (due to Tarski) of *satisfaction* on a given collection A and then the usual d... | https://mathoverflow.net/users/24883 | Some questions about *Set Theory* by Frank R. Drake | For your main question, you are right that there is a subtle issue
with the claim that $$V\models\phi\iff L\models\phi\text{ for
}\Delta\_1^{ZF}\text{ assertions }\phi.\qquad\qquad (\star)$$ But
your interpretation is not as strong as one can give here.
First, supporting your worries, let's point out that we cannot e... | 12 | https://mathoverflow.net/users/1946 | 101277 | 58,808 |
https://mathoverflow.net/questions/101270 | 9 | If $\Gamma\subseteq SL(n,\mathbb{R})$ is a lattice (i.e. discrete and finite covolume), does $\Gamma$ necessarily contain some $\mathbb{R}$-diagonalizable copy of $\mathbb{Z}^{n-1}$?
I know that the answer is yes if the lattice is cocompact, and that the answer is also yes in the case $\Gamma=SL(n,\mathbb Z)$. So I w... | https://mathoverflow.net/users/24880 | Lattices in $SL(n,\mathbb R)$ | The answer is yes. It is theorem [2.13] of the following paper of Prasad and Raghunathan:
Prasad, Gopal; Raghunathan, M. S. Cartan subgroups and lattices in semi-simple groups. Ann. of Math. (2) 96 (1972), 296–317.
There is also a lot of information in this paper:
<http://www.math.bgu.ac.il/~barakw/papers/clorbit.p... | 13 | https://mathoverflow.net/users/16143 | 101278 | 58,809 |
https://mathoverflow.net/questions/101274 | 7 | I would really like to know whether the following famous conjecture has been solved. I've read in a few places that it has been solved, but I have been unable to find a reference. I do know that there was once a very credible proof which was believed for a while and then turned out to be false. The statement is:
>
... | https://mathoverflow.net/users/35353 | Are coefficients of Maass forms of eigenvalue 1/4 known to be algebraic? | Your boxed statement is an open problem. Blasius and Ramakrishnan did not rely on a widely believed statement which turned out to be false. Their argument accidentally conflated two L-packets for $GSp\_4(\mathbb{R})$ which are in fact distinct, due to a miscalculation of the central character of one of the L-packets in... | 9 | https://mathoverflow.net/users/1464 | 101285 | 58,813 |
https://mathoverflow.net/questions/101168 | 3 | What is the dual of a norm that is the sum of two-norms? Specifically, say we have the following norm for $\mathbf{x}\in \mathbb{R}^n$ and $\mathbf{A}\_i \in \mathbb{R}^{m \times n}$
$\|\mathbf{x}\| = \displaystyle{ \sum\_{i=0}^{k} \|\mathbf{A}\_i \cdot \mathbf{x} \|\_2}$.
How would you then find
$\|\mathbf{y}\|\... | https://mathoverflow.net/users/24707 | Dual Norm For Sum of 2-Norms | Here is an answer. This is certainly the right answer theoretically (and almost a tautology, but I do not think much more can be said in general). I am really not sure it will be helpful numerically, sorry.
So the dual norm is given by
$$ \|y\|\_\* = \inf \{\max\_{i=1}^k \|y\_i\|\_2, y\_i \in \mathbf R^m, \sum\_i A\... | 2 | https://mathoverflow.net/users/10265 | 101291 | 58,818 |
https://mathoverflow.net/questions/101293 | 1 | The optimization problem is:
maximize $$\min(\sum\limits\_{i=1}^N \log\left(a\_{1,i}+\frac{b\_{1,i}}{c\_{1,i}+d\_{1,i}x\_i}\right),\sum\limits\_{i=1}^N \log\left(a\_{2,i}+\frac{b\_{2,i}}{c\_{2,i}+d\_{2,i}x\_i}\right))$$
subject to $x\_i\ge0, i=1,...,N$ and $\sum\limits\_{i=1}^N x\_i=C$, where $x\_i$ are variables, $a... | https://mathoverflow.net/users/5072 | what method can I employ to solve this optimization problem which involves \min? | A common transformation when faced with a problem of this type:
$${\rm maximize} \min (f(x), g(x))$$
is to instead solve the equivalent problem
$${\rm maximize} \ \ \ z $$
subject to
$$ z\le f(x); z\le g(x).$$
This can be helpful, for instance, in making the problem more tractable for some numerical optimizatio... | 4 | https://mathoverflow.net/users/20507 | 101295 | 58,819 |
https://mathoverflow.net/questions/101296 | 0 | Sorry if the following are stupid questions (i do not know much about the graph theory).
**1. Motivation**
we do not know the graph isomorphism problem in class P or NP complete and it is P in the case trees (see <http://en.wikipedia.org/wiki/Graph_isomorphism_problem>).
**2. Transformation a graph to tree**
... | https://mathoverflow.net/users/17901 | graph to tree and graph isomorphism problem | If two cycles of minimum length have a common edge, then it matters which is chosen. Try two triangles with a common edge, plus one more vertex joined to an apex of one of the triangles.
| 6 | https://mathoverflow.net/users/9025 | 101298 | 58,820 |
https://mathoverflow.net/questions/101306 | 1 | CR refers to *Methods of Representation Theory* by Charles Curtis and Irving Reiner.
Let $F$ be a finite extension of $\mathbb{Q}\_p$ with valuation ring $\mathcal{O}\_F$.
Let $G$ be a finite group and let $A$ be the group algebra $F[G]$.
Let $\Lambda$ be an $\mathcal{O}\_F$-order in $A$.
Let $K\_0(\Lambda)$ (resp. $... | https://mathoverflow.net/users/7443 | Triviality of SK_0(Lambda) for Lambda an order in a group algebra over a $p$-adic field | No, the kernel of the map $\varphi: K\_0(\Lambda)\longrightarrow K\_0(A)$ is not necessarily trivial for all orders $\Lambda$ in $A$. In fact, it is only trivial for all orders in $A$ if $A$ is a direct sum of fields and skew fields.
Assume the Wedderburn decomposition of $A$ has the matrix algebra $M\_n(D)$ as a si... | 2 | https://mathoverflow.net/users/17498 | 101312 | 58,825 |
https://mathoverflow.net/questions/101324 | 0 | Let $M$ be a not necessarily free module over a commutative unital ring. True or false: if every linear form assigns to a vector of $M$ zero, then the vector is the zero vector?
| https://mathoverflow.net/users/16425 | Characterization of zero vectors with linear forms | Counterexample: $R=\mathbb Z$ and $M=\mathbb Z/2$.
| 3 | https://mathoverflow.net/users/35353 | 101329 | 58,831 |
https://mathoverflow.net/questions/101331 | 8 | A colleague is refereeing a paper in which the following
lemma appears implicitly:
For any family $\mathcal G$ of nonempty sets let us call
a set $B$ a "selector" if $B$ meets all $F\in\mathcal G$.
Lemma: For every family $\mathcal G$ of nonempty finite sets there
is a minimal selector $B$. (That is, for all $... | https://mathoverflow.net/users/14915 | Minimal selector for a family of finite sets | I think "selector" usually refers to choosing just one element from each set in a family. For the concept you described here, I've seen names like "blocker" or "blocking set", but I haven't seen them so often that I'd call them standard.
The blockers of a family of finite sets (all included in some big set $X$) obvi... | 7 | https://mathoverflow.net/users/6794 | 101337 | 58,835 |
https://mathoverflow.net/questions/100798 | 12 | I've been reading:
math.stanford.edu/~conrad/249BPage/handouts/geomcft.pdf
in an attempt to shed some geometric light on class field theory. The last paragraph there reads:
*In case the ground field $k$ is perfect, the essential difficulty in the proof of class field theory – proving
that the Artin map kills certai... | https://mathoverflow.net/users/5309 | What makes Geometric CFT easier than CFT? | There is a lot here and it is hard to tell where you are confused. I'll try to fill in some steps along the way. For simplicity, I'll present the whole theory only for unramified extensions. I'll assume that you have gone through CFT for number fields in the ideal theoretic (not adelic) presentation, as in Janusz or Co... | 16 | https://mathoverflow.net/users/297 | 101339 | 58,837 |
https://mathoverflow.net/questions/101321 | 3 | Let $G$ be a non-amenable countable discrete group. How can I show that the group von Neumann algebra $L(G)$ has no injective direct summand?
| https://mathoverflow.net/users/9401 | Injective von Neumann algebra | Here is an adaptation of the standard proof that $G$ is amenable if $LG$ is injective. (I believe for instance that it is contained in the book of Brown and Ozawa).
Suppose $p \in LG$ is a non-zero central projection such that $p LG$ is injective. Thus, there exists a conditional expectation $E: \mathcal B(p \ell^2 G... | 11 | https://mathoverflow.net/users/6460 | 101340 | 58,838 |
https://mathoverflow.net/questions/101335 | 4 | It is well-known that if a reduced algebraic group $G$ acts on a *separated reduced* scheme $X$, and $G$ acts trivially on a dense open subscheme $U\subseteq X$, then the action is trivial.
If $X$ is non-reduced, the standard counterexample is $\def\AA{\mathbb A}$$\AA^1$ with an embedded point, with the group acting ... | https://mathoverflow.net/users/1 | Can a non-trivial action of a connected group on a reduced scheme be trivial on a dense open? | Let me work over an algebraically closed field $k$, let $X$ be a reduced $k$-scheme of finite type and let $G$ be a smooth connected $k$-group scheme acting on $X$, and acting trivially on an open dense subset $U$ of $X$. I will show that $G$ acts trivially on $X$.
I will only manipulate closed points. Since both $G$... | 7 | https://mathoverflow.net/users/2868 | 101341 | 58,839 |
https://mathoverflow.net/questions/101326 | 9 | Topical. I know there are good mathematical theories in which "Higgs" is used, in a geometrical sense. Would someone care to explain?
To clarify, I'd like to know about Higgs bundles on Riemann surfaces, rather than the standard model of high-energy physics.
| https://mathoverflow.net/users/6153 | What is the current state of the mathematics of Higgs fields? | Since you are asking about Higgs bundles, I can say a few words here.
These were introduced by Hitchin in the mid 1980's, although I'm not sure he used this term. One can look at the introduction to his paper "The self-duality equations on Riemann surfaces" for some of the motivation and background. In brief outline, ... | 20 | https://mathoverflow.net/users/4144 | 101347 | 58,842 |
https://mathoverflow.net/questions/101037 | 2 | It is well known that any unipotent algebraic group (over a field) can be embedded as a closed subgroup of $U\_n$ for some $n$, where $U\_n$ denotes the set of all $n \times n$ upper triangular matrices with $1$'s on the diagonal.
An algebraic subgroup $H$ of an algebraic group $G$ is called $\textbf{observable}$ if... | https://mathoverflow.net/users/19048 | Observable Unipotent Algebraic Subgroups of the Unipotent Upper Triangular Groups | To answer your comment, take the representation
$\left(\begin{array}{ccc} 1 & a & b \\ 0 & 1 & a \\ 0 & 0 & 1\end{array}\right) \to \left(\begin{array}{cc} 1 & b-a^2/2 \\ 0 & 1 \end{array}\right)$
If this were a pullback of a representation of the additive group, it would be a pull-back of a two-dimensional unipot... | 1 | https://mathoverflow.net/users/18060 | 101352 | 58,843 |
https://mathoverflow.net/questions/101348 | 7 | In [this question](https://mathoverflow.net/questions/101331) I asked for a reference for the following lemma:
Lemma X: For every family $\mathcal G$ of nonempty finite sets there
is a minimal "blocking set" $B$. By a "blocking set" $B$ I mean a set which meets each $F\in\mathcal G$, and "minimal" refers to the subs... | https://mathoverflow.net/users/14915 | Minimal blocks for a family of finite sets | This is only a partial answer: If you assume (with notation as in the question) that every point is in only finitely many sets from $\mathcal G$, then the existence of a minimal blocking set follows from the Boolean prime ideal theorem (BPI). One way to see this is via the compactness theorem for propositional logic (w... | 6 | https://mathoverflow.net/users/6794 | 101359 | 58,846 |
https://mathoverflow.net/questions/101342 | 4 | Following the standard conventions in the literature, the commutation relations of the Virasoro Lie algebra are given by
$$[L\_m,L\_n]=(m-n)L\_{m+n}+\delta\_{m,-n}\frac1{12}(m^3-m)c,$$
$$[c,L\_n]=0.$$
Similarly, following the standard conventions in the literature, the commutation relations of the Neveu-Schwarz super... | https://mathoverflow.net/users/5690 | Why is there a discrepancy between the normalizations of the central terms for the commutation relations of the Virasoro versus Neveu-Schwarz Lie algebras? | As far as I can tell, a $\sigma$-model with $d$ dimensional target space will have Virasoro central charge $d$ with bosonic strings, and $3d/2$ with supersymmetric strings. I believe the normalizations were chosen so that the constant $c$ reflects the dimension of spacetime in which the strings are propagating (even if... | 2 | https://mathoverflow.net/users/121 | 101365 | 58,848 |
https://mathoverflow.net/questions/101308 | 17 | I'll begin with some background that is unnecessary for the actual question, but that might be interesting to the reader:
*Topological modular forms ($TMF$) is a generalized cohomology theory whose coefficient ring $TMF^\*(pt)$ is closely related to the ring $MF\_\*:=\mathbb Z[c\_4,c\_6,\Delta]/c\_4^3-c\_6^2-1728\Del... | https://mathoverflow.net/users/5690 | Character of parity-twisted supersymmetric VOA module -- question inspired by the Stolz-Teichner program | Such an object is described in Dixon, Ginsparg, Harvey, *Beauty and the Beast: superconformal symmetry in a monster module* Comm. Math. Phys. Volume 119, Number 2 (1988), 221-241. A reasonably explicit construction is given in Huang's paper *[A nonmeromorphic extension of the moonshine module vertex operator algebra](h... | 11 | https://mathoverflow.net/users/121 | 101366 | 58,849 |
https://mathoverflow.net/questions/66895 | 6 | We are given an affine variety $V\subset \mathbb{A}^n\times\mathbb{A}^n$, and wish to know if it contains a product of the form $C\_1\times C\_2$, where $C\_1$ and $C\_2$ are two curves in $\mathbb{A}^n$. First, is there an algorithm to decide this? Second, is it true that if $V$ is of degree $d$ and does contain a pro... | https://mathoverflow.net/users/806 | Does a variety contain a cartesian product of two curves? | Yes, there is such an algorithm. There is an effectively computable constant $N$ such that if $V$ contains a product $S\times T$ where $S,T$ are $N$-point sets, then $V$ contains product of two curves. It is actually true even in the semialgebraic setting. The result is Theorem 1.9 from <http://arxiv.org/abs/1207.0705>... | 5 | https://mathoverflow.net/users/806 | 101384 | 58,860 |
https://mathoverflow.net/questions/101372 | 3 | Does there exist a Turing complete, cellular automata with universe and alphabet $\mathbb{Z}$ such that the only allowable configurations are permutations of $\mathbb{Z}$? Formally, consider $\tau : Sym(\mathbb{Z}) \to Sym(\mathbb{Z})$ satisfying the following property: there exist a finite subset $S \subset \mathbb{Z}... | https://mathoverflow.net/users/22871 | Turing-Complete Cellular Automata and Sym(Z) | Yes, your permutation automata concept can simulate Turing
machines.
You describe an update procedure $\tau$ that operates on a given
permutation of $\mathbb{Z}$, let us imagine the permutation
written out in a line, by local rearrangements: the new value at
position $n$ is determined by the previous values in the si... | 2 | https://mathoverflow.net/users/1946 | 101394 | 58,862 |
https://mathoverflow.net/questions/101393 | 7 | I have thought that Gödel introduced the concept of Primitive recursive functions in his seminal paper "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme" (I hope I got the title right...). This paper, to best of my knowledge, was published in 1931.
Ackermann published the famous Acke... | https://mathoverflow.net/users/nan | Who introduced the concept of Primitive recursive functions? | I believe the explicit use of definitions by primitive recursion goes back to Grassman, 1861.
Dedekind in 1888 not only highlighted such definitions but had a proof that they work as intended, i.e. define unique functions.
But it is probably Skolem who first clearly recognised the primitive recursive functions as... | 18 | https://mathoverflow.net/users/14111 | 101401 | 58,866 |
https://mathoverflow.net/questions/101389 | 2 | Let $X$ be a smooth, complex projective variety, and $G$ a finite abelian group. We want to study $G$-principal bundles over $X$, or, in other words, étale $G$-covers over $X$.
Topologically, these objects are classified (up to isomorphism of $G$-covers over $X$) by $H^1(X, G)$, or equivalently, by homomorphisms $\ph... | https://mathoverflow.net/users/23434 | Principal G-covers with G finite abelian | Note that there is a morphism of etale sheaves
$G \to \mathcal{H}om(G^\vee , \mathcal O\_X^\times)$
given by $g \mapsto (\chi \mapsto (\chi(g)))$, where we consider $\chi(g)\in \mathbb C^\times$ as a constant function.
I claim that this is in fact an isomorphism of sheaves. For cyclic groups, this follows from th... | 3 | https://mathoverflow.net/users/7762 | 101403 | 58,867 |
https://mathoverflow.net/questions/101399 | 2 | Let $X:=\mathbb{C}^n$, and let the symmetric group $S\_n$ act by permutation of coordinates in the obvious way; let $X\_n:=X/S\_n$ be the quotient by the group action. Now, $X\_n\simeq \mathbb{C}^n$, so we get a map $\pi: X\to X\_n$ with finite fibers. Away from the discriminant $\Delta:=\prod\_{i\lt j}(x\_i-x\_j)$, $\... | https://mathoverflow.net/users/1481 | Monodromy of covering map related to symmetric group | For every $k=1,\dots,n-1$, the monodromy group of $q\_k:X\_k\to X\_n$ is isomorphic to the symmetric group $S\_n$; one argument uses the fundamental theorem of Galois theory (or its analogue for topological covering spaces, if you prefer to work topologically).
First of all $q\_1:X\_1\to X\_n$ equals $\pi$, which is a ... | 8 | https://mathoverflow.net/users/13265 | 101406 | 58,869 |
https://mathoverflow.net/questions/101420 | 9 | Mathematical analysis of music started when Pythagoras made his observations about consonant intervals and ratios of string lengths.
**ADDED:**
In the paper **Mathematical Music Theory -- Status Quo 2000**, G. Mazzola, ETH Zürich, Departement GESS, and Universität Zürich, Institut für Informatik, available [here](h... | https://mathoverflow.net/users/22714 | Music: mathematical point of view (revised) | I have to disagree strongly with Gerald Edgar.
The controversial book The Topos of Music by Guerino Mazzola could constitute a very serious attempt to answer your question. This is not a book that would be accessible to a typical musician, or even a typical expert in music theory - it is definitely a mathematics book... | 16 | https://mathoverflow.net/users/10909 | 101425 | 58,878 |
https://mathoverflow.net/questions/101322 | 13 | Is it true that for every reductive algebraic $G$ over ${\mathbb C}$ with a Lie algebra $\mathfrak g$ there is an open neighborhood $U$ of the identity in $G$ and an algebraic function (in a sense of algebraic geometry) $L: U\to \mathfrak g$ which satisfies the following properties of logarithm:
(1) $L$ is $G$-equiva... | https://mathoverflow.net/users/23935 | Cayley Transform for all reductive groups a.k.a an algebraic logarithm | Bardsley and Richardson (Etale slices for algebraic transformation groups in characteristic p.
Proc. London Math. Soc. (3) 51 (1985), no. 2, 295–317) give a construction which seems to do what you want. It gives less than a "Cayley map" as in Borovoi's answer.
Let $G$ be connected and semisimple over a field of char.... | 7 | https://mathoverflow.net/users/4653 | 101427 | 58,880 |
https://mathoverflow.net/questions/101422 | 8 | If $A= \bigoplus\_{i\ge 0}A\_i$ is a graded commutative Noetherian algebra over a field, its Poincaré series is given by $P(t) = \sum\_{i\ge 0} \dim(A\_i)t^i$. Although the definition of $P(t)$ only depends on the graded vector space underlying $A$, the Krull dimension of the ring $A$ can be obtained from the Poincaré ... | https://mathoverflow.net/users/18571 | Which information can be obtained from Poincaré series ? | I don't know if it's possible to obtain further information on the ring structure in
general. However, if $A$ is the mod-p cohomology ring of a finite group, a result
of Benson and Carlson states that if $A$ is Cohen-Macaulay, then $P(t)$ satisfies
the functional equation
$$P(1/t) = (-1)^d P(t)\hspace{90pt}(\ast)$$... | 8 | https://mathoverflow.net/users/10194 | 101438 | 58,885 |
https://mathoverflow.net/questions/101400 | 1 | Suppose $A$ is a $n \times m$ matrix and $B$ is a $m \times n$ matrix. Then it is known that $det(I\_{n}+AB)=det(I\_{m}+BA)$.
Is there an analogous identity of the form $det(P\_{1}+AB)=det(P\_{2}+BA)$, where $P\_{1},P\_{2}$ are positive definite? Or something like it?
| https://mathoverflow.net/users/22051 | determinantal identity sought | Given $P$, $\det(P+AB)$ does not depend only on $BA$. For example, take $n=m=2$,
$P = \pmatrix{2 & 0\cr 0 & 1\cr}$, $A = \pmatrix{1 & t\cr 0 & 1\cr}$, $B = \pmatrix{0 & 1\cr
1 & -t\cr}$. Then $\det(P + AB) = 1 - t$ depends on $t$, but $BA = \pmatrix{0 & 1\cr 1 & 0\cr}$ doesn't depend on $t$. So any $P\_2$ such that $\d... | 6 | https://mathoverflow.net/users/13650 | 101440 | 58,886 |
https://mathoverflow.net/questions/101437 | -2 | Let $f$ and $g$ are binary relations (on some set $\mho$). The function $f\times^{C} g$ is defined by the formula: $(f\times^{C} g) a = g\circ a \circ f^{-1}$ (for every binary relation $a$ on $\mho$.
Suppose $f$ and $g$ are non-empty. Can we restore $f$ and $g$ knowing only the value of $f\times^{C} g$?
| https://mathoverflow.net/users/4086 | Restore multipliers from the product | Let $a\_{u,v} = \{(u,v)\}$. Then $g\circ a\_{u,v}\circ f^{-1} = f^{-1}(u)\times g(v)$. Assuming $g$ is nonempty, pick $v\in \text{dom}(g)$. Then $\text{dom} \left[(f\times^C g)a\_{u,v}\right] = f^{-1}(u)$, so varying $u$ allows you to recover $f$. Similarly picking $u\in\text{range}(f)$ and varying $v$ you can recover ... | 4 | https://mathoverflow.net/users/5963 | 101441 | 58,887 |
https://mathoverflow.net/questions/101077 | 2 | The following question is probably classic in Morse theory, so a reference to an existing result should be sufficient. I don't know much about Morse theory and I am dealing with the following situation. I have a compact manifold $X$ and I have a Morse function $f$ on it with a saddle point at $x\_0$. I don't like the p... | https://mathoverflow.net/users/17965 | Perturbation of Morse function | There is classic result (I think going back to H. Whitney) that states that if $f:M\to \mathbb{R}$ is a Morse function on a compact smooth manifold $M$, then there exist a neighborhood $\mathscr{N}$ of $f$ in $C^\infty(M)$ sucht that if $g\in \mathscr{N}$, then $g$ is equivalent to $f$, i.e., one can obtain $g$ from $f... | 8 | https://mathoverflow.net/users/20302 | 101449 | 58,894 |
https://mathoverflow.net/questions/101386 | 4 | Dear MOs,
I am now considering the following norm:
$$
||f||\_{H}^2 := \iint f(x) H(x,y) f(y) d x d y\:.
$$
where the integral is over the whole space $R^{2d}$ and $H(x,y)$ is some non-negative definite kernel. Denote the space of functions with finite $||\cdot ||\_{H}^2$ norm to be $L\_H^2(R^d)$. If $H(x,y)=\delt... | https://mathoverflow.net/users/36814 | Ask for theory about the weighted L^2(R^d) space. | Formally, we can define an operator $A$ on $L^2({\bf R}^d)$ by setting $$Af(x) = \int H(x,y)f(y) dy.$$ Then $$\langle Af, f\rangle = \int\int f(x)H(x,y)f(y) dxdy = \|f\|^2\_H$$ (I am assuming real scalars, as I think you are). That is, you are just defining $\|f\|\_H$ to be $\|A^{1/2}f\|$. The kernel $H(x,y) = \delta\_... | 3 | https://mathoverflow.net/users/23141 | 101453 | 58,896 |
https://mathoverflow.net/questions/101452 | 35 | Recall that complex $K$-theory is a cohomology theory on topological spaces, which can be described in several equivalent ways:
* Given a finite complex $X$, $K^0(X)$ is the Grothendieck group of vector bundles on $X$. $K^\*$ is even-periodic, and this determines the entire cohomology theory. Using the tensor produc... | https://mathoverflow.net/users/344 | What do loop groups and von Neumann algebras have to do with elliptic cohomology? | David Roberts mentioned in his comments the relationship
**K-theory : spin group
TMF : string groups**
Let me recommend the first 6 pages of my [unfinished article](http://www.staff.science.uu.nl/%7Ehenri105/PDF/TringWP.pdf)
for a uniform construction of $SO(n)$, $Spin(n)$, and $String(n)$,
which suggests the ex... | 22 | https://mathoverflow.net/users/5690 | 101455 | 58,897 |
https://mathoverflow.net/questions/101456 | 1 |
>
> **Possible Duplicate:**
>
> [Examples of algebraic closures of finite index](https://mathoverflow.net/questions/8756/examples-of-algebraic-closures-of-finite-index)
>
>
>
The question is in the title.
I can prove that if such field $F$ exist then the extension $\mathbb{R}/F$ cannot be of degree $2$ (ess... | https://mathoverflow.net/users/18698 | Is there a subfield $F$ of $\mathbb{R}$ such that $\mathbb{R}$ is a finite algebraic extension of $F$. | If the degree of $\mathbf R$ over $F$ were $d$, then the degree of $\mathbf C$ over $F$ would be $2d$, contradicting the [Artin-Schreier theorem](http://www.math.uconn.edu/~kconrad/blurbs/galoistheory/artinschreier.pdf).
| 7 | https://mathoverflow.net/users/1310 | 101457 | 58,898 |
https://mathoverflow.net/questions/101431 | 3 | Let $R$ be a commutative ring and let $M$ and $N$ be two $R$-modules. Suppose that for every $R$-module $P$, the modules $Hom\_R(M,P)$ and $Hom\_R(N,P)$ are isomorphic. Is it true that $M$ and $N$ are isomorphic?
| https://mathoverflow.net/users/24479 | Are there two non-isomorphic modules such that all the Hom-sets are isomorphic? | [K. Bongartz, "A generalization of a theorem of Auslander"](http://blms.oxfordjournals.org/content/21/3/255.full.pdf+html):
>
> Let R be a commutative ring and A an abelian R-linear category such
> that each morphism set in A has finite length as an R-module. Let C be a full
> subcategory of A closed under direc... | 9 | https://mathoverflow.net/users/460 | 101459 | 58,899 |
https://mathoverflow.net/questions/101464 | 9 | Let $p$ be a prime number. The number of monic irreducible polynomial $P\in{\mathbb F}\_p[X]$, in terms of the degree $d$, begins with
$${\rm irr}(1)=p,\qquad{\rm irr}(2)=\frac{p(p-1)}2,\qquad{\rm irr}(3)=\frac{p(p^2-1)}3,\qquad{\rm irr}(4)=\frac{5p^2(p^2-1)}{12}.$$
This seems to be the beginning of a nice sequence of ... | https://mathoverflow.net/users/8799 | The number of irreducible polynomials over ${\mathbb F}_p$ | The product of all monic irreducible polynomials of degree dividing $d$ in $\mathbb F\_q[x]$ is $$x^{q^d}-x,$$
so from Mobius inversion we get the number of irreducible polynomials of degree $d$ is
$$M \_d (q) = \frac{1}{d} \sum \_{k | d} \mu(k) q ^{d/k} $$
This is known as the [necklace polynomial](http://en.wikipedia... | 16 | https://mathoverflow.net/users/2384 | 101465 | 58,900 |
https://mathoverflow.net/questions/101395 | 35 | Many ODE's and PDE's arising in nature have a variational formulation. An example of what I mean is the following. Classical motions are solutions $q(t)$ to Lagrange's equation
$$
\frac{d}{dt}\frac{\partial L(q,\dot q)}{\partial\dot q}=\frac{\partial L(q,\dot q)}{\partial q},
$$
and these are critical points of the *fu... | https://mathoverflow.net/users/12156 | Which differential equations allow for a variational formulation? | Others give useful references that discuss what is known about the answer, but no statement of the answer itself. The relevant algebraic setting is the variational bicomplex, which is discussed in the works of Anderson and others. In this setting, there are two differentials, the horizontal differential $d\_H$ (represe... | 22 | https://mathoverflow.net/users/2622 | 101476 | 58,904 |
https://mathoverflow.net/questions/101472 | 10 | Given two (or more) quadratic forms (on the same vector space) consider the group of matrices that preserve these forms, i.e. $Q\_i=U Q\_i U^T$, $i=1,2..,k$ What is known about such groups? (at least for k=2 and the forms are symmetric and one is of full rank)
The keywords? Where to read?
| https://mathoverflow.net/users/2900 | Groups of matrices that preserve several quadratic forms | Let $V$ be the vector space you begin with. As you probably know, transformations $T \in \operatorname{End}(V)$ that preserve a symmetric form $(-,-)$ of full rank are called orthogonal, and the group of these transformations is denoted $O(V)$ (let me work with transformations instead of matrices).
Now, given any oth... | 11 | https://mathoverflow.net/users/24195 | 101480 | 58,907 |
https://mathoverflow.net/questions/101477 | 4 | When I say "Markov chain" I think of a directed positively weighted (finite) graph, such that the sum of all edges going out of a vertex equals 1. Also I assume that it is aperiodic and irreducible.
If the Markov chain happens to be equivalent to an "undirected Markov chain", i.e. for each directed arc (u,v) with we... | https://mathoverflow.net/users/11541 | Combinatorial descriptions of the stationary distribution of a Markov chain | "Markov Chain Tree Theorem" is what you should look up: it says essentially that the stationary probability of a given state/vertex is given by the sum of the weights of all spanning trees rooted at that vertex.
| 11 | https://mathoverflow.net/users/730 | 101486 | 58,911 |
https://mathoverflow.net/questions/101304 | 2 | Hi,
let $X$ be an algebraic variety over a field $k$. Let $G$ be an algebraic group over $k$ and $P$, $Q$ $G$-torsors over $X$. Assume I have a morphism $\phi:P\to Q$ of $G$ torsors. Let $A\to k$ be a ring extension with square zero kernel $I$, $X\_A$ a variety over $Spec(A)$ such that $X\_A\otimes k\cong X$ and assu... | https://mathoverflow.net/users/24893 | lifting morphism of torsors | Assuming that $G\_A$ is smooth over $A$ and that $X\_A$ is flat over $A$, the "naive" obstruction to lifting $\phi$ is an element in $H^1(X,\mathfrak{g}\_P)$, where $\mathfrak{g}\_P$ is the locally free $\mathcal{O}\_X$-module obtained from the trivial bundle $\mathfrak{g}\otimes\_k \mathcal{O}\_X$ by "twisting" via th... | 8 | https://mathoverflow.net/users/13265 | 101490 | 58,914 |
https://mathoverflow.net/questions/101488 | 5 | This question is motivated by the answers given to my previous [one](https://mathoverflow.net/questions/101464). In combinatorics, the [necklace polynomials](http://en.wikipedia.org/wiki/Necklace_polynomial) are given by
$$M(X,n)=\frac1n\sum\_{d|n}\mu\left(\frac{n}{d}\right)X^d,$$
where $\mu$ is the Möbius function.
... | https://mathoverflow.net/users/8799 | A combinatorial formula involving the necklace polynomial | By plugging in the definition of $M$ and rearranging the terms, your polynomial is
$$\sum\_dx^d\sum\_{m\le n/d}\left\lfloor\frac n{md}\right\rfloor\mu(m).$$
A basic property of Möbius inversion is that
$$\sum\_{m\le x}\left\lfloor\frac xm\right\rfloor\mu(m)=\begin{cases}1,&x\ge1,\\\\0,&x<1.\end{cases}$$
| 4 | https://mathoverflow.net/users/12705 | 101494 | 58,916 |
https://mathoverflow.net/questions/101497 | 3 | This is sort of a two part question:
1) In one construction of $BP$, the Brown-Peterson spectrum, one uses this idempotent map of Quillen's, $g:MU\_{(p)}\to MU\_{(p)}$ and from what I can tell, the image of this map is $BP$. I have not worked through the details of this carefully. I guess the main idea is just using ... | https://mathoverflow.net/users/11546 | Formal group laws arising from localizations of MU | Regarding (1), since the localisation map from $\mathbb{Z}$ to ${\mathbb{Z}}\_{(p)}$ is injective, and $MU\_\ast$ is free over $\mathbb{Z}$, localisation $MU \to MU\_{(p)}$ will be injective, and so the image of the localisation map will again be $MU$.
Regarding $(2)$, localisation does preserve complex orientations ... | 3 | https://mathoverflow.net/users/4649 | 101500 | 58,919 |
https://mathoverflow.net/questions/101501 | 1 | Let $\gamma\_{\varepsilon} \rightharpoonup \gamma$ in $W^{1,\infty}(0,1)$. Then for any fixed $s \in \mathbb (0,1)$ does the limit $\lim\_{\varepsilon \rightarrow 0} \frac{\gamma\_{\varepsilon}(s\varepsilon)}{\varepsilon}$ exist?
I am on the fence as to whether or not it does. I rewrite it as $\lim\_{\varepsilon \rig... | https://mathoverflow.net/users/24644 | Convergence of Difference Quotients | You must have assumed $\gamma(0)=0$. Let $f\_\epsilon\in L^\infty$ be the derivative of $\gamma\_\epsilon$. Then
$$\frac1\epsilon\gamma\_\epsilon(s\epsilon)=\int\_0^sf\_\epsilon(t\epsilon)dt.$$
Your assumption is that $f\_\epsilon$ converges weak-star to some $f\in L^\infty$. Take for instance
$$f\_\epsilon(x)=\sin\fra... | 3 | https://mathoverflow.net/users/8799 | 101502 | 58,920 |
https://mathoverflow.net/questions/101314 | 26 | Donaldson-Thomas invariants are the (virtual) Euler characteristics of moduli spaces of elements of the derived category of coherent sheaves (with some fixed Chern class, satisfying some stability condition, etc.) which bear some relation to "holomorphic Chern Simons theory", whatever that is.
>
>
> >
> > Should... | https://mathoverflow.net/users/4707 | Are Donaldson-Thomas invariants "A-model" or "B-model" ? | I got a (very) short answer to this question from Nikita Nekrasov who emailed me with "[DT theory is the] B model per se. The GW/DT correspondence is the duality between the A model and the B model (S-duality)".
I had not been aware that the GW/DT correspondence is an instance of S-duality. Recall that S-duality occ... | 21 | https://mathoverflow.net/users/9617 | 101504 | 58,921 |
https://mathoverflow.net/questions/101492 | 8 | Let $k$ be an algebraic number field. I understand that given a finite set of non-complex places $S\subset V(k)$ of even cardinality, there exists a unique quaternion algebra $Q$ over $k$ such that $Q$ ramifies at all $v\in S$ (i.e. $Q \otimes\_k k\_v$ is a division algebra) and $Q$ splits at all the other places. Is t... | https://mathoverflow.net/users/24932 | Explicit description of a quaternion algebra with a prescribed set of ramified places | To have zero-divisors, the norm of an an element must be $0$. The norm of an element is a three-variable quadratic form. You need a quadratic form that has zeroes at the split primes but not at the non-split primes.
Wikipedia provides an [explicit description](http://en.wikipedia.org/wiki/Quaternion_algebra) of all q... | 5 | https://mathoverflow.net/users/18060 | 101512 | 58,924 |
https://mathoverflow.net/questions/101506 | 1 | Let $p$ and $q$ be prime divisors of finite group $G$. Also let $n\_{p}$ be
the number of Sylow $p$-subgroups of $G$ . Is there any example such that
$n\_{p}=n\_{q}\neq 1$? Thanks in advance.
| https://mathoverflow.net/users/24936 | Question on the equal Sylow number | Unless I mis-computed, this happens in the group of affine transformations ($x \mapsto ax+b$ with $a\neq 0$) over the field of 7 elements. There seem to be 7 2-Sylow subgroups and 7 3-Sylow subgroups.
| 4 | https://mathoverflow.net/users/6794 | 101513 | 58,925 |
https://mathoverflow.net/questions/101507 | 2 | Suppose $u$ is a harmonic function of a domain $\Omega\subset \mathbb{R}^n$ and $u$ is continuous up to the boundary. If $\partial\Omega$ has an open smooth portion, can $u$ be extended to a harmonic function outside this smooth portion?
I have a very vague claim that if this portion is analytic, then we can extend $... | https://mathoverflow.net/users/22815 | Extension of harmonic function | Certainly not: think of $\Omega$ the unit disk, and $u$ the harmonic extension to $\Omega$ of any continuous, nowhere differentiable function on $\partial \Omega$.
| 8 | https://mathoverflow.net/users/6101 | 101515 | 58,927 |
https://mathoverflow.net/questions/101390 | 1 | Let $G$ be a discrete Abelian group and denote by $\widehat G$ the (compact) Pontryagin-Van Kampen dual of $G$. I was reading in a paper of Justin Peters that Fourier Transform induces a bijection between the following sets of functions:
(1) $L^1(G)^+\cap \mathcal P(G)$ (continuous, non-negative, positive-definite an... | https://mathoverflow.net/users/24891 | A bijective correspondence induced by Fourier Transform | I try to write down in full detail the answers of Nik and BS (many thanks to both!).
The references in the proof are to Folland's book "A Course in Abstract Harmonic Analysis" and Rudin's book "Fourier Analysis on Groups".
---
Let $G$ be an LCA group, then {$\widehat\phi:\phi\in L^1(G)^+\cap \mathcal P(G) $}$=... | 1 | https://mathoverflow.net/users/24891 | 101516 | 58,928 |
https://mathoverflow.net/questions/101531 | 52 | **Background:** The *Strassen Algorithm*, described [here](http://en.wikipedia.org/wiki/Strassen_algorithm), has a computational complexity of $\text{O}(n^{2.807})$ for the multiplication of two $n \times n$ matrices (the exponent is $\frac{\log7}{\log2}$). However, the constant is so large that this algorithm is in fa... | https://mathoverflow.net/users/18263 | How fast can we *really* multiply matrices? | There are currently no practical implications of any fast matrix multiplication algorithms besides Strassen's. The Coppersmith/Winograd algorithm and its descendants (Stothers, Williams) are very complex, depend on probabilistic constructions, etc. There's no theoretical obstacle to implementing them in the sense you'r... | 82 | https://mathoverflow.net/users/4720 | 101532 | 58,931 |
https://mathoverflow.net/questions/101503 | 3 | Let $G$ be a reductive affine algebraic $\mathbb{C}$-group (not necessarily connected). Suppose $X$ is an irreducible affine algebraic set over $\mathbb{C}$ where $G$ acts rationally. Suppose that $H$ is a reductive subgroup of $G$ (again not necessarily connected). Let $x\in X$.
If the orbit $G\cdot x$ is closed in... | https://mathoverflow.net/users/12218 | Closed reductive sub-orbits | $\def\smat#1{\left(\begin{smallmatrix}#1\end{smallmatrix}\right)}$
The answer to your question is "no". Let $G=GL(2,\mathbb C) \times GL(2,\mathbb C)$ act on $X=GL(2, \mathbb C)$ by $(A,B)\cdot C= ACB^{-1}.$ The orbit of the matrix $x=\smat{
1 & 1 \\
0 & 1 \\\
}$
is $X$. (Hence it is closed.) Let $H$ be the subgrou... | 8 | https://mathoverflow.net/users/23935 | 101533 | 58,932 |
https://mathoverflow.net/questions/101546 | 2 | Let $A$ be an artinian ring and $B$ be an $A$-algebra such that $B \otimes\_A B \to B$ is an isomorphism, i.e. that $A \to B$ is an epimorphism in the category of commutative rings, see the Seminar [Les épimorphismes d'anneaux](http://www.numdam.org/numdam-bin/browse?id=SAC_1967-1968__2_) for a detailed account. In Exp... | https://mathoverflow.net/users/2841 | Epimorphisms with artinian domain | I won't be surprised if you already know this, but here is a proof for $A$ noetherian and/or $B$ finite over $A$:
0) We can replace $A$ with its image in $B$ and assume $A\rightarrow B$ is injective.
1) The result is clear if $A$ is a field.
2) The isomorphism $B\otimes\_AB\rightarrow B$ descends to an isomorphis... | 2 | https://mathoverflow.net/users/10503 | 101552 | 58,940 |
https://mathoverflow.net/questions/101540 | 4 | Hey everybody,
I think this question might be just a simple oversight on my part, but this has been bugging me a few days.
I am reading Hatcher's [Spectral Sequences book](http://www.math.cornell.edu/~hatcher/SSAT/SSch2.pdf), and trying to understand his example where he computes $\pi\_\*^s$ for $p=2$ (page 21-23)... | https://mathoverflow.net/users/24021 | How do you know when something must die in the Adams Spectral Sequence for $\pi_*^s$ | The point is that the element $h\_3$ in odd degree lifts to an element (typically called $\sigma$) in $\pi\_7^s$ whose square must be 2-torsion. This is true for all elements in odd degree because the stable homotopy groups of spheres are graded-commutative. It does not necessarily have to be exactly 2-torsion a priori... | 5 | https://mathoverflow.net/users/360 | 101559 | 58,943 |
https://mathoverflow.net/questions/101558 | 11 | Suppose $T:X \to X$ is a homeomorphism of a compact metric space. The recurrent set is the set of all points $x \in X$ such that for every $\epsilon>0$ there exists an $n\in \mathbb{Z}$, $n \ne 0$, such that $d(T^nx,x)<\epsilon$. Does there exists a $T$-invariant probability measure $\mu$ on $X$ with support equal to t... | https://mathoverflow.net/users/24949 | Invariant measures and recurrent sets. | Just to clarify, the set of recurrent points may not be closed. There exists some transitive homeomorphism, whose uniquely ergodic measure is a Dirac measure. For example start with an irrational vector field $X$ on $\mathbb{T}^2$ and put a stop at $o\in\mathbb{T}^2$. That is, $Y=f\cdot X$ with $f(o)=0$. Then let $\phi... | 6 | https://mathoverflow.net/users/11028 | 101561 | 58,945 |
https://mathoverflow.net/questions/101527 | 1 | Given regular matrices $A\_i,B\_i \in \textrm{GL}\_n(\mathbb{R}),$ $i=1,2$.
Let $A\_1 = U\_1 B\_1 V\_1$ and $A\_2=U\_2 B\_2 V\_2$ where $U\_i,V\_i \in \textrm{O}\_n(\mathbb{R})$ $(i=1,2)$ are orthogonal matrices. This means that $A\_1$ and $B\_1$ resp. $A\_2$ and $B\_2$ have the same singular values.
Now let $A\_2 ... | https://mathoverflow.net/users/24296 | singular value decomposition | Let $A\_2A\_1^{-1}=U\_A DV\_A$ and $B\_2B\_1^{-1}=U\_B DV\_B$ be the two SVDs with the same $D$.
Set $U\_2'=U\_AU\_B'$, $U\_1=V\_B'V\_A$, $V=B\_1^{-1} U\_1 A\_1$.
The first equality is clear. The second one is proved by $$U\_2'B\_2V=U\_A U\_B'B\_2 B\_1^{-1} V\_B' V\_A A\_1=U\_ADV\_A A\_1=A\_2A\_1^{-1}A\_1=A\_2.$$
... | 2 | https://mathoverflow.net/users/1898 | 101565 | 58,947 |
https://mathoverflow.net/questions/101562 | 5 | Apologies for posting such a simple question to mathoverflow. I've have been stuck trying to solve this problem for some time and have posted this [same query](https://math.stackexchange.com/questions/167413/boyd-vandenberghe-question-2-31d-stuck-on-simple-problem-regarding-interi) to math.stackexchange (but have recei... | https://mathoverflow.net/users/24951 | Proving the interior of a dual cone is the set of vectors whose inner product is strictly positive on the cone | $\DeclareMathOperator\cl{cl}$Something along the following lines will work: note that $z$ satisfies $z^\top x>0$ for all $x\in \cl(K)$ iff $z$ satisfies $z^\top x>0$ for all $x\in \cl(K)$ s.t. $\lVert x\rVert=1$, i.e. for all $x\in \cl(K)\cap S^{n-1}$. Note that $U:=\cl(K)\cap S^{n-1}$ is compact, thus the function $x\... | 2 | https://mathoverflow.net/users/11100 | 101566 | 58,948 |
https://mathoverflow.net/questions/101554 | 9 | I am looking for a reference for the fact that the top cohomology $H^n(X;A)$ of an $n$-dimensional manifold $X$ is non-trivial precisely when $X$ is compact.
I tried to ask [this question on Math.Stackexchange](https://math.stackexchange.com/questions/167002/top-cohomology-detecting-compactness), but there was some i... | https://mathoverflow.net/users/11084 | Top cohomology detecting compactness | A connected $n$-manifold $M \neq \emptyset$ is compact iff $H^n (M;\mathbb{Z}) \neq 0$.
EDIT: my old reference to Bredon, Topology and Geometry, page 346 ff, does not completely settle the issue, as George pointed out. So let me give a proof here. For compact $M$, Corollary 7.14 of the cited book gives the answer.
... | 14 | https://mathoverflow.net/users/9928 | 101570 | 58,951 |
https://mathoverflow.net/questions/101577 | 4 | I am considering the Hecke operators $T\_n$ acting on the space $M\_k(\text{SL}\_2(\mathbb{Z}))$ of weight $k$ modular forms of level 1. Are their eigenvalues always real?
I have read somewhere that the Fourier coefficients of a normalized eigenform are real. The coefficients are precisely the eigenvalues right? Is t... | https://mathoverflow.net/users/21596 | Are the eigenvalues of the Hecke operator always real? | More simply, the eigenvalues are real because the Hecke operators are Hermitian for the Peterson scalar product (a fact which can be checked by a straightforward computation). See for example the introduction by Serre on modular forms in Cours d'arithmétique.
| 15 | https://mathoverflow.net/users/9317 | 101582 | 58,958 |
https://mathoverflow.net/questions/101557 | 8 | Is there an obvious reason why $p$-localization of spectra is a "finite" localization in the sense of Haynes Miller? In other words, is there an obvious reason why the localizing subcategory (of the stable homotopy category) consisting of the $p$-acyclic spectra is generated (as a localizing subcategory) by the finite ... | https://mathoverflow.net/users/1589 | Is there an obvious reason why p-localization of spectra is a finite localization? | The fiber of $S\to S\_{(p)}$ is equivalent to $\mathrm{hocolim} \Sigma^{-1}M(k)$, where the limit is taken over the poset of natural numbers prime to $p$ (using the relation of divisiblilty, so $M(k)\to M(kk')$). Thus, if $X$ is a spectrum such that $X\_{(p)}=0$, then $X\approx \mathrm{hocolim} \Sigma^{-1}M(k)\wedge X$... | 8 | https://mathoverflow.net/users/437 | 101586 | 58,962 |
https://mathoverflow.net/questions/101587 | 9 | As we have known, the Sato-Tate measure for GL(2) turned out to be the half circle measure
$\frac{1}{2\pi} \sqrt{4-x^2}dx$ on [-2,2],
which appears in various versions of equi-distribution problems in GL(2).
My question is what the corresponding measure for GL(3) should be.
Firstly we shall note that Hecke eige... | https://mathoverflow.net/users/2666 | Sato-Tate measure for GL(3) Automorphic forms | (2017-11-26 edit by j.c.: earlier versions of this answer consisted of David Hansen's screenshot of the following, with the text "Here is a screenshot of a semi-answer which froze my computer when I hit 'post'":)
Things get more complicated.
$\newcommand{\SU}{\mathop{\rm SU}\nolimits}
\newcommand{\C}{\bf C}
\newcomma... | 12 | https://mathoverflow.net/users/1464 | 101592 | 58,964 |
https://mathoverflow.net/questions/101591 | 3 | Let $f:X\times Y\mapsto R$ be jointly continuous, where $X$ is some topological space, $Y$ is some countably compact topological space. My question is whether or not the envelop $\phi(x) := \max\_{y\in Y} f(x,y)$ is continuous?
I know the answer is yes, if we strengthen countably compact to compact. And I tried to co... | https://mathoverflow.net/users/23306 | Continuity of the maxima | In general the answer is no. For showing this claim I must bring two theorems. You could find these theorems in the page 238 of the text "**General topology**" written by **Ryszard Engelking**.
The following theorem is due to **Isiwata, Nobel, Hager and comfort**:
**Theorem1**: For the Tychonoff spaces $X , Y$ the ... | 1 | https://mathoverflow.net/users/23317 | 101606 | 58,971 |
https://mathoverflow.net/questions/99001 | 6 | Although the question itself can be expressed succinctly, I couldn't come up with a nice self-explanatory title - suggestions are welcome.
Motivation/Background
---------------------
I was investigating whether it would be good idea to use [Gaussian Process Regression](http://www.gaussianprocess.org) in my applicat... | https://mathoverflow.net/users/24274 | Calculating the probability of an event defined by a condition on a Gaussian random process | Suppose your process is continuous. (By the Kolmogorov-Centsov continuity theorem, a sufficient condition for this is that $c(s,s) + c(t,t) - 2 c(s,t) \le C |s-t|^\gamma$ for some $C, \gamma > 0$.) Then your process induces a Gaussian measure on $C([0,T])$ and a theorem due to Fernique gives an asymptotic result:
**T... | 4 | https://mathoverflow.net/users/4832 | 101614 | 58,977 |
https://mathoverflow.net/questions/101610 | 3 | A complete or open Káhler manifold with positive definite Ricci
tensor is simply connected? is there any counterexample?
| https://mathoverflow.net/users/nan | complete or open Kähler manifold and simply connected | Let $S\subset \mathbb{P}^1$ denote a finite subset consisting of $n\geq 2$ points. The Fubini-Study metric on $\mathbb{P}^1$ induces a Kahler metric on $X= \mathbb{P}^1\backslash S$ with a positive-definite Ricci tensor. $X$ is open and Kahler, but $\pi\_1(X, \*)$ is free on $n-1$ generators. Maybe you want to take the... | 8 | https://mathoverflow.net/users/24525 | 101616 | 58,978 |
https://mathoverflow.net/questions/101598 | 17 | A random graph on $n$ vertices is defined by selectiung the edges according to some probability distribution, the simplest case being the one where the edge between any two vertices exists with probability $p = \frac{1}{2}$. I believe this is the [Erdős–Rényi](http://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93R%C3%A9nyi_... | https://mathoverflow.net/users/24958 | Homotopy of random simplicial complexes | Babson, Hoffman, and Kahle have written [a paper](http://arxiv.org/abs/0711.2704) on fundamental groups of random 2-complexes. They worked with the Linial-Meshulam model whereby you begin with a complete graph on $n$ vertices and then add independently uniformly random 2-simplices.
Babson has just written [a paper](h... | 12 | https://mathoverflow.net/users/353 | 101626 | 58,983 |
https://mathoverflow.net/questions/101630 | 1 | Hi Folks,
I just came across a few lines where a (sheaf-)cohomology group of a scheme is treated as a sheaf. I've never seen this in Hartshorne.
Could you give any reference for this?
Thanks
Steven
| https://mathoverflow.net/users/24978 | Cohomology groups interpreted as sheafs | I guess you have seen sheaf cohomology as being the right derived functor of the global section functor, taking a sheaf $\mathcal{F}$ on a space $X$ to the abelian group $\Gamma(X,\mathcal{F})$. Suppose $X$ is a $k$-scheme, where $k$ is any field, with structural morphism $f:X\to\mathrm{Spec}(k)$. Then you can consider... | 3 | https://mathoverflow.net/users/18238 | 101636 | 58,987 |
https://mathoverflow.net/questions/101634 | 3 | Consider a triangulated orientable surface with the following data: on each edge a vector with integer coordinates is written so that for each triangle the sum of the vectors corresponding to three edges of its boundary is $0$.
May such an object be interpret as a discrete version of some topological construction, sa... | https://mathoverflow.net/users/21620 | Discrete version of some topological object. | Let $\Sigma$ be your surface.
I'll construct a new surface $\tilde \Sigma$ as follows.
Take $\Sigma$ apart into individual triangles, and reglue them with a twist at each edge (the adjacency graphs for the faces of $\Sigma$, and for the faces of $\tilde\Sigma$ are the same). Note that the surfaces $\Sigma$ and $\til... | 1 | https://mathoverflow.net/users/5690 | 101637 | 58,988 |
https://mathoverflow.net/questions/101635 | 4 | My question is from the polymer field's famous literature: *The Equilibrium Theory of Inhomogeneous Polymers* by Glenn H.Fredrickson.
In its Appendix C, the C.2 Functional differentiation item, it says a Taylor-expanded form of a functional $F[f+\delta f]$ is:
$F[f+\delta f] = F[f] + \int\_a^bdx\Gamma\_1(x)\delta f... | https://mathoverflow.net/users/24979 | Functional differentation | In Dieudonne's book *Foundations of Modern Analysis* you will find a description of Taylor expansion appropriate to your situation. It involves certain multi-linear maps. There is a famous theorem due to *L. Schwartz* stating that under certain assumptions multilinear maps can be given integral descriptions involving *... | 5 | https://mathoverflow.net/users/20302 | 101641 | 58,991 |
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