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https://mathoverflow.net/questions/67962 | 11 | Consider the assertion:
Every connected, but not necessarily paracompact, n-manifold is of cardinality
$2^{\aleph\_0}$ (at least assuming the axiom of choice).
For n=1 this may be proved via enumeration of the short list of examples. The essential point is that while there is a Long Line, there is no Extra Long L... | https://mathoverflow.net/users/15819 | Cardinality of connected manifolds | A connected Hausdorff manifold with more than one point has cardinality $2^{\aleph\_0}$.
Here's a proof sketch.
For each point $x$ of the manifold, let $U\_x$ be an open Euclidean neighbourhood of $x$. Define a transfinite sequence of subsets $V\_\alpha$ of the manifold as follows. Choose some point $y$ of the mani... | 18 | https://mathoverflow.net/users/12362 | 67985 | 41,794 |
https://mathoverflow.net/questions/67992 | 1 | I am sorry for too naive and stupid question,
How can I express the 2nd Hirzebruch surface, $F\_{2}$ in terms of $SO(3)$. Can F\_{2} be realizable as the total space of a bundle over $\mathbb{R}\_{+}$ with fibre $SO(3)$.
I have to admit that my knowledge of Hirzebruch surfaces is limited to the few lines in the w... | https://mathoverflow.net/users/9534 | Hirzebruch surfaces | What you want almost works, but with a little twist. The group $SO(3)$ is acting on $F\_2$, so that there are two orbits that are $\mathbb CP^1$ and all other orbits are $SO(3)$. More precisely, $F\_2$ can be seen as a compactification of $\mathbb R\_+\times SO(3)$. Indeed, $F\_2$ can be seen as a compactificaton of $(... | 5 | https://mathoverflow.net/users/943 | 67993 | 41,798 |
https://mathoverflow.net/questions/67990 | 1 | Hello,
I remember reading that if $X/\mathbf F\_q$ is a projective smooth global complete intersection, then the characteristic polynomial of the $\mathbf F\_q$-linear Frobenius of $X$
on $H^i\_{et}(X\otimes\overline{\mathbf F\_q},\mathbf Q\_l)$, $l\nmid q$, has integer coefficients and is independent of $l$.
How d... | https://mathoverflow.net/users/36285 | global complete intersection and independence of $l$ | If you allow a proof using the Weil conjectures, there's an otherwise simple proof of this for any smooth projective variety in Deligne's first paper proving the Weil conjectures. See pages 276-277 of [Weil I](http://archive.numdam.org/ARCHIVE/PMIHES/PMIHES_1974__43_/PMIHES_1974__43__273_0/PMIHES_1974__43__273_0.pdf).
... | 2 | https://mathoverflow.net/users/15630 | 67994 | 41,799 |
https://mathoverflow.net/questions/66959 | 32 | This is from the category "problems I cannot believe that are still open". But then again, I don't know whether it is still open; it seems to have escaped the attention of most number theorists and algebraists except for those in olympiad circles. This is the reason I am posting it here.
Let $p$ be a prime. Define a ... | https://mathoverflow.net/users/2530 | Generalization of Tamarkin's ARO 1993, final round, problem 10/8: still a conjecture? | Ok, here's how to show that $(1+n^{p+1})^p$ always gives a counterexample for $p > 3$.
Set $d(k) = \lfloor \frac{k}{p-1} \rfloor - v\_p(k!)$. Note that $d(k)$ is equal to the sum of the base-$p$ digits of $k$, divided by $p-1$, and rounded down, so for instance if $k$ is a nonzero multiple of $p-1$ then we have $d(k)... | 16 | https://mathoverflow.net/users/2363 | 68000 | 41,802 |
https://mathoverflow.net/questions/67997 | 1 | It is easy to give examples of non-separable $L^p$ spaces by considering a measure on a big space. If one adds the condition that the space has to have total measure 1, the problem is not as easy.
The following equivalences are known, and not very useful:
1. $L^1(\Omega,A, P)$ is separable
2. for all $p \in [1,\inf... | https://mathoverflow.net/users/15828 | Does anyone know an example of non-separable $L^1$ of a probability space? | Philip is right. A family of events is *independent* if any finite subfamily is independent. In order to discuss an *uncountable* independent family of events, we need a probability space that is not separable in your pseudometric, or equivalently the $L^1$ space is not separable. As usual, we may model the situation b... | 5 | https://mathoverflow.net/users/454 | 68008 | 41,807 |
https://mathoverflow.net/questions/68009 | 0 | Is the following correct?
* If something is provable for small sets in ZFC with Axiom of Universes ("For any set x, there exists a universe U such that x∈U.") then it is provable for any sets in ZFC (without Axiom of Universes).
I want to work in ZFC with Axiom of Universes, but I wish my results be downgradable to... | https://mathoverflow.net/users/4086 | Downgrading from ZFC with universes to ZFC | No can do. ZFC+AU proves con(ZFC), ZFC doesn't.
| 9 | https://mathoverflow.net/users/nan | 68013 | 41,810 |
https://mathoverflow.net/questions/67961 | 16 | Given an immersion of the n-1-sphere into a (closed) n-manifold, when does it extend to an immersion of the n-disk?
Remark: If the sphere had dimension k smaller than n-1, then such an immersion would exist if and only if the corresponding map from the k-sphere to the Stiefel manifold is 0-homotopic. This is the Hirs... | https://mathoverflow.net/users/39082 | Codimension zero immersions | This is subtle, even for $n=2$. In this case, clearly the problem reduces to $S^2$ or $\mathbb{R}^2$ since every surface has one of these as a universal cover. [Samuel Blank](http://www.ams.org/mathscinet-getitem?mr=2616455) found a criterion to determine if a curve in $\mathbb{R}^2$ bounds an immersed disk. An exposit... | 17 | https://mathoverflow.net/users/1345 | 68021 | 41,817 |
https://mathoverflow.net/questions/68027 | 2 | Let $V$ be an $n$-dimensional vector space. Is the space of embeddings
$$
\coprod\_1^{k} V \to V
$$
path connected for large enough $n$? Clearly $n=1$ is not enough, but I feel like $n=2$ is enough for $1$-connected. Does the space become highly connected as $n\to \infty$? This feels like it is equivalent to a question... | https://mathoverflow.net/users/11300 | Embeddings of vector spaces | No, it is not connected: for example, if $k=1$ it has two path components, given by the two orientations with which $V$ can be embedded into itself.
In general, it has the homotopy type of $F\_k(V; O(n))$ the space of configurations of $k$ particles in $V$ with labels on the orthogonal group, which has $2^k$ path com... | 8 | https://mathoverflow.net/users/318 | 68030 | 41,822 |
https://mathoverflow.net/questions/68004 | 22 | Let $V$ be a finite-dimensional vector space over a field $k$, $v\_1, \dotsc v\_n \in V$ a set of vectors, and $f\_1, \dotsc f\_n \in V^{\ast}$ a set of covectors. Up to permutation, there seem to be at least two "natural" choices of pairing
$${\bigwedge}^n(V) \otimes {\bigwedge}^n(V^{\ast}) \to k.$$
One is given b... | https://mathoverflow.net/users/290 | "Natural" pairings between exterior powers of a vector space and its dual | I would like to give some details in order to make clear that one can give a proof with hardly any computations at all (I have never looked at the Bourbaki presentation but I guess they make the same point though, because they want to make all proofs only depend on previous material they might make a few more computati... | 12 | https://mathoverflow.net/users/4008 | 68033 | 41,824 |
https://mathoverflow.net/questions/67964 | 7 | Let $X$ be a proper singular variety over $k=\overline{\mathbb F}\_p,$ irreducible of dimension $d.$ Let $H^\*(X)$ and $IH^\*(X)$ be the $l$-adic cohomology groups and $l$-adic intersection cohomology groups of $X,$ resp. Then, is the natural map $H^\*(X)\to IH^\*(X)$ compatible with the cup-product on $H^\*(X)$ and th... | https://mathoverflow.net/users/370 | Does intersection pairing on `$IH^*(X)$` agree with cup-product on `$H^*(X)$`? | I think the answer to your question is yes, at least if $X$ is irreducible.
One may think of cup product in cohomology arising from the obvious pairing in the derived category $\mathbb{Q}\_{l,X} \otimes^L \mathbb{Q}\_{l,X} \to \mathbb{Q}\_{l,X}$. The pairing on intersection cohomology comes from a pairing $IC\_X \ot... | 3 | https://mathoverflow.net/users/519 | 68036 | 41,826 |
https://mathoverflow.net/questions/68043 | 7 | I've been told that the answer is no, but I'm having a hard time finding a reference. More precisely, I'm interested in the following. Let $\Omega \subset \mathbb{R}^n$ be a bonded open connected set, say with $C^{\infty}$ smooth boundary and
$L=-\sum\_{i,j}\frac{\partial}{\partial\_{x\_j}}\left(a\_{i,j}(x)\frac{\pa... | https://mathoverflow.net/users/15843 | Can the solution of an elliptic operator with smooth coefficients have zeros of infinite order? | Take a look at [this](http://www.jstor.org/pss/1993331) paper by Protter; there are certainly newer references (which you can easily find starting from this one), but basically the problem was already solved at the time.
| 7 | https://mathoverflow.net/users/7294 | 68044 | 41,827 |
https://mathoverflow.net/questions/68040 | 5 | Let $A$ and $B$ be two isogenous abelian varieties over a number field $K$ and let $\mathcal{A}$ and $\mathcal{B}$ denote their Neron models over $\mathcal{O}\_K$. Let $v \in M\_K^0$ denote a finite prime of $K$, $k\_v$ its residue field, $\mathcal{A}\_v = \mathcal{A} \times \_{\mathcal{O}\_K} k\_v$ the special fiber o... | https://mathoverflow.net/users/12668 | connected component of the identity section in the special fiber of the Neron model under isogenies | I think the answer is yes. This can be deduced for example from the results of SGA7, Expose IX (p.14-15) as follows:
Firstly, the dimension of the unipotent part, the toric part and the abelian part of the connected component of the special fibre are the same for $\mathcal{A}\_v^0$ and $\mathcal{B}\_v^0$. The toric ... | 6 | https://mathoverflow.net/users/519 | 68045 | 41,828 |
https://mathoverflow.net/questions/67991 | 4 | (ZF + Countable Choice)
Let $\langle A,\mathcal{S} \hspace{.02 in} \rangle$ and $\langle B,\mathcal{T} \hspace{.06 in} \rangle$ be second-countable Hausdorff spaces.
Let $\Sigma$ be a sigma-algebra on $A$ such that $\mathcal{S} \subseteq \Sigma$. Let $\mu : \Sigma \to [0,+\infty]$ be an outer regular meas... | https://mathoverflow.net/users/nan | Does pushforward preserve outer regularity? | For infinite measures $\mu$, the pushforward measure need not be outer regular,
even if $\Sigma$ is the $\sigma$-algebra of Borel sets.
For a simple counterexample, let $A$ be the real line $\mathbb{R}$,
and let $B$ be the rationals $\mathbb{Q}$ (with subspace topology).
Let $\mu$ be Lebesgue measure. Write the rat... | 4 | https://mathoverflow.net/users/13506 | 68046 | 41,829 |
https://mathoverflow.net/questions/68047 | 5 | Greetings friends,
Let $X$ be a smooth complex projective variety with canonical divisor $K$.
For $n \in \mathbb{N}$, let $\lambda\_n$ denote the rational map $X \to \mathbb{P}^M$ induced from $H^0(X, nK)$, and let $d\_n = dim(im(\lambda\_n))$. Let $\kappa\_1$ be the maximum value $d\_n$ attains as $n$ ranges over... | https://mathoverflow.net/users/15844 | Basic Question about Kodaira Dimension | Dear Robert, a reference is Lazarsfeld, *Positivity in Algebraic Geometry I*, Corollary 2.1.38. Note that the statement there concerns the Iitaka dimension of any line bundle, not just the canonical bundle.
The basic idea of the proof is as follows: given a line bundle $L$ on $X$, one can find a birational morphism ... | 4 | https://mathoverflow.net/users/nan | 68053 | 41,834 |
https://mathoverflow.net/questions/68042 | 4 | I have a discrete group $G$ and classes $x,y\in H^1(G;\mathbb{Q})$ (group cohomology with coefficients in the rationals viewed as a trivial $G$-module) such that the Massey product $$\alpha:=\langle x, x, y\rangle\in H^2(G;\mathbb{Q})$$ is defined and has zero indeterminacy.
I would like to find a short exact sequen... | https://mathoverflow.net/users/8103 | When is a Massey product the image of a Bockstein operator? | If $\mathbb{Q}G$ is the group algebra, group cohomology is just $H^{\ast}(G,\mathbb{Q})=\operatorname{Ext}^{\ast}\_{\mathbb{Q}G}(\mathbb{Q},\mathbb{Q})$. Now take any ring $R$, any pair of $R$-modules $A$ and $B$, and any class $\alpha\in \operatorname{Ext}^{n}(A,B)$. Take a Yoneda $n$-extension representing $\alpha$:
... | 4 | https://mathoverflow.net/users/12166 | 68059 | 41,838 |
https://mathoverflow.net/questions/68041 | 15 | Suppose we have a block-diagonal matrix $M$, but the block diagonal structure is not immediately apparent from looking at the matrix because the rows/columns are shuffled.
I wish to find a reordering of rows and columns, $M' = P M P^{-1}$, where $P$ is a permutation matrix, that will make the block structure apparent... | https://mathoverflow.net/users/8776 | Showing block diagonal structure of matrix by reordering | If the matrix corresponds to a $d$-regular graph, then one can make some easy observations. For instance, if it is a block matrix with $k$ components, then there will be a $k$-dimensional eigenspace of vectors with eigenvalue $d$. As you say, this gives significant information about the blocks. You can't say that an ar... | 6 | https://mathoverflow.net/users/1459 | 68061 | 41,839 |
https://mathoverflow.net/questions/67999 | 1 | I hope that somebody can help me with the following problem:
Let $A$ be a positive operator on $\mathbf{B}(\mathcal{H})$, ( $\mathcal{H}$ is a Hilbert space) with its spectral measure $E$. Show that for every Borel set $\mathbf{B}$ from the domain of $E(\cdot)$ the following equality holds
$$f(\| AE(\mathbf{B})\|) = ... | https://mathoverflow.net/users/15777 | Positive operators - norm equality | The $L^2$ view usually helps, but I don't think it makes things simpler in this case.
Note that since $E(B)$ is a spectral projection of $A$, you have $f(A)\,E(B)=f(A\,E(B))$ (easy to see since the relation holds for any monomial).
Then the question reduces to whether $\|f(A)\|=f(\|A\|)$ for a positive operator. S... | 1 | https://mathoverflow.net/users/3698 | 68072 | 41,847 |
https://mathoverflow.net/questions/68063 | 8 | This question is closely related to a question of Gowers: [Are there any very hard unknots?](https://mathoverflow.net/questions/53471/are-there-any-very-hard-unknots) .
I'm thinking about how to create interesting knots from small numbers of local moves on unlinks. The "standard embedded n-component unlink" (let's c... | https://mathoverflow.net/users/2051 | Are there any very hard unlinks? | I conjecture that the answer is yes, that you may undo a split link with polynomially many moves. This would yield another proof that unlinking is in NP, but it would be somewhat more satisfying, since the certificate would say that you could actually show someone how to tease apart the two components in polynomial tim... | 7 | https://mathoverflow.net/users/1345 | 68076 | 41,849 |
https://mathoverflow.net/questions/68056 | 15 | I'm looking for an introduction to holomorphic foliations and foliations of complex manifolds.
Any little helps, but I'm particularily interested in problems of the type where we have a hermitian manifold $(X,h)$ (not necessarily compact) and a foliation $\mathcal F$ of $X$, such that the restriction of $h$ to any le... | https://mathoverflow.net/users/4054 | References for holomorphic foliations | As far as I can remember right now, the great general introduction to the theory of holomorphic foliations is yet to be written. Anyway let me mention some of the books that
I know and which you may find useful. Let me warn you that none of them address your specific question.
1. Brunella - Birational geometry of fol... | 13 | https://mathoverflow.net/users/605 | 68087 | 41,851 |
https://mathoverflow.net/questions/68088 | 1 | I'm not especially well-versed in Hopf algebra theory, so apologies in advance if the following question has a very easy answer. Given a Hopf algebra $H$, let $\Delta$ denote the comultiplication, $\sigma$ the coinverse, and $\*$ the adjoint action of $H$ on itself. Then, explicitly, the adjoint action is given by $X\*... | https://mathoverflow.net/users/1528 | Compatibility of adjoint action with comultiplication in a Hopf algebra | Yes, this is true for any cocommutative Hopf algebra.
Let me rewrite your statement using my (or actually my professor's) brand of Sweedler notation: I write $x\_{(1)}\otimes x\_{(2)}$ for $\Delta\left(x\right)$, without sum sign. And I denote the antipode of the Hopf algebra $H$ by $S$. Also I will denote your actio... | 3 | https://mathoverflow.net/users/2530 | 68093 | 41,854 |
https://mathoverflow.net/questions/68081 | 4 | I know this is not the forum for this particular question, but the majority of users are immersed in the environment. I want to study differential geometry, but I need to know what courses would help guide me in that direction. I enjoy and/or also want to study the following as well: abstract algebra, real analysis, pa... | https://mathoverflow.net/users/15850 | Going to graduate school for mathematics next year, need some advice | The sociological and metamathematical aspects of this question are too often overlooked, I think. First, undergrad or grad students' discussions with their peers too often subtly veers into a "Lord of the Flies" scenario. Second, many "advisors" (whether undergrad or grad), have some weaknesses in communications skills... | 22 | https://mathoverflow.net/users/15629 | 68105 | 41,857 |
https://mathoverflow.net/questions/68110 | 5 | I'll start with example:
Let $X$ be a scheme, and $O\_X$ be its structure sheaf. It is defined at the moment on open sets of $X$, and it takes them to $Sets$. However, it is extendable to a sheaf on the Zariski site of $Sch$ by: Take a scheme $S$ to $\mathbb{G}\_a(S)$. Now that it is a functor $Sch \rightarrow Sets$,... | https://mathoverflow.net/users/5309 | When is a sheaf on a scheme extendable to a representable functor? | If $F$ is a coherent sheaf on a noetherian scheme $X$, there is a natural extension of $F$ to the large Zariski site of $X$: with an object $f\colon T \to X$, you associate the group of global sections of the pullback $f^\*F$. According to a result of Nitin Nitsure, this is representable if and only if $F$ is locally f... | 12 | https://mathoverflow.net/users/4790 | 68116 | 41,863 |
https://mathoverflow.net/questions/68086 | 2 | I would like to know if the odd Betti numbers of a projective bundle P(E) for some vector bundle E over say a compact complex smooth algebraic variety B are zero just as in the case for ordinary projective spaces over Spec(k), or more generally how to generalize standard calculations of the cohomology of projective spa... | https://mathoverflow.net/users/15852 | odd betti numbers of a projective bundle | If $E$ is of rank $r$ then $H^i(P\_B(E)) = \sum\_{t = 0}^{r-1} H^{i-2t}(B)$ (where the summands with negative $i - 2t$ are omitted). So $H^{odd}(P\_B(E)) = 0$ if and only if $H^{odd}(B) = 0$.
| 1 | https://mathoverflow.net/users/4428 | 68118 | 41,864 |
https://mathoverflow.net/questions/68140 | 1 | This is a simple question about notation: Given two generators $x,y$ how does one denote the vector space spanned by all finite **K**-polynomials in $x$ and all finite polynomials in $y$. If I use **K\*\*$[x] \oplus$ \**K*\*$[y]$, then I get two copies of \*\*K**. I could just quotient \**K*\*$[x] \oplus$ \**K*\*$[y]$ ... | https://mathoverflow.net/users/2612 | Notation: Vector space spanned by all finite polynomials in $x$ and all finite polynomials in $y$ | If it is clear from context that you are working in the ambient setting of $k[x,y]$, then you can write $k[x] + k[y]$. Otherwise, I would spell it out in words.
| 5 | https://mathoverflow.net/users/297 | 68142 | 41,879 |
https://mathoverflow.net/questions/68125 | 15 | The $2$-functor $\text{Qcoh} : \text{Sch}^{op} \to \text{Cat}$, which sends a scheme to its category of quasi-coherent modules, is a stack by Descent Theory. Is it actually an algebraic stack? If not, how far is it from being algebraic? For example, which fragment of Artin's criteria is satisfied? I expect that this sh... | https://mathoverflow.net/users/2841 | Qcoh(-) algebraic stack? | Artin's axioms do not apply in this case, because the stack is not limit-preserving. They only work with stacks that are locally finitely presented.
In any case, it is easy to give examples of quasi-coherent sheaves whose functor of automorphisms is not representable (for example, an infinite dimensional vector space... | 20 | https://mathoverflow.net/users/4790 | 68150 | 41,883 |
https://mathoverflow.net/questions/68145 | 29 | All the statements below are considered over local rings, so by regular, I mean a regular local ring and so on;
It is well-known that every regular ring is Gorenstein and every Gorenstein ring is Cohen-Macaulay. There are some examples to demonstrate that the converse of the above statements do not hold. For example,... | https://mathoverflow.net/users/13351 | Regular, Gorenstein and Cohen-Macaulay | I will argue that the examples you gave are "simplest" in some strong sense, so although they look unnatural, if Martians study commutative algebra they will have to come up with them at some point.
Let's look at the first one $A=k[[x,y,z]]/(x^2-y^2, y^2-z^2, xy,yz,zx)$. Suppose you want
>
> a $0$-dimensional G... | 41 | https://mathoverflow.net/users/2083 | 68151 | 41,884 |
https://mathoverflow.net/questions/68096 | 8 | The "most symmetric" Mukai-Umemura 3-fold with automorphism group $PGL(2,\mathbb{C})$ admits a Kaehler-Einstein metric according to Donaldson's result.
On the contrary, there are some arbitrarily small complex deformations of the above $3$-fold which do not admit Kaehler-Einstein metrics, as shown by Tian. All examp... | https://mathoverflow.net/users/15600 | Mukai-Umemura 3-fold and Kaehler-Einstein metrics | [Edited] Such a manifold cannot exist.
Indeed the small deformations of the "symmetric" Mukai-Umemura $3$-fold $X$ are described explicitly by Donaldson in [this paper](http://arxiv.org/pdf/0803.0985), pages 43-44. There he describes 5 classes of deformations. Classes 1,2,3 correspond to points in $H^1(X,TX)$ whose ... | 5 | https://mathoverflow.net/users/13168 | 68157 | 41,889 |
https://mathoverflow.net/questions/68106 | 7 | The set-up is this: Let $G$ be a finite group, and $H$ a subgroup. We are given an irreducible representation of $H, \rho: H\rightarrow GL\_n(K)$ (I will notationally identify $\rho$ with its character). I want to decompose $Ind^G\_H(\rho)$ into irreducibles. I am given character tables of both $G$ and $H$.
If $K$ we... | https://mathoverflow.net/users/3238 | How does one compute induced representations for modular representations? | Frobenius reciprocity for Brauer characters is a little more complicated (a lot more complicated if you don't have complete tables). You need the projective characters to compute the multiplicities. I don't use anything special about the character being induced, though sometimes you can leverage that information (espec... | 4 | https://mathoverflow.net/users/3710 | 68158 | 41,890 |
https://mathoverflow.net/questions/68149 | 10 | Suppose $f$ is a modular form of weight $k \ge 2$.
It's "well-known" that there are "periods" $\Omega\_-$ and $\Omega\_+ \in \mathbb{C}$, such that the $L$-values $L(f, \chi, j)$, for $\chi$ a Dirichlet character and $1 \le j \le k-1$, are $(2\pi i)^j$ times an algebraic multiple of one of $\Omega\_{\pm}$.
What is ... | https://mathoverflow.net/users/2481 | Periods and L-values of modular forms | Check out Manin's paper [*Periods of parabolic forms and $p$-adic Hecke series*](http://iopscience.iop.org/0025-5734/21/3/A02). He only deals with level 1 and even weight, but this is enough to know that Vatsal's statement is not the right one (though not enough to fully confirm Colmez's statement, though it's true as ... | 13 | https://mathoverflow.net/users/1021 | 68159 | 41,891 |
https://mathoverflow.net/questions/68037 | 4 | In "Was sind und was sollen die Zahlen?" Dedekind gives a noncircular
proof of the statement that a set is finite if and only if it cannot be
put in bijective correspondence with a proper subset. By "circular" I
mean in this context that you should not prove it by simply saying that a
proper subset of a finite set wil... | https://mathoverflow.net/users/11145 | Dedekind's theorem | Regarding the narrower interpretation of your question, the fact that the finite numbers are not equinumerous with
any proper subset is also expressed as the classical
[pigeon hole
principle](http://en.wikipedia.org/wiki/Pigeonhole_principle).
And for this, the linked Wikipedia article asserts that "the first
formaliza... | 8 | https://mathoverflow.net/users/1946 | 68160 | 41,892 |
https://mathoverflow.net/questions/68165 | 5 | Hello All
---
Consider a matrix with elements:
$A\_{i,j}=x\_i$ for $i=j$
$A\_{i,j}=1$ for $i\neq j$
Is there a closed form expression for the elements of $A^{-1}$?
Will be glad to know of any reference.
Thanks
HC
| https://mathoverflow.net/users/15871 | Is there a closed form expression for the inverse of the matrix with elements $A_{i,j}=x_i$ for $i=j$ and $A_{i,j}=1$ for $i\neq j$? | You can use the [Sherman-Morrison formula](http://en.wikipedia.org/wiki/Sherman-Morrison_formula).
In the notation of the Wikipedia article, let $u=v=(1,\ldots,1)'$ and $A$ (***not*** the same as your $A$) be the diagonal matrix with $(x\_{1}-1, \ldots, x\_{n}-1)$ on the diagonal.
Then, if I haven't made a mistake,... | 14 | https://mathoverflow.net/users/13602 | 68167 | 41,897 |
https://mathoverflow.net/questions/68154 | 4 | Assume I am blowing up an algebraic variety $X$ in an ideal sheaf $\mathcal{I}$, write $Y:=\mathrm{Bl}\_\mathcal{I}(X)$. Now, also assume I have a globally generated line bundle $\mathcal{O}\_X^k\twoheadrightarrow\mathcal{L}$. Denote by $h\_i\in\mathcal{L}(X)$ the images of the canonical base vectors, i.e. the global g... | https://mathoverflow.net/users/9947 | Strict Transform of a Line Bundle? | There are a number of issues with what you want mostly along the lines of the comments.
The main problem is that the *strict transform* is an operation on a *divisor* and not on a *divisor class*. A line bundle corresponds to a divisor class. So, basically you want to figure out a way to define the strict transform f... | 5 | https://mathoverflow.net/users/10076 | 68180 | 41,903 |
https://mathoverflow.net/questions/68176 | 3 | Thanks to Higham I know that $A f(BA) = f(AB) A$ for any two matrices whose sizes are compatible.
Now I believe that $A (BA)^D = (AB)^D A$, even though the Drazin inverse is not the same function (polynomial?) for $AB$ as for $BA$.
I have validated this relationship via numerical experiments with random matrices, ... | https://mathoverflow.net/users/15877 | An identity involving the Drazin inverse | [These notes](http://benisrael.net/GI-LECTURE-7.pdf) say that the Drazin Inverse is the matrix function corresponding to $f(z) = 1/z$, defined on the *nonzero* eigenvalues. Thus, by the theorem that you cite, the said equality should hold.
| 1 | https://mathoverflow.net/users/8430 | 68183 | 41,905 |
https://mathoverflow.net/questions/68163 | 5 | One of the lemmas at the foundation of Teichmuller theory is as follows. Let $Q(x,y)$ be a positive definite quadratic form. Then there exists unique $\lambda \in \mathbb{R}$ and $\mu \in \mathbb{C}$ with $\lambda > 0$ and $\|\mu\| < 1$ such that the following hold. Let $Q\_{\mu}(x,y)$ be the quadratic form
$$Q\_{\mu}(... | https://mathoverflow.net/users/15870 | Identification of conformal classes of pos def quadratic forms on R^2 with unit ball | This is a special case of a general fact, that the space of positive definite quadratic forms on $\mathbb{R}^n$ with determinant $=1$ is a symmetric space for the Lie group $PSL(n,\mathbb{R})$. A matrix acts on a quadratic form by change of basis, and a stabilizer is a conjugate of $SO(n,\mathbb{R})$. When $n=2$, the s... | 2 | https://mathoverflow.net/users/1345 | 68187 | 41,907 |
https://mathoverflow.net/questions/68193 | 5 | Do you know of any self-injective basic algebra $A$ over a field $k$ such that its enveloping algebra $A^{\mathrm{op}}\otimes\_k A$ is **not** self-injective?
The algebra $A$ cannot be finite-dimensional, since then $A$ is Frobenius and so is $A^{\mathrm{op}}\otimes\_k A$.
| https://mathoverflow.net/users/12166 | Self-injective basic algebras | [Nagata, Masayoshi. A conjecture of O'Carroll and Qureshi on tensor products of fields. Japan. J. Math. (N.S.) 10 (1984), no. 2, 375--377. MR0884425] proved that the Krull dimension of the tensor product $\mathbb C(x,y)\otimes\_{\mathbb C}\mathbb C(x,y)$ is $2$. If this tensor product is reduced (I think it is, but it ... | 5 | https://mathoverflow.net/users/1409 | 68197 | 41,911 |
https://mathoverflow.net/questions/68199 | 3 | Let $X=(V,E)$ be a finite, connected, $k$-regular graph. Let $avg(d^2)$ be the averaged square distance between vertices, as defined in [Average squared distance vs diameter in vertex-transitive graphs](https://mathoverflow.net/questions/67838/average-squared-distance-vs-diameter-in-vertex-transitive-graphs) . Is it tr... | https://mathoverflow.net/users/14497 | Average squared distance in $k$-regular graphs | The number of vertices in the ball of radius $c \log\_k(|V|)$ ($c<1$) is small compared to $|V|$, so most pairs of vertices are more than that apart.
| 6 | https://mathoverflow.net/users/1061 | 68200 | 41,913 |
https://mathoverflow.net/questions/68212 | 6 | Is there a simple way to calculate/estimate the volume of Minkowski sum of an n-dimensional unit ball and an n-dimensional ellipsoid? Even a simple ellipsoid like $\frac{x\_1^2}{a^2} + x\_2^2 + \ldots + x\_n^2 = 1$ will do.
| https://mathoverflow.net/users/15886 | Volume of Minkowski sum of a ball and an ellipsoid | I guess you know that the result can be written as a polynomial.
$$w\_0+w\_1{\cdot}r+\dots+w\_n{\cdot}r^n.$$
So your question is how to estmate the coefficients $w\_i$; this is so called "cross-sectional measures" and they can be defined for any convex body $K$.
* $w\_0$ is the volume of $K$,
* $w\_n$ is the volume ... | 6 | https://mathoverflow.net/users/1441 | 68214 | 41,919 |
https://mathoverflow.net/questions/68202 | 0 | Let
$x\_s = \sin(\theta+\frac{2\pi s}{3})$ and
$y\_s = 1+\cos(\frac{2\pi s}{3})$, $s=0,1,2$.
Define $f(\theta) = \sum\_{s=0}^2 x\_s\ln y\_s$.
Is there any method to derive roots of $f(\theta)$.
I have run a simulation on it, and found that $\theta=0$ is a solution.
But I am unable to see how to analytically obtain ... | https://mathoverflow.net/users/15873 | Roots of an entropy-like function | By trigonometry, $$f(\theta)= \log (2)\left( \sin\theta+\cos(\pi/6-\theta)-\cos(\pi/6+\theta) \right)=\log(4)\sin\theta.$$
---
For the revised problem, we have this: $f(\theta)$ is $2\pi/3$-periodic, and $f(\theta)$ is odd, so it suffices to find the roots between $0$ and $\pi/3$ (both of which are themselves roo... | 4 | https://mathoverflow.net/users/935 | 68215 | 41,920 |
https://mathoverflow.net/questions/61312 | 7 | Given a regular local ring $R$ and an $R$-algebras $S$, which is torsion free and finitely generated (even free if needed) as an $R$-module.
Assume we have a nontrivial surjective map $f: M \rightarrow T$, where $M$ is a projective $S$-module, finitely generated and torsion free, and $T$ is a torsion module over $S$.... | https://mathoverflow.net/users/3233 | Extensions of torsion modules | Actually, it would be easier to look at the other end of the exact sequence. Namely, your map $f^\*$ is trivial implies the map $g^\*: Hom\_S(T,Q) \to Hom\_S(M,Q)$ is an isomorphism.
Now, since $M$ is projective, the support of $Hom\_S(M,Q)$ is equal to the support of $Q$. Thus we have $Supp(Hom\_S(T,Q)) = Supp(Q)$,... | 4 | https://mathoverflow.net/users/2083 | 68216 | 41,921 |
https://mathoverflow.net/questions/68099 | 13 | Consider $0\leq \alpha\leq 1$, and let $A\_{\alpha}$ be the Toeplitz $n\times n$ matrix given by
$$
A\_\alpha := \begin{bmatrix}
1 & \alpha & \alpha^2 & \ldots &\alpha^{n-1} \\\
\alpha & 1 & \alpha & \ddots & \vdots \\\
\alpha^2 & \alpha & \ddots & \ddots & \alpha^2 \\\
\vdots & \ddots & \ddots & 1 & \alpha \\\
\al... | https://mathoverflow.net/users/13825 | Diagonalizing a Certain Real and Symmetric Toeplitz Matrix | There's likely no explicit diagonalization of $A\_\alpha$ except when
$n$ is very small or in special cases like $\alpha = 0$ and $\alpha
= 1$. Nevertheless each "limit moment" $\gamma\_k$ can be computed as
a rational function of $\alpha$, and this can be used to describe for
each $\alpha$ the distribution of eigenval... | 15 | https://mathoverflow.net/users/14830 | 68225 | 41,927 |
https://mathoverflow.net/questions/68170 | 1 | Hello,
Can anyone help me see how one can get from the following integral
$$\lambda\int\_{\beta=0}^{y}\int\_{\alpha=x}^{Q-\beta}e^{-\mu(\alpha-x)}f(\alpha,\beta)d\alpha d\beta+k\int\_{\alpha=s}^{x}f(\alpha,y)d\alpha$$
$$ = $$
$$\lambda\int\_{\beta=0}^{y}\int\_{\alpha=s}^{Q-\beta}e^{-\mu(\alpha-s)}f(\alpha,\beta)d\a... | https://mathoverflow.net/users/15874 | Writing an integral equation as a partial differential equation | Let $U =$
$$
\lambda \int\_{0}^{y} \int\_{x}^{Q - \beta} \operatorname{e} ^{\bigl(-\mu (\alpha - x)\bigr)} f (\alpha,\beta) d \alpha d \beta - \lambda \int\_{0}^{y} \int\_{s}^{Q - \beta} \operatorname{e} ^{\bigl(-\mu (\alpha - s)\bigr)} f (\alpha,\beta) d \alpha d \beta +
k \int\_{s}^{x} f (\alpha,y) d \alpha - k \int\... | 0 | https://mathoverflow.net/users/454 | 68232 | 41,933 |
https://mathoverflow.net/questions/67616 | 12 | Consider the Hopf fibrations $S^1\to S^{2n+1}\to CP^n$ and $S^3\to S^{4n+3}\to HP^n$. These are Riemannian submersions with totally geodesic fibers. Consider now their canonical variations (the so-called Berger spheres), i.e., for each $t>0$, let $S^{2n+1}\_t$ (resp $S^{4n+3}\_t$) be the Riemannian manifold $S^{2n+1}$ ... | https://mathoverflow.net/users/15743 | First eigenvalue of the Laplacian on Berger spheres | Tanno, the same author of the paper above mentioned in the case of 1-dim fibers, has another paper one year later [Tanno, Shûkichi. Some metrics on a $(4r+3)$-sphere and spectra.
Tsukuba J. Math. 4 (1980), no. 1, 99–105. ] in which he addresses exactly my original question, regarding the case of 3-dim fibers. As a cons... | 5 | https://mathoverflow.net/users/15743 | 68239 | 41,937 |
https://mathoverflow.net/questions/68208 | 40 | My question is somewhat similar to [this previous question](https://mathoverflow.net/questions/8097/number-theory-textbook-with-an-algebraic-perspective), but from a slightly different perspective. Is there any textbook on elementary number theory that develops the properties of $\mathbb{Z}$ as, say, the initial object... | https://mathoverflow.net/users/6856 | Elementary number theory text from a categorical perspective | I have to admit that this is not really an answer, but rather some sort of meta-answer with some very general remarks which I hope do not bore everyone reading this; it just seems to me that this is necessary to indicate that it is rather misguided, as Yemon already says in the comments and I strongly agree with, to as... | 29 | https://mathoverflow.net/users/2841 | 68244 | 41,939 |
https://mathoverflow.net/questions/68246 | 12 | Is the normalization of a Cohen-Macaulay domain necessarily Cohen-Macaulay? I suspect that the answer is no, but I don't have a counterexample. I am most interested in "geometric" situations, so one can place assumptions like excellence on the domain if it's relevant.
| https://mathoverflow.net/users/321 | Cohen-Macaulay domain with non-Cohen-Macaulay normalization? | No, take a normal non-Cohen-Macaulay $d$-dimensional projective variety $X \subseteq \mathbb{P}^n$ and do a series of generic projections from points not on the variety so that its image is a hypersurface $Y \subseteq \mathbb{P}^{d+1}$.
The generic hypersurface projection $Y$ will be Cohen-Macaulay (its a hypersurfa... | 19 | https://mathoverflow.net/users/3521 | 68251 | 41,943 |
https://mathoverflow.net/questions/68218 | 17 | Let's consider the moduli space of representations of $\pi=\pi\_1(\Sigma)$ (a surface group) into $G$ (a lie group). Call this $X=\operatorname{Hom}(\pi,G)$, and let $Y=\operatorname{Hom}(\pi,G)/\\!/G$, where $G$ acts by conjugation on $X$ (and we take the GIT quotient). Let's denote the quotient map by $f:X\to Y$. Gol... | https://mathoverflow.net/users/35353 | Is there an algebraic construction of the Quillen (determinant) Line Bundle? | For $G={\rm PGL}\_n({\mathbb C})$ there is no complex algebraic Quillen line bundle on the complex character variety $Y$ as the cohomology class $[\omega]=\alpha\_2\in H^2(Y;{\mathbb Q})$ is not pure by Proposition 4.1.8 in <http://arxiv.org/abs/math.AG/0612668>.
EDIT: It is instructive to think about the case $G={ G... | 14 | https://mathoverflow.net/users/1583 | 68266 | 41,956 |
https://mathoverflow.net/questions/68275 | 6 | Here is an interesting problem, which I could not solve and would appreciate any comment in solving it;
Assume that we have omitted infinitely many lines from $\mathbb{C}P^2$ to obtain $\mathbb{C}^2$ and now, consider the following lines in which $(z,w)$ is the coordinate of $\mathbb{C}^2$: $z=0, z=1, w=0, w=1, z=w.$... | https://mathoverflow.net/users/13351 | fundamental group of the complement of lines in C^2 | What you have is an arrangement of hyperplanes in $\mathbb C^2$ obtained by complexifying a real arrangement. There is a simple way to describe the fundamental group of its complement, due to Randell. You can find the details and references in the beautiful book *Arrangement of hyperplanes* by Peter Orlik and Hiroaki T... | 16 | https://mathoverflow.net/users/1409 | 68276 | 41,959 |
https://mathoverflow.net/questions/68217 | 1 | Let $X$ be a smooth complete (not-necessarily projective) complex algebraic threefold, and $D$ an effective divisor with one dimensional (stable?) base locus $Bs(|D|)=Z$, i.e. $D$ is base point free off a finite union of curves. Let $\pi: X' \rightarrow X$ be the blow up of $X$ along a smooth curve $C\_0 \nsubseteq Z$ ... | https://mathoverflow.net/users/15889 | base locus under blowup in a threefold | If you keep all of your assumptions, I don't think this will hold. Let $D'$ denote the pull-back of $D$ to $X'$. (I am assuming this is what you meant).
My first instinct was that I do not see why $mD'-E$ would even have a non-empty linear system. I guess that could actually happen, but generally it would not.
I th... | 2 | https://mathoverflow.net/users/10076 | 68277 | 41,960 |
https://mathoverflow.net/questions/68226 | 4 | Let $D\subset \mathbb{R}^n$ be a bounded domain.
An extension map is $E\_D: W^{p,k}(D)\to W^{p,k}(\mathbb{R}^n)$ satisfying:
```
(1) $E_D(f)(x)=f(x)$ for all $x\in D$,
(2) $\| E_D f\|_{W^{p,k}(\mathbb{R}^n)} \le K(D,p,k) \| f\|_{W^{p,k}(D)}$.
```
Thus, $K(D,p,k)$ is the norm of $E\_D$.
From the answer of Tapio R... | https://mathoverflow.net/users/15891 | Dependence of norm of extension map on Sobolev spaces and $(\epsilon,\delta)$ domains | One way to see that your intuition is correct (without using any theorems) is to estimate the $W^{2,k}$-norm of the extension radially. Because the extension is a Sobolev function it is absolutely continuous on almost every ray. Let $f$ be a constant function $1$. Now, taking into consideration only the value of the ex... | 1 | https://mathoverflow.net/users/11716 | 68280 | 41,963 |
https://mathoverflow.net/questions/67940 | 10 | (Warning: I'm not an expert in the topic) Let's work in a "geometric" category, for example the category $\mathfrak{Diff}$ of "manifolds" (without the requirements of connectedness and second countability and even of finite dimensionality), and consider also the subcategory of "usual" finite dimensional (Edit: possibly... | https://mathoverflow.net/users/4721 | Groupoids vs Pseudogroups | You've asked a mouthful. Let me do my best:
First, in your first remark, the Lie groupoid is NOT infinite dimensional. It is very finite dimensional. The arrows of $Germ(G)$ can be viewed as encoding a sheaf over $G\_0,$ and hence can be given the unique topology making the source map into a local homeomorphism. This... | 5 | https://mathoverflow.net/users/4528 | 68285 | 41,966 |
https://mathoverflow.net/questions/68284 | 0 | Is there any condtions in terms of coefficients, which is equivalent to two polynomials p and q having a common root inside the unit disc. More precisely, suppose that p and q are two complex polynomials. they have a common root inside the unit disc if and only if.....?
| https://mathoverflow.net/users/15620 | roots of polynomials | [This question](https://mathoverflow.net/questions/67049/roots-of-polynomials-outside-the-unit-disc)
Deals with the question of one polynomial having a root outside the disk. For the current question, apply the algorithms described in the answers to the greatest common divisor of $p$ and $q$.
| 1 | https://mathoverflow.net/users/11142 | 68286 | 41,967 |
https://mathoverflow.net/questions/68213 | 13 | I know (edit: three) families of smooth projective connected curves over $\bar{\mathbf{Q}}$ for which the Belyi degree is not hard to bound from above.
1. The modular curves $X(n)$. They are constructed by compactifying the quotient $Y(n) = \Gamma(n)\backslash \mathbf{H}$. The natural morphism $X(n) \longrightarrow ... | https://mathoverflow.net/users/4333 | Families of curves for which the Belyi degree can be easily bounded | Another example, like JSE's, that comes already equipped with a Belyi map but is not as familiar as modular curves and Fermat curves: For any relatively prime integers $m,n$ with $0<m<n$, and any subgroup $G$ of $S\_n$, the curve that parametrizes trinomials $x^n + a x^m + b$ up to scaling with Galois group contained i... | 12 | https://mathoverflow.net/users/14830 | 68298 | 41,973 |
https://mathoverflow.net/questions/68293 | 1 | Is there a good nerve-like functor from simplicial objects in categories to simplicial sets which takes level-wise equivalences of categories to weak equivalences?
To give this some context, I'd like to extract a simplicial set from the Waldhausen S-construction applied to a category with cofibrations, and I realize... | https://mathoverflow.net/users/9581 | Nerves of simplicial objects in categories/Waldhausen's S-construction | Here is an idea. Try the homotopy coherent nerve. (This was originally introduced, sort of, by Boardman and Vogt in a topological context and was formulated for simplicially enriched categories (and please do not use `simplicial category' as it is ambiguous!) by Cordier in 1980. The H.c. nerve is related to the bisimpl... | 3 | https://mathoverflow.net/users/3502 | 68306 | 41,976 |
https://mathoverflow.net/questions/68308 | 2 | Hello,
I would like to know whether, given an algebraic number $\alpha$ of degree $d$, the Dedekind Zeta function $\zeta\_{\mathbb{Q}(\alpha)}$ is always a function of the Selberg class of degree $d$ of not. I know that it is true when $\mathbb{Q}(\alpha)$ is an abelian extension of $\mathbb{Q}$, but what about the n... | https://mathoverflow.net/users/13625 | Algebraic numbers and Selberg class | Yes, even the broader class of Hecke L-functions (general L-functions for GL(1) over number fields) are in Selberg's class, for straightforward reasons coming from the functional equation proven by Hecke (and redone by Iwasawa-Tate).
Perhaps the least obvious part is the "order" requirement, but this is what follows ... | 5 | https://mathoverflow.net/users/15629 | 68310 | 41,977 |
https://mathoverflow.net/questions/68309 | 13 | I would like to learn more about the background of this [talk](http://www.raumzeitmaterie.de/veranstaltungen.php?evt=select&sqn=7987 "link"), but found no text on that theme. Do you know more? Edit: [An interesting talk](http://media.medfarm.uu.se/flvplayer/strings2011/video32 "link") by Miranda Cheng ([slides](http://... | https://mathoverflow.net/users/451 | "Modular forms from Feynman integrals "? | I can say a little about the work of Brown and Schretz, since Brown gave a talk at BIRS last month.
If you take a graph with $N$ edges and some restriction on valence (called a Feynman graph), there is a certain integral on an $N-1$-dimensional domain with boundary that in small cases appeared to yield linear combina... | 18 | https://mathoverflow.net/users/121 | 68314 | 41,979 |
https://mathoverflow.net/questions/68311 | 5 | Let $H$ ba a Hopf algebra with coaction $\Delta: H \to H \otimes H$. Denote the action of $\Delta$ by $\Delta (h) = h\_{(1)} \otimes h\_{(2)}$. I was wondering if every element of $H$ can arise as a $h\_{(2)}$, for some $h$. To be more precise, for any $g \in H$, does there exist a $h \in H$, such that $\Delta(h) = f \... | https://mathoverflow.net/users/2612 | Image of the Coproduct of a Hopf Algebra | Counitarity implies that $$(\varepsilon\otimes\mathrm{id}\_H\circ\Delta)(H)=H.$$ If $\Delta(H)\subseteq H\otimes H'$ for some subspace $H'\subseteq H$, then this tells us ---since $\varepsilon$ takes scalar values!--- that $H\subseteq H'$.
| 8 | https://mathoverflow.net/users/1409 | 68315 | 41,980 |
https://mathoverflow.net/questions/68307 | 3 | Fix a prime $p\geqslant 5$ and weight $k\leqslant p+1$, and let $f\in S\_k(N,\overline{\mathbb{Q}})$ be an eigenform.
Due to the congruence $E\_{p-1} \equiv 1 \mod p$, we know that $\overline{E\_{p-1}f}\in S\_{k+p-1}(N,\overline{\mathbb{F\_p}})$ is congruent to $f$ mod $p$, and it is a result (of Deligne-Serre?) that... | https://mathoverflow.net/users/2615 | If $f$ is an $p$-nonordinary eigenform of weight $k\leqslant p+1$ are there always two eigenforms in weight $k + p-1$ congruent to $f$? | It's deeper than theta cycles, I think.
I am going to assume that $N$ is prime to $p$ -- you don't say this in your question but most of my answer assumes this in a very serious way.
If $f$ is ordinary then the space of oldforms attached to $f$ at level $Np$ has dimension 2 and contains one ordinary and one non-ord... | 9 | https://mathoverflow.net/users/1384 | 68318 | 41,982 |
https://mathoverflow.net/questions/68319 | 5 | Let $E/F$ be a Galois extension of number fields, and $G$ a reductive group over $F$. If Langlands Base Change is known for $G/F$ and $G/E$, and moreover the eigenvarieties for $G/F$ and $G/E$ have been constructed, is there a rigid map between the eigenvarieties which interpolates the base change?
Assuming the answe... | https://mathoverflow.net/users/5513 | Base Change for Eigenvarieties | This is all a bit complicated -- the theory is still in its infancy and some arguments aren't quite as smooth as they should be.
If all you know is that "the eigenvarieties have been constructed" then you're in a hopeless situation -- in fact in some sense I don't even know what this statement means. You need to know... | 4 | https://mathoverflow.net/users/1384 | 68320 | 41,983 |
https://mathoverflow.net/questions/68323 | 5 | I have been having trouble understanding some statements regarding flatness in Hartshorne - in particular relating to some of the examples in the text. Any help would be appreciated!
Here is the issue:
* In example III.9.8.4 Hartshorne discusses an example of a family of twisted cubics arising from the projection o... | https://mathoverflow.net/users/1724 | Question on an example about flatness in Hartshorne | Enrique, I think what's happening is that $Y$ does not have an embedded point, only its special fiber does. Taking $Y\_{\mathrm red}$ does not change anything.
| 10 | https://mathoverflow.net/users/10076 | 68324 | 41,984 |
https://mathoverflow.net/questions/42672 | 8 | This question is inspired by the construction of the time evolution for endomotives as given by Connes and Marcolli in their book <http://www.alainconnes.org/docs/bookwebfinal.pdf>.
Let $M$ be a monoid (countable and discrete) acting on a locally compact Hausdorff space $X$ and consider the $C^\*$-algebra $A$ given b... | https://mathoverflow.net/users/5831 | Tomita-Takesaki theory for a simple class of crossed products | I'd like to take this opportunity to dispel this wrong beleif:
>
> "Recall that Tomita-Takesaki theory is only visible in the noncommutative case, in the commutative case the $\mathbb R$-action is trivial"
>
>
>
As mentioned in [this question](https://mathoverflow.net/questions/68270/tomita-takesaki-versus-... | 4 | https://mathoverflow.net/users/5690 | 68326 | 41,985 |
https://mathoverflow.net/questions/68325 | 2 | Let $(M,g)$ be a closed, smooth Riemannian manifold. Let $\Delta = -div\nabla$ be the Laplace-Beltrami operator. Let $h$ be a smooth function on $M$. Is there a condition on $h$ weaker than non-negativity such that $\Delta + h$ is a positive or non-negative operator?
I'm thinking of something akin to the following: ... | https://mathoverflow.net/users/15856 | Positivity of Second-Order Elliptic Differential Operators | This can't be right as stated: if $h$ takes a negative value at some point $p$ of $M$ then $\Delta+ch$ has a negative eigenvalue for sufficiently large $c$.
Proof: let $f: M \rightarrow {\bf R}$ be a nonnegative smooth function that's positive at $p$ and supported on a small enough neighborhood of $p$ that $f(q)=0$ w... | 9 | https://mathoverflow.net/users/14830 | 68327 | 41,986 |
https://mathoverflow.net/questions/68322 | 13 | Let $f\in\mathbb{Z}[X\_1,\ldots,X\_n]$ be a Diophantine equation which, for the purposes of this question, I will assume is homogeneous and nonsingular on $\mathbb{R}^n\setminus\{0\}$ (so that $\nabla f\not=0$). Supposing that it has infinitely many primitive integer zeros, we can posit that they are smoothly distribut... | https://mathoverflow.net/users/1004 | A heuristic for the density of solutions to Diophantine equations | You are on the way to redeveloping the singular series, which does indeed give the correct asymptotic for integral solutions to many flavors of Diophantine equation -- they key words here are "Hardy-Littlewood method" or "circle method," which you can read about in any text on analytic number theory, such as the book o... | 15 | https://mathoverflow.net/users/431 | 68332 | 41,988 |
https://mathoverflow.net/questions/68335 | 27 | There is a theorem of Deligne in SGA4 that a "coherent" topos (e.g. one on a site where all objects are quasi-compact and quasi-separated) has enough points (i.e. isomorphisms can be detected via geometric morphisms to the topos of sets). I've heard it said that this is a form of Goedel's completeness theorem for first... | https://mathoverflow.net/users/344 | What do coherent topoi have to do with completeness? | They are indeed formally equivalent. See for instance Johnstone: Topos theory, p. 243 but here is a quick explanation. Given a topos $T$ one may define a geometric theory associated to it consisting of formulas describing essentially the topos. More specifically a geometric morphism from another topos $S$ to $T$ is the... | 27 | https://mathoverflow.net/users/4008 | 68342 | 41,994 |
https://mathoverflow.net/questions/68312 | 7 | For this question let $G$ be a group, perhaps infinite, and let $H\_i$ for $i\in I$ be a (finite) family of subgroups closed under taking intersections. I am interested in the coset poset $\mathcal{C}(G,\{H\_i\})$ which is defined as the set of cosets $g H\_i$ with ordering by inclusion. Note that $g H\_i\subseteq g'H\... | https://mathoverflow.net/users/109 | Posets of cosets and contractibility | Here are some references that should be of use.
(i) H. Abels and S. Holz, Higher generation by subgroups , J. Alg, 160, (1993), 311– 341.
(ii) S. Holz, 1985, Endliche Identifizier zwishen Relationen , Ph.D. thesis, Univerist\"{a}t Bielefeld.
(iii) A. Bak, R. Brown, G. Minian and T.Porter, Global Actions, Groupoi... | 5 | https://mathoverflow.net/users/3502 | 68344 | 41,996 |
https://mathoverflow.net/questions/68347 | 0 | In his paper [Kronecker Graphs: An approach to modeling Networks](http://portal.acm.org/citation.cfm?id=1756039) Jure et Al, mention that an important property of networks are that they are heavy tailed.
I'm trying to get an insight on what this really means. Do you have good examples of real heavy-tailed distributio... | https://mathoverflow.net/users/15914 | Heavy Tailed Network | "Heavy tailed" is a bit tricky in the sense that there is no unique definition of what it really means. Usually, it refers to the MGF being infinite on the side of zero that the heavy tail is (i.e. right tail, MGF infinite for arguments $> 0$ etc.). However, as is pointed out by Mikosch in one of his papers (I forget w... | 1 | https://mathoverflow.net/users/15752 | 68350 | 42,000 |
https://mathoverflow.net/questions/68291 | 2 | Page 276 in the book Differential Topology and Quantum Field theory by C. Nash, describes a "generalization of determinant of linear map" as follows: for linear map
$O:{V} \to {W}$
its determinant is (No further description in the book)
$\det O \in {\left( {{\Lambda ^{\max }}V} \right)^\*} \otimes \left( {{\Lambd... | https://mathoverflow.net/users/15884 | about Generalized Determinant? | In order for such a definition to be multiplicative with respect to composition, the determinants between vector spaces of different dimensions has to be zero. Here is a proof.
>
> Let $f: \mathbb{R}^k \to \mathbb{R}^m$ be a linear map. If $k < m$, then we let $g: \mathbb{R}^m \to \mathbb{R}^m$ to be any projection... | 3 | https://mathoverflow.net/users/121 | 68357 | 42,003 |
https://mathoverflow.net/questions/68305 | 9 | The following question is based on some remarks in section I.2 of Deligne's book *Equations Différentielles à Points Singuliers Réguliers*.
Let $X$ be a smooth complex variety and $X\_1$ the first infinitesimal neighborhood of the diagonal in $X \times X$, so there is a natural morphism $X \to X\_1$. If we write $p\_... | https://mathoverflow.net/users/3544 | Grothendieck connections and jets | According to EGA IV, section 16.7 we may denote 1-jets by $\mathcal{P}^1\_X(V)$ (In fact you may even take for $V$ a coherent sheaf). There is a canonical morphism $V \to\mathcal{P}^1\_X(V)$. Notice that it is not $\mathcal{O}$-linear but just $\mathbb{C}$-linear. Now Proposition 16.8.4 tells us
$$
\mathcal{H}om\_X(\ma... | 3 | https://mathoverflow.net/users/6348 | 68359 | 42,004 |
https://mathoverflow.net/questions/68297 | 2 | Let $[X/G]$ be a quotient stack such that $X$ is irreducible and $G$ acts trivially on $X$ (I am just adding automorphisms to every point). Under which hypothesis is $[X/G]$ irreducible as an Artin stack?
| https://mathoverflow.net/users/33841 | Irreducibility of quotient stacks. | If you have a presentation $s, t:R \to U$ of a stack $[U/R]$, the set of points
of $[U/R]$ is just the equivalence classes of points in $|U|$ determined by
the equivalence relation given by the image of the map $|R| \to |U|\times |U|$.
In particular $|U| \to |[U/R]|$ is always surjective.
The topology on the underlyi... | 3 | https://mathoverflow.net/users/1084 | 68361 | 42,005 |
https://mathoverflow.net/questions/68360 | 2 | Let $C$ be a smooth projective connected curve of genus $g$ over $\bar{\mathbf{Q}}$. Fix a finite **non-empty** (Edit) set of closed points $S$ in $C$ and let $U$ be the complement of $S$ in $C$.
**Q1.** *(Algebraic formulation)* Does there exist a finite (surjective) morphism $\pi:C\longrightarrow \mathbf{P}^1\_{\ba... | https://mathoverflow.net/users/4333 | Can we construct rational functions with prescribed ramification on an algebraic curve over \Qbar | No, it is easy to construct examples where this is not possible (aside from trivial ones with $|S| < 3$). For example, if $g(C)>0$ one can find $S$ arbitrarily large so that the points of $S$ give linearly independent elements in $Pic(C)$. For such an $S$ there can be no map of the kind you want since the elements of $... | 3 | https://mathoverflow.net/users/519 | 68363 | 42,007 |
https://mathoverflow.net/questions/68356 | 6 | I have a rather stupid lattice theory question. Suppose $L$ is a root lattice that can be primitively embedded in the $ E\_8 $ lattice. Is the orthogonal complement of $ L$ in $E\_8$ unique up to isomorphism, or for different primitive embeddings could I get non-isomorphic complements?
| https://mathoverflow.net/users/4192 | Orthogonal Complements of Root Lattices in E_8 | You can get different orthogonal complements for different embeddings. There are two different embeddings of $A\_{7}$ in $E\_{8}$ so that for the first embedding the orthogonal complement is the lattice $A\_{1}$, and for the second embedding the orthogonal complement is the lattice $\langle 8 \rangle$.
| 7 | https://mathoverflow.net/users/439 | 68368 | 42,008 |
https://mathoverflow.net/questions/68364 | 3 | In the definition of a (weakly holomorphic) modular form we require a specific growth behaviour at all the cusps.
I assume that this requirement is not void, i.e. not automatically satisfied.
However, I have never seen an example of a modular form, which satisfies the growth condition at one cusp but not at another.
... | https://mathoverflow.net/users/3757 | Modular form with pole of infinite order | Let $\Gamma = \Gamma\_0(2)$, and let $\Delta$ be the usual weight 12 cusp form. Then $f(z) = \Delta(2z) / \Delta(z)$ is a meromorphic modular function of weight 0 and level $\Gamma$, holomorphic on the upper half-plane, and with a zero at the cusp $\infty$ and a pole at the cusp $0$. So $\exp(f(z))$ is an example of a ... | 7 | https://mathoverflow.net/users/2481 | 68371 | 42,011 |
https://mathoverflow.net/questions/68366 | 7 | Let $k$ be a commutative ring and let $G$ be a flat affine algebraic group scheme over $k$.
Let $G$ act by algebra automorphisms on the commutative $k$-algebra $A$. So $G(R)$ acts by $R$-algebra
automorphisms on $A\otimes\_k R$ for any commutative $k$-algebra $R$.
Let $N$ be the nilradical of $A$. Is $N$ always a $G$ ... | https://mathoverflow.net/users/4794 | Does the action of an affine group scheme preserve the nilradical of an algebra? | This is true if you assume that $G$ is smooth. Consider the coaction $A \to A \otimes\_k k[G]$; since $k[G]$ is a smooth $k$-algebra, the nilradical of $A \otimes\_k k[G]$ is $N \otimes\_k k[G]$; since $N$ is sent to the nilradical of $A \otimes\_k k[G]$, this implies the thesis.
Otherwise it is false in general, eve... | 14 | https://mathoverflow.net/users/4790 | 68374 | 42,014 |
https://mathoverflow.net/questions/68367 | 20 | I consider on $M\_n(\mathbb C)$ the normalized $2$-norm, i.e. the norm given by $\|A\|\_2 = \sqrt{\mathrm{Tr}(A^\* A)/n}$.
My question is whether a $k$-uple of hermitian matrices that are almost commuting (with respect to the $2$-norm) is close to a $k$-uple of commuting matrices (again with respect to the $2$-norm).... | https://mathoverflow.net/users/10265 | Are almost commuting hermitian matrices close to commuting matrices (in the 2-norm)? | There is a [recent paper](http://arxiv.org/abs/1002.3082) by Glebsky titled "Almost commuting matrices with respect to normalized Hilbert-Schmidt norm" which shows that this is indeed true for any $k$ for Hermitian matrices (and in fact also unitary and normal matrices).
| 14 | https://mathoverflow.net/users/7767 | 68379 | 42,016 |
https://mathoverflow.net/questions/68376 | 11 | I would like to know what is known about the spectrum of the Laplace-Beltrami operator on 2-dimensional negatively curved surfaces of constant curvature.
For instance,
* What is the spectrum of the Hyperbolic plane of constant curvature $-k$?
* What is the Laplace-Beltrami operator and its spectrum for a compact s... | https://mathoverflow.net/users/13825 | Laplace-Beltrami Operator on Surfaces | Probably you should try the following:
Chavel, "Eigenvalues in Riemannian geometry"
Buser "Geometry and spectra of compact Riemann surfaces"
and if you can read french: Berger, Gauduchon, Mazet "Le spectre d'une variete riemannienne"
| 6 | https://mathoverflow.net/users/15673 | 68381 | 42,017 |
https://mathoverflow.net/questions/68377 | 1 | Consider a real-valued Markov process $X$ with a transition density $f(x,y)$, i.e.
$$
\mathsf P[X\in A|X\_0 = x] = \int\limits\_A f(x,y)\,dy.
$$
For this process I want to find
$$
u(x) = \mathsf P[X\_n < 0\text{ for some }n\geq 0|X\_0 = x].
$$
Since for this problem does not matter the distribution of this process f... | https://mathoverflow.net/users/11768 | Modification of a Markov process on the real line | Here is a realization of the process $(Y\_n)$, which shows that $u(x)=v(x)$ for every $x\ge0$. Fix $x\ge0$, consider $(X\_n)$ starting from $X\_0=x$, introduce the stopping time $\tau=\inf\{n\ge0;X\_n<0\}$ and the function $\varphi$ such that $\varphi(y)=y$ for every $y\ge0$ and $\varphi(y)=a$ for every $y<0$, and fina... | 1 | https://mathoverflow.net/users/4661 | 68383 | 42,018 |
https://mathoverflow.net/questions/68386 | 12 | I'm looking for a theorem of the form
>
> If $R$ is a nice ring and $v$ is a reasonable element in $R$ then Kr.Dim$(R[\frac{1}{v}])$ must be either Kr.Dim$(R)$ or Kr.Dim$(R)-1$.
>
>
>
My attempts to do this purely algebraically are not working, so I started looking into methods from algebraic geometry. I thou... | https://mathoverflow.net/users/11540 | Can we say anything about the Krull dimension of a localization? | The dimension of $R[1/v]$ is the biggest height of some prime ideal $P$ such that $v\notin P$. So, let $I\_{d-1}$ be the intersection of all primes of height at least $d-1$ ($d= \dim R$), then
>
> $\dim R[1/v] \geq d-1$ if and only if $v\notin I\_{d-1}$.
>
>
>
Under a mild condition (all maximal ideals has he... | 10 | https://mathoverflow.net/users/2083 | 68395 | 42,025 |
https://mathoverflow.net/questions/68393 | 2 | This is a follow up to my [previous question](https://mathoverflow.net/questions/68154/strict-transform-of-a-line-bundle), and I have lowered my demands to a situation as follows:
Let $X$ be an algebraic variety, $\mathcal{I}$ a coherent sheaf of ideals and $\mathcal{L}$ a line bundle on $X$. Let $h\_1,\ldots,h\_k\in... | https://mathoverflow.net/users/9947 | Followup; Strict Transform of a Line Bundle | Jesko, you still need more. The condition $Z(\mathscr I)\subseteq Z(h\_i)$ ensures that the strict transform of these sections will be different from their pull-back, but you also need a condition that guarantees that they will be linearly equivalent.
Here is the point more concretely: Let $D\_i=Z(h\_i)$ and $\sum E... | 3 | https://mathoverflow.net/users/10076 | 68399 | 42,028 |
https://mathoverflow.net/questions/68316 | 6 | The first Hirzebruch surface $F\_1$ does not admit any Kaehler metric of constant scalar curvature. Yet it admits an extremal metric representing each Kaehler class as shown by Calabi. The two points blow-up of $\mathbb{CP}^2$ behaves similarly.
Do you know any other example of Fano manifold such that:
1. The anti-ca... | https://mathoverflow.net/users/15600 | Extremal Fano with non constant scalar curvature vs Kaehler-Einstein Fano manifolds | I'm not 100% sure what you're looking for because of the wording of your question. Do you want a Kähler manifold for which *every* Kähler class has an extremal representative, but no constant scalar curvature representative? Or examples of manifolds with just one such Kähler class?
If it's the latter, then note that ... | 5 | https://mathoverflow.net/users/380 | 68402 | 42,031 |
https://mathoverflow.net/questions/30325 | 1 | Consider $S\_{2k} (\Gamma\_0 (N))$ and let $S(N)$ denote the direct limit of the finite direct sums of the $S\_{2k}$. Since each $S\_{2k} (\Gamma\_0 (N))$ is also a Hilbert space w.r.t. the [Petersson inner product](http://en.wikipedia.org/wiki/Petersson_inner_product), $S(N)$ is as well, and we can consider the C\*-al... | https://mathoverflow.net/users/1847 | Bounded operators on direct limit of direct sums of spaces of cusp forms | That colimit of finite-dimensional spaces won't actually be a Hilbert space, but it will nevertheless be quasi-complete. Still, it won't be a representation space for $GL(2,R)$ or $GL(2,Q\_p)$, which is what has taken people in other directions, specifically, to look at (for example) $L^2$ completions of spaces of auto... | 4 | https://mathoverflow.net/users/15629 | 68409 | 42,035 |
https://mathoverflow.net/questions/68378 | 20 | This should be a very easy question, but the proof in Lawson/Michelson (Spin geometry) is wrong and I do not find a really correct and complete argument:
Let V be a nonzero real vector space with scalar product: Why is the Clifford-algebra (constructed from the tensor algebra by quotiening out an ideal) non-zero?
| https://mathoverflow.net/users/3816 | Clifford algebra non-zero | To show that an algebra constructed as a quotient of the tensor algebra of a vector space is nonzero, one of the main ways to go is to construct representations. We can do this for the Clifford algebra as follows.
Let $V$ be a vector space over a field $k$ and $(,):V\times V \to k$ a symmetric bilinear form on $V$. T... | 28 | https://mathoverflow.net/users/703 | 68411 | 42,036 |
https://mathoverflow.net/questions/68415 | 2 | Suppose $d$ is a constant $< n^2$, and $m > n$. I have a kind of factorial function where I subtract $d$ from each *pair* of terms.
Is there any simple upper bound on
$$
P = ( m (m-1) - d) \times ( (m-2) (m-3) - d ) \times \cdots \times \times ((n+1) n - d)
$$
I only need an polynomially tight upper bound. If it ma... | https://mathoverflow.net/users/9896 | Simple variation on factorial --- upper bound | Edit. Of course, I would find the mistake after posting. k ranges over the even numbers from 2 to 2h, so the result is not as nice. I will try rescuing the approach. End Edit
Each term is close to (m- k+ sqrt(d) + 1/2)(m - k- sqrt(d) + 1/2), where k ranges from 1 to h=(m-n)/2. (h may be off by 1.) Your product is clo... | 1 | https://mathoverflow.net/users/3206 | 68422 | 42,042 |
https://mathoverflow.net/questions/68427 | 6 | Dear All,
my following question may be known and ought to be known, so in case it is folklore please could you give me the references.
To start, it is obvious that growth of rational languages are always either polynomial or exponential. That is, if $L$ is a regular language, then the sequence $a\_n$, where $a\_n$ ... | https://mathoverflow.net/users/13070 | Growth zeta-functions of regular languages | Yes (though the standard term for these is *generating functions* rather than zeta functions); in fact, there's a relatively straightforward explicit construction for finding the generating function for a regular language given an unambiguous regular expression for it; replace null with $0$, any symbol with $x$, concat... | 9 | https://mathoverflow.net/users/7092 | 68431 | 42,046 |
https://mathoverflow.net/questions/68430 | 4 | Let $C$ be a convex polygon in the plane containing the origin, and let $r(\theta)$ for $\theta\in[0,2\pi)$ be a parametrization of its boundary. Is there a condition on $r$ that is equivalent to (or necessary for) convexity of $C$?
| https://mathoverflow.net/users/13363 | Polar interpretation of convexity | Convexity is equivalent to the function $r(\theta):[0,2\pi)\to\mathbb{R}^2$ being well-defined and satisfying the condition
$$| r(\lambda \theta \_1+(1-\lambda)\theta \_2)| \geq \left| \lambda r(\theta \_1)+(1-\lambda)r(\theta \_2)\right|$$ for all $\theta\_1,\theta\_2$ and $\lambda\in (0,1)$. I'm using vector valued f... | 4 | https://mathoverflow.net/users/2384 | 68434 | 42,048 |
https://mathoverflow.net/questions/68436 | 50 | On page 98 of Weibel's *[An Introduction to Homological Algebra](http://books.google.com/books?id=flm-dBXfZ_gC&q=continuum+hypothesis#v=snippet&q=continuum%2520hypothesis&f=false)* he mentions that the ring $R = \prod\_{i=1}^\infty \mathbb{C}$ has global dimension $\geq 2$ with equality iff the continuum hypothesis hol... | https://mathoverflow.net/users/11540 | What the heck is the Continuum Hypothesis doing in Weibel's Homological Algebra? | In [Osofsky, B. L. Homological dimension and cardinality. Trans. Amer. Math. Soc. 151 1970 641--649. MR0265411 (42 #321)] she proved that the global dimension of a countable product of fields is $k+1$ iff $2^{\aleph\_0}=\aleph\_{k}$. In particular, if the continuum hypothesis holds, so that $2^{\aleph\_0}=\aleph\_1$, t... | 54 | https://mathoverflow.net/users/1409 | 68439 | 42,050 |
https://mathoverflow.net/questions/68421 | 62 | If they are not proper, two complex algebraic varieties can be nonisomorphic yet have isomorphic analytifications. I've heard informal examples (often involving moduli spaces), but am not sure of the references.
>
> What are the simplest examples of nonisomorphic complex algebraic varieties with isomorphic analyti... | https://mathoverflow.net/users/299 | Simplest examples of nonisomorphic complex algebraic varieties with isomorphic analytifications | I believe the following is an elementary example: Let $X$ be an affine smooth curve of geometric genus at least one. Let $L$ be a non-trivial algebraic line bundle
on $X$ (easy to produce such things). Then $L$ is analytically trivial because $X$ is a Stein space ($H^1(X, O\_X)=0$) with trivial integral $H^2$. Hence, t... | 47 | https://mathoverflow.net/users/1826 | 68444 | 42,054 |
https://mathoverflow.net/questions/68438 | 10 | I recall reading somewhere that if a conformal class contains an Einstein metric then that metric is the unique metric with constant scalar curvature in its conformal class, with the exception of the case of the round sphere. Does this sound right? If it is true: where can I find the proof of this result?
| https://mathoverflow.net/users/15856 | Einstein metrics and conformal geometry | This is not the original reference, but the most general (i.e., applicable to the non-compact and pseudo-Riemannian cases) result I know is:
[Kühnel, W., & Rademacher, H. Conformal diffeomorphisms preserving the Ricci tensor. Proceedings of the American Mathematical Society, 123 (1995), no. 9, 2841–2848.](http://www.... | 13 | https://mathoverflow.net/users/2063 | 68458 | 42,064 |
https://mathoverflow.net/questions/68451 | 3 | Given $s$ successes in $n$ trials, where $p=\frac{s}{n}$, is there a standard way to determine if I have enough data to compute a meaningful statistic? For example, given $s=1, n=10, p=0.1$, the 95% confidence interval ranges from $0.002 < p < 0.445$.
It seems like I could just use the gap between the 95% confidence ... | https://mathoverflow.net/users/15910 | Standard way of determining if you have enough data to reliably compute success probability | It doesn't seem clear how you got your alleged 95% confidence interval. Whether you're going about it the wrong way would be easier to assess if you told us how you're going about it.
The most frequently taught method of finding a confidence interval for $p$ works reasonably well when the observed number of successes... | 2 | https://mathoverflow.net/users/6316 | 68460 | 42,066 |
https://mathoverflow.net/questions/68428 | 1 | I am looking at the description of LTI systems in the time domain.
Intuitively, I'd have guessed it would be the composition of the input function and some "system function".
$$ y(t) = f(x(t)) = (f\circ x)(t)$$
Where $x(t)$ is the input, $y(t)$ output and $f(x)$ a "system function".
Why is it not that way? Could su... | https://mathoverflow.net/users/15928 | Why is the output of an LTI system the convolution of the input funtion and the impulse response? | Well, it's just as simple as this: the output at any moment reflects the effect of the input at just that moment, plus the lingering effect after one second of the input from one second before, plus the lingering effect after two seconds of the input from two seconds before, plus the lingering effect after three second... | 1 | https://mathoverflow.net/users/3902 | 68463 | 42,069 |
https://mathoverflow.net/questions/68437 | 23 | The *divisor bound* asserts that for a large (rational) integer $n \in {\bf Z}$, the number of divisors of $n$ is at most $n^{o(1)}$ as $n \to \infty$. It is not difficult to prove this bound using the fundamental theorem of arithmetic and some elementary analysis.
My question regards what happens if ${\bf Z}$ is rep... | https://mathoverflow.net/users/766 | The divisor bound in number fields | As long as you allow a fixed number field $F = {\bf Q}(\alpha)$ you can prove $H^{o(1)}$ as you more-or-less suggest towards the end, by first showing that the number of *ideals* of $F$ that divide $n$ is $H^{o(1)}$ and then proving that any ideal has $O(\log^r H)$ generators of height at most $H$, where $r = r\_1 + r\... | 23 | https://mathoverflow.net/users/14830 | 68464 | 42,070 |
https://mathoverflow.net/questions/68080 | 14 | It is well known that the ordinary and exponential generating functions of a sequence of numbers are related by an integral transform (the [Borel transformation](http://en.wikipedia.org/wiki/Borel_summation)).
Does there exist a combinatorial theory of integral transforms? The example above indicates that something m... | https://mathoverflow.net/users/6779 | Combinatorial interpretations of integral transforms | I claim that there is a construction here similar to the one in "Lagrange inversion for species" by Gessel and Labelle, which can explain the general picture.
To every species $S:\mathcal{B}\to \mathcal{B}$ there should correspond a "labelled" version $L(S):\mathcal B\to \mathcal B$ whose exponential generating func... | 7 | https://mathoverflow.net/users/2384 | 68472 | 42,076 |
https://mathoverflow.net/questions/49490 | 0 | The following question was asked by me on the forum sci.math.research,
“An imprimitive group is a transitive permutation group with a non-trivial
equivalence relation compatible with the action of the group.
Suppose we have a transitive transformation semigroup with a non-trivial
equivalence relation compatible ... | https://mathoverflow.net/users/10833 | A Nomenclature Issue : Imprimitive Semigroup? | The name is not commonly used, but there is no reason why you shouldn't use it since it is as good a name as any. The term primitive transformation monoid has been used to mean no nontrivial congruences. You might also want make clear what you mean by transitive since there are two possible ways to generalize the group... | 2 | https://mathoverflow.net/users/15934 | 68480 | 42,082 |
https://mathoverflow.net/questions/68385 | 4 | I have a 3 x 4 projection matrix $P$ given that calculates a homogeneous 2-Vector ${\bf i}=(u,v,w)^T$ on some screen (e.g.) from a homogeneous 3-Vector ${\bf x}=(x,y,z,w)^T$ in world space by $P \cdot {\bf x} = {\bf i}$.
How can I calculate the position of the camera in world space from that?
| https://mathoverflow.net/users/15918 | Calculate camera position from 3x4 projection matrix | Sorry for answering my own question, but just now a colleague told me the solution and I want to share it - maybe it is of some use for anybody else some day.
1. Separate $P$ into a 3x3 matrix $P'$
(including the first three columns)
and a vector $\bf F'$ (the last
column).
2. Invert $P'$
3. The projection
reference ... | 6 | https://mathoverflow.net/users/15918 | 68488 | 42,087 |
https://mathoverflow.net/questions/68487 | 1 | Anyone with any ideas on this one? Mathematica and Matlab can't do it.
$\int\_0^\infty \frac{\sin(a y) \coth(y)}{(1+9y^2)^2}dy$
Here coth is the hyperbolic cotangent, and a is a positive parameter which will figure in the answer. It's giving me fits.
Greg
| https://mathoverflow.net/users/13358 | Difficult integral: $\int_0^\infty \frac{\sin(a y) \coth(y)}{(1+9y^2)^2}dy$ | Maple says "undefined". But if you replace $\coth(y)$ with the equivalent $\frac{e^y+e^{-y}}{e^y-e^{-y}}$ then an answer is produced in terms of the cosine integral Ci and the shifted sign integral Ssi
$$\frac{\alpha}{18}\cosh (\frac{\alpha}3) {\it Ci}( \frac{\alpha i}3) -
\frac{\alpha \pi i}{36}\cosh (\frac{\alpha}3... | 2 | https://mathoverflow.net/users/8008 | 68490 | 42,089 |
https://mathoverflow.net/questions/68418 | 3 | Is there some reference where the existence of local generalized action-angle variables is discussed in some detail for concrete examples of hamiltonian systems of mechanical type?
After Dazord and Delzant, by local generalized action-angle coordinates on a symplectic manifold $(M^{2n},\omega)$ I mean a locally trivi... | https://mathoverflow.net/users/12617 | On degenerate integrable hamiltonian systems | If I understand correctly what you are after, a good place to have a look at is this expository article by Fasso'
F. Fasso'. *Superintegrable Hamiltonian systems: geometry and perturbations*. Acta Appl. Math. 87 (2005),
in which he reviews the case of superintegrable Hamiltonian systems defined by Mischenko and Fom... | 2 | https://mathoverflow.net/users/13022 | 68491 | 42,090 |
https://mathoverflow.net/questions/68496 | 2 | Let $X(2)$ denote the compact Riemann surface obtained by compactifying $Y(2) = \Gamma(2)\mathfrak{h}$ by adding cusps.
The modular $\lambda$-function on the complex upper half plane $\mathfrak{h}$ induces an isomorphism $\lambda:X(2)\longrightarrow \mathbf{P}^1$. It has a $q$-expansion $$\lambda = \sum\_{j=0}^\inft... | https://mathoverflow.net/users/4333 | Convergence radius of the q-expansion of the modular lambda function | Sorry about the one-sentence answer. Here is an expanded version that uses the following facts: The modular function $\lambda$ is holomorphic on the upper half-plane, invariant under the action of $\Gamma(2)$, and non-singular at infinity.
From the invariance under the subgroup of $\Gamma(2)$ generated by $\binom{12}... | 3 | https://mathoverflow.net/users/121 | 68497 | 42,094 |
https://mathoverflow.net/questions/68505 | 1 | I've been stuck for quite a while on what is probably a trivial problem. Let $X\subset\mathbb{P}^n$ be a smooth projective curve, and let
$$\mathcal{I}=\{(p,q,r):p,q\in X,p\neq q,r\in\overline{pq}\}$$
(where $\overline{pq}$ is the line that joins $p$ and $q$) and let
$$\mathcal{J}=\{(p,r):p\in X,r\mbox{ lies on the ... | https://mathoverflow.net/users/14143 | Projecting projective curves | $\alpha(\mathcal I)$ is the *secant variety* of $X$. I think it is enough if you understand this one, the other is similar. Of course the actual **proof** that $\alpha(\mathcal I)\subsetneq \mathbb P^n$ is exactly the dimension count you're mentioning. So, I assume you're looking for intuition and/or a heuristic explan... | 3 | https://mathoverflow.net/users/10076 | 68507 | 42,098 |
https://mathoverflow.net/questions/68494 | 9 | Ramsey Theory says that every sufficently large (but finite) complete graph having $d-$coloured edges contains a monochromatic complete subgraph with $k$ vertices.
One could ask for asymptotics: Let $A(n,d,k)$ be the minimal number of
monochromatic complete subgraphs with $k$ vertices contained in any complete graph ... | https://mathoverflow.net/users/4556 | Asymptotics for Ramsey Theory | I'm not sure this is as interesting as you think. Here is a sketch that $\alpha=k$. For large enough $n$ we can apply Szemer\'edi's regularity lemma simultaneously to all the colour classes. The result is a collection of block graphs that add up to 1 everywhere (approximately). If the number of blocks is larger than th... | 10 | https://mathoverflow.net/users/1459 | 68510 | 42,099 |
https://mathoverflow.net/questions/68503 | 9 | If $f \colon C \to C'$ is a dominant morphism of smooth projective curves, there is a norm map $f\_\ast = \mathrm{Nm} \colon JC \to JC'$ between their Jacobians, and we can consider the abelian subvariety $Z = (\ker \mathrm{Nm})^0$ and the polarization on $Z$ induced from $JC$. In two particularly interesting cases thi... | https://mathoverflow.net/users/1310 | Has anyone studied the Prym map for double covers with two ramification points? | Hi, this is actually a part of my Ph.D. thesis. I am going to discuss it in 6 months. Here you can find a preprint of the work with my advisor <http://arxiv.org/abs/1010.4483>. It is not the final version, so there could be some minor mistakes.
We have proved that, with two ramification point, the Prym map is generic... | 13 | https://mathoverflow.net/users/6784 | 68528 | 42,109 |
https://mathoverflow.net/questions/68527 | 4 | The concepts of [overdetermined](http://eom.springer.de/O/o070660.htm) and [underdetermined](http://eom.springer.de/u/u095150.htm) PDE systems are well known. However, all sources I have so far looked into appear to avoid giving any name to PDE systems which are *neither* overdetermined *nor* underdetermined. Is there ... | https://mathoverflow.net/users/2149 | A name for PDE systems which are neither under- nor overdetermined? | determined: Bryant et. al, Exterior Differential Systems, p. 189
| 10 | https://mathoverflow.net/users/13268 | 68531 | 42,110 |
https://mathoverflow.net/questions/68533 | 5 | Kurepa Hypothesis says there is a Kurepa tree, which is a $\omega\_1$-tree has at least $\omega\_2$ many branches.
It is known that beginning from a model with an inaccessible cardinal $\kappa$, after collapes $\kappa$ to $\omega\_2$ using the Levy collape, then in the generic extension, Kurepa Hypothesis fails.
In abo... | https://mathoverflow.net/users/3692 | On the independence of the Kurepa Hypothesis | The answer is yes, and merely forcing over your model to add additional subsets of $\omega\_1$ will pump up the value of $2^{\omega\_1}$, while not creating Kurepa trees.
Specifically, let us start in $V$, where $\kappa$ is an inaccessible cardinal, and suppose also that the GCH holds. You mentioned the result of Si... | 6 | https://mathoverflow.net/users/1946 | 68538 | 42,116 |
https://mathoverflow.net/questions/68498 | 12 | Let $G$ denote the $\operatorname{Spin}(n)$ group with $n>4$ and let $\Gamma$ be a cyclic subgroup $G$ of a prime order $p >2$. When does the projection $G \to G/\Gamma$ induce a surjection
between cohomology groups $H^3$ with integral coefficients?
| https://mathoverflow.net/users/10086 | Cohomology of the quotient of a Lie group by a finite subgroup | Here is a solution for $G$ a compact, simple, connected, simply-connected Lie group and $\Gamma$ a subgroup of the center of $G$.
The group $H^3(G,Z)$ classifies $U(1)$-gerbes over $G$. A gerbe $\mathcal{G}$ is in the image of the pullback map $$H^3(G/\Gamma,Z) \to H^3(G,Z)$$ if and only if it admits a *$\Gamma$-equi... | 4 | https://mathoverflow.net/users/3473 | 68543 | 42,120 |
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