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https://mathoverflow.net/questions/88560 | 0 | I have a question regarding sets of finite perimeter. I feel that it should be true, but I didn't manage to prove or find a reference about it. Suppose $D$ is an open, bounded subset of $\Bbb{R}^n$, and define the perimeter of a measurable set $A \subset D$ as
$$P\_D(A)=\sup \left \lbrace \int\_D \chi\_A {\rm div} \... | https://mathoverflow.net/users/13093 | A property of sets of finite perimeter | I'm sorry that I answer my own question, but I found out the answer this morning from my teacher. There are examples of sets of finite perimeter with positive measure, which do not contain any open ball.
For example, take $D=B(0,1)$, the unit ball in $\Bbb{R}^2$ and denote $S=D \cap \Bbb{Q}^2=(x\_n)\_{n \geq 0}$. The... | 1 | https://mathoverflow.net/users/13093 | 88617 | 52,455 |
https://mathoverflow.net/questions/84749 | 4 | Let $M$ be a closed manifold, $f:M\to M$ be a homeomorphism, and $\phi\_t:M\to M$ be a flow. .
The map $f$ is said to be (point)-transitive if some orbit $\lbrace f^nx:n\in\mathbb{Z}\rbrace$ is dense in $M$.
The flow $\phi\_t$ is said to be (flow)-transitive if some flow line $\lbrace\phi\_tx:t\in\mathbb{R}\rbrace... | https://mathoverflow.net/users/11028 | Transitivity of a flow and its time-1 map | I think the answer for the second question is no. We can construct a counterexample as follows:
Let $f\colon M\to M$ be an arbitrary transitive homeomorphism and $u\colon M\to\mathbb (0,1/4)$ be an arbitrary non-constant continuous function. Then, let's define $$c(x):=u(x)-u(f(x))+1,\quad\forall x\in M,$$
and consid... | 2 | https://mathoverflow.net/users/889 | 88625 | 52,458 |
https://mathoverflow.net/questions/88621 | 2 | Roughly speaking, a theorem by Vershik says that every ergodic invertible measure-preserving transformation is isomorphic to some "adic" transformation on the spaces of paths of a Bratelli-Vershik graph. What is the adic transformation corresponding to a Bernoulli shift ?
| https://mathoverflow.net/users/21339 | What is the adic realization of a Bernoulli shift ? | The short answer is that nobody knows. The reason is that Vershik's proof uses Rokhlin's towers and is thus virtually non-constructive.
As far as I know, the only known examples of explicit adic realizations are substitutional dynamical systems and the irrational rotations of the circle. Even for a simple ergodic ro... | 4 | https://mathoverflow.net/users/8131 | 88626 | 52,459 |
https://mathoverflow.net/questions/87387 | 13 | I am looking for an example of a well-pointed space in which no (sufficiently small) neighbourhood of the base-point is contractible. As usual, a well-pointed space is a pointed space in which the inclusion of the base-point is a Hurewicz cofibration.
In the reverse direction, I am looking for conditions under which ... | https://mathoverflow.net/users/21095 | Well-pointed space which is not locally contractible | My initial answer was fully ignorant of the literature on the subject; I'm now recalling that it does exist. In 1934, Borsuk and Mazurkiewicz constructed a $2$-dimensional AR (=contractible, locally contractible $2$-dimensional compactum) $X$ that is not a countable union of smaller ARs (Sur les rétractes absolus indéc... | 12 | https://mathoverflow.net/users/10819 | 88628 | 52,460 |
https://mathoverflow.net/questions/88623 | 0 | It is a classical result of Ribet that if an eigenform has CM the its residual projective image is "small" (cyclic or dihedral.) Is the converse true, i.e, if f is a form whose associated residual Gal representation has "small" projective image then f has CM? Thanks.
| https://mathoverflow.net/users/5310 | On the image of the residual representation attached to a CM form | No. For instance there are plenty of modular forms that are not CM, but are congruent mod p to CM forms or to Eisenstein series, and thus whose residual Galois representations have small image.
| 5 | https://mathoverflow.net/users/2481 | 88629 | 52,461 |
https://mathoverflow.net/questions/88627 | 10 | In the paper *On the complex projective spaces*, Hirzebruch and Kodaira prove the following:
>
> If $X$ is compact Kähler manifold diffeomorphic to $\mathbb{CP}^n$, then $X$ is biholomorphic to $\mathbb{CP}^n$ if $n$ is odd or $n$ is even but $c\_1 \neq -(n+1)g$.
>
>
>
Here, $c\_1$ is the first Chern class of... | https://mathoverflow.net/users/15882 | On compact Kähler manifold diffeomorphic to complex projective space | The case when $n$ is even has been famously ruled out by Yau as a consequence of his proof of the Calabi Conjecture, see [his original paper](http://www.jstor.org/stable/67110) or [these notes](http://www.math.columbia.edu/%7Etosatti/cpn.pdf).
In fact, thanks to results of Novikov, you can replace diffeomorphism by h... | 24 | https://mathoverflow.net/users/13168 | 88633 | 52,462 |
https://mathoverflow.net/questions/88632 | 0 | Can I simplify:
\begin{equation}
\sum\_{x=x\_0}^{x\_1} \frac{1}{ax+b}
\end{equation}
| https://mathoverflow.net/users/19899 | Simplifying finite sum over 1/(ax+b) | Mathematica (or rather Wolfram Alpha) gives an answer in terms of the digamma function: [http://www.wolframalpha.com/input/?i=Sum[1%2F%28a+x%2Bb%29%2C{x%2Cx0%2Cx1}]](http://www.wolframalpha.com/input/?i=Sum[1%2F%28a%20x%2Bb%29%2C%7Bx%2Cx0%2Cx1%7D])
| 3 | https://mathoverflow.net/users/3400 | 88636 | 52,464 |
https://mathoverflow.net/questions/88624 | 17 | I do not know if this question is appropriate for this site, but I posted [here](https://math.stackexchange.com/questions/107804/coordinate-free-proof-of-the-hamiltonian-character-of-the-geodesic-flow) without having answers, so now I make this attempt.
Let be $(M,g)$ a pseudoriemannian manifold. Let us identify the ... | https://mathoverflow.net/users/12617 | Is there a coordinate-free proof of the hamiltonian character of the geodesic flow? | You may find an elegant proof of this fact on **Paternain's book "Geodesic Flows"** (Birkhauser), in the very first pages. For convenience, I will reproduce the main parts of the argument here:
The most important step is to understand the geometry of $TTM$, the tangent bundle to the tangent bundle of $M$. Henceforth ... | 13 | https://mathoverflow.net/users/15743 | 88642 | 52,469 |
https://mathoverflow.net/questions/88568 | 7 | Let $X$ be a smooth projective variety (say, over a field of characteristic zero).
Let us say that strong Kodaira vanishing holds for $X$ if
$$
H^q(X,\Omega^p\otimes L)=0
$$
for every $p\geq 0$, $q>0$ and an ample line bundle $L$ on $X$.
My questions are now these:
1) Does strong Kodaira vanishing hold for $X={\m... | https://mathoverflow.net/users/3891 | Strong Kodaira vanishing | It turns out that questions 1 and 2 are completely answered here <http://arxiv.org/abs/alg-geom/9508009> (and some technique for 3 is there as well). In particular, the statement is true
for ${\mathbb P}^N$ but not true for most flag varieties.
| 6 | https://mathoverflow.net/users/3891 | 88652 | 52,473 |
https://mathoverflow.net/questions/88660 | 5 | It is said, as far as I can tell that an arbitrary spectral space, i.e. a space that is $T\_0$, sober and quasi-compact whose collection of quasi-compact open sets forms a basis and is closed under finite intersections, must be $Spec(R)$ for some ring $R$. Is there a canonical (or any) way of reconstructing $R$ from it... | https://mathoverflow.net/users/11546 | Ring of a Spectral Space | Check out [Prime ideal structure in commutative rings](http://www.jstor.org/pss/1995344) by Melvin Hochester where the theorem you mentioned is proved and functoriality discussed.
| 8 | https://mathoverflow.net/users/15934 | 88661 | 52,477 |
https://mathoverflow.net/questions/88666 | 2 | Given the Diophantine equation$$ax^2+bxy+cy^2+dx+ey+f=0$$
if the coefficients $(a,b,c,d,e,f)$ are chosen among all the prime numbers, we have infinite equations. Is it possible to prove that the solutions of the infinite equations are infinite and countable?
| https://mathoverflow.net/users/21258 | Infinite solutions of a diophantine equation | Yes, if e.g. $a=e=2,\, b=5,\,c=d=3,$ and $f$ varies among all primes, that equation has the solution $x=-y=f$, which already makes infinitely many (distinct) solutions.
| 1 | https://mathoverflow.net/users/6101 | 88667 | 52,481 |
https://mathoverflow.net/questions/77632 | 11 | **Background**
I am interested in elementary embeddings from a model of set theory into itself. One way of producing such elementary embeddings is when the model is generated by indiscernibles; this idea is very closely related to the existence of sharps. Jech and Kanamori discuss $0^{\#}$ and $0^\dagger$ in detail b... | https://mathoverflow.net/users/3183 | Do indiscernibility embeddings exist for an initial segment of an inner model of many measurable cardinals? | My reading of this question was different from Andreas', because Norman asked for order preserving maps of the indiscernibles to extend to embeddings
$j:L[U]\_\theta \rightarrow$ $L[U] \_\theta$
i.e. the indiscernibles should be below $\theta$ as the ordinal height of the structures mentioned is $\theta$?
In that... | 9 | https://mathoverflow.net/users/6942 | 88674 | 52,484 |
https://mathoverflow.net/questions/88677 | 4 | Let $G$ be a discrete countable group which acts on a countable set $X$. This action defines a standard unitary representation, called the *permutation representation*, on the Hilbert space $l\_2(X)$ of all functions $X\to \mathbb C$ with finite $l\_2$ norm. The Hilbert space have an obvious Hilbert basis consisting of... | https://mathoverflow.net/users/6205 | Which unitary (permutation-)representations are isolated? | There is for sure no general theory. Your question is interesting already for $X=G$, i.e. the left regular representation of $G$. I am aware only of isolated examples where that has been studied, e.g. free groups. For example, there is a paper by Pytlik and Szwarc
<http://www.math.uni.wroc.pl/%7Eszwarc/pdf/acta.pdf>
... | 5 | https://mathoverflow.net/users/14497 | 88700 | 52,495 |
https://mathoverflow.net/questions/88694 | 4 | By Mazur's theorems (Reference:- Modular curves and the Eisenstein ideal, 1977),
we know that the only rational points of X\_0(N) for N any prime > 163 are
the two cusps (o) and (oo) (|X\_0(N)(Q)| = 2 ).
Is there any bound for |X\_0(N)(Q)|, where N is an arbitrary +ve integer?.
More Generally,
Is there a bound on... | https://mathoverflow.net/users/20754 | Bound for the number of rational points on the modular curve | Let me restrict throughout to prime $N \geq 23$. This ensures that, indeed, $|X\_0(N)(K)|$ is finite, for any number field $K$.
If your question is "For given $N$ and $K$, how big is $|X\_0(N)(K)|$?", then I don't really know. Certainly you'll have the two cusps, and possibly some "CM points" coming from CM elliptic... | 12 | https://mathoverflow.net/users/5744 | 88707 | 52,498 |
https://mathoverflow.net/questions/88716 | -1 | Hi everybody,
I need to know if there is a notion of the product of two categories, or what could substitute for this in general. Do you have any references please?
The definition I have in mind is the following:
DEFINITION : Let $\mathcal{A}$ and $\mathcal{B}$ be two categories. The product category, if it exist... | https://mathoverflow.net/users/21374 | What is the product of two categories? Or its substitute ? | Your example should work and there is a glitch in your proof. The product with an empty set is always again empty. In fact, you can construct the product of two categories $\mathcal{C}$ and $\mathcal{D}$ by taking pairs of morphisms $(f,g)$ with $f$ in $\mathcal{C}$ and $g$ in $\mathcal{D}$ and using the obvious compos... | 1 | https://mathoverflow.net/users/21479 | 88717 | 52,499 |
https://mathoverflow.net/questions/88725 | 4 | I am working with a class of matrices $A$ which are non-negative-definite, not symmetric, and have maximum eigenvalue less than 1. I am interested in the spectral properties of the matrix $H = A(I - A)^{-1}$.
Is there a spectral theorem for this context, which gives sufficient conditions on $A$ for the matrix $H$ to... | https://mathoverflow.net/users/238 | Spectral Properties of $A(I-A)^{-1}$ | If you take a Schur form $A=QTQ^T$, then $H=QT(I-T)^{-1}Q^T$, and you can ignore the orthogonal factors $Q$. You might also want to set $N=I-T$, so that $Q^THQ=N^{-1}-I$. Now the problem looks much simpler.
* $H$ diagonalizable $\Leftrightarrow$ $N^{-1}$ diagonalizable $\Leftrightarrow$ $N$ diagonalizable $\Leftright... | 7 | https://mathoverflow.net/users/1898 | 88727 | 52,506 |
https://mathoverflow.net/questions/88728 | 7 | Let $X,Y\in\mathbb{R}^{n\times n}$ be symmetric matrices. You may assume that $X$ is positive semidefinite and $Y$ negative semidefinite, if needed, but not that they are invertible.
I would like to find a way to factor the $2n\times 2n$ block matrix
$$
\begin{bmatrix}
X & I\\\\ I & Y
\end{bmatrix}
$$
into some form ... | https://mathoverflow.net/users/1898 | Factorizing a block symmetric matrix | Assuming that $X$ and $Y$ are invertible (if not, probably a perturbation argument will yield the generalization). Then, the following choice of $M$ works, i.e., $MDM^T$ equals your original matrix with $D$ being the anti-diagonal identity matrix as desired.
(Also, **note slightly different notation**, I write $-Y$ ... | 3 | https://mathoverflow.net/users/8430 | 88748 | 52,513 |
https://mathoverflow.net/questions/88746 | 2 | For a field $K$ let denote by $Tr\_1(d,K)$ the nilpotent group of all upper triangular $d\times d$-matrices over $K$ with each diagonal entry equal to 1. Let $\mathbb{Q}\_p$ the field of $p$-adic numbers and consider $Tr\_1(d,\mathbb{Q}\_p)$ with the $p$-adic topology (observe that this is a $p$-adic analytic group). T... | https://mathoverflow.net/users/15235 | Closed subgroups of $Tr_1(d,\mathbb{Q}_p)$ | Pick $\lambda \in \mathbb{Z}\_p \backslash \mathbb{Q}$ and define
$H = \left\lbrace\begin{pmatrix} 1 & 0 & a
\cr 0 & 1 & \lambda a \cr 0 & 0 & 1 \end{pmatrix} : a \in \mathbb{Z}\_p\right\rbrace$.
Then $H$ is a closed subgroup of $Tr\_1(3,\mathbb{Q}\_p)$ but $H \cap Tr\_1(3,\mathbb{Q})$ is the trivial group, hence... | 3 | https://mathoverflow.net/users/6827 | 88751 | 52,515 |
https://mathoverflow.net/questions/88752 | 0 | I'm looking for a sequence of smooth functions $f\_i(x)$ converging to [Sign](http://en.wikipedia.org/wiki/Sign_function)$(x)$, each of which additionally have the following property:
\begin{equation}
f\_i(x\_1+x\_2) = g\_i(x\_1, f\_i(x\_2))
\end{equation}
for some $g\_i$
Also, is there a name for a function that... | https://mathoverflow.net/users/19899 | Sequence of smooth functions converging to sgn(x) | Let $f\_i$ be any sequence of strictly increasing smooth functions that converge to the Sign function, such as $f\_i(x) = \tanh(ix)$, and let $g\_i$ be defined by $g\_i(x,y) = f\_i(x+f\_i^{-1}(y))$ for $y$s in the range of $f\_i$ (e.g., $-1 < y < 1$) and however you like elsewhere (since the problem posits no smoothnes... | 1 | https://mathoverflow.net/users/15837 | 88756 | 52,517 |
https://mathoverflow.net/questions/88711 | 0 | Let us consider the homogeneous continuous time Markov chain $(X\_t)\_{t\ge 0}$ with two states {0,1} and the intensity matrix
$Q=\begin{pmatrix}-\lambda& \lambda\\\ \mu& -\mu\end{pmatrix}$
Let $N\_t$ be the number of $1 \to 0$ transitions of $X\_t$ in the interval [0, t].
The main interesting question is to find... | https://mathoverflow.net/users/11272 | Number of transitions of a markov chain in a time interval | The usual infinitesimal analysis leading to a differential system applies, with a twist: for small $s>0$,
$$
p\_{i0}(k,t+s)=(1-\lambda s)p\_{i0}(k,t)+\mu sp\_{i1}(k-1,t)+o(s),
$$
and
$$
p\_{i1}(k,t+s)=(1-\mu s)p\_{i1}(k,t)+\lambda sp\_{i0}(k,t)+o(s),
$$
with the convention that $p\_{ij}(-1,t)=0$ for every $t\geqsla... | 3 | https://mathoverflow.net/users/4661 | 88757 | 52,518 |
https://mathoverflow.net/questions/88658 | 1 | Given $a\_1,b\_1,c\_1$ in $F\_2[t]$ with $\gcd(a\_1,b\_1,c\_1)=1$ it is known that there exists an element
$g$ of $SL(3, F\_2[t])$ (by explicit construction) such that $g$ has first line
$$
[a\_1^2, b\_1,c\_1],
$$
in particular this works when $a\_1^2$ is in the ideal $(b\_1,c\_1)$ generated by $a\_1$ and $b\_1$ in $F\... | https://mathoverflow.net/users/11016 | Completing unimodular vectors with $3$ entries in $F_2[t]$ to a $3$ by $3$ matrix with determinant equal to $1.$ | Okay, suppose $K$ is a field and you have three elements $a,b,c\in K[t]$ with $gcd(a,b,c)=1$.
I will show that the row $(a,b,c)$ can always be extended to a matrix in $SL(3,K[t])$.
Suppose that at least one of $b$ and $c$ is nonzero (otherwise $a\in K$ and extending the row is trivial).
Let $g:=gcd(b,c)$ and choose $... | 1 | https://mathoverflow.net/users/17498 | 88760 | 52,520 |
https://mathoverflow.net/questions/88784 | 14 | Hi,
Let $M$ be a compact Riemannian manifold of dimension $n$. Define
the integer-valued function $N(k)$ to be the number of eigenvalues of the Laplacian on $M$
which are less than or equal to $k$. Weyl's law states that the function $N(k)$ has asymptotic:
$$N(k) = C\_n Vol(M) k^n + O(k^{n-1}),$$
for some explicit c... | https://mathoverflow.net/users/21491 | Weyl's law on asymptotic of Laplacian vs Hilbert's theorem on degree of a projective variety | There should be a relationship like this, because the heat function proof of Hirzebruch-Riemmann-Roch uses a much more refined equality between hilbert functions and Laplacian spectra. Let me see if I can put this together.
Let $M$ be a $d$-dimensional smooth projective variety. Let $L^k$ be the line bundle on $M$ ob... | 13 | https://mathoverflow.net/users/297 | 88794 | 52,542 |
https://mathoverflow.net/questions/88785 | 6 | I am doing a course at university and it deals with Gaussian Processes mainly. We use them for fitting data and prediction, machine learning, regression, classification. Is there any particular reason why Gaussian Processes stand out from the millions other random processes you could have?
I suppose that's a bit like... | https://mathoverflow.net/users/21494 | Are Gaussian Processes more important than other stochastic processes? | Gaussian processes are much easier to analyze, so are useful to produce a first approximation to reality. A frequent mistake (see, eg, recent derivative mispricing debacles) is to confuse first approximation for the truth.
| 13 | https://mathoverflow.net/users/11142 | 88797 | 52,543 |
https://mathoverflow.net/questions/88798 | 15 | Let $X \to Y$ be a smooth projective morphism of smooth varieties over $\mathbb{C}$. Then the fibers $X\_y, y \in Y$ have locally constant Hodge numbers $H^q(X\_y, \Omega^p\_{X\_y})$. Namely, one can argue that the Hodge numbers are upper semicontinuous because they represent the kernels of the $\overline{\partial}$-La... | https://mathoverflow.net/users/344 | Hodge numbers in a family | Without flatness you have little chance for this to even get off the ground: Let $f:X\to Y$ be the blow up of a smooth (closed) point of $Y$. Then all fibers except one consist of a single point, while the special fiber is a $\mathbb P^n$. That will have non-zero Hodge numbers that the others can't even dream about.
... | 16 | https://mathoverflow.net/users/10076 | 88803 | 52,545 |
https://mathoverflow.net/questions/88782 | 2 | Hello,
$CR$ manifold for example $S^1\times C^{n-1}$ is every where levi flat. Can I have example of $CR$ manifold which has at least one non levi flat point.
I can't see what the happening in Non-Levi flat points. Sorry for vague question and if it is trivial.. .... but basically i want to understand the non levi ... | https://mathoverflow.net/users/16031 | What does non-levi flat point mean geometrically | Probably the most easy example to understand is the round sphere in $\mathbb{C}^2$. It is strictly convex, so that in particular it is strictly pseudo-convex, and very not Levy-flat.
For a submanifold $M$, Levy flatness is equivalent to the integrability of $TM\cap i TM$. If you compute this plane distribution in the... | 1 | https://mathoverflow.net/users/4961 | 88805 | 52,547 |
https://mathoverflow.net/questions/88824 | 3 | Given a commutative ring $R$, what are relations between w.gldim$(R)$ and w.gldim$(R[[x]])$ (gldim$(R)$ and gldim$(R[[x]])$)?
| https://mathoverflow.net/users/21505 | The global (weak) dimension of formal power series rings | Let $R$ be commutative.
1. If $R$ is Noetherian, then $\text{gl.dim}(R[[X]]) = 1 + \text{gl.dim}(R)$.
2. If $R[[X]]$ is coherent, then $\text{w-gl.dim}(R[[X]]) = 1 + \text{w-gl.dim}(R)$.
Now let $R$ be Noetherian. Hence $R[[X]]$ is Noetherian and since global and weak-global dimension agree for Noetherian rings, w... | 5 | https://mathoverflow.net/users/10194 | 88832 | 52,555 |
https://mathoverflow.net/questions/88777 | 13 | Let $n=p\_1^{e\_1}\cdots p\_k^{e\_k}$ be an integer with $k$ prime factors. We know that the number of integers less than $n$ and coprime to it is
$$\Phi(n)=n-\sum\_i\frac n{p\_i}+\sum\_{i \lt j}\frac n{p\_ip\_j}-\cdots+(-1)^{k}\frac n{n}=nr$$ where $r=\prod(1-\frac 1{p\_i})$.
For any positive integer $x$, the nu... | https://mathoverflow.net/users/8008 | Bound the error in estimating a relative totient function | First of all as you remark above, it is easy to see by elementary means that the error is at most $2^k$. It was an old conjecture of Erdos that this error term could be improved to $o(2^k)$. However this conjecture turns out not to be true!
In 1951 Vijayaraghavan proved that the error term $O(2^k)$ is best possible i... | 13 | https://mathoverflow.net/users/19368 | 88836 | 52,559 |
https://mathoverflow.net/questions/88813 | 4 | Hello Everybody,
Is there a proof of Sobolev embedding theorem without using the GNS or Morrey inequalities? If so, can you provide me with some references?
**Background:** I happened to attened a talk on Computational PDEs and Sobolev spaces. The speaker made a reference to a proof by S. L. Sobolev using polynom... | https://mathoverflow.net/users/7333 | Sobolev embedding proof without Gagliardo–Nirenberg–Sobolev inequality or Morrey's inequality | There is an English translation of Sobolev's book containing his original proof:
Sobolev, S. L. Some applications of functional analysis in mathematical physics. Translated from the third Russian edition by Harold H. McFaden. With comments by V. P. Palamodov. Translations of Mathematical Monographs, 90. American Math... | 3 | https://mathoverflow.net/users/12205 | 88839 | 52,561 |
https://mathoverflow.net/questions/88787 | 6 | $\DeclareMathOperator\Sym{Sym}$In order to define Heegaard Floer Homology for a connected, closed, oriented 3 manifold, we fix a generic path of nearly symmetric almost complex structure $J\_s$ over $\Sym^g(\Sigma)$. By transversality theorem we can say the moduli spaces $M\_{J\_s}$ are all smooth. Under certain cases ... | https://mathoverflow.net/users/21493 | Path of almost complex structure in the definition of Heegaard Floer homology | According to Proposition 3.9 of [Ozsvath and Szabo's original paper "Holomorphic disks and topological invariants for closed three-manifolds"](https://arxiv.org/abs/math/0101206) there are indeed topological conditions one can put on the homotopy class of discs to ensure that one can take a single almost complex struct... | 4 | https://mathoverflow.net/users/10839 | 88841 | 52,563 |
https://mathoverflow.net/questions/87552 | 2 | I consider a special kind of sets in $\mathbb{R}^n\_+$ given by $G\_t = $ {$x \in \mathbb{R}^n\_+ \mid g(x) < t$}, where $\nabla g > 0$ entrywise. Let's consider an integral
$$
f(t) = \int\limits\_{ G\_t } \mu(dx)
$$
If measure $\mu$ is absolutely continious with the density $a(x)$ and $\Omega$ is a $(n-1)$-form such... | https://mathoverflow.net/users/17896 | Derivative of the Lebesgue integral. Currents. | Denote by $\nu$ the pushforward of $\mu$ via the function $g$. More precisely $\nu$ is a measure on $\mathbb{R}$ and for any Borel subset $B\subset \mathbb{R}$ we have
$$ \nu(B)=\mu\bigl(\; g^{-1}(B)\;\bigr).$$
Then
$$f(t)= \nu\bigl(\;(-\infty,t)\;\bigr).$$
Now invoke Radon-Nicodym Theorem. The "derivative" of... | 3 | https://mathoverflow.net/users/20302 | 88843 | 52,564 |
https://mathoverflow.net/questions/88816 | 7 | Let $X\_1,\dots,X\_n$ be complete vector fields on $\mathbb R^n$ and suppose that $(X\_1(p),\dots,X\_n(p))$ is a basis for all $p \in \mathbb R^n$.
Question: Is it possible to choose a cube $C$ around the origin of $\mathbb R^n$ such that there is for every $p \in C$ a piecewise smooth curve $\alpha \subset C$ which ... | https://mathoverflow.net/users/20999 | Linearly independent vector fields | For a suitable nbd of the origin $U$, yes, even following the $n$ flows in a prescribed order. Assuming $X\_1,\dots,X\_n$ locally Lipschitz continuous, the corresponding flows $\phi\_1(t,x),\dots, \phi\_n(t,x)$ are $C^1$ maps (in the pair) so the map $\Phi$
$$(t \_ 1,\dots,t\_ n)\mapsto \phi \_ n (t \_ n, \phi\_{n-1}( ... | 7 | https://mathoverflow.net/users/6101 | 88846 | 52,566 |
https://mathoverflow.net/questions/88849 | 4 | The Laplace operator is the generator of an analytic semigroup on $L^p(\mathbb R^n)$
for $1 < p < \infty$. Is the same true for $L^1(\mathbb R^n)$? If it is, could someone give a reference? The proof must be different from the case $1 < p < \infty$.
| https://mathoverflow.net/users/17035 | Generator of Laplace operator as analytic semigroup on $L^1(\mathbb{R}^n)$ | [Arendt-Batty-Hieber-Neubrander](http://books.google.hu/books?id=sOnRZFgR374C&lpg=PP1&hl=de&pg=PP1#v=onepage&q&f=false): Vector valued Laplace transforms and Cauchy Problems, First Edition, Examle 3.7.6. The Gaussian semigroup. The proof is the same using Fourier multipliers using the excplicite convolution form of the... | 2 | https://mathoverflow.net/users/12898 | 88862 | 52,575 |
https://mathoverflow.net/questions/88871 | 0 | Why is ln($\pi^{\pi}$) rational?
| https://mathoverflow.net/users/21526 | Why is this rational? | It is not known if it is rational. It's suspected to be transcendental over $\mathbb Q$. It is known that $e^\pi$ is transcendental, but none of $e^e,\pi^e,\pi^\pi$ is known to be irrational or transcendental.
| 4 | https://mathoverflow.net/users/20692 | 88872 | 52,579 |
https://mathoverflow.net/questions/87551 | 9 | For the elliptic integral of first kind, $K(m)=\int\_0^{\pi/2}\frac{d\theta}{\sqrt{1-m^2sin^2\theta}} $, it is well-known that $K(m)$ can be expressed in what Chowla and Selberg call "finite terms" (i.e. algebraic numbers and a finite product of Gamma functions of rational values) whenever $i\frac{K(\sqrt{1-m^2})}{K(m)... | https://mathoverflow.net/users/29783 | Can elliptic integral singular values generate cubic polynomials with integer coefficients? | The flurry of comments did not yet produce an answer to the question
concerning the complete elliptic integrals
$$
I\_1 := \frac12
\int\limits\_0^\infty\dfrac{dt}{\sqrt{t(t+1)(t+\frac{8+3\sqrt{7}}{16})}}
$$
and
$$
I\_2 := \int\limits\_0^\infty\dfrac{dt}{\sqrt{t(t^2+21t+112)}}.
$$
It turns out that (i) Yes, $I\_1$ can b... | 11 | https://mathoverflow.net/users/14830 | 88874 | 52,580 |
https://mathoverflow.net/questions/88853 | 3 | Consider the general polynomial $P\_1(t) = \prod\_{j=1}^n (t+x\_j)$.
Construct $P\_k(t) = \prod\_{\sigma \subset [n], |\sigma|=k} (t+x\_{\sigma\_1}x\_{\sigma\_2}\cdots x\_{\sigma\_k})$
where the product is over all subsets of size $k$ of the numbers $1,2,\dots,n.$
The coefficients of $P\_1(t)$ will be the elementary sy... | https://mathoverflow.net/users/1056 | Constructing new polynomials by product of roots | In the language of symmetric functions, you are computing the *plethysm* $e\_j[e\_k]$ (also denoted $e\_k\circ e\_j$) of elementary symmetric functions. In terms of $\mathrm{GL}(n,\mathbb{C})$ representations, you are looking at the representation $\Lambda^j(\Lambda^k(\mathbb{C}^n))$. Some information is at [What is kn... | 8 | https://mathoverflow.net/users/2807 | 88876 | 52,581 |
https://mathoverflow.net/questions/83154 | 9 | Let $\{G\_{n}\}$ be a sequence of $k$-regular [expander graphs](http://en.wikipedia.org/wiki/Expander_graph). For each $n$ assume that each pair of nodes in the graph is transmitting a unit load of traffic and the traffic goes through the minimum path between nodes (if there is more than one minimum path we choose one ... | https://mathoverflow.net/users/13825 | Number of Geodesic Paths Passing Through a Vertex in an Expander Graph | My short answer: I think not much is known. But: here is the state of the art on related problems, as far as I am aware.
[Aldous and Bhamidi](http://arxiv.org/pdf/0708.0555v1.pdf) consider the following model. Place independent exponential edge weights on the edges of the complete graph $K\_n$; we view the weights a... | 3 | https://mathoverflow.net/users/3401 | 88878 | 52,582 |
https://mathoverflow.net/questions/88863 | 9 | In SGA 1, chapter 4, there is a corollary describing a flatness condition for a module of finite type over a local noetherian integral ring. The corollary is number 4.4, and states:
Suppose that $A$ is a local noetherian integral ring with maximal ideal $I$ and residue field $k = A/I$ and field of fractions $K$. Let ... | https://mathoverflow.net/users/21523 | flatness condition for local noetherian ring without nilpotent elements | I assume you are referring to [page 78 here](http://arxiv.org/pdf/math/0206203v2.pdf). I think the author is only implying that you may drop the condition of $A$ being an integral domain (in favor of it being a ring without nilpotent elements). You should still assume that $A$ is noetherian and local with maximal ideal... | 7 | https://mathoverflow.net/users/17498 | 88881 | 52,584 |
https://mathoverflow.net/questions/88858 | 6 | What conditions would be sufficient for a generalization of Cauchy-Davenport for simple groups? I can see two possible difficulties with a generalization for general groups:
1. The sets could both be part of a subgroup of the group.
2. The sets could both be cosets of a normal subgroup. This is impossible for simple ... | https://mathoverflow.net/users/21519 | Conditions for an analogue of Cauchy-Davenport for simple groups | First, a slightly tangential comment regarding what I assume you mean by 'Cauchy--Davenport fails';
I include it for reader potentially unfamiliar with it and since there is also a somewhat common other way to generalize it; cf below.
The Cauchy--Davenport Theorem asserts that for $G$ a prime cyclic group of order $p... | 5 | https://mathoverflow.net/users/nan | 88885 | 52,587 |
https://mathoverflow.net/questions/88875 | 2 | this is probably a very common question, but i couldn't find the answer on my books.
is every darboux function the derivative of a function? even the nowhere continuous ones?
| https://mathoverflow.net/users/20692 | antiderivative of a darboux function | No. Unfortunately, there is no such simple criterion for a derivative. At least none is known.
Ref: Bruckner, Andrew, Differentiation of real functions. Second edition. CRM Monograph Series, 5. American Mathematical Society, Providence, RI, 1994. xii+195 pp. ISBN: 0-8218-6990-6, MR1274044 (94m:26001)
| 2 | https://mathoverflow.net/users/454 | 88887 | 52,589 |
https://mathoverflow.net/questions/88729 | 5 | I wonder - is there a general characterization of the object in the Fukaya category that is mirror to the dualizing sheaf on the other side?
This question has two cases:
1. CY
2. Non-CY
In 1. what I know is that by Polischuk-Zaslow the mirror of the dualizing sheaf in case of the 2-torus is a linear Lagrangian tor... | https://mathoverflow.net/users/21485 | Mirror to the dualizing sheaf | I'll comment on the related question "what is the Serre functor for the Fukaya category?"
**Calabi-Yau setting**
The Serre functor $S$, by definition, satisfies $\mathsf{Hom}(X,SY) \cong \mathsf{Hom}(Y,X)^\vee$; since it's characterized categorically, it's preserved by the derived equivalences which arise in mirro... | 13 | https://mathoverflow.net/users/2356 | 88892 | 52,593 |
https://mathoverflow.net/questions/88899 | 3 | If we consider an induced representation $Ind\_{H}^{G}1\_{H}$ of finite groups, where $1\_{H}$ is the trivial caracter of $H$, how we decompose this representation into irreducible ? There is a decomposition $Ind\_{H}^{G}1\_{H}=V\oplus W$, where $V$ is formed of canstant functions on $G$ and $W$ is formed of functions ... | https://mathoverflow.net/users/6849 | Decomposition into irreducible of induced representations | Since it has not really been made completely explicit in the answers above, I point out that the specific question of whether your invariant subspace $W$ is irreducible or not is easier to check, at least in the case of finite groups. It rarely is. As is implicit in the earler answers, there is a formula due to Mackey ... | 7 | https://mathoverflow.net/users/14450 | 88909 | 52,600 |
https://mathoverflow.net/questions/88903 | 5 | If a closed smooth manifold $M$ admits a smooth free involution $T$, then it bounds. In fact, the mapping cylinder of the quotient map $M \to M/T$ is the manifold whose boundary is $M$.
Is the converse true? If not, then could someone give an example of a closed smooth manifold which bounds but does not admit any free ... | https://mathoverflow.net/users/20620 | Does a manifold which bounds always admit a free involution? | The answer is no. In your example, $M$ is the boundary of a twisted $I$-bundle over another manifold. That's the level of generality in which there's an if and only if statement.
If you want a closed smooth manifold which bounds, but which does not admit a free involution (you can go further and say it does not admi... | 10 | https://mathoverflow.net/users/1465 | 88912 | 52,602 |
https://mathoverflow.net/questions/88860 | 1 | For each $n \in \mathbb{N}$ let $X\_n$ be a random variable taking its values in a finite set $E\_n$ with $P(X\_n=x\_n)>0$ for all $x\_n \in E\_n$. Say that $X\_n$ is asymptotically degenerate if $\min\_{x\_n \in E\_n}\Pr(X\_n \neq x\_n) \to 0$.
Denoting by $h(X)$ the entropy of (the law of) a discrete random variable ... | https://mathoverflow.net/users/21339 | asymptotic behaviour of the entropy and degeneracy | Of course, not. Your "asymptotic degeneracy" condition means that the weight of the maximal atom of the distribution of $X\_n$ goes to 1, which is a pretty strong condition. If the sizes of your sets $E\_n$ go to infinity, take a sequence $\epsilon\_n\to 0$. Then the sequence of measures $\nu\_n$ with two atoms of weig... | 2 | https://mathoverflow.net/users/8588 | 88919 | 52,608 |
https://mathoverflow.net/questions/88861 | 2 | Let $f:X\to Z$ be an etale morphism of integral schemes with $Z$ normal, and let $k(Z)\subseteq L\subseteq k(X)$ be an intermediate field between the associated function fields. Can we find a scheme $Y$ with $k(Y)=L$ so that $f$ factorises into an etale tower $X\to Y\to Z$? Can anything be said if $f$ is only assumed t... | https://mathoverflow.net/users/19757 | `Normalisation' in etale towers? | Yes, you can take $Y$ to be the image of $X$ in the normalization of $Z$ in $L$. You need to show that if in the composite $X \to Y \to Z$ all schemes are normal, $X\to Y$ is surjective and $X \to Z$ is étale, then $X \to Y$ is also étale. By base-changing along $X\to Z$ you can assume that $X \to Z$ is an isomorphism,... | 3 | https://mathoverflow.net/users/4790 | 88928 | 52,614 |
https://mathoverflow.net/questions/88935 | 4 | The function ${}\_2F\_1\Big(\frac{a-b}{2},\frac{a+b-1}{2};c;y\Big)$ can be transformed (as reported by A. Erdélyi) by the following formula
${}\_2F\_1\Big(\frac{a-b}{2},\frac{a+b-1}{2};c;y\Big)=
(1-z)^{1-b}{}\_2F\_1\Big(a,a-1;b;z\Big)
$
for
$y =4z(1-z)$, and $\quad a,b,c \in\mathbb{R}$
The problem is that solv... | https://mathoverflow.net/users/19493 | Quadratic Transformation of the Hypergeometric Function 2F1 | Here is a general explanation (which I can make more precise once you fix the formulas): the identity will hold "everywhere" except on the branch cut for the hypergeometric function (i.e. on the real line, from 1 to $+\infty$). On the branch cut, the choice of sign will flip the orientation of the cut, so that you get ... | 1 | https://mathoverflow.net/users/3993 | 88936 | 52,615 |
https://mathoverflow.net/questions/88941 | 3 | Let $(M,g)$ be a closed, compact Riemannian manifold. Let $P$ be a $2r$th order pseudo-differential operator, where $r \in \Bbb{R}\_+$. Suppose that the differential equation $Pu=f$ has a unique $H^r(M)$ solution for all $f$ in the dual space of $H^r(M)$. Does it follow that $P$ is elliptic?
| https://mathoverflow.net/users/15856 | ellipticity and invertible differential operators | Yes, this is correct. You can find a more general result in
>
> Topics in pseudo-differential operators. 1969 Pseudo-Diff. Operators (C.I.M.E., Stresa, 1968) pp. 167–305 Edizioni Cremonese, Rome
>
>
>
More precisely see Corollary 1, Chap. IV, page 251 in the above reference.
| 4 | https://mathoverflow.net/users/20302 | 88942 | 52,617 |
https://mathoverflow.net/questions/88880 | 16 | In a short talk, I had to explain, to an audience with little knowledge in geometry or algebra, the three different ways one can define the tangent space $T\_x M$ of a smooth manifold $M$ at a point $x \in M$ and more generally the tangent bundle $T M$:
* Using equivalent classes of smooth curves through $x$
* Using ... | https://mathoverflow.net/users/21522 | An easy way to to explain the equivalence definitions of tangent spaces? | What all three definitions have in common is that they each try to capture the first order behavior of a smooth function on $M$.
1. The derivative of a smooth function $f$ along a curve $\gamma$ with $\gamma(0) = p$ depends on $\gamma$ only insofar as it depends on $\gamma'(0)$, and indeed it recovers the directional... | 16 | https://mathoverflow.net/users/4362 | 88947 | 52,620 |
https://mathoverflow.net/questions/88939 | 11 | Can anyone give me a relatively simple proof or Some reference for the following fact.(I know that there is a proof of this theorem in Gerard J. Murphy'book: "$C^\*$-Algebras and Operator Theory", but I'm sure that there should be a simple proof of this.
**Every hereditary C\*-subalgebra of a simple $C^\*$-algebra is... | https://mathoverflow.net/users/21462 | Question about hereditary $C^*$-algebra | I don't know much about C\* algebras, but I would guess that there shouldn't be a simple proof of the theorem you state. Here's an algebraic example which seems to be a counterexample to the theorem once the modifier C\* is removed from the statement. Let A be the Weyl algebra, i.e. the algebra generated by x,y with th... | 0 | https://mathoverflow.net/users/2669 | 88961 | 52,629 |
https://mathoverflow.net/questions/88955 | 4 | Hi, I apologise if this is the wrong place for this question but i need to ask it somewhere.
The question is whether the right, (perhaps left?), adjoint to $Hom(-,A)$ exists in $\bf Set$ and how to deduse it.
I would very much like it to be the disjoint union $-\oplus A $ but im not quite sure how to deduce it or if ... | https://mathoverflow.net/users/16435 | Is there a right adjoint to the contravariant functor Hom(-,B) in the category of Sets | To speak of right (or left) adjoints for a contravariant functor $F: C\to D$, one needs to decide whether to view it as a functor from $C^{op}$ to $D$ or as a functor from $C$ to $D^{op}$. What the one viewpoint calls a left adjoint, the other will call a right adjoint. One therefore often speaks instead of two contrav... | 10 | https://mathoverflow.net/users/6794 | 88974 | 52,637 |
https://mathoverflow.net/questions/88968 | 3 | Let $G$ be an abelian group and let $R$ be a $G$-graded commutative ring, i.e., $R=\oplus\_{g\in G} R\_g$ with $R\_gR\_h\subseteq R\_{g+h}$. Let $M$ be a $G$-graded $R$-module
i.e. $M=\oplus\_{g\in G}M\_g$ and $R\_g M\_h\subseteq M\_{g+h}$. We will say that a morphism of $G$-graded $R$-modules $f:M\rightarrow N$ has d... | https://mathoverflow.net/users/11765 | When does grading pass to (co)-homology? | In short, everything works fine in the graded case, i.e. there are projective and free resolutions and a graded tensor product. Eventually, since kernels and quotients of graded modules are graded, there is also a natural grading on $\operatorname{Tor}$.
In more detail: I don't know of a reference for graded homolog... | 2 | https://mathoverflow.net/users/10194 | 88976 | 52,639 |
https://mathoverflow.net/questions/88719 | 5 | Assume you have a simple and infinite graph. Choose $x\_{0}$ an arbitrary vertex and consider
$$
G\_{n}:=\{x\in G:d(x\_0,x)\leq n\}
$$
with the graph metric (hop metric). Now for each pair of nodes there is a unit flow that travels through the minimum path between nodes (if there is more than one minimum path it split... | https://mathoverflow.net/users/13825 | Asymptotic Geodesic Flow on Planar Graphs | You can't do better. Lipton and Tarjan's [planar separator theorem](http://en.wikipedia.org/wiki/Planar_separator_theorem) says that any $n$-node planar graph $G=(V,E)$ contains a set $S$ of $O(\sqrt{n})$ vertices whose removal separates the graph into components all of which have size at most $2n/3$. We can then parti... | 7 | https://mathoverflow.net/users/3401 | 88982 | 52,642 |
https://mathoverflow.net/questions/87659 | 2 | Let $G$ be a nice topological group, say a compact connected Liegroup.
Then one can construct a model of its classifying space as $EG/G$ where $EG$ is any contractible space with free $G$ action.
On the other hand, one could take the geometric realization of the simplicial construction of the classifying space of t... | https://mathoverflow.net/users/2837 | Classifying space commutes with geometric realization - reference request | I think you should be able to prove this roughly as follows: first consider the loop space of your construction. For nice simplicial spaces, the loop space can be calculated level-wise (see May's Geometry of Iterated Loop Spaces, for instance), and hence the loop space of your construction is homotopy equivalent to the... | 1 | https://mathoverflow.net/users/4042 | 88985 | 52,644 |
https://mathoverflow.net/questions/88491 | 6 | This is probably has an obvious proof or a straightforward counterexample, but I'm having trouble finding either.
Let $p:E \to B$ be a fibre bundle, with fibre $F$. Assume that there is a spectrum $X$ and a homotopy equivalence of spectra
$$f: X \wedge \Sigma^{\infty} B\_+ \to \Sigma^{\infty} E\_+$$
(in particula... | https://mathoverflow.net/users/4649 | Stable triviality of fiber bundles | This is often not true. Here's an example where all components are finite complexes.
Let $E = S^3 \times S^2$ and $B = S^2$, with the map given by $S^3 \times S^2 \to S^3 \to S^2$ where the latter is the Hopf fibration. The suspension spectrum is equivalent to $\Sigma^\infty\_+ S^3 \wedge \Sigma^\infty\_+ S^2$, but t... | 3 | https://mathoverflow.net/users/360 | 88989 | 52,646 |
https://mathoverflow.net/questions/88364 | 9 | Suppose I have three identically-distributed homogeneous continuous-time discrete state space Markov chains $X\_1(t), X\_2(t), X\_3(t)$, $t\geq 0$. They evolve independently but share a common random variable $X\_0$ as an initial condition.
I let $$X\_1=X\_1(t\_1), \ \ \ X\_2=X\_2(t\_2), \ \ \ X\_3=X\_3(t\_3)$$ for som... | https://mathoverflow.net/users/7949 | Is this a situation where triple mutual information is always non-negative? | My conjecture in the original question is false.
Let $X\_i(t)$ be continuous time Markov chains with two states 0 and 1, such that the rate of transition from 0 to 1 and from 1 to 0 is $1$. Let $X\_0$ be 0 with probability $0.9$ and 1 with probability $0.1$. Choose $t\_1=t\_2=t\_3$ such that the probability that $X\_... | 3 | https://mathoverflow.net/users/7949 | 88990 | 52,647 |
https://mathoverflow.net/questions/88991 | 2 | If we fix a reductif algebraic group $G$ defined over a local field $F$, an we put $\mathbf{G}$ the group of rationnel point of $G$, we denote $\mathcal{R}(\mathbf{G})$ the category of smooth representations of $G$, $\mathcal{S}(\mathbf{G})$ the category of $\mathbf{G}$-simplicial complex of finite dimensional, that is... | https://mathoverflow.net/users/6849 | exact functor and representations of p-adic groups | I do not think that your procedure gives the whole ${\mathcal R}({\mathbf G})$. In particular you'll have difficulties to get *irreducible representations* different from the Steinberg representation. However I do not know how to prove this fact!
To see examples you may read my papers:
*Representations of ${\rm PGL}(... | 1 | https://mathoverflow.net/users/4767 | 89002 | 52,651 |
https://mathoverflow.net/questions/88988 | 2 | We note $G$ a connected algebraic group, $T$ a maximal torus in $G$, $B$ a Borel subgroup containing $T$. We put also, $N=N\_{G}(T)$ the normalizer of $T$ in $G$. We know that $(B,N)$ is a BN-pair of $G$. How we prove that every parabolic subgroup of $G$ containing $B$ is of the form $P=BW\_{T}B$, where $W$ is the Weyl... | https://mathoverflow.net/users/6849 | Parabolic subgroups and BN-pairs | Combine Theorem 6.43 (1) pg. 315 in Abramenko-Brown "Buildings" together with Proposition 6.36 (6) on pg. 310.
These are two steps:
As you mention, $BW\_T B$ is a subgroup (Proposition 6.36).
$BW\_TB$ contains $B$, any subgroup containing the Borel subgroup is parabolic, and any parabolic is conjugate to a subgroup... | 6 | https://mathoverflow.net/users/10400 | 89003 | 52,652 |
https://mathoverflow.net/questions/88938 | 1 | **Theorem 12** of the following [link](http://empslocal.ex.ac.uk/people/staff/rjchapma/etc/dirichlet.pdf) asserts the following:
>
> $\textbf{Theorem.}$ Let $\chi \in X\_{N}$ with $\chi \neq \epsilon$. There exists $C > 0$ such that $$L(s,\chi) = L(1,\chi) + O(s-1)$$ as $s \to 1^{+}$. In particular, $$\lim\_{s \to ... | https://mathoverflow.net/users/1483 | Doubt in the proof of $\lim_{s \to 1^{+}} L(s,\chi) = L(1,\chi)$ | Since you edited the question but did not say that it is clear now, I assume you are hoping for some details in addition to what Ralph said. So:
Let $f\_n(s) = \frac{1}{n^s}- \frac{1}{(n+1)^{s}}$ the derivative of this with respect to $s$ is as you computed $\frac{-\log n}{n^s}+ \frac{\log(n+1)}{(n+1)^{s}}$. This is ... | 3 | https://mathoverflow.net/users/nan | 89007 | 52,655 |
https://mathoverflow.net/questions/88715 | 3 | Given a Legendrian $\Lambda$ in a contact manifold $(Y,\alpha)$, suppose one has a rigid J-holomorphic curve into the symplectization $(M,\omega)=(Y\times\mathbb{R},\mathrm{d}(e^t\alpha))$. As considered in Legendrian contact homology, the domain of the curve should be the boundary punctured unit disc, at boundary punc... | https://mathoverflow.net/users/21481 | Persistence of boundary punctured holomorphic curves | Yes, if the linearized operator is also surjective. If the linearized operator is not surjective (i.e. you don't have transversality), what I say is not applicable.
To prove this, you want to use an implicit function theorem argument. You can, for instance, set up the problem so that you are looking for sections of t... | 2 | https://mathoverflow.net/users/477 | 89012 | 52,658 |
https://mathoverflow.net/questions/88993 | 11 | This is related to this posting:
[complex cobordism from formal group laws?](https://mathoverflow.net/questions/5166/complex-cobordism-from-formal-group-laws)
but not entirely the same. I'd like to know if there are any proofs of Quillen's theorem that $\pi\_\* (MU)$ is the Lazard ring (home of the universal formal... | https://mathoverflow.net/users/4649 | Alternate proofs of Quillen's theorem on formal group laws and MU | Quillen's proof of this in the paper
*Elementary proofs of some results of cobordism theory using Steenrod operations*. Advances in Math. 7 1971 29–56 (1971)
does not make use of Adams spectral sequences or the structure of $H^\ast(MU)$ over the Steenrod algebra. In fact the only result from homotopy which is assum... | 10 | https://mathoverflow.net/users/8103 | 89014 | 52,659 |
https://mathoverflow.net/questions/88430 | 1 | Let K be a finite field and let R,P be groups (with R a subgroup of P). I know that the irreducible KP-modules have dimensions 1,4 and 16 over K. I have a KP-module M, and I know that M has dimension at most 5 over K. I also know that M does not have a quotient of dimension 1 over K. Moreover, if I consider M as a modu... | https://mathoverflow.net/users/19783 | Restrictions of Modules and Dimensions | I see no reason in general if $\operatorname{Ext}\_P^1(4,1) \neq 0$ that, on restriction to R the four dimensional simple can't drop down and have a 2-dimensional submodule. I.e. On restriction to R it looks like:
`\begin{equation}\begin{array}{c}
2\\ 2
\end{array} \oplus 1
\end{equation}`
If you knew on restric... | 1 | https://mathoverflow.net/users/15672 | 89015 | 52,660 |
https://mathoverflow.net/questions/88996 | 23 | Not every complex manifold is a Kähler manifold (i.e. a manifold which can be equipped with a Kähler metric). All Riemann surfaces are Kähler, but in dimension two and above, at least for compact manifolds, there is a necessary topological condition (i.e. the odd Betti numbers are even). This condition is also sufficie... | https://mathoverflow.net/users/21564 | Non-compact complex surfaces which are not Kähler | Following David Speyer's suggestion, let $X=\mathbb{C}^2-\{0\}/\lbrace(x,y)\mapsto (2x,2y)\rbrace$
be the standard Hopf surface. The image, $E$, of the $x$-axis is an elliptic curve.
Remove a point of $X-E$ to get $Y$. The second Betti number $b\_2(Y)=0$ because it is homeomorphic to $S^3\times S^1-pt$. If $Y$ were Kä... | 29 | https://mathoverflow.net/users/4144 | 89018 | 52,663 |
https://mathoverflow.net/questions/89021 | 1 | Let $G$ be a simply-connected Lie group endowed with a biinvariant Riemannian metric. Can you please tell me, if the following is true: Is the Riemann curvature tensor $R(X, Y)Z$ of the above metric always parallel, i.e. do we have $\bigtriangledown R(X, Y)Z=0$? I wonder this because the situation looks so much like a ... | https://mathoverflow.net/users/20818 | Question on simply-connected Lie group. | To elaborate a bit on Liviu's answwer: If a Riemannian metric $g$ on $G$ is bi-invariant, then it easily follows that $g$ is invariant under the inversion map $\iota: G\to G$, and then bi-invariance implies that, for any $a\in G$, the map $\iota\_a:G\to G$ defined by $\iota\_a(b) = a(b^{-1})a$ also leaves $g$ invariant... | 5 | https://mathoverflow.net/users/13972 | 89031 | 52,668 |
https://mathoverflow.net/questions/89039 | 0 | How to prove that if $\theta \_1,\theta \_2,\theta \_3$ be the arguments of the primitive roots of unity, $\sum \cos p\theta = 0$ when $p$ is a positive integer less than $\dfrac {n} {abc\ldots k}$, where $a, b, c, \ldots, k$ are the different constituent primes of n and when $p=\dfrac {n} {abc\cdots k}$, $\sum \cos p\... | https://mathoverflow.net/users/18929 | 1895 Math Trip problem on primitive roots of unity | This problem seems to use deliberately confusing notation. Let $Z\_n$ be the set of primitive $n$-th roots of unity. Since $\cos p \theta$ is just the real part of $e^{i p \theta}$, the problem is to evaluate $\mathrm{Re} \sum\_{\zeta \in Z\_n} \zeta^p$. The real part is just a read herring, since the sum is real anywa... | 10 | https://mathoverflow.net/users/297 | 89041 | 52,672 |
https://mathoverflow.net/questions/89040 | 6 | Suppose we have a (commutative) ring $R$ and an $R$-algebra $S$. Furthermore, suppose that $S\cong R^n$ as $R$-modules, that is, $S$ is free of rank $n$ as an $R$-module. Can we always choose $1$ to be a basis element for $S$? Equivalently, is it necessary that $S/R \cong R^{n-1}$?
If not, how about in the case that ... | https://mathoverflow.net/users/1474 | Does $S$ being a free rank-$n$ $R$-algebra imply that $S/R$ is free rank $n-1$? | This is not true in general. For example, assume that $P$ is a projective module on $R$ that is not free, but such that $P \oplus R$ is free (there are many such examples). Set $S= R \oplus P$, and give $S$ an $R$-algebra structure by taking the products of two elements of $P$ to be 0.
On the other hand, it is true w... | 9 | https://mathoverflow.net/users/4790 | 89045 | 52,675 |
https://mathoverflow.net/questions/88986 | 14 | There is a classical result which says that the assignment $$M \mapsto C^{\infty}\left(M\right)$$ is an embedding of the category of (paracompact Hausdorff) smooth manifolds into the opposite category of $\mathbb{R}$-algebras. However, all of the proofs I have seen first establish this for manifolds of the form $\mathb... | https://mathoverflow.net/users/4528 | Are all manifolds affine? | The functor is not an embedding if we remove the paracompactness assumption.
I will need some preliminary definitions. Let $R$ be the long ray, i.e., the topological space given by $\omega\_1 \times [0, 1)$ equipped with the order topology induced by lexicographic order, and let $L$ be the long line obtained by glui... | 14 | https://mathoverflow.net/users/2926 | 89049 | 52,678 |
https://mathoverflow.net/questions/89050 | 1 | Let $M$ be a noncompact $C^\infty$ manifold, let $X$ be a complete $C^\infty$ vector field on $M$, and take $f\in C^\infty\big(M;(0,\infty)\big)$ a strictly positive function.
Question: Does anyone know sufficient conditions on the function $f$ implying the completeness of the vector field $fX$ ?
(When $M$ is compa... | https://mathoverflow.net/users/21080 | completeness of a vector field fX, with X complete and f>0 | If $f$ is bounded, then the rate at which one travels along the integral curves of $X$ is only increased by a bounded factor, so it still takes infinite time to get all the way along each integral curve,so $fX$ is complete.
| 2 | https://mathoverflow.net/users/13268 | 89051 | 52,679 |
https://mathoverflow.net/questions/89033 | 2 | This question arises because I'm reading Frenkel and Ben Zvi's book "vertex algebras and algebraic curves" at the moment.
Let $X$ be a curve, $\Delta \subset X \times X$ the usual diagonal embedding, and $\mathcal{O}$ the sheaf of functions on $X$. A way to define the sheaf $\Omega$ of differential $1$-forms on $X$ i... | https://mathoverflow.net/users/18540 | Differential forms with poles on the diagonal | The first point to observe is that question is equivalent to showing that the line bundle
$$\Omega \boxtimes \Omega(2 \triangle) \mid\_{\triangle}$$ is canonically trivial.
Indeed, given any line bundle $L$ on $X \times X$, we have an exact sequence of sheaves
$$
L(-\triangle) \to L \to \triangle\_\ast (L \mid\_{\t... | 5 | https://mathoverflow.net/users/14681 | 89066 | 52,687 |
https://mathoverflow.net/questions/89069 | 121 | As every MO user knows, and can easily prove, the inverse of the matrix $\begin{pmatrix} a & b \\\ c & d \end{pmatrix}$ is $\dfrac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$. This can be proved, for example, by writing the inverse as $ \begin{pmatrix} r & s \\ t & u \end{pmatrix}$ and solving the resul... | https://mathoverflow.net/users/1050 | Should the formula for the inverse of a 2x2 matrix be obvious? | EDIT (8/14/2020): A couple people have suggested that this answer should come with a warning -- this is a pretty fancy approach to an elementary question, motivated by the fact that I know the OP's interests. Some of the other answers below are probably better if you just want to invert some matrices :). I've also fixe... | 78 | https://mathoverflow.net/users/6950 | 89074 | 52,691 |
https://mathoverflow.net/questions/88927 | 26 | The surreal numbers are sometimes introduced as a place where crazy expressions like $(\omega^2+5\omega-13)^{1/3-2/\omega}+\pi$ (to use the nLab's example) make sense. The problem is, there seem to be varying definitions of the exponential in the surreal numbers and since I can't find any recent reference that covers t... | https://mathoverflow.net/users/5583 | Surreal exponentiation -- are the varying definitions equivalent? If not, is there agreement on which ones are better? | There is only one official definition of surreal exponentiation in the literature, the one due to Martin Kruskal. It was rediscovered by Harry Gonshor (with hints from Kruskal) and incorporated into his book (An Introduction to the Theory of Surreal Numbers) where important results on surreal exponentiation that go bey... | 26 | https://mathoverflow.net/users/18939 | 89081 | 52,694 |
https://mathoverflow.net/questions/89064 | 1 | I have a set, $A$, of $m \times n$ matrices with certain properties and a subset $B$ of $A$. I would like to say that when randomly selecting such a matrix, I am "almost always" never in $B$. I can show that $A$ is the disjoint union of subsets, $A\_j$, such that for each $A\_j$, the measure of $A\_j\cap B$ is 0. Howev... | https://mathoverflow.net/users/21586 | What does it mean to say "almost always" ? | I see some contradiction in your hypotheses: since the $A\_j$'s are disjoint and have non empty interiors, the union should be at most countable (there can't exist more $A\_j$'s than the cardinal of the set of rational points in $\mathbb{R}^{m\times n}$).
| 1 | https://mathoverflow.net/users/19603 | 89089 | 52,699 |
https://mathoverflow.net/questions/89032 | 6 | Suppose we color a $X \times X$ finite plane by red and blue arbitrarily. How large does X need to be to guarantee a monochromatic combinatorial square $k \times k$
1 0 1 0 1 1
1 1 1 1 1 1
1 0 1 0 1 1
1 1 1 1 1 1
The above figure shows a combinatorial $2 \times 2$ filled square filled by zeros.
| https://mathoverflow.net/users/21575 | The Problem about 2-coloring finite plane | The exact answer, $15$, to this question is the content of my paper with Shalom Eliahou:
Here a copy of the corresponding entry of Math-Review:
Bacher, Roland; Eliahou, Shalom
Extremal binary matrices without constant 2-squares
J. Comb. 1 (2010), no. 1, [ISSN 1097-959X on cover], 77–100.
05D10 (11B75)
Summary: "In ... | 9 | https://mathoverflow.net/users/4556 | 89091 | 52,700 |
https://mathoverflow.net/questions/89096 | 4 | It is well known that
1) if there exists a non-trivial automorphism of a graph $G$ with corresponding permutation matrix $P$ then if $(v,\lambda)$ is an eigenvector-eigenvalue pair of the graph Laplacian $L(G)$ then $(Pv,\lambda)$ is also an eigenvector-eigenvalue pair (if $v$ and $Pv$ are linearly independent then t... | https://mathoverflow.net/users/21598 | Does graph asymmetry imply all eigenvalues of the graph Laplacian are simple? | This is false.
There are strongly regular graphs with trivial automorphism group; these will have many repeated eigenvalues (both for adjacency matrix and Laplacian).
You can find some examples in the answers to this question:
[Are "almost all" strongly regular graphs rigid?](https://mathoverflow.net/questions/41... | 11 | https://mathoverflow.net/users/1492 | 89099 | 52,701 |
https://mathoverflow.net/questions/88037 | 5 | The question is stated in the title. I think BCnrd states in a comment here
[Is every (Artin/DM) algebraic stack fibered in sets an algebraic space?](https://mathoverflow.net/questions/41833/is-every-artin-dm-algebraic-stack-fibered-in-sets-an-algebraic-space)
that while the answer is not found in Laumon & Moret-Ba... | https://mathoverflow.net/users/1231 | Reference request: an algebraic stack whose closed points have no automorphisms is an algebraic space | In case you're still looking for a reference I Lemma 2.3.9 of Abramovich and Hassett stable varieties with a twist should do it (consider the map to a point, which has no automorphisms, which by assumptions is injective on automorphism groups)
<http://arxiv.org/pdf/0904.2797v1.pdf>
| 3 | https://mathoverflow.net/users/16857 | 89112 | 52,706 |
https://mathoverflow.net/questions/89118 | 4 | Are Schur functors and categorification somehow related ?
If yes, probably looking on Schur functors (which I know) one can illustrate on this example why "categorification" (which I do not know) is so important/popular now ?
(I am intersted to learn somehting about "categorification", but I would prefer to have
som... | https://mathoverflow.net/users/10446 | Categorification and Schur functors | Categorification can be thought of as the process of replacing equalities with isomorphisms (in some category). A basic example is replacing a numerical combinatorial identity such as
$$2^n = \sum\_{k=0}^n {n \choose k}$$
with a bijection between two sets (an isomorphism in $\text{Set}$); in this case, between the se... | 7 | https://mathoverflow.net/users/290 | 89122 | 52,710 |
https://mathoverflow.net/questions/89120 | 0 | Let $u(x, t)$ be a solution of $u\_t=u\_{xx}$ in the domain $x>0, t>0$. We also have the initial condition $u(x,0)=g(x)$ and the boundary condition $u(0,t)=h(t)$. Do we have maximum principle in this case? Can we conclude that $u(x,t)$ is bounded if we assume both $g$ and $h$ are bounded? If not, what additional condit... | https://mathoverflow.net/users/1777 | Maximum principle for heat equation on infinite domain | You need essentially the same condition as in the case of the domain $x\in\mathbb R$. That is, $$u(x,t)=o(e^{\epsilon|x|^2})$$
for every $\epsilon>0$.
**Edit**. Tikhonov provided an example of a non-trivial solution of the heat equation on the domain $\mathbb R$, with zero data. Take either its odd part, or the deriv... | 4 | https://mathoverflow.net/users/8799 | 89123 | 52,711 |
https://mathoverflow.net/questions/89097 | 20 | This question is out of plain curiosity. The first sentence of Deligne's
*Les corps locaux de caractéristique $p$, limites de corps locaux de caractéristique* $0$ (1984) reads (in rough translation) as follows :
>
> D. Kazhdan has introduced the
> principle that the representation
> theory of a reductive group ov... | https://mathoverflow.net/users/2821 | Representation theory of reductive groups in characteristic $p$ as a limit of the theories in characteristic $0$ | I think, although it's dated later than Deligne's paper that you mentioned, that the first written instance of Kazhdan's principle is in the paper "Representations of groups over close local fields", Journal d'Analyse Math\'ematique, vol. 47,1986, pp.175--179.
This is in the same journal issue as "Cuspidal Geometry o... | 14 | https://mathoverflow.net/users/3545 | 89126 | 52,712 |
https://mathoverflow.net/questions/89117 | 1 | Let A be a *G-group*, i.e. a set on which G acts on, has a group structure and satisfies $^s(xy)=^s x ^s y$ for all $x,y \in A \ , s \in G$. A *homogeneous principal space* P is a non-empty G-set on which A acts on the right, in a manner competible with G, and satisfy the following: $\forall x,y \in P : \exists ! a \in... | https://mathoverflow.net/users/21601 | Isomorphism between the set of classes of Principal Homogeneous spaces and non-Abelian H^1(G,A) cohomology | As far as I can see, $P\_a$ should be the set $A$ (i.e., the underlying set of the given group $A$), considered not as a group but as a set with an action of $G$, namely the action in the "twisted" formula. In addition to this action of $G$, there is also an action of the group $A$ on the set $P\_a$, namely right trans... | 2 | https://mathoverflow.net/users/6794 | 89128 | 52,713 |
https://mathoverflow.net/questions/89124 | 3 | Suppose I'm trying to estimate the spectral radius of a square $n \times n$ matrix $A$,
and let $N$ be a distribution over Gaussian i.i.d. vectors of length $n$.
Is the following lemma true:
If the spectral radius of $A$ is larger than $\epsilon$ then with probability at least
$1/poly(n)$, a vector $v$ sampled accor... | https://mathoverflow.net/users/21604 | Estimating spectral radius with a Gaussian vector | No: given $\eta > \epsilon > 0$, there are $n \times n$ symmetric matrices $A\_n$
with spectral radius $> \eta$, such that $Pr\left[|v^TA\_nv|/\|v\|^2 > \epsilon\right] < e^{-cn}$ for some $c > 0$.
I assume a standard Gaussian distribution, with mean $0$ and covariance matrix $I$.
Consider an $n \times n$ diagonal ma... | 5 | https://mathoverflow.net/users/13650 | 89131 | 52,715 |
https://mathoverflow.net/questions/89110 | 4 | Jacquet, Piateski-Shapiro, and Shalika defined new vectors for generic representations of $GL(n,F)$, where $F$ is a non-archimedean local field. I know that this notion has been extended to $GSp(4,F)$. Is there an extension to other $p$-adic groups?
| https://mathoverflow.net/users/6849 | New vectors for $p$-adic groups | The theory of new vectors for ${\rm GSp}(4)$ has been written by Schmidt and Roberts :
Local Newforms for GSp(4).
Springer Lecture Note in Mathematics, vol. 1918 (2007)
See also Schmidt's webpage :
<http://www2.math.ou.edu/~rschmidt/>
The definition is trickier than in the case of ${\rm GL}(N)$
By the way : ... | 7 | https://mathoverflow.net/users/4767 | 89133 | 52,716 |
https://mathoverflow.net/questions/89132 | 11 | The Eilenberg-Mazur swindle shows that the Grothendieck group of an additive category with countable coproducts is trivial. The strategy is to observe that any "Euler characteristic" $\chi$ on such a category must be zero, because for any object $P$, we have $$P \oplus \bigoplus\_{i=1}^\infty P \simeq \bigoplus\_{i = 1... | https://mathoverflow.net/users/344 | Eilenberg-Mazur swindle for higher K groups | Yes, there is an analogue, see Weibel's book, chapter V, \S 1.9 (<http://www.math.rutgers.edu/~weibel/Kbook.html>). He calls an exact category $A$ "flasque" if there is a functor $\infty: A \to A$ such that $A \coprod \infty(A) \cong \infty (A)$ and proves that the Quillen K-theory space $K(A)$ of a flasque category $A... | 14 | https://mathoverflow.net/users/9928 | 89135 | 52,717 |
https://mathoverflow.net/questions/89116 | 1 | (I first recall the definitions, but specialists can probably go directly to the question.)
A *twist map* of the annulus $A=(\mathbb R/\mathbb Z)\times \mathbb R$ is an orientation preserving homeomorphism $f=(f\_1,f\_2):A\to A$ that satisfies the "twist condition": for every $x\_1$, the function $f\_1(x\_1,x\_2)$ is... | https://mathoverflow.net/users/4129 | Twist maps of the annulus | I assume that the expression "that preserves the measure" means that preserves area (a.k.a. the Lebesgue, or Haar measure) on $\mathbb{R}/\mathbb{Z}\times\mathbb{R}$.
In this case the answer is no. The most simple example of a twist map which is not topologically conjugate to an area-preserving one is given by a diss... | 3 | https://mathoverflow.net/users/889 | 89138 | 52,719 |
https://mathoverflow.net/questions/89127 | 4 | Given positive integers $d$ and $v$ with $v \geq d+1$, does there always exist a (convex) vertex-transitive $d$-polytope with $v$ vertices? It seems that the answer should be "obviously" true, but I don't know of any natural constructions. A couple of points:
1) If the question is changed to the existence of regular ... | https://mathoverflow.net/users/21606 | Vertex-transitive polytopes in any dimension with any number of vertices? | There is no three dimensional vertex transitive polyhedron with 7 vertices. The rotation groups of three dimensional polyhedra are finite quaternion groups. The only finite quaterionion groups whose order is divisible by 7 are associated with polygons with 7 or a multiple of 7 sides. So there have to be another element... | 7 | https://mathoverflow.net/users/1098 | 89139 | 52,720 |
https://mathoverflow.net/questions/89136 | 3 | Take the following small model for the category of finite-dimensional vector spaces and isomorphisms: The set of objects is $\mathbb{N}$ and the set of morphisms $Mor(n,m)$ is empty, if $n \neq m$ and $U(n)$ otherwise. If we take the classifying space of that, we get $\coprod\_{n \in \mathbb{N}} BU(n)$.
>
> What h... | https://mathoverflow.net/users/3995 | classifying space of linear embeddings | Let $\mathcal{A}$ be your category. This is a skeleton of the category $\mathcal{B}$ of all nonzero finite-dimensional complex Hilbert spaces and isometric embeddings, so $B\mathcal{A}$ is homotopy equivalent to $B\mathcal{B}$. Now let $\mathcal{B}\_0$ be the larger category where we allow the zero space. We have an in... | 8 | https://mathoverflow.net/users/10366 | 89141 | 52,722 |
https://mathoverflow.net/questions/89144 | 7 | I encountered an interesting function which is called "Eulerian" by the Wolfram's MathWorld:
$$\phi(q)=\prod\_{k=1}^{\infty} (1-q^{k})$$
It is interesting because it is claimed that roots of any polynomial can be expressed in this function and elementary functions. Is this true and how the roots of arbitrary polyno... | https://mathoverflow.net/users/10059 | Can roots of any polynomial be expressed using Eulerian function? | This Euler function is essentially the same as Dedekind's eta function ([Wikipedia](http://en.wikipedia.org/wiki/Dedekind_eta_function), [Mathworld](http://mathworld.wolfram.com/DedekindEtaFunction.html)). The usual use of the $\eta$ function is to express various modular forms. In particular, you should be able to rew... | 17 | https://mathoverflow.net/users/297 | 89149 | 52,723 |
https://mathoverflow.net/questions/89146 | 9 | The cardinal equation $\kappa^{\aleph\_0}=2^\kappa$ is satisfied by $\kappa=\aleph\_0$.
It is also satisfied by any $\kappa$ for which $MA(\kappa)$ holds.
Under $GCH$, the equation is satisfied by $\kappa$ if and only if $cof(\kappa)=\aleph\_0$.
So my question is:
>
> Is it consistent with $ZFC$ that the
> ... | https://mathoverflow.net/users/17836 | Is there always an uncountable $\kappa$ such that $\kappa^{\aleph_0}=2^\kappa$? | No, it is provable in ZFC that there are many $\kappa$ for which $2^\kappa=\kappa^\omega$. For example, let $\kappa=\beth\_\omega$, and the same argument will work with any strong limit cardinal of cofinality $\omega$. Observe that every subset of $\kappa=\beth\_\omega$ is determined by the sequence of its intersection... | 13 | https://mathoverflow.net/users/1946 | 89154 | 52,727 |
https://mathoverflow.net/questions/67308 | 5 | While analysing the average runtime of an algorithm, I came across the following identity, and would like to know if anybody knows of any references for it?
For $i \in \mathbb{N}$, let $\bar{s}(i)$ denote the square-free part of $i$, eg., $\bar{s}(12) = 3$ (and $\bar{s}(1)=1$). Then
$$
\lim\_{n \rightarrow \infty} \... | https://mathoverflow.net/users/15056 | Reference requested for $\lim_{n \rightarrow \infty} \frac{\sum_{i=1}^{n} \bar{s}(i)}{n^2} = \frac{\pi^2}{30}$ | The question asked for a reference, not a proof. A reference is Karl Greger, Square divisors and square-free numbers, Mathematics Magazine 51, No. 4 (Sept. 1978), 211-219. I make no claim that the result was unknown before Greger's paper.
| 1 | https://mathoverflow.net/users/3684 | 89158 | 52,730 |
https://mathoverflow.net/questions/89030 | 2 | Assume we have two "nice" schemes $X$ and $Y$ over $k=\mathbb{C}$, a finite flat map $f:X\rightarrow Y$ and a k-algbera $A$. Then we get an induced finite flat map $f\_A:X\times\_k A \rightarrow Y\times\_k A$.
Now given an $O\_{X\times\_k A}$-module $M$, flat over $A$ and an $A$-module $N$.
Is $(f\_A)\_{\*}(M\otime... | https://mathoverflow.net/users/3233 | Direct image sheaf and tensor product (is the projection formula an isomorphism?) | Most of this becomes obvious when translated using affine charts. For the first question, use a presentation of $N$ and the fact that $(f\_A)\_\*$ is exact to reduce to the case $N=A$. The projection formula is stated in EGA I (new edition), Corollaire 9.3.9.
| 1 | https://mathoverflow.net/users/2035 | 89164 | 52,733 |
https://mathoverflow.net/questions/89170 | 1 | The following problem arises in a particular machine learning problem:
Assume that we have $n$ independent Bernoulli random variables with parameters $p\_i$, e.g. $n=5$ and the $p$ vector is $(0.2, 0.3, 0.7, 0.6, 0.3)$. All possible realizations of the random variables form the corners of the $\lbrace 0,1\rbrace^n$-h... | https://mathoverflow.net/users/21614 | Hamming distance distribution induced by binary hypercube | You can certainly phrase this question more simply. Without loss of generality you can take $p\_i\leq 1/2$, so that the most likely corner is at 0. Then you are looking for the distribution of $\sum\_{i=1}^n X\_i$ where $X\_i$ are independent Bernoulli$(p\_i)$.
A simple way to calculate the probabilities you're afte... | 4 | https://mathoverflow.net/users/5784 | 89181 | 52,738 |
https://mathoverflow.net/questions/89180 | 1 | Let L be a 2-dimensional lattice and P- a lattice polygon. Suppose, it is triangulated into lattice tiangles. What are restrictions on their areas? For instance, can a lattice triangle of even area always be divided into lattice triangles of area 1? Is there any general approach to such questions?
| https://mathoverflow.net/users/21620 | Triangulations of lattice polygons | By Pick's Theorem, the area of a lattice polytope is related to the number $i$ of lattice points in the interior and the number $b$ of lattice points on the boundary, according to the formula:
$$A = i + \frac{b}{2} - 1$$
Note that if $b$ is odd, then the area must be equivalent to $1/2$ mod $1$. This implies that a... | 4 | https://mathoverflow.net/users/15331 | 89183 | 52,739 |
https://mathoverflow.net/questions/89182 | 1 | why the concept of compactly (or well) generated in triangulated categories is introduced?
| https://mathoverflow.net/users/21505 | why the concept of compactly (or well) generated in triangulated categories is introduced?. | Compactly generated ones are the biggest class of triangulated categories where the [Brown representability theorem](http://en.wikipedia.org/wiki/Brown%27s_representability_theorem) holds and can be checked using the same proof as for the stable homotopy category. This class of triangulated categories is not closed und... | 2 | https://mathoverflow.net/users/12166 | 89187 | 52,741 |
https://mathoverflow.net/questions/89188 | 2 | Norimatsu's lemma says that on a smooth projective complex variety $X$ of dimension $n$, then we have $H^i(X,\mathcal O\_X(-A-E))=0$ for $i < n$ when $A$ is an ample divisor and $E$ is a simple normal crossings (SNC) divisor. Does this statement remain true if $E$ is an effective divisor with SNC support? In other word... | https://mathoverflow.net/users/11661 | Kodaira type vanishing for an ample divisor + effective divisor with SNC support | Here is a counterexample.
Let $X$ be a smooth cubic surface in $\mathbf{P}^3$ and $E$ a line on it; moreover take $A=-K\_X$.
Then by Serre duality $$H^1(X, -A-aE)=H^1(X, K\_X + A + aE)=H^1(X, aE).$$
On the other hand $h^0(X, aE)=1$ since $E^2=-1$ and $h^2(X, aE)=h^0(X, K\_X-aE)=0$ since $K\_X$ is not effective.
... | 5 | https://mathoverflow.net/users/7460 | 89194 | 52,746 |
https://mathoverflow.net/questions/89177 | 4 | I'll call a polynomial in $z\_1,..,z\_N$ cyclic if it is invariant under cyclic permutation of the indices. I hope that's standard terminology.
I have N complex numbers $(z\_1,...,z\_N)$. I want to be able to compute what they are up to cyclic permutation, given the value of some set of cyclic polynomials. For exampl... | https://mathoverflow.net/users/13464 | cyclic polynomials and their solutions | Actually, if $N>2$, it's impossible to find $N$ "cyclic polynomials" as you desire. That's the same as asking if the cyclic invariants are a polynomial ring, which is impossible by the Chevalley-Shepard-Todd theorem
I suspect in general, the right thing to look at is the sum of the $z\_i$'s and all monomials in $p\_h... | 5 | https://mathoverflow.net/users/66 | 89196 | 52,747 |
https://mathoverflow.net/questions/89142 | 4 | Let $L \subseteq A^\star$ be a formal language over $A$ generated by a context-free grammar, and $L' = A^\star - L$ be the relative complement in $A^\star$.
If $L$ and $L'$ are both context-free, are they necessarily deterministic context-free?
| https://mathoverflow.net/users/12823 | Are context-free languages with context-free complements necessarily deterministic context-free? | It seems that the answer to your question is no. See [here](https://cstheory.stackexchange.com/questions/4263/the-class-cfl-cap-co-cfl).
| 6 | https://mathoverflow.net/users/15934 | 89197 | 52,748 |
https://mathoverflow.net/questions/89195 | 9 | I have the conjecture stated below, which would be very helpful for a cryptographic security proof I'm working on for several months now. At first glance, it seemed a pretty natural question to me, but after asking a lot of people and also searching the internet I'm still without a solution or any helpful hints. This i... | https://mathoverflow.net/users/21623 | Estimating the distance from a variety by evaluating the polynomial | You want this: <http://en.wikipedia.org/wiki/%C5%81ojasiewicz_inequality>
The exponent might not be the degree, though.
| 7 | https://mathoverflow.net/users/2290 | 89209 | 52,753 |
https://mathoverflow.net/questions/89213 | 6 | Let $C$ be a category with cofibrations in the sense of (Waldhausen, Algebraic K-Theory of Spaces) and denote by $F\_n(C)$ the category with cofibrations consisting of sequences of $n$ cofibrations $A\_0 \rightarrowtail A\_1 \rightarrowtail \dotsc \rightarrowtail A\_n$ in $C$. A cofibration in $F\_n(C)$ is a commutativ... | https://mathoverflow.net/users/2841 | When is a cube of cofibrations are "lattice"? | The notion you are looking for is well-known in homotopy theory under the name *Reedy cofibration*, but for some reason this name doesn't show up in papers about Waldhausen categories, even though the concept is used all the time.
To keep things close to your question let's say that $J$ is a finite poset (in general ... | 8 | https://mathoverflow.net/users/12547 | 89215 | 52,756 |
https://mathoverflow.net/questions/89198 | 1 | I have a matrix in which each element contains the coordinates of a 3D surface. Sometimes, some points will be "out of line" meaning that they will not conform to the general shape. For example you would have a slightly curved plane and all of a sudden a point with coordinates which are very different from the rest whi... | https://mathoverflow.net/users/21494 | Smooth a matrix | If I understand your question, you want to reconstruct a surface from a sample of points with noise. It is an active area of research, I do not now much about this but you can get started by looking to the web page <http://quentin.mrgt.fr/> of a colleague of mines, Quentin Mérigot. In particular I would guess that the ... | 1 | https://mathoverflow.net/users/4961 | 89216 | 52,757 |
https://mathoverflow.net/questions/89217 | 8 | Is the fibered K-theoretic farrell-jones conjecture true for cat(0)-groups?
| https://mathoverflow.net/users/19318 | The K-theoretic Farrell-Jones conjecture for cat(0) groups | Yes. More precisely, both the L-theory and K-theory Farrell-Jones conjectures with coefficients in any additive category have recently been proved for CAT(0) groups:
* Arthur Bartels, Wolfgang Lück, “The Borel Conjecture for hyperbolic and CAT(0)-groups”, Annals of Mathematics 175 (2012), 631–689, <http://dx.doi.org/... | 14 | https://mathoverflow.net/users/21636 | 89221 | 52,759 |
https://mathoverflow.net/questions/89178 | 11 | Let $G$ and $\Gamma$ be discrete groups, and let $\phi\colon\thinspace G\to \Gamma$ be a homomorphism.
Define its *cohomological dimension* $\operatorname{cd}\phi$ to be the least integer $d$ such that $\phi^\ast\colon H^i(\Gamma;M)\to H^i(G;M)$ is the zero homomorphism for all $i>d$ and all $\Gamma$-modules $M$ (whe... | https://mathoverflow.net/users/8103 | Cohomological dimension of a homomorphism | How about this? Let $\Gamma$ be free abelian of rank $2$. Poincare duality in the torus identifies $H^2(\Gamma;M)$ with $H\_0(\Gamma;M)$, so that in particular there is a natural surjection $M\to H^2(\Gamma;M)$.
Let $F$ be the particular $\Gamma$-module $\mathbb Z\Gamma$, free module of rank one for the group ring. ... | 4 | https://mathoverflow.net/users/6666 | 89225 | 52,762 |
https://mathoverflow.net/questions/89086 | 20 | [Generating Compiler Optimizations from Proofs](http://cseweb.ucsd.edu/~rtate/publications/proofgen/) is a wonderful paper. The authors say that they were faced with the problem, got stuck, then tried reasoning about it using category theory. They took the obvious tack, isolated the new idea, designed the abstract algo... | https://mathoverflow.net/users/756 | Examples of algorithms that came from category theory? | I'm glad you liked the paper. I thought you might like to know that the algorithms for both of my PLDI papers were actually designed using a category-theoretic framework I made for existential types (half of which turned out to be opfibrations, but I didn't know about those at the time). Others have asked to see it, so... | 15 | https://mathoverflow.net/users/21638 | 89227 | 52,764 |
https://mathoverflow.net/questions/89098 | 1 | **Setting and question**
Let $X$ be a variety over an algebraically closed field of null characteristic, and let $C$ be a (regular if you want) curve included in $X$.
Consider $X'$ the normalization of $X$ and $C^\*$ the sheaf-theoric pull-back of $C$ in $X$. Assume that $C^\*$ is reduced, or even regular if you want... | https://mathoverflow.net/users/19205 | Is this function field extension a Galois extension ? | As requested by Francois: No, this is typically not correct. In affine space with coordinates $x$,$y$ and $z$, consider the variety cut out by the single equation $y^3−3yz^2−xz^3$. This is not normal; the normalization is obtained by adjoining the fraction $u=y/z$. The normalization is itself isomorphic to a hypersurfa... | 6 | https://mathoverflow.net/users/13265 | 89235 | 52,768 |
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