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https://mathoverflow.net/questions/52171 | 4 | The following paragraph appears in Analytic Number Theory (Iwaniec, Kowalski):
---
The Siegel-Walfisz theorem asserts that:
$\displaystyle \hspace{5cm} \psi(x;q,a) = \frac{x}{\phi(q)} + O(x(\log x)^{-A})$
for any $q\geq 1, (a,q)=1, x\geq 2$ and $A\geq 0$. **Notice that this estimate is non-trivial only if $q ... | https://mathoverflow.net/users/12160 | When is the Siegel-Walfisz theorem non-trivial? | The $\ll$ in this context means that there is some constant $C > 0$ such that $q \leq C (\log{x})^A$. If $q$ goes to infinity (with $x$) much faster than this then the main term itself can be bounded by the error term (in which case the result would be "trivial").
| 9 | https://mathoverflow.net/users/2627 | 52175 | 32,704 |
https://mathoverflow.net/questions/52177 | 4 | If K is a field, v a discrete valuation, and k the residue field, there is a residue map $\partial: K^M\_n(K) \to K^M\_{n - 1}(k)$. All the definitions I have seen for this map involve two pages of combinatorics of different cases (e.g. the book of Fesenko Vostokov).
Does anyone know of a clean, concise, compact defi... | https://mathoverflow.net/users/8324 | Is there a clean definition of the residue map in Milnor K-theory? | Look at Lemma 2.1 in [Milnor's original paper](http://0-www.springerlink.com.fama.us.es/content/t025u1152j330325/), $\partial$ is characterized by the following properties: given $\pi, u\_i \in K$ with $v(\pi)=1$ and $v(u\_i)=0$,
$$\partial(u\_1 \cdots u\_n)=0,$$
$$\partial(\pi\cdot u\_2 \cdots u\_n)=\bar{u}\_2\cdo... | 4 | https://mathoverflow.net/users/12166 | 52179 | 32,705 |
https://mathoverflow.net/questions/52176 | 11 | Let $G$ be a (discrete) group, and $1/G$ the corresponding groupoid with one object. Consider the diagram in (the 2-category) Groupoids with one vertex, labeled $1/G$, the one arrow from that vertex to itself, given by the identity map.
$$ \begin{matrix} 1/G \\ {\huge \circlearrowleft} \\ \scriptstyle \mathrm{id} \e... | https://mathoverflow.net/users/78 | How to interpret topologically that the equalizer in Groupoids of ${\rm id}, {\rm id}: BG \rightrightarrows BG$ is $G/G$ (adjoint action)? | You should think of the limit of that diagram as "loops in $BG$." A loop in $BG$ is a principal $G$-bundle on $S^1$. Every principal $G$-bundle on the circle comes from taking the trivial principal $G$-bundle on $[0,1]$ and identifying the fibers over $0$ and $1$ (making this the basepoint). If you like left principal ... | 11 | https://mathoverflow.net/users/66 | 52181 | 32,706 |
https://mathoverflow.net/questions/52178 | 3 | Given a matrix $M$, I want to find a nontrivial vector in the kernel of $M$ that also lies in the first orthant, if such a vector exists. That is, I want to simultaneously solve
$$Mx = 0$$
$$x \geq 0$$
$$x \neq 0$$
I'm having trouble phrasing this problem in a way that can be efficiently solved numerically. One app... | https://mathoverflow.net/users/6542 | finding an element of a vector subspace contained in the first orthant | @Ben's answer is within $\epsilon$ of correct. The problem with it is that (depending on how you interpret the constrains $x\_i \geq 1$) there might be no solution with either a specific $x\_{i\_0} > 0,$ or *all* $x\_i>0,$ and as in the original question, cycling through all the indices is inefficientInstead you use Go... | 3 | https://mathoverflow.net/users/11142 | 52187 | 32,711 |
https://mathoverflow.net/questions/52191 | 9 | Let $X \to B$ be a smooth, proper, dominant map of schemes over $\text{Spec }k$ an algebraically closed field of characteristic zero with $B$ integral. We have the generic fibre $\overline{F}$ defined over $\text{Spec }\overline{K(B)}$ and by base-changing along $\text{Spec }\overline{K(B)} \to \text{Spec }k$, we obtai... | https://mathoverflow.net/users/7690 | Is a fibration in algebraic geometry a fibre bundle? | The answer here is a resounding no. I think the most important point is that you're applying the wrong topological intuition here. A variety shouldn't be thought of as like a manifold, but as like a complex manifold, and the corresponding theorem to the "submersion=fiber bundle" theorem in smooth manifolds is just fals... | 17 | https://mathoverflow.net/users/66 | 52195 | 32,716 |
https://mathoverflow.net/questions/52198 | 0 | I am reading Local fields and see Serre using $v\_{\mathfrak{p}}(\mathfrak{a})$ where $\mathfrak{a}$ is a fractional ideal of the Dedekind domain $A$ and $v\_{\mathfrak{p}}$ is the valuation associated to the discrete valuation ring $A\_{\mathfrak{p}}$. Serre did not really define this in the book, I looked it up on th... | https://mathoverflow.net/users/12312 | What is the definition of the valuation of a fractional ideal? | Let $d'$ be any other nonzero element of $R$ such that $d' \mathfrak{a} \subset R$. Then
$v\_{\mathfrak{p}}(d' \mathfrak{a}) + v\_{\mathfrak{p}}(d) = v\_{\mathfrak{p}}(dd' \mathfrak{a}) =
v\_{\mathfrak{p}}(d') + v\_{\mathfrak{p}}(d\mathfrak{a})$.
So
$v\_{\mathfrak{p}}(d'\mathfrak{a}) - v\_{\mathfrak{p}}(d') = ... | 4 | https://mathoverflow.net/users/1149 | 52200 | 32,719 |
https://mathoverflow.net/questions/52197 | 11 |
>
> Let $f \colon R \to S$ be an epimorphism of commutative rings, where $R$ and $S$ are integral domains. Suppose that $\mathfrak{p} \subset S$ is a prime such that $f^{-1}(\mathfrak{p}) = 0$. Does it follow that $\mathfrak{p} = 0$?
>
>
>
If answer is "yes", then it follows that for any epimorphism of commutati... | https://mathoverflow.net/users/5094 | Is Krull dimension non-increasing along ring epimorphisms? | Yes. Letting $k$ be the field of fractions of $R$, we have the following commutative diagram.
$$
\begin{array}{ccc}
R&\stackrel{f}{\rightarrow}&S\\\\
\downarrow\scriptstyle{}&&\downarrow\scriptstyle{}\\\\
k&\stackrel{g}{\rightarrow}&S\_{\mathfrak{p}}
\end{array}
$$
However, $f$ and the localization $S\to S\_{\mathfrak{... | 17 | https://mathoverflow.net/users/1004 | 52204 | 32,722 |
https://mathoverflow.net/questions/52186 | 9 | Is there an example of a self-dual complete category that is not a partially-ordered set?
| https://mathoverflow.net/users/3711 | Self-dual Complete Category | The category of suplattices (complete join-semilattices) is complete and self-dual. It is complete because it is monadic over Set (the monad is the covariant powerset monad). The duality $Sup^{\mathrm{op}}\to Sup$ sends a suplattice $X$ to the opposite poset $X^{\mathrm{op}}$, which is again a suplattice since every su... | 11 | https://mathoverflow.net/users/49 | 52210 | 32,724 |
https://mathoverflow.net/questions/52215 | 4 | STPL := soundness theorem for predicate logic
(see [this](http://books.google.com/books?id=Y87XKUfGlCUC&pg=PA149&lpg=PA149&dq=predicate+logic+soundness+theorem&source=bl&ots=9-5JeqFp2v&sig=4iCLsI8m6RQWQ5QV7lslsGlzWZo&hl=en&ei=EYQyTYuSOYPQsAOPqdGDBg&sa=X&oi=book_result&ct=result&resnum=2&sqi=2&ved=0CB4Q6AEwAQ#v=o... | https://mathoverflow.net/users/nan | Soundness Theorem in reverse mathematics | First, a caveat: Simpson treats the Soundness Theorem in SOSOA, but not in the way you intend it. Simpson defines (II.8.3) a model $M$ as having a truth valuation for all sentences in the language of $M$ augmented with a constant for each element of $M$. When models are defined in this way, the Soundness Theorem is pro... | 5 | https://mathoverflow.net/users/2000 | 52228 | 32,733 |
https://mathoverflow.net/questions/52224 | 10 | Let $p\colon X\to B$ be a fibration. Let $G$ be a topological group acting continuously on $X$ and $B$, and assume that the map $p$ is $G$-equivariant.
We can apply the Borel functor $EG\times\_G-$ for a contractible, free $G$-space $EG$. This gives a map $1\times\_G p\colon EG\times\_G X\to EG\times\_G B$.
>
> ... | https://mathoverflow.net/users/8103 | Does the Borel functor take equivariant fibrations to fibrations? | If $\pi:EG \to BG$ is a numerable $G$-principal bundle and if $p$ is a Hurewicz fibration, then the map $1 \times\_G p$ is a Hurewicz fibration.
Proof: Let $U \subset BG$ be open such that $EG \to BG$ is trivial over $U$. Observe that $\pi^{-1}(U) \times\_G B \cong U \times B$. Likewise for $X=E$ instead of $B$ and t... | 12 | https://mathoverflow.net/users/9928 | 52234 | 32,735 |
https://mathoverflow.net/questions/52232 | 8 | Define the axiom constants $p\_{A,B}^{((A\rightarrow B)\rightarrow A)\rightarrow A}$ as realizers of the Peirce formula, and $f\_A^{\bot\rightarrow A}$ as realizers of the Ex Falso Quodlibet. Then $p\_{A\vee\lnot A,\bot}^{(\lnot (A\vee\lnot A)\rightarrow(A\vee\lnot A))\rightarrow(A\vee\lnot A)}\lambda\_{u^{\lnot(A\vee\... | https://mathoverflow.net/users/3118 | What fails when using call/cc as realizer of the Peirce formula | I'm not sure what you mean by 'fail' here.
It is true, as you say, that classical proofs can exhibit behaviour that constructive proofs can't. In general, it's no longer true that a term of type A reduces to a constructive proof of the proposition A -- but that's only to be expected. Proof-terms in classical logic ar... | 11 | https://mathoverflow.net/users/4262 | 52244 | 32,740 |
https://mathoverflow.net/questions/52248 | 7 | 1. Over an (algebraically closed) characteristic $p$ field, is it known that the cohomological equivalence of cycles relation (with respect to $\mathbb{Q}\_l$-adic \'etale cohomology) does not depend on the choice of a prime $l$ (distinct from $p$). At least, is it known that: if the numerical equivalence of cycles rel... | https://mathoverflow.net/users/2191 | Questions on standard (motivic) conjectures | The answer to both questions in 1 is NO. For example, Clozel has shown that for an abelian variety over the algebraic closure of a finite field, there are infinitely many l for which numerical and homological equivalence coincide, but this doesn't help with proving the statement for all l (or even the independence of h... | 5 | https://mathoverflow.net/users/12252 | 52255 | 32,744 |
https://mathoverflow.net/questions/52105 | 14 | I ask this, although I don't conduct any research in the area, or even plan to. -- There seems to be general agreement that the axioms for higher categories grow very rapidly in complexity as the order $n$ of the category grows to infinity. I am referring to the ''algebraic style'' axioms here, axiomatising a ''maximal... | https://mathoverflow.net/users/8624 | Are the axioms for higher category-theory effectively computable? | **Yes.**
…at least, for Leinster’s reformulation of Batanin’s definition of globular operadic weak ω-category (and hence also for the finite-dimensional versions of this). Showing this is essentially a matter of repeatedly applying one lemma: if $\mathbf{T}$ is an essentially algebraic theory with a computable presen... | 5 | https://mathoverflow.net/users/2273 | 52256 | 32,745 |
https://mathoverflow.net/questions/52241 | 46 | Let E be an elliptic curve, let $L(s) = \sum a\_n n^{-s}$
denote its L-function, and set
$$ f(x) = \sum a\_n \frac{x^n}{n}. $$
Then Honda has observed that
$$ F(X,Y) = f^{-1}(f(X) + f(Y)) $$
defines a formal group law.
The formal group law of an elliptic curve has applications to
the theory of torsion points, a... | https://mathoverflow.net/users/3503 | Formal group laws and L-series | Okay, here's a few words about the relation between the $L$-series and the formal group. In general, if $F(X,Y)$ is the formal group law for $\hat G$, then there is an associated formal invariant differential $\omega(T)=P(T)dT$ given by $P(T)=F\_X(0,T)^{-1}$. Formally integrating the power series $\omega(T)$ gives the ... | 50 | https://mathoverflow.net/users/11926 | 52260 | 32,746 |
https://mathoverflow.net/questions/52257 | 7 | Given two long binary strings of length N, it's easy to find the Hamming distance between them. If you're allowed to cyclically shift one of the strings, you'll get N different Hamming distances when comparing the two. What is an efficient way to find the maximum Hamming distance over all N shifts?
This question is m... | https://mathoverflow.net/users/12242 | Cross correlation detection in binary Hamming distance | I'd suggest that you start by encoding your signal in terms of the symbols +1 and -1 rather than 0 and 1. If you have two signals x and y, then take elementwise products of x and y and sum to get a measure of the distance between x and y. If the signals are identical, then the sum will be n. If the signals differ in ea... | 6 | https://mathoverflow.net/users/9022 | 52269 | 32,753 |
https://mathoverflow.net/questions/52272 | 2 | Let $\Sigma$ be a symmetric positive definite matrix. Then the Cholesky decomposition gives us $\Sigma=LL'$ where $L$ is lower triangular and unique.
Under what conditions (if any) does there exist a second symmetric positive definite matrix $\Omega$ which is NOT diagonal that satisfies $\Sigma=\hat{L} \Omega \hat{L}... | https://mathoverflow.net/users/12064 | Is it possible to decompose a symmetric, positive definite matrix in this way? | It seems to me that if you look at <http://en.wikipedia.org/wiki/Cholesky_decomposition>
the "Cholesky outer product algorithm" writes $L = L\_1 \dots L\_k,$ so if you write $\Lambda\_i = L\_i\dots L\_k,$ then $\Omega=\Lambda\_i \Lambda\_i^\prime$ should work for most values of $i.$
| 1 | https://mathoverflow.net/users/11142 | 52275 | 32,757 |
https://mathoverflow.net/questions/52270 | 8 | Let $W\_p$ be the Weil group of $\mathbf{Q}\_p$. What is the Galois cohomology group $H^2(W\_p,\mathbf{C}^{\times} )$ (with trivial action)? Is it zero, or something huge and complicated?
(This group comes up, at least for me, when you want to compare two Weil group representations whose projectivizations agree.)
| https://mathoverflow.net/users/1464 | Is H^2(W_p,C^times) well-known? | It is known that $H^2(W, C^\times)$ is trivial, when $W$ is the Weil group of a global or local field, with the trivial action on $C^\times$, and the cohomology is taken in the sense of Moore (measurable cochains). This is the main result of C.S. Rajan, "On the vanishing of the measurable cohomology groups of Weil grou... | 11 | https://mathoverflow.net/users/3545 | 52278 | 32,760 |
https://mathoverflow.net/questions/52073 | 17 | While working on a problem of differential topology I stumbled on a question of algebraic geometry that seems pretty basic, but that I'm unable to answer because I know very little algebraic geometry. I hope somebody could help:
It's known that if $X$ and $Y$ are (algebraic) subvarieties of $d$-dimensional (complex) ... | https://mathoverflow.net/users/1516 | Intersection of subvarieties in grassmannian space | Well, after a crash course on cohomology of the grassmannians, Schubert calculus, etc, I think I've found the answer to the second question: If the codimension of $X$ in $G\_m(\mathbb{C}^n)$ is less than or equal to $m-k$ then $X$ should intersect the (Schubert cell) $Y$.
Luckily this should be sufficient for the app... | 1 | https://mathoverflow.net/users/1516 | 52293 | 32,770 |
https://mathoverflow.net/questions/15476 | 6 | Fix an integer $d \ge 2$ and let $M\_d$ be the space of real $d \times d$ matrices. Let $E$ be a vector subspace of $M\_d$. We say that $E$ is *transitive* if $E \cdot \mathbb{R}^d\_\* = \mathbb{R}^d$, that is, for all vectors $v \in \mathbb{R}^d\_\* = \mathbb{R}^d-\{0\}$ and $w \in \mathbb{R}^d$ there exists a matrix ... | https://mathoverflow.net/users/1516 | Algebraic characterization of transitive spaces of matrices | Here is the real answer:
A space of matrices is transitive iff its "orthogonal complement" contains no matrix of rank one.
The idea was not mine; it is I found in Sec. 4 from the paper below (See also some more modern and more readable papers that cite it):
Azoff, E.A. On finite rank operators and preannihilators... | 4 | https://mathoverflow.net/users/1516 | 52295 | 32,772 |
https://mathoverflow.net/questions/52296 | 1 | Let $p$ be an odd prime number.
Ramanujan's tau function satisfies:
(a)
$$
\tau(p^{n+1}) = \tau(p^n)\tau(p)-p^{11}\tau(p^{n-1})
$$
for all positive integers $n>0.$
So $\tau(p)=0$ implies
(b)
$$
\tau(p^{2r+1})=0,
$$
and
(c)
$$
\tau(p^{2r})=(-1)^rp^{11r}
$$
for all nonnegative integers $r \geq 0.$
Assume now that (... | https://mathoverflow.net/users/11016 | If Ramanujan's tau function has a prime power zero then $\ldots$ | Theorem 2 of Lehmer's paper "The vanishing of Ramanujan's function $\tau(n)$" says that the smallest $n$ for which $\tau(n)=0$ is prime.
More generally, this paper should be of interest to you given your question.
| 7 | https://mathoverflow.net/users/5743 | 52300 | 32,774 |
https://mathoverflow.net/questions/52299 | 8 | Hello everybody.
I'm looking for an "easy" example of a (non-zero) holomorphic function $f$ with almost everywhere vanishing radial boundary limits: $\lim\limits\_{r \rightarrow 1-} f(re^{i\phi})=0$.
Does anyone know such an example.
Best
CJ
| https://mathoverflow.net/users/10893 | Holomorphic function with a.e. vanishing radial boundary limits | I am not sure if this would qualify as 'easy' but the first example of such a function was constructed by Lusin. It can be found in N. Lusin, J. Priwaloff, [Sur l'unicité et la multiplicité des fonctions analytiques](http://archive.numdam.org/ARCHIVE/ASENS/ASENS_1925_3_42_/ASENS_1925_3_42__143_0/ASENS_1925_3_42__143_0.... | 9 | https://mathoverflow.net/users/5371 | 52301 | 32,775 |
https://mathoverflow.net/questions/52246 | 0 | I got this puzzle some time ago and it has been bugging me since, I cant solve it - but it is supposedly solvable, I am interested in a solution or any tips on how to proceed.
In front of you is an entity named Adam. Adam is a
solid block with a single speaker, through which
he hears and communicates. For all propo... | https://mathoverflow.net/users/12249 | Seemingly complex logic/set-theoretic puzzle | First, observe that you can get around the difficulty that
you don't know if high means yes or low in the following
way. If you really want to ask the question $\varphi$, you
should instead ask the question *"high means yes for this round if and
only if $\varphi$"*. If high means yes, then this is the
same as asking $\... | 17 | https://mathoverflow.net/users/1946 | 52303 | 32,776 |
https://mathoverflow.net/questions/52291 | 13 | I have a specific concrete example of a complete, finite volume, cusped hyperbolic 3-manifold $M$, and I am trying to determine whether or not $M$ is a cover of the figure-8 knot complement (call this $F\_8$).
The manifold $M$ is described combinatorially, and all the numbers work out; for instance I know $M$ and $F... | https://mathoverflow.net/users/12260 | Detecting a cover of the figure-8 knot complement | **Edited, in light of description of manifold in comments (at end)**
**Added #2: The answer is no, details at the end**
In principle this is doable, but caution is necessary. Given a manifold tiled by
ideal simplices, its fundamental group is a subgroup $H$ of the full group $G$ of isometries of finite index in the... | 19 | https://mathoverflow.net/users/9062 | 52305 | 32,778 |
https://mathoverflow.net/questions/52302 | -2 | Topological Methods in Group Theory witten by Ross Geoghegan
What about this book?
| https://mathoverflow.net/users/12264 | How about this book Topological Methods in Group Theory | I can just review the first half of the book. In order, the short introduction of general topology is too short, a good reference would have been preferable. For the following, the theory of CW complexes requires a tough notation and technicalities in proofs and definitions. Furthermore, proofs are not so easy to read,... | 1 | https://mathoverflow.net/users/47274 | 52308 | 32,781 |
https://mathoverflow.net/questions/52318 | 3 | The question is in the title, and I do not really have anything to add. Nevertheless I had to write something here in order to be able to ask the question. Thanks.
| https://mathoverflow.net/users/4800 | Is there always, for a given prime $p$, a prime $\ell<p$ that is not a quadratic residue mod $p$? | Of course. Take a quadratic nonresidue $1\leq n\leq p-1$, then some prime divisor $\ell$ of $n$ will be a quadratic nonresidue.
See [this MO question](https://mathoverflow.net/questions/52211/effective-chebotarev-density/52237#52237) for what is known about number fields.
| 17 | https://mathoverflow.net/users/11919 | 52319 | 32,786 |
https://mathoverflow.net/questions/52286 | 42 | I recently heard the following fact :
Up to the $15$th skeleton, the classifying space $BE\_8$ and $K(\mathbb{Z},4)$ are homotopy equivalent?
I have two questions on this :
(1) Is there any easy way to see this? Of course, knowing the first fourteen homotopy groups of $E\_8$ is enough but then the question is how... | https://mathoverflow.net/users/1993 | How are the classifying space of $E_8$ and $K(\mathbb{Z},4)$ related? | Given a simple Lie group $G$, you can check how far $G$ is from beeing a $K(\mathbb Z,3)$ by looking at the place where the affine vertex gets glued onto the Dynkin diagram, and measuring the length of that tail. For $E\_8$, it's the longest, and so $E\_8$ is the best possible approximation to a $K(\mathbb Z,3)$.
$$\... | 54 | https://mathoverflow.net/users/5690 | 52321 | 32,788 |
https://mathoverflow.net/questions/52310 | 2 | Hello all
As is probably well-known to most, in the upper halfplane we have a natural action of $SL\_2(\mathbb{R})$ through linear fractional transformations, and a $2$-form $\frac{dx dy}{y^2}$ which is invariant under this action.
In the case of the upper half-space $H\_3 = \{ (z,t) \ | \ z\in \mathbb{C} \ , \ t>... | https://mathoverflow.net/users/12266 | invariant 2-form in hyperbolic 3-space | In upper half space, the thing that's analogous to the invariant 2-form (which measures hyperbolic area) is the hyperbolic volume form $1/z^3 dx dy dz$.
If you want to understand 2-forms invariant by a particular discrete group, they are
the same thing as 2-forms on the quotient orbifold. In the case of $SL\_2$ of th... | 8 | https://mathoverflow.net/users/9062 | 52324 | 32,789 |
https://mathoverflow.net/questions/52290 | 1 | I'm pretty sure that this is a minor error, but I could use some help here. So the book I'm referring to in the title is [this](https://www.springer.com/us/book/9783764327354) book (MR1164870).
On pg. 16-17, he is proving that the space of almost complex structures on a compact smooth surface without boundary is a sm... | https://mathoverflow.net/users/10046 | Is there an Error on pg. 17 of Tromba's "Teichmuller Theory in Riemannian Geometry"? | A small typo: your definition of $\mathcal{M}$ is incorrect; shouldn't it be $\det^{-1}(1)$?
Anyway, I think you are mixing up two parts of the proof. The first is that $\mathcal{M}$ is a smooth submanifold. Indeed, using your computation, it is easy to see that taking $H= J$ you have $D\det(J)H = \mathop{tr}(J^{-1}... | 4 | https://mathoverflow.net/users/3948 | 52325 | 32,790 |
https://mathoverflow.net/questions/52145 | 10 | I've decided it's time to start learning how to use a computer to do calculations... I've used Singular to some small extent so far, but I want to start relying on computer algebra systems more.
### Question
Which computer algebra system is best for what, and what is the easiest/most fun(?) way to learn how to deal... | https://mathoverflow.net/users/5309 | Roadmap to Computer Algebra Systems Usage for Algebraic Geometry | Your question is "Which computer algebra system is best for what, and what is the easiest/most fun(?) way to learn how to deal with them?"
* Regarding *community*, I think Sage (<http://sagemath.org>) is the best CAS, since Sage is completely open and free, and there are about a dozen mailing lists, and thousands of... | 17 | https://mathoverflow.net/users/8441 | 52340 | 32,795 |
https://mathoverflow.net/questions/52322 | 19 | Let $G$ be some (dimension $1$, to simplify) formal group over a characteristic $0$ field $K$. The law of $G$ is denoted by $\oplus$. If $w(X) \in K[[X]] dX$ is a differential form, let $F\_w(X)$ be the unique power series such that $dF\_w=w$ and $F\_w(0)=0$. Let $F\_w^2(X,Y) = F\_w(X \oplus Y) - F\_w(X) - F\_w(Y)$. Sa... | https://mathoverflow.net/users/5743 | De Rham cohomology of formal groups | I have put an updated copy of my formal groups notes here:
<http://neil-strickland.staff.shef.ac.uk/courses/formalgroups/fg.pdf>
They are not really finished, but the relevant material is discussed in Section 18.
| 18 | https://mathoverflow.net/users/10366 | 52354 | 32,804 |
https://mathoverflow.net/questions/52353 | 8 | Let $X$ be a topological space. When I call a set nowhere dense, meagre or similar without qualification, I mean that it has this property as a subset of $X$. Call a subset of $X$ *weager* (for weakly meagre) if it is the union of a chain (wrt containment) of nowhere dense sets. Using that finite unions of nowhere dens... | https://mathoverflow.net/users/12281 | Wanted: chain of nowhere dense subsets of the real line whose union is nonmeagre, or even contains intervals | **Theorem.** There is no chain of nowhere dense subsets of $\mathbb{R}$ whose union contains an interval.
Proof. Suppose there was such a chain $\{\ B\_i \mid i\in I\ \}$, where $\langle I,\lt\rangle$ is a linear order and $i\lt j$ implies $B\_i\subset B\_j$. First, I claim that this chain cannot have countable cofin... | 15 | https://mathoverflow.net/users/1946 | 52363 | 32,811 |
https://mathoverflow.net/questions/52341 | 2 | $K$ a field, $A\subset K$ a subring, $M\subset K$ an $A$-submodule. Define
$$(A:\_{K}M):= \lbrace s\in K|sM\subset A\rbrace$$
Then it is easy to see that
$$M\subset A\Longleftrightarrow A\subset (A:\_{K}M),$$
$$A\subset M\Longrightarrow (A:\_{K}M)\subset A$$
But I couldn't show the reverse implication of the sec... | https://mathoverflow.net/users/5292 | Related to fractional ideals | This is not true, you are guessing right. Here is a counter-example: take $k$ a field, $A=k[X,Y]$, $K=k(X,Y)$.
Let $M=XA+YA$. Then using the fact that $A$ is a UFD one see easily that $(A:\_K M)=A$.
(indeed, let $P/Q \in (A:\_K M)$ with $P,Q \in A$, $Q \neq 0$. We have $XP/Q \in A$ so
$Q$ divides $XP$. Similarly $Q$ di... | 6 | https://mathoverflow.net/users/9317 | 52373 | 32,817 |
https://mathoverflow.net/questions/52381 | 4 | I was thinking a bit more about the setting of my recent [question](https://mathoverflow.net/questions/52353/wanted-chain-of-nowhere-dense-subsets-of-the-real-line-whose-union-is-nonmeagre) about unions of chains of nowhere dense subsets of the reals and got stuck almost immediately on a follow-up question. I suspect t... | https://mathoverflow.net/users/12281 | Unions of chains closed under finite unions? Under unions of chains? | Let $X$ be the disjoint union of a countable set $A$ and $\omega\_1$, the first uncountable ordinal. Let $B$ be the family of sets of the form $a\cup c$ where $a$ is a finite subset of $A$ and $c$ is a countable subset of $\omega\_1$. $B$ is closed under subsets and finite unions. The union of a chain in $B$ has the fo... | 5 | https://mathoverflow.net/users/6085 | 52386 | 32,823 |
https://mathoverflow.net/questions/52383 | 12 | In the abstract of
>
> Singularités irrégulières Correspondance et documents
>
> Pierre Deligne, Bernard Malgrange, Jean-Pierre Ramis
>
> Documents mathématiques 5 (2007), xii+188 pages ([link](http://smf4.emath.fr/en/Publications/DocumentsMathematiques/2007/5/html/smf_doc-math_5.php))
>
>
>
there is a... | https://mathoverflow.net/users/4177 | What is the wild fundamental group? | Consider the category of algebraic integrable connections on a smooth connected algebraic variety $X$ with a base point $x$ (over a field of characteristic zero). This is a Tannakian category with a fibre functor given by taking fibres over $x$ and is hence equivalent to the representations of some pro-algebraic group.... | 20 | https://mathoverflow.net/users/4008 | 52388 | 32,824 |
https://mathoverflow.net/questions/52397 | 6 | Let $X$ be an Enriques surface. A $(-2)$-curve is an irriducible rational curve on X such that $C^2 = -2$. By Proposition [VIII,16.1] from Barth-Peters-Van de Ven, we have that if $D^2 = -2$, then it is a $(-2)$-curve, but do such curves exist?
| https://mathoverflow.net/users/1937 | Are there (-2)-curves on an Enriques surface? | As explained in J.C. Ottem's answer, the generic Enriques surface contains no smooth rational curves at all.
However, it can happen that some special Enriques surface $X$ contains $(-2)$-curves, and also infinitely many of them (see [this paper](http://www.springerlink.com/content/x175482343514234/) by Cossec and Dol... | 14 | https://mathoverflow.net/users/7460 | 52401 | 32,830 |
https://mathoverflow.net/questions/52406 | 13 | The famous Harer stability theorem asserts that the homology group $H\_d(\mathcal{M}\_{g,n},\mathbf{Z})$ is independent of *g* and *n* in the range $0 \leq 2d < g-1$. This is proven by analyzing the maps of mapping class groups $\Gamma\_{g,n}\to \Gamma\_{g+1,n}$ given by gluing a torus with a disk removed to a boundary... | https://mathoverflow.net/users/1310 | Is there Harer stability for moduli of curves with level structure? | This is a hard open problem. Essentially nothing is known except for linear congruence subgroups. Denoting by $Mod\_{g,n}(L)$ the level $L$ linear congruence subgroup, the desired result is only known for $H\_1(Mod\_{g,n}(L);\mathbb{Q})$ (which is due to Hain) and for $H\_2(Mod\_{g,n}(L);\mathbb{Q})$ (which is due to m... | 15 | https://mathoverflow.net/users/317 | 52407 | 32,832 |
https://mathoverflow.net/questions/52410 | 1 | I want to determine if a given graph is a minimal 3-connected graph. That is, deletion of any edge will reduce the vertex connectivity to 2.
My approach right now, is to look at every edge where both endpoints have degree 4 or more, remove the edge and see if the vertex connectivity has decreased to 2 in the whole gr... | https://mathoverflow.net/users/1539 | Testing connectivity when deleting an edge | I would say **yes**. If the connectivity between $x$ and $y$ has decreased, then obviously the connectivity of $G$ has decreased. On the other hand, if $G - e$ is not 3-connected, then there are a pair of subgraphs $(A,B)$ such that $G-e=A \cup B$, and $|V(A) \cap V(B)|=2$. Since $G$ is 3-connected, this implies that $... | 2 | https://mathoverflow.net/users/2233 | 52412 | 32,835 |
https://mathoverflow.net/questions/52404 | 18 | I have started studying some étale cohomology and I am trying to build up some intuition about the concept of *local for the étale topology*. I can understand some nice examples (like Kummer exact sequence) but I am still quite confused by some "easy" notions such as locally constant sheaves.
I believe that an étale ... | https://mathoverflow.net/users/973 | Locally constant sheaves for the étale topology, lack of intuition about "étale-localness" | $Isom(F,G)$ is indeed an etale sheaf. If we take $F = \mathbb Z/n$ and $G = \mu\_n$,
then $G$ is a sheaf of $F$-modules, and so evaluation at the global section $1$ gives an isomorphism of sheaves $Hom(\mathbb Z/n,\mu\_n) \cong \mu\_n$, which identifies $Isom(\mathbb Z/n,\mu\_n)$ with the subsheaf of $\mu\_n$ whose sec... | 16 | https://mathoverflow.net/users/2874 | 52416 | 32,836 |
https://mathoverflow.net/questions/52339 | 2 | Let $S\_a^d$ be the $(d-1)$-dimensional sphere of radius $a$ in $\mathbb{R}^d$. Let $r>0$ be a constant and $R=\nu r$ where $\nu>1$ (some constant). Are there any known upper bounds on the number of disjoint $S\_r^d$ that can be completely contained in $S\_R^d$?
| https://mathoverflow.net/users/2011 | Sphere packing in a sphere | Although the book "Sphere packings, lattices and groups" by Conway and Sloane deals mostly with infinite packings, it has an extensive bibliography that gives some references for the problem you're interested in too. Here are some entries that looked relevant, although I haven't read them:
* Gritzmann and Wills, "Fin... | 2 | https://mathoverflow.net/users/11108 | 52421 | 32,839 |
https://mathoverflow.net/questions/52417 | 10 | Let $P(m,n)$ mean that there is a number, $M$, such that starting with $M$ there are $m$ consecutive numbers each having exactly $n$ distinct prime factors. Is it obvious that $P(m,n)$ is true for all $m$ and $n$? My gut says "obviously" and $P(4,4)$ and $P(5,5)$ are definitely true (for 134043 and 129963314 respective... | https://mathoverflow.net/users/12295 | Consecutive numbers with n prime factors | $P(4,1)$ is true because of $2,3,4,5$ but I strongly doubt that $P(5,1)$ is true. Indeed $P(4,1)$ is only true that once and even $P(3,1)$ happens for the last time at $7,8,9$.
Of any $210$ consecutive integers one is divisible by $2,3,5,7$ so $P(210,3)$ fails. I'm sure that can be improved. It does generalize .
As n... | 10 | https://mathoverflow.net/users/8008 | 52422 | 32,840 |
https://mathoverflow.net/questions/52419 | 14 | Let $S$ be a complex surface of general type. Are there infinitely many smooth rational curves on $S$? And more general, what if $V$ is a variety of general type?
| https://mathoverflow.net/users/5328 | Rational curves on varieties of general type | I think that the best result in this direction is the following theorem due to Bogomolov:
>
> **Theorem.** Let $S$ be a surface of general type with $c\_1^2(S) > c\_2(S)$. Then for any $g$ the curves of geometric genus $g$ on $S$ form a bounded family.
>
>
>
In particular, since a surface of general type cann... | 15 | https://mathoverflow.net/users/7460 | 52424 | 32,841 |
https://mathoverflow.net/questions/52433 | 6 | What is the complexity of the theory of addition (Presburger arithmetic) augmented by a unary predicate that recognizes powers of 2?
| https://mathoverflow.net/users/10909 | Theory of addition and a predicate that recognizes powers of 2 | The theory of the natural numbers with addition and $x\mapsto 2^x$ is decidable.
One reference is the Cherlin-Point paper "On extensions of Presburger arithmetic".
It can be found on Francoise Point's webpage:
<http://www.logique.jussieu.fr/~point/papiers/cherlin_point86.pdf>
| 7 | https://mathoverflow.net/users/5849 | 52459 | 32,857 |
https://mathoverflow.net/questions/52457 | 6 | A theorem of Chevalley, Shepard, and Todd states that if $G$ is a finite group and $\rho: G \rightarrow GL\_n(\mathbb{C})$ a representation so that $\rho(G)$ is generated by pseudoreflections, then $\mathbb{C}[z\_1,\dots,z\_n]^G$ (the subring of $G$-invariant polynomials) is again a polynomial ring.
From my understan... | https://mathoverflow.net/users/12306 | Finite groups generated by pseudo-reflections | It's a property of the representation. For example, under the 1-dimensional representation of $\newcommand{\Z}{\mathbb Z} \Z/2$ given by $x\mapsto -x$, $\Z/2$ is generated by pseudoreflections, but under the 2-dimensional representation $(x,y)\mapsto(-x,-y)$, it's not.
It makes sense to ask if there is an abstract fi... | 7 | https://mathoverflow.net/users/1 | 52460 | 32,858 |
https://mathoverflow.net/questions/52169 | 30 | Motivated by the apparent lack of possible classification of integer matrices up to conjugation ([see here](https://mathoverflow.net/questions/51942/analogue-of-smith-normal-form-for-matrices-over-mathbb-zt)) and by a question about possible complete graph invariants ([see here](https://mathoverflow.net/questions/52141... | https://mathoverflow.net/users/8176 | Adjacency matrices of graphs | Yes.
Consider the adjacency matrices
$$ A = \left[\begin{array}{rrrrrrrrrrr}
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\
1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\
0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\
0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\
0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\
0 & 0 ... | 41 | https://mathoverflow.net/users/7936 | 52461 | 32,859 |
https://mathoverflow.net/questions/52426 | 3 | Let G be an algerbraic group, and X be a G-variety (that I will assume to be smooth). Then one can consider a simlicial variety whose terms are $G^i\times X$. This simplicial variety yields a 'complex of varieties' $\dots\to G\times G\times X\to G\times X\to X$ (the arrows are formal alternating sums of the correspondi... | https://mathoverflow.net/users/2191 | For a G-variety, what could one say about the motif of the corresponding simplicial variety | The simplicial variety you write is a presentation of the stack $X/G$ as a simplicial sheaf, and the motive you obtain is just be the motive of $X/G$, no? Of course you might use this as the definition of the motive of $X/G$, so maybe this is orthogonal to your question.
It's a motive over $BG$, or equivalently corresp... | 2 | https://mathoverflow.net/users/582 | 52472 | 32,864 |
https://mathoverflow.net/questions/52467 | 10 | Let $\mathcal{P}$ denote the set of primes. Define the function $r\_2(N)$ to be the number of ways to write $N$ as a sum of two not necessarily distinct primes (where order matters). Then the famous Goldbach Conjecture can be phrased as following:
$r\_2(2N) > 0$ for all $N \geq 1$.
A related function, $r\_3(N)$ wh... | https://mathoverflow.net/users/10898 | A question about primes as an additive basis | I think the heuristic evidence suggests quite the opposite, that $r\_2(N)$ increases without bound. See <http://www.ieeta.pt/~tos/goldbach.html> for both theoretical background and computational results.
| 6 | https://mathoverflow.net/users/3684 | 52473 | 32,865 |
https://mathoverflow.net/questions/45320 | 10 | Let $P$ be a probability distribution on a finite set $\mathcal{X}$ and let $X\_1, X\_2, \ldots, X\_n$ be drawn i.i.d. according to $P$. Define the empirical distribution:
$\hat{P\_n}(x) = \frac{1}{n} \sum\_{i=1}^{n} 1\_{X\_i = x}$
Let $d\_H(P,Q)$ be the Hellinger distance:
$d\_H(P,Q) = \left( \frac{1}{2} \sum\_{... | https://mathoverflow.net/users/1185 | Convergence of an empirical distribution w.r.t. the Hellinger distance | it is possible to show that $\mathrm {E}d(P,\hat{P\_n})\sim \frac{C}{\sqrt{n}}$ and specify the value of $C$.
let
$$D\_n^2 =\sum\_{x \in \mathcal{X}} \left( \sqrt{P(x)} - \sqrt{\hat{P\_n}(x)} \right)^2 = 2d^2(P,\hat{P\_n}). $$
$4nD\_n^2$ is known in statistics [for reasons unclear to me] as the freeman-tukey good... | 5 | https://mathoverflow.net/users/8977 | 52474 | 32,866 |
https://mathoverflow.net/questions/52455 | 3 | I am interested in (Chow) motives over (algebraically closed) characteristic $p>0$ fields. For $H$ being $\mathbb{Q}\_l$-adic cohomology, one can consider $Ch\_l(M)=\sum (-1)^i\dim\_{\mathbb{Q}\_l} H^i(M)$.
Does this Euler characteristic depend on the choice of $l\neq p$?
| https://mathoverflow.net/users/2191 | Do $Q_l$-etale Euler characteristics of Chow motives coincide for all $l$? | I might be missing something (seeing that I don't know what a Chow motive is), but I think the answer is yes. It is a result of G.Laumon proved in Comparaison de caractéristiques d’Euler-Poincaré en cohomologie $\ell$-adique, C. R. Acad. Sc. Paris, t. 292 (1981), Série I, 209-212 that for $X$ a separated scheme of fini... | 3 | https://mathoverflow.net/users/2284 | 52475 | 32,867 |
https://mathoverflow.net/questions/52448 | 7 | I was wondering if someone could recommend a reference that deals with time integrals of diffusion processes.
Suppose $X$ is an Ito diffusion process with dynamics
$dX\_t = \mu(X\_t)dt + \sigma(X\_t)dW\_t$. The process I'm interested in is $Y\_t = \int\_0^t X\_s ds$. I haven't seen any treatment of the properties of ... | https://mathoverflow.net/users/9564 | Time integrals of diffusion processes | One can adapt the argument used to show that, for a standard Brownian motion $W$, the laws of $W$ and $\sigma W$ on any interval $[0,t]$ with $t > 0$ and $\sigma^2\ne1$ are singular.
For every positive $v$, let $E\_v$ denote the space of $C^1$ real valued functions defined on $[0,t]$ such that the quadratic variatio... | 7 | https://mathoverflow.net/users/4661 | 52483 | 32,872 |
https://mathoverflow.net/questions/52470 | 2 | Hello,
Let $K/{\mathbb Q}$ be a finite extension which is not necessarily Galois, and ${\mathcal O}$ be the ring of integers of $K$. Let $p$ be a prime in ${\mathbb Q}$ and let
$p {\mathcal O}={\mathfrak p}\_1^{k\_1} \cdots {\mathfrak p}\_r^{k\_r}$ be its prime factorization in $K$.
Next let ${\mathcal O}\_i$ be the... | https://mathoverflow.net/users/10458 | A subring question (revised) | Assume $\mathcal{O}\_p=\mathbb{Z}\_p\times\mathbb{Z}\_p$.
The only subring of finite index in $\mathbb{Z}\_p$ is $\mathbb{Z}\_p$: indeed such a ring must contain $n\mathbb{Z}\_p$ for some $n$, hence $p^k\mathbb{Z}\_p$ for some $k$, hence corresponds to a subring of $\mathbb{Z}/p^k\mathbb{Z}$. (Of course I take it th... | 3 | https://mathoverflow.net/users/7666 | 52487 | 32,876 |
https://mathoverflow.net/questions/43022 | 5 | Let $A\_{n,d}$ be the space of polynomials of degree $d$ or less whose coefficients are real $n\times n$ matrices --- or, if you prefer, the space of matrices whose entries are degree-$d$ polynomials. Given $A(z),B(z)\in A\_{n,d}$, there are $P(z),Q(z)\in A\_{n,d}$ so that $P(z)A(z)=Q(z)B(z)$ (proof: consider the $2dn\... | https://mathoverflow.net/users/1898 | annihilator/common left multiple of matrix polynomials | In case anyone is interested, I have found a counterexample to (2) by delving into some old papers: it's in Gohberg, I.; Kaashoek, M. A.; Lerer, L.; Rodman, L. *Common multiples and common divisors of matrix polynomials. II. Vandermonde and resultant matrices.* Linear and Multilinear Algebra 12 (1982/83), no. 3, 159–20... | 2 | https://mathoverflow.net/users/1898 | 52488 | 32,877 |
https://mathoverflow.net/questions/52489 | 14 | In a paragraph written by a person emphasizing that rigour is not everything in mathematics (I wish I had written down the details), it was stated that the moon landings would have been impossible without the use of nonrigorous algorithms (I believe he was referring to the calculations of the trajectories of the spacec... | https://mathoverflow.net/users/4692 | On the non-rigorous calculations of the trajectories in the moon landings | The existence of [the Arenstorf Orbits](http://en.wikipedia.org/wiki/Richard_Arenstorf) was discovered in 1963 on the basis of numerical computations. The Arenstorf orbits appear as periodic solutions to the equations for the plane restricted three body problem (see the [original paper](http://www.jstor.org/pss/2373181... | 29 | https://mathoverflow.net/users/5371 | 52490 | 32,878 |
https://mathoverflow.net/questions/52479 | 1 | Suppose X is a curve.
Under sufficiently nice conditions we have that every line bundle on X corresponds to an equivalence class of divisors modulo principal divisors, with tensor product of bundles corresponding to addition of divisors.
Given a line bundle L on X, I will call the Euler Characteristic of L minus th... | https://mathoverflow.net/users/7935 | Effect of tensor product on euler characteristic of line bundles | I think there is a complete answer to your question in
SGA 6 Exp X $\S$ 5 :
for a quasi-compact scheme there is a natural isomorphism (first Chern class) ${\rm Pic} X \simeq Gr^1(X)$ where $Gr^1(X)$ is defined by the filtration of the Grothendieck group $K\_.(X)$ of coherent sheaves on $X$ given by the dimension of su... | 5 | https://mathoverflow.net/users/11682 | 52495 | 32,881 |
https://mathoverflow.net/questions/52454 | 7 | Let $\mathcal H$ be a Hilbert space, and let $a \in \mathcal B(\mathcal H)$ satisfy $\mathrm{Tr}(a)=0$. If $a$ is self-adjoint, then we can find a vector $\xi \in \mathcal H$ such that $\langle \xi | a \xi \rangle=0$. Is this true in general? Is it always possible to find an orthonormal basis of such vectors?
| https://mathoverflow.net/users/2206 | Vanishing Trace | The answers to both your questions are yes. (ie for any trace class operator $a$ on a Hilbert space $H$ with $Tr(a)=0$, there exists an orthonormal basis of $H$ consisting of vectors $\xi$ such that $\langle a \xi,\xi\rangle=0$).
First note that the fact that there exists an orthonormal basis of vectors $\xi$ such th... | 6 | https://mathoverflow.net/users/10265 | 52498 | 32,884 |
https://mathoverflow.net/questions/52492 | 3 | I'm reading Switzer's "Algebraic Topology", which talks about the (homology) ASS in chapter 19. His ASS (where he puts $S^0$ in the first slot) either converges to $\pi\_n(Y)$, or to $\pi\_n(Y)/\cap\_{s\geq 0}F^{s,n+s}$ (which happens e.g. when our theory $E$ is the spectrum $H\mathbb{F}\_p$ representing $H^\*(-;\mathb... | https://mathoverflow.net/users/303 | localization at a homology theory and the Adams spectral sequence | If you're talking about strong convergence, your statement that the ASS converges to a quotient of $\pi\_\*(Y)$ is not correct, even for $Y=S^0$ and $E=H\mathbb{F}\_p$, where it converges to the p-adic homotopy groups $\pi\_\*(S^0) \otimes \mathbb{Z}\_p$. Which agrees with the localization in this case. Weak convergenc... | 8 | https://mathoverflow.net/users/4183 | 52500 | 32,885 |
https://mathoverflow.net/questions/52396 | 23 | $\DeclareMathOperator\End{End}\newcommand\Id{\mathrm{Id}}\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\SO{SO}$I start with some background, but people familiar with the subject may jump directly to the question.
Let $M^{4n}$ be a compact oriented smooth manifold. Recall that an *almost hypercomplex structure* on $M... | https://mathoverflow.net/users/10675 | Are there topological obstructions to the existence of almost quaternionic structures on compact manifolds? | Thomas' [proof](https://mathoverflow.net/a/52436) for the fact that $S^{4n}$ has no almost quaternionic structure is correct, but I have found an alternative argument for this statement using the twistor space. Indeed, if $S^{4n}$ has an almost quaternionic structure $Q$, then the twistor space $S(Q)$ is an $S^2$-bundl... | 7 | https://mathoverflow.net/users/10675 | 52510 | 32,890 |
https://mathoverflow.net/questions/52513 | 4 | I know this is true with etaleness added (SGA I, 5.1; in that case the morphism is an open immersion) or if the morphism is in addition proper (since by ZMT the map is then finite and the fibers are all either isomorphic to the residue field or empty, so the morphism is a closed immersion). Is this true in general? I d... | https://mathoverflow.net/users/344 | Does unramified + radicial imply that a morphism is an immersion? | No, this is not true, things can go wrong with the topology. For example, consider a smooth curve $X$ with a point $p$, and the natural morphism $(X \smallsetminus \{p\}) \sqcup \{p\} \to X$.
| 9 | https://mathoverflow.net/users/4790 | 52514 | 32,893 |
https://mathoverflow.net/questions/52509 | 21 | Let $E$ be a closed subspace of $L^2[0,1]$. Suppose that $E\subset{}L^\infty[0,1]$. Is it true that $E$ is finite dimensional?
PS. This is actually a question from the real analysis qualifier. I came across it as I was teaching qualifier preparation course, and was solving problems from old qualifiers. So, though it ... | https://mathoverflow.net/users/10714 | Subspace of $L^2$ that lies in $L^\infty$ | Another solution: as Mikael wrote, $||f||\_{\infty} \leq C ||f||\_2$ for every $f \in E$.
Let $f\_1,\ldots,f\_n$ be an orthonormal family in your subspace.
Then for every $x \in [0,1]$, $f\_1(x)^2+\ldots+f\_n(x)^2 \leq ||f\_1(x)f\_1+\ldots+f\_n(x)f\_n||\_{\infty} \leq C \|f\_1(x)f\_1+\ldots+f\_n(x)f\_n\|\_2$ $$=C \sqrt... | 29 | https://mathoverflow.net/users/5735 | 52525 | 32,903 |
https://mathoverflow.net/questions/52526 | 1 | Alfred Tarski, in his paper "Ueber unerreichbare Kardinalzahlen" Fund. Math. vol 30 (1938) pp 68-89 proves the followig theorem of ZFC "If the cardinal of the set Y is equal to the cardinal of the set of the subsets of Y that are not equipotent with Y, then the cardinal of Y is (strongly) inaccessible". The proof of th... | https://mathoverflow.net/users/30395 | Tarski's caracterisation of inaccessible cardinals | In modern notation, it says, "if $\kappa$ is a cardinal and $\kappa ^{< \kappa} = \kappa$, then $\kappa$ is strongly inaccessible." This isn't entirely true since the antecedent holds for $\kappa = \omega$ but $\omega$ isn't considered strongly inaccessible, but that's not a big deal. More importantly, under CH the ant... | 2 | https://mathoverflow.net/users/7521 | 52531 | 32,907 |
https://mathoverflow.net/questions/52532 | 4 | Given a index set $S$ together with a ultrafilter $\mu$ on $S$ (such that no set of cardinality $< |S|$ has measure $1$). Let the ordered field $\mathbb{R}(S,\mu)$ denote the ultrapower of $\mathbb{R}$ with respect to $S$, i.e.
$\mathbb{R}(S,\mu):=\mathbb{R}^S/\sim$, where two maps $f,g$ are called equivalent, if $\m... | https://mathoverflow.net/users/3969 | cardinality of local bases in the non-standard reals | Your question is essentially equivalent to the one [here](https://mathoverflow.net/questions/15596/is-there-a-version-of-the-archimedean-property-which-does-not-presuppose-the-natu); for what is known see in particular the bottom of Joel David Hamkins' answer there.
To see the equivalence, note that choosing such a l... | 2 | https://mathoverflow.net/users/5963 | 52536 | 32,909 |
https://mathoverflow.net/questions/52438 | 9 | Hello,
Let $F$ and $G$ be two functions belonging in the Selberg class, $\Lambda\_{F}$ and $\Lambda\_{G}$ the complete L-functions associated to $F$ and $G$. I would like to know whether this assertion is true or not:
"$\Lambda\_{F}$ and $\Lambda\_{G}$ have the same zeroes if and only if $F=G$ or $F=\overline{G}$."... | https://mathoverflow.net/users/11542 | Zeroes of complete L-functions | Expanding on Stopple's comment above, I believe the following argument based on Landau's explicit formula answers the question.
Here is a generalization of Landau's explicit formula for the zeros of the Riemann zeta-function which is exercise 8.4.8 in M. Ram Murty's book *Problems in Analytic Number Theory*: Let $F$ ... | 7 | https://mathoverflow.net/users/3659 | 52537 | 32,910 |
https://mathoverflow.net/questions/52538 | 3 | My question relates to the proof of the Atiyah-Singer Index Theorem for families of elliptic operators, as presented in "The Index of Elliptic Operators: IV", M. F. Atiyah and I. M. Singer.
Let $A$ be a compact Hausdorff space and $q:A\times \mathbb{C}^n\rightarrow A$ be the projection, then we obtain the induced Tho... | https://mathoverflow.net/users/12329 | Short question relating to the proof of the Atiyah-Singer Index Theorem for families | The answer should probably go something like this:
Both $q\_!$ and $ind$ are $K(A)$-module maps. Since $q\_!$ is an isomorphism, to check that these are the same maps, it suffices to check they are the same on a generator; if $u\_A\in K\_{Cpt}(A\times C^n)$ is the unique element such that $q\_!(u\_A)=1$, then we need... | 5 | https://mathoverflow.net/users/437 | 52540 | 32,911 |
https://mathoverflow.net/questions/52543 | 2 | Alfred Tarski, in his paper "Ueber unerreichbare Kardinalzahlen" Fund. Math. 1938 proves the following ZFC theorem "if the cardinal of the set Y is equal to the cardinal of the set of subsets of Y that are not equipotent to Y, then the cardinal of Y is REGULAR." The proof of the paper is rather long and involved.
Quest... | https://mathoverflow.net/users/30395 | Tarski's caracterisation of REGULAR cardinals | Let me prove the contraposition: If $\kappa$ is not regular, then it has more than $\kappa$ subsets of size strictly less than $\kappa$.
Suppose $\kappa$ is not regular. Let $\lambda$ be the cofinality of $\kappa$.
Note that the number of subsets of $\kappa$ of size $\lambda$ is $\kappa^\lambda$. It is therefore enou... | 5 | https://mathoverflow.net/users/7743 | 52546 | 32,913 |
https://mathoverflow.net/questions/52541 | 13 | You are and your friend are given a list of $N$ distinct integers and are told this:
Six distinct integers from the list are selected at random and placed one at each side of a cube. The cube is placed in the middle of a rectangular room in front of its only door, with one face touching the floor, its 6 sides paralle... | https://mathoverflow.net/users/12265 | Hard Cube Puzzle | This is a variation on a classic card trick (audience pick 5 cards, magician A removes a card of his choosing and hands the rest to magician B, who then names the missing card.)
There are two ways you can view the process - from the point of view of the cube-orienter, or from the point of view of the final guesser.
... | 29 | https://mathoverflow.net/users/11108 | 52547 | 32,914 |
https://mathoverflow.net/questions/52493 | 11 | Let $S$ be a circle with unit circumference. Suppose that $n$ random points are chosen independently uniformly from $S$; choosing one arbitrarily as $x\_1$, label the rest $x\_2, \dots, x\_n$ in clockwise order. What is the expected value of
$$
\max |x\_{i}-x\_{i-1}|
$$
(where $x\_0$ is interpreted as $x\_n$)? Or even ... | https://mathoverflow.net/users/5091 | Expected second moment for random points on a circle | The problem is equivalent to choosing $n-1$ points at random on the unit interval and considering the length of the longest resulting subinterval. Given that, the distribution of the maximum and its expected value are in my answer to this recent math.SE question on [Average Length of the Longest Segment](https://math.s... | 5 | https://mathoverflow.net/users/9716 | 52548 | 32,915 |
https://mathoverflow.net/questions/52555 | 4 | The cost of solving a linear system ("exactly") with Gauss Elimination and other similar methods with a few right hand side and where the matrix has no structure is $\mathcal{O}(N^3)$ where $N$ is the system size.
I am wondering about the lower bound for solving a linear system. An obvious lower bound is $\mathcal{\O... | https://mathoverflow.net/users/nan | Lower Bound on the Cost of Solving Linear System | Any $O(N^t)$ algorithm for matrix multiplication yields a corresponding $O(N^t)$ algorithm for matrix inversion. There are not any non-trivial lower bounds for matrix multiplication. It is believed that there exist $O(N^{2 + \epsilon})$ algorithms for any $\epsilon > 0$, but as far as I know there are not believed to b... | 4 | https://mathoverflow.net/users/9896 | 52557 | 32,918 |
https://mathoverflow.net/questions/52553 | 9 | $\DeclareMathOperator\GL{GL}\DeclareMathOperator\Frob{Frob}$In his paper, [Duke paper](https://projecteuclid.org/journals/duke-mathematical-journal/volume-54/issue-1/Sur-les-repr%C3%A9sentations-modulaires-de-degr%C3%A9-2-de-GalQQ/10.1215/S0012-7094-87-05413-5.short), Serre consider continuous, odd Galois representatio... | https://mathoverflow.net/users/12312 | On the determinant of an odd, continuous Galois representation | $\det (\rho)$ is a one dimensional rep of the absolute Galois group of the rationals, i.e., it is a character. All such characters can be described by class field theory or, more simply, by the Kronecker-Weber theorem. So is a Dirichlet character and, by the hypotheses, its conductor divides $pN$. Factor it as a charac... | 7 | https://mathoverflow.net/users/2290 | 52558 | 32,919 |
https://mathoverflow.net/questions/52445 | 9 | I quote Proposition 2.3, page 14 lines -3 and -4 of Michael Rosen's book
Number Theory in Function Fields:
Let $b\_n$ be the number of square-free monics in $A= \mathbb{F}\_q[t]$ of degree $n.$
Then $b\_1=q$ and for $n>1$, $b\_n=q^n-q^{n-1}.$
Using the proposition it is easy to prove that any polynomial $P \in A$... | https://mathoverflow.net/users/11016 | All polynomials over a finite field are sums of $2$ square-free polynomials | Alright, putting together all the comments it appears to still be true when $q = 2$, depending on how you like your square-free polynomials.
First, to quote Sonia's comment, If $P$ is not sq.free, then for each sq.free polynomial $Q$ of degree $n$, we have a non-zero $P−Q$, of degree $\leq n−1$, $2^{n−1}$ in all. The... | 6 | https://mathoverflow.net/users/9542 | 52562 | 32,922 |
https://mathoverflow.net/questions/52544 | 0 | I find a definition of "exact neighborhood" (the book is online (\*1) Definition 1.5 on p.10) confusing.
Do I understand it right that neighborhood that contains the global optimum and no other local optimums/ma is what they call "exact neighborhood" in this book ?
Is this terminology accepted anywhere, or just in ... | https://mathoverflow.net/users/12330 | definition of "exact neighborhood" [optimization] | It's important that you understand the definition of a neighborhood used in this book. This is definition 1.3 on page 7. $N$ is not a set but rather a function that maps a solution to the problem to a subset of the entire set of solutions. You can also speak of the neighborhood of a particular solution, s, with respect... | 3 | https://mathoverflow.net/users/9022 | 52566 | 32,925 |
https://mathoverflow.net/questions/52550 | 6 | Let $X$ be a normal projective complex variety.
A theorem of Fujita-Zariski says that if $L$ is a Cartier divisor on $X$ such that
the base locus $Bs(|L|)$ is a finite set then $L$ is semiample.
It seems to me that by using this theorem it is possible to prove that the stable base locus $\mathbb{B}(L):=\bigcap\_{m \in ... | https://mathoverflow.net/users/6430 | Stable base loci cannot contain isolated points | Yes, you are right, and the result is almost immediate using the FZ theorem. A proof can be found in the recent paper [Restricted volumes and base loci of linear series.](http://www.ams.org/mathscinet-getitem?mr=2530849) by Ein, Lazarsfeld,Mustaţă, Nakamaye and Popa. For the sake of completeness I sketch the argument h... | 8 | https://mathoverflow.net/users/3996 | 52570 | 32,926 |
https://mathoverflow.net/questions/31266 | 7 | A stupid question: which statements in section 5 of BBD will fail if we replace $\overline{\mathbb{Q}\_l}$-sheaves by just $\mathbb{Q}\_l$-ones? I am especially interested in Proposition 5.1.15.
BBD = Beilinson A., Bernstein J., Deligne P., Faisceaux pervers, Asterisque 100, 1982, 5-171.
| https://mathoverflow.net/users/2191 | Which statements in section 5 of BBD will fail if we consider $\mathbb{Q}_l$-adic sheaves there? | I think that all the statements are true, except for 5.3.9 (ii). Remark 5.3.10 says that all the statements in 5 up to and including 5.3.8 are true for $\mathbb{Q}\_\ell$-coefficients with the same proof, and that 5.3.9 (i) is still true but with a slightly more complicated proof. I am pretty sure that the proof of cor... | 5 | https://mathoverflow.net/users/12336 | 52571 | 32,927 |
https://mathoverflow.net/questions/52554 | 6 | I'm reading a paper, in which we have $M^n$ an n-dimensional compact hypersurface embedded in $\mathbb{R}^{n+1}$. We take the scalar cuvature $R$ to be the elementary symmetric polynomial of degree 2 in the principal curvatures of $M$. We know that $R$ is constant.
The author then says "As $M$ has one elliptic point,... | https://mathoverflow.net/users/11266 | Hypersurfaces and Elliptic Points | Elliptic point is, by definition, a point where all the principal curvatures are positive, hence $R$ is positive. A point of maximal distance from some far-away basepoint is elliptic.
| 4 | https://mathoverflow.net/users/11142 | 52573 | 32,928 |
https://mathoverflow.net/questions/52576 | 14 | I was reading about some of the zero density Theorems for my Analytic Number Theory Topics course. While looking over some more complicated results and proofs a few simple questions came up:
First, let $$N(\sigma,T)=|\{ \rho=\beta+i \gamma \text{ }:\text{ } \zeta(\rho)=0, \text{ } 0< \gamma < T, \text{ } \sigma\leq\b... | https://mathoverflow.net/users/12176 | Zeta Function: Zero Density Theorems. | If you state the *Density Hypothesis* as
$$ N(\sigma,T) \ll T^{ 2(1-\sigma)+\epsilon}$$
the factor $1-\sigma$ seems natural because the exponent $2(1-\sigma)$ is 1 when $\sigma=1/2$, where there are $
\gg T\log T$ zeros, and is $2(1-\sigma)$ is 0 when $\sigma=1$ where there are no zeros. So as a linear function of $\si... | 11 | https://mathoverflow.net/users/3659 | 52577 | 32,930 |
https://mathoverflow.net/questions/52579 | 1 | Fulton's Book on intersection theory (Pg.223, theorem 12.3) asserts the following result:
For r pure dimensional schemes in P^n, whose co-dimensions add to at most n, the product of their degrees is at least as great as the sum of the degrees of the irreducible components of their intersection.
Under what condition... | https://mathoverflow.net/users/5218 | A Theorem in Intersection theory. | Call the schemes to be intersected $(X\_i)$, where $X\_i$ has pure codimension $r\_i$ in ${\mathbb P}^n$. Let $R = \sum r\_i$. (Edited so as not to restrict to $R = n$ unnecessarily.)
Definitely, every component of the intersection has codimension at most $R$. If the codimensions are all exactly $R$, *and the schemes... | 5 | https://mathoverflow.net/users/391 | 52582 | 32,932 |
https://mathoverflow.net/questions/52578 | 10 | Suppose we have a matrix $M$ such that $M$ is non-symmetric real and has positive eigenvalues. Do we have a relation between eigenvalues/eigenvectors of $(M+M^T)$ and those of $M$?
What if $M$ and $(M+M^T)$ both are of low rank?
Suppose, $M = AP$ where $A$ is a positive semi-definite matrix and $P$ is an orthogonal p... | https://mathoverflow.net/users/12341 | Eigenvalues of sum of a non-symmetric matrix and its transpose $(A+A^T)$ | Let $N:=(M+M^T)/2$. besides the obvious equality $Tr(N)=Tr(M)$ which is an equality of the sums of eigenvalues, you have the following. Let $\lambda\_\pm$ be the smallest/largest eigenvalues of $N$. Then every eigenvalue of $M$ satisfies $\Re\lambda\in[\lambda\_-,\lambda\_+]$. In addition, if $w(M):=\max\{\lambda\_+,-\... | 13 | https://mathoverflow.net/users/8799 | 52588 | 32,934 |
https://mathoverflow.net/questions/52587 | 17 | Let $V$ be a finite-dimensional vector space and let $\mathfrak g \subset \mathfrak{gl}(V)$ be a representation of a semisimple Lie algebra on $V$. Let $e\_1, \dots, e\_n$ be a basis for $V$. Let $e\_1', \dots, e\_n'$ be the dual basis of $\{e\_i\}$ under the Killing form $B\_V(X,Y) = \mathrm{trace}(X \circ Y)$. The *C... | https://mathoverflow.net/users/2362 | Basis-free definition of Casimir element? | The Casimir element is dual to the Killing form. (I think. I am somewhat uncertain about this because nobody has ever said this to me, even though it seems like the right thing to say, and frankly I don't know why Lie algebra textbooks don't just say this.) That is, the nondegeneracy of the Killing form is equivalent t... | 20 | https://mathoverflow.net/users/290 | 52594 | 32,937 |
https://mathoverflow.net/questions/52535 | 10 | Consider the rational curve (conic) given by image of the map
$$ u([z,w])=[z^2,-z^2,w^2,-w^2,zw] \in \mathbb{P}^4 $$
which lies in quintic 3-fold $X: x\_1^5+\cdots+x\_5^5- x\_1\cdots x\_5=0$.
By Grothendick theorem and the fact that $X$ is Calabi-Yau, we know that $u^\*N\_C^X= O(a) \oplus O(b)$, for some $a+b=-2$... | https://mathoverflow.net/users/5259 | Calculating the decomposition of a vector bundle over rational curve | Let $(z,w) \mapsto (f\_1(z,w),\dots,f\_5(z,w)$, $\deg f\_i = s$, be a map $P^1 \to P^4$ and $g(x\_1,\dots,x\_5)$, $\deg g = d$, be an equation of a hypersurface containing the image. Then the normal bundle is the middle cohomology of the following complex
$$
O(1)^2 \to O(s)^5 \to O(ds)
$$
where the first map is given b... | 8 | https://mathoverflow.net/users/4428 | 52597 | 32,940 |
https://mathoverflow.net/questions/52607 | 6 | If I am not mistaken there is a theorem that says any curve $C$ can be embedded in $\mathbf{P}^3$. What can be said about surfaces? Do we have a theorem like:
>
> All surfaces can be embedded in $\mathbf{P}^{N}$ for some fixed $N$.
>
>
>
And if not what is the simplest counterexample? A counterexample would b... | https://mathoverflow.net/users/8811 | Embedding of algebraic surfaces | All smooth projective surfaces can be embedded in $\mathbb{P}^5$ using a (linear) projection. This is a classical theorem in algebraic geometry. In general, it is known that any smooth projective variety of dimension $n$ can be embedded into $\mathbb{P}^{2n+1}$.
The proof runs as follows: The secant variety $Sec(X)$... | 18 | https://mathoverflow.net/users/3996 | 52608 | 32,944 |
https://mathoverflow.net/questions/52551 | 5 | I am interested in compactification of the moduli space of elliptic curves, and I heard that Log geometry is very important for the problem.
I am developping the same technique for quantum geometry.
My question was that:
1-Why Log structure can give us a better way to understand degeneration of Elliptic curves? W... | https://mathoverflow.net/users/12333 | Log structure and degeneration | 1. Read the introduction in the Kato-Usui book. It's got some nice motivating examples, including a degenerating family of elliptic curves with pictures.
2. There is no problem with defining a group object in the category of log schemes.
3. This depends on what you want to do with such a structure, i.e., what propertie... | 3 | https://mathoverflow.net/users/121 | 52610 | 32,945 |
https://mathoverflow.net/questions/52581 | 1 | Assume that $X$ is the complement of a plane algebraic curve $C$ in $\mathbb{C}^2$ and Y is the complement of the union of $C$ and a line $L$ (not contained in $C$). Assume that $Y$ is $K(\pi, 1)$. Is it true that $X$ is $K(\pi, 1)$? Why or why not?
| https://mathoverflow.net/users/2348 | Homotopy type of complement of a plane algebraic curves. | The answer to this question is no. Let me explain why, the explanation uses two facts.
**1)** The complement to the collection of $4$ generic lines in $\mathbb CP^2$ is not $K(\pi,1)$.
Indeed, it is not hard to see that the fundamental group of this complement is $\mathbb Z^3$. At the same time the complement has h... | 7 | https://mathoverflow.net/users/943 | 52612 | 32,947 |
https://mathoverflow.net/questions/52604 | 2 | Hi,
I'm trying to construct some coordinates on Minkowski spacetime based on a world line, $C$, ($\dot{C}\cdot\dot{C}=-1$) and forward light cone. I want the "time" coordinate of a point, $p$, to be the "retarded time", i.e the time $t(p)$ at the (unique when it exists) point $C(t(p))$ on $C$ joining $p$ by a null ge... | https://mathoverflow.net/users/4890 | Retarded coordinates on (flat) spacetime | Let me first rephrase your construction of the coordinate system. Given your curve $C$, foliate the space-time by future null-cones emanating from points on $C$. Call each of the cones $\mathcal{N}\_t$, indexed by the time-coordinate on the base point. This gives the $t$ coordinate of your points. Let $p$ be a point an... | 4 | https://mathoverflow.net/users/3948 | 52617 | 32,950 |
https://mathoverflow.net/questions/52599 | 3 | Let $\mathcal{A}$ be a $k$-linear abelian symmetric tensor category with unit $\mathcal{O}\_A$; here $k$ is a comm. ring. By that I assume implicitly that $\otimes$ is finitely cocontinuous in each variable.
Assume that $\mathcal{L}$ is a line object on $\mathcal{A}$, i.e. it is invertible with respect to $\otimes$ a... | https://mathoverflow.net/users/2841 | Epimorphisms between line objects | Assuming that $\otimes$ preserves finite colimits in each separate argument, I think it's true in general.
Since it's very easy to get twisted around here, let me abstract the situation a bit. Recall that if $\alpha: F' \to F$ is a map between left adjoints in a bicategory (with right adjoints $G'$ and $G$ respectiv... | 2 | https://mathoverflow.net/users/2926 | 52623 | 32,952 |
https://mathoverflow.net/questions/52609 | 3 | Denote by $A$ the full ring of polynomials in one variable $t$ over the finite field with $q$ elements.
For any monic polynomial $P \in A$ define
$$
\sigma(P) = \sum\_{d \mid P, d\, \text{monic}} d.
$$
A monic polynomial $P \in A$ is called `perfect` if
$$
P = \sigma(P).
$$
Let $q=2.$
Two polynomials $P,Q \in A$ are ... | https://mathoverflow.net/users/11016 | Can two Consecutive Polynomials both be perfect ? | The answer is no. This follows from the following claims (which I will prove below):
**1.** If $P$ is perfect and $P(0)\ne 0$ or $P(1)\ne 0$ then $P$ is a square.
**2.** A perfect polynomial which is a square has no roots in $\mathbb{F}\_2$.
Assume that $P$ and $P+1$ are both perfect. From Claim 1, either $P=Q^2$... | 6 | https://mathoverflow.net/users/10675 | 52630 | 32,955 |
https://mathoverflow.net/questions/52642 | 6 | Let $A=(a\_{ij}) \in \mathbb{R}^{n \times n}$ be a matrix with $a\_{ij} = a\_{ji} \geq 0$ and
$B=(b\_{ij})$ with $b\_{ij} = \sqrt{a\_{ij}}$.
Is $B$ positive-definite whenever $A$ is?
In other words:
$\sum\_{1 \leq i,j \leq n} c\_i c\_j a\_{ij} > 0 \iff \sum\_{1 \leq i,j \leq n} c\_i c\_j \sqrt{a\_{ij}} > 0$
for e... | https://mathoverflow.net/users/12360 | Is the componentwise square-root of a positive-definite matrix also pos.-def.? | Counterexample:
$B=\left(\begin{array}{ccc} 1 & 0 & 1 \\\\ 0 & 1 & 1 \\\\ 1 & 1 & \sqrt3 \end{array}\right)$.
This is for $n=3$, and easily extends to all $n\geq 3$.
For $n=2$ it holds, though.
| 8 | https://mathoverflow.net/users/2530 | 52644 | 32,959 |
https://mathoverflow.net/questions/52625 | 12 | The following question came up in a discussion with a colleague about local Galois representations:
>
> To what extent is the classification of continuous $p$-adic representations
> of $G\_{\mathbf{Q}\_{\ell}}$ for $\ell\neq p$ similar to the classification of
> tamely ramified $p$-adic representations for $\el... | https://mathoverflow.net/users/2215 | Tamely ramified p-adic Galois representations | It's true that if $\rho$ is tamely ramified, then $\rho$ is de Rham. In fact, it's even potentially crystalline with all Hodge-Tate weights equal to 0.
First, note that $\rho(I\_{\mathbb Q\_p})$ is finite. The reason is that the image of $\rho$ lands in $GL\_n(\mathbb Z\_p)$, which has a pro-$p$ subgroup of finite in... | 16 | https://mathoverflow.net/users/1729 | 52645 | 32,960 |
https://mathoverflow.net/questions/52631 | 12 | Is the converse to the [Bishop-Gromov Inequality](http://en.wikipedia.org/wiki/Bishop%E2%80%93Gromov_inequality) true?
In other words, if, for a complete $n$-dimensional Riemannian manifold $M$, there is $k \in \mathbb{R}$, such that defining $V\_k(R)$ to be the size of a ball of radius $R$ in the standard space $S^... | https://mathoverflow.net/users/1540 | Converse to Bishop-Gromov Inequality | This is false for $n=3$, and for $k<0$. [One can get a different pinching condition](http://www2.math.uic.edu/~agol/blog/031028.pdf), $R(g)g-Ric \geq -4g$ which is weaker than $Ric \geq -2g$ ($R(g)$ is the scalar curvature), and which gives the same upper bound on volume as Gromov's inequality (for a manifold with nega... | 11 | https://mathoverflow.net/users/1345 | 52651 | 32,963 |
https://mathoverflow.net/questions/37655 | 2 | Suppose I have a diagram $D$ over a category $C$, where $D$ (as a graph) is a single-rooted directed acyclic graph, and all 'joins' in this DAG are actually colimits. Let the root of this diagram be the object $c$ of $C$. Suppose I also have a (new) arrow $a : c\rightarrow e$ where $e$ is not in diagram $D$. From this ... | https://mathoverflow.net/users/3993 | Pushout over a whole diagram | I've written up some details pushouts of diagrams at <http://r6research.livejournal.com/23849.html>. The short answer is that you can form a category of diagrams by considering the lax slice category over $C$ where $C$ is your category that you want to take diagrams of. In the lax slice category over $C$ the objects ar... | 2 | https://mathoverflow.net/users/4085 | 52652 | 32,964 |
https://mathoverflow.net/questions/30049 | 6 | This is probably well-known:
Does every nonempty profinite space occur as the underlying space of a profinite group? If not, which conditions have to be imposed?
~~- Is every profinite group isomorphic to the Galois group of some Galois extension?~~ (yes, see the comments)
| https://mathoverflow.net/users/2841 | profinite spaces coming from profinite groups | (Note: This was intended to be a comment to unknown (google)'s answer - but as I'm new here I can't post comments.)
As Pete L. Clark points out, unknown (google)'s answer is false as stated.
However, this is only because of the omission of the word "infinite".
A correct statement is:
An infinite profinite group $G$... | 4 | https://mathoverflow.net/users/12362 | 52657 | 32,966 |
https://mathoverflow.net/questions/52656 | 13 | Let $G$ be the compact Lie group $SO(n)$. There are some classical constructions of the classifying bundle of $G$ based upon on direct limits of Grassmann and Stiefel manifolds:
$$BG \simeq \underset{m \to \infty}{\lim} SO(m+n)/SO(m) \times SO(n)$$
$$EG \simeq \underset{m \to \infty}{\lim} SO(m+n)/SO(m)$$
with... | https://mathoverflow.net/users/2926 | Can a homotopy inverse of the map from a Lie group to loops on its classifying space be given by holonomy? | Yes, certainly. The model for BG as you described it has a canonical, universal connection
for its $SO(n)$ bundle: just the induced Riemannian connection from Euclidean space.
As you
move an $n$-dimensional plane in $\mathbb E^{n+m}$, the induced connection
is the limit of compositions
of orthogonal projections betw... | 12 | https://mathoverflow.net/users/9062 | 52664 | 32,971 |
https://mathoverflow.net/questions/52666 | 2 | I was reading various proofs of the irrationality of $\sqrt{2}$ including a geometric proof by Richard Guy involving similar right triangles. Since then, after thinking about it, I wonder why we are ever taught about the irrationality of square roots of any one particular non-square natural at all, as the proof that al... | https://mathoverflow.net/users/12363 | irrationality of square roots of all non perfect square naturals | The point is that you use this fact from number theory:
$p$ and $q$ are relatively prime $\Longrightarrow$ $p^2$ and $q^2$ are relatively prime.
While this is not hard, this is not trivial either. You either need unique prime factorization or the Euclidean algorithm.
For $n=2$, you just have to prove that if the ... | 4 | https://mathoverflow.net/users/2530 | 52668 | 32,972 |
https://mathoverflow.net/questions/52665 | 16 | Does anybody knows about good overview on intersection theory.
The book of Fulton has very hard language. Does there exist simple overview on this topic with many examples?
| https://mathoverflow.net/users/4246 | Survey article on Intersection Theory | Dear Klim, when you say "the" book, i suppose you mean *Intersection Theory* published by Springer . However Fulton has written a [much more elementary overview](http://books.google.fr/books?id=WK1vYJK9m2oC&printsec=frontcover&dq=fulton+intersection+theory&hl=fr&ei=44Q4Tdi_Ds6r8AOvkpnECA&sa=X&oi=book_result&ct=result&r... | 20 | https://mathoverflow.net/users/450 | 52672 | 32,974 |
https://mathoverflow.net/questions/52403 | 4 | Let $A \subset \mathbb{C}[x\_1,x\_2,\ldots,x\_n]$ - be finitely generated graded algebra and $B$ be its subalgebra. How to prove that $A=B.?$
Unfortunalelly I know only one method to do it - to compare theirs Poincare series.
Anybody know more?
| https://mathoverflow.net/users/9645 | I am interested in collecting different methods of proofs that a subalgebra coincides with whole algebra. | This question is posed way too generically in order to obtain an answer that is useful to you by more than mere coincidence, but here are three things that I found of use:
(1) Your algebra is graded, thus in particular filtered. Try induction. Generally, if $A=\bigcup\limits\_{n\geq 0} A\_n$ is a filtered ring and $B... | 2 | https://mathoverflow.net/users/2530 | 52675 | 32,976 |
https://mathoverflow.net/questions/52620 | 7 | So it is relatively easy to show that there exists only one smooth structure on
the real line $\mathbb{R}$. So here are 2 natural questions:
Q1: Up to equivalence, is there only one real analytic structure on $\mathbb{R}$? If so,
then do we have a simple proof of that?
Q2: Where can I find the simplest proofs tha... | https://mathoverflow.net/users/11765 | Smooth and analytic structures on low dimensional euclidian spaces | Regarding Q1, put an analytic Riemann metric on your 1-manifold. Integrating a unit speed vector field gives an analytic diffeomorphism to $\mathbb R$. Another way to prove analytic structures are unique is to notice the same argument that one uses to prove that the group of $C^k$-diffeomorphisms of $\mathbb R$ has the... | 2 | https://mathoverflow.net/users/1465 | 52682 | 32,980 |
https://mathoverflow.net/questions/52667 | 5 | Given a smooth projective variety $X$ over an algebraically closed field $k$. Now given a another projective variety $T$ and a coherent $O\_{X\times T}$-module $F$, which is flat over $T$.
Given $r,s \in T$, and let $F\_r$ and $F\_s$ be the pullback of $F$ to the fibers over $r$ and $s$.
Do we have $ch(F\_r)=ch(F\_... | https://mathoverflow.net/users/3233 | Chern classes in flat families | The answer depends on the version of the Chern classes you are using.
For example, if you consider Chern classes with values in the Chow ring then the answer is negative. The simplest example is the following, take $X = T$ to be an elliptic curve, and $F$ to be the line bundle corresponding to the diagonal. Then $c\_1(... | 14 | https://mathoverflow.net/users/4428 | 52689 | 32,985 |
https://mathoverflow.net/questions/52654 | 14 | In Shannon's 1948 paper "[A Mathematical Theory of Communication](http://cm.bell-labs.com/cm/ms/what/shannonday/shannon1948.pdf)", early on he derives the equation $$N(t)=N(t-t\_1)+N(t-t\_2)+\ldots+N(t-t\_n).$$
He then says "according to a well-known result in finite differences, $N(t)$ is then asymptotic to $X\_0^t$... | https://mathoverflow.net/users/11054 | Shannon's communication paper and finite differences | When the $t\_i$ are incommensurable in the sense that they generate a dense subgroup, $N(t)=CX\_0^t+o(X\_0^t)$ for a given constant $C$. This is a consequence of the standard renewal theorem and needs no hypothesis on the monotonicity of the function $t\mapsto N(t)$.
To see this, let $(\xi\_k)$ denote some i.i.d. ran... | 9 | https://mathoverflow.net/users/4661 | 52693 | 32,987 |
https://mathoverflow.net/questions/52692 | 23 | It's possible this question is trivial, in which case it will be answered quickly. In any case, I realized that it's a basic question the answer to which I should know but do not.
Everybody loves knots — one-dimensional compact manifolds mapped generically into three-dimensional compact manifolds — and it's natural t... | https://mathoverflow.net/users/78 | Can surfaces be interestingly knotted in five-dimensional space? | If $M$ is a connected compact $2$-manifold, then it unknots in $\Bbb R^5$. More generally, $k$-connected $n$-manifolds embed in $\Bbb R^{2n-k}$, provided $k<\frac{n-2}2$, and unknot in $\Bbb R^{2n-k+1}$, provided $k<\frac{n-1}2$. This was proved around 1961 - by Roger Penrose, J.H.C. Whitehead, and Zeeman in the PL cat... | 26 | https://mathoverflow.net/users/10819 | 52697 | 32,991 |
https://mathoverflow.net/questions/52661 | 8 | A metric on $n$ points $N$ can be represented as a vector $x \in \mathbb{R}\_+^{n \choose 2}$.
For each pair of distinct $i, j \in N$, we have $d(i,j) = d(j,i) = x\_{i,j}$. The set of all metrics is the set of points which lie inside the cone defined by the (triangle) inequalities:
$$x\_{i,j} - x\_{i,k} - x\_{j,k} \l... | https://mathoverflow.net/users/3027 | Estimating the Volume of the Metric Polytope | Some googling reveals:
1. What you call a *metric polytope* is also called a *semimetric polytope* (see for example the standard reference: [Geometry of cuts and metrics](http://books.google.com/books?id=Lx9dYMW70J0C&lpg=PR1&ots=SAFSPMtHru&dq=deza%2520geometry&pg=PR1#v=onepage&q&f=false) )
2. In the same book, [see h... | 7 | https://mathoverflow.net/users/8430 | 52698 | 32,992 |
https://mathoverflow.net/questions/52677 | 2 | Let $d$ be a positive integer greater than 2. Define an equivalence relation on monic integer polynomials of degree $d$: $f\sim g \iff f(k\_1 x+k\_2)=g(k\_3 x+k\_4)$ for some integers $k\_1,...,k\_4$.
>
> Is there a number $m$ such that for any $m$ distinct integers, there is at most one equivalence class that atta... | https://mathoverflow.net/users/2024 | Can finitely many values of a polynomial determine it? | You are asking for a condition on $f$ and $g$ such that the curve:
$$C: f(y) - g(x) = 0$$
has a uniformly bounded number of rational points.
Note that if $f$ and $g$ are equivalent under an affine transformation,
then $C$ is divisible by a linear factor and is not reducible. The converse is
*almost* true. Namely, as lo... | 7 | https://mathoverflow.net/users/nan | 52711 | 33,000 |
https://mathoverflow.net/questions/30311 | 10 | I'll freely admit that I have a rather hard time keeping straight different notions of purity of etale sheaves, and I think part of the problem is the lack of counterexamples.
For example, it's a theorem that the stalks of intersection cohomology sheaves (with coefficients in $\overline{\mathbb{Q}}\_\ell$, say) are p... | https://mathoverflow.net/users/66 | What's an example of an intersection cohomology sheaf whose stalks are pure but not pointwise pure? | Here is an example. Sorry it's so complicated. (There's probably a simpler one, but my mind works in complicated ways, it seems.)
Consider a Siegel modular threefold $U$, i.e., a Shimura
variety for the general symplectic group $GSp(4)$ with some level $n\geq 3$.
(So that $U$ is smooth and quasi-projective over $\mat... | 11 | https://mathoverflow.net/users/12336 | 52712 | 33,001 |
https://mathoverflow.net/questions/52726 | 4 | There are two non-abelian groups of order $p^3$, namely, semi-direct product of $\mathbb Z /p \mathbb Z \times \mathbb Z /p \mathbb Z$ by $\mathbb Z /p \mathbb Z$ and semi-direct product of $\mathbb Z /p^2 \mathbb Z$ by $\mathbb Z /p \mathbb Z$. What are the automorphism groups of these groups?
| https://mathoverflow.net/users/6761 | Automorphisms of non-abelian groups of order $ p^3$ | For the latter group, the answer is *Bidwell, J. N. S.; Curran, M. J.*, [**The automorphism group of a split metacyclic $p$-group**](https://doi.org/10.1007/s00013-006-1899-z), Arch. Math. 87, No. 6, 488–497 (2006). [Zbl 1116.20016](https://zbmath.org/?q=an:1116.20016).
| 8 | https://mathoverflow.net/users/12310 | 52730 | 33,008 |
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