parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
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https://mathoverflow.net/questions/74642 | 60 | I've been told that it's important to know modern physics, Differential Geometry and Algebraic Topology for understanding higher structures. Is there any other prerequisite for understanding Lurie's work? Since the title of the book indicates, I guess Algebraic Geometry is also important. Please tell me if I'm wrong. M... | https://mathoverflow.net/users/17617 | If I want to study Jacob Lurie's books "Higher Topoi Theory", "Derived AG", what prerequisites should I have? | To read Higher Topos Theory, you'll need familiarity with ordinary category theory
and with the homotopy theory of simplicial sets (Peter May's book "Simplicial Objects in Algebraic Topology" is a good place to learn the latter). Other topics
(such as classical topos theory) will be helpful for motivation.
To read "H... | 195 | https://mathoverflow.net/users/7721 | 74643 | 45,377 |
https://mathoverflow.net/questions/74633 | 2 | Let $L=\mathbb{P}^l\subset\mathbb{P}^N$ be a fixed linear space, $l\geq0$, and let $M=\mathbb{P}^{N-l-1}$
be a linear space skew to $L$, i.e. $L\cap M=\emptyset$ and $\langle L, M\rangle=\mathbb{P}^N$.
Let $X\subseteq\mathbb{P}^N$ be a closed irreducible variety not contained in $L$
and let
$$
\pi\_L:X\dashrightarr... | https://mathoverflow.net/users/15606 | On quasi-finite and unramified linear projections. | The answer is **yes** in characteristic $0$.
In fact, take the non-empty Zariski open set $X^0$ where $\pi \colon X \to M$ has finite fibres, and let $M^0 \subset M$ be the image of $X^0$. Then $M^0$ is a Zariski open set of $M$ and the restriction $\pi^0 \colon X^0 \to M^0$ is a finite map. Passing to function field... | 3 | https://mathoverflow.net/users/7460 | 74647 | 45,379 |
https://mathoverflow.net/questions/74644 | 2 | Hi all. There is a conjecture by Erdős that states "There doesn't exist an integer covering system with all moduli odd AND distinct." The link is <http://en.wikipedia.org/wiki/Covering_system>
I think that, if a covering exists with all moduli distinct, not only the lcm of the moduli must be even, it must also be abu... | https://mathoverflow.net/users/17614 | On integer covering systems with all moduli distinct | Each modulus $m$ covers a proportion $1/m$ of the integers. To cover all the integers, we need $\sum 1/m\ge1$, where the sum is over all the moduli. Multiply by the lcm, $L$, to get $\sum(L/m)\ge L$. But all the terms in the sum are distinct, proper divisors of $L$, so $L$ is abundant (unless we have equality, but it's... | 5 | https://mathoverflow.net/users/3684 | 74648 | 45,380 |
https://mathoverflow.net/questions/74630 | 1 | Are there a *non-abelian* nilpotent Lie algebra $\mathfrak{n}$ over $\mathbb{R}$ and an automorphism $\alpha: \mathfrak{n} \to \mathfrak{n}$ such that:
* $\alpha$ is periodic,
* the fixed subspace of $\alpha$ is the origin, and
* there is an $\alpha$-invariant lattice $L \subset \mathfrak{n}$ ?
**REMARK:** If $\mat... | https://mathoverflow.net/users/16862 | Periodic automorphism of nilpotent Lie algebra | There is no example with eigenvalues $-1$. More generally, suppose that $\mathfrak{g}$ is a Lie algebra, and $\alpha$ is an automorphism of order $2$ whose fixed subspace is trivial. Then I claim that $\mathfrak{g}$ is abelian.
Proof: Since $\alpha^2=\mathrm{Id}$ and $\alpha$ has no fixed points, we must have $\alpha... | 5 | https://mathoverflow.net/users/297 | 74656 | 45,382 |
https://mathoverflow.net/questions/74653 | 0 | I am reading a book on time series analysis and I am having problems understanding the section about outlier detection.
The authors say that when you want to know whether at a certain time $T$ there was an outlier, you should use a certain test statistic and a test with size less than $\alpha$. But when you don't kno... | https://mathoverflow.net/users/17619 | Bonferroni for outlier detection? | If you do $n$ tests of size $\alpha/n$, then $\alpha$ is the Bonferroni bound on at least one of the tests succeeding. It is conservative because it is the worst possible bound without any further information about dependency between the tests. It is only exact if the tests are disjoint (i.e. at most one can be true at... | 2 | https://mathoverflow.net/users/9025 | 74658 | 45,384 |
https://mathoverflow.net/questions/74655 | 6 | Let $(X,d)$ be a metric space, let $B(x,r)$ be the open ball of radius $r$ about $x$ and $N(x,r)$ be the set of elements $y\in X$ such that $d(x,y)\leq r$. It is well-known that it is not always true that $N(x,r)$ is the closure of $B(x,r)$.
I need, for some research, to restrict my attention to metric spaces for whi... | https://mathoverflow.net/users/13809 | Is there a name for the class of metric spaces such that the closure of the open ball of radius $r$ around each point $x$ is the set of elements $y$ such that $d(x,y)\leq r$ ? | I'm not sure what they're called, but according to [this](http://www.mathkb.com/Uwe/Forum.aspx/math/4535/Closure-of-open-balls) site an equivalent characterization of spaces $X$ where $\overline{B(x,r)} = N(x,r)$ is: for all $p\in X$, the
only local minimum of the function $x \rightarrow d(x,p)$ is at $x=p$. The proof ... | 5 | https://mathoverflow.net/users/11540 | 74659 | 45,385 |
https://mathoverflow.net/questions/60174 | 25 | It is known (see Theorem 4.1.7 in R. Horn & C. Johnson) that every matrix $A\in M\_n(\mathbb R)$ (*real* entries) can be written as the product $HK$ of two Hermitian matrices (*complex* entries). Of course, the pair $(H,K)$ is far from being unique, because the real dimension of $\mathbb H\_n\times\mathbb H\_n$ is $2n^... | https://mathoverflow.net/users/8799 | Factorization of a real matrix into Hermitian x Hermitian. Is it stable ? | Surprisingly (at least to me) the answer is no when $n\ge 3$. This was proved by Yves Benoist and me after I mentioned the problem in a talk at MSRI and Yves came up with a great idea.
It is enough to show that there is no uniform bound when $n=3$.
Here is an elementary argument that involves little computation. I d... | 17 | https://mathoverflow.net/users/2554 | 74672 | 45,388 |
https://mathoverflow.net/questions/74634 | 5 | We know that principal congruence subgroups are characteristic in $SL(n,\mathbb Z)$.
Suppose $\Gamma$ is a finite index subgroup of $SL(n,\mathbb Z)$ and $\Gamma\_m$ is a principal congruence subgroup of level m contained in $\Gamma$. Will it be characteristic in $\Gamma$?
| https://mathoverflow.net/users/13835 | Suppose $\Gamma_m$ is a principal congruence subgroup of level m contained in a finite index subgroup $\Gamma$ of $SL(n,\mathbb Z)$. Is $\Gamma_m$ characteristic in $\Gamma$? | This is false. Consider the case $n=2$, and let $p$ be a prime. Let $A=\left[\begin{array}{cc}0 & -1 \\\ p & 0\end{array}\right]
$ be an Atkin-Lehner involution (considered as an element of $PGL\_2(\mathbb{Q})$), and consider the subgroup
$\Gamma\_0(p) = \{ \left[\begin{array}{cc}a & b \\\ c & d\end{array}\right]\in SL... | 6 | https://mathoverflow.net/users/1345 | 74675 | 45,390 |
https://mathoverflow.net/questions/74676 | 1 | Hey,
For personal exercising purposes I try to give a proof, that a U(1)-principal-bundle has curvature $\alpha$ iff the cohomology class of $\alpha$ is integral:
By the Cech-deRham-isomorphism a $[\alpha] \in H^2\_{DR}(M, \mathbb{Z})$ iff $[\alpha] \in H^2\_{Cech}(M, \mathbb{Z})$. Then we can use the isomorphism $... | https://mathoverflow.net/users/17047 | Principal Bundle with given Curvature | EDIT: I'm not very happy the the exposition below. But I hope you get the idea.
Your proof is correct. Given a principal bundle with connection having the given curvature, and a second connection on a possibly different with the same curvature, the difference between the two is a bundle with a flat connection (note t... | 3 | https://mathoverflow.net/users/4177 | 74697 | 45,399 |
https://mathoverflow.net/questions/74693 | 2 | So consider the $\mathbb{Q}$-vector space $V$ of functions which satisfy the following conditions
(1) $f:\mathbb{H}\rightarrow\mathbb{C}$ is holomorphic. Here $\mathbb{H}$ stands for the
upper half plane.
(2) $f(z+1)=f(z)$
(3) The Fourier series of $f$ at infinity has the form $\sum\_{n\geq 1} a\_nq^n$
where $q=... | https://mathoverflow.net/users/11765 | On pseudo rational modular forms of weight 2 and level N | To summarize some remarks in the comments. The function $f \cdot d \tau$ will be a differential on $Y\_N:=\mathbf{H}/\Gamma$, where $\Gamma\_N$ is the group generated by the two matrices
$$\left( \begin{matrix} 1 & 1 \\\ 0 & 1 \end{matrix} \right) \ , \
\left( \begin{matrix} 0 & -1 \\\ N & 0 \end{matrix} \right)$$
If ... | 2 | https://mathoverflow.net/users/17277 | 74699 | 45,400 |
https://mathoverflow.net/questions/74689 | 52 | Weyl's theorem states that any finite-dimensional representation of a finite-dimensional semisimple Lie algebra is completely reducible. In my mind, the "natural" way to prove this result is by way of Lie groups. However, as a student, I first encountered Weyl's theorem in the textbook by Humphreys, in which he gives a... | https://mathoverflow.net/users/3106 | Motivating the Casimir element | I think that there are two things to be motivated: one is the Casimir, and the other is the proof of semi-simplicity.
First, for the Casimir, it might help to note that there is a ``formula-free" construction. A symmetric bilinear form $\kappa$ defines $\mathfrak{g}\simeq \mathfrak{g}^{\vee}$, and the Casimir is the ... | 56 | https://mathoverflow.net/users/15630 | 74701 | 45,401 |
https://mathoverflow.net/questions/74698 | 10 | I've read that one way to formulate the Langlands program is the following:
Let $\mathcal{L}\_ {\mathbb{Q}}$ be the conjectural Langlands group. Then the category of semi-simple (continuous) representations of $\mathcal{L}\_{\mathbb{Q}}$ that are algebraic(!) is equivalent to the category of motives over $\mathbb{Q}$... | https://mathoverflow.net/users/5756 | How does the conjectural Langlands group fit into the Tannakian point of view? | The Langlands group is not meant to be the motivic Galois group; rather, it is larger (in Langlands's original formulation), or alternatively not an algebraic group, but a locally compact group which has some kind of underlying algebraic avatar (this is the more recent, indeed current, formulation, due to Kottwitz), so... | 11 | https://mathoverflow.net/users/2874 | 74702 | 45,402 |
https://mathoverflow.net/questions/74716 | 3 | The Collatz conjecture is known to all. Has this question been approached by methods related to statistics? I think of Collatz iterates as a time series, and the question of whether we always get the number 1 in the end then becomes a question related to stationarity of corresponding time series.
So the question is t... | https://mathoverflow.net/users/17614 | Collatz conjecture and stationarity of time series | I don't think statistics alone could be strong enough to resolve the Collatz conjecture, as statistics deals with expected properties of random processes, whereas Collatz concerns the actual properties of a deterministic process. There is a heuristic argument that assuming the iterates in the Collatz sequences are inde... | 8 | https://mathoverflow.net/users/7106 | 74723 | 45,414 |
https://mathoverflow.net/questions/74724 | 19 | It is known that there are non-Hausdorff spaces which admit unique limits for all convergent sequence (see [here](http://www.sciencedirect.com/science/article/pii/0166864193901476)) and it is also known that unique limits for nets implies Hausdorff.
What I am wondering is, if there is a (somehow weak) condition which... | https://mathoverflow.net/users/9652 | Unique limits of sequences plus what implies Hausdorff? | [First countable](http://en.wikipedia.org/wiki/First-countable_space) is enough. Let $x\neq y$ be two points in your space that cannot be separated by neighborhoods. Let $O\_1,O\_2,\ldots$ form a neighborhood base of $x$ and let $U\_1,U\_2,\ldots$ form a neighborhood base for $y$. Choose a sequence $(z\_n)$ such that $... | 19 | https://mathoverflow.net/users/35357 | 74727 | 45,416 |
https://mathoverflow.net/questions/74728 | 6 | Dear community,
I have the following combinatorial question which I will explain in short first and then with some more detail. At the end you will find a very simple example.
Short version
-------------
Le $A \in \mathbb{N}\_0^{n \times n}$ be a symmetric matrix with zeros on the diagonal, whose row- and column-... | https://mathoverflow.net/users/12366 | Counting the (additive) decompositions of a quadratic, symmetric, empty-diagonal and constant-line matrix into permutation matrices | In general the number of decompositions depends on the structure of the matrix, not just on its size and row sum. This is even so in the case of 0-1 matrices, where the question is equivalent to 1-factorization of regular bipartite graphs. Even very simple-looking cases are difficult, for example if the matrix is full ... | 9 | https://mathoverflow.net/users/9025 | 74733 | 45,417 |
https://mathoverflow.net/questions/74742 | 2 | One example that I always have in mind is that the plane nodal (or even the plane cuspidal) cubic curve $X$ is obtained by an appropirate 2-dim linear subsystem of $|\mathcal{O} (3)|$ on $\mathbb{P}^1$. If one takes the full linear system $|\mathcal{O}\_{\mathbb{P}^1} (3)|$ then we get the twisted cubic $\tilde{X}$ in ... | https://mathoverflow.net/users/17638 | Line bundles, linear systems and normalization | It is in fact true that the normalization of a projective variety is projective, as J.C. Ottem discusses in the comments.
It is not true that if a normal variety is mapped to a projective space by a linear series $V\subset H^0(L)$ then some larger linear series $W\supset V$ has image isomorphic to the normalization.
... | 7 | https://mathoverflow.net/users/7399 | 74750 | 45,423 |
https://mathoverflow.net/questions/74682 | 6 | Let $g$ a finite-dimensional complex simple Lie algebra and $\sigma$ a finite order Dynikin diagram automorphism of $g$.
Consider $\tilde g$ as the loop algebra associated to $g$, and $\tilde g^\sigma$ as the twisted affine Lie algebra (associated to $g$) in the spirit of the theory developed in the book ''Infinite ... | https://mathoverflow.net/users/40886 | twisted affine algebras | To add to what Carnahan has posted. There is a difference between the representation theories of untwisted and twisted affine algebras. But they seem to be unified through vertex algebra theory.
Vertex operator algebras themselves are "untwisted" but admit twisted modules. Haisheng Li has some results that make this ... | 3 | https://mathoverflow.net/users/17263 | 74754 | 45,426 |
https://mathoverflow.net/questions/74755 | 3 | Hi,
this is more or less a "reference" question.
Suppose $D$ a redueced irreducibile divisor in $X$ and I take $f:Y\rightarrow D$ a desingularization of his.
What information can i get from the support of the higher direct imagese $R^if\_\*\mathcal{O}\_Y$?
for example if they are all empty then the divisor has ra... | https://mathoverflow.net/users/6949 | Higher direct images and singularities | One place is a paper of Kovacs: [Irrational Centers][1]
In particular, the associated primes/points of the higher direct images are defined to be "irrational centers", which can then be used to obtain depth estimates. In the case you are interested in (ie no boundary case) then this was looked at previously by Alexee... | 6 | https://mathoverflow.net/users/3521 | 74767 | 45,434 |
https://mathoverflow.net/questions/74771 | 1 | Let $X \to \mathbb{A}^1\_k$ be a smooth morphisms of varieties over a field $k$. Its generic fiber is a smooth morphism $X\_\eta \to \eta = \text{Spec }k(t)$. Is it true that we have an injection
$H^2(X\_{et},\mathbb{G}\_m) \to H^2((X\_\eta)\_{et},\mathbb{G}\_m)$?
Where does it come from and why does this hold?
| https://mathoverflow.net/users/1107 | Brauer group and smoothness | For a regular, integral, quasi-compact scheme $X$, we have $H^2(X\_{et}, \mathbb{G}\_m) \hookrightarrow H^2(K(X)\_{et},\mathbb{G}\_m)$ (use the Leray spectral sequence for the inclusion of the generic point), and you can factor this injection into $H^2(X\_{et}, \mathbb{G}\_m) \to H^2((X\_\eta)\_{et}, \mathbb{G}\_m) \ho... | 2 | https://mathoverflow.net/users/nan | 74773 | 45,437 |
https://mathoverflow.net/questions/74721 | 7 | I am currently reading a monograph by Jose Seade, " On the topology of isolated singularities in analytic spaces".
I have following questions but before asking questions I recall the definition of algebraic knots/ links
**Definition :** Let $ f : (\mathbb{C}^2,0) \rightarrow (\mathbb{C},0)$ be a holomorphic functio... | https://mathoverflow.net/users/5538 | Examples of Non-algebraic Fibered Knots? | Let me elaborate a bit on my comments.
First of all, algebraic knots (up to **isotopy**) have been classified. People refer to an unpublished paper by Bonahon and Siebenmann, *The classification of algebraic links*, and there appears to be a discussion of this fact in the book *Three-dimensional link theory and invar... | 6 | https://mathoverflow.net/users/13119 | 74774 | 45,438 |
https://mathoverflow.net/questions/74744 | 5 | I have been computing eigenvalues of adjacency matrices for several directed (not necessarily strongly connected) graphs and one remarkable property seemed to hold (each graph that I have examined contained at least one cycle, but this need not to be a necessary condition):
"If $\lambda$ is an eigenvalue of an adjace... | https://mathoverflow.net/users/17481 | Complex Eigenvalues of Directed Graphs | Let $D$ be the Paley tournament on seven vertices. Its vertices are the integers mod seven
and there is an arc from $i$ to $j$ is $j-i$ is a non-zero square mod seven. The characteristic polynomial of the adjacency matrix is $(x-3)(x^2+x+2)^3$. The only real eigenvalue is 3, the remaining eigenvalues are equal to $(-1\... | 9 | https://mathoverflow.net/users/1266 | 74787 | 45,443 |
https://mathoverflow.net/questions/74708 | 15 | If I'm studying classical mechanics, we might start by viewing propositions as true/false valued questions on points of phase space.
Then, if I'm interested in a proposition-oriented view of things, I might flip things around and ask what points of phase space correspond to propositions, and observe a Boolean algebr... | https://mathoverflow.net/users/nan | What is the analog of a topos in quantum logic? | While the other two answers referred to very interesting connections of topos theory and quantum physics I think the following is going more into the direction the OP was imagining: In non-commutative topology one considers quantales, which are, roughly, an axiomatization of what you get when you replace open sets with... | 10 | https://mathoverflow.net/users/733 | 74789 | 45,444 |
https://mathoverflow.net/questions/74791 | 1 | Let $G$ be $SL\_2({\mathbb C})$ and for $a,b\in G$ let $[a,b]=aba^{-1}b^{-1}$ be the commutator bracket. Let $n$ be a natural number $\ge 2$ and let
$X\subset G^{2n}$ be the set of all $g\in G^{2n}$ such that
$$
[g\_1,g\_2]\cdots[g\_{2n-1},g\_{2n}]=1.
$$
The first question is, whether $X$ is connected.
If not, can one... | https://mathoverflow.net/users/nan | Do representations of Fuchsian groups have unitary deformations? | $X$ is the $SL\_2(\mathbb{C})$--representation variety of the surface group, and, by Goldman's thesis, it is irreducible, and so connected.
See
Goldman, Topological components of spaces of representations.
Invent. Math. 93 (1988), no. 3, 557–607.
If you take $G$ to be $PSL\_2(\mathbb{C})$, then there are two co... | 3 | https://mathoverflow.net/users/1335 | 74793 | 45,445 |
https://mathoverflow.net/questions/74788 | 0 | For a smooth test function \eta and some constant C is it possible to have an estimate like the following?
|grad \eta|^2 < C {\eta}^2 ?
Thanks.
| https://mathoverflow.net/users/17648 | Test function . | No. Take any line through the support of $\eta$. Along such a line, you would have
$|d\eta/ds|\le C|\eta|$. But since $\eta=0$ on a part of the line, you get $\eta=0$ everywhere by Gronwall's inequality.
| 2 | https://mathoverflow.net/users/12120 | 74794 | 45,446 |
https://mathoverflow.net/questions/74756 | 16 | Let $E$ be the total space of the sphere bundle $S^k\to E\to M$, is it true that there exists a disk bundle $D^{k+1}\to N\to M$ such that $E=\partial N$? (where $D^{k+1}$ is the unit disk in $\mathbb R^n$)
| https://mathoverflow.net/users/16750 | Is it true that all sphere bundles are boundaries of disk bundles? | If you don't specify what structure group you want the disc bundle $D^{k+1} \to N \to M$ to have, then it is always true: you just take the fibrewise cone on the original family.
If you want to know whether smooth $S^k$-bundles always bound smooth disc bundles, this is true iff
$$O(k+1) \to \mathrm{Diff}(S^k)$$
is a ... | 13 | https://mathoverflow.net/users/318 | 74801 | 45,450 |
https://mathoverflow.net/questions/74737 | 5 | Given a functor from a small category to $Set$, we can describe the colimit set as a quotient of the disjoint union of image sets by an equivalence relation arising from morphisms in the source category (as seen in [Wikipedia](http://en.wikipedia.org/wiki/Limit_%28category_theory%29#Existence_of_limits), or Kashiwara-S... | https://mathoverflow.net/users/121 | Where can I find an explicit description of the pseudocolimit of a small pseudofunctor to Cat? | An answer can more or less be extracted from Kelly's *Elementary Observations on 2-categorical limits*, at least if you already know that it's there. (-:
First, as Kelly notes in section 6, it would suffice to construct what we may call *strict pseudo-colimits*, that is pseudo-colimits in your sense for which the fun... | 8 | https://mathoverflow.net/users/49 | 74804 | 45,453 |
https://mathoverflow.net/questions/74770 | 14 | It is well known that the formal group law $F\_U$ of complex cobordism, expressing the Euler class of a tensor product of complex line bundles, is universal.
Also, the formal group law $F\_O$ of unoriented cobordism, expressing the Euler class of a tensor product of real line bundles, is universal among formal group ... | https://mathoverflow.net/users/8103 | Formal group law of unoriented cobordism | I'm fairly sure you just get the same formula, with $\mathbb{C}P^k$ replaced by $\mathbb{R}P^k$, and $H\_{ij}$ replaced by the corresponding real hypersurface in $\mathbb{R}P^i\times\mathbb{R}P^j$. The proof of the equivalent formula
$$ \left(\sum [\mathbb{R}P^r]\;X^r\right)
\left(\sum [\mathbb{R}P^s]\;Y^s\right)
F\... | 8 | https://mathoverflow.net/users/10366 | 74810 | 45,458 |
https://mathoverflow.net/questions/74806 | 19 | To put this question in precise language, let $X$ be an affine scheme, and $Y$ be an arbitrary scheme, and $f : X \rightarrow Y$ a morphism from $X$ to $Y$. Does it follow that $f$ is an affine morphism of schemes? While all cases are interesting, a counterexample that has both $X$ and $Y$ noetherian would be nice.
| https://mathoverflow.net/users/5473 | Are morphisms from affine schemes to arbitrary schemes affine morphisms? | No, here is an example of a morphism $f:X\to Y$ which is not affine although $X$ is affine.
Take $X=\mathbb A^2\_k$, the affine plane over the field $k$ and for $Y$ the notorious plane with origin doubled: $Y=Y\_1\cup Y\_2$ with $Y\_i\simeq \mathbb A^2\_k$ open in $Y$ and $Y\setminus Y\_i= \lbrace O\_i\rbrace$, a clo... | 35 | https://mathoverflow.net/users/450 | 74812 | 45,460 |
https://mathoverflow.net/questions/74790 | 2 | Ivic writes, [at the beginning of chapter 13 of his The Riemann Zeta Function](http://books.google.com/books?id=jT9gjGipNDUC&pg=PA352&cad=4#v=onepage&q&f=false), about a method of expressing the principal terms of the Dirichlet Divisor Problem as polynomials of $log\\ n $ with coefficients built from arrangements of th... | https://mathoverflow.net/users/12498 | Principal term of the Dirichlet Divisor problem, from the work of A.F. Lavrik? | I assume you're asking how you can explicitly calculate the polynomials $P\_{k-1}(x)$. The answer is in equation 13.4:
$$
P\_{k-1}(\log x) = \mathop{\rm Res}\_{s=1} \big( x^{s-1} \zeta(s)^k s^{-1} \big).
$$
To calculate this residue, expand everything as a Laurent series at $s=1$:
$$
x^{s-1} \zeta(s)^k s^{-1} = \bigg( ... | 5 | https://mathoverflow.net/users/5091 | 74819 | 45,464 |
https://mathoverflow.net/questions/74816 | 5 | The title more or less says it all.... Let $V$ be a vector space (over your favorite field; $V$ not necessarily finite dimensional), and let $S$ be a subset of $V$. A maximal linearly independent subset of $S$ is exactly that: a subset of $S$ that is linearly independent yet not properly contained in any other linearly... | https://mathoverflow.net/users/5091 | Is there a standard name for the intersection of all maximal linearly independent subsets of a given set in a vector space? | For a [matroid](http://en.wikipedia.org/wiki/Matroid) the elements that are contained in every basis are called coloops, dual to the notion of a loop, which is an element not contained in any basis. Since you are interested in linearly independent sets perhaps adopting the language of matroids is not such a bad idea.
... | 10 | https://mathoverflow.net/users/2384 | 74820 | 45,465 |
https://mathoverflow.net/questions/74796 | 14 | Let $G$ be a compact Lie group and $T$ a maximal torus of $G$. Then the flag manifold $G/T$ is a complex manifold and a symplectic manifold. One way to see the symplectic structure is to view $G/T$ as the co-adjoint orbit of a generic element $F\_0 \in Lie(T)^\*$. Then the symplectic structure is given by
$$
\omega\_F(... | https://mathoverflow.net/users/4622 | Complex structure on flag manifolds | This is essentially a more "condensed" version of Johannes Ebert's answer.
From the root space decomposition
$$ \mathfrak g /\mathfrak t \otimes \mathbb C = \oplus\_{\alpha \in \Phi} R\_\alpha, $$
one can see that a choice $\Phi^+$ of positive roots gives rise to a $G$-invariant almost complex structure on $G/T$. Ind... | 13 | https://mathoverflow.net/users/430 | 74822 | 45,466 |
https://mathoverflow.net/questions/74418 | 4 | The weak approximation theorem states that given a field $F$ and nontrivial inequivalent absolute values $|\cdot|\_1,\ldots,|\cdot|\_n,$ and letting $F\_i$ denote $F$ with the topology from $|\cdot|\_i$, then the diagonal in $F\_1 \times \ldots \times F\_n$ is dense.
So suppose now we have the same setup, except now ... | https://mathoverflow.net/users/5583 | Does the weak approximation theorem hold for general topological fields? | Well, I feel silly -- this is answered early on in Wiesław's "Topological Fields" now that I look. The answer is no, distinct (non-discrete) topologies on a field need not be independent, they can be comparable, or even incomparable but still dependent.
For a simple example, take two values on the rationals; the topo... | 2 | https://mathoverflow.net/users/5583 | 74834 | 45,471 |
https://mathoverflow.net/questions/71110 | 8 | I'm essentially reopening [this old question](https://mathoverflow.net/questions/37849/closedness-of-finite-dimensional-subspaces) of Ricky Demer which was never fully answered.
Essentially the original question: Suppose we have a topological field $F$ which is complete, Hausdorff, and non-discrete, and we put a Haus... | https://mathoverflow.net/users/5583 | Finite dimensional vector spaces over a complete but not-necessarily-valued field | Well, now I feel silly -- on looking through Wieslaw again, I see he does give examples of non-discrete, non-straight fields, just not in that section. For instance, take two absolute values on the rationals; the topology they generate together still make the rationals a topological field, is not discrete, and is obvio... | 2 | https://mathoverflow.net/users/5583 | 74836 | 45,472 |
https://mathoverflow.net/questions/74837 | 8 | hi, does anyone know a good book or some lecture notes on the theory of frechet manifolds ?
| https://mathoverflow.net/users/17656 | frechet manifolds book | There is the book by Kriegl and Michor called "Convenient setting of global analysis" published by the AMS. It goes much beyond Fréchet and really gives a big panorama. However, it is not easy reading and requires really some work. But I guess that is due to the subject...
| 8 | https://mathoverflow.net/users/12482 | 74838 | 45,473 |
https://mathoverflow.net/questions/34833 | 2 | Hello,
I am stuck with the following (hopefully not too trivial) problem.
I want to know, if the map
$${\cal D}(\mathbb{R}^2)\to L^2(H\_m,d\Omega\_m)\qquad f \mapsto \hat{f}|\_{H\_m}$$
has dense range.
Here $H\_m$ is the "upper mass shell" $\{ p\in \mathbb{R}^2:p\_0>0, p^2=m \}$ in the 2-dimensional Minkowski space... | https://mathoverflow.net/users/8230 | Fourier Transforms restricted to mass shell | I don't have any book in front of me at the moment, so I am a bit improvising here, but you'll find this in eg. Reed Simon 2,
Streater Wightman, or Josts book.
Let us call the map $$E:{\cal D}(\mathbb{R}^2)\to H=L^2(H\_m,d\Omega\_m)\qquad f \mapsto \hat{f}|\_{H\_m}.$$ I claim that already $E(\mathcal D(O))$ with $O$,... | 1 | https://mathoverflow.net/users/10718 | 74850 | 45,482 |
https://mathoverflow.net/questions/74853 | 4 | Let $f:E\to D^\*$ be a family of complex elliptic curves parametrized by the punctured open disk $D^\*.$ Assume that the monodromy on $H^1$ is trivial (i.e. $R^1f\_\*\mathbb Z$ is a constant sheaf on $D^\*$). Does this imply that $f$ extends to a family of elliptic curves over the full disk $D?$
Here's an attempt, w... | https://mathoverflow.net/users/370 | Analogue of Shafarevich-Ogg's theorem over complex numbers | The answer is, as you expected, yes:
Choosing a symplectic isomorphism of $R^1f\_\* \mathbb{Z}$ with the constant sheaf $\mathbb{Z}^2$ (i.e. full level structure) gives an analytic map $c:D^\* \to \mathfrak{h}$ where $\mathfrak{h}$ is the upper half plane (viewed as the moduli space of elliptic curves with full level... | 3 | https://mathoverflow.net/users/519 | 74875 | 45,499 |
https://mathoverflow.net/questions/74863 | 50 | From time to time, I pretend to be an algebraic topologist. But I'm not really *hard-core* and some of the deeper mysteries of the subject are still ... mysterious. One that came up recently is the exact role of CW-complexes. I'm very happy with the mantra "CW-complexes Good, really horrible pathological spaces Bad." b... | https://mathoverflow.net/users/45 | What does actually being a CW-complex provide in algebraic topology? | Those of us who do algebraic topology too much should remember occasionally that topological spaces are, in general, terrible to work with.
CW-complexes have a lot of properties that make them nice to work with in homotopy theory, such as being amenable to study by homotopy groups and such as being able to define map... | 47 | https://mathoverflow.net/users/360 | 74877 | 45,500 |
https://mathoverflow.net/questions/74866 | 2 | Here is what I mean exactly:
Let $A=(a\_1,a\_2)$ and $B=(b\_1,b\_2)$ be two points in the real plane (for simplicity, but general finite dimensions would also be nice), and define the $\ell\_p$-metric as $\|A\|\_p=(|a\_1|^p+|a\_2|^p)^{1/p}$ for $1\leq p<\infty$ and set $\|A\|\_\infty=\max(a\_1,a\_2)$.
What I want ... | https://mathoverflow.net/users/17662 | Is there an elementary proof for preserving inequalities under the change of l_p metrics? | The inequality is not true in higher dimensions.
For instance, let $A$ have two components equal to 1/2 and 200 components eqaul to 1/100, and the rest zeros, and let $B$ have ten components equal to 1/4 and the rest zeros.
We compute
$$\|A\|\_1=3,\|B\|\_1=5/2,\|A\|\_2=0.7211,\|B\|\_2=0.7906,\|A\|\_3=0.6301,\|B\|\_3=0.... | 2 | https://mathoverflow.net/users/12120 | 74886 | 45,506 |
https://mathoverflow.net/questions/74892 | 1 | Let $A\_n$ be the $n\times n$ matrix whose $(i,j)$-element is $1/(i+j-1)$. This is a famous matrix in linear algebra and has some nice properties (like, its inverse is integral).
Does anybody remember the name of this matrix? I am sure it was named after somebody but I don't remember. And I need the name for some rea... | https://mathoverflow.net/users/5259 | Looking for name of a famous matrix | <http://en.wikipedia.org/wiki/Cauchy_matrix> ?
| 3 | https://mathoverflow.net/users/nan | 74894 | 45,510 |
https://mathoverflow.net/questions/74908 | 5 | Consider any sequence consisting of n A's and n B's so that in any of its initial partial segments, the number of B's never exceed the number of A's. It is well known that the number of such sequences is the Catalan number $\frac{1}{n+1}\binom{2n}{n}$.
Now consider sequences consisting of n A's, n B's, and n C's, so ... | https://mathoverflow.net/users/11299 | Generalizing the Catalan number (enumerative combinatorics) | Yes. This is the generalized ballot problem.
The directed walks on $\mathbb Z\_{+}^k$ starting from the origin and ending at $(\lambda\_1,\dots,\lambda\_k)$, that satisfy $0\le x\_1\le \cdots \le x\_k$ at every point are in bijection with the number of standard Young tableaux of shape $\lambda$, and this can be found... | 16 | https://mathoverflow.net/users/2384 | 74913 | 45,520 |
https://mathoverflow.net/questions/74910 | 6 | I have some questions about homology, manifolds and bordism. First of all, if X is a smooth manifold, in general an integral homology class in X cannot be represented by a smooth embedded submanifold, as Thom proved.
1) If X is a topological manifold, does the same result hold? Are there in general singular homology ... | https://mathoverflow.net/users/10758 | Smooth and topological bordism and homology | (3) Yes, this is smooth approximation theory. See Hirsch's "Differential Topology" textbook.
I believe (1) and (2) were effectively answered by Larry Siebenmann in his ICM paper "Topological Manifolds". You can find it on Ranicki's webpage.
In particular, the topological bordism groups are a direct sum of the smoo... | 4 | https://mathoverflow.net/users/1465 | 74916 | 45,522 |
https://mathoverflow.net/questions/74855 | 3 | If $\varphi$ is an automorphism of $G = \langle x\_1, \ldots, x\_n; \mathbf{r}\rangle$ such that there exists an automorphism of $F(x\_1, \ldots, x\_n)$, $\overline{\varphi}$, with $$x\_i\varphi=\_G x\_i\overline{\varphi}$$ for all $1\leq i\leq n$ then $\varphi$ is said to be a free automorphism on $x\_1, \ldots, x\_n$... | https://mathoverflow.net/users/6503 | Free Automorphisms | Such automorphisms are sometimes called "tame". I would recommend asking Vladimir Shpilrain whose email address can be easily found on the Internet (also see his and Gupta's survey Gupta, C. K., Shpilrain, V. Lifting automorphisms: a survey. Groups '93 Galway/St. Andrews, Vol. 1 (Galway, 1993), 249–263, London Math. So... | 6 | https://mathoverflow.net/users/nan | 74919 | 45,525 |
https://mathoverflow.net/questions/74899 | 3 | The forgetful functor from the category of $\lambda$-rings to that of rings has a right adjoint in the form of the universal $\lambda$ functor $\Lambda$, which is isomorphic to the big Witt vectors functor. But, does the forgetful functor have a left adjoint? Some kind of free $\lambda$-ring functor?
EDIT: I see thi... | https://mathoverflow.net/users/10206 | Left Adjoint to the Forgetful Functor on $\lambda$-rings? | The edit suggests to me that you want not only the *existence* of the free $\lambda$-ring on any given ring (which is immediate from general results about adjoints of forgetful functors between varieties of algebras) but a construction using terms. Given a ring R, first form the set of all formal $\lambda$-ring terms o... | 5 | https://mathoverflow.net/users/6794 | 74927 | 45,531 |
https://mathoverflow.net/questions/74906 | 6 | Let $E$ and $E'$ be non-isogenous elliptic curves over a field $k$ (characteristic 0) such that $Gal(k(E[p^{\infty}])/k)=Gal(k(E'[p^{\infty}])/k) = SL\_2(\mathbb{Z}\_p)$ with $p \geq 5$ (where $E[p^{\infty}]$ is the set of $p^n$ torsion points of $E$ for all $n$). Then is it true that $k(E[p^{\infty}])\cap k(E'[p^{\inf... | https://mathoverflow.net/users/16858 | Intersection of field extensions of torsion points of non-isogenous elliptic curves | Since both fields $K(E\_{l^\infty})$ and $K(E'\_{l^\infty})$ contain the $l$-adic cyclotomic extension of $K$, your expectation cannot hold.
However, this is almost the only obstruction.
In *Propriétés galoisiennes des points d'ordre fini des courbes elliptiques*,
Invent. Math. 15, 259--331 (1972), J-P. Serre
prove... | 9 | https://mathoverflow.net/users/10696 | 74930 | 45,534 |
https://mathoverflow.net/questions/74876 | 3 | Usually in a first course on differential geometry we learn some classical results on the geometry of curves and surfaces in the ordinary euclidean space, and just later in more advanced courses we learn systematically the concepts and the tools of the analysis on manifolds, one of whose pillars is the Frobenius' Theor... | https://mathoverflow.net/users/12617 | What are elementary applications of the Frobenius'Theorem in the Classical Differential Geometry? | First, anything that is proved using the Frobenius theorem can also be proved using the existence and uniqueness theorem for ODE's and the fact that partials commute. The theorem is used in differential geometry to show that local geometric assumptions imply global ones. Here are a few examples that come to mind:
1) ... | 15 | https://mathoverflow.net/users/613 | 74931 | 45,535 |
https://mathoverflow.net/questions/74799 | 13 | Let $U$ be a connected open subset of $\mathbb R^3$. Furthermore, we have:
1. $\mathbb R^3\setminus U$ has exactly two connected components (thus by Alexander duality, $H\_2(U;\mathbb Z)=\mathbb Z$).
2. $U$ may be "very complicated": we make no assumptions on the regularity of $\partial U$.
I would like to understa... | https://mathoverflow.net/users/35353 | Incompressible surfaces in an open subset of R^3 | Yes, I think if I understand your question, what you're asking for is true. Take minimal area representatives for your surfaces $F\_1,\ldots, F\_p$, and take the boundary of the component of the complement of $F\_1\cup\cdots \cup F\_p$ which contains infinity (I think this is the same as your surface $F$). If this surf... | 16 | https://mathoverflow.net/users/1345 | 74935 | 45,538 |
https://mathoverflow.net/questions/74938 | 7 | This isn't really a research question, but at least it's research-level mathematics. I'm talking with some other people about the first uncountable ordinal, and I want some facts to inform this discussion. Specifically, what useful or interesting foundations of mathematics do or don't allow one to prove the existence o... | https://mathoverflow.net/users/8508 | Foundations: Existence of uncountable ordinals. | For the existence of an uncountable set with a definable well-ordering, it suffices to have $\mathcal P(\mathcal P(\mathbb N))$, where $\mathcal P$ means the power set. Any countable order-type is represented by a well-ordering of $\mathbb N$, and can therefore be coded by a subset of $\mathbb N$. Call two such codes e... | 18 | https://mathoverflow.net/users/6794 | 74940 | 45,540 |
https://mathoverflow.net/questions/74885 | 6 | Let $X$ be a smooth variety over $\mathbb{C}$.
Let $D \subset X$ be an effective Cartier divisor.
>
> **Question 1.** What is the definition of the logarithmic differential sheaf $\Omega^1\_X (\log D)$ ?
>
>
>
I saw the definition in the book by Esnault-Viehweg (meromorphic form $\alpha$ such that $\alpha, ... | https://mathoverflow.net/users/12390 | exact sequence of logarithmic differential sheaves associated to an effective Cartier divisor on a smooth variety | Here is a way to define this sheaf algebraically over any field of characteristic zero. Let $\mathrm{T}\_{X}$ denote the tangent sheaf on $X$. Choose a local equation $\phi\_U$ for $D$ on $U$. Consider the following submodule:
$\mathrm{T}\_X(-\log\phi\_U)=\ ${$\partial\in\mathrm{Der}(\mathcal{O}\_X(U))\mid \partial\p... | 7 | https://mathoverflow.net/users/10941 | 74947 | 45,542 |
https://mathoverflow.net/questions/74941 | 5 | Just a curiosity:
>
> Is there an assertion of which a proof (formalizable, say, in ZFC) is not known but a proof that it's *not* undecidable (in ZFC) *is* known?
>
>
>
Edit: after the comments, I think the actual question was
>
> Is there an ("interesting") assertion of which neither a proof (formalizabl... | https://mathoverflow.net/users/4721 | Is there an "undecided" assertion of which a proof that it's not undecidable is known? | If it's known that some statement $S$ is decidable in ZFC, then you can just run a computer program that enumerates all ZFC-proofs and stops when it finds a proof of $S$ or a proof of $\neg S$. By hypothesis, this algorithm is guaranteed to terminate. Therefore, the only possible obstacle separating decidable statement... | 12 | https://mathoverflow.net/users/3106 | 74956 | 45,547 |
https://mathoverflow.net/questions/74944 | 1 | So let $D\subseteq \mathbb{C}^n$ be a **bounded** connected open set with a transitive action of its group of biholomorphisms (which we denote by $Hol(D)$). Note that **I'm not** assuming that $D$ is symmetric. We thus have that $D$ is "homeomorphic" to $Hol(D)/K$ where $K=Stab(d\_0)$ for some $d\_0\in D$.
In the spe... | https://mathoverflow.net/users/11765 | On bounded homogeneous connected domains of C^n | Re question 3: a bounded homogeneous domain is biholomorphic to a Siegel domain, which is contractible. See e.g. [Siegel domain](http://www.encyclopediaofmath.org/index.php?title=Siegel_domain) and references therein (those references probably answer question 2 as well). Another useful link is [Homogeneous bounded doma... | 2 | https://mathoverflow.net/users/2349 | 74959 | 45,549 |
https://mathoverflow.net/questions/74915 | 5 | Let X be a smooth projective variety over the complex numbers, of dimension at least two. $D$ is an ample divisor on X. Then we know for $m>>0$, $H^i(mD)=0$. Now suppose $E$ is another divisor that is algebraiclly equivalent to $mD$, i.e. the line bundle $\mathcal{L}(E-mD)\in Pic^0(X)$ lies in the identity component of... | https://mathoverflow.net/users/1877 | Deforming ample line bundles vs cohomology group | What you want follows easily from the Kodaira vanishing theorem:
If $m$ is sufficiently large then $mD - K$, where $K$ is the canonical divisor, is ample (this is true for $K$ replaced by any divisor). Since ampleness is preserved by algebraic equivalence, for example by Kleiman's criterion, it follows that $E - K$ i... | 5 | https://mathoverflow.net/users/519 | 74975 | 45,557 |
https://mathoverflow.net/questions/74843 | 6 | Let $X$ be a projective (or affine) variety over $\mathbb{C}$ defined by some homogenous ideal $I = (f\_1,\ldots,f\_n)$. How can we interpret the Euler characteristic of $X$ other than as just an invariant to distinguish non isomorphic objects? What I mean is what could it tell us if anything about either $X$ or the po... | https://mathoverflow.net/users/12402 | How to Interpret the Euler Characteristic of Complex Algebraic Varieties | Hi Dori! I think that you will find Paolo Aluffi's paper ["Computing characteristic classes of projective schemes"](http://arxiv.org/abs/math/0204230) useful, if not for the exact formulas, at least for the algorithm.
Among other things it proves the version of the formula in J.C. Ottem's answer for singular hypersur... | 2 | https://mathoverflow.net/users/2384 | 74978 | 45,558 |
https://mathoverflow.net/questions/74970 | 3 | I have [asked this question on Stack Exchange](https://math.stackexchange.com/questions/62723/defect-groups-and-subgroups) but had no response; it's been bugging me for a few days. I am struggling to see how to apply Mackey's theorem to prove a certain Lemma in *Local representation theory* by JL Alperin.
**Lemma** L... | https://mathoverflow.net/users/15632 | Defect groups and subgroups | This is an exercise in writing out the definitions:
since the defect group of $B$ is $E$, we have $B|(B\_{\delta(E)})^{G\times G}$. So by assumption,
$$
b\;|\;B\_{H\times H}\;|\;\left((B\_{\delta(E)})^{G\times G}\right)\_{H\times H}=\bigoplus\_{(g\_1,g\_2)\in \delta(E)\backslash G\times G/H\times H}\left(B\_{\delta(E)^... | 7 | https://mathoverflow.net/users/35416 | 74980 | 45,560 |
https://mathoverflow.net/questions/74889 | 5 | Ramanujan introduced mock theta functions and described them by an "order" which he did not define. As a result of the work of Zwegers and others we now have a better understanding of mock theta functions. They appear as the holomorphic projection of weight 1/2 harmonic Maass forms and in the theta expansions of meromo... | https://mathoverflow.net/users/10475 | What is the modern understanding of the order of a mock theta function? | I'm afraid there isn't very much in the literature about the order. If you really want to know, you'll have to check it yourself...
I think Ramanujan actually only defined the order for the functions of order 3, 5 and 7 (in his last letter to Hardy). Mock theta functions of different order show up in his Lost Notebook,... | 6 | https://mathoverflow.net/users/17687 | 74982 | 45,562 |
https://mathoverflow.net/questions/74955 | 11 | This is a slightly pedantic question about the "2-categorical" nature of localization. Recall the definition:
**Definition**. A localization of a category $\cal C$ with respect to a class of morphisms $W$ is a category ${\cal C}[W^{-1}]$ together with a functor $q:{\cal C}\to {\cal C}[W^{-1}]$ such that
1. $q$ send... | https://mathoverflow.net/users/1148 | Strict categorical localization is automatically a "2-localization"? | Let $\widetilde{\mathcal{C}}$ be the category with same objects as $\mathcal{C}$ and exactly one morphism between each pair of objects. Then there exists a unique functor $F:\mathcal{C} \to \widetilde{\mathcal{C}}$ that is the identity on objects. It sends all morphisms in $W$ to isomorphisms (since all morphisms of $\... | 8 | https://mathoverflow.net/users/12547 | 74992 | 45,567 |
https://mathoverflow.net/questions/74968 | 2 | Whenever, people talk about singular plane curves they talk about their Newton polytope which is obviously coordinate dependent. I understand that with some conditions over the singular curve, some invariantes can be calculated from the Newton polytope e.g the multiplier ideal of the monomial ideal $(x^p,y^q)$ is the s... | https://mathoverflow.net/users/17602 | Does the Newton polytope characterize the equisingular i.e topological type? | The answer is **yes**.
More precisely, if two *nondegenerate* Newton singularities $f(x,y)=0$ and $g(x,y)=0$ have the same Newton polygon, then they are topologically equivalent.
For a reference, look at Takamura's book "Splitting deformations of degenerations of complex curves III", [pag. 138](http://books.google... | 3 | https://mathoverflow.net/users/7460 | 74996 | 45,570 |
https://mathoverflow.net/questions/74985 | 4 | If $G$ is a finite $p$-group of order $p^n$, then it is well known that for ($1\leq m\leq n$), number of subgroups of order $p^m$ is $1$(mod $ p$).
***Question:*** Is it true that number of subgroups of order $p^m$, which are isomorphic within themselves, is $0$(mod $ p$) or $1$(mod $p$).
It looks to be true for ... | https://mathoverflow.net/users/17456 | Number of Subgroups of p-Groups | This is just a matter of searching for a counterexample!
You will find that in GAP or Magma, SmallGroup(81,7) has four subgroups of order 27, two of which are isomorphic, but the other two are not isomorphic to any others.
It is possible that the answer might be yes for abelian groups.
| 4 | https://mathoverflow.net/users/35840 | 75003 | 45,572 |
https://mathoverflow.net/questions/75001 | 2 | Let $D$ be a big and nef divisor on a smooth complex projective minimal surface and let $\phi\_D$ be the induced rational map. Is it true that $\phi\_D$ is generically finite? Otherwise does someone know a counterexample?
Thank you
| https://mathoverflow.net/users/15415 | Rational map associated to a big and nef divisor | No, you can only conclude that *for some multiple* $mD$, with $m$ large enough, the map is generically finite. In fact, "big" is actually equivalent to this condition.
For a counterexample to your question, take a minimal surface $S$ of general type with $p\_g(S)=2$ (there are lots of them). Then $K$ is big and nef, ... | 7 | https://mathoverflow.net/users/7460 | 75006 | 45,574 |
https://mathoverflow.net/questions/75002 | 5 | I want as many 80-bits words as possible with the constraint that the hamming distance between any couple of words is exactly 40. How many can I generate? Is there a generic formula telling me how many n-bits words I can generate with the constraint that any couple of words is at hamming distance exactly n/2? Any gener... | https://mathoverflow.net/users/17693 | binary code with constant hamming distance | There cannot be more than $n$ such words. To see this, consider the words as elements of $\{-1,1\}^n$. If two such words have Hamming distance $\Delta$, then their scalar product in $\mathbb R^n$ is $n-2\Delta$. In particular, if they all have distance $n/2$, then they are orthogonal, hence linearly independent, hence ... | 11 | https://mathoverflow.net/users/12705 | 75014 | 45,577 |
https://mathoverflow.net/questions/75009 | 2 | I've just started learning these things and so probably my questions will be very easy. Please forgive me.
A metric space $(X,d)$ is called locally finite if every bounded set is finite.
A metric space is said to have coarse bounded geometry if there is $\Gamma\subseteq X$ such that
1) there exists $c>0$ such that ... | https://mathoverflow.net/users/13809 | Local finiteness and coarse bounded geometry | Q2--no. Let $A\_n$ have cardinality $n+1$ for $n=0,1,...$. Specify all distances between distinct points in the same $A\_n$ to be one, and the distance between a point in $A\_n$ to a point in $A\_m$ to be $n+m$ when $n\not= m$.
This gives a simple example for Q1 as welll.
| 2 | https://mathoverflow.net/users/2554 | 75016 | 45,579 |
https://mathoverflow.net/questions/75017 | 7 | Dear community,
in his 2005 Inventiones Paper "On motivic decompositions arising from the method
of Białynicki-Birula" P. Brosnan deduced from the classical (?) theorem of Bialynicki-Birula on decomposition of smooth projective varieties with $\mathbb{C}^\*$-action as a cellular variety by invoking a theorem of Karpe... | https://mathoverflow.net/users/17695 | Decomposition of Motives of cellular varieties | The varieties considered by Brosnan and Karpenko are not cellular over the base field but become cellular over the algebraic closure. For a cellular variety over any field the Chow groups are freely generated by the closures of the cells and are "equal" to the cohomology. A Kunneth type formula also holds so the motive... | 4 | https://mathoverflow.net/users/519 | 75020 | 45,581 |
https://mathoverflow.net/questions/75013 | 10 | The Scholz reflection principle says, among other things, that if $D < 0$ is a negative fundamental discriminant, not $-3$, then the 3-ranks of the class group of $\mathbb{Q}(\sqrt{D})$ is either equal to that of $\mathbb{Q}(\sqrt{-3D})$, or one larger.
Does anyone know of (and recommend) any introductory reading on ... | https://mathoverflow.net/users/1050 | Introductory reading on the Scholz reflection principle? | This is simple class field theory plus Galois theory. Consider a quadratic number field $K$
with class number divisible by $3$. For constructing an unramified cyclic cubic extension $L/K$, adjoin the cube root of unity, and denote the resulting field by $K'$. The Kummer generator of the Kummer extension $L' = K'(\sqrt[... | 10 | https://mathoverflow.net/users/3503 | 75027 | 45,584 |
https://mathoverflow.net/questions/75030 | 11 | This is a spiritual successor to a question that Peter Shor answered here:
[Generalized Euclidean TSP](https://mathoverflow.net/questions/57017/generalized-euclidean-tsp)
Are there any results known on the asymptotic behavior of clique sizes in a unit disk graph with uniformly sampled points? That is, suppose I sam... | https://mathoverflow.net/users/11828 | Clique sizes in a unit disk graph | Yes, a lot is known about this. See Mathew Penrose's book, "Random Geometric Graphs." He discusses subgraph counts for random geometric graphs on distributions with bounded, measurable, density functions --- this includes uniform distributions on any convex body as a special case. He also discusses maximum clique size.... | 9 | https://mathoverflow.net/users/4558 | 75032 | 45,586 |
https://mathoverflow.net/questions/74777 | 3 | For integer $n$, $1 \le n \le N$, consider the random variables
$X\_n = \cos[t \omega\_n]$
For any fixed $N$, we can take the mean
$Y\_N = \frac{1}{N} \sum\_{n=1}^N X\_n$
and define a (cumulative) distribution by averaging over long times:
$P(Y\_N \le y) = \lim\_{T \to \infty} \frac{1}{2 T} \lambda(\{t \in [-... | https://mathoverflow.net/users/5789 | What conditions on a probability distribution defined by long-time averaging do I need to satisfy a central limit theorem? | In your example, algebraic manipulation gives
$$
2^{-M} \sum\_{n=0}^{2^M-1} X\_n^{(M)} = \prod\_{j=0}^{M-1} \cos t h\_j.
$$
If the $h\_j$'s are linearly independent over $\Bbb Q$, then, as you point out, the random variables $\cos t h\_j$ approach independence as $t$ is chosen over larger and larger intervals. Therefo... | 3 | https://mathoverflow.net/users/17657 | 75045 | 45,595 |
https://mathoverflow.net/questions/74567 | 3 | We have a given positive martingale ρt, with the dynamics:
$$\textrm{d}\rho\_t = \lambda\_t \rho\_t \textrm{d}W\_t$$
where $W\_t$ is a standard Brownian motion. Now we have an "exponentially dampened" process $p\_t$:
$$p\_t=\int\_0^t \exp(-\int\_0^s r\_u \textrm{d} u)\textrm{d} \rho\_s$$
where $r\_u \geq 0$. If needed ... | https://mathoverflow.net/users/3160 | Stochastic integrals as honest martingales — exponential damping | Yes, in this case it is true that $p$ is a proper martingale! Note that your integrand $\exp\left(-\int\_0^tr\_u du\right)$ is an adapted, continuous, and decreasing process bounded by 1. So, the following statement implies that $p$ is a martingale.
>
> Let $\rho$ be a cadlag martingale and $\xi$ be an adapted left... | 4 | https://mathoverflow.net/users/1004 | 75047 | 45,596 |
https://mathoverflow.net/questions/75053 | 5 | Given a Riemannian metric $g$ on a smooth manifold $M$, one defines an
$L^2$-inner product on the space $\bigwedge^\ast(M)$ of differential
forms by
$$
\langle \alpha, \beta \rangle\_g = \int\_M \alpha \wedge \ast\_g \beta,
$$
where $\ast\_g$ denotes the Hodge-star operator relative to $g$, and
$\alpha, \beta$ are... | https://mathoverflow.net/users/5706 | Inner products on differential forms | Fixing a $k$ for simplicity, there are many inner products on $\bigwedge^k (M)$ (which I would usually denote $\Omega^k (M)$). Since $\bigwedge^k T^\* M$ is a vector bundle, there are, for example, Sobolev $H^s$ inner products on its space of smooth sections for any natural number $s$. See, for example, Palais, *Founda... | 6 | https://mathoverflow.net/users/2063 | 75056 | 45,599 |
https://mathoverflow.net/questions/75059 | 9 | The [Catalan numbers](http://en.wikipedia.org/wiki/Catalan_number) are the moments of the [Wigner semicircle distribution](http://en.wikipedia.org/wiki/Wigner_semicircle_distribution).
$$ \frac{1}{2\pi} \int\_{-2}^2 x^{2n} \sqrt{4 - x^2} dx = \frac{1}{n+1} \binom{2n}{n} $$
[Motzkin numbers](http://en.wikipedia.org/w... | https://mathoverflow.net/users/1358 | Recognizing a measure whose moments are the motzkin numbers | Motzkin numbers are a very popular sequence. A lot of identities and formulas are already recorded at [OEIS](http://oeis.org/A001006). The analogous integral representation for Motzkin numbers is
$$M\_n=\frac{1}{2\pi} \int\_{-1}^3 x^n\sqrt{(3-x)(1+x)}dx.$$
---
A few words about the general picture. There is a we... | 17 | https://mathoverflow.net/users/2384 | 75062 | 45,603 |
https://mathoverflow.net/questions/75064 | 4 | I'm interested in finding an algebra with elements x,y which are identified by every finite-dimensional module. I'm primarily interested in the case that everything is over the complex field, but answers over other fields would also be interesting.
| https://mathoverflow.net/users/799 | Algebra with elements x, y such that r(x)=r(y) for all finite-dimensional modules r | The Weyl algebra, generated by $x$ and $y$ subject to the single relation $xy-yx=1$, is an example. A silly one, one may add: it does not have any finite dimensional representation!
There are more interesting examples: let $\mathfrak g$ be the Lie algebra with basis $x\_i$, $y\_i$, with $i\in\mathbb Z$, and $z$, such... | 12 | https://mathoverflow.net/users/1409 | 75065 | 45,604 |
https://mathoverflow.net/questions/75051 | 3 | I would like to solve the following optimization problem in $k$-vector $w\_i$
$$ \min\_{w\_i} \quad \left\|P\_i - X \mbox{diag} (w\_i) Y^T \right\|\_F^2 $$
where $P\_i$ is a $6 \times 6$ matrix, $X$ and $Y$ are $6 \times k$ matrices, and $\mbox{diag}(w\_i)$ is a (square) diagonal matrix whose main diagonal is $w\_i... | https://mathoverflow.net/users/17707 | How do I optimize over (or take derivative wrt) a square diagonal matrix? | Your notation is somewhat confusing, in that you apply the subscript $i$ to $w$, and have a vector $w\_{i}$, but don't use $i$ in any meaningful way in your problem. I'm going to take the liberty of rewriting the problem as
$\min\_{w} \| P-X \mbox{diag}(w) Y^{T} \|\_{F} $.
You may have a whole bunch of these probl... | 5 | https://mathoverflow.net/users/9022 | 75066 | 45,605 |
https://mathoverflow.net/questions/75075 | 1 | Here is an elemantary example:
Define $f:S^1\times \mathbb{C}\rightarrow\mathbb{C}$ by $f(\zeta, z)=\zeta\cdot z^n$, where $n\geq 2$ is an integer, then $f$ is a smooth map, every fiber of $f$ is a smooth submanifold of $S^1\times\mathbb{C}$ and is diffeomorphic to $S^1$. However, $(S^1\times \mathbb{C}, \mathbb{C}, ... | https://mathoverflow.net/users/15289 | The smoothness of fiber and fiber bundle | By definition, a morphism $f \colon X \to Y$ between smooth projective varieties is smooth if $f$ is flat and all fibres are smooth (in the scheme-theoretical sense). In particular, $f$ is a smooth submersion when it is considered as a differentiable map between real manifolds.
Then the answer to your question is *ye... | 4 | https://mathoverflow.net/users/7460 | 75081 | 45,610 |
https://mathoverflow.net/questions/75079 | 0 | Hi,
I have the following definition of what an Abelian Variety $A$ over an arbitrary field $k$ is:
it is a geometrically integral and proper group scheme over $Spec(k)$.
By a group scheme I understand a scheme over $k$ which has $k-$ morphisms
$m: A\times A \rightarrow A$,
$i: A \rightarrow A$,
$e: Spec(k) ... | https://mathoverflow.net/users/16876 | Definition of abelian variety | You cannot expect any group structure on $A(\bar k)$ or any homomorphism $A(\bar k)\to B(\bar k)$ to come from morphisms, just consider an elliptic curve over $k=\bar k=\mathbb C$ where $A(\bar k)\cong\mathbb R^2/\mathbb Z^2$ as groups (for *any* group scheme structure on $A$).
However, two morphisms $f,g\colon A\to ... | 2 | https://mathoverflow.net/users/2035 | 75086 | 45,615 |
https://mathoverflow.net/questions/75088 | 3 | If $\phi$ is any formula of set theory with just one free variable $x$, the abstraction term $A\_{\phi}=\lbrace x | \phi(x) \rbrace$ is either a set or a proper class. Assume that ZFC is consistent, or any large cardinal axiom you like. Then my question is, are there two formulas
$\phi$ and $\psi$ such that ZFC+($A\_{... | https://mathoverflow.net/users/2389 | mutually incompatible abstraction terms? | If by "consistent" you mean "consistent relative to the consistency of ZFC," then there is a simple example. Let $\phi$ be the [Rosser sentence](http://en.wikipedia.org/wiki/Rosser%27s_trick#The_Rosser_sentence) for ZFC, and let $\psi$ be its negation. Then, in any model of ZFC, $A\_\phi$ is either $V$ (if $\phi$ is tr... | 5 | https://mathoverflow.net/users/2000 | 75091 | 45,617 |
https://mathoverflow.net/questions/75089 | 2 | If factoring is in $P$ (with a blazing fast polynomial time in $P$), would it affect the [index calculus algorithm](http://en.wikipedia.org/wiki/Index_calculus_algorithm) used for Discrete Log calculation in any serious way?
**Other connections**
$1.)$ "Number field cryptography" Johannes Buchmann Tsuyoshi Takagi U... | https://mathoverflow.net/users/16007 | Factoring and Index Calculus and duality between DL and factoring via compuational problems made easy through them | Nobody knows. It's striking that the best algorithms for factoring and finite field discrete log are so closely analogous, and it hints at a deeper relationship between the problems (as joro pointed out in the comments), but no efficient reduction is known in either direction. In particular, it might just be a coincide... | 4 | https://mathoverflow.net/users/4720 | 75108 | 45,625 |
https://mathoverflow.net/questions/75104 | 4 | Let $X\subset\mathbb{P} ^ {14} \_{\mathbb{C}}$ be the image of the 2-uple embedding of $\mathbb{P}^4 \_{\mathbb{C}}$ in $\mathbb{P}^{14} \_{\mathbb{C}}$.
What is the secant variety $ Sec(X)=\overline{ \bigcup\_{x\_1,x\_2\in X\atop x\_1\neq x\_2}\langle x\_1, x\_2\rangle }$ of $X$? What is its degree?
Thanks.
| https://mathoverflow.net/users/15606 | Secant variety of the 2-uple embedding $\mathbb{P}^4\hookrightarrow\mathbb{P} ^ {14}$. | This is the variety of $5 \times 5$ symmetric matrices of rank $\leq 2$. It is studied in Chapter 6.3 of Weyman's [Cohomology of Vector Bundles and Syzygies](http://books.google.com/books?id=t_jdqfMMtnYC&lpg=PA229&ots=M21MRTokWP&dq=%2522rank%2520variety%2522%2520symmetric&pg=PA175#v=onepage&q=symmetric%2520matrices&f=f... | 16 | https://mathoverflow.net/users/297 | 75110 | 45,626 |
https://mathoverflow.net/questions/75109 | 1 | is there any definite method or algorithm,software to get exact function or expression from series.e.g we get series solution of differential equation and we want exact expression rather than approximation series ?
| https://mathoverflow.net/users/17720 | method for getting function from power series/perturbation series | In some cases yes, see:
<http://www.reduce-algebra.com/docs/qsum.pdf>
and references therein...
**EDIT** A canonical reference is Petkovsek-Zeiberger's A=B.
| 2 | https://mathoverflow.net/users/11142 | 75112 | 45,628 |
https://mathoverflow.net/questions/75049 | 130 | (This question was posed to me by a colleague; I was unable to answer it, so am posing it here instead.)
Let $f: {\bf R}^n \to {\bf R}^n$ be an everywhere differentiable map, and suppose that at each point $x\_0 \in {\bf R}^n$, the derivative $Df(x\_0)$ is nonsingular (i.e. has non-zero determinant). Does it follow t... | https://mathoverflow.net/users/766 | Does the inverse function theorem hold for everywhere differentiable maps? | The usual reference to the proof is A. V. Cernavskii in "Finite-to-one open mappings of manifolds", Mat. Sb. (N.S.), 65(107) (1964), 357–369 and "Addendum to the paper "Finite-to-one open mappings of manifolds"", Mat. Sb. (N.S.), 66(108) (1965), 471–472. If I remember it correctly, he does not state it explicitly, but ... | 70 | https://mathoverflow.net/users/17725 | 75118 | 45,633 |
https://mathoverflow.net/questions/75119 | 0 | What is known about the quadruple layer potential in 3D (on closed smooth surfaces)? In terms of jump relations, continuity on Hölder Spaces (and/or Sobolev spaces), and Calderon-type identities (regularization). I'm interested in the Laplace and also the Helmholtz case (the acoustic problem). Thanks
| https://mathoverflow.net/users/16450 | Properties of the quadruple layer potential | Here's a reference to a paper by Shidong Jiang which may be useful as regards jump relations
: <http://web.njit.edu/~jiang/Papers/jump.pdf>
| 2 | https://mathoverflow.net/users/14740 | 75120 | 45,634 |
https://mathoverflow.net/questions/13899 | 6 | Consider the Hecke algebra $H\_n$ of type $A\_{n-1}$ with standard basis $T\_w$, $w \in S\_n$ with the quadratic relations $(T\_s - u) (T\_s + u^{-1}) = 0$ and braid relations. The unsigned canonical basis $C'\_w$, $w \in S\_n$ gives rise to a basis for the irreducible $H\_n$-module $M\_\lambda$ of shape $\lambda$: fix... | https://mathoverflow.net/users/3318 | Signed and unsigned Hecke algebra canonical basis | Not much seems to be known about this matrix in general, and it does become the identity at $u=0$. I show this in the paper *Quantum Schur-Weyl duality and projected canonical bases* <http://arxiv.org/abs/1102.1453>
using quantum Schur-Weyl duality and its compatibility with canonical bases. This only proves it in typ... | 2 | https://mathoverflow.net/users/3318 | 75121 | 45,635 |
https://mathoverflow.net/questions/74519 | 1 | Let $(G,\cdot)$ be a group and $\phi:G\rightarrow\mathbb R$ bounded. Let me say that the pair $(G,\phi)$ is amenable if there is a finitely additive probability measure $\mu$ on $G$ such that for all $y\in G$
$$
\int \phi(x)d\mu(x)=\int \phi(x\cdot y)d\mu(x)=\int\phi(y\cdot x)d\mu(x)
$$
>
> **Question:** Does the... | https://mathoverflow.net/users/13809 | Amenability with respect to a function | The answer is no, no such non-amenable group can exist. It follows from Justin Moore's answer to his own [question](https://mathoverflow.net/questions/60247/when-is-non-amenablity-witnessed-by-a-single-non-measurable-set) that a single characteristic function can witness the non-amenability of a group.
| 2 | https://mathoverflow.net/users/8176 | 75123 | 45,637 |
https://mathoverflow.net/questions/69468 | 13 | Does anyone know who was the first to coin the term "Lie group"?
The following thesis from 1928 suggests that the term was already in use by that time: "Systems of Two Differential Equations from the Lie-Group Standpoint"
(<http://genealogy.math.ndsu.nodak.edu/id.php?id=6129>)
I've also found the term in the book... | https://mathoverflow.net/users/1355 | When did the term "Lie group" first appear? | (Just to get this off the un-answered list, I'm copying Qiaochu's comment as an answer.)
<http://jeff560.tripod.com/l.html> suggests it is 1891, in the paper "Sur une application des groupes de M. Lie" by L. Autonne, with first *English language* appearance in 1897 with an [article by Lovett](http://www.jstor.org/sta... | 8 | https://mathoverflow.net/users/3948 | 75126 | 45,638 |
https://mathoverflow.net/questions/75038 | 36 | Let $\omega$ be a closed non-exact differential $k$-form ($k \geq 1$) on a closed orientable manifold $M$.
**Question**: Is there always a Riemannian metric $g$ on $M$ such that $\omega$ is $g$-harmonic, i.e., $\Delta\_g \omega = 0$?
Here $\Delta\_g$ is the Laplace-deRham operator, defined as usual by
$\Delta\_g ... | https://mathoverflow.net/users/5706 | When is a closed differential form harmonic relative to some metric? | A closed $k$-form is called *intrinsically harmonic* if there is some Riemannian metric with respect to which it is harmonic. E. Calabi (*Calabi, Eugenio*, An intrinsic characterization of harmonic one-forms, Global Analysis, Papers in Honor of K. Kodaira 101-117 (1969). [ZBL0194.24701](https://zbmath.org/?q=an:0194.24... | 30 | https://mathoverflow.net/users/9471 | 75136 | 45,645 |
https://mathoverflow.net/questions/75131 | 3 | Let $J$ be an arc in $\mathbb{S}^{1}\subset\mathbb{C}$ (no matter open or
closed) and $\alpha\in(0,2\pi)$ be an angle such that $\alpha/\pi$ is
irrational. Consider in $\mathbb{S}^{1}$ the sequence $z\_{n}=e^{in\alpha}$.
Then this sequence is dense in $\mathbb{S}^{1}$ by Kronecker's Theorem or by
ergodicity. Let's asso... | https://mathoverflow.net/users/14441 | How to detect frequency? | It seems likely to me that $\alpha$ can be computed by calculating the frequencies of subwords of the coding sequence, but in a manner which depends on certain parameters. For example, if $\alpha<\min\{|J|,2\pi-|J|\}$ then the interval $J \setminus J +\alpha$ has length precisely $\alpha$, and it follows easily that $\... | 5 | https://mathoverflow.net/users/1840 | 75139 | 45,647 |
https://mathoverflow.net/questions/74821 | 1 | Let $F\_2[x]$ denote the ring of polynomials over the field of 2 elements.
Richard Brent has a page on finding primitive trinomials in $F\_2[x]$ of huge degree at <http://maths.anu.edu.au/~brent/trinom.html>.
My problem is different, I want to find primitive polynomials none of whose multiples--which are of course ... | https://mathoverflow.net/users/17773 | Minimum Growth Rate of Hamming Weight of Multiples of Primitive Polynomials | Scratch the use of the parity check matrix. Combine Felipe's idea with a counting argument similar to the scratched solution.
Assume $n\ge4.$ Let $\alpha$ be a root of $f(x)$. The set $P=\{ \alpha^i\mid n+2\lt i\lt N \}$ contains $N-n-3\ge 8$ distinct elements of the field $GF(2^n)$. Therefore it must intersect non-t... | 1 | https://mathoverflow.net/users/15503 | 75146 | 45,652 |
https://mathoverflow.net/questions/75150 | 9 | Say we are given a complex manifold $X$ and an $\mathcal{O}\_X$-module $\mathcal{F}$. Assume that for any point $P\in X$ the stalk $\mathcal{F}\_P$ is a free $(\mathcal{O}\_X)\_P$-module of finite rank. Does it imply that $\mathcal{F}$ is locally free? If not, what do you need to know additionally about $\mathcal{F}$ t... | https://mathoverflow.net/users/17732 | Sheaf with free stalks | Just looking at stalks is not enough:
Suppose that $X$ is a nontrivial complex manifold. Let
$i\_x:x\to X$ denote the inclusion, and set
$$\mathcal{F} =\bigoplus\_{x\in X} i\_{x\*}\mathcal{O}\_x$$
Notice that
it is naturally an $\mathcal{O}\_X$-module with $\mathcal{F}\_x\cong \mathcal{O}\_x$,
and yet it is certainly... | 8 | https://mathoverflow.net/users/4144 | 75161 | 45,661 |
https://mathoverflow.net/questions/75160 | 13 | Let $k$ be a field. Can each degree $n$ polynomial $P(t) \in k[t]$ be written as the determinant of the matrix $A + tB$, where $A$ and $B$ are two *symmetric* $(n \times n)$-matrices with entries in $k$?
Over an algebraically closed field this is pretty obvious, but is this also true for non-closed fields?
| https://mathoverflow.net/users/1107 | Which polynomials are determinants of a symmetric matrix with linear entries? | Here is a simple argument showing that you can get any polynomial up to a constant factor. As Rob Israel and my comments above show, you might not be able to get rid of that constant.
If $\det(A\_1 t + B\_1) = f\_1(t)$ and $\det(A\_2 t + B\_2) = f\_2(t)$, then $\det \left( \begin{smallmatrix} A\_1 t+B\_1 & 0 \\ 0 & A... | 10 | https://mathoverflow.net/users/297 | 75189 | 45,674 |
https://mathoverflow.net/questions/75196 | 0 | I have an expression: $E[(b+X)^2|Y]$ where $X$ and $Y$ are normally distributed random variables, being two components of a final unknown outcome $Z$ ($Y$ is known, $X$ is the noise component):
$Y$ = $Z$ + $X$.
~~Their distributions are normal and independent: $X$ ~ $N(0,\sigma\_x^2)$ and $Y$ ~ $N(0,\sigma\_y^2)$... | https://mathoverflow.net/users/17745 | Conditional expectation of a product | Hi Apeirohedron,
your expectation is $\sigma\_x^2 + b^2$, since $E[(b+X)^2|Y] = E[X^2|Y] + 2bE[X|Y] + b^2 = \sigma\_x^2 + 2bE[X] + b^2 = \sigma\_x^2 + b^2$. $E[X|Y] = E[X]$ holds since $X$ and $Y$ are independent.
**In the general case:**
You know that $X$ and $Z$ are Gaussian. In particular, you know that $X\sim ... | 1 | https://mathoverflow.net/users/17017 | 75199 | 45,679 |
https://mathoverflow.net/questions/74405 | 14 | Commutative Poisson algebras $A$ can be thought of as commutative algebras equipped with a first-order deformation into a noncommutative algebra given by the Poisson bracket. A simple example is the symmetric algebra $S(\mathfrak{g})$ of a Lie algebra, which can be deformed into the universal enveloping algebra $U(\mat... | https://mathoverflow.net/users/290 | Poisson algebras as deformations vs. Poisson algebras in algebraic topology | Let me start by rephrasing what is already in the answers of David Ben-Zvi and Theo Johnson-Freyd. The DG $\mathbb{Q}$-linear operad $\mathbb{E}\_n:=C\_{-\bullet}(E\_n,\mathbb{Q})$ is filtered. For $n\geq2$ the filtration is the degree filtration, and thus $gr(\mathbb{E}\_n)=H\_{-\bullet}(E\_n,\mathbb{Q})={\rm Pois}^n$... | 7 | https://mathoverflow.net/users/7031 | 75201 | 45,680 |
https://mathoverflow.net/questions/75192 | 20 | Is there a ring with $\mathbb{Z}$ as its group of units?
More generally, does anyone know of a sufficient condition for a group to be the group of units for some ring?
| https://mathoverflow.net/users/12631 | Ring with Z as its group of units? | The example provided by Noam answers the first question. The second question is very old and, indeed, too general. See e.g. the notes to Chapter XVIII (page 324) of the book "László Fuchs: Pure and applied mathematics, Volume 2; Volume 36". In particular, rings with cyclic groups of units have been studied by RW Gilmer... | 18 | https://mathoverflow.net/users/14653 | 75209 | 45,684 |
https://mathoverflow.net/questions/75202 | 2 | If $f(X) \in \mathbb{F}\_2((T))[X]$ and $f(\mathbb{F}\_2[[T^2,T^3]]) \subseteq \mathbb{F}\_2[[T^2,T^3]]$, then does it follow that $f'(0) \in \frac{1}{T}\mathbb{F}\_2[[T]]$?
This might seem like an unmotivated question, but if it is true then I can show that the ring consisting of all such polynomials $f(X)$ is not ... | https://mathoverflow.net/users/17218 | Polynomials mapping the twisted cubic power series ring to itself | Counterexample: $f(X)=T^{-2}X+T^{-5}X^2+T^{-9}X^4+(T^{-2}+T^{-5}+T^{-9})X^8$
| 3 | https://mathoverflow.net/users/2035 | 75217 | 45,685 |
https://mathoverflow.net/questions/59814 | 21 | Let $X \subset \mathbb{P}^n$ be a complete intersection of two quadrics. It is classical that, if $X$ contains a line, then it is rational. The proof is very simple and basically it is given by taking the projection off the line.
It is also stated in a paper by Colliot-Thélène, Sansuc and Swinnerton-Dyer that if $X$ ... | https://mathoverflow.net/users/4096 | Rationality of intersection of quadrics | I've thought about this and discussed about this with someone else. A big part from what will follow is not my own contribution (in particular I hadn't thought about invoking Amer's theorem).
In fact it suffices to prove that any complete intersection of two quadrics containing a curve of odd degree must contain a li... | 8 | https://mathoverflow.net/users/1107 | 75229 | 45,691 |
https://mathoverflow.net/questions/75231 | 7 | Let $G$ an algebraic (reductive) group. $T$ a maximal torus, $B$ a Borel subgroup containing $T$, and $w\_0$ the longest element of the Weyl group.
I'm looking for a reference explaining why when you conjugate $B$ by $w\_0$, the result is the opposite Borel subgroup $B^-$.
Is there a proof involving roots of $G$ re... | https://mathoverflow.net/users/15404 | Longest element of a Weyl group | The proof depends on how you're setting things up. In my opinion the cleanest approach is the Lie algebraic one, and it goes as follows. Your Borel subalgebra $\mathfrak b$ determines a choice of simple roots $\Delta$ and consequently a choice of positive roots $\Phi^+$: $\mathfrak b = \mathfrak t \oplus \bigoplus\_{\a... | 8 | https://mathoverflow.net/users/430 | 75239 | 45,696 |
https://mathoverflow.net/questions/75222 | 3 | Suppose we have a connected graph $H$ with $m$ edges and $n$ vertices, and we add an edge to it. How can one bound the number of spanning trees of $H \cup e$ in terms of $H$?
The following formula seems very plausible: if $\kappa(H) = \binom{m'}{n-1}$, then $\kappa(H \cup e) \leq \binom{m' + 1}{n-1}$.
In particul... | https://mathoverflow.net/users/9896 | Spanning trees of $H \cup e$ in terms of $H$ | I get the following counterexample: Let $H$ be the graph on $12$ vertices, called $u\_1$, $u\_2$, ..., $u\_{6}$, $v\_1$, $v\_2$, ..., $v\_{6}$ with the following edges: $(u\_i, u\_j)$ and $(v\_i, v\_j)$ for all $1 \leq i < j \leq 6$, and $(u\_1, v\_1)$. Let the additional edge $e$ connect $(u\_2, v\_2)$.
The graph $H... | 3 | https://mathoverflow.net/users/297 | 75242 | 45,697 |
https://mathoverflow.net/questions/75245 | 3 | Hi
A circle graph is defined as the intersection graph of a set of chords of a circle.
I'm interested in any information which might help to enumerate connected circle graphs.
Thanks
Andy
| https://mathoverflow.net/users/44243 | Enumerating Connected Circle Graphs | You can find a table enumerating the connected circle graphs up to $n=12$ at this
Univ. Bergen link: [Database of Circle Graphs](http://www.ii.uib.no/~larsed/circle/). For example, there are 892,278,076 connected
circle graphs on 12 vertices. The paper that explains how the enumeration was
computed is "Interlace Polyno... | 4 | https://mathoverflow.net/users/6094 | 75247 | 45,701 |
https://mathoverflow.net/questions/4442 | 7 | I have come across a bit of folklore(?) which goes something like "given any finite sequence of numbers, there is more than one 'valid' way of continuing the sequence". For example see [here](http://www.guardian.co.uk/books/2005/feb/05/featuresreviews.guardianreview13). I would like to know if this is actually stated a... | https://mathoverflow.net/users/262 | Is there a theorem that says that there is always more than one way to "continue a finite sequence"? | Since you mentioned Wittgenstein's paradox, I thought someone should explain what that is. Warning: As you suspected, what follows is really philosophy and not mathematics. Nevertheless, it seems worth clarifying what people mean by Wittgenstein's paradox so that you don't go chasing wild geese.
In Wittgenstein's *Ph... | 29 | https://mathoverflow.net/users/3106 | 75260 | 45,710 |
https://mathoverflow.net/questions/75262 | 4 | Crossposted from [math.stackexchange](https://math.stackexchange.com/questions/64016/periodicity-of-the-nullspace-of-a-module) since I'm not getting any answer.
Let $W$ be the finite $\mathbb{Z}$-module obtained from $\mathbb{Z}\_q^n$ with addition componentwise where $\mathbb{Z}\_q$ is the integers mod $q$. Let $V$ ... | https://mathoverflow.net/users/17500 | Double orthogonal complement of a finite module | The answer is yes. The easiest way for me is to appeal to character theory. If $\zeta$ is
a complex $q$-th root of 1 then the map from $\mathbb{Z}\_q^n$ to $\mathbb{C}$ given by
$$
x \mapsto \zeta^{a^Tx},\qquad (a\in\mathbb{Z}\_q^n)
$$
is a character of the abelian group $W=\mathbb{Z}\_q^n$. The set of characters obta... | 4 | https://mathoverflow.net/users/1266 | 75268 | 45,715 |
https://mathoverflow.net/questions/75255 | 23 | There are many known results proving convergence of finite element method for elliptic problems under certain assumptions on underlying mesh [e.g., Braess,2007]. Which of these common assumptions are indeed necessary? Can anyone recommend any exact reference to an example of a sequence of triangulations on which the fi... | https://mathoverflow.net/users/14639 | Convergence of finite element method: counterexamples | The maximum and minimum angle conditions for meshes are needed to prove various bounds on the error of interpolation. In other words, the solution of the PDE is a secondary concern; what goes wrong is that one cannot control the interpolation error.
Of the two, the minimum angle condition is less restrictive. What o... | 18 | https://mathoverflow.net/users/14740 | 75270 | 45,717 |
https://mathoverflow.net/questions/75265 | 4 | First a word of warning: I am not a trained algebraic geometer, so it is possible (likely) that these questions are inappropriate for MO, if so: my apologies.
Said this: As far as I understand the tangent bundle $TX$ of a scheme $X$ is defined as the spectrum of the symmetric algebra of its sheaf of differentials, an... | https://mathoverflow.net/users/675 | Vector space structure on the tangent bundle of a scheme and relation to the tangent sheaf | The tangent bundle of a scheme is not in general a vector bundle. You need your sheaf of differentials to be locally free. This is automatic if the scheme is smooth over a field.
However, the additive group structure arises canonically from the symmetric algebra construction of the tangent scheme with no smoothness h... | 7 | https://mathoverflow.net/users/121 | 75276 | 45,720 |
https://mathoverflow.net/questions/75219 | 3 | Consider the inclusion $k\subset A$ of the field $k$ in the domain $A$ and the fraction field $K=Frac(A)$ of $A$.
Obviously if a family $(a\_i)\_{i\in I}$ of elements $a\_i \in A$ is algebraically independent over $k$ it will remain algebraically independent in $K$.
Consider however a family $(\alpha \_i) \_{i \i... | https://mathoverflow.net/users/450 | Is the transcendence degree of a domain over a subfield the same as that of the fraction field of that domain? | As already said in the comments, there is a general statement (analogous to extending bases in linear algebra) that for a field extension $K/k$ and subsets $A'\subseteq A\subseteq K$ such that $A'$ is algebraically independent and $K$ is algebraic over $k(A)$ there is a transcendence basis $A'\subseteq B\subseteq A$.
... | 4 | https://mathoverflow.net/users/2035 | 75285 | 45,723 |
https://mathoverflow.net/questions/75284 | 5 | This question may be utterly trivial, or not, but as someone with hardly any knowledge of algebraic geometry I thought there could be a chance I get lucky.
Let X be a rational surface obtained by n blows up of $\mathbb{P}^1(\mathbb{C})\times\mathbb{P}^1(\mathbb{C})$. I write $H\_x$ and $H\_y$ for the lines x=constant... | https://mathoverflow.net/users/17768 | (Anti)Canonical divisor of a blow up | If p:Y->X is the blowup of the surface X at a point x, and E is the exceptional divisor, then $K\_Y=p^\*K\_X+E$. Hence the formula for the blowup of P^2 at 9 points. In the $P^1 \times P^1$ case, the canonical divisor is $-2(H\_x+H\_y)$, and you can compute the canonical divisor on the blowup easily. You can find all t... | 6 | https://mathoverflow.net/users/1939 | 75289 | 45,726 |
https://mathoverflow.net/questions/75283 | 1 | Is it always possible to find an isogeny from a hyperelliptic curve of genus 4, to a 'normal' elliptic curve (genus 1), or a product of elliptic curves?
Are such isogenies easy to compute?
This question is motivated by a particular instance of a discrete logarithm problem. On hyperelliptic curves, computing group ... | https://mathoverflow.net/users/17769 | Isogenies from hyperelliptic to elliptic curves | This is only a partial answer.
Let $C$ be a hyperelliptic curve and $E$ an elliptic curve. Let $i\_1:C\to \mathbb{P}^1$ and $i\_2: E\to\mathbb{P}^1$ be the double cover maps. Let $Q\_1,\dots,Q\_{2g+2}$ be the critical values of $i\_1$ (i.e., the images of the Weierstrass points) and $P\_1,\dots,P\_4$ be the critical ... | 5 | https://mathoverflow.net/users/8621 | 75290 | 45,727 |
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