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https://mathoverflow.net/questions/40796 | 22 | Here is my precise question. Let $M, \omega$ be a symplectic manifold and let $H: M \to \mathbb{R}$ be any smooth function. The symplectic form gives rise to an isomorphism between the tangent bundle and cotangent bundle of $M$, and in this way we can associate to the 1-form $dH$ a vector field $X\_H$ which is characte... | https://mathoverflow.net/users/4362 | When is the time evolution of a Hamiltonian system described by the geodesic flow on a Riemannian manifold? | My reading of the question is this: we're given $H\in C^\infty(M)$ with $M$ symplectic, and we want to know whether there's a submanifold $L\subset M$, a Riemannian metric $g$ on $L$, and a symplectomorphism $T^\ast L \cong M$ under which $H$ pulls back to the norm-square function. And we want to know if $(L,g)$ is uni... | 10 | https://mathoverflow.net/users/2356 | 40871 | 26,126 |
https://mathoverflow.net/questions/40857 | 8 | If we have a Serre fibration $p: E \rightarrow B$ with fiber of homotopy type $S^{k-1}$, then we can create a fibration with contractible fiber by first taking the mapping cylinder $M\_p$ of $p$ to get a map $M\_p \rightarrow B$ (not necessarily a fibration) with contractible "fiber" and then applying the "space of pat... | https://mathoverflow.net/users/1874 | Is the mapping cylinder of a Serre fibration also a Serre fibration? | Waldhausen, Jahren and myself proved a
fiber gluing lemma for Serre fibrations,
in the context of simplicial sets, that
may be useful. In Propositions 2.7.10
and 2.7.12 of
"Spaces of PL manifolds and categories
of simple maps"
<http://folk.uio.no/rognes/papers/plmf.pdf>
we prove that given:
* a diagram of
simplic... | 9 | https://mathoverflow.net/users/9684 | 40873 | 26,128 |
https://mathoverflow.net/questions/39567 | 6 | Suppose we have an odd composite $N$ and want to find numbers $a\_1,\ldots,a\_k$ such that each $a\_i^2$, reduced mod $N$, is $b$-smooth. Of course we can use the quadratic sieve algorithm (minus the matrix step) to find such $a\_i$. With a factorization oracle, they could be found directly by factoring small squares—t... | https://mathoverflow.net/users/6043 | Speeding the quadratic sieve with an oracle | If we know the factorization of $N$ then we can take square roots of small numbers that are quadratic residues mod all primes dividing $N$. Knowing a partial factorization of size $M\sim N^\alpha$, square a number close to $N^{1/2−\alpha/2}$ mod $N/M$, find a root mod $M$, then use crt to find the root mod $N$. We have... | 2 | https://mathoverflow.net/users/2024 | 40881 | 26,133 |
https://mathoverflow.net/questions/40882 | 1 | We know that Morrey's inequality says $W^{1,p} \subset C^{0,\gamma}$ for $\gamma = 1 - n/p$ where $n$ is the dimension. However, in 1D, following the proof of Evans "Partial Differential Equations" (first edition, pp. 267), we have for any function $f$ in $W^{1,p}(0,1)$, and $x, y \in (0,1)$
$u(x) - u(y) = \int\_0^1 ... | https://mathoverflow.net/users/9754 | In 1D, is a $W^{1,p}$ function always Lipschitz, for $p\ge1$? | This does not look right. You have $u(x)-u(y) = \int\_y^x Du(s)(x-y)ds$ but the correct expression is clearly $\int\_y^x Du(s)ds$. I think a change of variables went wrong somewhere.
A counterexample: take $u(x) = x^{3/4}$ on $(0,1)$, with $p=2$. Then $Du(x) = \frac{3}{4}x^{-1/4} \in L^2(0,1)$, so $u \in W^{1,2}(0,1)... | 5 | https://mathoverflow.net/users/4832 | 40884 | 26,135 |
https://mathoverflow.net/questions/40828 | 2 | Let $X$ be a nonsingular complex projective variety. Suppose $X$ is embedded as a nonsingular real subvariety of complex projective space ${\mathbb{CP}}^n$.
When can we embed ${\mathbb{CP}}^n$ in some larger complex projective space ${\mathbb{CP}}^N$ such that the image of $X$ is now a nonsingular *complex* subvarie... | https://mathoverflow.net/users/nan | Obstruction for real subvariety to be embedded as complex subvariety | I am assuming from your comment that you are demanding that the embedding $X \to \mathbb{CP}^N$ preserves the complex structure that $X$ originally had (although it is still not completely clear). In this case, the real embedding $X \to \mathbb{CP}^n$ has to be a complex embedding. Otherwise, the actions of multiplicat... | 1 | https://mathoverflow.net/users/121 | 40909 | 26,150 |
https://mathoverflow.net/questions/40893 | 3 | By Tychonoff Theorem $\prod\_{\mathbb R} [0,1]$ is compact and since $\mathbb R=2^{\omega}$, if for $\alpha \in 2^{\omega}$, $x\_n(\alpha)=\alpha(n)$ then if we consider a subsequence $x\_{n\_0}, x\_{n\_1}, x\_{n\_2},...$ where $ n\_0 < n\_1< n\_2<... $ and where for $\alpha \in 2^{\omega}, \alpha(n\_0)=1, \alpha(n\_1)... | https://mathoverflow.net/users/3859 | A sequence with no convergent subsequence without choice | I think this question is fairly clear now, in its edited form. Assuming AC,
$[0,1]^{\mathbb R}$ is not sequentially compact, witnessed by the sequence specified in the question. What happens without AC? Is there a sequence with a limit point that has no convergent subsequence? I think this is an interesting question.
... | 4 | https://mathoverflow.net/users/7743 | 40915 | 26,155 |
https://mathoverflow.net/questions/40807 | 0 | For a surface $z(x,y)$, let $p = z\_x$ and $q = z\_y$. I have a pair of coupled semilinear PDEs in p and q.
PDE1: $p\_x - a p\_y = b q$.
PDE2: $q\_x - a q\_y = -b p$.
Note that $a$ and $b$ are functions of $(x,y)$. Is there a general way to solve such coupled systems of semilinear PDEs? (We may or may not ass... | https://mathoverflow.net/users/9728 | Coupled semilinear PDEs | First, disregard the constraint $p\_y=q\_x$. Consider the integral curves of the vector field $V:=(1,-a)$, namely the solutions of the ODE
$$\frac{dy}{dx}=-a(x,y).$$
They are parametrized by $x$. Your equations are ODEs along these curves,
$$\dot p=bq,\qquad\dot q=-bp.$$
Set $q+ip=\rho e^{i\theta}$. Then the ODEs becom... | 3 | https://mathoverflow.net/users/8799 | 40916 | 26,156 |
https://mathoverflow.net/questions/40895 | 7 | The question is about the specific case of reflection theorems (copied straight from Franz Lemmermeyer's "Class Groups of Dihedral Extensions"):
>
> Let $k^+ = \mathbb{Q}(\sqrt{m})$ with $m\in \mathbb{N}$, and put $k^- = \mathbb{Q}(\sqrt{-3m})$; then the 3-ranks $r\_3^+$ and $r\_3^-$ of $Cl(k^+)$ and $Cl(k^-)$ sati... | https://mathoverflow.net/users/2024 | Explicit map for Scholz reflection principle | One way to think the reflection principle, which is similar to what you are proposing in your question, is a relation between the index 3 subgroups of $Cl(k^{+})$, which I'll call $I\_{3}(m)$, and the subgroups of $Cl(k^{-})$ order 3 which I'll call $S\_{3}(-3m)$. It is not difficult to see that $$|S\_{3}(-3m)|=\frac{3... | 8 | https://mathoverflow.net/users/2089 | 40917 | 26,157 |
https://mathoverflow.net/questions/40922 | 0 | I want to maximize $||x-y||$ with $x$ and $y$ in $C$ where $C$ is the intersection of some discs. We assume the intersection is nonempty, and closed. I am thinking, how to formulate it as a semidefinite programmimg problem? Does anyone know how?
| https://mathoverflow.net/users/9557 | max length or size of a convex set | I hope that the word "disc" indicates that we work in 2-dimensional Euclidean space.
Then an intersection of discs would be something like a polygon, just that the sides are not straight lines but arcs of a circle. It should be possible to save the intersection in some data structure. And then one can compute the maxi... | 1 | https://mathoverflow.net/users/3969 | 40925 | 26,159 |
https://mathoverflow.net/questions/40920 | 99 | The title of the question is also the title of a talk by Vladimir Voevodsky, available [here](http://video.ias.edu/voevodsky-80th).
Had this kind of opinion been expressed before?
**EDIT.** Thanks to all answerers, commentators, voters, and viewers! --- Here are three more links:
[Question arising from Voevodsk... | https://mathoverflow.net/users/461 | What if Current Foundations of Mathematics are Inconsistent? | The talk in question was given as part of a celebration (this past weekend) of the 80th anniversary of the founding of the Institute for Advanced Study in Princeton. As you might guess there were quite a few very well-known mathematicians and physicists in the audience. (To name just a few, Jack Milnor, Jean Bourgain, ... | 81 | https://mathoverflow.net/users/7311 | 40927 | 26,160 |
https://mathoverflow.net/questions/40918 | 3 | I heard it said that the cohomology rings of some Lie groups and Grassmannians can be read from the Dynkin graph. Can someone give me any reference?
| https://mathoverflow.net/users/8152 | the relation between cohomology and Dynkin graphs of lie groups | Your question involves a complex (semi)simple Lie group, I guess, and its Dynkin diagram. The topology of Lie groups and their homogeneous spaces $G/P$ (such as Grassmannians) is an old and rich subject (E. Cartan, Stiefel, Samelson, Bott, Kostant, Chevalley, Borel, ...) which can be approached from a number of viewpoi... | 6 | https://mathoverflow.net/users/4231 | 40929 | 26,162 |
https://mathoverflow.net/questions/40934 | 1 | Hello,
this may be a trivial question, but I am not very familiar with the topic.
Let (M,g) be a Riemannian Manifold. (In fact, we don't need the metric here.)
What exactly does it take for two k-submanifolds $S$ and $S'$ to lie in the same homology class? And why does that imply $~ \int\_S ~ \omega = \int\_{S'} ~ ... | https://mathoverflow.net/users/9762 | When do submanifolds lie in the same homology class? | By triangulating each manifold $S$ and $S'$, you can regard them as sums of singular simplices. (A singular simplex is a continuous map $f\colon \Delta^n\to M$.) So the very simple answer is that $S$ and $S'$ are the same homology class if and only if they are the boundary of some sum of singular simplices. This can ha... | 5 | https://mathoverflow.net/users/9417 | 40937 | 26,164 |
https://mathoverflow.net/questions/40944 | 5 | Let $G\_0$ be a finitely generated group, and suppose there are groups $G\_i$ and $K\_i$ as in the following short exact sequences
$1\to K\_i\to G\_{i+1}\to G\_i\to 1$
with $K\_i$ free and nonabelian (you may assume finitely generated), and $G\_i$ commutative transitive. (If $a$ is nontrivial and $b$ and $c$ both c... | https://mathoverflow.net/users/2225 | Ranks of iterated extensions of a group by free groups. | I assume that you consider the infinite cyclic group to be free. Then take the free nilpotent group $G\_n$ of class $c\gg 1$ with 2 generators. It has an infinite cyclic central subgroup $K\_n$, the factor-group $G\_{n-1}=G\_n/K\_n$ is nilpotent and torsion-free, it has an infinite cyclic subgroup $K\_{n-1}$, and so on... | 11 | https://mathoverflow.net/users/nan | 40946 | 26,169 |
https://mathoverflow.net/questions/23981 | 1 | This question is related to [On a positivity of a matrix with trace entries.](https://mathoverflow.net/questions/16983/on-a-positivity-of-a-matrix-with-trace-entries)
Let $A\_1, \cdots, A\_m$ be strictly contractive $n\times n$ complex matrices .Is it true that
$$\left(\begin{array}{cccc}Tr\{(I-A\_1^\*A\_1)^{-1}\}&T... | https://mathoverflow.net/users/3818 | A matrix with trace entries. | I guess, in the meanwhile you might have already proved that this matrix is **not** positive-semidefinite. I ran a *brute force* experiment, using $2 \times 2$ symmetric, real matrices, which shows that the above conjecture is not true.
I tried different values of $m$, and indeed, the smaller the $m$, the lower the (... | 4 | https://mathoverflow.net/users/8430 | 40956 | 26,177 |
https://mathoverflow.net/questions/40960 | 9 | Let $X$ be an integral scheme of finite type over a field. Then there is a surjective finite map $\tilde{X} \to X$ from the normalization $\tilde{X}$ of $X$.
Is this going to be bijective?
In the simplest non-normal case, namely the spectrum of $k[x^2, x^3] = k[t,u]/(t^3 -u^2)$, the map is bijective, because the c... | https://mathoverflow.net/users/344 | Is the normalization map bijective? | Your example is not really the simplest case, in that a cusp is not the simplest possible curve singularity. Rather, the simplest curve singularity is a node, e.g. as in
$y^2 = x^3 - x^2$. There are then two branches passing through the singularity, and
the normalization map separates them. (Exercise: The normalizatio... | 20 | https://mathoverflow.net/users/2874 | 40963 | 26,181 |
https://mathoverflow.net/questions/40945 | 78 | This question is inspired by [an answer of Tim Porter](https://mathoverflow.net/questions/40005/generalizing-a-problem-to-make-it-easier/40010#40010).
Ronnie Brown pioneered a framework for homotopy theory in which one may consider multiple basepoints. These ideas are accessibly presented in his book Topology and... | https://mathoverflow.net/users/2051 | Compelling evidence that two basepoints are better than one | Here is an interesting example where groupoids are useful. The mapping class group $\Gamma\_{g,n}$ is the group of isotopy classes of orientation preserving diffeomorphisms of a surface of genus $g$ with $n$ distinct marked points (labelled 1 through n). The classifying space $B\Gamma\_{g,n}$ is rational homology equiv... | 36 | https://mathoverflow.net/users/4910 | 40965 | 26,183 |
https://mathoverflow.net/questions/40887 | 9 | This question - so far as I know - has no broader mathematical significance, but it occurred to me a while ago and I haven't been able to make any headway.
Any knot diagram $D$ splits the plane into a finite number of pieces. For example, in a standard diagram for a trefoil (e.g., <http://en.wikipedia.org/wiki/File:T... | https://mathoverflow.net/users/8133 | Polygons arising from knot diagrams | Colin Adams, Reiko Shinjo and Kokoro Tanaka have a paper (<http://arxiv.org/abs/0812.2558>) that shows that for any knot you can find a diagram which has only regions with 2, 4 and 5 sides.
| 10 | https://mathoverflow.net/users/9717 | 40966 | 26,184 |
https://mathoverflow.net/questions/40938 | 13 | Well, the title is self-explaining, I guess - I am referring to the complete Segal space model structure of Theorem 7.2 in Rezk's article "[A model for the homotopy theory of homotopy theories](http://www.math.uiuc.edu/~rezk/rezk-ho-models-final-changes.pdf)".
Has anyone tried to find out at all, whether this model cat... | https://mathoverflow.net/users/733 | Is the model category of Complete Segal Spaces right proper? | The model structure for complete Segal spaces is not right proper. To see this, one can first prove that the model structure for quasi-categories is not right proper: for instance, the map $\delta^1\_2:\Delta\_1\to\Delta\_2$ is a fibration between fibrant objects in the Joyal model category (because it is the nerve of ... | 19 | https://mathoverflow.net/users/1017 | 40967 | 26,185 |
https://mathoverflow.net/questions/40930 | 11 | Suppose $\pi$ and $\rho$ are cuspidal automorphic representations on $GL(n)$ and $GL(m)$ respectively. Then the L-function $L(s,\pi \times \rho)$ has a pole iff and $m=n$ and $\pi$ is isomorphic to the contragradient of $\rho$ by some twist. Does anyone know some reference containing the proof of this fact?
I checked... | https://mathoverflow.net/users/1832 | reference help needed on a fact about poles of L-functions | It seems to be in the Cogdell-PS paper "Remarks on Rankin-Selberg Convolutions" in the Shalika volume, though I haven't read through it myself.
| 5 | https://mathoverflow.net/users/6518 | 40980 | 26,193 |
https://mathoverflow.net/questions/40962 | 6 | Suppose that I have $n$ unknown variables $x\_1,\ldots,x\_n$. I wish to compute their sum:
$$Sum(x) = \sum\_{i=1}^nx\_i$$
However, the only access to these variables is through products: that is, for any subset $S \subset [n]$ I may compute:
$$P(S) = \prod\_{i \in S}x\_i$$
That is, I wish to find some number of subse... | https://mathoverflow.net/users/9769 | How many products specify a sum? | Here is an extreme case: I will tell you that either every variable is zero or possibly a single one of them is 1. So your task is to decide if the sum is 0 or it is 1. Any product of more than one term gives no information at all. To rule out all zero you need to check each variable.
| 30 | https://mathoverflow.net/users/8008 | 40986 | 26,199 |
https://mathoverflow.net/questions/36988 | 2 | I am not familiar with much semigroup theory, but this question came up in my research and I've been unable to find much on it.
| https://mathoverflow.net/users/8434 | what conditions can one place on a finitely generated periodic semigroup that will ensure the semigroup is finite? | This question has been studied extensively more than 20 years ago. Most results are about semigroups satisfying identities. the first such result is by Morse and Hadlund: there exists an infinite 3-generated (infinite 2-generated) semigroup satisfying $x^2=0$ (resp. $x^3=0$). The latest results are in Sapir, M. V. Prob... | 5 | https://mathoverflow.net/users/nan | 40990 | 26,201 |
https://mathoverflow.net/questions/40997 | 16 | Given a large number $q$ (say, a prime) and a number $a$ between 2 and $q-1$ what is the fastest algorithm known for computing the inverse of $a$ in the group of residue classes modulo $q$?
| https://mathoverflow.net/users/2344 | Fast computation of multiplicative inverse modulo q | The fast Euclidean algorithm runs in time $O(M(n)\log n)$, where $n=\log q,$ and $M(n)$ is the time to multiply two $n$-bit integers. This yields a bit-complexity of
$$
O(n\log^2 n\log\log n)
$$
for the time to compute the inverse of an integer modulo $q$, using the standard Schonhage-Strassen algorithm for multiplying... | 30 | https://mathoverflow.net/users/4757 | 41004 | 26,210 |
https://mathoverflow.net/questions/40989 | 2 | I am reading "Fundamentals of Diophantine Geometry" by Serge Lang.
Let V be a (absolute) variety, W be a simple subvariety of V. Then we know that the local ring of W is a discrete valuation ring, hence induces a discrete valuation v. But why is v well-behaved (in the sense of Lang's book)?
Can anyone help me find... | https://mathoverflow.net/users/4621 | Why is an absolute value generated by a simple subvariety of a variety V well-behaved? | According to Lang the valuation $v$ of the field $K$ is well-behaved, if for every finite extension $E/K$ the equation
$
[E:K] =\sum\limits\_{w|v} [E\_w:K\_v]
$
holds, where the summation runs over all extension $w$ of $v$ to $E$, and $K\_v$, $E\_w$ are the completions of the fields $K$, $E$ with respect to the val... | 2 | https://mathoverflow.net/users/3556 | 41007 | 26,211 |
https://mathoverflow.net/questions/41006 | 4 | Let $G$ be a transitive permutation group on a set of size $n$, and suppose $Z(G)=1$ (for instance $G$ is a direct power of a non-abelian simple group). What can we say about the centraliser $K$ of $G$ in $Sym(n)$? I'm interested firstly if there are any restrictions on $K$ independent of degree, and secondly on what r... | https://mathoverflow.net/users/4053 | Centralisers of transitive permutation groups | Of course, there is the classical result that $C\_{Sym(n)}(G)$ is a semi-regular subgroup of $Sym(n)$ of cardinality $|Fix(G\_{0})|$, where $G\_{0}$ is the stabilizer of a point and $Fix(G\_{0})$ is the set of points fixed by $G\_{0}$.
| 6 | https://mathoverflow.net/users/4706 | 41012 | 26,214 |
https://mathoverflow.net/questions/40939 | 8 | Let $X$ be a regular integral noetherian scheme of dimension 2 and let $D$ be a simple normal crossings divisor in $X$.
EDIT: Let $U = X-D$.
Consider a finite etale morphism $V\longrightarrow U$ with $V$ connected. Let $\pi:Y\longrightarrow X$ be the normalization of $X$ in the function field of $V$. So $Y$ is a $... | https://mathoverflow.net/users/4333 | Is this finite surjective flat morphism of 2 dimensional schemes a local complete intersection | Let $k$ be a field of characteristic prime to $n$ and $\zeta\in k$ a primitive $n$-th root of unity. Let $a\in \mathbb{Z}$ and let $Y$ be the quotient of $\mathop{\rm{Spec}}k[u,v]$ by the automorphism $(u,v)\mapsto (\zeta u,\zeta^a v)$.
Then $Y$ is a tame cyclic cover of $X=\mathop{\rm{Spec}}k[u^n,v^n]$ etale away f... | 7 | https://mathoverflow.net/users/5480 | 41014 | 26,216 |
https://mathoverflow.net/questions/41024 | 7 | Given any group $G$, is there an amenable group $A(G)$ together with a morphism $G\rightarrow A(G)$, such that every other morphism $G\rightarrow A'$ to another amenable group $G'$ uniquely factorizes through $A'$?
That is the question. My approach would be to consider the set of normal subgroups with amenable quotie... | https://mathoverflow.net/users/3969 | Is there an amenabilization of groups ? | Infinite cartesian product of amenable groups is not necessarily amenable. For example, the free group is a subgroup of the infinite (cartesian) product of finite groups. In general, there are finitely generated residually finite just infinite groups (for example, lattices in semi-simple Lie groups of higher ranks, say... | 12 | https://mathoverflow.net/users/nan | 41029 | 26,223 |
https://mathoverflow.net/questions/41005 | 7 | Given a compact Riemannian manifold (with a fixed metric) and a Morse function on it (also fixed). Is there a bound (depending on the metric and the Morse function) on the length of the Morse trajectories? (You can assume the Morse-Smale condition if helpful.)
EDIT (In response to Dick's answer): The Morse function a... | https://mathoverflow.net/users/3509 | Is there a bound on the length of the longest Morse trajectory? | As explained by Bill Thurston, on a given Riemann manifold $(M,g)$ you may really have Morse functions with gradient lines of any length. However, as now you are asking for a bound in terms of the Morse function $f$, the following argument shows why lengths of all Morse trajectories (MT) are bounded, and how to get a q... | 15 | https://mathoverflow.net/users/6101 | 41038 | 26,229 |
https://mathoverflow.net/questions/41033 | 4 | Let $\mathfrak{g}$ = $\mathfrak{gl}\_{\infty}$.
To each positive integer $k$ one can associate the level $k$ Fock space $\mathcal{F}\_{k}$.
For a dominant weight $\lambda$ of level $k$, one can define an action of $\mathfrak{g}$ on $\mathcal{F}\_{k}$
so that it has a highest weight vector of weight $\lambda$ wh... | https://mathoverflow.net/users/316 | Realizing higher level Fock spaces | For $\mathfrak{gl}\_{\infty}$ the answer is easy, you realize the Fock space as a direct limit of polynomial representations of finite $\mathfrak{gl}\_n$ modules. You can read about the construction [here](http://www.ams.org/mathscinet/search/publdoc.html?pg1=IID&s1=842325&vfpref=html&r=3&mx-pid=2473335). I worked on t... | 2 | https://mathoverflow.net/users/4366 | 41053 | 26,237 |
https://mathoverflow.net/questions/41042 | 6 | For simplicity fix a base field $k$ of characteristic zero, and consider smooth affine algebraic $k$-groups. (It is understood that unipotent groups in positive characteristic are more complicated, as one might have interesting non-smooth ones.)
Question 1: forms of unipotent groups
If $k$ is algebraically closed, ... | https://mathoverflow.net/users/9776 | unipotent groups, their forms and representations | As Torsten points out, unipotent groups correspond naturally to nilpotent Lie algebras in characteristic 0. This is dealt with nicely on the scheme level, for example, in IV.2.4 of Demazure-Gabriel *Groupes algebriques*. They also treat in Chapter IV some questions about prime characteristic, which get quite tricky out... | 4 | https://mathoverflow.net/users/4231 | 41071 | 26,245 |
https://mathoverflow.net/questions/41043 | 23 | Let me take this question again from the top.
I would like to know what a special parahoric subgroup is.
I think this is a "real" question, though not an especially good one -- it indicates my complete lack of expertise in this area, but it is certainly a question of interest to research mathematicians (who else?).... | https://mathoverflow.net/users/1149 | What is a special parahoric subgroup? | I'd recommend first that you and your friend spend more time with Tits :), "Reductive groups over local fields", from the Corvallis volume (free online, last time I checked). Undoubtably there are other references, like papers of Prasad-Raghunathan mentioned by Greg Kuperberg, and any paper that treats Bruhat-Tits theo... | 20 | https://mathoverflow.net/users/3545 | 41073 | 26,246 |
https://mathoverflow.net/questions/41039 | 5 | Let $B(x)$ be infinitely differentiable with respect to $x$.
Drop the use of parentheses on $B$ to delimit the argument $x$
and use them instead to hold the order of the derivative with respect to $x$.
i.e. $B(0) = B$,
$B(1) = dB/dx$, etc.
Let parentheses on $x$ hold the order of the derivative of $x$ with respect to... | https://mathoverflow.net/users/8789 | Formula for n-th iteration of dx/dt=B(x) | These expansions can be described in terms of [rooted trees](https://en.wikipedia.org/wiki/Tree_(graph_theory)). The first few coefficients are easy to derive by hand, but rooted trees provide you a way of generating the coefficients to arbitrary order, see [here](http://en.wikipedia.org/wiki/Butcher_group). You start ... | 11 | https://mathoverflow.net/users/3909 | 41082 | 26,253 |
https://mathoverflow.net/questions/41054 | 2 | Firstly, I have to say that I don't understand Padé approximation well.
But I discovered that, it is more precise than Taylor series.
I have to create approximation for these functions: Log(x) and Tanh. And I have to create iterative algorithms (I must compute result with variable precision).
So my questions a... | https://mathoverflow.net/users/9787 | Padé approximation - usability in iterative algorithms | A qualitative reason for using rational approximations (e.g. Padé) instead of polynomial ones (e.g. Taylor) is that rational approximations can exhibit behavior (e.g. poles and asymptotes) that polynomials are hard-pressed to emulate; they thus tend to be slightly more accurate (there are always exceptions, though).
... | 4 | https://mathoverflow.net/users/7934 | 41085 | 26,255 |
https://mathoverflow.net/questions/41027 | 11 | Given any space $X$ of Lebesgue dimension at most $n$. Suppose a group $G$ acts on $X$ continuously. Can the dimension of the quotient $G\backslash X$ exceed the dimension of $X$?
I know examples, where quotient maps increase the dimension. But I don't know an example, where this quotient is given by dividing out a g... | https://mathoverflow.net/users/3969 | Can dividing out a group action can increase the Lebesgue dimension ? | There are examples of dimension-raising orbit maps arising from actions of p-adic groups. Quoting from the MathSciNet review of the following paper: Raymond, Frank; Williams, R. F. Examples of $p$-adic transformation groups. Ann. of Math. (2) 78 1963 92--106, review by P. Conner: "The authors of this paper show that if... | 9 | https://mathoverflow.net/users/1822 | 41089 | 26,258 |
https://mathoverflow.net/questions/41103 | 9 | I have the following situation: let $m,n$ be integers such that $m|n$ and let $\zeta\_m$, $\zeta\_n$ denote primitive $m$ and $n$th roots of unity. Then we have the inclusion of fields
$$\mathbb{Q}\subset \mathbb{Q}(\zeta\_m) \subset \mathbb{Q}(\zeta\_n)$$
Now suppose we also have primes (where $(p,n)=1$)
$$(p)\subse... | https://mathoverflow.net/users/9769 | Congruences mod primes in Galois extensions | Sure. $a\equiv b\pmod{\mathfrak{P}}$ just means $a-b\in\mathfrak{P}$. Taking norms to any subfield $K$ of $\mathbb{Q}(\zeta\_n)$ (e.g., $\mathbb{Q}$ or $\mathbb{Q}(\zeta\_m)$) gives you $N\_{\mathbb{Q}(\zeta\_n)/K}(a-b)\in N\_{\mathbb{Q(\zeta\_n)}/K}\mathfrak{P}.$
For $K=\mathbb{Q}$, the latter norm is just $p^f$ wh... | 5 | https://mathoverflow.net/users/35575 | 41105 | 26,268 |
https://mathoverflow.net/questions/41032 | 5 | Given a once-punctured surface $F$ and an orientation preserving self homeomorphism $\phi$, let $M\_\phi$ be the bundle over $S^1$ with fiber $F$ and monodromy $\phi$.
In Sakuma's survey article [The topology, geometry and algebra of unknotting tunnels](http://ams.org/mathscinet/search/publdoc.html?extend=1&pg1=IID&s... | https://mathoverflow.net/users/4325 | Unknotting tunnels in surface bundles | [Scharlemann and Thompson](http://www.ams.org/mathscinet-getitem?mr=1990938) proved that one can isotope/slide a tunnel
to be disjoint from a minimal genus Seifert surface (which was what I had in mind
in my comment above). Their argument does not depend on it being a knot complement.
I also found a [citation for a pap... | 1 | https://mathoverflow.net/users/1345 | 41107 | 26,270 |
https://mathoverflow.net/questions/41117 | 2 | I have been reading about Riemann Zeta function $\zeta(s)$ and have been thinking about it for some time. I did some calculations, and reached a conclusion where $\Re(\rho) \le \log\_2(3) - 1$ as $\Im(\rho) \to \infty$ where $\rho$'s are the roots of Riemann Zeta function in the critical strip. Anyways, I know its not ... | https://mathoverflow.net/users/2865 | Upper bound for real part of Riemann Zeta function zeros | There is no known non-trivial (less than 1) bound for real parts of Zeta zeros (I guess, it is even called "weak Riemann conjecture" to find such a bound). So, your result is very-very interesting, maybe the most interesting result in mathematics for many years.
| 9 | https://mathoverflow.net/users/4312 | 41120 | 26,279 |
https://mathoverflow.net/questions/41138 | 2 | Let $A$ be a complete regular local noetherian ring of dimension $d>1$ and $B$ an $A$-algebra, finite and free as $A$-module. Assume moreover that there exists an open subset $U$ of $\textrm{Spec}\ A$ of primes of height $d-1$ such that there is exactly one prime in $\textrm{Spec}\ B$ over a prime of $U$. Is it then tr... | https://mathoverflow.net/users/2284 | A weaker form of Zariski's connectedness principle | Yes. Assume not. Since $B$ is finite flat over $A$, each irreducible component of $\mathrm{Spec}(B)$ is pure $d$-dimensional, finite and *surjective* over $\mathrm{Spec}(A)$. If $Y\neq Z$ are two of them, then the image of $Y\cap Z$ in $\mathrm{Spec}(A)$ is a proper closed subset, hence does not contain $U$. Every poin... | 3 | https://mathoverflow.net/users/7666 | 41140 | 26,287 |
https://mathoverflow.net/questions/41141 | 61 | I want to cite a paper which is on arxiv.org but is not published or reviewed anywhere, and no publication or review seems to be in the pipeline. Would citing this arxiv.org paper be bad? Should I wait for a paper to be peer reviewed before I cite it?
Added:
I don't actually know whether a 'real' publication is in th... | https://mathoverflow.net/users/9501 | Should I not cite an arxiv.org paper which otherwise seems to be unpublished? | [It is] Not really [bad to cite an arXiv paper]\*. If the paper on arXiv provides the result you want, you are free to cite it. Before the arXiv, citing "private communication" or "pre-print" is not unheard of. On the other hand, since it hasn't been peer reviewed, you probably should double check and make sure you und... | 70 | https://mathoverflow.net/users/3948 | 41146 | 26,290 |
https://mathoverflow.net/questions/41136 | 1 | Given $S$ a $K3$ surface and $M$ the moduli space of simple sheaves of rank 2 and fixed Chern classes on $S$, under which conditions does a universal family on $S\times M$ exist?
| https://mathoverflow.net/users/33841 | Universal family over the moduli space of simple sheaves on a K3 surface | Let us introduce the Mukai pairing on the cohomology $H^\*(X,\mathbb{Z})=H^0\oplus H^2\oplus H^4$ of a K3 surface X, as
$<(r,l,s).(r',l',s')>=l.l'-rs'-r's$
where $l.l'$ is the intersection pairing on $H^2$.
The Mukai vector of a sheaf $E$ is defined to be
$v(E)=ch(E).\sqrt{Td(X)}=(rk(E),c\_1(E),rk(E)+c\_1^2(E)/2... | 2 | https://mathoverflow.net/users/5714 | 41147 | 26,291 |
https://mathoverflow.net/questions/41137 | 2 | Hello,
Is there any software for calculating Green polynomials (of type A)? Or, at least, where can I find tables of Green polynomials? Also, I would be interested in some formulas for Green polynomials in simple cases.
| https://mathoverflow.net/users/6772 | Green polynomials | I am not really an expert but it looks like the Morita paper [Decomposition of Green polynomials of type A and Springer modules for hooks and rectangles](http://dx.doi.org/10.1016/j.aim.2006.07.007) could be of some help here. In particular, at page 483 one finds an explicit formula for these polynomials for one of the... | 3 | https://mathoverflow.net/users/2149 | 41148 | 26,292 |
https://mathoverflow.net/questions/41128 | 5 | Given a smooth function $f:M\rightarrow \mathbb R$ on a manifold, its local homology at a critical point $x$ is the group
$$ C\_\star(x) := H\_\star ( M\_{ < c} \cup \{ x \} , M\_{ < c} ) ,$$
where $H\_\star$ denotes singular homology (with any coefficient group), $c=f(x)$, and $M\_{ < c}$ is the space of those points... | https://mathoverflow.net/users/9736 | Local homology of degenerate critical points | $(x^2+y^2)z^2-c(x^2+y^2+z^2)^2$ for small positive $c$.
More generally $f-cr^{2d}$ where $f\ge 0$ is a homogeneous polynomial function of degree $2d$ in $n$ variables. The local homology at the origin should be essentially the homology of the set of points in $S^{n-1}$ where $f=0$.
| 5 | https://mathoverflow.net/users/6666 | 41156 | 26,298 |
https://mathoverflow.net/questions/41158 | 19 | Let $f:X\to Y$ be a morphism of complex analytic spaces. Assume $f$ is flat (or, more generally, that there is a coherent sheaf on $X$ with support $X$ which is $f$-flat). Is $f$ an open map?
The rigid-analytic analogue is true (via Raynaud's formal models): see Corollary 7.2 in S. Bosch, Pure Appl. Math. Q., 5(4) :1... | https://mathoverflow.net/users/7666 | Are flat morphisms of analytic spaces open? | The answer is **yes**.
In fact, there is the following result, see
Banica-Stanasila, *Algebraic methods in the global theory of Complex Spaces*, Theorem 2.12 p. 180.
>
> **Theorem.**
> Let $f \colon X \to Y$ be a morphism of complex spaces and let $\mathscr{F}$ be a coherent analytic sheaf on $X$, which is flat... | 19 | https://mathoverflow.net/users/7460 | 41163 | 26,303 |
https://mathoverflow.net/questions/41122 | -1 | The Frobenius, or Hilbert-Schmidt, norm of an $n$ by $n$ matrix $A$ is defined as $\|A\|\_2 = \sqrt{\sum\_{i,j=1}^n |A\_{ij}|^2}$. The absolute value of $A$ is the unique positive matrix $|A|$ satisfying $|A|^2 = A^\* A$. Are there any known relations between $\| |A| \|\_2$ and $\|A\|\_2$?
| https://mathoverflow.net/users/9807 | Absolute values and Frobenius norm | It holds that
|||A|||\_2 =||A||\_2
Proof: Let the singular value decomposition of A be given by $A=U\Sigma V^H$. Since the HS-norm is invariant under unitary transformations, $||A||\_2= ||\Sigma||\_2$ holds. As for $|A|$, we obtain $|A|= |A^HA|=|V\Sigma U^HU\Sigma V^H|=|V\Sigma^2 V^H|= V\Sigma V^H$. Again, unitary ... | 0 | https://mathoverflow.net/users/9810 | 41169 | 26,308 |
https://mathoverflow.net/questions/41167 | 10 | Suppose we are working in a category of schemes over a scheme $S$. The scheme $S$ itself is geometrically a ``point''. Let $G$ be a group scheme that acts on a scheme $X$. The quotient stack $[X/G]$ looks in a way as a factor of $X$ by a free action of $G$, i.e. if the action is indeed free and there is a scheme $Y$ su... | https://mathoverflow.net/users/2234 | Classifying stacks and homotopy type of a point | This is all best explaied by working with groupoids in schemes. By this, I mean, groupoid objects in the category of schemes over $S$. A groupoid object $\mathcal G$ consists of two schemes $\mathcal G\_0$ (the objects) and $\mathcal G\_1$ (the morphisms) and a whole bunch of maps between them:
• a map $s:\mathcal G... | 17 | https://mathoverflow.net/users/5690 | 41172 | 26,310 |
https://mathoverflow.net/questions/41177 | 20 | I am stuck on solving an apparently simple ODE. I have checked numerous texts, references, software packages and colleagues before posting this...
$$y(t)^n+a(t)\frac{dy(t)}{dt}=ba(t)$$
If the RHS had a $y$ term it would simply be Bernoulli's equation. Does the $n$ term prevent a solution?
| https://mathoverflow.net/users/3754 | A nonlinear first order ordinary differential equation: $y(t)^n+a(t)\frac{dy(t)}{dt}=ba(t)$ | First, rewrite your equation as
$$
\frac{dy(t)}{dt}=b+f(t)(y(t))^n, \qquad \qquad \qquad \qquad \qquad (\*)
$$
where $f(t)=-1/a(t)$.
This is a special case of the so-called [Chini equation](http://www.maplesoft.com/support/help/Maple/view.aspx?path=odeadvisor/Chini)
(Equation 1.55 in the Kamke's book mentioned below... | 31 | https://mathoverflow.net/users/2149 | 41181 | 26,315 |
https://mathoverflow.net/questions/41183 | 11 | Is it true that any manifold homotopy equivalent to a k-dimensional CW-complex admits a proper Morse function with critical points all of index <= k? I believe this is not true, so I would like to see a counterexample.
| https://mathoverflow.net/users/9800 | index of morse functions and homotopical dimension | Take a contractible $3$-manifold which is not homeomorphic to $\mathbb R^3$ -- like the Whitehead manifold.
If such a Morse function existed on the Whitehead manifold, it would be a Morse function with only one critical point, the minimum, and therefore the Whitehead manifold would be an open 3-ball. The proof of th... | 10 | https://mathoverflow.net/users/1465 | 41184 | 26,316 |
https://mathoverflow.net/questions/41180 | 4 | What's the easiest (by which I mean uses the least fancy machinery) proof of the direct summand conjecture in dimension 2?
Recall that the direct summand conjecture says that:
**Conjecture** (Hochster): If $R$ is a regular ring and $S$ is a module finite integral extension, then $R \to S$ splits as a map of $R$-mod... | https://mathoverflow.net/users/3521 | Reference request, direct summand conjecture in dimension 2 | You may assume that $R,S$ are complete and $S$ is a domain. Now take the integral closure $T$ of $S$, which is $S$-finite. Since we are in dimension $2$, $T$ is maximal Cohen-Macaulay module over $R$, so $T$ is $R$-free. Thus the composition map $R\to T$ splits (as it takes $1$ to $1$) whence the map $R\to S$ splits.
... | 6 | https://mathoverflow.net/users/2083 | 41185 | 26,317 |
https://mathoverflow.net/questions/41150 | 31 | This question is prompted by a recent MO question on explicit computations of Weyl group invariants for certain exceptional simple Lie algebras:
[37602](https://mathoverflow.net/questions/37602/polynomial-invariants-of-the-exceptional-weyl-groups). Like some others who started graduate study in the 1960s with almost no... | https://mathoverflow.net/users/4231 | What was Casimir's precise role in describing the center of the universal enveloping algebra of a semisimple Lie algebra? | At the first glance it appears that he more or less just gave the first nontrivial example(s) of what was later called the Casimir operators.
His [obituary](http://dx.doi.org/10.1088/0143-0807/22/4/320) says:
*On 1 May 1931 he wrote a letter from The Hague to the famous Gottingen mathematician
Hermann Weyl and anno... | 16 | https://mathoverflow.net/users/2149 | 41193 | 26,322 |
https://mathoverflow.net/questions/41195 | 1 | More accurately, let $\displaystyle A=\sum\_{i=0}^{\infty}A\_i$ be a finitely generated graded algebra over say $\mathbb{Q}$ but $\dim A\_i=\infty$ for each $i.$ Is it possible?
| https://mathoverflow.net/users/9645 | Need an example of finitely generated graded algebra such that each its graded subspace has infinite dimension. | $\mathbb Q[x,y]$ with $x$ in degree 0, and $y$ in degree 1.
If you want your generators to be in positive degrees, then what you're asking for is impossible.
| 7 | https://mathoverflow.net/users/5690 | 41196 | 26,323 |
https://mathoverflow.net/questions/41118 | 12 | Does the usual development of category theory (within Goedel-Bernays set theory, for example) require the axiom of replacement? I would have asserted that this was obviously true, but it seems to be common wisdom that the axiom of replacement is an exotic axiom not used outside of axiomatic set theory. Also, sadly the ... | https://mathoverflow.net/users/3711 | Axiom of Replacement in Category Theory | There’s one issue underlying a lot of the discrepancies between people’s answers, I think:
>
> How are we defining “$f$ is a function $s \to V$”, where $s$ is a set and $V$ is a (possibly proper) class?
>
>
>
(hence also, how we define subsequent things like “a small-category-indexed diagram of sets”) There ar... | 9 | https://mathoverflow.net/users/2273 | 41200 | 26,326 |
https://mathoverflow.net/questions/41202 | 3 | I'm looking for information about how representations of $S\_n$ decompose under restriction.
I know about the branching rule: That is, in characteristic 0, irreducible modules $L(\lambda)$ for $S\_n$ decompose into a direct sum of irreducible modules of $S\_{n-1}$ and there is a nice combinatorial rule for determini... | https://mathoverflow.net/users/9821 | Decompositions for Symmetric Groups | To elaborate on Alexander Woo's comment, the multiplicity of $L(\nu) \otimes L(\mu)$ in $L(\lambda)$ restricted is the Littlewood-Richardson coefficient $c^\lambda\_{\nu, \mu}$. See <http://en.wikipedia.org/wiki/Littlewood-Richardson_coefficient> for the statement. For a proof (i.e., why this is related to symmetric fu... | 4 | https://mathoverflow.net/users/321 | 41209 | 26,332 |
https://mathoverflow.net/questions/41217 | 21 | This question arises from the talk by Voevodsky mentioned in
[this recent MO question](https://mathoverflow.net/questions/40920/what-if-current-foundations-of-mathematics-are-inconsistent). On one of his slides, Voevodsky says that
>
> a general formula even with one free variable describes a subset of
> natural... | https://mathoverflow.net/users/1587 | Question arising from Voevodsky's talk on inconsistency | Let $S$ be a first order definable Martin-Löf random set such as Chaitin's $\Omega$. If Peano Arithmetic, or ZFC, or any other theory with a computable set of axioms, proves infinitely many facts of the form $n\in S$ or $n\not\in S$ then it follows that $S$ is not immune or not co-immune and hence not ML-random after a... | 21 | https://mathoverflow.net/users/4600 | 41220 | 26,337 |
https://mathoverflow.net/questions/41127 | 7 | Suppose $p\_n$ is $n$-th prime, $g\_n=p\_{n+1}-p\_n$ is the corresponding [prime gap](http://en.wikipedia.org/wiki/Prime_gap). What is the highest number $C$, such that $p\_N>C$ can be proven for $N=\min\{n\mid g\_n\geq 1.609\cdot 10^{18}\}$.
**Motivation:** I've read about [Goldbach's weak conjecture](http://en.wiki... | https://mathoverflow.net/users/8134 | What is the best known estimate for the place of the prime gap with length 1.609*10^18? | The title and body ask different questions, so I'll address both.
A reasonable estimate for the first prime gap of length $L$ is $e^{\sqrt L}$, so no prime gap this large would be expected below $\exp(1.268\cdot10^9)$, a 550,886,759-digit number.
As for a lower bound, Dusart 2010 [1] shows that for $x\ge396,738$, t... | 8 | https://mathoverflow.net/users/6043 | 41222 | 26,338 |
https://mathoverflow.net/questions/41173 | 3 | This is a followup to [an earlier question](https://mathoverflow.net/questions/33597/) on a taxonomy for quantum algorithms in which I ultimately concluded in a comment that all known nontrivial quantum algorithm speedups (in [Jordan's quantum zoo](http://www.its.caltech.edu/~sjordan/zoo.html)) could be regarded as ari... | https://mathoverflow.net/users/1847 | Amplitude amplification as a quantum walk algorithm | To understand how amplitude amplification naturally fits in as a particularly simple quantum walk, I'd recommend understanding properly how Grover search works as a quantum walk. Try reading [Andrew Childs' lecture notes](http://www.math.uwaterloo.ca/~amchilds/teaching/w08/co781.html), particularly lectures 14 and 15. ... | 5 | https://mathoverflow.net/users/8075 | 41233 | 26,347 |
https://mathoverflow.net/questions/41229 | 1 | A *weakly initial set* in a category C is a set of objects I of C such that every object a of C has at least one arrow from an object contained in I.
The question is then, does Fields have a weakly initial set? This is equivalent to the collection of prime fields being a set.
The converse is, is there a (fairly nat... | https://mathoverflow.net/users/4177 | Weakly initial sets - examples and nonexamples | Regarding the second question: I'm not sure what should count as "natural", but couldn't you just work with examples where the solution set condition in an adjoint functor theorem fails? The solution set *is* a weakly initial set in a comma category.
For example, there is no left adjoint to the underlying-set functo... | 1 | https://mathoverflow.net/users/2926 | 41235 | 26,348 |
https://mathoverflow.net/questions/41194 | 18 | I have heard through the academic rumor mill (my advisor heard from so-and-so about a result they heard from big-name who saw it in some journal, etc.) of the following theorem:
**Theorem:** Almost all [strongly regular graphs](http://en.wikipedia.org/wiki/Strongly_regular_graph) have trivial automorphism group.
Th... | https://mathoverflow.net/users/4167 | Are "almost all" strongly regular graphs rigid? | The article *Random strongly regular graphs?* by Peter Cameron <http://www.maths.qmul.ac.uk/~pjc/preprints/randsrg.pdf> provides some information about what is known and why someone might make that claim.
First an example: There are 11,084,874,829 strongly regular graphs with parameters SRG(57,24,11,9) which arise fr... | 11 | https://mathoverflow.net/users/8008 | 41240 | 26,352 |
https://mathoverflow.net/questions/41174 | 32 | I am trying to find the precise statement of the correspondence between stable Higgs bundles on a Riemann surface $\Sigma$, (irreducible) solutions to Hitchin's self-duality equations on $\Sigma$, and (irreducible) representations of the fundamental group of $\Sigma$. I am finding it a bit difficult to find a reference... | https://mathoverflow.net/users/83 | What is the precise statement of the correspondence between stable Higgs bundles on a Riemann surface, solutions to Hitchin's self-duality equations on the Riemann surface, and representations of the fundamental group of the Riemann surface? | See Ó. Garcia-Prada's appendix to the third edition of R.O. Wells' book Differential Analysis on Complex Manifolds. It addresses most of what you are asking quite explicitly, in the context of Riemann surfaces, and has references to the original papers. Also the book of Lübke and Teleman on the Kobayashi-Hitchin corres... | 8 | https://mathoverflow.net/users/9471 | 41241 | 26,353 |
https://mathoverflow.net/questions/41214 | 59 | Mathematics has undergone some rather nice developments recently with the adoption of new techologies, things like on-line journals, the arXiv, this website, etc. I imagine there must be many further developments that could be quite useful.
What I'm thinking of is a website where anyone can contribute formal proofs o... | https://mathoverflow.net/users/1465 | Has anyone thought about creating a formal proof wiki with verifier? | There are lots of sites for formal proofs, but no wiki that I am aware of.
Typical examples are:
archive of formal proofs at <https://www.isa-afp.org/>
Mizar <http://mizar.org/>
Lots of proofs are contained in the distributions of various interactive theorem provers: Isabelle, Hol, hol light, Coq, acl2 etc etc
... | 34 | https://mathoverflow.net/users/9831 | 41242 | 26,354 |
https://mathoverflow.net/questions/41212 | 16 | I would be interested to learn if the following generalization of the classical [Looman-Menchoff theorem](https://en.wikipedia.org/wiki/Looman%E2%80%93Menchoff_theorem) is true.
---
Assume that the function $f=u+iv$, defined on a domain $D\subset\mathbb{C}$, is such that
1. $u\_x$, $u\_y$, $v\_x$, $v\_y$ exist ... | https://mathoverflow.net/users/5371 | The Cauchy–Riemann equations and analyticity | No.
Let $c$ be the [Cantor function](https://en.wikipedia.org/wiki/Cantor_function) on $[0,1]$, so that $c$ is continuous, $c' = 0$ a.e., but $c$ is not constant. Then take $u(x+iy) = v(x+iy)=c(x)c(y)$. We have $u\_x=u\_y=v\_x=v\_y=0$ a.e. so the Cauchy–Riemann equations are trivially satisfied, and $f(z)=u(z)+iv(z)$... | 27 | https://mathoverflow.net/users/4832 | 41268 | 26,366 |
https://mathoverflow.net/questions/41254 | 1 | Suppose $\mathcal{A}$ is a quasi coherent sheaf of algebras over a group scheme $\mathcal{G}$. Suppose it is generated by global section. Then , what can we say about the external tensor product $\mathcal{A}\boxtimes\mathcal{A}$? Will this sheaf also be generated by the tensor product of the global section with itself?... | https://mathoverflow.net/users/9492 | External tensor product of sheaves | What do you mean by $A\boxtimes A$? If this is the sheaf $p\_1^\*A\otimes p\_2^\*A$ on $G\times G$ then certainly it is globally generated by the tensor square of global sections. It follows easily from right-exactness of the pullback and of the tensor product --- let $V$ be the space of global sections, then $V\otimes... | 4 | https://mathoverflow.net/users/4428 | 41269 | 26,367 |
https://mathoverflow.net/questions/41264 | 4 | Given a diagram $X\_1 \rightarrow X\_{12} \leftarrow X\_2$ of spaces (though I think the question applies more generally), there are two cosimplicial resolutions which I've seen used to compute the homotopy pullback. The first one, which I'll call $\mathcal A$ is given by $$\mathcal A^p = X\_1 \times X\_{12}^{\times p}... | https://mathoverflow.net/users/4466 | Equivalence of cosimplicial models for homotopy pullbacks | The first one has codegeneracies as well as cofaces, yes? And when you say that it computes the htpy pullback basically by definition I suppose you mean that its Tot is homeomorphic to the space whose points are triples consisting of a point in $X\_1$ a point in $X\_2$ and a path connecting their images in $X\_{12}$? T... | 5 | https://mathoverflow.net/users/6666 | 41272 | 26,370 |
https://mathoverflow.net/questions/41262 | 10 | What can be said about the structure of maximal ideals of $R=k[\{x\_i\}\_{i \in I}]$, or geometric properties of $\text{Spm } k[\{x\_i\}\_{i \in I}]$? Here $k$ is an arbitrary field and $I$ is an *infinite* set. Kernels of evaluation homomorphisms yield an injective map
$\overline{k}^I / Aut(\overline{k}/k) \to \text... | https://mathoverflow.net/users/2841 | maximal ideals of $k[x_1,x_2,...]$ | If $|k| > |I|$ then the usual cheap proof of Nullstellensatz still works: let $K$ be a residue field. Then $\dim\_k K \le \dim\_kR = |I|$, but if $t\in K$ is transcendental over $k$, the elements $1/(t-a)$ for $a\in k$ are $k$-linearly independent. So $K/k$ is algebraic.
| 16 | https://mathoverflow.net/users/5480 | 41275 | 26,372 |
https://mathoverflow.net/questions/41279 | 1 | Let $D$ be a set, $\mathbb{N\_0}$ the set of natural numbers including zero. Let $P$ be the set of all functions from $D$ to $\mathbb{N\_0}$, i.e. $P = \lbrace m \mid m: D \rightarrow \mathbb{N\_0} \rbrace$.
Let $f, g \in P$. Then we define that $f \leq g$ holds iff $\forall d\in D f(d) \leq g(d)$. So $\leq$ is a pa... | https://mathoverflow.net/users/9711 | A complete lattice of functions | Once one adds infinity, it is easy to check that $(P', \leq)$ is indeed a complete lattice. For any subset $A$ of $P'$, it is easy to see that $\vee A$ is just the pointwise supremum of members of $A$. In the case that $A$ is empty, we have that $\vee A$ is the zero function. So, $(P', \leq)$ is a complete join semi-la... | 2 | https://mathoverflow.net/users/2233 | 41281 | 26,376 |
https://mathoverflow.net/questions/41234 | 18 | Fix some $n \geq 3$. It's hopeless to classify the torsion elements in $\text{GL}\_n(\mathbb{Z})$, but I have a couple of less ambitious questions. It's well-known that for any odd prime $p$, the map $\phi\_p : \text{GL}\_n(\mathbb{Z}) \rightarrow \text{GL}\_n(\mathbb{Z}/p)$ is ``injective on the torsion''. In other wo... | https://mathoverflow.net/users/317 | Torsion in GL_n(Z) | Here is a loose collection of partial answers, most of which address a simpler question than that Andy actually asked. I am particularly focusing on Andy's question in the comments "can we classify all liftable elements of ${\rm GL}\_n(\mathbb{Z}/p\mathbb{Z})$ of order $k$ for some fixed $k$?". Please take all this wit... | 11 | https://mathoverflow.net/users/35416 | 41290 | 26,378 |
https://mathoverflow.net/questions/41273 | 4 | Suppose we have $n$ normal variable $X\_1,X\_2,\dots,X\_n$, with corresponding mean $\mu\_1,\dots,\mu\_n$ and sd $\sigma\_1,\dots,\sigma\_n$. What is the probability of $X\_1 < X\_2 < \dots < X\_n$, i.e. $P(X\_1 < X\_2 < \dots < X\_n)$.
Numerical method is also okay. Actually I have thought about how to do the multi-... | https://mathoverflow.net/users/9836 | How to calculate the probability of N normal variable being in increasing order? | We can use multivariate normal distribution to solve this problem. Denote the difference between consecutive elements with a (n-1)-dimensional random vector $Y=(X\_1-X\_2,X\_2-X\_3,\dots, X\_{n-1}-X\_n)$, what we need to find is $Y<(0,0,...,0)$.
Random vector $Y$ has a multivariate normal distribution.
The mean vec... | 5 | https://mathoverflow.net/users/9836 | 41293 | 26,379 |
https://mathoverflow.net/questions/41283 | 3 | Given a pizza, represented by the unit disk $D\_1(0,0)=\{(x,y)\in\mathbb{R}^2\mid \|(x,y)\|\leqslant 1\}$, and given $N$ slices of $r$-pepperoni, represented by disks $D\_r(a\_i,b\_i)=\{(x,y)\in\mathbb{R}^2\mid \|(x,y)-(a\_i,b\_i)\|\leqslant r\}$, for $i\in[1..N]$.
Assume that the $N$ slices are randomly distributed ... | https://mathoverflow.net/users/9839 | Is always possible to slice a pizza in a fair way | Intermediate Value Theorem (for two people).
We draw a $x$-axis through the centre of the disk. For an $\theta \in R$ we draw an axix which makes an angle of $\theta$-degrees with the x-axis (note that the axix for $\theta+\pi$ gives exactly the opposite direction).
Lets look now at the two half disks defined by th... | 2 | https://mathoverflow.net/users/9498 | 41295 | 26,380 |
https://mathoverflow.net/questions/41257 | 4 | For given continuous real functions $f$ and $g$ defined on $[-1,1]$, let's define
$$
D(f,g) = \sup\_{x \in [-1,1]} \left|{\frac{f(x)-g(x)}{f(x)}}\right|
$$
(in this context, let's take $0/0$ to be $0$ and $x/0 = \infty$ for every $x \not= 0$).
I am looking for a theory of approximation by polynomials with respect to th... | https://mathoverflow.net/users/9355 | Relative error approximation by polynomials | According to Johan's answer, the problem is not well posed if the function $f$ has zeroes in $[-1,1]$ (if the number of zeroes is finite, perhaps something could be done). In the following I assume that $f(x)\ne0$ for all $x\in[-1,1]$. Let $n$ be a positive integer and let $\mathbb{P}(n)$ be the set of all real polynom... | 5 | https://mathoverflow.net/users/1168 | 41298 | 26,382 |
https://mathoverflow.net/questions/41211 | 26 | Let $X$ be a finite-dimensional Banach space whose isometry group acts transitively on the set of lines (or, equivalently, on the unit sphere: for every two unit-norm vectors $x,y\in X$ there exist a linear isometry from $X$ to itself that sends $x$ to $y$). Then $X$ is a Euclidean space (i.e., the norm comes from a sc... | https://mathoverflow.net/users/4354 | Easy proof of the fact that isotropic spaces are Euclidean | It is a famous problem (due to Banach and Mazur) whether a separable infinite dimensional Banach space which has a transitive isometry group must be isometrically isomorphic to a Hilbert space. Of course, if every two dimensional subspace has a transitive isometry group, then the space is a Hilbert space since then the... | 17 | https://mathoverflow.net/users/2554 | 41302 | 26,384 |
https://mathoverflow.net/questions/41285 | 10 | By the Riemann-Hilbert correspondence, there is an equivalence between
(1)
$\mathcal{D}\operatorname{-mod}(X)$
, the (derived) category of holonomic D-modules on a complex variety X, and
(2)
$D^b\_c(X)$
, the (derived) category of constructible sheaves on X.
There is a "naive" t-structure we can put on bot... | https://mathoverflow.net/users/3593 | How does one interpret the naive t-structure on constructible sheaves as a t-structure on D-modules? | The t-structure is described in
[this paper of Kashiwara](http://arxiv.org/abs/math/0302086).
It looks essentially like Donu and t3suji suggest in their comments: defined by conditions that look like middle-perversity support conditions.
| 13 | https://mathoverflow.net/users/2628 | 41304 | 26,385 |
https://mathoverflow.net/questions/41310 | 13 | The question is the following: Can one create two nonidentical loaded 6-sided dice such that when one throws with both dice and sums their values the probability of any sum (from 2 to 12) is the same. I said nonidentical because its easy to verify that with identical loaded dice its not possible.
Formally: Let's say ... | https://mathoverflow.net/users/9834 | Any sum of 2 dice with equal probability | You can write this as a single polynomial equation
$$p(x)q(x)=\frac1{11}(x^2+x^3+\cdots+x^{12})$$
where $p(x)=p\_1x+p\_2x^2+\cdots+p\_6x^6$ and similarly for $q(x)$.
So this reduces to the question of factorizing $(x^2+\cdots+x^{12})/11$
where the factors satisfy some extra conditions (coefficients positive,
$p(1)=1$ e... | 12 | https://mathoverflow.net/users/4213 | 41311 | 26,388 |
https://mathoverflow.net/questions/41296 | 17 | In his 1951 report [Sur la théorie du corps de classes](https://projecteuclid.org/euclid.jmsj/1261734944), Weil writes that
>
> *La recherche d'une interprétation de* $C\_k$ *si* $k$ *est un corps de
> nombres*, analogue en quelque manière
> à l'interprétation par un groupe de
> Galois quand $k$ est un corps de
... | https://mathoverflow.net/users/2821 | L'un des problèmes fondamentaux de la théorie des nombres | Interpreting "progress" in a different (perhaps more controversial!) way than in Tony Scholl's answer, one could also mention that Langlands (followed by Kottwitz and perhaps others) has introduced a hypothetical group $L\_k$ which should be even bigger than $W\_k$, normally referred to as the Langlands group, which sh... | 13 | https://mathoverflow.net/users/2874 | 41318 | 26,392 |
https://mathoverflow.net/questions/41314 | 7 | Let $X$ be a separable Banach space, $M \subset X$ a linear subspace. Must $M$ be a Borel set in $X$?
I believe the answer is "no," since I have seen authors who are careful to talk about "Borel subspaces". But I have not been able to find a counterexample.
If the answer is indeed "no", does every infinite-dimensio... | https://mathoverflow.net/users/4832 | Non-Borel subspace of Banach space | This is an answer, but not the "right" answer".
Presumably you mean that $X$ is infinite dimensional and hence has Hamel dimension the continuum $c$. For every subset of a given Hamel basis you get the linear subspace spanned by the subset, and these subspaces are different for different subsets of the basis. Thus $... | 11 | https://mathoverflow.net/users/2554 | 41319 | 26,393 |
https://mathoverflow.net/questions/41289 | 8 | This question consists of two parts. I will try to be as short and clear as possible.
Let $S$ be a Dedekind scheme of characteristic zero. The main examples are $\mathbf{P}^1\_k$, with $k$ a field of characteristic 0 and $\textrm{Spec} \ \mathbf{Z}$.
A fibered surface is a projective flat morphism $X\longrightarrow... | https://mathoverflow.net/users/4333 | Rational singularities for fibered surfaces | Ariyan,
EDIT: This contains some substantial edits and added references.
Lipman has defined the following notion (EDIT: twice):
**Definition** (Lipman ; Section 9 of "*Rational singularities with applications to algebraic surfaces and factorization*"): If $X$ is 2-dimensional and normal, $X$ has two *pseudo-ratio... | 7 | https://mathoverflow.net/users/3521 | 41321 | 26,394 |
https://mathoverflow.net/questions/41253 | 61 | I am teaching a course leading up to Tate's thesis and I told the students last week, when defining ideles, that the first topology that was put on the ideles was not so good (e.g., it was not Hausdorff; it's basically the profinite topology on the ideles, so archimedean components don't get separated well). You can fi... | https://mathoverflow.net/users/3272 | who fixed the topology on ideles? | I know nothing about work of ``idelic nature'' by Von Neumann or Pruefer. Already in the 1930's
Weil understood that Chevalley was wrong to ignore the connected component, because Weil understood already then that Hecke's characters were the characters of the idele class group for the right topology on that. I don't kn... | 161 | https://mathoverflow.net/users/9848 | 41332 | 26,398 |
https://mathoverflow.net/questions/41266 | 4 | An [Enriques surface](http://en.wikipedia.org/wiki/Enriques_surface) (in characteristic zero) is an algebraic surface which is the quotient of a K3 surface by a fixed-point-free involution. Such a surface has a rank 10 lattice of divisors.
>
>
> >
> > (1) What are the ample and effective cones of an Enriques sur... | https://mathoverflow.net/users/4707 | What is known about the ample and effective cones of an Enriques surface? | The answer to question 2 is negative thanks to Kamawata-Morrison conjecture about nef cones of Calabi-Yau varieties that is proven for surfaces (a recent reference is here <http://arxiv.org/abs/1008.3825>), and to the comment of Damiano below.
Indeed, it is known that on a minimal two-dimesnional surface of Kodaira d... | 2 | https://mathoverflow.net/users/943 | 41340 | 26,403 |
https://mathoverflow.net/questions/41288 | 4 | Let $G$ be a finite soluble group, let $x$ be a non-identity element of $G$, let $v$ be an $n$-tuple in $G$, and let $w$ be a word in $n+2$ variables. Does there exist $(G,x,v,w)$ satisfying the following equations?
$w(1,1,v)=w(x,1,v)=w(1,x,v)=1; w(x,x,v)=x.$
As for why the soluble condition is there: let $w(a,b,c)... | https://mathoverflow.net/users/4053 | Can a soluble group compute OR? | After thinking about it some more, I realise it can't be done. Suppose $x$ is contained in a normal subgroup $M$ of $G$, but not in $M'=[M,M]$. Then modulo $M'$ we have
$w(x,1,v) = w(1,1,v)m\_1; w(1,x,v) = w(1,1,v)m\_2; w(x,x,v) = w(1,1,v)m\_1m\_2$
where $m\_1,m\_2 \in M$ (just 'pull through' all occurrences of the... | 3 | https://mathoverflow.net/users/4053 | 41341 | 26,404 |
https://mathoverflow.net/questions/41343 | 8 | Let $X$ be a real smooth manifold, and $M$ a locally-finitely-generated sheaf of $\mathcal C^\infty(X)$-modules. (If $X$ is not compact, I will also insist that there be a global bound on the number of generators I might need in different regions; maybe this is part of the usual meaning of the words "locally finitely g... | https://mathoverflow.net/users/78 | Is there a "Hilbert syzygy theorem" for smooth manifolds? Or: does every finitely generated $C^\infty$ module have a finite-length resolution in vector bundles? | No. Let $X$ be $\mathbb R$. In the ring $C^{\infty}(X)$ let $I$ be the ideal of all functions vanishing to infinite order at $0$. The module $C^{\infty}(X)/I$ does not have a finite resolution by finitely generated projective modules.
Edit:
Still no if you want the finitely generated module to be contained in a f... | 13 | https://mathoverflow.net/users/6666 | 41344 | 26,406 |
https://mathoverflow.net/questions/41349 | 6 | If X and Y are non-isomorphic objects, then "[is / is not] isomorphic to [ X / Y ]" are invariants that distinguish X and Y. You can also do things like take an object Z that is not isomorphic to Y, and then "is isomorphic to X or isomorphic to Z" is another invariant that distinguishes X and Y. Similarly, if W is isom... | https://mathoverflow.net/users/nan | non-isomorphic objects with no known nontrivial distinguishing invariants | Hereditarily indecomposable Banach spaces are strange objects that fail to be isomorphic to any of their proper subspaces. However, in a certain sense those subspaces are not "interestingly different" from the spaces themselves. So there is no hope of finding an invariant to distinguish between the space and a subspace... | 15 | https://mathoverflow.net/users/1459 | 41355 | 26,409 |
https://mathoverflow.net/questions/41353 | 9 | This must be common knowledge.
Where exactly in the development of homological algebra does one need the axiom that makes ~~additive~~pre-abelian and abelian categories different? (I mean this statement: for every morphism $u: X \to Y$, the canonical morphism $\bar{u}: \mathrm{Coim}\ u \to \mathrm{Im}\ u$ is an isomo... | https://mathoverflow.net/users/2234 | abelian categories vs. additive categories | A good example of the situation you are thinking about is the category of filtered modules (over some ring). As Yemon Choi notes in a comment, Banach spaces give another example (and in general, filtrations behave very similarly to topologies, and are a closely related notion); but (at least for me) it is a bit easier ... | 14 | https://mathoverflow.net/users/2874 | 41381 | 26,422 |
https://mathoverflow.net/questions/41369 | 4 | I've heard it said that the reason why the homology groups of a space are a *computable* invariant is because they are a *stable invariant* in the sense that they are stable under suspension.
I'm familiar with the standard computation which shows that for reduced homology ${\tilde H\_n}(S X)\simeq {\tilde H\_{n-1}}(X... | https://mathoverflow.net/users/1148 | Homology is computable because it is stable under suspension | Stability alone surely does not give you much. It gives you just the values on the spheres if you have the value on $S^0$. But many interesting spaces (eg all smooth manifolds) are CW-complexes and can be built out of spheres. And if you have a way to handle cofiber sequence $X\to Y \to CX\cup\_X Y$ with your theory, y... | 9 | https://mathoverflow.net/users/2039 | 41382 | 26,423 |
https://mathoverflow.net/questions/40801 | 15 | Is the following fact true?
>
> Let $v\_1,\ldots, v\_k \in \mathbb{R}^2$, $\|v\_i\|\leq 1$, be vectors that add up to zero. Does there exist a permutation $\sigma\in S\_k$ and vectors $w\_1,\ldots, w\_k \in \mathbb{R}^2$, $\|w\_i\|\leq 1$, such that $v\_{\sigma(i)}=w\_i-w\_{i-1}$? (here, I assume $w\_0=w\_k$)
>
>... | https://mathoverflow.net/users/8134 | Representation of vectors in $\mathbb{R}^2$ via differences of small vectors. | I have a counterexample in $\mathbb R^2$.
Here's how it goes.
Pick two numbers $n$ and $N$, with $N>>n>>1$.
The collection {$v\_i$} consists of:
* $N(n-1)$ times the vector $(\frac{n-2}{n},\frac{1}{N(2n-3)})$
* $N(n-2)$ times the vector $(-\frac{n-1}{n},\frac{1}{N(2n-3)})$
* The vector $(0,-1)$ once.
The sm... | 8 | https://mathoverflow.net/users/5690 | 41384 | 26,424 |
https://mathoverflow.net/questions/41351 | 3 | Everything below is defined over $\mathbb C$.
Let $T$ be a smooth affine variety, $\pi : \mathscr X \to T$ be a smooth family of smooth
projective varieties, and $\mathcal F$ be a locally free coherent sheaf on $\mathscr X$.
As usual write $\mathscr X\_t$ for the fiber of $\pi$ over $t$ and $\mathcal F\_t$ for the
... | https://mathoverflow.net/users/605 | Existence of non-zero sections | No, this is not true. Here is the simplest example. Take $T = \mathbb A^2$, $X = \mathbb A^2 \times \mathbb P^1$. The morphims $\pi \colon X \to T$ is the projection. We will denote by $\cal O(d)$ the pullback of $\cal O\_{\mathbb P^1}(d)$ to $X$. Let $x$ and $y$ be the coordinates on $\mathbb A^2$, and $u$ and $v$ the... | 7 | https://mathoverflow.net/users/4790 | 41392 | 26,427 |
https://mathoverflow.net/questions/41388 | 5 | Let $\Lambda$ denote the ring of symmetric functions in variables $x\_1,x\_2,\dots$ and with coefficients in $\mathbf{Q}$. Then $\Lambda$ is freely generated as an $\mathbf{Q}$-algebra by $p\_1,p\_2,\dots$, where $p\_n$ denotes the $n$-th power sum function $x\_1^n+x\_2^n+\cdots$. Let $\Delta^+$ and $\Delta^{\times}$ d... | https://mathoverflow.net/users/1114 | Are the Schur functions the minimal basis of the ring of symmetric functions with the following properties? | There is another simple set of symmetric functions fulfilling your properties 1 to 4: the set of products of power sums,
$$
p\_1, p\_2, p\_1^2, p\_3, p\_1 p\_2, ...
$$
The convex cone they generate is not contained in the cone generated by the Schur functions. For instance, $p\_2=s\_2-s\_{1,1}$.
EDIT: and the Schur ... | 4 | https://mathoverflow.net/users/6768 | 41401 | 26,432 |
https://mathoverflow.net/questions/41397 | 2 | When I work out the James construction for a discrete pointed space, it appears that the
induced map $\pi\_0 (J(X)) \to \pi\_0( \Omega\Sigma X)$ is the inclusion of the free monoid on $\pi\_0(X)$ into the free group on $\pi\_0(X)$, so $J(X) \to \Omega\Sigma X$
is not a homotopy equivalence; and it seems clear that th... | https://mathoverflow.net/users/3634 | James Construction for Disconnected Spaces | No. Let $X$ be the disjoint union of a point and a circle, and compare the two different $J(X)$ that you get according to what point you call the base point. Now think about $\pi\_1(\Omega \Sigma X)=\pi\_2(S^1\vee S^2)$. Same as $\pi\_2$ of the universal covering space ...
| 5 | https://mathoverflow.net/users/6666 | 41402 | 26,433 |
https://mathoverflow.net/questions/41287 | 10 | Let $G$ be the mapping class group of a closed surface $S\_{g}$. Bestvina-Bromberg-Fujuwara <https://arxiv.org/abs/1006.1939> recently constructed a finite index subgroup $B$ of $G$ such that for every essential closed simple curve $\gamma$ on $S\_{g}$ and every $h\in B$ either $h\gamma=\gamma$ or $h\gamma$ intersects ... | https://mathoverflow.net/users/nan | A finite index subgroup of the Mapping Class Group | Regarding the new question, you're correct that $f\in \mathcal{I}$ doesn't imply $f(\gamma)$ meets $\gamma$: for example, if $f$ is a product of disjoint bounding pairs and $\gamma$ is a (nonseparating) curve meeting each curve-defining-$f$ in at most 1 point, then $f\in\mathcal{I}$ but $f(\gamma)$ is disjoint from $\g... | 5 | https://mathoverflow.net/users/250 | 41408 | 26,437 |
https://mathoverflow.net/questions/41373 | 5 | Hi
I really need a proof for the following statement by Baumgartner:
There exists a stationary subset of $[\omega\_2]^{\omega}$ of size $\aleph\_2$.
This is Exercise 38.15. in Jechs Book (2003) and you can find a hint there which goes like this: For each $\alpha < \omega\_2$, let $f\_{\alpha} : \alpha \to \omega... | https://mathoverflow.net/users/4753 | Why is this set stationary? | Given $F:[\omega\_2]^{<\omega}\to[\omega\_2]^{\aleph\_0}$ as above, we first claim the existence of an ordinal $\omega\_1\leq\alpha<\omega$ that is closed under $F$, i.e., $s\in [\alpha]^{<\omega}$ implies $F(s)\subseteq\alpha$. For this, let $\alpha$ be the limit of the sequence $\omega\_1=\alpha\_0<\alpha\_1<\cdots$ ... | 10 | https://mathoverflow.net/users/6647 | 41422 | 26,448 |
https://mathoverflow.net/questions/41429 | 2 | Given two smooth elliptic curves $C\_1$ and $C\_2$ over $\mathbb{C}$. Assume they are not isogenous. I'm interested in the structure of $Pic(A)$ and $Pic^{0}(A)$ for $A:=C\_1 \times C\_2$.
Reading Birkenhake/Lange - Complex Abelian Varieties, i think this has to do with correspondences of curves. Since an elliptic cu... | https://mathoverflow.net/users/3233 | Line bundles on special abelian surfaces | Yes: in fact $Pic^0(C\_1\times C\_2)=Pic^0(C\_1)\times Pic^0(C\_2)$ for any pair of curves. The fact that $C\_1$ and $C\_2$ are not isogenous in your case only affects the Neron-Severi group $Pic/Pic^0$ of $C\_1\times C\_2$, exactly for the reasons you describe.
| 2 | https://mathoverflow.net/users/5480 | 41431 | 26,451 |
https://mathoverflow.net/questions/41389 | 20 | For my finals, I am digging through the book by Varadarajan [An introduction to harmonic analysis on semisimple Lie groups](http://books.google.com/books?id=seycdRenNfoC&lpg=PP1&dq=varadarajan%2520harmonic%2520analysis%2520on%2520semisimple%2520lie%2520groups&pg=PP1#v=onepage&q&f=false%20%22Varadarajan%20-%20An%20intro... | https://mathoverflow.net/users/6818 | Harmonic analysis on semisimple groups - modern treatment | Speaking as a nonexpert, I'd emphasize that the subject as a whole is deep and difficult. Even leaving aside the recent developments for $p$-adic groups, the representation theory of semisimple Lie groups has been studied for generations in the spirit of harmonic analysis. So there is a lot of literature and a fair num... | 7 | https://mathoverflow.net/users/4231 | 41434 | 26,454 |
https://mathoverflow.net/questions/41391 | 4 | Let $I =\langle f\_1,\cdots,f\_m\rangle \subset K[x\_1,\cdots,x\_n]$be an ideal,
where $f\_k\in K[x\_1,\cdots,x\_n].$
$K[e\_1,\cdots,e\_n]$ the polynomial algebra generated by the elementary symmetric polynomials
$e\_1,\cdots,e\_n\in K[x\_1,\cdots,x\_n].$
Is there any method(algorithm) to compute the K-algebra $I ... | https://mathoverflow.net/users/8152 | Finding generators of subalgebra of polynomial algebra $K[x_1,\cdots,x_n]$ that are invariant under the action of symmetric group | UPDATE: Sorry! As pointed out in a comment, my previous answer was incorrect. So I've edited my answer. The following simpler algorithm seems to me that it should work (at least assuming I'm understanding the question\dots).
From the geometric perspective, the inclusion map $k[e\_1, \dots, e\_n]\subseteq k[x\_1, \dot... | 5 | https://mathoverflow.net/users/4 | 41436 | 26,455 |
https://mathoverflow.net/questions/41259 | 3 | Let $\mathcal{P}(\mathbb{N})$ be the set of all probability mass functions on $\mathbb{N}=\{1,2,\dots \}$. Let $E$ be a closed(with respect to pointwise convergence, or equivalently the total variation metric) subset of $\mathcal{P}(\mathbb{N})$ and $Q\notin E$. Let $0<\beta<1$.
Now $\displaystyle \sum\_{x\in N} P(x)^{... | https://mathoverflow.net/users/7699 | Tightness of probabilty distributions | I think this conjecture is false, that is, there does not necessarily exist a subsequence that converges to a true probability distribution. Consider the following situation:
Let $Q=(1,0,0,0,...)$, i.e. the probability distribution with all mass at $x=1$.
Define the distribution $R\_n$, for $n=2,3,...$, as $$R\_n(1... | 2 | https://mathoverflow.net/users/8938 | 41438 | 26,456 |
https://mathoverflow.net/questions/41428 | 5 | If I have a Deligne-Mumford stack $\Pi : X \to (\mathrm{Sch}/k)$ for some field $k$, can it be reconstructed from $\Pi^{-1}(C) \subset X$ for some "small" subcategory $C \subset (\mathrm{Sch}/k)$? For example, let's say we assume the fppf topology (to be concrete), then does the restriction to the fully faithful subcat... | https://mathoverflow.net/users/5935 | Can Deligne-Mumford stacks be characterized by their restriction to a small subcategory? | Restricting to Aff is certainly enough, but Aff isn't small (there are e.g., polynomial algebras on arbitrary sets). If your DM stack is *finitely presented* over $k$ (which is probably good to include in the definition, to avoid these issues), then it is determined by it's restriction to finitely-presented affines (wh... | 7 | https://mathoverflow.net/users/1921 | 41443 | 26,459 |
https://mathoverflow.net/questions/41390 | 10 | Let $f:X \rightarrow Y$ be a morphism between two smooth projective varieties $X,Y$ which are defined over an algebraically closed field $k$. I am looking for some criteria which guaranties the projectivity of $f$.
For instance if $f$ is finite it is projective. Here we don't need the projectivity of the varieties $... | https://mathoverflow.net/users/5286 | Morphism between projective varieties | Here is an attempt to prove Angelo's comment (it seems too simple to use a reference for it):
$X,Y$ defined over $S$. If they are both proper over $S$, then so is $f$ by Hartshorne, II.4.8(e). In particular $f$ is separated and universally closed.
If $X$ is projective over $S$ then for some $n$ there exists $\iota:... | 9 | https://mathoverflow.net/users/10076 | 41446 | 26,461 |
https://mathoverflow.net/questions/41329 | 0 | (This might look like just a post to you and you might think I shouldn't have submitted it as a question here but in reality it is some questions put together, so I hope you don't close it)
I only started studying about this topic two days ago and with my math not being really strong (yet) I am struggling to survive.... | https://mathoverflow.net/users/3550 | A few questions about Computational Problems Complexity Classification | 1) It depends what you mean by "exist", but probably the answer is no: we can't build an actual, physical, non-deterministic machine. (Of course, we can't literally build a Turing machine either, since the definition of a Turing machine generally requires infinite amounts of memory; but even if you're willing to fake t... | 5 | https://mathoverflow.net/users/8991 | 41452 | 26,464 |
https://mathoverflow.net/questions/41447 | 6 | Definitions
===========
I believe set theorists have studied all of the following three notions in the context of forcing extensions of a model of ZFC, $M$ (hopefully the terminology is the standard one).
1. A function $f:\mathbb N\rightarrow\mathbb N$ is *eventually different* if for each function $g:\mathbb N\rig... | https://mathoverflow.net/users/4600 | An eventually different function adding no Solovay real nor dominating function? | Hi Bjørn, and congratulations to you and Bonnie!
The answer to I is yes. In fact, there is a standard way of doing this, with the "eventually different forcing ${\mathbb E}$". This notion does not add random or dominating reals, and adds an eventually different function.
Conditions have the form $(s, A)$ where $s\... | 6 | https://mathoverflow.net/users/6085 | 41461 | 26,470 |
https://mathoverflow.net/questions/41286 | 8 | Suppose we are given a finite collection of finite binary strings $\mathcal{S}$, of various lengths. Our task is to express any binary sequence $x\in 2^\mathbb{N}$ as juxtaposition of strings taken from $\mathcal{S}:$
$$x=\sigma\_1\sigma\_2\sigma\_3\dots$$
For any such sequence of "bricks", $\sigma\in\mathcal{S}^\mathb... | https://mathoverflow.net/users/6101 | Breaking efficiently a binary sequence into given strings | Not an answer, but perhaps of some use:
It would help if S, the set of blocks, were finite. In any case, if S contains all words of length k, you know that for sufficiently long words L will have k as a lower bound.
(Computing k for a given S closed under consecutive substrings is feasible, but may not be
polynomial... | 3 | https://mathoverflow.net/users/3568 | 41463 | 26,472 |
https://mathoverflow.net/questions/41484 | 6 | Let G be a semisimple Lie group and let k be some number field. Let A be a finite index subgroup of some group B which is discrete and Zariski dense in G. Suppose A is in G(k). Then can we expect that B is in G(k') where k' is a finite Galois extension of k? When can we expect k'=k?
| https://mathoverflow.net/users/9891 | Finite index subgroup of discrete subgroup | I'll prove this when $G=SL(n,\mathbb{C})$. So assume $B\leq SL(n,\mathbb{C})$ is discrete, and $A\leq B$ is a finite-index subgroup. Let $A'\lhd A$ such that $A'\lhd B$ is the core, which is also finite index (by considering the kernel of the action on cosets of $A$ in $B$). Suppose that $A\subset SL(n,k)$ where $k$ is... | 7 | https://mathoverflow.net/users/1345 | 41485 | 26,484 |
https://mathoverflow.net/questions/41427 | 4 | In which sense is it possible to solve $\Delta u=0$, $\partial\_\nu u=\phi$, for $\int\phi=0$ on a closed domain, say a ball $B^3\subset\mathbb R^3$?
For example would a $\phi\in L^p(\partial B^3)$, $1< p<2$ make sense?
In other words, is $W^{1,p}$ really the right trace space, or else, which is?
Where can I f... | https://mathoverflow.net/users/5628 | Trace space and Neumann boundary condition | You can solve the problem with even less regularity than in Rekalo's answer. If $u\in W^{1,p}(\Omega)$, it does not have a normal trace in general. But if you assume in addition that $\Delta u\in L^p(\Omega)$, then the normal trace is well-defined and belongs to $W^{-1/p',p}(\partial\Omega)$, where $p'$ is the conjugat... | 6 | https://mathoverflow.net/users/8799 | 41487 | 26,486 |
https://mathoverflow.net/questions/41464 | 1 | The short form of my question is: Can we find two formulae (in the multiplicative fragment of linear logic (MLL), that is, without additives or exponentials) A and B such that (1st) A is provable and B is provable, (2nd) A ⅋ B (A par B) is provable?
The context of the question is this: If we can find two such formula... | https://mathoverflow.net/users/9881 | Can the Multiplicative Fragment of Linear Logic be shown to be non-truth-functional? | I don't have a complete proof, but I'm rather skeptical of the existence of such formulas. Certainly in the unit-free fragment of MLL it's hopeless, by applying a proof net criterion. For example, using the Danos-Regnier criterion for validity of proof nets, if you have a sequent $\to A \wp B$ (which is equivalent to $... | 2 | https://mathoverflow.net/users/2926 | 41498 | 26,491 |
https://mathoverflow.net/questions/41041 | 4 | Let $I,J$ be homogeneous ideals in the algebra of polynomials in $n$ variables over the complex numbers.
Let $V(I)$ and $V(J)$ be the **affine** algebraic varieties that are determined by $I$ and $J$ (**not** the projective varieties). Suppose that $V(I)$ and $V(J)$ are isomorphic as algebraic varieties. By this I mean... | https://mathoverflow.net/users/1193 | If two "homogeneous" algebraic varieties are isomorphic, are they necessarily related by a linear map? | Trying to generalize Torsten's answer: It seems that if the cones are isomorphic then the isomorphism can indeed be chosen to preserve the origin.
For a projective variety $V$ let's denote the affine cone by $C(V)$. Torsten says that if $V$ is not a (projective) cone then in $C(V)$ the "cone point" $0$ is the unique ... | 1 | https://mathoverflow.net/users/6666 | 41509 | 26,496 |
https://mathoverflow.net/questions/41497 | 1 | Let $A$ be a $n \times n$ matrix all of whose entries has modulus 1.
Suppose the matrix $A$ is singular.
We will assume without loss of generality that all the entries in the first row and the first column of the matrix are 1.
Observe when $n=2$ the matrix $A$ can be then singular if and only if $a\_{2,2}=1$ a... | https://mathoverflow.net/users/6766 | structure of singular matrices whose entries have modulus one | Your first calculation for $3\times3$ matrices applies in full generality: If your matrix writes blockwise $[1 \quad e^T ; e \quad M]$, with $e^T=(1,\ldots,1)$, and if $M$ is non-singular (implied by your assumption), then the property that $\det A=0$ is equivalent to $e^T\hat M e=\det M$, that is $e^TM^{-1}e=1$, or to... | 2 | https://mathoverflow.net/users/8799 | 41518 | 26,502 |
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