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/222104 | 5 | Let $A[x]$ be the algebra of polynomials with coefficients in a $k$-algebra $A$. Assume that, for any simple $A[x]$-module $M$, we have $\text{End}\_{A[x]} M = k$. Does it follow that any element of $J(A)$ is nilpotent?
| https://mathoverflow.net/users/nan | Does it follow that any element of $J(A)$ is nilpotent? | Let $a \in J(A)$, and suppose for a contradiction that the left ideal $A[x](1 - xa)$ is proper.
We can then choose a maximal left ideal $J$ of $A[x]$ containing $A[x](1-xa)$, so that $M := A[x] / J$ is a simple $A[x]$-module. Let $v := 1 + J$ be the canonical $A[x]$-module generator of $M$; then $(1 - xa)\cdot v = 0$... | 4 | https://mathoverflow.net/users/6827 | 226106 | 105,619 |
https://mathoverflow.net/questions/226101 | 3 | Can we say in general, [*Treewidth*](https://en.wikipedia.org/wiki/Treewidth) of every graph is less than or equal to number of vertices in that graph?
Is there any other general relation between *Treewidth* and [*Order*](http://mathworld.wolfram.com/GraphOrder.html) of graphs?
| https://mathoverflow.net/users/83880 | What is the relation between Treewidth and Order of graph? | The treewidth of an $n$-vertex graph is always at most $n-1$ (because it is defined as one less than the cardinality of the largest bag in an optimal tree decomposition, and each bag is a set of $n$ or fewer vertices). It is exactly $n-1$ in the case of a complete graph, and can be as small as $0$ for an independent se... | 4 | https://mathoverflow.net/users/440 | 226114 | 105,620 |
https://mathoverflow.net/questions/156444 | 11 | We encountered polynomials defined by the recursive relations for the coefficients $b\_k>0$ as defined below:
$$p\_{n}(x)=\sum\_{k=0}^{n}\binom{2n}{2k}b\_k x^k$$
$$\frac{b\_k^2}{b\_{k-1}b\_{k+1}}=1+\frac{\pi}{31(k+1/2)}=\frac{k+A}{k+B}>1$$
These polynomials showed up when we tried to find a polynomial approximation t... | https://mathoverflow.net/users/33672 | Zeros of polynomials related to Jensen polynomial associated with Riemann xi function $\xi(x)$ | Just two pointers for this problem (sorry, no solution).
1. Hurwitz used a version of Sturm's theorem on numerators of
continued fractions to study the zeros of
the Bessel functions, and Watson's treatise on Bessel functions (1944) has
this in section 9.7. The series in question can be written as
$$\sum\_{k=0}^\infty... | 2 | https://mathoverflow.net/users/84137 | 226119 | 105,622 |
https://mathoverflow.net/questions/226044 | 9 | This question assumes familiarity with combinatorial cardinal characteristics of the continuum.
Let $\mathcal{E}$ be the $\sigma$-ideal generated by closed measure zero subsets of the real line. It is known that
$$\operatorname{cov}(\mathcal{N})\cdot\operatorname{cov}(\mathcal{M})\le
\operatorname{cov}(\mathcal{E})\... | https://mathoverflow.net/users/2415 | How many closed measure zero sets are needed to cover the real line? | Since we can cover the reals by at most $\mathfrak{r}$ (reaping number) closed null sets, we can do a countable support iteration of rational perfect set forcing over a model of CH to get a model of $\omega\_1 = \mathfrak{r} < \mathfrak{d} = \omega\_2$.
To see that $\operatorname{cov}(\mathcal{E}) \leq \mathfrak{r}$... | 7 | https://mathoverflow.net/users/2689 | 226131 | 105,624 |
https://mathoverflow.net/questions/226064 | 8 | What is known about the structure of $J\_{0}(N)$ over $\mathbb{Q}[\mu\_{p^{\infty}}]$?
More generally, what do we know about $J\_{0}(N)$ over
$\mathbb{Q}[\mu\_{p^{\infty}},k^{1/p^{n}}]$, where $k\in\mathbb{Z}$?
Does it have infinite rank? Does it have finite torsion?
I am especially interested in the case where ... | https://mathoverflow.net/users/70751 | What do we know about the structure of $J_{0}(N)$ over $\mathbb{Q}[{\mu}_{{p}^{\infty}},{{k}}^{\frac{{1}}{{p}^{n}}}])$? | Theorem 14.4 of Kato's paper in Asterisque 295 (2004), on page 236, says:
>
> Let $A$ be an abelian variety over $\mathbf{Q}$ such that there is a surjective homomorphism $J\_1(N) \to A$ for some integer $N$. Then for any $m \ge 1$, $\bigcup\_n A(\mathbf{Q}(\mu\_{m^n}))$ is a finitely-generated abelian group.
>
> ... | 6 | https://mathoverflow.net/users/2481 | 226135 | 105,626 |
https://mathoverflow.net/questions/225011 | 11 | I would like to know what is the recent progress about the group homomorphism
$$ \mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})\rightarrow \mathrm{Out}(\hat{F\_{2}})$$
* $\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$ is the absolute Galois group of $\mathbf{Q}$.
* $\mathrm{Out}(\hat{F\_{2}})$ is the group of outer ... | https://mathoverflow.net/users/82229 | Dessins d'enfants and absolute Galois group | For the sake of getting this off the unanswered stack...
Please refer to the following reference:
>
> Guillot, P. *An elementary approach to dessins d'enfants and the Grothendieck-Teichmüller group*, Enseign. Math. vol 60, 2014. [[arXiv version](http://arxiv.org/abs/1309.1968) | [journal version](http://www.ems-p... | 7 | https://mathoverflow.net/users/801 | 226149 | 105,629 |
https://mathoverflow.net/questions/226144 | 5 | Let $\gamma\_n: \mathbb{R}^n\to\mathbb{R}$ be the Gaussian distribution function defined by
$$
\gamma\_n(x):=(2 \pi)^{-\frac{n}{2}} e^{-\frac{|x|^2}{2}}.
$$
Let $d\gamma\_n$ denote the following measure (weight) $\gamma\_n(x) dx$ and consequently $L^2(\mathbb{R}^n,d\gamma\_n)$ and $H^1(\mathbb{R}^n,d\gamma\_n)$ be the... | https://mathoverflow.net/users/84149 | Rellich-Kondrachov compacteness Theorem for the Euclidean space with Gaussian measure | I'll start with several known facts.
>
> **Proposition 1.** Suppose that $E$ is a real Hilbert space with norm $\Vert -\Vert$ and $K: E\to E$ is a a compact, selfadjoint
> positive operator. Denote by $R(K)$ the range of $K$. Equip the
> range with the norm
>
>
> $$ \Vert x\Vert\_K:=\Vert K^{-1} x\Vert,\;\;x\i... | 5 | https://mathoverflow.net/users/20302 | 226161 | 105,635 |
https://mathoverflow.net/questions/226155 | 5 | Let $X$ be a compact Hausdorff topological space, and $C(X)$ denote the ring of complex-valued, continuous functions on $X$. Let $A$ be a matrix with entries from $C(X)$ of size $m\times n$ and $b\in \mathbb{C}^{m\times 1}$. Suppose that for each $x\in X$, the equation $A(x) v=b$ has a solution $v\in \mathbb{C}^n$. The... | https://mathoverflow.net/users/84154 | Continuity of solutions to $Av=b$ | No. Take $n=m=2$, $A(x)=\pmatrix{1&0\\x&x^2}$, $x\in \mathbb{R}$, $b=\pmatrix{1\\0}$. For $x\ne 0$ the only solution of $AV=b$ is $V(x)=\pmatrix{1\\-1/x}$, it has discontinuity at 0 for any choice of $V(0)$.
| 7 | https://mathoverflow.net/users/4312 | 226164 | 105,637 |
https://mathoverflow.net/questions/226158 | 6 | I was recently thinking about the proof of a theorem where Waldhausen compared the Segal's delooping machinery with his, in the case when the cofibration is splittable (sec.1.8 in 'Algebraic $K$-theory of space').
Assume $\mathcal{C}$ is a Waldhausen category. By forgetting some structures, one obtains a category wit... | https://mathoverflow.net/users/82649 | Waldhausen and Segal's delooping machinery | There is no hope of showing that $w N. E(\mathcal C) \to w N. \mathcal C \times w N. \mathcal C$ is a homotopy equivalence without assuming that the cofibrations of $\mathcal C$ are splittable. To see that, take a simple example and look at the direct sum Grothendieck groups which appear as $\pi\_1$ of these K-theory s... | 9 | https://mathoverflow.net/users/15247 | 226168 | 105,639 |
https://mathoverflow.net/questions/225954 | 5 | **What kind of definitions of [t-structures](http://ncatlab.org/nlab/show/t-structure#in_stable_categories)
on stable model categories have been investigated in the literature?**
Of course, one can always define a t-structure on a stable model category as a t-structure on its homotopy category;
this is analogous to h... | https://mathoverflow.net/users/402 | Reference for t-structures on stable model categories |
>
> one can always define a t-structure on a stable model category as a t-structure on its homotopy category
>
>
>
This is a definition, but it's extremely unsatisfying and completely unenlightening.
The [paper](http://arxiv.org/abs/1408.7003) you link from the nLab page proves that t-structures in a stable in... | 6 | https://mathoverflow.net/users/7952 | 226172 | 105,642 |
https://mathoverflow.net/questions/226082 | 20 | Nowadays there is a lot of talk about derived algebraic geometry, but not so much about the related subject of spectral algebraic geometry.
Now I'm curious what future is there for spectral algebraic geometry. That is, what problems could be possibly attacked using this formalism, rather than arithmetic algebraic geo... | https://mathoverflow.net/users/82193 | Motivation and potential applications of spectral algebraic geometry | This is not really an answer to your question, just an attempt to address your question from the comments.
There are various flavours of homotopical or higher algebraic geometry that are commonly considered, which have different levels of connectivity, linearity, or strictness of commutativity.
These include:
1) si... | 23 | https://mathoverflow.net/users/2503 | 226189 | 105,646 |
https://mathoverflow.net/questions/226205 | 0 | Where can I find references to proofs/can anyone supply me a quick proof of the following facts?
* If the $n$-dimensional manifold $M$ can be immersed in $\mathbb{R}^{n+1}$, then each $w\_i(M)$ is equal to the $i$-fold cup product $w\_1(M)^i$.
* If $\mathbb{RP}^n$ can be immersed in $\mathbb{R}^{n+1}$, then $n$ must ... | https://mathoverflow.net/users/78773 | Immersing spaces in $\mathbb{R}^{n+1}$, Stiefel-Whitney classes | Read Milnor Stasheff: Characteristic classes.
For the first question:
The total normal Stiefel Whitney class of $M$ is $1-w\_1(M)$, hence the inverse is
$W(M) = 1 + w\_1(M) + w\_1(M)^2 +... .$
For the second Question one uses that the tandent bundle $\tau$ of $RP^n$ satisfies the equality $\tau \oplus \epsilon$ is ... | 12 | https://mathoverflow.net/users/36950 | 226208 | 105,649 |
https://mathoverflow.net/questions/226200 | 4 | Let $H$ be the subgroup of $\text{GL}(n + k, \mathbb{R})$ consisting of matrices whose lower $n \times k$ block is empty; i.e. consisting of matrices of the form$$\begin{pmatrix} A & \* \\ 0 & B\end{pmatrix},$$where $A \in \text{GL}(n, \mathbb{R}$), $B \in \text{GL}(k, \mathbb{R})$, and $\*$ is arbitrary. How do I see ... | https://mathoverflow.net/users/74305 | $\text{GL}(n + k, \mathbb{R})$ is a principal $H$ bundle over the Grassmann manifold $G_n(\mathbb{R}^{n+k})$? | The first thing to note is that the Grassman manifold is a homogeneous manifold, since the group $G=GL(n+k,\mathbb R)$ naturally acts on it transitively. This implies that $G\_n(\mathbb R^{n+k})=G/H$, where $H$ is a stabilizer of (any) point $x\in G\_n(\mathbb R^{n+k})$.
Now take $x=\mathrm{span}(e\_1,\dots e\_n)$, w... | 3 | https://mathoverflow.net/users/40950 | 226211 | 105,650 |
https://mathoverflow.net/questions/226210 | 2 | Motivation of my question: Let $A$ be a bounded selfadjoint operator with spectral measure $E$ and $x$ a vector. Then it is easily seen that the closed linear span of all $A^nx$ ($n\in\mathbb N$) coincides with that of all $E(\Delta)x$, where $\Delta$ runs through all Borel sets. I would like to know if the same holds ... | https://mathoverflow.net/users/84182 | Pointwise convergence of polynomials to a function on a compact set K that is 1 on some disc D and zero outside D | Yes, using Runge's theorem. I'll assume wlog your $D$ is the unit disk
$\{z: |z| \le 1\}$. Given positive integer $n$, take
$$ K\_n = \{z: (|z| \le 1 \ \text{or}\ 1 + 1/n \le |z| \le n)\ \text{and}
(\text{Im}(z) \le 0 \ \text{or}\ \text{Im}(z) \ge 1/n)\}$$
Note that $\mathbb C \backslash K\_n$ is connected, and ther... | 2 | https://mathoverflow.net/users/13650 | 226212 | 105,651 |
https://mathoverflow.net/questions/226065 | 4 | I would like to ask about an explicit suggestion/reference for the following type of heat processes:
Roughly, assume we have a "wedge" $W$ of the following form - a domain in $\mathbb{R}^n$ with a tip at 0, having some hyperplanes (passing through 0) as sides and, say, the piece of the unit cyllinder/sphere as a bott... | https://mathoverflow.net/users/61506 | Reference for a Heat Process in a Wedge | Lemma 1 of [Brownian motion in cones](http://www.math.purdue.edu/~banuelos/Papers/cones.pdf#page=6) by Banuelos and Smits gives the Dirichlet heat kernel for generalized cones in $\mathbb{R}^n$ in terms of Bessel functions and spherical harmonics. The paper references page 379 of [Conduction of Heat in Solids](https://... | 2 | https://mathoverflow.net/users/15192 | 226215 | 105,652 |
https://mathoverflow.net/questions/226214 | 0 | Let $p>2$ be prime. By the classification of quadratic forms, there are $8$ pairwise non-equivalent isotropic orthogonal groups in $4$ variables. Is there a concrete classification of orthogonal groups of these groups? Could they be isomorphism? The analogous question for $3$ variables has been asked and answered in a ... | https://mathoverflow.net/users/3635 | $p$-adic orthogonal groups in four variables | There is a concrete description of the various forms in a paper by Harris, Soudry, and Taylor, "l-adic representations associated to modular forms over imaginary quadratic fields," Inventiones, 1993. The paper is available here: <https://eudml.org/doc/144109>
| 0 | https://mathoverflow.net/users/10458 | 226222 | 105,653 |
https://mathoverflow.net/questions/226204 | 3 | I have an optimization problem of the form:
\begin{align}
&\text{maximize}\quad f(\mathbf{x}) = \dfrac{\sum\limits\_{n=1}^{N}x\_na\_n}{1+\sum\limits\_{n=1}^{N}x\_nb\_n}\\
& \text{subject to}\quad \sum\limits\_{n=1}^{N}x\_n\leqslant M,\\
& \quad\quad\quad\quad\quad\;\mathbf{x}=\left[x\_1,\ldots,x\_N\right]^\top\in\{0,... | https://mathoverflow.net/users/76839 | Can we say that this problem is NP-hard? | Short answer: with negative $b\_i$'s, it's NP-hard, but otherwise it's polynomial time.
If you allow negative $b\_i$ then it is hard (set all the $a\_i=1$, so the goal is to get the chosen subset of $b\_i$'s to sum to a number close to but greater than $-1$ in order to make the denominator as small as possible — this... | 4 | https://mathoverflow.net/users/440 | 226231 | 105,654 |
https://mathoverflow.net/questions/226218 | 2 | Let $H$ be the subgroup of $\text{GL}(n + k, \mathbb{R})$ consisting of matrices whose lower $n \times k$ block is empty; i.e. consisting of matrices of the form$$\begin{pmatrix} A & \* \\ 0 & B\end{pmatrix},$$where $A \in \text{GL}(n, \mathbb{R}$), $B \in \text{GL}(k, \mathbb{R})$, and $\*$ is arbitrary. I know that $... | https://mathoverflow.net/users/74305 | $E \times_H \mathbb{R}^n$ is isomorphic to the total space of the tautological bundle $\gamma^n$ over $G_n(\mathbb{R}^{n+k})$? | Write an element of $GL(n+k,\mathbb R)$ as $\begin{pmatrix}C & D\\ E& F\end{pmatrix}$, then the principal $H$-bundle sends this element to $\rm{Im}\begin{pmatrix}C \\ E\end{pmatrix}$ in the Grassmannian.
Write elements of $E\times\_H\mathbb R^n$ as $\left[ \begin{pmatrix}C & D\\ E& F\end{pmatrix}, x\right]$.
Write... | 3 | https://mathoverflow.net/users/83633 | 226232 | 105,655 |
https://mathoverflow.net/questions/226230 | 4 | Let $P$ be a smooth projective variety of dimension $4$ and let $Z$ be an irreducible subvariety of dimension $2$ ($Z$ is not necessarily smooth, but you can assume it).
>
> Is there a smooth, irreducible *ample* hypersurface $H \subseteq P$ such that $H$ contains $Z$?
>
>
>
I first tried to mimic Hartshorne... | https://mathoverflow.net/users/29730 | A Bertini-type result for hypersurfaces containing a subvariety | For non-smooth $Z$ the answer is in general **no**.
Take $P=\mathbb{P}^4$ and let $Z \subset \mathbb P^4$ be a surface with a non-normal double point $p$ (i.e. a singularity locally analytically isomorphic to the one given by two planes intersecting in a single point, it is no difficult to construct irreducible examp... | 9 | https://mathoverflow.net/users/7460 | 226234 | 105,656 |
https://mathoverflow.net/questions/226192 | 6 | I already asked this question [here](https://math.stackexchange.com/questions/1397339/another-equivalent-characterization-of-schwartz-function "Another equivalent characterization of Schwartz function?") on MSE, didn't get an answer, and I'm still stuck with it.
---
Suppose I have a smooth function $\psi$ from $... | https://mathoverflow.net/users/56727 | Is this function Schwartz? | I think the bound is sufficient to prove that $u$ is a Schwartz function.
Your assumption implies that $x^\alpha \Delta^p u$ is in $L^2$ for all $\alpha,k$. Thus for the Fourier transform $v$ of $u$ we have that $D^\alpha(|x|^{2p}v)$ is in $L^2$ for all $\alpha,p$. Take $\alpha=0$: we get that $|x|^{2p}v$ is in $L^2$... | 1 | https://mathoverflow.net/users/7294 | 226236 | 105,657 |
https://mathoverflow.net/questions/226219 | 2 | Can anyone provide an idea of the proof or a reference of the fact that a complex structure on the once punctured torus extends to one on the torus?
In other words, the Teichmuller space of the punctured torus is the same as that of the Torus (=upper half plane)?
| https://mathoverflow.net/users/12395 | Complex structure on a punctured torus giving a complex structure on the torus? | There's a subtlety you may be missing. By definition of a punctured Riemann surface, your "complex structure on the once-punctured torus" extends to a complex structure on the torus (it is not just a complex structure on the torus minus a point). So there's nothing to show. I suppose a less confusing terminology would ... | 7 | https://mathoverflow.net/users/25590 | 226237 | 105,658 |
https://mathoverflow.net/questions/226136 | 8 | Let $X$ be a smooth algebraic variety over $\mathbb{C}$. Is the sheaf of smooth functions on $X$ flat as an $\mathcal{O}\_X$ module?
| https://mathoverflow.net/users/58609 | Is the sheaf of smooth functions flat? | Yes, it is. First of all, the ring of germs of holomorphic functions $\mathcal{O}^h\_x$ is flat over the ring of germs regular functions $\mathcal{O}\_x$ at some point $x \in X$, see for example Taylor, "Several complex variables with Connections to Algebraic Geometry and Lie Groups", Theorem 13.3.5.
Secondly, $\math... | 14 | https://mathoverflow.net/users/49151 | 226238 | 105,659 |
https://mathoverflow.net/questions/226209 | 0 | Let $G$ be a finite group and let $M$ be a perfect $\mathbb{Q}[G]$-complex.
Suppose that $M\otimes\_{\mathbb{Q}[G]}\mathbb{Q}$ is quasi-isomrphic to $0$ can we conclude that $M$ is quasi-isomorphic to $0$ ? It will be nice if one provide a concrete counterexample in the negative case.
| https://mathoverflow.net/users/82229 | Perfect $Q[G]$-complex | Since all $\mathbb{Q}[G]$-modules are projective, every bounded complex of finitely generated modules is perfect. For $M\otimes\_{\mathbb{Q}[G]}\mathbb{Q}$ to be acyclic you just need that the homology of $M$ has no nonzero trivial summand. So, for example, take $G=C\_2$, and $M$ the one-dimensional $\mathbb{Q}[G]$ mod... | 4 | https://mathoverflow.net/users/22989 | 226239 | 105,660 |
https://mathoverflow.net/questions/226111 | 0 | Tanaka's theorem ([wikipedia](https://en.wikipedia.org/wiki/Tanaka%27s_formula)) implies that $X\_t = |B\_t|$ is a weak solution to the SDE
$dX\_t = dW\_t + dL\_t^0(X\_t)$,
where $W\_t$ is a Brownian motion and $L\_t^0(X\_t)$ is the local time of $X\_t$ at $0$. I have two related questions:
1. Does there exist a ... | https://mathoverflow.net/users/63913 | Weak existence for modified Tanaka SDE | For the first see ch 6 prob 2.24 of Revuz & Yor
| 0 | https://mathoverflow.net/users/nan | 226242 | 105,661 |
https://mathoverflow.net/questions/226241 | 0 | Let $G=(V,E)$ be a finite, simple, undirected graph. We call $D\subseteq V$ *cycle-intersecting* if for every [simple cycle](https://en.wikipedia.org/wiki/Cycle_(graph_theory)) $C\subseteq V$ we have $C\cap D \neq \emptyset$.
Is there a graph $G$ such that for every cycle-intersecting subset $D$ and for every simple ... | https://mathoverflow.net/users/8628 | Cycle-intersecting subsets | No. And this is not about cycles at all.
Take minimal cycle-intersecting subset. If it has at least two vertices from any cycle, remove any vertex from it and get a smaller cycle-intersecting subset.
| 3 | https://mathoverflow.net/users/4312 | 226244 | 105,662 |
https://mathoverflow.net/questions/226233 | 2 | Is the following statement true? If yes, how can find the solutions?
The equation
$$3x^2+8xy+7y^2\equiv-1\pmod p$$
has an integral solution for every prime $p>5$.
| https://mathoverflow.net/users/40723 | A quadratic Diophantine equation | Call $q$ your quadratic form, and $M$ the $\mathbf Z$-quadratic space $(\mathbf Z^2,q)$. It's discriminant is $-20$. Now let $p$ be a prime. Then $V\_p:=M\otimes \mathbf F\_p$ is a $2$-dimensional quadratic space over $\mathbf F\_p$, whose discriminant is the image of $-20$ in $\mathbf F\_p$. Thus this quadratic space ... | 4 | https://mathoverflow.net/users/39552 | 226249 | 105,664 |
https://mathoverflow.net/questions/226229 | 0 | In page-$13$ of [Graph minors. $X$. Obstructions to tree-decomposition](http://www.sciencedirect.com/science/article/pii/009589569190061N), $\gamma(G)$ introduced as maximum size of an edge. What is the definition of size of an edge$?$ I think it may be number of edges that incident on terminals of an edge.
| https://mathoverflow.net/users/83880 | What is the definition of size of an edge$?$ | At the beginning of the section in the paper they let $G$ be a *hypergraph*. The size of a (hyper)edge is the number of vertices it contains.
| 1 | https://mathoverflow.net/users/51668 | 226250 | 105,665 |
https://mathoverflow.net/questions/226228 | 3 | Let $k$ be fixed and consider the sum $$F(k,n)=\sum\_{p\_1<p\_2<\cdots<p\_k\leq n~:~p\_1 p\_2\cdots p\_k\leq n} \frac{1}{p\_1 p\_2 \cdots p\_k}.$$
Are tight upper and lower bounds on $F(k,n)$ known as $n\rightarrow \infty$? One approximation to the sum might be $$\left( \sum\_{p\leq n^{1/k}} \frac{1}{p}\right)^k\approx... | https://mathoverflow.net/users/17773 | Upper and lower bounds on reciprocals of restricted prime products | Let $\pi\_k(x)$ be the number of integers $\leq x$ with exactly $k$ prime factors. In the range $k<e\log\log x$, an asymptotic formula for $\pi\_k(x)$ was given by Sathe (J. Indian Math. Soc. (N.S.) 17, (1953), 63–82). Selberg (J. Indian Math. Soc. (N.S.) 18, (1954), 83–87) gave a much simpler proof. Sifting out non-sq... | 3 | https://mathoverflow.net/users/37555 | 226251 | 105,666 |
https://mathoverflow.net/questions/226253 | 6 | It is trivial that there are a lot of minimal surfaces in the flat $R^3$: for example, for any point, any 2-plane containing this point,
and any two othogonal vectors in this plane, and any negative number $K$ there exists a (in fact, infinitely many) minimal suface tangent to the plane at that point such that main cu... | https://mathoverflow.net/users/14515 | How many minimal surfaces do we have if the metric in the target space is not flat | The existence certainly remains true if the ambient metric is real-analytic, and this follows from the Cartan-Kähler Theorem since all you are asking for is local minimal surfaces.
In the case of a smooth metric, because the minimal surface equation is determined elliptic, the existence of a local solution attaining... | 5 | https://mathoverflow.net/users/13972 | 226263 | 105,668 |
https://mathoverflow.net/questions/226252 | 2 | Let $A = kQ/I$ be a bound quiver algebra for some algebraically closed field $k$, $Q$ a finite connected quiver without oriented cycles, and $I$ an admissible ideal. Say that $I'$ is also an admissible ideal such that $I \subseteq I'$.
My question is this: if $A$ is representation-finite, is $A' = kQ/I'$ also represe... | https://mathoverflow.net/users/84175 | Is a quotient of a bound quiver algebra of finite representation type also representation-finite? | There is indeed an easy way to see this.
Let A be a ring and let I be an ideal of A.
If A has only finitely many isomorphism classes of indecomposable modules then the ring A/I also has only finitely many isomorphism classes of indecomposable modules. The point is that an A/I-module is indecomposable iff it is indeco... | 2 | https://mathoverflow.net/users/1148 | 226265 | 105,669 |
https://mathoverflow.net/questions/225774 | 2 | Assume that given $n$ i.i.d samples $(X\_1, X\_2, ..., X\_n)$ drawn from $p\_X$, an unknown probability mass function defined over a finite alphabet $\mathcal{X}$, one wants to estimate $p\_X(x)$ for each $x \in \mathcal{X}$.
Assume that one estimates $(p\_X(x))\_{x \in \mathcal{X}}$ via the empirical distribution. I... | https://mathoverflow.net/users/nan | Literature question on the convergence rate of the empirical distribution | The minimax rate (in $\ell\_1$) for estimating the empirical distribution on an alphabet of size $d$ is $\Theta(\sqrt{d/n})$, where $n$ is the number of samples. See here for more details:
<http://arxiv.org/abs/1411.1467>
| 1 | https://mathoverflow.net/users/12518 | 226268 | 105,670 |
https://mathoverflow.net/questions/226198 | 0 | I was wondering about the following problem:
Assume we have a state space $S:=\mathbb{Z}$ and a Markov chain, such that we can go from any state $x$ to some state $y$ with positive probabilities, i.e. $p\_t(x,y)>0$ for any $t >0 $ and $x,y \in S.$
Let $T\_0^x$ be the hitting time to go from state $x$ to $0$.
If ... | https://mathoverflow.net/users/84176 | Finite hitting time implies hits at any finite time? | It's not clear what condition you are imposing, but I think the following example indicates that the answer is no, there are positive times so that you don't have to get to $0$ with uniformly bounded probability:
Arrange the states in a tree with root $0$ with disjoint paths of length $n$ connected to the root for e... | 0 | https://mathoverflow.net/users/2954 | 226271 | 105,671 |
https://mathoverflow.net/questions/226280 | 5 | What is the easiest/quickest way to see the following?
>
> If $n + 1 = 2^rm$ with $m$ odd, then there do not exist $2^r$ vector fields on the projective space $\mathbb{P}^n$ which are everywhere linearly independent.
>
>
>
Thanks!
| https://mathoverflow.net/users/nan | $n + 1 = 2^rm$ with $m$ odd $\implies$ do not exist $2^r$ vector fields on $\mathbb{P}^n$ that are everywhere linearly independent? | The standard way to treat statements like this one is to analyze the Stiefel-Whitney classes of $\mathbb P^n$. The full Stiefel-Whitney class of the tangent bundle is
$$w(T\mathbb P^n)=(1+u)^{n+1},$$
where $u\in H^1(\mathbb P^n,\mathbb Z\_2)$ is the generator.
In your particualr case $(n+1)=2^rm$ and let $p=2^r(m-1)$... | 5 | https://mathoverflow.net/users/40950 | 226283 | 105,675 |
https://mathoverflow.net/questions/226281 | 7 | I am currently learning the theory of Shimura varieties. Out of curiosity, is it known which number fields can occur as reflex fields? More precisely, can one find, for any number field, a positive dimensional Shimura variety which has this field as its reflex field?
| https://mathoverflow.net/users/39954 | Reflex fields of Shimura varieties | The answer depends on your definition of a Shimura pair $(G,X)$.
Look in Section 2.1 of [Deligne's paper](https://publications.ias.edu/sites/default/files/34_VarietesdeShimura.pdf).
If you assume only axioms (2.1.1.1), (2.1.1.2) and (2.1.1.3), then any number field $F$ can occur as the reflex field $E(G,X)$ with $X$ ... | 8 | https://mathoverflow.net/users/4149 | 226286 | 105,676 |
https://mathoverflow.net/questions/226270 | 3 | Let $G$ be a compact Lie group and $\mathfrak g$ its Lie algebra.
For any $f$ in the dual space $\mathfrak g^\*$, we can define a skew-symmetric bi-linear form on $\mathfrak g$ by $(A,B)\mapsto f([A,B])$.
Let us say that a differential 2-form on $G$ is a *commutator 2-form* if it has this structure at every point.
We c... | https://mathoverflow.net/users/55893 | Commutator 2-forms on Lie groups | Let $X$, $Y$, be left invariant vector fields, and let $f$ be a left invariant one-form. By Cartan's formula,
$$(df)(X,Y)=X(\underbrace{f(Y)}\_{\text{const}})-Y(\underbrace{f(X)}\_{\text{const}})-f([X,Y])=-f([X,Y])\;,$$
so your constant commutator $2$-forms are exact.
In the nonconstant case, Cartan's formula together ... | 5 | https://mathoverflow.net/users/70808 | 226294 | 105,680 |
https://mathoverflow.net/questions/226074 | 8 | Is it true that any compact piecewise linear homology manifold is homotopically equivalent to a (smooth?) manifold of the same dimension?
---
Let me say bit more since my question was wrongly understood.
1. Any link of homology manifold has to be a homoplogy sphere.
2. By double suspension every point on a simp... | https://mathoverflow.net/users/10330 | Any PL-homology-manifold is homotopy equivalent to a manifold | On the revised question:
I am not sure what you mean by doing the same in smooth category but
there are PL -manifolds that are not homotopy equivalent to smooth ones, see e.g. [M. Davis and J-C. Hausmann, Aspherical manifolds without smooth or PL structure, Springer Lecture Notes in Math. 1370, (1989), 135--142] avai... | 3 | https://mathoverflow.net/users/1573 | 226297 | 105,682 |
https://mathoverflow.net/questions/226278 | 16 | Let $\mu(n)$ denote the Möbius function and $\varphi(n)$ the Euler-phi function. What is known about $f(x) = \sum\_{n \leq x} \mu(n) \varphi(n)$? For example:
1. Is it known that $f(x)$ grows without bound?
2. Is it known that $-f(x)$ grows without bound?
3. Is it known that $f(x)$ crosses the origin infinitely often... | https://mathoverflow.net/users/44797 | What is known about $\sum_{n \leq x} \mu(n) \varphi(n)$? | Fleshing out Ofir Gorodetsky's comment: if we define $G(s) = \sum\_{n=1}^\infty \mu(n)\phi(n)n^{-s}$, then we have $G(s) = F(s)/\zeta(s-1)$ where
$$
F(s) = \prod\_p \bigg( 1 - \frac1{p^s-p} \bigg)
$$
is absolutely convergent for $\Re s>1$. The rightmost singularities of $G(s)$ are therefore at the points $1+\rho$ where... | 14 | https://mathoverflow.net/users/5091 | 226298 | 105,683 |
https://mathoverflow.net/questions/226299 | 11 | Let $A$ be an abelian surface and $\text{Km}(A)$ be the Kummer surface of $A$. If I remember correctly, the Picard number $\rho(\text{Km}(A))$ is equal to $16+\rho(A)$.
>
> Does anyone know any reference or proof for this fact?
>
>
>
| https://mathoverflow.net/users/84225 | The Picard number of the Kummer surface of an abelian surface | In the fourth section of this paper <http://www.staff.uni-bayreuth.de/~bt270951/quart9h.pdf> they sketch the proof.
The idea is that the orthogonal complement of the subspace generated by the sixteen $-2$ curves in $\mathrm{H}^2(Km(A),\mathbb{Z})$ is isomorphic to $\mathrm{H}^2(A,\mathbb{Z})$.
| 8 | https://mathoverflow.net/users/82672 | 226303 | 105,684 |
https://mathoverflow.net/questions/213149 | 2 | Suppose that $(X\_i)\_{i\in I}$ is a family satisfying the covering property $\Omega \choose \text{T}$ (for the definition of this covering property, see [this post](https://mathoverflow.net/questions/209944/implications-between-different-covering-properties-of-spaces)).
Does $\prod\_{i\in I} X\_i$ necessarily satisf... | https://mathoverflow.net/users/8628 | Is the covering property $\Omega \choose \text{T}$ closed under products? | The answer is "No." This is yet another instance where selection principles are useful. One uses knowledge on more understood, related properties, to answer questions on less understood ones.
First, if you allow $I$ to be infinite, then notice that $\mathbb{N}$ (with the discrete topology) satisfies $\binom{\Omega}{\... | 2 | https://mathoverflow.net/users/2415 | 226316 | 105,688 |
https://mathoverflow.net/questions/226323 | 5 | Let $X$ and $Y$ be complex Banach spaces and $B(X,Y)$ be the Banach space
of all bounded operators. An operator $T\in B(X,Y)$ is weakly compact if
$T(\{ x\in X;\; \| x\| \leq 1\})$ is relatively compact in the weak topology
of $Y$. If $X$ or $Y$ is reflexive, then every operator in $B(X,Y)$ is weakly
compact. I guess t... | https://mathoverflow.net/users/64556 | Weakly compact operators between Banach spaces | The fact that each $T\in B(X,Y)$ is weakly compact does not imply $X$ or $Y$ reflexive. For example, every non-weakly compact operator $T:\ell\_\infty\to Y$ is an isomorphism on a subspace isomorphic to $\ell\_\infty$ (See Prop. 2.f.4 in Classical Banach spaces I, by Lindenstrauss ans Tzafriri).
Thus if $Y$ is a non... | 11 | https://mathoverflow.net/users/39421 | 226326 | 105,693 |
https://mathoverflow.net/questions/226328 | 6 | A pair of vector bundles over a base space $X$ is a pair $(E,F)$ where $E$ is a vector bundle over $X$ and $F$ is a sub-bundle of $E$. Two pairs $(E\_{1},F\_{1})$ and $(E\_{2}, F\_{2})$ are isomorphic if there is an isomorphism from $E\_{1}$ to $E\_{2}$ which send $F\_{1}$ onto $F\_{2}$. The direct sum of two pairs has... | https://mathoverflow.net/users/36688 | Relative Characteristic classes | You obviously have all characteristic classes of $E$, of $F$, and of $E/F$,
with some relations because $E\cong F\oplus E/F$. To see if there are more, consider the classifying space for your pairs. If you are working over $\Bbbk$, $\mathrm{rk}(F)=k$, $\mathrm{rk}(E)=\ell$, it is the colimit over $n$ of the space of pa... | 8 | https://mathoverflow.net/users/70808 | 226329 | 105,694 |
https://mathoverflow.net/questions/226257 | 12 | let $G$ be a transitive permutation group acting on $\{1, \ldots, n\}$, and let $d(G)$ be the minimal number of generators of $G$. Is it true, that for $n\rightarrow\infty$ we have $\frac{d(G)\log|G|}{n^2}\rightarrow 0$? If this is true, can you give a complete list of groups with $\frac{d(G)\log |G|}{n^2}\geq \frac{\l... | https://mathoverflow.net/users/37555 | Can a large transitive permutation group need many generators? | Let $G$ be a permutation group of degree n, and let $r>1$ be the minimal block size of $G$. Also, let $s:=n/r$, so that G may be viewed as a subgroup in the wreath product $R\wr S$, where $R\le Sym(r)$ is primitive, and $S=\pi(G)\le Sym(s)$ is transitive ($\pi$ here denotes projection $G\rightarrow Sym(s)$). Then
$$d(... | 11 | https://mathoverflow.net/users/84245 | 226337 | 105,697 |
https://mathoverflow.net/questions/225224 | 3 | It's stated in Väisälä's 'Lectures on n-dimensional quasiconformal mappings' (p. 20) that, in the geometric definition of a quasiconformal mapping, that the modulus of a family of curves associated to a ring is unaffected by considering admissible functions $\rho$ which are *continuous* and not just measurable. No furt... | https://mathoverflow.net/users/83625 | Assuming admissible functions $\rho$ are continuous in definition of conformal modulus | Gehring shows (for p=n=3) in the first equality of Theorem 1 in his paper
Extremal length definitions for the conformal capacity of rings in space.
<https://projecteuclid.org/euclid.mmj/1028998672>
that the conformal modulus of a ring domain $R$ (defined via measurable functions) can equivalently be defined as the ... | 2 | https://mathoverflow.net/users/66777 | 226344 | 105,700 |
https://mathoverflow.net/questions/221962 | 5 | It seems to me that in a statically typed, object oriented language, there is a striking similarity to wiring diagrams. Wires (objects) of type $X$ go into functions (boxes) of input type $X$. Is the corresponding string calculus that of a symmetric monoidal category?
Edit: it has been suggested that I formulate a mo... | https://mathoverflow.net/users/10007 | Monoidal cats and string diagrams for a semantics of object oriented programming languages | No full answer but just some random thoughts on the issue:
Concerning the string diagram calculus: It should be that of a mutlicategory (see David Spivaks paper on the Wiring Operad: <http://arxiv.org/abs/1305.0297>).
Plus I think there's a hidden issue here: Private member variables are only of real concern if the... | 4 | https://mathoverflow.net/users/1261 | 226354 | 105,706 |
https://mathoverflow.net/questions/226343 | 18 | Let $A$ be a commutative ring, and $L, M, N$ be $A$-modules. Then is it true that
$$\text{Hom}\_A (L, M)\otimes\_A N \cong \text{Hom}\_A (L, M\otimes\_A N)$$
as $A$-modules?
(Note that there is a natural morphism from the left to the right, I think it's not easy to check it is injective or surjective, but I didn't r... | https://mathoverflow.net/users/42571 | Does module Hom commute with tensor product in the second variable? | The example $A=M=\mathbb{Z}$, $L=N=\mathbb{Q}$ shows that the answer is negative: we have $\text{Hom}(L,M)=0$ so $\text{Hom}\_A(L,M)\otimes\_AN=0$, but $\text{Hom}\_A(L,M\otimes\_AN)=\text{Hom}\_{\mathbb{Z}}(\mathbb{Q},\mathbb{Q})=\mathbb{Q}\neq 0$
| 26 | https://mathoverflow.net/users/10366 | 226358 | 105,707 |
https://mathoverflow.net/questions/226375 | 5 | Suppose $f(x,t)\in\mathbb{Q}(t)[x]$ is an irreducible polynomial with Galois group G. For any rational number $a$ we may consider the polynomial $f(x,a)\in\mathbb Q[x]$ and its corresponding Galois group $G\_a$, which is a subgroup of $G$. By the Hilbert Irreducibility Theorem, the groups $G$ and $G\_a$ are the same ou... | https://mathoverflow.net/users/46987 | Exceptional specializations of Galois groups in the Hilbert Irreducibility Theorem | In some sense it can be described quite explicitly. There is a finite set of curves $C\_i$ and maps $\phi\_i:C\_i\to\mathbb P^1$ of degree at least $2$ defined over $\mathbb Q$ such that the desired thin set is contained in the union
$$ \bigcup\_{i=1}^n \phi\_i\bigl(C\_i(\mathbb Q)\bigr).$$
And I'm pretty sure that one... | 7 | https://mathoverflow.net/users/11926 | 226379 | 105,714 |
https://mathoverflow.net/questions/226383 | 28 | Let $K$ be a number field, and let
$$\zeta\_{K}(s):= \sum\_{0
\neq I \text{ ideal of }O\_K} \frac{1}{N\_{K/\mathbb{Q}}(I)^s} = \sum\_{n \ge 1} \frac{a\_n}{n^s}$$
be the Dedekind zeta function of $K$. The quantity $s\_K(x):=\sum\_{n \le x} a\_n$ counts ideals of $O\_K$ of norm up to $x$.
$\zeta\_K$ is analytic in $s... | https://mathoverflow.net/users/31469 | What's special about the circle problem? | There is nothing special about the circle problem (except of course that it goes back to Gauss)! The problem for number fields has been extensively investigated, and goes back to Landau. If the number field has degree $k$, then the problem is quite analogous to the error term in the $k$-divisor problem. One expects tha... | 28 | https://mathoverflow.net/users/38624 | 226389 | 105,716 |
https://mathoverflow.net/questions/226371 | 1 | I'm trying to understand the proof of Theorem II.1.7 in Kollar's Rational curves on algebraic varieties. In particular, there is a claim in there that I can't make sense of.
The setting is the following (we slightly simplify the statement). Let $C/S$ be a flat projective curve without embedded points and $Y/S$ a smoo... | https://mathoverflow.net/users/61541 | Conormal bundle of the strict transform | There are definitely some typos in that proof, but for your question you may assume that $S=\{s\}$ which will take care most of them.
If you write down the restriction of differential forms sequence (the dual of the normal bundle sequence) for both $\Gamma\subset X$ and $\Gamma'\subset X'$ and compare them, then you ... | 2 | https://mathoverflow.net/users/10076 | 226391 | 105,717 |
https://mathoverflow.net/questions/226169 | 38 | This question does *not* concern the comparative merits of standard (SA) and nonstandard (NSA) analysis but rather a comparison of different approaches to NSA. What are the concrete advantages of the abstract approaches to NSA (e.g., via the compactness theorem), as compared to the more concrete approach using ultrapow... | https://mathoverflow.net/users/28128 | What are the advantages of the more abstract approaches to nonstandard analysis? | To my way of thinking, there are at least three distinct
perspectives one can naturally take on when undertaking work in
nonstandard analysis. In addition, each of these perspectives can
be varied on two other dimensions, independently. Those dimensions are, first,
the order of nonstandardness (whether one
wants nonsta... | 29 | https://mathoverflow.net/users/1946 | 226393 | 105,718 |
https://mathoverflow.net/questions/226394 | 3 | Let a compact Lie group $G$ acts on a closed symplectic manifold $(M,\omega)$. If the action is Hamiltonian with $\mu$ the moment map, then the integral $$\int\_M e^{i\mu (X)+\omega}$$ is equal to the first term in the stationary phase approximation.
My question is if there is a similar integral formula for a compact... | https://mathoverflow.net/users/18261 | Duistermaat-Heckman integral formula on compact manifold with boundary | There is the paper by [E. Prato and and S. Wu](http://arxiv.org/pdf/alg-geom/9307005.pdf), 1993 which deals with (at least) a special case of this.
| 1 | https://mathoverflow.net/users/11142 | 226397 | 105,719 |
https://mathoverflow.net/questions/226356 | 5 | If $k$ is an ordered field, the least ordinal $s(k)$ which doesn't embed in $(k,<)$ is regular. This is because every interval of an ordered field embeds in every infinite interval so given a strictly increasing map $f: \alpha < s(k) \rightarrow s(k)$ you can define an embedding $(\sup(f(\alpha)),\in) \rightarrow (k,<)... | https://mathoverflow.net/users/45005 | Ordinals which embed in surreal subfields | I claim that $s(\text{No}(\lambda))=\lambda^+$.
To see this, first let me mention that it is a standard exercise
in elementary set theory to prove that there is no increasing or
decreasing $\lambda^+$-sequence in the set ${}^\lambda 2$ of all
binary $\lambda$-sequences, ordered in the lexical order, so that
$s<t$ jus... | 4 | https://mathoverflow.net/users/1946 | 226399 | 105,720 |
https://mathoverflow.net/questions/225724 | 6 | A result of Helgason's is that any self-intersecting geodesic in a Riemannian globally symmetric space is simple and closed. To what extent does this generalise to Riemannian naturally reductive homogeneous spaces with the canonical connection?
| https://mathoverflow.net/users/14454 | Self-Intersecting Geodesics in Homogeneous Spaces | Every geodesic loop on a compact homogeneous space is a closed geodesic. Let c be such a loop c(L)=c(0). Take (n-1) Killing vector fields $X\_i$ whose value at p are a basis of $c'(0)^\perp$. Then $X\_i$ restricted to c is a Jacobi field and hence $<X\_i(c(t),c'(t)>$ is linear. Since X has bounded length, $<X\_i(c(t),c... | 6 | https://mathoverflow.net/users/84274 | 226400 | 105,721 |
https://mathoverflow.net/questions/226388 | 5 | A simple random walk $S\_n = X\_1 +\cdots +X\_n$, where $P(X\_i = 1) = p \not = 0.5$ and $P(X\_i=-1)= q \triangleq 1-p$, admits the following probability
$$P(S\_n \textrm{ reaches } a \textrm{ before} -b) = \left(\frac{1-\left(\frac{q}{p}\right)^{b}}{1-\left(\frac{q}{p}\right)^{a+b}}\right),$$
for $a$ and $b$ positiv... | https://mathoverflow.net/users/51757 | Random walk with continuously distributed steps on [-1,1] | As already pointed out by Anthony in a comment, you can't really expect explicit formulae. Let's write $p(x)$ for the probability that the RW starting at $x\in (a,b)$ hits $a$ before it hits $b$. Then, by the same argument as in the discrete case (condition on what happens on the very next step),
$p(x) = \int p(x+t)f(t... | 4 | https://mathoverflow.net/users/48839 | 226404 | 105,722 |
https://mathoverflow.net/questions/226417 | 4 | Suppose that $\{a\_{i}\}\_{i\ge1}$ and $\{b\_{i}\}\_{i\ge1}$ are sequences
of natural numbers such that for each $x\in\overline{\mathbb{F}\_{3}}^{\*}$
there is a natural number $I\_{x}$ satisfying that $x^{a\_{i}}+x^{b\_{i}}+1=0$
for each $i\ge I\_{x}$. Does it follows that for each $x\in\overline{\mathbb{F}\_{3}}^{\*}... | https://mathoverflow.net/users/4948 | A toy question on solutions to a sequence of trinomials over $\mathbb{F}_3$ | For any natural $k$, non divisible by 3, there exists $n$ such that each root of $x^{k}-1$ is a root of $x^{a\_i}+x^{b\_i}+1$. Since $f\_k(x)=x^k-1$ does not have double roots (its derivative $f\_k'=kx^{k-1}$ is coprime to $f\_k$), we conclude that $f\_k$ divides $x^{a\_i}+x^{b\_i}+1$. Reduction of $x^a$ modulo $x^k-1$... | 2 | https://mathoverflow.net/users/4312 | 226420 | 105,727 |
https://mathoverflow.net/questions/226365 | 6 | Suppose that $A = kQ/I$ is a bound quiver algebra for $k$ an algebraically closed field, $Q=(Q\_0, Q\_1)$ a finite connected quiver with no oriented cycles with no multiple edges or self-loops, and $I$ the ideal of commutative relations when viewing $Q$ as a commutative diagram (say of vector spaces and linear maps ove... | https://mathoverflow.net/users/84175 | Is this modified bound quiver algebra necessarily representation-finite? | The quiver
$$\begin{array}{ccccc}
&&1&\to&2\\
&&\uparrow&&\uparrow&\\
3&\to&4&&5\\
&&\uparrow&&\\
&&6&&
\end{array}$$
is a Dynkin quiver of type $D\_6$, so has finite representation type. But if you add an arrow from vertex $4$ to vertex $5$, together with commutation relations, it has infinite representation type, sin... | 9 | https://mathoverflow.net/users/22989 | 226423 | 105,729 |
https://mathoverflow.net/questions/226434 | 3 | Everything over $\Bbb{C}$. Say we have a smooth curve $C$ of genus $10$ which is a double cover of a smooth plane cubic curve. Therefore $C$ admits a 1-dimensional family of pencils of degree 4 (arising from the involutions on the cubic).
Can we deduce from this that $C$ does not admit any pencil of degree $3$ ? (in ... | https://mathoverflow.net/users/40038 | Existence of pencils on some special curves of genus 10 | For every integer $p\_a>4$, there does not exist a smooth, projective, geometrically connected curve of genus $p\_a$ that admits both a degree $2$, finite, flat morphism, $f:C\to E$, to a smooth plane cubic $E$ and a degree $3$, finite, flat morphism, $g:C\to \mathbb{P}^1$, to the projective line. Probably this can be ... | 5 | https://mathoverflow.net/users/13265 | 226437 | 105,733 |
https://mathoverflow.net/questions/226427 | 8 | Let $X$ be Polish. It is known that every analytic and coanalytic subset of $X$ is universally measurable. The [Wikipedia article](https://en.wikipedia.org/wiki/Universally_measurable_set) about universally measurable sets notes that (assuming projective determinance) every set in the projective hierarchy is universall... | https://mathoverflow.net/users/58682 | Relation between projective hierarchy and universally measurable sets | Under the continuum hypothesis, or even just when
$2^{\aleph\_0}<2^{\aleph\_1}$, there must be universally measurable
sets that are not projective. The reason is that Hausdorff proved
that there are always universally null sets of size $\aleph\_1$,
and since any subset of such a set is also universally null and
hence u... | 10 | https://mathoverflow.net/users/1946 | 226440 | 105,735 |
https://mathoverflow.net/questions/226436 | 2 | I have 2 symplectic matrices $X\_{1},X\_{2} \in \mathbb{R}^{2n\times2n}$. The matrix $X=X\_{1} \cdot X\_{2}$ is also symplectic.
**Question:** Are there any theorems which allow me to express eigenvalues of $X$ if I know eigenvalues of each $X\_i$?
**P.S.** Actually I am interested in some numerical algorithms. I... | https://mathoverflow.net/users/84292 | Eigenvalues of product of symplectic matrices | Answering to your edit: there are numerical methods to compute the eigenvalues of a matrix product without forming it first. This has stability advantages with respect to forming the product explicitly. There is an excellent review by D.S. Watkins, *Product eigenvalue problems*, 2005, Siam review. As far as I know ther... | 2 | https://mathoverflow.net/users/1898 | 226448 | 105,742 |
https://mathoverflow.net/questions/226455 | 8 | Let $M$ be a complete and noncompact Riemannian manifold. Fix a point $p$ in $M$. Let $\gamma$: $[0, L]\rightarrow M$
(parametrized by its arc length) be a geodesic starting from $p$. Denote by $d(\cdot, \cdot)$ the distance function on $M$ induced by
the Riemannian metric. $\gamma$ is minimal if $d(\gamma(s\_1), \gam... | https://mathoverflow.net/users/83838 | Can a complete manifold have an uncountable number of ends? | The answer is yes. Consider a hyperbolic pair of pants where all three boundary circles are of the same size. Glue countably many of them together such each new one is glued to the existing manifold along exactly one boundary circle. Then the manifold looks like the boundary of a fattened tree.
Fix $p$ in one of the ... | 17 | https://mathoverflow.net/users/70808 | 226458 | 105,744 |
https://mathoverflow.net/questions/226449 | 15 | While browsing the ***Database of Number Fields***, I came across ***[17T8](http://galoisdb.math.upb.de/search/display?req=p413)***. It only had four equations, one of which is,
$$\small{x^{17} - 5x^{16} + 40x^{15} - 140x^{14} + 610x^{13} - 1622x^{12} + 4870x^{11} - 10220x^{10} + 22720x^9 - 38080x^8 + 63500x^7 - 8410... | https://mathoverflow.net/users/12905 | What's so special about these $17$th deg equations? | Many of the polynomials in the Klueners-Malle database and also in my
[database with John Jones](http://arxiv.org/abs/1404.0266) come from families in the way you correctly
describe. So you have "reverse engineered" the source family.
This particular source family is the first of two similar families
described in Sec... | 23 | https://mathoverflow.net/users/84296 | 226460 | 105,745 |
https://mathoverflow.net/questions/214878 | 5 | Let $\mathsf{d}^\star$ be the asymptotic upper density, defined on the power set of positive integers $\mathbf{N}^+$, so that
$$
\mathsf{d}^\star\colon \mathcal{P}(\mathbf{N}^+) \to\mathbf{R}\colon X\mapsto \limsup\_{n\to \infty} \frac{|X\cap [1,n]|}{n}.
$$
It is easy to verify that if $k\cdot \mathbf{N}^++h:=\{kx+h\co... | https://mathoverflow.net/users/32898 | Additivity of upper densities with respect to arithmetic progressions of integers | The answer is in the negative.
Let $f$ and $g$ be two upper densities (in the sense of the OP), and let $\alpha \in [0,1]$ and $q \in [1,\infty[$. Then the function
$$h := (\alpha f^q + (1-\alpha) g^q)^{\frac{1}{q}}$$
is an upper density too (in particular, condition (F3) follows from Minkowski's inequality, which ... | 2 | https://mathoverflow.net/users/16537 | 226461 | 105,746 |
https://mathoverflow.net/questions/226451 | 5 | How do I see that the set $\mathfrak{N}\_4$ consisting of all unoriented cobordism classes of smooth closed $4$-manifolds contains at least four distinct elements?
| https://mathoverflow.net/users/nan | Does $\mathfrak{N}_4$ contain at least four distinct elements? | Thom proved that the unoriented bordism is a polynomial ring over $\mathbb{F}\_2$ generated by elements $x\_i$ with $i$ running over all numbers not of the form $2^k-1$. Thus, $\mathfrak{N}\_4 = \mathbb{F}\_2 \cdot \{x\_2^2, x\_4\}$. Thom moreover proves that the even $x\_i$ can be represented by $\mathbb{RP}^i$.
Th... | 14 | https://mathoverflow.net/users/2039 | 226462 | 105,747 |
https://mathoverflow.net/questions/226450 | 1 | (I have asked a similar question in [MSE](https://math.stackexchange.com/questions/1572152) around a week ago, but did not receive any responses. I have therefore cross-posted it to this site, hoping to get some answers.)
An odd perfect number $N$ is said to be given in **Eulerian form** if $N = {q^k}{n^2}$ where $q$... | https://mathoverflow.net/users/10365 | If $N = {q^k}{n^2}$ is an odd perfect number given in Eulerian form, is $n$ squarefree? | No, if an odd perfect number exists, then $n$ must contain a square factor. This is a 1937 result of Steuerwald:
R. Steuerwald, "Verschärfung einer notwendigen Bedingung für die Existenz einer
ungeraden vollkommenen Zahl," S.-B. Math.-Nat. Abt. Bayer. Akad. Wiss., 1937, pp. 68-73.
A very nice recent paper on this t... | 7 | https://mathoverflow.net/users/16510 | 226466 | 105,749 |
https://mathoverflow.net/questions/226503 | 5 | Is there a repository of cospectral non-isomorphic graphs available somewhere?
I am looking for list of $0/1$ adjacency matrix pairs that can be input data in tools such as MATLAB.
| https://mathoverflow.net/users/10035 | Database of adjacency matrices on cospectral non-isomorphic graph pairs | The simplest source of cospectral graphs is lists of strongly regular graphs, lots of which are easily available from Ted Spence's web page at <http://www.maths.gla.ac.uk/~es/srgraphs.php>.
Otherwise you can use Sage to generate small graphs (up to 10 or so vertices) and then filter out cospectral pairs or groups. I ... | 6 | https://mathoverflow.net/users/1492 | 226506 | 105,769 |
https://mathoverflow.net/questions/226508 | 1 | Let $\Omega$ be a bounded smooth domain in $\mathbb{R}^d$ and let $L$ be a uniformly elliptic second order partial differential operator:
$$Lu(x,t)=-\sum\_{i,j=1}^{d}{a\_{ij}(x,t)u\_{x\_{i}x\_{j}}(x,t)}+\sum\_{i=1}^{d}{b\_{i}(x,t)u\_{x\_i}(x,t)}$$
where $x\in\Omega$ and $t>0.$
My question is:
Does exist a positive eig... | https://mathoverflow.net/users/83215 | Eigenfunction of an uniformly elliptic second order operator | This happens for operators ```in divergence form'', i.e., Laplace operators associated to a Riemann metric on $\Omega$. If the Riemann metric is described by the tensor $(g\_{ij}(x))\_{1\leq i,j \leq d}$ satisfying the positivity condition: $\exists c>0$ $\newcommand{\bR}{\mathbb{R}}$
$$ \sum\_{ij}g(ij)(x)\xi\_i\xi\_... | 1 | https://mathoverflow.net/users/20302 | 226509 | 105,771 |
https://mathoverflow.net/questions/225840 | 0 | I dont really understand how to reduce the index of DAEs ?
Does Reducing the index of DAE result in an ODE ?
How would I reduce the index of the DAE by Hand ?
Say I have :
$$
\begin{matrix}
E & 0 & 0 \\
0 & M & C' \\
0 & C & 0 \\
\end{matrix}
\*
\begin{bmatrix}
\dot{y} \\
\ddot{y} \\
-\lambda
\end{bmatri... | https://mathoverflow.net/users/83151 | Index Reduction of Differential Algebraic Equations by Hand | One intuitive way to understand a [DAE](http://www.scholarpedia.org/article/Differential-algebraic_equations) is to interpret it as a dynamical system which can be controlled by some input signals, whose output signals have to satisfy some (equational) constraints. For a typical multibody system, the input signals are ... | 3 | https://mathoverflow.net/users/20781 | 226517 | 105,772 |
https://mathoverflow.net/questions/226525 | 3 | Qing Liu's "Algebraic Geometry and Arithmetic Curves" page 299 COrollary 7.4.41 gives the following result.
Let $X$ be a smooth, connected, projective curve over an algebraically closed field $k$, of genus $g$. Let $Pic^0(X)$ denote the subgroup of $Pic(X)$ consisting of divisors of degree $0$. Let $n\in \mathbb{Z}$ ... | https://mathoverflow.net/users/24965 | Is $Pic^0(X)$ of a curve of genus $\geq 1$ over a non-algebraically closed field still non-finitely generated? | As Will Sawin says, the key phrase here is Mordell-Weil theorem. Also, this isn't really a theorem about Picard groups of curves, it's a theorem about abelian varieties. Here is a fairly general statement:
**Theorem** (Mordell-Weil-Lang-Neron) *Let $K$ be a field that is of finite type over its prime field (where the... | 9 | https://mathoverflow.net/users/11926 | 226530 | 105,779 |
https://mathoverflow.net/questions/226528 | 1 | The usual fixed point property can be interpreted in terms of non empty intersection of the graph of all maps with the graph of the identity map.
This motivates us to consider the following "weak fixed point property" by replacing the identity map with another continuous map:
**A weak fixed point property $"P"$:**
... | https://mathoverflow.net/users/36688 | A weak fixed point property | Your property is actually equivalent to the usual FPP. For, if a space $X$ does not satisfy FPP, that is there is a $\phi:X\to X$ with no fixed points, then, for any $F:X\to X$ the map $g:=\phi\circ F$ coincide with $F$ at no point: for any $x\_0$, $g(x\_0)=\phi(F(x\_0))\neq F(x\_0)$, so $X$ fails to satisfy property $... | 4 | https://mathoverflow.net/users/6101 | 226533 | 105,780 |
https://mathoverflow.net/questions/226534 | 7 | What is known about combinatorial structure of the rational maps of degree 2 over finite fields? From some general reasons I think it was studied. For being more specific, consider the field $\mathbb{F}\_{2^n}$ and the map $x\rightarrow f(x)=x+1/x$. We may consider such a map as oriented graph with outdegrees 1 (except... | https://mathoverflow.net/users/4312 | Algebraic dynamics in finite fields | There is a growing body of literature on dynamics of rational maps over finite fields. The following paper would seem to be relevant.
* Ugolini, S., Graphs associated with the map $x\mapsto x+x^{-1}$ in finite fields of characteristic two, *Theory and applications of finite fields*, Contemp. Math. **579**, 197-204, ... | 14 | https://mathoverflow.net/users/11926 | 226535 | 105,781 |
https://mathoverflow.net/questions/226243 | 7 | Let $X$ be a Banach space and let $p\in (1,\infty)$. If $q$ denotes the conjugate exponent to $p$, then $L\_q(X^\*)$ is easily seen to be isometric to a subspace of $(L\_p(X))^\*$ via the map $$f\mapsto \int\limits\_{[0,1]}\langle \cdot(\omega), f(\omega)\rangle\,{\rm d}\omega\quad (f\in L\_q(X^\*)).$$ One of the numer... | https://mathoverflow.net/users/15129 | Is $L_q(X^*)$ complemented in $(L_p(X))^*$? | Still no answers? Perhaps we should try the following.
Let $g : [0,1] \to X^\*$ be weak\* measurable, in the sense that, for every $x \in X$, the function
$$
\omega \mapsto \langle x, g(\omega)\rangle
$$
is measurable. Then there exist weak\* measurable functions $g\_1, g\_2 : [0,1] \to X^\*$ such that
(1) $g$ is... | 3 | https://mathoverflow.net/users/454 | 226538 | 105,782 |
https://mathoverflow.net/questions/226555 | 7 | The matrix $\begin{bmatrix}1 & 0 \\ 0 & -1\end{bmatrix}$ is orthogonal and indefinite.
$\begin{bmatrix}1 & 0 \\ 0 & 2\end{bmatrix}$ is positive definite and not orthonormal.
and the Identity matrix $I$ is of course both orthogonal and positive definite. Let $S$ be the intersection of orthogonal matrices and positive ... | https://mathoverflow.net/users/29887 | The space of positive definite orthogonal matrices | You may find the Cayley transform to be useful here:
As is well-known and easy to prove, every orthogonal $n$-by-$n$ matrix $R$ that does not have $-1$ as an eigenvalue can be written uniquely in the form
$$
R = (I-A)(I+A)^{-1}
$$
for some anti-symmetric matrix $A$ for which $I+A$ is invertible, and, conversely, if $... | 14 | https://mathoverflow.net/users/13972 | 226567 | 105,791 |
https://mathoverflow.net/questions/124242 | 2 | Let's consider an asymmetric Random Walk on $Z$, with transition probabilities $p\_{i, i+1}=p$, $~~p\_{i, i+1}=q$, $\forall i \in \mathcal{Z}$, $p+q=1$ and $p>q$.
I am interested in the probability of the first hitting time of a bareer in $i=n$, assuming that the walk started from $i=0$ at time $t=0$. Is there an exp... | https://mathoverflow.net/users/26798 | asymmetric random walk, hitting time probability | The explicit formula is: $P[N\_m=n]=(m/n)P[S\_n=m]$, where $P[N\_m=n]$ is the
probability the position $m$ is hit after exactly $n$ steps,
$S\_n = X\_1+X\_2+\dots X\_n$ and $P[S\_n=m]$ is the probablity after $n$ steps
the path to be at the position $m$. This last is well-known and is given
by
$$P[S\_n=m]=\frac{n!}{... | 4 | https://mathoverflow.net/users/84354 | 226578 | 105,797 |
https://mathoverflow.net/questions/226583 | 10 | Given a graph $G$ with $n$ vertices, that has a perfect matching $M$, what is the maximal number of edges that $G$ can have without contradicting the uniqueness of $M$?
Are examples of such extremal graphs known?
| https://mathoverflow.net/users/31310 | Densest Graphs with Unique Perfect Matching | Let $G$ be a graph with $2n$ vertices with a unique perfect matching $M$. Choose any $2$ edges from $M$ and consider the $4$ vertices present. In addition to the $2$ edges from the matching there can be at most $2$ more edges if we want the matching to be unique. Thus the number of edges must be less than or equal to $... | 12 | https://mathoverflow.net/users/51668 | 226587 | 105,799 |
https://mathoverflow.net/questions/226537 | 22 | This is maybe a little basic for MathOverflow, but I'm hoping it will get some interesting answers.
Let $\unrhd$ be the dominance order on partitions of $n \in \mathbb{N}$.
For partitions $\lambda$ and $\mu$ of $n$, the *Kostka Number* $K\_{\lambda\mu}$ is the number of semistandard Young tableaux of shape $\lambda$ ... | https://mathoverflow.net/users/7709 | Is there a short proof that the Kostka number $K_{\lambda \mu}$ is non-zero whenever $\lambda$ dominates $\mu$? | I think the following is a simple combinatorial argument which constructs the most dominant semistandard $\lambda$-tableau of content $\mu$ whenever $\lambda\trianglerighteq\mu$. (n.b. I haven't followed the reference given in Richard Stanley's comment, so I don't know whether I'm duplicating what's done there.)
In a... | 15 | https://mathoverflow.net/users/6771 | 226600 | 105,807 |
https://mathoverflow.net/questions/226579 | 0 | Goldbach's conjecture asserts that every even integer greater than $3$ is the sum of two primes, while de Polignac's one says every even positive integer is a prime gap infinitely often. My question is thus: which even positive integers are known to be both prime gaps (at least once) and the sum of 2 primes? Can we pro... | https://mathoverflow.net/users/13625 | Which even numbers are known to be both prime gaps and the sum of 2 primes? | Goldbach's Conjecture has been verified for all numbers up to $4 \times 10^{17}$ by [Tomás Oliveira e Silva](http://sweet.ua.pt/tos/goldbach.html). On the other hand, [Thomas R. Nicely](http://www.trnicely.net/gaps/gaplist.html) has shown that all even numbers up to $2000$ occur as prime gaps. This includes a list of a... | 11 | https://mathoverflow.net/users/2233 | 226605 | 105,809 |
https://mathoverflow.net/questions/226586 | 3 | There is a fairly rich classification on graphs with respect to the existence of Hamiltonian cycles either in unmodified graphs or after certain small modifications.
Do there also exist such classifications with respect to perfect matchings?
Specifically, I would like to know, whether there exist
* "Hypo-Matching"... | https://mathoverflow.net/users/31310 | "Hypo" and "Hyper" for Perfect Matching | A hypo-matching graph cannot exist. Assume $G$ does not contain a perfect matching, but $G - \{u,v\}$ does. This means $uv$ is not an edge is $G$ or else $G$ has a perfect matching. This cannot hold for an arbitrary pair of vertices since that would force $G$ to be edgeless.
A hyper-matching graph definitely exists. ... | 4 | https://mathoverflow.net/users/51668 | 226609 | 105,812 |
https://mathoverflow.net/questions/226403 | 11 | On a (simply connected) domain $\Omega$ for a smooth vector field $F\colon \Omega \to \mathbb{R}^3$, when does $\nabla\times(\nabla\times F)=0$ imply $\nabla \times F=0$. I know that $n\cdot(\nabla\times F)=0$ on $\partial\Omega$ is sufficient, and also $t\cdot(\nabla\times F)=0$ is sufficient ($t$ the tangential). Is ... | https://mathoverflow.net/users/75786 | When does $\nabla\times(\nabla\times F)=0$ imply $\nabla \times F=0$ | Here are some basic thoughts. Let $G$ be a vector field which is of the form $\nabla \times F$, and also obeys $\nabla \times G = 0$.
Since $\nabla \times G = 0$, the vector field $G$ is locally of the form $\nabla h$ for some scalar valued function $h$. The condition that $G = \nabla \times F$ imples that $\nabla \c... | 5 | https://mathoverflow.net/users/297 | 226628 | 105,817 |
https://mathoverflow.net/questions/226568 | 12 | Let $\mathbb{R}$ be the real field. For any homogeneous polynomial $f(X\_1,\cdots,X\_n)$ in $\mathbb{R}[X\_1,\cdots,X\_n]$, we use $S\_f(X\_1,\cdots,X\_n)$ to denote the following homogeneous symmetric polynomial:
$$S\_f(X\_1,\cdots,X\_n)=\sum\_{\sigma=[i\_1,\cdots,i\_n]\in S\_n}f(X\_{i\_1},\cdots,X\_{i\_n}).$$
Here th... | https://mathoverflow.net/users/58096 | An interesting inequality | Alas, this is false at least for even $n\geqslant 6$, I do not know about $n=4$.
A similar question about symmetrization of $$f\_{a,b}:=\prod\_{1\leq i\leq a,a+1\leq j \leq a+b} (X\_i-X\_j)$$
may be asked (of course, it equals to 0 if $ab$ is odd). I asked this for $a=1$, $b=n-1$ [here](https://mathoverflow.net/quest... | 11 | https://mathoverflow.net/users/4312 | 226637 | 105,819 |
https://mathoverflow.net/questions/226277 | 35 | Y. Sergeyev developed a positional system for representing infinite numbers using a basic unit called a "grossone", as well as what he calls an "infinity computer". The mathematical value of this seems dubious but numerous articles have already appeared in refereed *research* journals. Thus, there are currently 23 such... | https://mathoverflow.net/users/28128 | What is... a grossone? | I do not understand what the bounty on this question is for, as it seems to me that the other answers were already rather devastating. Here is a semi-reasoned technical answer.
According to G. Lolli (the paper you cite) "Sergeyev is wary of the axiomatic method because he thinks that by adopting it we would be tied t... | 54 | https://mathoverflow.net/users/1176 | 226644 | 105,821 |
https://mathoverflow.net/questions/226643 | 5 | I came across the following definite of the Lapse Function:
$N=\sqrt{\frac{1}{2}g(L,\overline{L})}$
where $L,\overline{L}$ are the null geodesic vector fields. Further, I have been looking at this review paper of the 3+1 formalism <http://arxiv.org/pdf/gr-qc/0703035v1.pdf> , where the lapse function is defined on p... | https://mathoverflow.net/users/51137 | Intuition behind the "Lapse Function" | I find this [presentation](http://www.aei.mpg.de/~rezzolla/lnotes/IMPRS_2008/intro_to_numrel.pdf) quite understandable (search for "intuitive interpretation of the lapse").
The Einstein equations allow for a certain arbitrariness when one chooses to foliate space time into a three-dimensional spatial hypersurface wi... | 7 | https://mathoverflow.net/users/11260 | 226650 | 105,823 |
https://mathoverflow.net/questions/226651 | 5 | Consider a random variable $X$ whose variance is large. As a contrast to Markov's or Chebyshev's inequality, both of which measure the concentration of a probability distribution, is there a measure of how "spread out" a distribution is?
more specifically, I would like to have an inequality of the following sort:
$$
\P... | https://mathoverflow.net/users/74799 | A measure of how "spread out" a probability measure is | Here is one crude calculation. The claim is that if the distribution of $X$ has bounded density $f$ then $\sup\_c \mathsf{P}\{|X - c| \le 1\} = O(1/\sqrt n)$. The assumption is probably far too strong, but the $1/\sqrt{n}$ asymptotics is clearly optimal, by comparing to the Gaussian.
Let $\gamma\_{\sigma^2}$ be the G... | 5 | https://mathoverflow.net/users/22758 | 226655 | 105,824 |
https://mathoverflow.net/questions/226652 | 0 | I am interested in finding out what is known about the following generalization of balanced incomplete block designs (BIBDs):
"What is the maximum size of a collection $B$ of $v$-dimensional *unit* real vectors with the following property: there exists a constant $\lambda$ such that $\forall x,y\in B$: $x\ne y \impl... | https://mathoverflow.net/users/84389 | Vector version of balanced incomplete block designs | The Gram matrix of your vectors is equal to $(1-\lambda)I +\lambda J$
(where $J$ is the all-ones matrix). If $\lambda\ne -1/(v-1)$, this matrix is invertible, whence your set of vectors is linearly independent and there are at most $v$ of them. If $\lambda = -1/(v-1)$, then the Gram matrix has rank $v-1$ and your vecto... | 1 | https://mathoverflow.net/users/1266 | 226657 | 105,825 |
https://mathoverflow.net/questions/226634 | 2 | Suppose $y\sim N(0,\Sigma)$ is an $n-$dimensional vector. I'm interested in an upper bound for $\Pr(\max\_{1\leq i\leq n} y\_i > k)$ for $k$ large. I know a little about $\Sigma$: $\sigma\_{ii}=\sigma\_{jj}$ for any $i, j$ and $\sigma\_{ij}\geq 0$. I can also bound the pairwise correlations from above, but it's rather ... | https://mathoverflow.net/users/12064 | Bounding exceedance probabilities for correlated normal variables | Slepian's inequality allows you to dominate the probability for the case you ask about by the same with $\Sigma=\sigma\_{11} I$.
| 3 | https://mathoverflow.net/users/35520 | 226669 | 105,829 |
https://mathoverflow.net/questions/226663 | 0 | I have that a sequence of random matrices, $M\_n$, converges almost surely to a diagonal matrix, $D$, with finite real entries on its diagonal. During convergence, the off-diagonals are not necessarily zero. How can I go about showing that the smallest eigenvector of $M\_n$ converges almost surely to the smallest eigen... | https://mathoverflow.net/users/84393 | Almost sure convergence of smallest eigenvector of diagonal matrix | The set of symmetric matrices that have multiple eigenvalues has Lebesgue measure 0. If the probability measure you use on the space of matrices is absolutely continuous w.r.t. the Lebesgue measure, then the probability that a random matrix has multiple eigenvalues is zero. I assume that this is the case. Assume that t... | 1 | https://mathoverflow.net/users/20302 | 226674 | 105,831 |
https://mathoverflow.net/questions/226645 | 4 | For integer $N$ consider the mapping $$f : (0,1)^N \to \mathbb{R}, \quad x \mapsto \min\_{b \in \{0,1\}^N} \left\{ x^b + x^{1-b} \right\},$$
where $x^b = x\_1^{b\_1} \cdots x\_N^{b\_N}$ and $1-b = (1-b\_1, \ldots, 1-b\_N)$. Note that $x$ is a vector of reals and $b$ is a binary vector.
Is there a faster than $O(2^N)... | https://mathoverflow.net/users/56620 | Complexity of this minimization | To minimize $x^b + x^{1-b}$, we must find $b$ such that $x^b$ and $x^{1-b}$ are as close together as possible.
Taking logs, this is equivalent to partitioning $(\log(x\_1), \log(x\_2), \dots, \log(x\_n))$ into two subsets whose sums are as close together as possible. This is the well-known subset-sum problem, which i... | 5 | https://mathoverflow.net/users/39142 | 226676 | 105,833 |
https://mathoverflow.net/questions/226681 | 1 | Let $B$ be a smooth projective complex variety and $\pi:X\to B$ a smooth projective map whose fibres $X\_b$ are abelian varieties. Let $\psi:Y\to B$ be the naturally associated bundle such that the fibres $Y\_b$ are dual to $X\_b$ (more precisely, $Y$ is the relative Picard scheme $Pic^0(X/B)$). Is it always true that ... | https://mathoverflow.net/users/37059 | Derived equivalence of families of dual abelian varieties | Yes if $X$ is an abelian scheme over $B$: the Fourier-Mukai functor provides an equivalence of the derived categories. This is Theorem 1.1 in Mukai's *Fourier functor and its application to the moduli of bundles on an abelian variety*, in Algebraic geometry, Sendai, 1985, pp. 515-550; Adv. Stud. Pure Math., 10, North-H... | 3 | https://mathoverflow.net/users/40297 | 226686 | 105,836 |
https://mathoverflow.net/questions/218010 | 3 | Let $\mathscr{L}$ be a recursive language. Let $\varphi$ be a $\mathscr{L}\_{\omega\_1 \omega}$-sentence and $\varphi \in L\_{\omega\_1^\emptyset}$. (Let $\varphi$ be a computably infinitary formula.) Let $\text{Mod}(\varphi)$ denote the set of countable models of $\varphi$.
Is there anything inherently wrong with t... | https://mathoverflow.net/users/43354 | Scott Rank of Models of Infinitary Sentences | If $\phi$ is a counterexample to Vaught's Conjecture, then the Scott ranks of the models of $\phi$ include every limit ordinal below $\omega\_{2}$ and above the quantifier depth of $\phi$. This follows from Theorem 10.6 in the following paper of mine:
<http://www.users.miamioh.edu/larsonpb/scott_proc_b1o.pdf>
I attri... | 3 | https://mathoverflow.net/users/31807 | 226697 | 105,840 |
https://mathoverflow.net/questions/226680 | 4 | Let A be a finite dimensional algebra and $S\_1,S\_2,...,S\_n$ the simple $A$-modules and $P\_1,..,P\_n$ the indecomposable projective $A$-modules. For $i=1,...,n$, define the standard module $\Delta\_i$ as the largest factor module of $P\_i$ with no composition factors $S\_j$ for a $j>i$. For $i=1,...,n$, define the p... | https://mathoverflow.net/users/61949 | Homological characterisation of standardly stratified algebras using Ext | Such a characterization was already given in the paper where standardly stratified algebras were first defined:
>
> Ágoston, István; Dlab, Vlastimil; Lukács, Erzsébet. Stratified algebras. C. R. Math. Acad. Sci. Soc. R. Can. 20 (1998), no. 1, 22--28
>
>
>
[Link](http://www.cs.elte.hu/~agoston/papers/stratified... | 5 | https://mathoverflow.net/users/18756 | 226698 | 105,841 |
https://mathoverflow.net/questions/226693 | 4 | Let $f:X\to B$ and $g:Y\to B$ be smooth morphisms of complex projective varieties. Assume that for every closed point $b\in B$, the fibres $X\_b=X\times \kappa(b)$ and $Y\_b$ are derived equivalent. *Assume furthermore that the Fourier-Mukai transforms are induced by complexes of sheaves $F\_b \in D^b(X\_b \times Y\_b)... | https://mathoverflow.net/users/37059 | Does derived equivalence of the fibres imply derived equivalence of the total spaces? | With the added assumptions the answer is yes. In other words, one can check equivalences by looking at fibres.
This is Proposition 2.15 of <http://arxiv.org/pdf/math/0610319.pdf> (it might be that when everything is smooth this was already known).
| 2 | https://mathoverflow.net/users/73972 | 226700 | 105,842 |
https://mathoverflow.net/questions/226699 | 8 | For any positive integer $n$, let
$$A\_n=\idotsint\limits\_{\substack{x\_1+\cdots+x\_n+y\_1+\cdots+y\_n\leq1\\x\_1,\cdots,x\_n,y\_1,\cdots,y\_n\geq0}}\prod\_{i,j=1}^n(x\_i-y\_j)dx\_1\cdots dx\_ndy\_1\cdots dy\_n.$$
It is easy to prove that $A\_n=0$ when $n$ is odd.
I conjecture that $A\_n>0$ when $n$ is even.
To p... | https://mathoverflow.net/users/58096 | An interesting integration | Let's prove a bit more, namely that the integral over $x\_i>0, y\_j>0, \sum\_i x\_i+\sum\_j y\_j=S$ with respect to the $2n-1$-dimensional Lebesgue measure is positive. Note that due to the obvious scaling $x\_i,y\_j\mapsto tx\_i,ty\_j$, this integral depends on $S$ in a trivial way (as some pure power function of $S$)... | 19 | https://mathoverflow.net/users/1131 | 226705 | 105,844 |
https://mathoverflow.net/questions/226673 | 0 |
>
> Update: I've marked this question as answered. If you are thinking "What the heck are floretions?", go right to the answer provided by the Grinch. I definitely should have added clearer information on the algebra as opposed to "hiding" it in the links. Yes, of course I will continue to study the space. The whole ... | https://mathoverflow.net/users/81135 | 4th Order Floretions: Floret's Equation | $\def\F{\mathbb{F}}$
$\def\C{\mathbf{C}}$
$\def\H{\mathbb{H}}$
**Note: this answer is not really intended for the OP, but rather for regular mathoverflow readers who must be thinking "what the heck are the floretions?" and who doesn't want to bother reading up about them. Maybe this question will be closed, but I'm k... | 6 | https://mathoverflow.net/users/84419 | 226710 | 105,846 |
https://mathoverflow.net/questions/226478 | 19 | Recently, I've been trying to understand Jacob Lurie's 2-equivariant elliptic cohomology a bit better than I had in the past.
From what I can tell, the fragment of the story that only deals with **finite groups**, with **characteristic zero** stuff, and only talks about the **degree zero** part of the cohomology theo... | https://mathoverflow.net/users/5690 | elliptic curves and group cohomology | As Charles indicates, "the moduli stack of $G$-bundles on $E$" is not quite the right thing to consider, especially if you're not working over $\mathbf{C}$. This is for two (unrelated) reasons:
1) The geometric object $M\_{G}$ that you associate to a group $G$ isn't something that you can access directly (at least by... | 18 | https://mathoverflow.net/users/7721 | 226716 | 105,847 |
https://mathoverflow.net/questions/226695 | 7 | For each prime $p\geq 3$ let $\alpha\_p:S^{2p}\to S^3$ denote a representative of $\pi\_{2p}S^3$ of order $p$. Berstein and Hilton showed that for each $p$ the homotopy cofiber $C\_{\alpha\_p}$ of $\alpha\_p$ is a co-H-space which does not have the homotopy type of a suspension space.
The maps $\alpha\_p$ give rise ... | https://mathoverflow.net/users/40835 | Is it known whether this space is a suspension space? | Isn't the localization of a 2-connected suspension also a suspension? Then this cannot be a suspension. (Brayton Gray pointed this out to me.)
| 10 | https://mathoverflow.net/users/6872 | 226729 | 105,851 |
https://mathoverflow.net/questions/226742 | 0 | This is cross-posted on MSE: <https://math.stackexchange.com/q/1584519/9464>
Let $\mathcal{V}$ be the space (without topology)
$$\displaystyle \mathcal{V}=\{u\in C\_0^\infty(\Omega)\mid \nabla\cdot u=0\}$$
where $\Omega$ is a nonempty open connected subset of $\mathbb{R}^n$.
It is said in the *Navier-Stokes Equ... | https://mathoverflow.net/users/nan | $H_0^1(\Omega)$ in the study of the Navier-Stokes Equations | Yes, it is the same. I think that Roger (my advisor) wanted only to emphazise that because $\cal V$ is included in $H^1\_0$, and the latter is a closed subspace of $H^1$, the closure of $\cal V$ is contained in $H^1\_0$, hence its elements satisfy the boundary condition $u=0$.
| 4 | https://mathoverflow.net/users/8799 | 226743 | 105,857 |
https://mathoverflow.net/questions/226736 | 65 | I am asking this question starting from two orders of considerations.
Firstly, we can witness, considering the historical development of several sciences, that certain physical entities "disappeared": it is the case of luminiferous aether with the surge of [Einstein's relativity](https://en.wikipedia.org/wiki/Lumini... | https://mathoverflow.net/users/84431 | Do mathematical objects disappear? | I certainly can't think of examples similar to your physics examples of concepts that were just wrong so effectively became extinct.
Two extremes which are present in mathematics are
1) Things which become too simple to have their name retain prominence as commonly known terminology.
For example: For Aristotle, ... | 38 | https://mathoverflow.net/users/8008 | 226748 | 105,860 |
https://mathoverflow.net/questions/226775 | 2 |
>
> "A *translation surface* is a union of polygons with pairs of parallel edges identified by translation, up to cut and paste equivalence."
>
>
>
I take that succinct (and not fully precise) definition from a paper by
Lelievre & Weiss.1
My question is whether the computational complexity is known
for recogni... | https://mathoverflow.net/users/6094 | Complexity of recognizing equivalent translation surfaces | Decidable? A translation surface is just a (special kind of) a singular Euclidean surface. If you define it as a union of polygons as above, you can start with a triangulation (by triangulating each polygon), then get a canonical (Delaunay) triangulation (with respect to the singularities) by edge flipping, then the tw... | 3 | https://mathoverflow.net/users/11142 | 226779 | 105,875 |
https://mathoverflow.net/questions/226780 | 3 | For $A,B\subseteq \omega$ we write $A \subseteq^\* B$ if $A\setminus B$ is finite. We call ${\cal T}\subseteq {\cal P}(\omega)$ a *tower* if it is linearly quasiordered with respect to $\subseteq^\*$.
Using Zorn's Lemma, it is easy to see that every tower is contained in a maximal tower.
Does every maximal tower ha... | https://mathoverflow.net/users/8628 | Cardinalities of maximal towers in ${\cal P}(\omega)$ | The answer is yes, for I claim that every maximal chain has size continuum.
Suppose that $C$ is a chain of subsets of $\mathbb{N}$ which is
maximal with respect to almost inclusion. Let's work in the
quotient, so we consider only one member from each equivalence
class.
First, notice that $C$ must be dense as a line... | 6 | https://mathoverflow.net/users/1946 | 226782 | 105,877 |
https://mathoverflow.net/questions/226778 | 2 | Given a quasi-ordered set $(Q,\leq)$ the *interval topology* on $Q$ is generated by
$$\{Q\setminus\downarrow x : x\in Q\} \cup \{Q\setminus\uparrow x : x\in Q\},$$
where $\downarrow x = \{y\in Q: y\leq x\}$ and $\uparrow x = \{y\in Q: y\geq x\}$.
Let $\mathbb{N}^\mathbb{N}$ denote the set of all functions $f:\mathbb{... | https://mathoverflow.net/users/8628 | Is the interval topology of $(\mathbb{N}^\mathbb{N}, \leq^*)$ connected? | I claim that this topology has the stronger property of being [hyperconnected](https://en.wikipedia.org/wiki/Hyperconnected_space), i.e., the intersection of any two nonempty open sets is nonempty.
Indeed, any open set contains a set of the form
$$ U=\bigcap\_{i=1}^m \{x:x\not\le^\*y\_i\}\bigcap\bigcap\_{j=1}^n \{x... | 2 | https://mathoverflow.net/users/37103 | 226783 | 105,878 |
https://mathoverflow.net/questions/226784 | 2 | It is known that $$\sum\_{p\leq x} \frac{\log p}{p}=\log x+c.$$
Are any tight bounds on
$$\sum\_{p\leq x} \frac{\log \log p}{p}$$ known?
I haven't managed to find anything in the literature. Trying to approximate via $p\_k\approx k \log k$ doesn't give a tractable integral.
| https://mathoverflow.net/users/17773 | estimate sum of $\log \log p/p$ | Let $S(t):=\sum\_{p\leq t}\frac{\log p}{p}$. By Mertens' theorem, $S(t)=\log t+T(t)$, where $T(t)$ is bounded. It follows that
$$ \sum\_{p\leq x}\frac{\log\log p}{p}=\int\_{2-}^x\frac{\log\log t}{\log t}dS(t)=\int\_{2}^x\frac{\log\log t}{\log t}\cdot\frac{dt}{t}+\int\_{2-}^x\frac{\log\log t}{\log t}dT(t).$$
On the righ... | 8 | https://mathoverflow.net/users/11919 | 226791 | 105,883 |
https://mathoverflow.net/questions/226407 | 14 | There is the famous Serre intersection formula in algebraic geometry using the Tor functor (see for example [here](https://en.wikipedia.org/wiki/Intersection_theory#Intersection_multiplicities)). I would like to know if there is such a formula in analytic (i.e. complex) geometry. Thanks.
In somewhat more detail: giv... | https://mathoverflow.net/users/84277 | Is there a Serre intersection formula in analytic geometry? | Serre's formula works in the analytic category as well. If X is a smooth complex manifold, there is a ring structure on the Grothendieck group $K(X)$ of coherent sheaves given by the usual formula
$$ [\mathcal{F}] . [\mathcal{G}]=\sum\_{i \geq 0} (-1)^i [\mathrm{Tor}^i\_{\mathcal{O}\_X}(\mathcal{F}, \mathcal{G})].$$
T... | 6 | https://mathoverflow.net/users/13503 | 226798 | 105,885 |
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